Common factors, trends, and cycles in large datasets*
Abstract:
JEL classification: C32, C38, C55, E0.
Keywords: Non-stationary Approximate Dynamic Factor Model; Trend-Cycle Decomposition; Quasi Maximum Likelihood; EM Algorithm; Kalman Smoother; Gross Domestic Output; Output Gap.
1 Introduction
This paper is about two stylized facts of macroeconomic time series: co-movements and non-stationarity Lippi & Reichlin (1994a). More precisely, this paper is about disentangling long-run co-movements (common trends) from short-run co-movements (common cycles) in a large dataset of non-stationary US macroeconomic indicators.
Since the seminal work of Beveridge & Nelson (1981), the issue of decomposing GDP into a trend and a cycle has been a central question in both time series econometrics and policy analysis. This is not surprising, as long-run trends are mainly influenced by supply-side factors, while short-run cycles are mainly associated with demand-side factors, and therefore different estimates of the trend and of the cycle can lead to different policy recommendations. Given the relevance of the issue, in the last 30 years, many papers have suggested different ways to obtain a Trend-Cycle (TC) decomposition of GDP. Roughly speaking, those works can be grouped under two main approaches: one based on univariate methods (e.g. Lippi & Reichlin, 1994b; Morley et al., 2003; Watson, 1986; Dungey et al., 2015), and another using multivariate, but low-dimensional, time series techniques (e.g. Stock & Watson, 1988; Creal et al., 2010; Garratt et al., 2006; Lippi & Reichlin, 1994a; Gonzalo & Granger, 1995).
In this paper we use a novel approach to decompose GDP into a trend and cycle based on large datasets. We first disentangle common and idiosyncratic dynamics by using a Non-Stationary Approximate Dynamic Factor Model (DFM), and then we disentangle common trends from common cycles by applying a non-parametric TC decomposition to the latent common factors. Our methodology builds on four points: first, focusing on a high-dimensional setting is crucial, as only in a high-dimensional setting it is possible to disentangle common from idiosyncratic dynamics in a consistent way Forni et al. (2000); Stock & Watson (2002); Bai & Ng (2002) -- i.e., we can separate macroeconomic fluctuations from sectoral dynamics and measurement error only in a high-dimensional setting. Second, assuming the existence of a factor structure is a realistic and convenient way to represent co-movements in large macroeconomic datasets. Third, considering non-stationary data is necessary to account for the presence of common trends or, equivalently, cointegration Bai & Ng (2004); Bai (2004); Barigozzi et al. (2016b,a). And fourth, by using a non-parametric TC decomposition we do not have to make assumptions on the law of motion of either the trend, or the cycle. Our approach is deliberately reduced form, and therefore our empirical analysis is conducted "without pretending to have too much a priori economic theory" Sargent & Sims (1977), thus letting the data speak as freely as possible.
The first contribution of this paper is methodological. Namely, we propose a Quasi Maximum Likelihood estimator of the non-stationary DFM based on the Expectation Maximisation (EM) algorithm combined with the Kalman Filter and the Kalman Smoother estimators of the factors. The theoretical properties of this approach in the large stationary DFM case have been studied in Doz et al. (2012,2011), and here we extend their results to the non-stationary case by proving consistency and by providing rates of convergence for the factors and the parameters of the model. Compared to the non-stationary principal component estimator Bai & Ng (2004), the estimator proposed in this paper is more efficient, and it is more flexible in that, thanks to the use of the Kalman Filter, it allows us to explicitly model the idiosyncratic dynamics, and to impose economically meaningful restrictions.
The second contribution of this paper is to show how to isolate common trends and common cycles in large macroeconomic datasets. In detail, we use a non-parametric approach that identifies the common trends as those linear combinations of the factors obtained by the leading eigenvectors of the long-run covariance matrix (Bai (2004); Peña & Poncela (2006)), and the common cycles as deviations from the long-run equilibria, which coincide with the space orthogonal to that of the common trends -- i.e., the cointegration space Zhang et al. (2016). Because our approach is non-parametric, we are not imposing any particular form to the trend, which is not constrained to be a random walk, or to the cycle. This is what differentiates our approach from the standard state-space, which normally is applied on a handful of variables and where the trend and the cycle dynamics are explicitly specified and jointly estimated with the parameters of the model (Harvey (1990)).
Our final contributions are empirical. Specifically, we employ our methodology to analyse a large panel of US quarterly macroeconomic time series with the goal of estimating the cyclical position of the economy and the observation error. With the expression "estimating the observation error," we mean estimating aggregate real output. With the expression "estimating the cyclical position of the economy," we mean decomposing aggregate real output into potential output and output gap. To the best of our knowledge, Fleischman & Roberts (2011) and Aruoba et al. (2016) are the only works that, so far, have used (small) factor models to estimate aggregate real output. On the other hand, a few papers have used low-dimensional factor models to estimate the cyclical position of the economy (e.g. Fleischman & Roberts, 2011; Jarocinski & Lenza, 2016), and a few more to estimate long-run trends (e.g. Antolin-Diaz et al., 2016). Finally, Aastveit & Trovik (2014) and Morley & Wong (2017) have used a high-dimensional setting for estimating the output gap by means of a factor model and a large Bayesian VAR, respectively. However, in both works the variables are transformed to stationarity prior to model estimation.
The first part of our empirical analysis is about estimating aggregate real output, to which we refer as Gross Domestic Output (GDO). We first show that our model naturally produces an estimate of GDO as that part of GDP/GDI that is driven by the macroeconomic (common) shocks. We then compare our estimate of GDO with "the average of GDP and GDI" released by the Bureau of Economic Analysis, and with "GDPplus" proposed by Aruoba et al. (2016) and released by the Philadelphia Fed. Our results show that these three measures are very similar, which is not surprising, as they are attempting to estimate the same thing. However, we estimate that since 2010 quarterly annualized GDO growth was on average 1/2 of a percentage point higher than estimated by the BEA or the Philadelphia Fed, thus pointing out that -- based on the commonality in the data -- the US economy grew at a faster pace than measured by national account statistics.
The second part of our empirical analysis is about estimating the output gap. To this end, we use the above-mentioned TC decomposition in order to separate long-run from short-run co-movements, and in particular we focus on the decomposition derived for GDO. We compare our estimate with the one produced by the Congressional Budget Office (CBO), which estimates potential output as that level of output consistent with current technologies and normal utilisation of capital and labour, and the output gap as the residual part of output. Although these two estimates are obtained in completely different ways, in practice they look very similar. The two estimates are comparable for most of the sample considered, but from the late nineties to the financial crisis, when our measure suggests that a greater part of the produced output was driven by transitory factors. In particular, according to our estimate between 2001:Q1 and 2005:Q4 the output gap was on average 2 1/2 percentage points higher than estimated by the CBO.
The rest of this paper is structured as follows. In Section 2 we discuss representation of large non-stationary panels of time series. In this section we first present the non-stationary dynamic factor model, and we define the concepts of commonality -- i.e., the common factors. Then we discuss how to disentangle long-run co-movements from short-run co-movements -- i.e., we define what common trends and common cycles are. In Section 3 we discuss estimation. We first introduce in Section 1 the static representation of the DFM, which is just a convenient way to approach estimation of the dynamic model presented in Section 2. We then present in Section 2 our estimator, we discuss its properties, and we compare it with existing methods. Finally, in Section 3 we present the non-parametric TC decomposition that we use in the empirical section. Then, Section 4 presents the empirical analysis. This section is split in two, with the first part presenting our estimate of GDO (Section 1), and the second part presenting our estimate of the output gap (Section 2). To conclude, in Section 5 we discuss our findings and the advantages and limitations of our methodology, and we propose directions for further research. In the Appendix we report all technical proofs and the description of the data used and their transformation.
Notation
A vector $$ \mathbf z_t$$ is $$ I(1)$$ if the higher-order of integration among all its components is 1, thus under this definition some components of $$ \mathbf z_t$$ can be stationary. Eigenvalues are always considered as ordered from the largest to the smallest, so for a given set of eigenvalues $$ \{\mu_j\}_{j=1}^m$$, we have $$ \mu_1\ge\mu_2\ge\ldots\ge\mu_{m-1}\ge \mu_m$$. Therefore, the spectral norm of $$ A$$ is defined as $$ \Vert A\Vert^2=\mu_1^{A'A}$$. The j-th largest eigenvalue of a spectral density matrix at frequency $$ \omega$$ is denoted as $$ \mu_j(\omega)$$. The generic $$ (i,j)$$-th entry of a matrix $$ A$$ is denoted as $$ [ A]_{ij}$$. We denote by L the lag operator, such that $$ L^k y_t =y_{t-k}$$, for any $$ k\in\mathbb{Z}$$ and we use the notation $$ \Delta y_t:=(1-L)y_t$$. Finally, we let $$ M,M_0,M_1\ldots$$ denote generic positive and finite constants that do not depend on the panel dimensions n or T, and whose value may change from line to line.
2 Representation of non-stationary panels of time series
Let us assume to observe a vector of n time series $$ \{\mathbf y_t =(y_{1t}\cdots y_{nt})':t=1,\ldots ,T\}$$ such that
where $$ \mathcal D_{it}$$ is a deterministic component -- e.g., a linear trend -- and $$ \mathbf x_t=(x_{1t}\cdots x_{nt})'$$ is such that $$ \mathbf x_t\sim I(1)$$. We also assume that $$ \mathsf{E}[x_{it}]= 0$$, for any i and t, therefore, $$ \mathbf x_{t}$$ contains all the stochastic trends but no deterministic component. Throughout, the spectral density matrix of $$ \Delta \mathbf x_t$$ is assumed to exist.
In a high-dimensional setting, it is reasonable to assume that there are common trends and common cycles, but also idiosyncratic terms. Thus, for each variable xit we write
where $$ \mathcal T_{it}\sim I(1)$$ is the trend component, $$ \mathcal C_{it}\sim I(0)$$ is the cycle component, and $$ \xi_{it}$$ is the idiosyncratic component, which is allowed to be either $$ I(1)$$ (in presence of idiosyncratic trends) or $$ I(0)$$ (e.g. measurement errors). The trend and the cycle are capturing the common dynamics across series, and thus constitute the common component defined as $$ \chi_{it}={\mathcal T}_{it} + {\mathcal C}_{it}$$. Hence, (2) is also written as
We define the vectors of common and idiosyncratic components as $$ \chi_t=(\chi_{1t}\cdots \chi_{nt})'$$ and $$ \xi_t=(\xi_{1t}\cdots \xi_{nt})'$$, respectively. Finally, notice that consistently with the data considered in this paper: (i) some (but not all) components of $$ \mathbf x_t$$ are allowed to be stationary, and (ii) the deterministic components $$ \mathcal D_{it}$$ are not common to all series -- i.e., there are no common deterministic trends.
We assume that the co-movements in $$ \chi_t$$ are driven by q "structural" shocks, with $$ q\ll n$$, which are collected in a weak white noise vector process $$ \mathbf u_t=(u_{1t}\cdots u_{qt})'$$. Then, for a given q, we decompose each element of $$ \mathbf x_t$$ as
| $$\displaystyle x_{it}$$ | $$\displaystyle = b_i'(L) f_t+\xi_{it},$$ | (4) |
| $$\displaystyle \Delta f_t$$ | $$\displaystyle = \mathbf C(L)\mathbf u_t,$$ | (5) |
where from (3) the common component is given by $$ \chi_{it}= b_i'(L) f_t$$ and the following properties hold:
- [A1.] $$ \mathbf u_t\stackrel{w.n.}{\sim} (\mathbf 0_q,\mathbf I_q)$$, with q is independent of n;
- [A2.] $$ \mathsf{E}[u_{jt} \xi_{is}]=0$$, for any $$ j=1,\ldots q$$, $$ i=1,\ldots, n$$, and $$ s,t=1,\ldots,T$$;
- [A3.] $$ \mathbf B(L)=( b_1'(L)\cdots b_n'(L))'$$ is an $$ n\times q$$ one-sided, matrix polynomial matrix of finite order s, $$ f_t\sim I(1)$$ of dimension q;
- [A4.] $$ \mathbf C(L)=( c_1'(L)\cdots c_q'(L))'$$ is a $$ q\times q$$ one-sided, infinite matrix polynomial with square-summable coefficients and such that $$ \mathsf {rk}(\mathbf C(1))=(q-d)$$ with $$ 0<d< q$$;
- [A5.] the q-th largest eigenvalue
$$ \mu_q^{\Delta \chi}(\omega)$$ of the spectral density matrix of
$$ \Delta \chi_t$$ is such that
$$\displaystyle M_1\le\lim\!\!\!\inf_{n\to\infty} n^{-1}{\mu_q^{\Delta \chi}(\omega)}\le \lim\!\!\sup_{n\to\infty} n^{-1}{\mu_q^{\Delta \chi}(\omega)}\le M_2,\qquad \omega$$-a.e.$$\displaystyle \in [-\pi,\pi], $$while the largest eigenvalue $$ \mu_{1}^{\Delta \xi}(\omega)$$ of the spectral density matrix of $$ \Delta \xi_t$$ is such that$$\displaystyle M_3\le \lim\!\!\!\inf_{n\to\infty}\mu_{1}^{\Delta \xi}(\omega)\le \lim\!\!\sup_{n\to\infty}\mu_{1}^{\Delta \xi}(\omega)\le M_4, \qquad \omega$$-a.e.$$\displaystyle \in [-\pi,\pi]. $$
Condition A5 is crucial and it allows for identification of the common component by defining it according to its spectral properties. An explanation for A5 in the time domain is provided by Hallin & Lippi (2013) who show that this condition is equivalent to defining the common and idiosyncratic component by asking that for any dynamic aggregation scheme given by an n-dimensional vector of weights $$ \mathbf a_k$$ such that $$ \sum_{k\in\mathbb{Z}}\mathbf a_k'\mathbf a_k=1$$, the following holds
| $$\displaystyle 0< \lim_{n\to\infty}\mathsf{Var}\left(\frac 1n\sum_{k=-\infty}^\infty \mathbf a_k'\Delta \chi_{t-k}\right)\le M\;$$ and $$\displaystyle \;\lim_{n\to\infty} \mathsf{Var}\left(\frac 1n\sum_{k=-\infty}^{\infty} \mathbf a_k'\Delta \xi_{t-k}\right)= 0.$$ | (6) |
The following asymptotic conditions for the eigenvalues $$ \mu_{i}(\omega)$$ of the spectral density of $$ \Delta \mathbf x_t$$ are a direct consequence of A4, A5, and Weyl's inequality:
- [B1.] for
$$ \omega$$-a.e.$$ \in [-\pi,\pi]$$ the following holds:
$$ M_1\le \lim\inf_{n\to\infty}n^{-1}{\mu_q(\omega)}\le\lim\sup_{n\to\infty}n^{-1}{\mu_q(\omega)} \le M_2$$,
$$ M_3\le \lim\inf_{n\to\infty}\mu_{q+1}(\omega)\le \lim\sup_{n\to\infty}\mu_{q+1}(\omega)\le M_4$$; - [B2.] for
$$ \omega = 0$$ the following holds:
$$ M_1\le \lim\inf_{n\to\infty}n^{-1}{\mu_{q-d}(0)}\le\lim\sup_{n\to\infty}n^{-1}{\mu_{q-d}(0)} \le M_2$$,
$$ M_3\le \lim\inf_{n\to\infty}\mu_{q-d+1}(0)\le \lim\sup_{n\to\infty}\mu_{q-d+1}(0)\le M_4$$.
Condition A4 allows for the presence of $$ (q-d)$$ common trends in the factors $$ f_t$$. In line with our empirical results in Section 4 we rule out the degenerate cases d=0 or d=q. This implies that the vector $$ f_t$$ admits a VECM representation with d cointegration relations (Engle & Granger (1987)), as well as the factor representation (Escribano & Peña (1994)):
where $$ \Psi$$ is $$ q\times (q-d)$$ and $$ \tau_t$$ is the vector of $$ (q-d)$$ common trends with components $$ \tau_{jt}\sim I(1)$$ for $$ j=1,\ldots, (q-d)$$, while $$ \gamma_t$$ is a q-dimensional stationary vector.1 Notice that (7) is different from the common trends representation (or multivariate Beveridge-Nelson decomposition) of Stock & Watson (1988) in that the trend $$ \tau_t$$ is not constrained to be a vector random walk, a property advocated for by many authors (e.g. Lippi & Reichlin (1994a)).
For a given choice of $$ \Psi$$, the $$ (q-d)$$ common trends can then be obtained by linear projection onto the space spanned by the columns of $$ \Psi$$:
| $$\displaystyle \tau_t = (\Psi '\Psi )^{-1}\Psi ' f_t=\Psi ' f_t.$$ |
where the second equality holds because, without loss of generality, we can always assume the identifying constraint $$ \Psi'\Psi=\mathbf I_{(q-d)}$$.
Different choices of $$ \Psi$$ lead to different definitions of common trends. Here we opt for a non-parametric approach and we identify the elements of $$ \tau_t$$ as the first $$ (q-d)$$ principal components of $$ f_t$$, as proposed by Bai (2004) and Peña & Poncela (2006) (see Section 3 for details on estimation). Given this definition, the columns of $$ \Psi$$ are orthonormal and therefore there exists a $$ q\times d$$ matrix $$ \Psi_{\perp}$$ such that $$ \Psi_{\perp}'\Psi_{\perp}=\mathbf I_d$$ and $$ \Psi_{\perp}'\Psi=\mathbf 0_{d\times (q-d)}$$. Now, consider the d-dimensional process obtained by projecting $$ f_t$$ onto the space orthogonal to the common trends
| $$\displaystyle c_t = (\Psi_{\perp}'\Psi_{\perp})^{-1}\Psi_\perp' f_t=\Psi_\perp' f_t=\Psi_\perp' \gamma_t.$$ |
It is straightforward to see that $$ { c}_t\sim I(0)$$, that its components are d common cycles in the sense of Vahid & Engle (1993), and that the columns of $$ \Psi_{\perp}$$ are a basis of the cointegration space of $$ f_t$$, thus these common cycles represent deviations from long-run equilibria -- see also e.g. Johansen (1991) and Kasa (1992) for similar definitions.2
According to our definition, common trends and common cycles are orthogonal by construction, and we have the TC decomposition of the factors:
| $$\displaystyle f_t = \Psi \Psi ' f_t + \Psi_\perp \Psi_\perp ' f_t= \Psi \tau_t + \Psi_\perp c_t,$$ | (8) |
and therefore, by combining (1), (4) and (8), we have the TC decomposition of the data:
| $$\displaystyle y_{it} = \mathcal D_{it}+ b_i'(L) \Psi\tau_t + b_i'(L) \Psi_\perp c_t + \xi_{it}= \mathcal D_{it}+\mathcal T_{it} +\mathcal C_{it}+\xi_{it}.$$ | (9) |
3 Estimation
In order to estimate (9), we need to estimate the factors, $$ f_t$$ and their TC decomposition. We opt for a two-step approach, where we first extract the common factors and then we estimate their TC decomposition. In particular, we first introduce a convenient re-parametrization of the DFM based on its static state-space representation (Section 1), which is then used for retrieving the factors space by means of the EM algorithm (Section 2). Then, in a second step we use principal component analysis for extracting common trends and cycles (Section 3). Notice that compared to the classical state-space approach (e.g. Fleischman & Roberts (2011)) or from the Bayesian approach (e.g. Jarocinski & Lenza (2016)) in which the trend and the cycle are estimated in one-step together with the parameters of the models, our approach has the advantage that it does not require us to specify a law of motion for the trend and the cycles.
For simplicity of exposition we assume in this section that there is no deterministic component and we refer to Section 4 and to Appendix D for the treatment of these terms in practice.
1 The static representation of dynamic factor models
Consider the state-space form of the DFM in (4)-(5) (Stock & Watson (2005); Forni et al. (2009)):
| $$\displaystyle x_{it}$$ | $$\displaystyle = \lambda_i'\mathbf F_t + \xi_{it},$$ | (10) |
| $$\displaystyle \Delta\mathbf F_t$$ | $$\displaystyle = \mathbf D(L)\mathbf u_t,$$ | (11) |
where from (3) the common component is now given by $$ \chi_{it}=\lambda_i'\mathbf F_t$$ and $$ \mathbf u_t$$ is the same as in (5). We assume that A1, A2 and A5 still hold and in addition we require:
- [C1.] $$ \mathbf D(L)=( d_1'(L)\cdots d_r'(L))'$$ is an $$ r\times q$$ one-sided, infinite matrix polynomial with square-summable coefficients and such that $$ \mathsf {rk}(\mathbf D(1))=(q-d)$$ with $$ 0<d< q$$;
- [C2.] $$ \Lambda=(\lambda_1 \cdots\lambda_n)'$$ is an $$ n\times r$$ loadings matrix such that $$ \lim_{n\to\infty}\Vert n^{-1}\Lambda'\Lambda-\mathbf I_r\Vert=0$$ and $$ \vert[\Lambda]_{ij}\vert<M$$, for any $$ i=1,\ldots, n$$ and $$ j=1,\ldots, r$$;
- [C3.] $$ \mathbf F_t\sim I(1)$$ of dimension r, with $$ \mathsf{E}[\Delta\mathbf F_t\Delta\mathbf F_t']$$ positive definite.
Let us stress once more the fact that here the DFM and the related TC decomposition are our focus, while the static representation is just a convenient way to approach estimation of the dynamic model. In particular, for (10)-(11) to be equivalent to (4)-(5) we need the following restrictions to hold:
- [R1.] there exists an invertible $$ r\times r$$ matrix $$ \mathbf K$$ such that $$ \mathbf F_t=\mathbf K ( f_t'\cdots f_{t-s}')'$$ and $$ \lambda_i' = ( b_{i0}'\cdots b_{is}')\mathbf K^{-1}$$, for any $$ i=1,\ldots, n$$, where $$ b_{ik}$$, for $$ k=0,\ldots, s$$, are the coefficients of $$ b_i(L)$$ defined in A3;
- [R2.] the dimension of $$ \mathbf F_t$$ is $$ r=q(s+1)$$;
- [R3.] the cointegration rank of $$ \mathbf F_t$$ is d.
Restriction R2 offers an alternative way to determine r with respect to the typical methods available in the literature based on the behavior of the eigenvalues of the covariance matrix of $$ \Delta \mathbf x_t$$ and therefore on C2, C3, and A5 (e.g. Bai & Ng (2002)). Specifically, by virtue of restriction R2, once we set q using B1, we can choose r such that the share of variance explained by the static factors $$ \mathbf F_t$$ coincides with the share of variance explained by the q dynamic factors $$ f_t$$ -- see also D'Agostino & Giannone (2012).
Finally, restriction R3 tells us that the autoregressive representation for (11) is a VECM with d cointegration relations (a proof is in Appendix A). Moreover, since the vector $$ \mathbf F_t$$ is singular, the autoregressive representation has a finite order (Barigozzi et al. (2016a)). However, in the next section we do not estimate a VECM, rather we estimate an unrestricted VAR in the levels (Sims et al. (1990)). We use the knowledge of the cointegration rank to determine the dimension of the common cycles space (see Section 3).
Summing up, by not fully imposing R1 and R3 when estimating the factors, we opt for simplicity of estimation versus complexity of a more realistic representation, which implies that the model considered is deliberately mis-specified. The effects of such mis-specification will appear clear in Section 3, when we consider TC decompositions of $$ \mathbf F_t$$ as opposed to those of $$ f_t$$.
2 Estimating the space of factors and loadings
We consider the following state-space form of (10)-(11) in which we assume a VAR(2) for the static factors as in the empirical analysis of Section 4:
| $$\displaystyle x_{it}$$ | $$\displaystyle = \lambda_i'\mathbf F_t + \xi_{it},$$ | (12) |
| $$\displaystyle \mathbf F_t$$ | $$\displaystyle = \mathbf A_1\mathbf F_{t-1}+\mathbf A_2\mathbf F_{t-2}+\mathbf H\mathbf u_t,$$ | (13) |
| $$\displaystyle \xi_{it}$$ | $$\displaystyle =\rho_i\xi_{it-1}+e_{it}.$$ | (14) |
We estimate (12)-(14) via the EM algorithm (Dempster et al. (1977)), combined with the Kalman Filter (KF) and the Kalman Smoother (KS) estimators of the factors (Harvey (1990); Anderson & Moore (1979)). In the stationary, low-dimensional -- i.e., finite n -- setting, estimation of a factor model by means of the EM algorithm can be found in Shumway & Stoffer (1982) and Watson & Engle (1983), while the asymptotic properties of this factors' estimator are studied by Doz et al. (2012,2011) under the joint limit $$ n,T\to\infty$$.3 In the non-stationary case, applications of the EM algorithm can be found in Quah & Sargent (1993) and Seong et al. (2013) in a low-dimensional setting. Here, we study the theoretical properties in the non-stationary case when $$ n,T\to\infty$$.
In order to run the KF-KS it is necessary to make some additional assumptions on the idiosyncratic component. Let $$ \mathbf R$$ be the covariance matrix of the vector $$ e_t=(e_{1t}\cdots e_{nt})'$$ of the idiosyncratic innovations in (14), then we assume:
- [D1.] $$ \rho_i=1$$ if $$ \xi_{it}\sim I(1)$$ or $$ \rho_i=0$$ if $$ \xi_{it}\sim I(0)$$;
- [D2.] $$ e_{t}\stackrel{w.n.}{\sim} \mathcal N(\mathbf 0_n, \mathbf R)$$, with $$ [\mathbf R]_{ii}>0$$ and $$ [\mathbf R]_{ij}=0$$ for any $$ i\neq j$$ and $$ i,j=1,\ldots, n$$;
- [D3.] $$ \mathbf u_t\stackrel{w.n.}{\sim} \mathcal N(\mathbf 0_q, \mathbf I_q)$$.
We collect all unknown parameters of the model into the vector $$ \Theta:=($$vec$$ (\Lambda)' \;$$vec$$ (\mathbf A_1)'\;$$vec$$ (\mathbf A_2)'\;$$ vec$$ (\mathbf H)'\;$$diag$$ (\mathbf R)')'$$. We denote by Q the dimension of $$ {\Theta}$$, then we assume that the true values of the parameters satisfy:
- [D4.] $$ \Theta \in$$ int$$ (\Omega)$$, with $$ \Omega\subseteq \mathbb{R}^Q$$ and compact.
The EM algorithm is based on the iteration of two steps. In the E-step, for a given estimator of the parameters $$ \widehat {\Theta}_k$$, we compute the expected likelihood conditional on all observed data $$ \{\mathbf x_1,\ldots,\mathbf x_T\}$$. This is in turn a function of the first and second conditional moments of the static factors, which are computed by means of the KS when using $$ \widehat {\Theta}_k$$.
Note that, under the assumption of normality, as in D2 and D3, and for a given value of the parameters $$ \Theta$$, the KF-KS give the conditional expectations:
| $$\displaystyle {\mathbf F}_{t\vert t-1} := \mathsf{E}_{{\Theta}}[\mathbf F_t \vert\mathbf x_1,\ldots,\mathbf x_{t-1}],\qquad {\mathbf F}_{t\vert t} := \mathsf{E}_{{\Theta}}[\mathbf F_t \vert\mathbf x_1,\ldots,\mathbf x_t],\qquad$$ | $$\displaystyle {\mathbf F}_{t\vert T} := \mathsf{E}_{{\Theta}}[\mathbf F_t \vert\mathbf x_1,\ldots,\mathbf x_T],$$ |
with the associated covariance matrices denoted as $$ {\mathbf P}_{t\vert t-1}$$, $$ {\mathbf P}_{t\vert t}$$, and $$ {\mathbf P}_{t\vert T}$$, respectively. These are therefore optimal estimators of the static factors since they minimize the associated Mean-Square-Error (MSE) for a given value of the parameters.
In the M-step a new estimator of the parameters $$ \widehat {\Theta}_{k+1}$$ is computed by maximizing the expected likelihood. At convergence of the EM algorithm, say at iteration $$ k^*$$, we obtain the estimator of the parameters, which we denote by $$ \widehat {\Theta}:= \widehat {\Theta}_{k^*}$$. The estimator of the factors is then obtained by running the KS a last times using $$ \widehat {\Theta}$$ and it is denoted by $$ \widehat {\mathbf F}_{t}:=\mathsf{E}_{\widehat {\Theta}}[\mathbf F_t \vert\mathbf x_1,\ldots,\mathbf x_T]$$. The estimated common and idiosyncratic components are then given by $$ \widehat {\chi}_{it}=\widehat { \lambda}_i'\widehat {\mathbf F}_{t}$$ and $$ \widehat {\xi}_{it}=x_{it}-\widehat {\chi}_{it}$$. Details of the EM algorithm, as well as closed form expressions for all the estimators, are in Appendix B.
To initialise the EM algorithm we use as initial estimator of the loadings the r leading eigenvectors of the covariance of $$ \Delta \mathbf x_t$$, from which we have an estimator of the static factors as the integrated principal components of $$ \Delta \mathbf x_t$$ (Bai & Ng (2004)). This factors' estimator is in turn used to: (i) initialize the KF, together with a diffuse prior for the factors' covariance (Koopman (1997); Koopman & Durbin (2000)) and (ii) estimate the VAR parameters (Barigozzi et al. (2016b)). Define as $$ \mathbf V$$ the $$ n\times r$$ matrix having as columns the r leading normalised eigenvectors of the covariance of $$ \Delta \chi_t$$, then the following identifying assumptions are convenient for proving consistency:
- [E1.] $$ \Lambda= \sqrt n\, \mathbf V$$ with $$ [ \Lambda]_{1j}>0$$ for all $$ j=1,\ldots, r$$;
- [E2.] $$ \mathbf F_t=n^{-1/2}\,\mathbf V'\chi_t$$ with $$ \mathbf F_0=\mathbf 0_r$$.
We have the following consistency result.
Define $$ \mathbf F_t^\dagger:=( f_t'\cdots f_{t-s}')'$$ and $$ \lambda_i^\dagger:=( b_{i0}'\cdots b_{is}')'$$. Then, there exists an invertible $$ r\times r$$ matrix $$ \mathbf K$$ such that, as $$ n,T\to\infty$$, for all $$ \, \bar t (T) \le t\le T$$ and any given $$ i=1,\ldots n$$,
| $$\displaystyle \sqrt T\,\Vert\widehat { \lambda}_i-\mathbf K^{-1'}{\lambda_i^\dagger} \Vert=O_p(1),$$ | (16) | |
| $$\displaystyle \min(\sqrt n, \sqrt T)\,\Vert\widehat {\mathbf F}_{t}-\mathbf K \mathbf F_t^\dagger\Vert=O_p(1),$$ | (17) | |
| $$\displaystyle \min(\sqrt n, \sqrt T)\,\vert\widehat {\chi}_{it}-\chi_{it}\vert=O_p(1).$$ | (18) |
Proposition 1 states that under the assumptions presented before, we can consistently estimate the common component, as well as the spaces spanned by the dynamic factors $$ f_t$$ and the corresponding dynamic loadings which are the coefficients of $$ b_i(L)$$ defined A3.
Our proof, which is presented in detail in Appendix C, is based on the same approach followed by Poncela & Ruiz (2015) in the one-factor case, and it is made of two main parts which we summarize here.
Population results. We first show that, when the parameters are known the one-step-ahead factors' MSE, $$ \mathbf P_{t\vert t-1}$$, converges to a steady state, while both the MSEs of the KF, $$ \mathbf P_{t\vert t}$$, and of the KS, $$ \mathbf P_{t\vert T}$$, tend to zero as $$ n\to\infty$$ (Lemmas 4 and 5). Notice that this is true also when initializing with a diffuse prior since this has an effect only for a finite number of initial periods, say t0 (Koopman (1997)). In particular, convergence to the steady state is exponentially fast Anderson & Moore (1979), hence our result holds for any $$ t\ge \bar t(T)>t_0$$, where $$ \bar t(T)$$ satisfies condition (15), which asymptotically requires $$ \bar t(T)=O(\log T)$$. In practice, though, the steady state is reached very quickly as shown in Figure 1, where we report the trace of $$ \mathbf P_{t\vert t-1}$$ (solid line), $$ \mathbf P_{t\vert t}$$ (dashed line) and $$ \mathbf P_{t\vert T}$$ (dashed-dotted line), computed for the data analysed in Section 4.
Estimation results. In the second step of the proof, consistency of the KF and KS estimators of the static factors when using estimated parameters is proved (Lemma 7). This is done by taking into account an additional parameter estimation error which has two components: (i) the error of the QML estimator of the parameters for the case of known factors, say $$ \widehat {\Theta}^*$$ (Lemma 6) and (ii) the error due to the numerical approximation of $$ \widehat {\Theta}$$ to $$ \widehat {\Theta}^*$$ which is related to the stopping rule of the EM algorithm (Meng & Rubin (1994), and Lemma 9). In particular, the latter error is shown to be negligible with respect to the former one. Therefore the rate of convergence of the loadings estimated via the EM algorithm is the same that one would obtain by QML estimation, were the true factors observable, and moreover, because of assumption D2 the loadings are estimated equation by equation, thus such error depends only on T. Results similar to (16) hold also for all other estimated parameters in $$ \widehat {\Theta}$$. On the other hand the rate of convergence for the estimated static factors is standard in the literature.
Figure 1: Conditional Mean Squared Errors
This figure reports $$ \mathsf {tr}(\mathbf P_{t\vert s})$$ when using $$ \widehat {\Theta}$$, computed for the data analyzed in Section 4, where: $$ s=t-1$$ is the one-step-ahead conditional MSE (solid line); s=t is KF conditional MSE (dashed line); s=T is the KS conditional MSE (dashed-dotted line).
The results in Proposition 1 extend those by Doz et al. (2012,2011) to the non-stationary case. A major difference between the EM algorithm in levels proposed in this paper, and the EM algorithm in first differences proposed by Doz et al. (2012), is relative to the way idiosyncratic components are modelled. Indeed, while by considering first differences it is implicitly assumed that all idiosyncratic components have a unit root, in our case we can distinguish between stationary and non-stationary idiosyncratic components -- i.e., we can allow for idiosyncratic trends only in some variables. This is not a minor difference, as it has substantial implications for the properties of the estimators.
First of all we model non-stationary idiosyncratic components as additional latent states rather than differencing them, thus improving efficiency (see also Remark 2 below). Second, when $$ \xi_{it}\sim I(0)$$, under D1 and D2 the QML estimator of the loadings of the i-th variable is obtained by minimizing the sample variance of $$ \xi_{it}$$. In this case this is not the same as differencing before estimation, since in that case the loadings would be estimated by minimizing the sample variance of $$ \Delta\xi_{it}$$. The resulting common component of the i-th variable has therefore different empirical properties: compared to our non-stationary approach, the common component estimated in first differences is likely to provide a better fit of the first differenced data, but not necessarily of the levels. Conversely, the common component obtained with our approach is likely to provide a better fit of the levels thus capturing better the lower frequencies -- and so the long-run trends -- and resulting in a smoother estimator, which however might have a worse fit of the differenced data.
We conclude this section by briefly discussing the possible mis-specifications introduced by assumptions D1, D2 and D3. In particular, we assume the vector of idiosyncratic shocks $$ e_t$$ to be i.i.d. Gaussian, thus imposing four restrictions on: (1) the cross-sectional dependence; (2) the variances; (3) the serial dependence; (4) the distribution. Let us consider the implications for the properties of the estimators of each of these restrictions -- see also Doz et al. (2011) for a similar discussion.
This is better illustrated by a simple example showing the properties of the KF (an analogous argument holds for the KS). Denote as $$ \mathbf P$$ the steady state of $$ \mathbf P_{t\vert t-1}$$ then it can be shown that $$ \mathbf P=\mathbf H\mathbf H'$$ (Lemma 4). Consider the case in which the parameters are given, $$ \xi_{t}\sim I(0)$$, and r=q, so that $$ \mathbf P$$ is invertible, then for $$ t\ge \bar t(T)$$ the KF estimator is such that
| $$\displaystyle \mathbf F_{t\vert t}$$ | $$\displaystyle = \mathbf F_{t\vert t-1} + \mathbf P \Lambda'(\Lambda\mathbf P\Lambda'+\mathbf R)^{-1}(\mathbf x_t-\Lambda\mathbf F_{t\vert t-1})$$ | |
| $$\displaystyle =\mathbf F_{t\vert t-1} +(\Lambda'\mathbf R^{-1}\Lambda+\mathbf P^{-1})^{-1}\Lambda'\mathbf R^{-1}(\mathbf x_t-\Lambda\mathbf F_{t\vert t-1})$$ | ||
| $$\displaystyle = (\Lambda'\mathbf R^{-1}\Lambda)^{-1}\Lambda'\mathbf R^{-1}\mathbf x_t + O(n^{-1})$$ | ||
| $$\displaystyle =\mathbf F_t + (\Lambda'\mathbf R^{-1}\Lambda)^{-1}\Lambda'\mathbf R^{-1}\xi_t+ O(n^{-1})$$ | ||
| $$\displaystyle =\mathbf F_t +O_p(n^{-1/2}),$$ |
where we used (in order) the Woodbury formula, assumption C2, the definition of $$ \mathbf x_t$$ in (12), and assumption A5. Clearly consistency of the KF does not depend on the specific assumption for $$ \mathbf R$$, as long as it is invertible. However for finite n the KF depends on $$ \mathbf R$$ and modeling also its out of diagonal terms could in principle improve its efficiency (e.g. Bai & Liao (2016)).
3 Trend and cycles
We now turn to estimation of common trends and common cycles. Notice that since we do not fully impose R1, the dynamic factors $$ f_t$$ are not identified and instead we have to deal with a TC decomposition of the static factors $$ \mathbf F_t$$, which can be carried out analogously to the one described in Section 2 for $$ f_t$$. Because of assumption C1 and restriction R3, for given values of q and d, the vector $$ \mathbf F_t$$ admits the factor representation:
| $$\displaystyle {\mathbf F}_t$$ | $$\displaystyle = \Phi_1\mathbf T_t + \Gamma_t,$$ |
where $$ \Gamma_t\sim I(0)$$, $$ \Phi_1$$ is $$ r\times (q-d)$$ and $$ \mathbf T_t$$ is the vector of $$ (q-d)$$ common trends with components $$ T_{jt}\sim I(1)$$ for $$ j=1,\ldots, (q-d)$$. Hence, in general the common trends admit the MA representation:
As a consequence of the results by Peña & Poncela (1997) and Proposition 1 above, given the estimated factors $$ \widehat {\mathbf F}_t$$, it is clear that, as $$ n,T\to\infty$$,
| $$\displaystyle \widehat {\mathbf S}:=\frac 1{T^2}\sum_{t=1}^{T} \widehat {\mathbf F}_t\widehat {\mathbf F}_{t}'\stackrel{}{\Rightarrow} \Phi_1\,{\mathcal B}(1)\,\Sigma_{\eta}^{1/2}\left(\int_0^1{\mathcal W}(u){\mathcal W}(u)'\mathrm d u\right)\Sigma_{\eta}^{1/2}\,{\mathcal B}(1)' \,\Phi_1',$$ | (19) |
where convergence is in the sense of weak convergence of the associated probability measures and $$ \{{\mathcal W}(u),\ 0\le u\le 1\}$$ is a $$ (q-d)$$-dimensional standard Wiener process. Hence, by virtue of (19), we can estimate the common trends $$ \mathbf T_t$$ as the first $$ (q-d)$$ principal components of the estimated static factors $$ \widehat {\mathbf F}_t$$ (Bai (2004); Peña & Poncela (2006)). Specifically, we denote by $$ (\widehat { \Phi}_1\;\widehat { \Phi}_0)$$ the $$ r\times r$$ matrix with columns given by the normalized eigenvectors of $$ \widehat {\mathbf S}$$, ordered according to the decreasing value of the corresponding eigenvalues, and such that $$ \widehat { \Phi}_1$$ is $$ r\times (q-d)$$ and $$ \widehat { \Phi}_0$$ is $$ r\times (r-q+d)$$. This leads to the estimator of common trends as the projection:
| $$\displaystyle \widehat {\mathbf T}_t = \widehat {\Phi}_1'\widehat {\mathbf F}_t.$$ |
As for the common cycles, notice first that, by projecting $$ \widehat {\mathbf F}_t$$ onto the columns of $$ \widehat { \Phi}_0$$, we obtain the $$ (r-q+d)$$-dimensional process
| $$\displaystyle \widehat {\mathbf G}_t=\widehat {\Phi}_{0}'\widehat {\mathbf F}_t,$$ |
which, by construction, is orthogonal to $$ \widehat {\mathbf T}_t$$. Moreover, $$ \widehat {\mathbf G}_t$$ is stationary since it belongs to the cointegration space of $$ \widehat {\mathbf F}_t$$ Zhang et al. (2016). However, by R3 we know that the cointegration space must have dimension d, but we do not impose R3 when estimating the static factors. Thus, we face the problem of identifying d cycles from the higher-dimensional stationary process $$ \widehat {\mathbf G}_t$$.
In order to identify the common cycles we then look for the d-dimensional projection of $$ \widehat {\mathbf G}_t$$ with maximum spectral density. In the empirical analysis of Section 4, we consider the VAR(2):
| $$\displaystyle \widehat {\mathbf G}_t = {\mathcal A}_1\widehat {\mathbf G}_{t-1}+{\mathcal A}_2\widehat {\mathbf G}_{t-2}+ v_t,$$ | (20) |
where $$ v_t\stackrel{w.n.}{\sim} (\mathbf 0_{r-q+d},\Sigma_v)$$ and $$ \det(\mathbf I_{r-q+d}- {\mathcal A}_1z-{\mathcal A}_2z^2)\neq 0$$ for $$ \vert z\vert\le 1$$. Once we estimate (20) we have its residuals $$ \widehat { v}_t$$ and their covariance matrix $$ \widehat {\Sigma}_v$$. Denote as $$ \widehat {{\mathcal H}}$$ the $$ (r-q+d)\times d$$ matrix having as columns the leading d normalized eigenvectors of $$ \widehat {\Sigma}_v$$. We then define the estimated cycle component as the d-dimensional projection:
| $$\displaystyle \widehat {\mathbf C}_t = \widehat {{\mathcal H}}'\widehat {\mathbf G}_t.$$ |
The estimated TC decomposition is then given by
| $$\displaystyle \widehat {\mathbf F}_t$$ | $$\displaystyle = \widehat {\Phi}_1 \widehat {\Phi}_1'\widehat {\mathbf F}_t+ \widehat {\Phi}_{0} \widehat {\Phi}_{0}'\widehat {\mathbf F}_t$$ | |
| $$\displaystyle =\widehat {\Phi}_1 \widehat {\mathbf T}_t + \widehat {\Phi}_{0}\widehat {\mathbf G}_t$$ | ||
| $$\displaystyle =\widehat {\Phi}_1 \widehat {\mathbf T}_t + \widehat {\Phi}_{0}\widehat {{\mathcal H}}\widehat {{\mathcal H}}'\widehat {\mathbf G}_t + \widehat {\Phi}_{0}\widehat {{\mathcal H}}_\perp\widehat {{\mathcal H}}'_\perp\widehat {\mathbf G}_t$$ | ||
| $$\displaystyle =\widehat {\Phi}_1 \widehat {\mathbf T}_t + \widehat {\Phi}_{0}\widehat {{\mathcal H}}\widehat {\mathbf C}_t + \widehat {\Phi}_{0}(\widehat {\mathbf G}_t-\widehat {{\mathcal H}}\widehat {\mathbf C}_t),$$ | (21) |
where $$ \widehat {{\mathcal H}}_\perp$$ is $$ (r-q+d)\times (r-q)$$ and such that $$ \widehat {{\mathcal H}}_\perp'\widehat {{\mathcal H}}=\mathbf 0_{(r-q)\times d}$$. The last term on the right-hand-side of (21) appears due to the mis-specification caused by not fully imposing R1 and R3 and in particular it has covariance of rank $$ (r-q)$$ and since $$ r>q$$ it is in general not zero.
Figure 2: Spectral Densities of Common Trends and Common Cycles
This figure reports for the data analyzed in Section 4 the spectral densities of the common trend $$ \Delta\widehat {\mathbf T}_t$$ (black line), the common cycles $$ \Delta\widehat {\mathbf C}_t$$ (red lines), and the residual cycles $$ (\Delta\widehat {\mathbf G}_t-\widehat {{\mathcal H}}\Delta\widehat {\mathbf C}_t)$$ (blue lines). On the horizontal axis we report periods $$ \tau _j$$ measured in years and corresponding to frequencies $$ \omega _j=\frac {2\pi }{4\tau _j}$$ (the data considered is quarterly).
To appreciate the meaning and the appropriateness of decomposition (21), in Figure 2 we show the spectral densities of the first differences of the three components of $$ \widehat {\mathbf F}_t$$ for the data analyzed in Section 4, where r=6, q=3, and d=2. As expected the estimated common trend $$ \widehat {\mathbf T}_t$$ (black line) contributes most at the lowest frequencies -- i.e., lower than $$ \frac \pi {10}$$ -- which correspond to periods higher than five years. Once we remove the common trend, of the remaining five processes $$ \widehat {\mathbf G}_t$$, the two estimated common cycles $$ \widehat {\mathbf C}_t$$ (red lines) capture most of the variation for almost all frequencies: one cycle dominates at periods longer than two years -- i.e., frequencies lower than $$ \frac\pi 4$$ -- and the other cycle dominates at periods shorter than two years -- i.e., frequencies higher than $$ \frac\pi 4$$. With respect to those two cycles, the residual three cycles $$ (\widehat {\mathbf G}_t-\widehat {{\mathcal H}}\widehat {\mathbf C}_t)$$ (blue lines) give a negligible contributions to the total variation. Given this empirical result, the extra term in (21) can be neglected and treated as a mis-specification error.
Finally, from (21), the estimated TC decomposition of the data immediately follows:
| $$\displaystyle x_{it}$$ | $$\displaystyle = \widehat {\lambda}_i' \widehat {\Phi}_1 \widehat {\mathbf T}_t +\widehat {\lambda}_i' \widehat {\Phi}_{0} \widehat {{\mathcal H}}\widehat {\mathbf C}_t +\widehat {\lambda}_i' \widehat {\Phi}_{0}(\widehat {\mathbf G}_t-\widehat {{\mathcal H}}\widehat {\mathbf C}_t)+\widehat {\xi}_{it},$$ |
which is the estimated counterpart of the representation given in (9).
4 Estimating the cyclical position of the economy
and the observation error
We now use our model to estimate the cyclical position of the US economy and the observation error. In particular, in Section 1 we will estimate "the observation error" by estimating the non-stationary approximate DFM as explained in Section 2. And, in Section 2 we will estimate "the cyclical position of the economy" by decomposing the common factors into common trends and common cycles using the TC decomposition discussed in Sections 2 and 3.
The following analysis is carried out on a large macroeconomic dataset comprising n=103 quarterly series from 1960:Q1 to 2017:Q1 describing the US economy. The complete list of variables and transformations is reported in Appendix D.
Compared to the papers that use small DFMs to estimate the cyclical position of the economy, which typically estimate the output gap using only high level variables such as GDP, the unemployment rate, and PCE price inflation, we include several other indicators, thus being able to capture information coming from a wider spectrum of the economy. Specifically, our datasets includes national account statistics, industrial production indexes, various price indexes including CPIs, PPIs, and PCE price indexes, various labor market indicators including indicators from both the household survey and the establishment survey as well as labor cost and compensation indexes, monetary aggregates, credit and loans indicators, housing market indicators, interest rates, the oil price, and the S&P500 index. Broadly speaking, all the variables that are $$ I(1)$$ are not transformed, while all the variables that are $$ I(2)$$ are differenced once. Notice that some variables should from a theoretical economic point of view always be considered as $$ I(0)$$ (e.g. inflation rates, unemployment rate, and interest rates) but since they exhibit a great deal of persistence are here treated as $$ I(1)$$. Finally, a linear trend is estimated where necessary before applying our methodology, thus accounting for the deterministic component in (1).
A thorough empirical analysis requires tackling two main preliminary problems. First, we need to determine the number of common trends $$ (q-d)$$, of common shocks q, and of static factors r. To determine the number of common trends $$ (q-d)$$ we use the criterion by Barigozzi et al. (2016b), which exploits the behaviour of the eigenvalues described in condition B2. This criterion indicates the presence of $$ (q-d)=1$$ common trend, which is in line with many theoretical models assuming a common productivity trend as the sole driver of long-run dynamics (e.g. Del Negro et al., 2007). To determine the number of common shocks q we use the test by Onatski (2009) and the criterion by Hallin & Liška (2007), which exploit the behaviour of the eigenvalues described in condition B1. Both methods indicate the presence of q=3 common shocks. Having determined q, as we explained in Section 1 by virtue of R2 we can set the number of static factors r according to their explained variance. By looking at Table 1 we can clearly see that $$ r\simeq2q$$, and therefore in our benchmark specification we set q=3 and r=6.4
Table 1: Percentage of explained variance
This table reports the percentage of total variance explained by the q largest eigenvalues of the spectral density matrix of $$ \Delta \mathbf x_t$$ and by the r largest eigenvalues of the covariance matrix of $$ \Delta \mathbf x_t$$.| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |
|---|---|---|---|---|---|---|---|---|---|---|
| q | 33.4 | 45.8 | 53.3 | 58.9 | 63.6 | 67.4 | 70.6 | 73.4 | 75.8 | 77.9 |
| r | 23.4 | 33.9 | 42.1 | 47.9 | 51.8 | 55.3 | 58.2 | 60.6 | 62.7 | 64.9 |
Second, we need to choose which idiosyncratic components to model as random walk, and which as white noises. Following the methodology proposed by Bai & Ng (2004), we can explicitly test the null-hypothesis H0: $$ {\rho}_i=1$$, and if we do not reject H0, we set $$ \rho_i=1$$, while if we reject H0, we set $$ \rho_i=0$$. This approach is applied to all variables in the dataset except GDP, GDI, unemployment rate, Federal funds rate, and CPI, core CPI, PCE, and core PCE inflation, for which we impose a priori $$ \rho_i=0$$. That is, while for most of the variables in the dataset we let the data determine what is driving their long run dynamics, we impose that the long-run dynamics of GDP, GDI, unemployment rate, Federal funds rate, and CPI, core CPI, PCE, and core PCE inflation are driven exclusively by macroeconomic shocks, with the idiosyncratic shocks accounting only for short-run movement.
1 Measuring Gross Domestic Output
A fundamental issue in economics is the measurement of aggregate real output, henceforth Gross Domestic Output (GDO). Historically, GDO has been measured mainly by the Gross Domestic Product (GDP), but GDP, which tracks all expenditures on final goods and services produced, is just an estimate of GDO. An equally acceptable estimate of the concept of GDO is represented by the Gross Domestic Income (GDI), which tracks all income received by those who produced the output. GDP is almost always preferred to GDI, the main reason being that it is released before GDI.5 However it has been shown that GDI reflects the business cycle fluctuations in true output growth better than GDP and moreover GDI is better than GDP in recognising the start of a recession (Nalewaik (2012,2010)).
In recent years, there has been interest in combining GDP and GDI to come up with a better estimate of GDO, where the rationale for doing so is that the difference between GDP and GDI is exclusively the result of measurement error -- using the NIPA table definition "statistical discrepancy" -- as these two statistics are in fact measuring the same thing. For example, starting from November 4, 2013, the Philadelphia Fed releases an estimate of GDO, called "GDPplus" proposed by Aruoba et al. (2016), which is defined as the common component of a bivariate one-factor model built with GDP and GDI growth rates. Similarly, and starting from July 30, 2015, the Bureau of Economic Analysis (BEA) releases "the average of GDP and GDI", which the Council of Economic Advisers refers to as GDO Council of Economic Advisers (2015).
Our approach differs from those mentioned above in that our estimate of GDO is not obtained by combining GDP and GDI, rather it is obtained by using all the 103 variables included in our dataset. In detail, we define GDO as that part of GDP/GDI that is driven by the macroeconomic (common) shocks, i.e., $$ \textrm{GDO}_t=\chi_t^\textrm{GDP}=\chi_t^\textrm{GDI}$$. To estimate GDO in this way, we estimate a constrained version of model (12)-(13), where we impose the restriction of equal common components: $$ \chi_t^\textrm{GDP}=\chi_t^\textrm{GDI}$$. This restriction is indeed corroborated by the data, as even if we do not impose it, the estimated $$ \chi_t^\textrm{GDP}$$ and $$ \chi_t^\textrm{GDI}$$ are nearly identical. In numbers, the standard deviation of $$ (\Delta y_t^\textrm{GDP}-\Delta y_t^\textrm{GDI})$$ is 1.93, while the standard deviation of $$ (\Delta \chi_t^\textrm{GDP}-\Delta \chi_t^\textrm{GDI})$$ is reduced to 0.28.
Figure 3: Gross Domestic Output
This figure reports different estimates of GDO. Black line: "the average of GDP and GDI" released by the BEA; blue line: "GDPplus" released by the Philadelphia Fed; red line: our estimate.
Figure 3 shows our proposed estimate of GDO (red line) together with "GDPplus" (blue line) and the "the average of GDP and GDI" released by the BEA (black line). Overall, the three measures are very similar, which is not surprising, as they are attempting to estimate the same quantity. However, three important differences emerges.
First, our estimate of GDO is smoother than the other two. This is not surprising. Compared to "GDPplus" and "the average of GDP and GDI", our estimate of GDO is constructed to contain a larger low frequency component, because it is estimated on data in levels rather than on growth rates. Moreover, because it is derived under the assumption that the idiosyncratic components of GDP and GDI are stationary, by construction our estimate of GDO captures all the low frequency movements of GDP and GDI.
Second, our estimate of GDO does not show any kind of residual seasonality in the last fifteen years, where the term "residual seasonality" refers to the presence of "lingering seasonal effects even after seasonal adjustment processes have been applied to the data" Moulton & Cowan (2016). Mainly motivated by the fact that since 2010 GDP growth in Q1 has been on average more than 1 percentage point lower than in the other quarters (NW plot of Figure 4), in recent years there has been lots of discussion on whether US GDP exhibit residual seasonality or not. The profession is not in agreement on this issue, as some authors (e.g. Gilbert et al., 2015; Lengermann et al., 2017) conclude that US GDP does not exhibit residual seasonality, while others (e.g. Lunsford, 2017; Rudebusch et al., 2015) find evidence of residual seasonality -- see Moulton & Cowan (2016) for a technical discussion on causes and remedies for residual seasonality in US GDP. Figure 4 shows average real GDO growth by quarter for our estimate of GDO (SE plot), "GDPplus" (SW plot), and "the average of GDP and GDI" (NE plot). As can be clearly seen, our estimate of GDO exhibits no residual seasonality whatsoever in the last 15 years.
Figure 4: Residual Seasonality
This figure reports average growth at an annual rate by quarter for GDP, "the average of GDP and GDI" released by the BEA, the Philadelphia Fed estimate of GDO (GDPplus), and our estimate of GDO (BL).
Third, our estimate of GDO in the recent years gives a different signal about the economy than the one given by `GDPplus" and "the average of GDP and GDI". According to our estimate, since 2010 quarterly annualized GDO growth was on average 1/2 of a percentage point higher than estimated by the BEA or the Philadelphia Fed, where this difference comes mainly from our estimate of GDO growth in the first quarter (see Figure 4), and therefore from the fact that our measure do not suffer of residual seasonality. In other words, based on the commonality in the data, the US economy grew at a faster pace than measured by national account statistics.
2 Measuring the output gap
Decomposing aggregate real output into potential output and output gap is a critical task for both monetary and fiscal policy, as the former is a key input for long-term projections, and the latter can be an important gauge of inflationary pressure. There exist many definitions of potential output and of output gap -- see Kiley (2013), for a survey of different methods and definitions. Here we use the definition implied by the TC decomposition discussed in Sections 2 and 3. Among the many existing approaches the most similar to ours are Fleischman & Roberts (2011) and Jarocinski & Lenza (2016), who use small dynamic factor models, Aastveit & Trovik (2014), who use a large stationary dynamic factor model combined with the Hodrick Prescott filter, and Morley & Wong (2017), who use a large stationary BVAR combined with the Beveridge and Nelson decomposition.
We compare our output gap estimate with the one produced by the Congressional Budget Office (CBO). The CBO estimates potential output and the output gap by using the so-called "production function approach" according to which potential output is that level of output consistent with current technologies and normal utilisation of capital and labour, and the output gap is the residual part of output. Specifically, the CBO model is based upon a textbook Solow growth model, with a neoclassical production function. Labour and productivity trends are estimated by using a variant of the Okun's law, so that actual output is above its potential (the output gap is positive), when the unemployment rate is below the natural rate of unemployment, which is in turn defined as the non-accelerating inflation rate of unemployment (NAIRU), i.e., that level of unemployment consistent with a stable inflation -- for further details see Congressional Budget Office (2001).
In Figure 5, we compare our measure of the output gap (red line) with the one produced by the CBO (blue line), where the left plot shows the level of the output gap, while the right plot shows the 4-quarter percentage change of the output gap. The main result emerging from Figure 5 is that our estimate of the output gap is remarkably similar to that of the CBO. However, there are a few periods in which the two estimates diverge, among which the main one is from the late nineties to the financial crisis. In particular, while according to the CBO the level of the output gap was negative between 2001:Q1 and 2005:Q4, according to our estimate in that same period the output gap was positive -- on average 2 1/2 percentage points higher than estimated by the CBO. Therefore, according to our estimate the level of the output gap right before the great financial crisis in 2007:Q4 was 1.3%, while according to the CBO was -0.7%, and hence we estimate that the level of slack in the economy at the trough of the crisis in 2009:Q2 was -4.5%, approximately 1 3/4 percentage points higher than estimated by the CBO.
Figure 5: Output gap
The left plot shows the level of the output gap estimated by the CBO (blue line) together with our estimate (red line). The right plot shows the 4-quarter percentage change of the output gap.
To conclude, let us emphasize that the fact that our estimate of the output gap is close to that of the CBO is a remarkable result, particularly so because our estimate of the output gap is very different from that of the CBO from both a technical and an interpretational point of view. Indeed, while the CBO constructs the output gap so that its level has a specific economic meaning, our measure of the output gap is simply the transitory/stationary part of the common component of output -- i.e., that part of aggregate real output that will disappear in the long-run.6 Therefore, our output gap estimate provides different and complementary information on the cyclical position of the economy than that contained in the CBO estimate. In particular, our estimate of the output gap seems more suitable to answer the question "which part of current growth is due to temporary factors?", while the measure of the CBO is certainly more suitable as a gauge of inflation pressure. This can explain in part the divergence of the two estimates in the 2000s. This period is characterized by stable and low inflation -- on average core CPI inflation between 2001:Q1 and 2007:Q4 was approximately 2.1%. Accordingly, the CBO estimates that slack is positive (i.e., the output gap negative). By contrast, our measure, which is not specifically affected by inflation, but it is more broadly influenced by the co-movement in the data, estimates that a part of the aggregate real output was transitory. This makes sense given that the years before the crisis were characterized by several factors that proved indeed transitory, such as the housing boom, a historically high share of sub-prime loan origination Haughwout & Okah (2009), and a large amount of equity withdrawal from housing Fuster et al. (2017). And, since our model includes a large number of variables, including housing indicators as well as loan and credit indicators, these transitory factors are captured by our model.
5 Discussion and conclusions
In this paper we disentangle long-run co-movements (common trends) from short-run co-movements (common cycles) in large datasets. To this end, we first estimate a non-stationary dynamic factor model by means of a Quasi Maximum Likelihood estimator based on the Expectation Maximisation algorithm, combined with the Kalman Filter and the Kalman Smoother estimators of the factors. We then disentangle common trends from common cycles by applying a non-parametric Trend-Cycle decomposition to the latent common factors and based on eigenanalysis of their long-run covariance. The asymptotic properties of this estimator are derived and discussed in the paper.
We estimate our model on a large panel of US quarterly macroeconomic time series with the goal of estimating the cyclical position of the economy and the observation error. After backing out the observation error, we show that our model naturally produces an estimate of aggregate real output, which we refer to as Gross Domestic Output (GDO). According to our estimate of GDO, since 2010 the US economy grew at a faster pace than measured by national account statistics.
We then use a Trend-Cycle decomposition to estimate the output gap. We compare our estimate of the output gap, which is entirely data-driven, with that produced by the Congressional Budget Office (CBO), which is instead based on theoretical economic models. It turns out that our estimate of the output gap is remarkably similar to that of the CBO except from the late nineties to the financial crisis, when our measure suggests that a greater part of the produced output was driven by transitory factors.
There are a number of aspects of our model that we have not fully developed in our empirical analysis and that are left for future research. First, due to the use of the Kalman Filter, our factor estimator is in principle able to handle both mixed frequency and missing data (e.g. Jungbacker et al., 2011; Mariano & Murasawa, 2003; Banbura & Modugno, 2014) and, therefore, it can be used for real-time analysis Giannone et al. (2008). This aspect is well-known to be particularly relevant when estimating the output gap, since as shown by Orphanides & van Norden (2002), end-of-sample revisions of GDP are of the same order of magnitude as the gap itself. Second, the use of the Kalman Filter makes our model suitable for scenario and counterfactual analysis based on conditional forecasts Banbura et al. (2015). Third, as shown in equation (21), our model naturally produces a Trend-Cycle decomposition for each variable in the dataset, and therefore it is possible to estimate other policy-relevant indicators, such as the unemployment gap (in our framework, the cycle component of the unemployment rate) or trend inflation (in our framework, the trend component of core CPI or the core PCE price indexes).
Our approach has been so far deliberately entirely data driven, and we have been careful in imposing the least possible amount of restrictions to let the data speak freely. This approach has undeniably some important merits, as estimation of GDO seems to fit naturally in our framework, and the Trend-Cycle decomposition that we obtain for GDO is economically sensible. However, we believe that imposing the statistical restrictions described in Section 1, thus eliminating the miss-specification error when computing the Trend-Cycle decomposition, as well as imposing economically meaningful constraints, seems to be an essential step forward. Our view is that one way to proceed is to consider Bayesian estimation of the model, so that our economic and statistical knowledge of the data can be included by means of suitable priors. All this is the subject of our current research.
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Appendix A Representation results
Hereafter, and throughout all appendices, we consider restriction R2 when s=1 as found empirically in Section 4. Therefore, $$ r=2q$$.
A.1 Proof of restriction R3
For the dynamic factors consider the VECM(2)
| $$\displaystyle \Delta f_t = - \mathbf a\mathbf b' f_{t-3} + \Gamma_1 \Delta f_{t-1}+ \Gamma_2 \Delta f_{t-2} +\mathbf u_t,$$ | (A1) |
where $$ \mathbf a$$ and $$ \mathbf b$$ are $$ q\times d$$ and for simplicity we consider just the case of two lags since this will imply a VECM(1) and therefore a VAR(2) for the static factors as implemented in (13).
First assume that in R1 we have $$ \mathbf K=\mathbf I_r$$. Our aim is then to find the correct VECM representation for $$ \mathbf F_t=( f_t'\; f_{t-1}')'$$ when the VECM in (A1), and restrictions R1 and R2 hold. Since we model $$ \mathbf F_t$$ as a VAR(2) we know that we must have a VECM(1) with reduced rank innovations by R1, hence
| $$\displaystyle \Delta \mathbf F_t = -\alpha\beta'\mathbf F_t+ \mathbf M\Delta\mathbf F_{t-1} + \mathbf H\mathbf u_t,$$ | (A2) |
where $$ \alpha$$ and $$ \beta$$ are $$ r\times c$$ with $$ c<r$$ and $$ \mathbf H$$ is $$ r\times q$$. Moreover, from Barigozzi et al. (2016a) we have $$ d\le c\le (r-q+d)$$. We are then interested in finding c, and the expressions of $$ \mathbf M$$, $$ \alpha$$, $$ \beta$$, and $$ \mathbf H$$ as functions of the parameters $$ \mathbf a$$, $$ \mathbf b$$, $$ \Gamma_1$$, and $$ \Gamma_2$$ in (A1). Let us write $$ \alpha=(\alpha_1'\;\alpha_2')'$$ and $$ \beta=(\beta_1'\;\beta_2')'$$ where $$ \alpha_1$$, $$ \alpha_2$$, $$ \beta_1$$, $$ \beta_2$$ are all $$ q\times c$$. We also denote as $$ \mathbf M_{ij}$$ for $$ i,j=1,2$$ the four $$ q\times q$$ blocks of $$ \mathbf M$$ and as $$ \mathbf H_1$$ and $$ \mathbf H_2$$ the two $$ q\times q$$ blocks of $$ \mathbf H$$. Following Proietti (1997), we define the $$ (2r+c)$$-dimensional vector
| $$\displaystyle \mathbf G_t=\left(\begin{array}{l} \Delta \mathbf F_t\\ \Delta \mathbf F_{t-1}\\ \beta'\mathbf F_{t-2} \end{array} \right)=\left(\begin{array}{l} \Delta f_t\\ \Delta f_{t-1}\\ \Delta f_{t-1}\\ \Delta f_{t-2}\\ \beta_1' f_{t-2}+ \beta_2' f_{t-3} \end{array} \right).$$ |
Then, the state-space form of (A2) is given by
| $$\displaystyle \Delta\mathbf F_t$$ | $$\displaystyle = {\mathbf Z} \mathbf G_t,$$ | |
| $$\displaystyle \mathbf G_t$$ | $$\displaystyle ={\mathbf T}\mathbf G_{t-1} +{\mathbf Z}'\mathbf H\mathbf u_t,$$ | (A3) |
with the $$ r\times (2r+c)$$ matrix $$ {\mathbf Z}=(\mathbf I_r \;\mathbf 0_r\;\mathbf 0_{r\times c})$$. Then,
| $$\displaystyle {\mathbf Z}'\mathbf H= \left(\begin{array}{l} \mathbf I_r \\ \mathbf 0_r\\ \mathbf 0_{c\times r} \end{array} \right)\left(\begin{array}{l} \mathbf H_1\\ \mathbf H_2 \end{array} \right) = \left(\begin{array}{l} \mathbf H_1\\ \mathbf H_2\\ \mathbf 0_q\\ \mathbf 0_q\\ \mathbf 0_{c\times q} \end{array} \right).$$ |
and the $$ (2r+c)\times (2r+c)$$ matrix $$ {\mathbf T}$$ is given by
| $$\displaystyle {\mathbf T} = \left(\begin{array}{lll} \mathbf M & -\alpha\beta' ... ...athbf 0_{c\times q} & \beta_1'& \beta_2' & \mathbf I_{c}\\ \end{array} \right).$$ |
Now using these definitions into (A3) we have five q-dimensional equations. The first one is
| $$\displaystyle \Delta f_t$$ | $$\displaystyle =\mathbf M_{11} \Delta f_{t-1}+ \mathbf M_{12} \Delta f_{t-2}-\alpha_1\beta_1' f_{t-2}-\alpha_1\beta_2' f_{t-3}+\mathbf H_1\mathbf u_t,$$ |
which is equivalent to (A1) when
| $$\displaystyle \mathbf M_{11}=\Gamma_1,\quad \mathbf M_{12}=\Gamma_2, \quad \alpha_1=\mathbf a,\quad\beta_1 =\mathbf 0_{q\times c},\quad\beta_2 =\mathbf b, \quad \mathbf H_1=\mathbf I_q,\quad c=d.$$ | (A4) |
The second equation is
| $$\displaystyle \Delta f_{t-1}$$ | $$\displaystyle =\mathbf M_{21}\Delta f_{t-1}+\mathbf M_{22}\Delta f_{t-2}-\alpha_2\beta_1'\Delta f_{t-2} -\alpha_2\beta_2'\Delta f_{t-3} -\alpha_2 \beta_1' f_{t-3}-\alpha_2 \beta_2' f_{t-4}+\mathbf H_2\mathbf u_t,$$ |
from which we see that we must also have
| $$\displaystyle \mathbf M_{21}=\mathbf I_q,\qquad \mathbf M_{22}=\mathbf 0_q, \qquad \alpha_2 =\mathbf 0_{q\times c}, \qquad \mathbf H_2=\mathbf 0_q.$$ | (A5) |
Under (A4) and (A5) the third, fourth and fifth equation in (A3) are just identities.
By imposing these restrictions we have the mapping between the VECM(1) for $$ \mathbf F_t$$ in (A2) and the VECM(2) for $$ f_t$$ in (A1)
| $$\displaystyle \mathbf M = \left(\begin{array}{ll} \Gamma_1 &\Gamma_2\\ \mathbf I_q &\mathbf 0_q \end{array} \right),\qquad \alpha = \left(\begin{array}{l} \mathbf a\\ \mathbf 0_{q\times d} \end{array} \right),\qquad \beta=\left(\begin{array}{l} \mathbf 0_{q\times d}\\ \mathbf b \end{array} \right), \qquad \mathbf H=\left(\begin{array}{c} \mathbf I_q\\ \mathbf 0_q \end{array} \right).$$ | (A6) |
If we now consider a generic $$ \mathbf K$$ in R1, then (A2) holds for $$ \mathbf F_t= \mathbf K( f_t'\; f_{t-1}')'$$ and (A6) becomes
| $$\displaystyle \mathbf M = \mathbf K\left(\begin{array}{ll} \Gamma_1 &\Gamma_2\\ \mathbf I_q &\mathbf 0_q \end{array} \right)\mathbf K^{-1},\qquad \alpha = \mathbf K\left(\begin{array}{l} \mathbf a\\ \mathbf 0_{q\times d} \end{array} \right),\qquad \beta={\mathbf K^{-1}}'\left(\begin{array}{l} \mathbf 0_{q\times d}\\ \mathbf b \end{array} \right), \qquad \mathbf H=\mathbf K\left(\begin{array}{c} \mathbf I_q\\ \mathbf 0_q \end{array} \right).$$ |
The cointegration rank c of $$ \mathbf F_t$$ is given by $$ \mathsf{rk}(\alpha\beta')=d$$.
A.2 Reduced and structural form of the state-space representation
Consider (12)-(13) written in matrix notation and using the companion form of the VAR
| $$\displaystyle \mathbf x_{t}$$ | $$\displaystyle =\Lambda \mathbf F_t + \xi_{t},$$ | (A7) |
| $$\displaystyle \left(\begin{array}{l} \mathbf F_t\\ \mathbf F_{t-1} \end{array} \right)$$ | $$\displaystyle =\left( \begin{array}{cc} \mathbf A_1&\mathbf A_2\\ \mathbf I_r &\mathbf 0_r \end{array} \right) \left(\begin{array}{c} \mathbf F_{t-1}\\ \mathbf F_{t-2} \end{array} \right) +\left(\begin{array}{c} \mathbf H\\ \mathbf 0_{r\times q} \end{array} \right) \mathbf u_t,$$ | (A8) |
with $$ \Lambda=(\lambda_1 \cdots\lambda_n)'$$ the $$ n\times r$$ loadings matrix. We call (A7)-(A8) the reduced form of the model. Similarly consider the structural form, where, for convenience, in the VAR we write twice the same equation:
| $$\displaystyle \mathbf x_{t}$$ | $$\displaystyle = \mathbf B_0 f_t+\mathbf B_1 f_{t-1}+ \xi_{t},$$ | (A9) |
| $$\displaystyle \left(\begin{array}{c} f_{t}\\ f_{t-1}\\ f_{t-1}\\ f_{t-2} \end{array} \right)$$ | $$\displaystyle = \left(\begin{array}{cccc} \Pi_1&\Pi_2&\mathbf 0_q &\Pi_3\\ \mathbf I_q &\mathbf 0_q&\mathbf 0_q&\mathbf 0_q\\ \mathbf I_q &\mathbf 0_q&\mathbf 0_q&\mathbf 0_q\\ \mathbf 0_q &\mathbf I_q&\mathbf 0_q&\mathbf 0_q\\ \end{array} \right) \left(\begin{array}{c} f_{t-1}\\ f_{t-2}\\ f_{t-2}\\ f_{t-3} \end{array} \right)+\left(\begin{array}{c} \mathbf I_q\\ \mathbf 0_q\\ \mathbf 0_q\\ \mathbf 0_q \end{array} \right) \mathbf u_t,$$ | (A10) |
where $$ \mathbf B_0=( b_{01}\cdots b_{0n})'$$, $$ \mathbf B_1=( b_{11}\cdots b_{1n})$$ are both $$ n\times q$$. Because of R1 there exists an invertible $$ r\times r$$ matrix $$ \mathbf K$$ such that
| $$\displaystyle \mathbf F_t = \mathbf K ( f_t'\, f_{t-1}')',$$ | $$\displaystyle \qquad ( f_t'\, f_{t-1}')'= \mathbf K^{-1}\mathbf F_t,$$ | (A11) |
| $$\displaystyle \Lambda = (\mathbf B_0\,\mathbf B_1)\mathbf K^{-1},$$ | $$\displaystyle \qquad (\mathbf B_0\,\mathbf B_1)=\Lambda \mathbf K.$$ | (A12) |
By comparing (A7)-(A8) with (A9)-(A10) and using (A11)-(A12), we have the parameters of the reduced form
| $$\displaystyle \mathbf A_1 =\mathbf K\left(\begin{array}{cc} \Pi_1&\Pi_2\\ \mathbf I_q&\mathbf 0_q \end{array}\right)\mathbf K^{-1},\qquad \mathbf A_2 =\mathbf K\left(\begin{array}{cc} \mathbf 0_q&\Pi_3\\ \mathbf 0_q&\mathbf 0_q \end{array}\right)\mathbf K^{-1},\qquad \mathbf H$$ | $$\displaystyle = \mathbf K\left(\begin{array}{c} \mathbf I_q\\ \mathbf 0_q \end{array}\right).$$ | (A13) |
The relations (A11), (A12) and (A13) are used throughout the following. Moreover, since a VAR(2) of dimension r can always be written as a VAR(1) of dimension $$ 2r$$, to avoid introducing further notation hereafter we consider the case of a VAR(1) for $$ \mathbf F_t$$, where $$ \mathbf A\equiv \mathbf A_1$$.
A.3 Properties of the structural and reduced form of the linear system
Proof. Equations (A9)-(A10) define a linear system with $$ r=2q$$ latent states $$ ( f_t'\; f_{t-1}')'$$. We say that a linear system is stabilizable if its unstable (non-stationary) states are controllable and all uncontrollable states are stable (see Anderson & Moore (1979), page 342), where stability is dictated by the eigenvalues of the matrix of VAR coefficients, which we denote as
| $$\displaystyle \widetilde{\mathbf A}={\left(\begin{array}{cc} \Pi_1&\Pi_2\\ \mathbf I_q &\mathbf 0_q \end{array} \right)}$$ | (A14) |
Because of cointegration, $$ \widetilde{\mathbf A}$$ has $$ (q-d)$$ unit eigenvalues corresponding to $$ (q-d)$$ unstable states. Moreover, $$ (\mathbf I_q-\Pi_1-\Pi_2)=\mathbf a\mathbf b'$$, where $$ \mathbf a$$ and $$ \mathbf b$$ have full column-rank $$ q\times d$$ matrices, so that $$ \mathsf{rk}(\mathbf a\mathbf b')=d$$. Define the $$ q\times (q-d)$$ matrices $$ \mathbf a_{\perp}$$ and $$ \mathbf b_{\perp}$$ such that $$ \mathbf a_{\perp}'\mathbf a=\mathbf b_{\perp}'\mathbf b=\mathbf 0_{(q-d)\times d}$$. Then, since $$ \mathsf{rk}(\mathbf a_{\perp}'\mathbf I_q)=(q-d)$$, the unstable states are controllable because they satisfy the Popov-Belevitch-Hautus rank test (see Franchi (2017), Theorem 2.1, and Antsaklis & Michel (2007), Corollary 6.11, page 249).
Now, by looking at (A10), we see that $$ \widetilde{\mathbf A}$$ has also $$ (r-q+d)=(q+d)$$ eigenvalues which are smaller than one in absolute value. Of these q correspond to states which are uncontrollable because they are not driven by any shock, but are also stable since have no dynamics (see the second equation in (A10)). The remaining d states follow a stable VAR, hence are controllable.
Similarly, we say that a linear system is detectable if its unstable states are observable and all unobservable states are stable (see Anderson & Moore (1979), page 342). First, notice that $$ \mathsf{rk}(\mathbf B_0)=q$$ and $$ \mathsf{rk}(\mathbf B_1)=q$$ because of C2 and (A12), therefore $$ \mathsf{rk}(\mathbf B_0 \mathbf b_{\perp})=(q-d)$$ and $$ \mathsf{rk}(\mathbf B_1 \mathbf b_{\perp})=(q-d)$$, which implies that the unstable states are observable because they satisfy the Popov-Belevitch-Hautus rank test (see Franchi (2017), Theorem 2.1, and Antsaklis & Michel (2007), Corollary 6.11, page 249). Since $$ \mathbf B_0$$ and $$ \mathbf B_1$$ have full column-rank there are no unstable unobservable states. This completes the proof.
Appendix B Details of estimation
This appendix provides details on estimation of factors and parameters which are necessary to introduce the notation required in the proofs in Appendix C. The model considered is (12)-(14), where for simplicity of exposition we consider a VAR(1) for the factors. As explained in the text without loss of generality we can assume $$ \xi_t\sim I(0)$$, thus considering as latent states only the r static factors.
Adding $$ I(1)$$ idiosyncratic components as latent states does not increase the dimension of the parameter space but it increases the dimension of the latent states vector. However, since the idiosyncratic components are assumed to be orthogonal (see D2), and moreover they are orthogonal to the static factors (see A2), the results in Appendix C can be generalized to this case by treating each new state separately.
B.1 Expectation Maximization algorithm
In what follows we denote the whole sample of observed data as $$ {\mathcal X}_T:= (\mathbf x_1\cdots \mathbf x_T)'$$ and the whole history of the unknown factors as $$ {\mathcal F}_T:= (\mathbf F_1\cdots\mathbf F_T)'$$. Recall that the vector of parameters is given by $$ {\Theta}:=($$vec$$ ({\Lambda})' \;$$vec$$ ({\mathbf A})'\;$$ vec$$ ({\mathbf H})'\; \;$$diag$$ ({\mathbf R}))'$$. To avoid heavier notation we use $$ \Theta$$ to indicate both a generic value of the parameters and the true value, whether we refer to one or the other is either clearly implied by the context or explicitly stated.
The joint pdf of data and factors is denoted as $$ f({\mathcal X}_T,{\mathcal F}_T;\Theta)$$ and the corresponding joint log-likelihood is denoted as $$ \ell({\mathcal X}_T,{\mathcal F}_T;\Theta):= \log f({\mathcal X}_T,{\mathcal F}_T;\Theta)$$ and it is such that
| $$\displaystyle \ell({\mathcal X}_T,{\mathcal F}_T;\Theta)=\ell({\mathcal X}_T\vert{\mathcal F}_T;\Theta)+\ell({\mathcal F}_T;\Theta),$$ | (B1) |
where $$ \ell({\mathcal X}_T\vert{\mathcal F}_T;\Theta)$$ is the log-likelihood of the data conditional on the factors and $$ \ell({\mathcal F}_T;\Theta)$$ is the marginal log-likelihood of the factors. Because of D2 and D3 all log-likelihoods are Gaussian and in particular
| $$\displaystyle \ell({\mathcal X}_T\vert{\mathcal F}_T;\Theta)$$ | $$\displaystyle =-\frac{nT}2\log(2\pi)-\frac T2\log\det (\mathbf R)-\frac 12\mathsf{tr}\left[({\mathcal X}_T-{\mathcal F}_T\Lambda')\mathbf R^{-1}({\mathcal X}_T-{\mathcal F}_T\Lambda')'\right].$$ | (B2) |
We first briefly review the steps of the EM algorithm, while in Section B.1.4 we prove that the values of the parameters obtained at convergence of the EM algorithm converge to the QML estimator.
B.1.1 Initialization
The EM algorithm is initialised with estimated parameters
| $$\displaystyle \widehat {\Theta}_0:=($$vec$$\displaystyle (\widehat {\Lambda}_0)' \;$$vec$$\displaystyle (\widehat {\mathbf A}_0)'\;$$ vec$$\displaystyle (\widehat {\mathbf H}_0)'\; \;$$diag$$\displaystyle (\widehat {\mathbf R}_0))'.$$ | (B3) |
These are obtained as follows. From the integration of the first r principal components of $$ \Delta \mathbf x_t$$ we have an estimator of the factors, $$ \widetilde {\mathbf F}_t$$, and then of the loadings, $$ \widehat {\Lambda}_0$$. The VAR parameters, $$ \widehat {\mathbf A}_0$$ are obtained by fitting a VAR on the estimated factors $$ \widetilde {\mathbf F}_t$$ and the columns of $$ \widehat {\mathbf H}_0$$ are given by the q leading eigenvectors of the covariance matrix of VAR residuals. Finally, the diagonal entries of $$ \widehat {\mathbf R}_0$$, are obtained as sample variances of $$ \widehat {\xi}_{it,0}=x_{it}-\widehat {\Lambda}_0\widetilde {\mathbf F}_t$$. Consistency of these estimators is discussed in Section C.3.
B.1.2 E-step
At iteration $$ k\ge 0$$, given $$ {\mathcal X}_T$$ and an estimate of the parameters $$ \widehat {\Theta}_k$$, we compute the expected log-likelihood as function of a generic value of the parameters $$ \Theta$$, where the expectation is computed with respect to the conditional distribution of $$ {\mathcal F}_T$$ given $$ {\mathcal X}_T$$ and when using $$ \widehat {\Theta}_k$$:
| $$\displaystyle \mathcal Q(\Theta;\widehat {\Theta}_k) := \int_{\mathbb{R}^{r\times T}} \ell({\mathcal X}_T, {\mathcal F}_T;\Theta) f({\mathcal F}_T\vert{\mathcal X}_T;\widehat {\Theta}_k) \mathrm d {\mathcal F}_T=\mathsf{E}_{\widehat {\Theta}_k} [\ell({\mathcal X}_T,{\mathcal F}_T;\Theta)\vert{\mathcal X}_T].$$ | (B4) |
In the Gaussian case (B4) depends on the conditional mean of the factors and their conditional second moments, which are obtained with the KS when using the parameters $$ \widehat {\Theta}_k$$ and are given by (see Section B.2 for details)
| $$\displaystyle {\mathbf F}_{t\vert T,k} := \mathsf{E}_{\widehat {\Theta}_k}[\mathbf F_t \vert{\mathcal X}_T],\qquad{\mathbf P}_{t\vert T,k} := \mathsf{E}_{\widehat {\Theta}_k}[(\mathbf F_t-{\mathbf F}_{t\vert T,k})(\mathbf F_t-{\mathbf F}_{t\vert T,k})' \vert{\mathcal X}_T].$$ | (B5) |
B.1.3 M-step
A new estimator of the parameters is obtained by maximising the expected log-likelihood over all possible values of the parameters:
| $$\displaystyle \widehat {\Theta}_{k+1} = \arg\!\!\!\!\!\max_{\Theta\in\Omega\subseteq \mathbb{R}^Q}\; \mathcal Q(\Theta;\widehat {\Theta}_k).$$ | (B6) |
Thus, maximizing the conditional expectation of (B2) and using (B5), we have the loadings estimator
Similarly we can obtain estimates of the other parameters $$ \widehat {\mathbf A}_{k+1}$$ and $$ \widehat {\mathbf R}_{k+1}$$ (see e.g. Banbura & Modugno (2014), for their expressions). The columns of $$ \widehat {\mathbf H}_{k+1}$$ are obtained as the q leading eigenvectors of the matrix
| $$\displaystyle \widehat {\Sigma}_{k+1} = \frac 1 T\left(\sum_{t=1}^T \mathsf{E}_{\widehat {\Theta}_k}[ \mathbf F_t\mathbf F_t' \vert{\mathcal X}_T] -\widehat {\mathbf A}_{k+1}\sum_{t=1}^T\mathsf{E}_{\widehat {\Theta}_k}[ \mathbf F_t\mathbf F_{t-1}' \vert{\mathcal X}_T] \right),$$ |
which is an estimator of the covariance of the VAR residuals, and where the second expectation can also be computed from the output of the KS.
B.1.4 Convergence
Denote the QML estimator of the parameters as
| $$\displaystyle \widehat {\Theta}^*:=($$vec$$\displaystyle (\widehat {\Lambda}^*)' \;$$vec$$\displaystyle (\widehat {\mathbf A}^*)'\;$$ vec$$\displaystyle (\widehat {\mathbf H}^*)'\; \;$$diag$$\displaystyle (\widehat {\mathbf R}^*))',$$ | (B7) |
then by definition we have
| $$\displaystyle \widehat {\Theta}^*= \arg\!\!\!\!\!\max_{\Theta\in\Omega\subseteq \mathbb{R}^Q}\;\ell({\mathcal X}_T;\Theta).$$ | (B8) |
where $$ \ell({\mathcal X}_T;\Theta)$$ is the log-likelihood of the data such that
| $$\displaystyle \ell({\mathcal X}_T;\Theta) = \ell({\mathcal X}_T, {\mathcal F}_T;\Theta)-\ell({\mathcal F}_T\vert{\mathcal X}_T;\Theta),$$ | (B9) |
where the first term on the rhs is given by (B1) and the second can be computed using the output of the KS for a given value of $$ \Theta$$. Define the expectation
| $$\displaystyle \mathcal H(\Theta;\widehat {\Theta}_k)$$ | $$\displaystyle := \int_{\mathbb{R}^{r\times T}}\ell({\mathcal F}_T\vert{\mathcal X}_T;\Theta) f({\mathcal F}_T\vert{\mathcal X}_T;\widehat {\Theta}_k) \mathrm d {\mathcal F}_T=\mathsf{E}_{\widehat {\Theta}_k}[\ell({\mathcal F}_T\vert{\mathcal X}_T;\Theta)\vert{\mathcal X}_T],$$ | (B10) |
and recall the definition of $$ \mathcal Q(\Theta;\widehat {\Theta}_k)$$ in the E-step in (B4). Since the lhs of (B9) does not depend on $$ {\mathcal F}_T$$, by taking its expectation with respect to the conditional distribution of $$ {\mathcal F}_T$$ given $$ {\mathcal X}_T$$ and when using $$ \widehat {\Theta}_k$$, for any $$ \Theta\in \Omega$$, we have
| $$\displaystyle \ell({\mathcal X}_T;\Theta)$$ | $$\displaystyle =\mathcal Q(\Theta;\widehat {\Theta}_k)-\mathcal H(\Theta;\widehat {\Theta}_k).$$ | (B11) |
Now, by definition of Kullback-Leibler divergence, we have (see also Lemma 1 in Dempster et al. (1977))
| $$\displaystyle \mathcal H(\widehat {\Theta}_{k+1};\widehat {\Theta}_k)$$ | $$\displaystyle \leq \mathcal H(\widehat {\Theta}_k;\widehat {\Theta}_k).$$ | (B12) |
Hence, from (B11) and (B12), for any k,
| $$\displaystyle \lim_{k\to\infty}\ell({\mathcal X}_T;\widehat {\Theta}_{k})=\ell({\mathcal X}_T;\widehat {\Theta}^*),\qquad \lim_{k\to\infty}\widehat {\Theta}_{k} = \widehat {\Theta}^*.$$ | (B13) |
The previous result holds in the limit $$ k\to\infty$$, but in practice we can run the EM algorithm only for a finite number of iterations $$ k_{\max}$$. Define, for any k,
| $$\displaystyle \Delta\ell_{k^*}<\eta, \;$$ but $$\displaystyle \; \Delta\ell_{k^*-1}\ge \eta.$$ |
Once we find $$ k^*$$, our estimator of the parameters is defined as $$ \widehat {\Theta}:= \widehat {\Theta}_{k^*}$$. The corresponding estimator of the factors is then defined as $$ \widehat {\mathbf F}_{t}:= {\mathbf F}_{t\vert T,k^*}$$, thus running the KS once last time using $$ \widehat {\Theta}_{k^*}$$. The rate of convergence of $$ \widehat {\Theta}$$ to $$ \widehat {\Theta}^*$$ in (B13) is studied in Lemma 9 below.
B.2 Kalman filter and Kalman smoother
For ease of notation assume to know the true parameter collected in the vector $$ \Theta$$. When using the KF-KS in the EM algorithm at a given iteration k, the factors' estimators given below are obtained by replacing $$ \Theta$$ with $$ \widehat {\Theta}_k$$ throughout this section. We denote the conditional expectation and covariance of the factors as
| $$\displaystyle \mathbf F_{t\vert s} := \mathsf{E}_{\Theta}[\mathbf F_t\vert{\mathcal X}_s], \qquad \mathbf P_{t\vert s} := \mathsf{E}_{\Theta}[(\mathbf F_t-\mathbf F_{t\vert s})(\mathbf F_t-\mathbf F_{t\vert s})'\vert{\mathcal X}_s],$$ | (B14) |
where $$ {\mathcal X}_s:=(\mathbf x_1\cdots \mathbf x_s)'$$. Under Gaussianity (D2 and D3) these can be computed with the KF-KS. Specifically, when $$ s=t-1$$ we have the optimal one-step-ahead prediction, when s=t we have the optimal in-sample estimator, when s=T we have the optimal smoother. The KF gives the first two cases while the KS gives the latter. In particular, we denote the KF-KS estimators respectively as: $$ \mathbf F_{t\vert t}$$ and $$ \mathbf F_{t\vert T}$$ when using the true value $$ \Theta$$ and as $$ {\mathbf F}_{t\vert t,k}$$ and $$ {\mathbf F}_{t\vert T,k}$$ when using $$ \widehat {\Theta}_k$$ (see also (B5)).
B.2.1 Forward iterations - Filtering
For given initial conditions $$ \mathbf F_{0\vert}$$ and $$ \mathbf P_{0\vert}$$, the KF is based on the forward iterations for $$ t=1,\ldots, T$$:
| $$\displaystyle \mathbf F_{t\vert t-1} = \mathbf A \mathbf F_{t-1\vert t-1},$$ | (B15) | |
| $$\displaystyle \mathbf P_{t\vert t-1} = \mathbf A\mathbf P_{t-1\vert t-1} \mathbf A' + \mathbf H\mathbf H',$$ | (B16) | |
| $$\displaystyle \mathbf F_{t\vert t} =\mathbf F_{t\vert t-1}+\mathbf P_{t\vert t-1}\Lambda'(\Lambda\mathbf P_{t\vert t-1}\Lambda'+\mathbf R)^{-1}(\mathbf x_t-\Lambda\mathbf F_{t\vert t-1}),$$ | (B17) | |
| $$\displaystyle \mathbf P_{t\vert t} =\mathbf P_{t\vert t-1}-\mathbf P_{t\vert t-1}\Lambda'(\Lambda\mathbf P_{t\vert t-1}\Lambda'+\mathbf R)^{-1}\Lambda\mathbf P_{t\vert t-1}.$$ | (B18) |
Moreover, by combining (B16) and (B18), we obtain the Riccati difference equation
| $$\displaystyle \mathbf P_{t+1\vert t}-\mathbf A\mathbf P_{t\vert t-1}\mathbf A'+\mathbf A\mathbf P_{t\vert t-1}\Lambda'(\Lambda\mathbf P_{t\vert t-1}\Lambda'+\mathbf R)^{-1}\Lambda\mathbf P_{t\vert t-1}\mathbf A'=\mathbf H\mathbf H'.$$ | (B19) |
The KF is started with given values of $$ {\mathbf F}_{0\vert}$$ and $$ {\mathbf P}_{0\vert}$$. The latter can be obtained with a diffuse prior run for $$ t<0$$ (see Koopman (1997), and Koopman & Durbin (2000), for details).
B.2.2 Backward iterations - Smoothing
The KS is then based on the backward iterations for $$ t=T,\ldots, 1$$:
| $$\displaystyle \mathbf F_{t\vert T}$$ | $$\displaystyle =\mathbf F_{t\vert t}+\mathbf P_{t\vert t}\mathbf A'\mathbf P_{t+1\vert t}^{-1}(\mathbf F_{t+1\vert T}-\mathbf F_{t+1\vert t}),$$ | (B20) |
| $$\displaystyle \mathbf P_{t\vert T}$$ | $$\displaystyle =\mathbf P_{t\vert t} + \mathbf P_{t\vert t} \mathbf A' \mathbf P_{t+1\vert t}^{-1} (\mathbf P_{t+1\vert T}-\mathbf P_{t+1\vert t})\mathbf P_{t+1\vert t}^{-1} \mathbf A \mathbf P_{t\vert t}.$$ | (B21) |
The KS iterations in (B20) require T inversions of $$ \mathbf P_{t\vert t-1}$$ and in the singular case $$ r>q$$ these matrices are likely to be singular (see also Lemma 5). There are two possible solutions to this problem. Kohn & Ansley (1983) suggest to use a generalized inverse of $$ \mathbf P_{t\vert t-1}$$, like the Moore-Penrose one. Alternatively, it can be proved that (B20) can be written in an equivalent way, which does not require matrix inversion, and which is defined by the backward iterations for $$ t=T,\ldots, 1$$:
| $$\displaystyle \mathbf F_{t\vert T}=\mathbf F_{t\vert t-1}+\mathbf P_{t\vert t-1}\mathbf r_{t-1},$$ | (B22) | |
| $$\displaystyle \mathbf r_{t-1}=\Lambda'(\Lambda\mathbf P_{t\vert t-1}\Lambda'+\mathbf R)^{-1}(\mathbf x_t-\Lambda\mathbf F_{t\vert t-1})+\mathbf L'_t\mathbf r_t,$$ | (B23) | |
| $$\displaystyle \mathbf P_{t\vert T}=\mathbf P_{t\vert t-1}-\mathbf P_{t\vert t-1}\mathbf N_{t-1}\mathbf P_{t\vert t-1},$$ | (B24) | |
| $$\displaystyle \mathbf N_{t-1}=\Lambda'(\Lambda\mathbf P_{t\vert t-1}\Lambda'+\mathbf R)^{-1}\Lambda+\mathbf L_t'\mathbf N_t\mathbf L_t,$$ | (B25) | |
| $$\displaystyle \mathbf L_t= \mathbf A-\mathbf A \mathbf P_{t\vert t-1} \Lambda' (\Lambda\mathbf P_{t\vert t-1}\Lambda'+\mathbf R)^{-1} \Lambda,$$ | (B26) |
where $$ \mathbf r_T=\mathbf 0_{r\times 1}$$, $$ \mathbf N_T=\mathbf 0_{r}$$ and by consturction $$ \mathbf A\mathbf P_{t\vert t}=\mathbf L_t\mathbf P_{t\vert t-1}$$ (see also Durbin & Koopman (2001), pp.70-73). Although numerically no appreciable differences emerge with respect to the chosen method, (B22)-(B26) are particularly useful for our proofs.
C.1 Preliminary results
Proof. Recall the Woodbury forumla
| $$\displaystyle ( C A C'+ B)^{-1}= B^{-1}- B^{-1} C( A^{-1}+ C' B^{-1} C)^{-1} C' B^{-1}.$$ | (C28) |
Denote $$ D=( A^{-1}+ C' B^{-1} C)^{-1}$$ then from (C28) the lhs of (C27) is equivalent to
| $$\displaystyle A C'\left[ B^{-1}- B^{-1} C D C' B^{-1}\right] = A\left[ C' B^{-1}- C' B^{-1} C D C' B^{-1}\right]= A\left[ I- C' B^{-1} C D\right] C' B^{-1}.$$ |
Then, (C27) becomes
Now notice that
| $$\displaystyle D = ( A^{-1}+ C' B^{-1} C)^{-1}$$ | $$\displaystyle = A( I+ A C' B^{-1} C)^{-1}.$$ | (C30) |
Substituting (C30) in (C29) and multiplying both sides on the left by $$ A^{-1}$$
Therefore (C27) is correct provided that $$ A C' B^{-1} C= C' B^{-1} C A$$ which is always true since both $$ A$$ and $$ C' B^{-1} C$$ are symmetric.
- (a)
- an $$ m\times m$$ matrix $$ A$$ symmetric and positive definite with $$ \mu_j^{A}\le M$$ for $$ j=1,\ldots,m$$;
- (b)
- an $$ n\times n$$ matrix $$ B$$ symmetric and positive definite with $$ \mu_j^{B}\le M$$ for $$ j=1,\ldots,n$$;
- (c)
- an $$ n\times m$$ matrix $$ C$$ such that $$ C' C$$ is positive definite with $$ \mu_j^{C'C}=M_j n$$ for $$ j=1,\ldots,m$$;
Proof. First notice that for two matrices $$ K$$ and $$ H$$ we have
| $$\displaystyle ( H+ K)^{-1}$$ | $$\displaystyle = ( H + K)^{-1}- K^{-1} + K^{-1}= ( H+ K)^{-1}( K - ( H + K)) K^{-1}+ K^{-1}$$ | |
| $$\displaystyle = ( H + K)^{-1}(- H) K^{-1} + K^{-1}= K^{-1}- ( H + K)^{-1} H K^{-1}.$$ | (C32) |
Then setting $$ K= C' B^{-1} C$$ and $$ H= A^{-1}$$ from (C32) we have
| $$\displaystyle ( A^{-1}+ C' B^{-1} C)^{-1}=( C' B^{-1} C)^{-1}-( A^{-1}+ C' B^{-1} C)^{-1} A^{-1}( C' B^{-1} C)^{-1}.$$ |
which implies
| $$\displaystyle ( A^{-1}+ C' B^{-1} C)^{-1} C' B^{-1} C = \mathbf I_m - ( A^{-1}+ C' B^{-1} C)^{-1} A^{-1}.$$ | (C33) |
Now consider the second term on the rhs of (C33)
| $$\displaystyle \Vert ( A^{-1}+ C' B^{-1} C)^{-1} A^{-1}\Vert^2$$ | $$\displaystyle \le\Vert ( A^{-1}+ C' B^{-1} C)^{-1}\Vert^2\,\Vert A^{-1}\Vert^2$$ | |
| $$\displaystyle \le\Vert ( C' B^{-1} C)^{-1}\Vert^2\,(\mu_n^A)^{-1}\le \Vert ( C' B^{-1} C)^{-1}\Vert^2\, M_1^{-1},$$ | (C34) |
where we use norm sub-additivity and the fact that by condition (a) $$ A$$ and $$ A^{-1}$$ are positive definite and therefore $$ \mu_n^{A}\geq M_1>0$$ and moreover $$ \mu_n^{A^{-1}}\geq M_2>0$$ thus by Weyl's inequality
| $$\displaystyle \Vert ( C' B^{-1} C)^{-1}\Vert^2\le \mathsf{tr}\left[( C' B^{-1} C)^{-2}\right]=\sum_{j=1}^m \frac 1{{(\mu_j^{C'B^{-1}C})}^2}=O(n^{-2}).$$ | (C35) |
Indeed, the m eigenvalues of $$ C' B^{-1} C$$ are also the m non-zero eigenvalues of $$ B^{-1/2} C C' B^{-1/2}$$, which are all $$ O(n)$$ by conditions (b) and (c). By using (C34) and (C35) in (C33) we prove the Lemma.
C.2 Consistency of KF and KS using the true value of the parameters
Proof. Define $$ \widetilde{\mathbf P}_{t\vert t-1}$$ as the conditional covariance matrix for the vector $$ ( f_t'\, f_{t-1}')'$$ given $$ {\mathcal X}_{t-1}$$. Then, due to stabilizability and detectability proved in Lemma 1, there exists a steady state for the structural model denoted as $$ \widetilde{\mathbf P}$$ solving the algebraic Riccati equation (ARE) and such that (see Anderson & Moore (1979), pp.76-77, and Harvey (1990), pp.118-119)
| $$\displaystyle \widetilde{\mathbf P}-\widetilde{\mathbf A}\widetilde{\mathbf P}\widetilde{\mathbf A}'+\widetilde{\mathbf A}\widetilde{ \mathbf P}\widetilde{\mathbf B}'(\widetilde{\mathbf B}\widetilde{ \mathbf P}\widetilde{\mathbf B}'+\mathbf R)^{-1}\widetilde{\mathbf B}\widetilde{ \mathbf P}\widetilde{\mathbf A}'=\left(\begin{array}{cc} \mathbf I_q&\mathbf 0_q\\ \mathbf 0_q&\mathbf 0_q \end{array}\right),$$ | (C36) |
where
| $$\displaystyle \widetilde{\mathbf A}=\left(\begin{array}{cc} \Pi_1&\Pi_2\\ \mathbf I_q&\mathbf 0_q \end{array} \right), \qquad \widetilde{\mathbf B} =(\mathbf B_0\,\mathbf B_1).$$ | (C37) |
Now since the structural model has only q controllable and observable states (see Lemma 1) and $$ \widetilde{\mathbf P}$$ is the steady state covariance of those states, $$ \mathrm{rk}(\widetilde{\mathbf P})=q$$. Define as $$ \mathbf V$$ the $$ r\times r$$ matrix of eigenvectors of $$ \widetilde{\mathbf P}$$ and as $$ \mathbf D$$ the $$ q\times q$$ diagonal matrix of its non zero eigenvalues, then
| $$\displaystyle \widetilde{\mathbf P}=\mathbf V\left(\begin{array}{cc} \mathbf D&\mathbf 0_q\\ \mathbf 0_q&\mathbf 0_q \end{array}\right)\mathbf V'=\mathbf V\left(\begin{array}{cc} \mathbf D^{1/2}&\mathbf 0_q\\ \mathbf 0_q&\mathbf 0_q \end{array}\right)\left(\begin{array}{cc} \mathbf D^{1/2}&\mathbf 0_q\\ \mathbf 0_q&\mathbf 0_q \end{array}\right)\mathbf V'= \mathbf W\left(\begin{array}{cc} \mathbf I_q&\mathbf 0_q\\ \mathbf 0_q&\mathbf 0_q \end{array}\right) \mathbf W',$$ | (C38) |
with
| $$\displaystyle \widetilde{\mathbf B}\widetilde{\mathbf P}\widetilde{\mathbf B}'=\widetilde{\mathbf B}\mathbf W \left(\begin{array}{cc} \mathbf I_q&\mathbf 0_q\\ \mathbf 0_q&\mathbf 0_q \end{array}\right)\mathbf W'\widetilde{\mathbf B}' = (\mathbf B_0^*\,\mathbf B_1^*) \left(\begin{array}{cc} \mathbf I_q&\mathbf 0_q\\ \mathbf 0_q&\mathbf 0_q \end{array}\right) \left(\begin{array}{c} {\mathbf B_0^*}'\\ {\mathbf B_1^*}' \end{array}\right) =\mathbf B_0^*\mathbf {B_0^*}',$$ | (C39) |
and
| $$\displaystyle \left(\begin{array}{cc} \mathbf I_q&\mathbf 0_q\\ \mathbf 0_q&\mathbf 0_q \end{array}\right)\mathbf W'\widetilde{\mathbf B}' = \left(\begin{array}{cc} \mathbf I_q&\mathbf 0_q\\ \mathbf 0_q&\mathbf 0_q \end{array}\right) \left(\begin{array}{c} {\mathbf B_0^*}'\\ {\mathbf B_1^*}' \end{array}\right)= \left(\begin{array}{c} {\mathbf B_0^*}'\\ \mathbf 0_{q\times n} \end{array}\right).$$ | (C40) |
From (C38), (C39), (C40), Lemmas 2 and 3, we have
| $$\displaystyle \left(\begin{array}{cc} \mathbf I_q&\mathbf 0_q\\ \mathbf 0_q&\mathbf 0_q \end{array}\right)\mathbf W'\widetilde{\mathbf B}'(\widetilde{\mathbf B}\widetilde{\mathbf P}\widetilde{\mathbf B}'+\mathbf R)^{-1}\widetilde{\mathbf B}\mathbf W\left(\begin{array}{cc} \mathbf I_q&\mathbf 0_q\\ \mathbf 0_q&\mathbf 0_q \end{array}\right)$$ | $$\displaystyle = \left(\begin{array}{c} {\mathbf B_0^*}'({\mathbf B_0^*}{\mathbf B_0^*}'+\mathbf R)^{-1}\\ \mathbf 0_{q\times n} \end{array}\right)(\mathbf B_0^*\,\mathbf B_1^*)\left(\begin{array}{cc} \mathbf I_q&\mathbf 0_q\\ \mathbf 0_q&\mathbf 0_q \end{array}\right)$$ | |
| $$\displaystyle = \left(\begin{array}{cc} ({\mathbf B_0^*}'\mathbf R^{-1}{\mathbf B_0^*}+\mathbf I_q)^{-1}{\mathbf B_0^*}'\mathbf R^{-1}{\mathbf B_0^*}&\mathbf 0_{q}\\ \mathbf 0_{q} & \mathbf 0_{q} \end{array}\right)$$ | ||
| $$\displaystyle =\left(\begin{array}{cc} \mathbf I_q +O(n^{-1})&\mathbf 0_q\\ \mathbf 0_q &\mathbf 0_q \end{array} \right).$$ | (C41) |
Notice that we can apply Lemma 3 to the top left $$ q\times q$$ block of (C41) since: $$ \mathbf I_q$$ trivially satisfies condition (a), $$ \mathbf R^{-1}$$ satisfies condition (b) because of D2 and $$ {\mathbf B_0^*}'\mathbf B_0^*$$ satisfies condition (c). Indeed, from definition (A12) we have
| $$\displaystyle \widetilde{\mathbf P} = \left(\begin{array}{cc} \mathbf I_q &\mathbf 0_q\\ \mathbf 0_q&\mathbf 0_q \end{array} \right) + O(n^{-1}).$$ | (C42) |
Now, notice that by construction $$ \mathbf P_{t\vert t-1} =\mathbf K\widetilde {\mathbf P}_{t\vert t-1}\mathbf K'$$ and since $$ \mathbf K$$ is full-rank then also the reduced form system is stabilizable and detectable, thus it has a steady state $$ \mathbf P$$ such that
Moreover, since $$ \mathbf K$$ does not depend on t nor n, we have $$ \mathbf P=\mathbf K\widetilde {\mathbf P}\mathbf K'$$ and the result follows directly from (C42). Last, from the definition of $$ \mathbf H$$ in (A13) we have also $$ \mathbf P=\mathbf H\mathbf H'$$.
| $$\displaystyle \sqrt n\, \Vert\mathbf F_{t\vert t}-\mathbf F_t\Vert = O_p(1),$$ | ||
| $$\displaystyle \sqrt n\,\Vert\mathbf F_{t\vert T}-\mathbf F_t\Vert = O_p(1).$$ |
Proof. By Lemma 4, the conditional covariance $$ \mathbf P_{t\vert t}$$ of the static factors given $$ {\mathcal X}_{t}$$ has a steady state $$ \mathbf S$$ such that (see (B18))
| $$\displaystyle \mathbf S = \mathbf P-\mathbf P\Lambda'(\Lambda\mathbf P\Lambda'+\mathbf R)^{-1}\Lambda\mathbf P.$$ | (C44) |
Then, notice that by Lemma 4 and (A12)
| $$\displaystyle \mathbf P\Lambda'$$ | $$\displaystyle = \mathbf K\left(\begin{array}{cc} \mathbf I_q &\mathbf 0_q\\ \mathbf 0_q&\mathbf 0_q \end{array} \right)\mathbf K'(\mathbf K')^{-1}\left(\begin{array}{c}\mathbf B_0'\\ \mathbf B_1'\end{array}\right)+O(n^{-1})=\mathbf K\left(\begin{array}{c} \mathbf B_0'\\ \mathbf 0_q \end{array} \right)+O(n^{-1}),$$ | (C45) |
| $$\displaystyle \Lambda\mathbf P\Lambda'$$ | $$\displaystyle = (\mathbf B_0\,\mathbf B_1)\mathbf K^{-1}\mathbf K\left(\begin{array}{cc} \mathbf I_q &\mathbf 0_q\\ \mathbf 0_q&\mathbf 0_q \end{array} \right)\mathbf K'(\mathbf K')^{-1}\left(\begin{array}{c}\mathbf B_0'\\ \mathbf B_1'\end{array}\right)+O(n^{-1})=\mathbf B_0\mathbf B_0'+O(n^{-1}).$$ | (C46) |
Using (C45) and (C46) and by applying Lemmas 2 and 3 we have
| $$\displaystyle \mathbf P\Lambda'(\Lambda\mathbf P\Lambda'+\mathbf R)^{-1}\Lambda\mathbf P$$ | $$\displaystyle = \mathbf K\left(\begin{array}{cc} \mathbf B_0'(\mathbf B_0\mathbf B_0'+\mathbf R)^{-1}\mathbf B_0&\mathbf 0_q\\ \mathbf 0_q&\mathbf 0_q \end{array}\right) \mathbf K'+O(n^{-1})$$ | |
| $$\displaystyle =\mathbf K\left(\begin{array}{cc} (\mathbf B_0'\mathbf R^{-1}\mathbf B_0+\mathbf I_q)^{-1}\mathbf B_0'\mathbf R^{-1}\mathbf B_0&\mathbf 0_q\\ \mathbf 0_q&\mathbf 0_q \end{array}\right) \mathbf K'+O(n^{-1})$$ | ||
| $$\displaystyle =\mathbf K\left(\begin{array}{cc} \mathbf I_q+O(n^{-1})&\mathbf 0_q\\ \mathbf 0_q&\mathbf 0_q \end{array}\right) \mathbf K'+O(n^{-1}).$$ | (C47) |
Notice that we can apply Lemma 3 to the top $$ q\times q$$ block of (C47) since: $$ \mathbf I_q$$ trivially satisfies condition (a), $$ \mathbf R^{-1}$$ satisfies condition (b) because of D2 and $$ {\mathbf B_0}'\mathbf B_0$$ satisfies condition (c) because of assumption C2 and definition (A12) (see also (C41) in the proof of Lemma 4). By substituting (C47) into (C44) and because of Lemma 4, we have
| $$\displaystyle \mathbf S = \mathbf K\left(\begin{array}{cc} \mathbf I_q &\mathbf 0_q\\ \mathbf 0_q&\mathbf 0_q \end{array} \right)\mathbf K' + O(n^{-1}) - \mathbf K\left(\begin{array}{cc} \mathbf I_q&\mathbf 0_q\\ \mathbf 0_q&\mathbf 0_q \end{array}\right) \mathbf K'+O(n^{-1})=O(n^{-1}).$$ | (C48) |
By substituting (C43) in (B18), from (C44) we have
Therefore, by substituting (C48) into (C49) and letting $$ n=T^{\gamma}$$ for $$ \gamma>0$$ and $$ \bar t\equiv \bar t (T)$$, because of (15) for $$ \bar t\le t \le T$$ we have
Now, let us consider $$ \mathbf P_{t\vert T}$$ defined in (B24). From (B18)
| $$\displaystyle \mathbf P_{t\vert t-1} = \mathbf P_{t\vert t}+\mathbf P_{t\vert t-1}\Lambda'(\Lambda\mathbf P_{t\vert t-1}\Lambda'+\mathbf R)^{-1}\Lambda\mathbf P_{t\vert t-1}.$$ | (C51) |
By substituting (C51) and (B25) in (B24) we have
| $$\displaystyle \mathbf P_{t\vert T}$$ | $$\displaystyle =\mathbf P_{t\vert t}+\mathbf P_{t\vert t-1}\mathbf L_t'\mathbf N_{t}\mathbf L_t\mathbf P_{t\vert t-1}.$$ | (C52) |
Since $$ \mathbf N_t$$ is function of $$ \mathbf P_{t\vert t-1}$$, because of Lemma 4, it has a steady state $$ \mathbf N$$ such that $$ \Vert\mathbf N\Vert=O(1)$$ and
Now, since $$ \mathbf A\mathbf P_{t\vert t}=\mathbf L_t\mathbf P_{t\vert t-1}$$, using (C50) and (C53), because of (15) for $$ \bar t\le t \le T$$ we have
| $$\displaystyle \mathbf P_{t\vert t-1}\mathbf L_t'\mathbf N_{t}\mathbf L_t\mathbf P_{t\vert t-1}=\mathbf P_{t\vert t}\mathbf A'\mathbf N_t\mathbf A\mathbf P_{t\vert t}=O(n^{-2}).$$ | (C54) |
By using (C50) and (C54) into (C52), for $$ \bar t\le t \le T$$ we have
By the law of iterated expectations, for $$ \bar t\le t \le T$$ we have (see also the definitions in (B14))
| $$\displaystyle \mathsf{E}_{\Theta}[(\mathbf F_t-\mathbf F_{t\vert t})(\mathbf F_t-\mathbf F_{t\vert t})'] = \mathsf{E}_{\Theta}[\mathsf{E}_{\Theta}[(\mathbf F_t-\mathbf F_{t\vert t})(\mathbf F_t-\mathbf F_{t\vert t})'\vert {\mathcal X}_t] ]= \mathsf{E}_{\Theta}[\mathbf P_{t\vert t}]=O(n^{-1}),$$ | ||
| $$\displaystyle \mathsf{E}_{\Theta}[(\mathbf F_t-\mathbf F_{t\vert T})(\mathbf F_t-\mathbf F_{t\vert T})'] = \mathsf{E}_{\Theta}[\mathsf{E}_{\Theta}[(\mathbf F_t-\mathbf F_{t\vert T})(\mathbf F_t-\mathbf F_{t\vert T})'\vert {\mathcal X}_T] ]= \mathsf{E}_{\Theta}[\mathbf P_{t\vert T}]=O(n^{-1}),$$ |
which imply mean-square convergence of the KF and KS when the parameters are known and for all $$ \bar t\le t \le T$$:
| $$\displaystyle \mathsf{E}_{\Theta}[\Vert\mathbf F_t-\mathbf F_{t\vert t}\Vert^2] = \sum_{j=1}^r\mathsf{E}_{\Theta}[(F_{j,t}-F_{j,t\vert t})^2] = \mathsf{tr}\left\{\mathsf{E}_{\Theta}[\mathbf P_{t\vert t}]\right\} = O(n^{-1}),$$ | ||
| $$\displaystyle \mathsf{E}_{\Theta}[\Vert\mathbf F_t-\mathbf F_{t\vert T}\Vert^2] = \sum_{j=1}^r\mathsf{E}_{\Theta}[(F_{j,t}-F_{j,t\vert T})^2] = \mathsf{tr}\left\{\mathsf{E}_{\Theta}[\mathbf P_{t\vert T}]\right\} = O(n^{-1}).$$ |
The result follows from Chebychev's inequality.
C.3 Consistency of KF and KS using estimated parameters
| $$\displaystyle \sqrt T\,\Vert\widehat {\lambda}_{i}^{*}-\lambda_i \Vert = O_p(1), \quad i=1,\ldots, n,$$ | ||
| $$\displaystyle \sqrt T\,\Vert\widehat {\mathbf A}^*-\mathbf A\Vert = O_p(1),$$ | ||
| $$\displaystyle \sqrt T\,\Vert\widehat {\mathbf H}^*-\mathbf H\Vert = O_p(1),$$ | ||
| $$\displaystyle \sqrt T\,\vert[\widehat {\mathbf R}]_{ii}^*-[\mathbf R]_{ii}\vert = O_p(1), \quad i=1,\ldots, n.$$ |
Proof. The QML estimator of the loadings, for any $$ i=1,\ldots, n$$, is given by
| $$\displaystyle \widehat {\lambda}_i^{*'}=\left(\sum_{t=1}^T x_{it} {\mathbf F}_{t}'\right) \left(\sum_{t=1}^T {\mathbf F}_{t}{\mathbf F}_{t}'\right)^{-1}.$$ | (C56) |
We know that $$ \mathbf F_t$$ is driven by $$ (q-d)$$ common trends (see C2), therefore we can find an orthonormal linear basis of dimension $$ (q-d)$$ such that the projection of $$ \mathbf F_t$$ onto this basis span the same space as the common trends. Collect the elements of this basis in the $$ r\times (q-d)$$ matrix $$ \gamma$$, and denote as $$ \gamma_\perp$$ the $$ r\times (r-q+d)$$ matrix such that $$ \gamma_\perp'\gamma=\mathbf 0_{(r-q+d)\times (q-d)}$$. Then, consider the $$ r\times r$$ linear transformation
| $$\displaystyle {\mathcal D}\mathbf F_t = \left(\begin{array}{c} \gamma'\mathbf F_t\\ \gamma_\perp'\mathbf F_t\\ \end{array} \right)=\left(\begin{array}{c} \mathbf Z_{1t}\\ \mathbf Z_{0t} \end{array}\right),$$ | (C57) |
where $$ \mathbf Z_{1t}$$ has all $$ (q-d)$$ components which are $$ I(1)$$ while $$ \mathbf Z_{0t}\sim I(0)$$ and is of dimension $$ (r-q+d)$$. Moreover, for $$ \mathbf Z_{1t}$$ we have the MA representation
$$ \zeta_t\stackrel{w.n.}{\sim} (\mathbf 0_{q-d},\Sigma_{\zeta})$$ with $$ \Sigma_{\zeta}$$ positive definite and $$ {\mathbf Q}(L)$$ is a $$ (q-d)\times (q-d)$$ one-sided, infinite matrix polynomial with square-summable coefficients and $$ \mathsf{rk}({\mathbf Q}(1))=(q-d)$$.
Because of orthonormality $$ {\mathcal D}'{\mathcal D}=\mathbf I_r$$. Then, the corresponding transformation of the loadings gives $$ \lambda_i'{\mathcal D}'=(\lambda_{i1}'\; \lambda_{i0}')$$ such that $$ x_{it}=\lambda_{i1}'\mathbf Z_{1t}+\lambda_{i0}'\mathbf Z_{0t}+\xi_{it}$$ and we also have $$ \widehat {\lambda}_i^{*'}{\mathcal D}'=(\widehat {\lambda}_{i1}^{*'}\;\widehat {\lambda}_{i0}^{*'})$$. Recall that $$ \mathbf Z_{1t}$$ and $$ \mathbf Z_{0t}$$ are orthogonal by construction, then we have
| $$\displaystyle \left(\begin{array}{c} \widehat {\lambda}_{i1}^{*'}-\lambda_{i1}'\\ \widehat {\lambda}_{i0}^{*'}-\lambda_{i0}' \end{array}\right)$$ | (C59) | |
| $$\displaystyle =\left(\begin{array}{cc} \left(\frac 1{T^2} \sum_{t=1}^T \xi_{it} {\mathbf Z}_{1t}'\right) \left(\frac 1{T^2}\sum_{t=1}^T {\mathbf Z}_{1t}{\mathbf Z}_{1t}'\right)^{-1}&\mathbf 0_{(q-d)\times (r-q+d)}\\ \mathbf 0_{(r-q+d)\times (q-d)}& \left(\frac 1{T} \sum_{t=1}^T \xi_{it} {\mathbf Z}_{0t}'\right) \left(\frac 1{T}\sum_{t=1}^T {\mathbf Z}_{0t}{\mathbf Z}_{0t}'\right)^{-1} \end{array} \right).$$ |
By Theorem 1 in Peña & Poncela (2006); Peña & Poncela (1997), under C1 and C3, and from (C58), as $$ T\to\infty$$,
| $$\displaystyle \frac 1{T^2}\sum_{t=1}^T {\mathbf Z}_{1t}{\mathbf Z}_{1t}'\Rightarrow \mathbf Q(1)\Sigma_{\zeta}^{1/2}\left(\int_0^1{\mathcal W}(u){\mathcal W}(u)'\mathrm d u\right)\Sigma_{\zeta}^{1/2}\mathbf Q(1)',$$ | (C60) |
where $$ {\mathcal W}(\cdot)$$ is a $$ (q-d)$$-dimensional standard Wiener process. Thus this term is $$ O_p(1)$$ and positive definite therefore invertible. Last, from (C58) we see that each component of $$ \Delta \mathbf Z_{1t}$$ has an MA representation with square summable coefficients ( $$ \Delta \mathbf Z_{1t}\sim I(0)$$ by construction), therefore $$ \mathsf{Var}(t^{-1}Z_{1jt})= O(1)$$ for any $$ j=1,\ldots, (q-d)$$ and any $$ t=1,\ldots, T$$. Thus, by using Gaussianity (see D2 and D3) and by A2 also independence of factors and idiosyncratic components, we can prove that, as $$ T\to\infty$$,
| $$\displaystyle \frac 1 {T^2}\sum_{t=1}^T\xi_{it} {\mathbf Z}_{1t}' = O_p\left({T}^{-1}\right).$$ | (C61) |
Moreover, from C1 it is easy to see that $$ {\mathbf Z}_{0t}$$ has an MA representation with square summable coefficients (it is stationary) and because of A2 and C3 we have, as $$ T\to\infty$$,
| $$\displaystyle \frac 1{T} \sum_{t=1}^T \xi_{it} {\mathbf Z}_{0t}' = O_p(T^{-1/2}),\qquad \frac 1{T} \sum_{t=1}^T \mathbf Z_{0t} {\mathbf Z}_{0t}' = \mathsf{E}[\mathbf Z_{0t} {\mathbf Z}_{0t}']+O_p(T^{-1/2}) = O_p(1).$$ | (C62) |
From (C59), (C60), (C61), and (C62), and since $$ {\mathcal D}$$ does not depend on T, we obtain the result.
Consider the VAR
| $$\displaystyle {\mathcal D}\mathbf F_t = ({\mathcal D}\mathbf A{\mathcal D}'){\mathcal D}\mathbf F_{t-1}+{\mathcal D}\mathbf H\mathbf u_t.$$ | (C63) |
such that $$ {\mathcal D}\mathbf H\mathbf u_t=(\mathbf e_{1t}'\;\mathbf {e}_{0t}')'$$ where $$ \mathbf e_{1t}$$ and $$ \mathbf e_{0t}$$ are white noise processes of dimensions $$ (q-d)$$ and $$ (r-q+d)$$, respectively. Then, similarly to (C59) we have
| $$\displaystyle {\mathcal D}(\widehat {\mathbf A}^*-\mathbf A){\mathcal D}'=$$ | (C64) | |
| $$\displaystyle =\left(\begin{array}{cc} \left(\frac 1{T^2} \sum_{t=1}^T \mathbf ... ...m_{t=1}^T {\mathbf Z}_{0t-1}{\mathbf Z}_{0t-1}'\right)^{-1} \end{array} \right)$$ |
Then, using the fact that $$ \mathbf e_{1t}$$ and $$ \mathbf e_{0t}$$ are white noise, it can be shown that
| $$\displaystyle \frac 1{T^2} \sum_{t=1}^T \mathbf e_{1t} {\mathbf Z}_{1t-1}' = O_p(T^{-1}),$$ | $$\displaystyle \frac 1{T^2} \sum_{t=1}^T \mathbf e_{1t} {\mathbf Z}_{0t-1}' = O_p(T^{-1/2}),$$ | (C65) | ||
| $$\displaystyle \frac 1{T} \sum_{t=1}^T \mathbf e_{1t} {\mathbf Z}_{0t-1}' = O_p(T^{-1}),$$ | $$\displaystyle \frac 1{T} \sum_{t=1}^T \mathbf e_{0t} {\mathbf Z}_{0t-1}' = O_p(T^{-1/2}).$$ |
Substituting (C60), (C62), and (C65) into (C64) and since $$ {\mathcal D}$$ does not depend on T, we have the result for the VAR parameters. Similar results can be proved for all other parameters.
| $$\displaystyle \mathbf F_{t\vert t}^* = \mathsf{E}_{\widehat {\Theta}^*}[\mathbf F_t\vert{\mathcal X}_t], \qquad \mathbf P_{t\vert t}^* = \mathsf{E}_{\widehat {\Theta}^*}[(\mathbf F_t-\mathbf F_{t\vert t}^*)(\mathbf F_t-\mathbf F_{t\vert t}^*)'\vert{\mathcal X}_t],$$ | ||
| $$\displaystyle \mathbf F_{t\vert T}^* = \mathsf{E}_{\widehat {\Theta}^*}[\mathbf F_t\vert{\mathcal X}_T],\qquad \mathbf P_{t\vert T}^* = \mathsf{E}_{\widehat {\Theta}^*}[(\mathbf F_t-\mathbf F_{t\vert T}^*)(\mathbf F_t-\mathbf F_{t\vert T}^*)'\vert{\mathcal X}_T].$$ |
Then, under condition (15) in the text, the following hold, for all $$ \bar t\le t \le T$$ and as $$ n,T\to\infty$$,
| $$\displaystyle \min(\sqrt n,\sqrt T)\, \Vert\mathbf F_{t\vert t}^*-\mathbf F_t\Vert = O_p(1),$$ | ||
| $$\displaystyle \min(\sqrt n,\sqrt T)\,\Vert\mathbf F_{t\vert T}^*-\mathbf F_t\Vert = O_p(1),$$ | ||
| $$\displaystyle \min( n,\sqrt T)\,\Vert\mathbf P_{t\vert t}^*\Vert = O_p(1),$$ | ||
| $$\displaystyle \min( n,\sqrt T)\,\Vert\mathbf P_{t\vert T}^*\Vert = O_p(1).$$ |
Proof. We start with three preliminary results. First, from (C50) in the proof of Lemma 5 we have
Second, because of Lemma 5 and (C66), we have
since $$ \mathbf u_t\stackrel{w.n.}{\sim} (\mathbf 0_q,\mathbf I_q)$$ and $$ \xi_t\sim I(0)$$.
Third, from transformation (C57), defined in the proof of Lemma 6, $$ {\mathcal D}\mathbf F_t=(\mathbf Z_{1t}'\;\mathbf Z_{0t}')'$$, we have $$ \Vert\mathbf Z_{1t}\Vert=O_p(\sqrt T)$$ and $$ \Vert\mathbf Z_{0t}\Vert=O_p(1)$$. Then, since $$ {\mathcal D}'{\mathcal D}=\mathbf I_r$$, as a consequence of Lemma 6 (see in particular (C59) and (C64)), the following hold
| $$\displaystyle (\widehat {\mathbf A}^*-\mathbf A)\mathbf F_t={\mathcal D}'{\mathcal D}(\widehat {\mathbf A}^*-\mathbf A){\mathcal D}'{\mathcal D}\mathbf F_t=O_p(T^{-1/2}),$$ | (C68) | |
| $$\displaystyle (\widehat {{\lambda}}_i^{*'}-{{\lambda}}_i')\mathbf A\mathbf F_t=(\widehat {{\lambda}}_i^{*'}-{{\lambda}}_i'){\mathcal D}'{\mathcal D}\mathbf A{\mathcal D}'{\mathcal D}\mathbf F_t=O_p(T^{-1/2}), \quad i=1,\ldots, n.$$ | (C69) |
Now, we compare the KF iterations, (B15)-(B18), with those obtained when using $$ \widehat {\Theta}^*$$:
| $$\displaystyle \mathbf F_{t\vert t-1}^* = \widehat {\mathbf A}^* \mathbf F_{t-1\vert t-1}^*,$$ | (C70) | |
| $$\displaystyle \mathbf P_{t\vert t-1}^* = \widehat {\mathbf A}^* \mathbf P_{t-1\vert t-1}^* \widehat {\mathbf A}^{*'} + \widehat {\mathbf H}^*\widehat {\mathbf H}^{*'},$$ | (C71) | |
| $$\displaystyle \mathbf F_{t\vert t}^* =\mathbf F_{t\vert t-1}^*+\mathbf P_{t\vert t-1}^*\widehat {\Lambda}^{*'}(\widehat {\Lambda}^*\mathbf P_{t\vert t-1}^*\widehat {\Lambda}^{*'}+\widehat {\mathbf R}^*)^{-1}(\mathbf x_t-\widehat {\Lambda}^*\mathbf F_{t\vert t-1}^*),$$ | (C72) | |
| $$\displaystyle \mathbf P_{t\vert t}^* =\mathbf P_{t\vert t-1}^*-\mathbf P_{t\vert t-1}^*\widehat {\Lambda}^{*'}(\widehat {\Lambda}^*\mathbf P_{t\vert t-1}^*\widehat {\Lambda}^{*'}+\widehat {\mathbf R}^*)^{-1}\widehat {\Lambda}^*\mathbf P_{t\vert t-1}^*.$$ | (C73) |
From (C70) we have
| $$\displaystyle \mathbf F_{t\vert t-1}^*-\mathbf F_{t\vert t-1}$$ | $$\displaystyle =\mathbf A(\mathbf F_{t-1\vert t-1}^*-\mathbf F_{t-1\vert t-1})+(\widehat {\mathbf A}^*-\mathbf A)(\mathbf F_{t-1\vert t-1}^*-\mathbf F_{t-1\vert t-1})+(\widehat {\mathbf A}^*-\mathbf A)\mathbf F_{t-1\vert t-1}$$ | |
| $$\displaystyle =\mathbf A(\mathbf F_{t-1\vert t-1}^*-\mathbf F_{t-1\vert t-1})+(\widehat {\mathbf A}^*-\mathbf A)(\mathbf F_{t-1\vert t-1}^*-\mathbf F_{t-1\vert t-1})+O_p(T^{-1/2}),$$ | (C74) |
since
| $$\displaystyle (\widehat {\mathbf A}^*-\mathbf A)\mathbf F_{t-1\vert t-1}$$ | $$\displaystyle =(\widehat {\mathbf A}^*-\mathbf A)(\mathbf F_{t-1\vert t-1}-\mathbf F_{t-1})+(\widehat {\mathbf A}^*-\mathbf A)\mathbf F_{t-1}$$ | |
| $$\displaystyle =O_p(T^{-1/2})O(e^{-(t-1)/2})+O_p(T^{-1/2})O(n^{-1/2})+O_p(T^{-1/2}),$$ |
because of Lemma 6, (C66) and (C68). Similarly, from (C71) we have
| $$\displaystyle \mathbf P_{t\vert t-1}^*$$ | $$\displaystyle -\mathbf P_{t\vert t-1} =\mathbf A(\mathbf P_{t-1\vert t-1}^*-\mathbf P_{t-1\vert t-1})\mathbf A'+(\widehat {\mathbf A}^*-\mathbf A)(\mathbf P_{t-1\vert t-1}^*-\mathbf P_{t-1\vert t-1})\mathbf A'$$ | (C75) |
| $$\displaystyle \;+(\widehat {\mathbf A}^*-\mathbf A)(\mathbf P_{t-1\vert t-1}^*-\mathbf P_{t-1\vert t-1})(\widehat {\mathbf A}^*-\mathbf A)'+\mathbf A(\mathbf P_{t-1\vert t-1}^*-\mathbf P_{t-1\vert t-1})(\widehat {\mathbf A}^*-\mathbf A)'$$ | ||
| $$\displaystyle \;+(\widehat {\mathbf A}^*-\mathbf A)\mathbf P_{t-1\vert t-1}\mathbf A'+(\widehat {\mathbf A}^*-\mathbf A)\mathbf P_{t-1\vert t-1}(\widehat {\mathbf A}^*-\mathbf A)'+\mathbf A\mathbf P_{t-1\vert t-1}(\widehat {\mathbf A}^*-\mathbf A)'$$ | ||
| $$\displaystyle \;+(\widehat {\mathbf H}^*-{\mathbf H})\mathbf H'+(\widehat {\mathbf H}^*-{\mathbf H})(\widehat {\mathbf H}^*-{\mathbf H})'+\mathbf H(\widehat {\mathbf H}^*-{\mathbf H})'$$ | ||
| $$\displaystyle =\mathbf A(\mathbf P_{t-1\vert t-1}^*-\mathbf P_{t-1\vert t-1})\mathbf A'+(\widehat {\mathbf A}^*-\mathbf A)(\mathbf P_{t-1\vert t-1}^*-\mathbf P_{t-1\vert t-1})\mathbf A'$$ | ||
| $$\displaystyle \;+(\widehat {\mathbf A}^*-\mathbf A)(\mathbf P_{t-1\vert t-1}^*-\mathbf P_{t-1\vert t-1})(\widehat {\mathbf A}^*-\mathbf A)'+\mathbf A(\mathbf P_{t-1\vert t-1}^*-\mathbf P_{t-1\vert t-1})(\widehat {\mathbf A}^*-\mathbf A)'+O_p(T^{-1/2}),$$ |
since
| $$\displaystyle (\widehat {\mathbf A}^*-\mathbf A)\mathbf P_{t-1\vert t-1}\mathbf A'=\mathbf A\mathbf P_{t-1\vert t-1}(\widehat {\mathbf A}^*-\mathbf A)'=O_p(T^{-1/2})O(e^{-(t-1)})+O_p(T^{-1/2})O(n^{-1}),$$ | ||
| $$\displaystyle (\widehat {\mathbf A}^*-\mathbf A)\mathbf P_{t-1\vert t-1}(\widehat {\mathbf A}^*-\mathbf A)'=O_p(T^{-1})O(e^{-(t-1)})+O_p(T^{-1})O(n^{-1}).$$ |
because of Lemma 6 and (C50) in the proof of Lemma 5 and
| $$\displaystyle \widehat {\mathbf H}^*\widehat {\mathbf H}^{*'}-{\mathbf H}{\mathbf H}'=(\widehat {\mathbf H}^*-{\mathbf H})\mathbf H'+\mathbf H(\widehat {\mathbf H}^*-{\mathbf H})'+(\widehat {\mathbf H}^*-{\mathbf H})(\widehat {\mathbf H}^*-{\mathbf H})'=O_p(T^{-1/2}),$$ |
because of Lemma 6.
Define
| $$\displaystyle \mathbf F_{t\vert t}^*-\mathbf F_{t\vert t}$$ | $$\displaystyle =\mathbf F_{t\vert t-1}^*-\mathbf F_{t\vert t-1}+(\widehat {{\mathcal K}}^*_t-{{\mathcal K}}_t)(\mathbf x_t-\Lambda \mathbf F_{t\vert t-1})$$ | |
| $$\displaystyle \;+(\widehat {{\mathcal K}}^*_t-{{\mathcal K}}_t)(\Lambda \mathbf F_{t\vert t-1}-\widehat {\Lambda}^* \mathbf F_{t\vert t-1}^*)+{{\mathcal K}}_t(\Lambda \mathbf F_{t\vert t-1}-\widehat {\Lambda}^* \mathbf F_{t\vert t-1}^*).$$ | (C76) |
Moreover, because of (C69)
| $$\displaystyle \frac{\widehat {\Lambda}^* \mathbf F_{t\vert t-1}^*-{\Lambda} \mathbf F_{t\vert t-1}}{\sqrt n}$$ | $$\displaystyle =\frac{\Lambda}{\sqrt n}(\mathbf F_{t\vert t-1}^*-\mathbf F_{t\vert t-1})+\left(\frac{\widehat {\Lambda}^*-\Lambda}{\sqrt n}\right)(\mathbf F_{t\vert t-1}^*-\mathbf F_{t\vert t-1})+\left(\frac{\widehat {\Lambda}^*-\Lambda}{\sqrt n}\right)\mathbf F_{t\vert t-1}$$ | |
| $$\displaystyle =\frac{\Lambda}{\sqrt n}(\mathbf F_{t\vert t-1}^*-\mathbf F_{t\vert t-1})+\left(\frac{\widehat {\Lambda}^*-\Lambda}{\sqrt n}\right)(\mathbf F_{t\vert t-1}^*-\mathbf F_{t\vert t-1})+O_p(T^{-1/2}),$$ | (C77) |
since
| $$\displaystyle \left(\frac{\widehat {\Lambda}^*-\Lambda}{\sqrt n}\right)\mathbf F_{t\vert t-1}$$ | $$\displaystyle =\left(\frac{\widehat {\Lambda}^*-\Lambda}{\sqrt n}\right)\mathbf A\mathbf F_{t-1\vert t-1}=\left(\frac{\widehat {\Lambda}^*-\Lambda}{\sqrt n}\right)\mathbf A(\mathbf F_{t-1\vert t-1}-\mathbf F_{t-1})+\left(\frac{\widehat {\Lambda}^*-\Lambda}{\sqrt n}\right)\mathbf A\mathbf F_{t-1}$$ | |
| $$\displaystyle =O_p(T^{-1/2})O(e^{-(t-1)/2})+O_p(T^{-1/2})O(n^{-1/2})+O_p(T^{-1/2}),$$ |
because of Lemma 6, (C66), (C69) and since $$ \Vert\mathbf A\Vert=O(1)$$. Similarly, from (C73)
| $$\displaystyle \mathbf P_{t\vert t}^*-\mathbf P_{t\vert t}$$ | $$\displaystyle =\mathbf P_{t\vert t-1}^*-\mathbf P_{t\vert t-1}^*-\left[(\widehat {{\mathcal K}}_t^*-{{\mathcal K}}_t)\Lambda\mathbf P_{t\vert t-1}\right .$$ | |
| $$\displaystyle +\;\left .(\widehat {{\mathcal K}}_t^*-{{\mathcal K}}_t)(\widehat {\Lambda}^*\mathbf P_{t\vert t-1}^*-\Lambda\mathbf P_{t\vert t-1}) + {{\mathcal K}}_t(\widehat {\Lambda}^*\mathbf P_{t\vert t-1}^*-\Lambda\mathbf P_{t\vert t-1})\right].$$ | (C78) |
Moreover,
| $$\displaystyle \frac{\widehat {\Lambda}^*\mathbf P_{t\vert t-1}^*-\Lambda\mathbf P_{t\vert t-1}}{\sqrt n}$$ | $$\displaystyle =\frac{\Lambda}{\sqrt n}(\mathbf P_{t\vert t-1}^*-\mathbf P_{t\vert t-1})+\left(\frac{\widehat {\Lambda}^*-\Lambda}{\sqrt n}\right)(\mathbf P_{t\vert t-1}^*-\mathbf P_{t\vert t-1})+\left(\frac{\widehat {\Lambda}^*-\Lambda}{\sqrt n}\right)\mathbf P_{t\vert t-1}$$ | |
| $$\displaystyle =\frac{\Lambda}{\sqrt n}(\mathbf P_{t\vert t-1}^*-\mathbf P_{t\vert t-1})+\left(\frac{\widehat {\Lambda}^*-\Lambda}{\sqrt n}\right)(\mathbf P_{t\vert t-1}^*-\mathbf P_{t\vert t-1})+O_p(T^{-1/2}),$$ | (C79) |
since
| $$\displaystyle \left(\frac{\widehat {\Lambda}^*-\Lambda}{\sqrt n}\right)\mathbf P_{t\vert t-1}$$ | $$\displaystyle =\left(\frac{\widehat {\Lambda}^*-\Lambda}{\sqrt n}\right)\left(\mathbf A\mathbf P_{t-1\vert t-1}\mathbf A'+\mathbf H\mathbf H'\right)$$ | |
| $$\displaystyle =O_p(T^{-1/2})O(e^{-(t-1)})+O_p(T^{-1/2})O(n^{-1})+O_p(T^{-1/2}),$$ |
because of Lemma 6 and (C50) in the proof of Lemma 5 and since $$ \Vert\mathbf A\Vert=O(1)$$ and $$ \Vert\mathbf H\Vert=O(1)$$. Following the same reasoning we also have
| $$\displaystyle \frac{\Lambda\mathbf P_{t\vert t-1}}{\sqrt n}$$ | $$\displaystyle =\frac{\Lambda}{\sqrt n}\left(\mathbf A\mathbf P_{t-1\vert t-1}\mathbf A'+\mathbf H\mathbf H'\right)=\frac{\Lambda}{\sqrt n}\left(O(e^{-(t-1)})+O(n^{-1})+\mathbf H\mathbf H'\right).$$ | (C80) |
Now, set t=1. Then, by noticing that $$ \mathbf F_{0\vert}^*=\mathbf F_{0\vert}$$ and $$ \mathbf P_{0\vert}^*=\mathbf P_{0\vert}$$, from (C74) and (C75) at t=1 we have
| $$\displaystyle \mathbf F_{1\vert}^*-\mathbf F_{1\vert} = O_p(T^{-1/2}), \qquad \mathbf P_{1\vert}^*-\mathbf P_{1\vert} = O_p(T^{-1/2}).$$ | (C81) |
Then, because of (C67), (C81) and Lemma 6, at t=1 we have
| $$\displaystyle \sqrt n(\widehat {{\mathcal K}}^*_1-{{\mathcal K}}_1)\left(\frac{\mathbf x_1}{\sqrt n}-\frac{\Lambda \mathbf F_{1\vert}}{\sqrt n}\right)=$$ | (C82) | |
| $$\displaystyle =\left[\mathbf P_{1\vert}^*\frac{\widehat {\Lambda}^{*'}}{\sqrt n}\left(\frac{\widehat {\Lambda}}{\sqrt n}\mathbf P_{1\vert}^*\frac{\widehat {\Lambda}^{*'}}{\sqrt n}+\frac{\widehat {\mathbf R}^*}n\right)^{-1}-\mathbf P_{1\vert}\frac{{\Lambda}'}{\sqrt n}\left(\frac{\Lambda}{\sqrt n}\mathbf P_{1\vert}\frac{\Lambda'}{\sqrt n}+\frac{\mathbf R}n\right)^{-1}\right]\left(\frac{\mathbf x_1}{\sqrt n}-\frac{\Lambda \mathbf F_{1\vert}}{\sqrt n}\right)$$ | ||
| $$\displaystyle =O_p(T^{-1/2}).$$ |
Moreover, from (C80)
| $$\displaystyle \sqrt n(\widehat {{\mathcal K}}_1^*-{{\mathcal K}}_1)\frac{\Lambda\mathbf P_{1\vert}}{\sqrt n}=O_p(T^{-1/2}).$$ | (C83) |
From (C76) and using (C74), (C77), (C81), and (C82) at t=1 we have
| $$\displaystyle \mathbf F_{1\vert 1}^*-\mathbf F_{1\vert 1}$$ | $$\displaystyle =\mathbf F_{1\vert}^*-\mathbf F_{1\vert}+(\widehat {{\mathcal K}}^*_1-{{\mathcal K}}_1)(\mathbf x_1-\Lambda \mathbf F_{1\vert})$$ | |
| $$\displaystyle \;+(\widehat {{\mathcal K}}^*_1-{{\mathcal K}}_1)(\Lambda \mathbf F_{1\vert}-\widehat {\Lambda} \mathbf F_{1\vert}^*)+{{\mathcal K}}_1(\Lambda \mathbf F_{1\vert}-\widehat {\Lambda} \mathbf F_{1\vert}^*)= O_p(T^{-1/2}).$$ | (C84) |
Similarly, from (C78) and using (C75), (C79), (C81), and (C83) at t=1 we have
| $$\displaystyle \mathbf P_{1\vert 1}^*-\mathbf P_{1\vert 1}$$ | $$\displaystyle =\mathbf P_{1\vert}^*-\mathbf P_{1\vert}^*-\left[(\widehat {{ K}}_1^*-{{ K}}_1)\Lambda\mathbf P_{1\vert}\right .$$ | |
| $$\displaystyle +\;\left .(\widehat {{\mathcal K}}_1^*-{{\mathcal K}}_1)(\widehat {\Lambda}^*\mathbf P_{1\vert}^*-\Lambda\mathbf P_{1\vert}) + {{\mathcal K}}_1(\widehat {\Lambda}^*\mathbf P_{1\vert}^*-\Lambda\mathbf P_{1\vert})\right]=O_p(T^{-1/2}).$$ | (C85) |
Then substituting (C84) into (C76) and (C85) into (C78) we have
| $$\displaystyle \mathbf F_{2\vert 1}^*-\mathbf F_{2\vert 1}=O_p(T^{-1/2}), \qquad \mathbf P_{2\vert 1}^*-\mathbf P_{2\vert 1}=O_p(T^{-1/2}).$$ | (C86) |
Then, because of (C67), (C86) and Lemma 6, at t=2 we have
| $$\displaystyle \sqrt n(\widehat {{\mathcal K}}^*_2-{{\mathcal K}}_2)\left(\frac{\mathbf x_2}{\sqrt n}-\frac{\Lambda \mathbf F_{2\vert 1}}{\sqrt n}\right)= O_p(T^{-1/2}),$$ | (C87) |
and from (C80)
| $$\displaystyle \sqrt n(\widehat {{\mathcal K}}_2^*-{{\mathcal K}}_2)\frac{\Lambda\mathbf P_{2\vert 1}}{\sqrt n}=O_p(T^{-1/2}).$$ | (C88) |
From, (C76) and using (C74), (C77), (C86), and (C87), at t=2 we have
| $$\displaystyle \Vert\mathbf F_{t\vert t}^*-\mathbf F_{t\vert t} \Vert = O_p(T^{-1/2}),\qquad \Vert\mathbf P_{t\vert t}^*-\mathbf P_{t\vert t} \Vert = O_p(T^{-1/2}),$$ | (C89) |
and also
| $$\displaystyle \Vert\mathbf F_{t\vert t-1}^*-\mathbf F_{t\vert t-1} \Vert = O_p(T^{-1/2}),\qquad \Vert\mathbf P_{t\vert t-1}^*-\mathbf P_{t\vert t-1} \Vert = O_p(T^{-1/2}).$$ | (C90) |
Because of Lemma 5 and (C89), we have for $$ \bar t\le t \le T$$,
| $$\displaystyle \Vert\mathbf F_{t\vert t}^*-\mathbf F_{t} \Vert \le \Vert\mathbf F_{t\vert t}^*-\mathbf F_{t\vert t} \Vert + \Vert\mathbf F_{t\vert t}-\mathbf F_{t} \Vert = O_p(T^{-1/2}) + O(n^{-1/2}),$$ | ||
| $$\displaystyle \Vert\mathbf P_{t\vert t}^* \Vert \le \Vert\mathbf P_{t\vert t}^*-\mathbf P_{t\vert t} \Vert + \Vert\mathbf P_{t\vert t} \Vert = O_p(T^{-1/2}) + O(n^{-1}).$$ |
Now compare the KS iterations, (B22)-(B26), with those obtained when using $$ \widehat {\Theta}^*$$:
| $$\displaystyle \mathbf F_{t\vert T}^*=\mathbf F_{t\vert t-1}^*+\mathbf P_{t\vert t-1}^*\mathbf r_{t-1}^*,$$ | (C91) | |
| $$\displaystyle \mathbf r_{t-1}^*=\widehat {\Lambda}^{*'}(\widehat {\Lambda}^*\mathbf P_{t\vert t-1}^*\widehat {\Lambda}^{*'}+\widehat {\mathbf R}^*)^{-1}(\mathbf x_t-\widehat {\Lambda}^*\mathbf F_{t\vert t-1}^*)+\mathbf L^{*'}_{t}\mathbf r_{t}^*,$$ | (C92) | |
| $$\displaystyle \mathbf P_{t\vert T}^*=\mathbf P_{t\vert t-1}^*-\mathbf P_{t\vert t-1}^*\mathbf N_{t-1}^*\mathbf P_{t\vert t-1}^*,$$ | (C93) | |
| $$\displaystyle \mathbf N_{t-1}^*=\widehat {\Lambda}^{*'}(\widehat {\Lambda}^*\mathbf P_{t\vert t-1}^*\widehat {\Lambda}^{*'}+\widehat {\mathbf R}^*)^{-1}\widehat {\Lambda}^*+\mathbf L_{t}^{*'}\mathbf N_{t}^*\mathbf L_{t}^*,$$ | (C94) | |
| $$\displaystyle \mathbf L_{t}^*= \widehat {\mathbf A}^*-\widehat {\mathbf A}^* \mathbf P_{t\vert t-1}^* \widehat {\Lambda}^{*'} (\widehat {\Lambda}^*\mathbf P_{t\vert t-1}^*\widehat {\Lambda}^{*'}+\widehat {\mathbf R}^*)^{-1} \widehat {\Lambda}^*,$$ | (C95) |
where $$ \mathbf r_{T}^*=\mathbf 0_{r\times 1}$$, $$ \mathbf N_{T}^*=\mathbf 0_{r}$$. First notice that obviously at t=T both KF and KS give the same result hence (C89) applies also in this case, and because of Lemma 6, (C67), (C77), and (C90), we have
| $$\displaystyle \mathbf r_{T-1}^*-\mathbf r_{T-1}=O_p(T^{-1/2}),\qquad \mathbf N_{T-1}^*-\mathbf N_{T-1}=O_p(T^{-1/2}).$$ | (C96) |
Moreover, from (C95), because of Lemma 6 and (C90), we have
| $$\displaystyle \mathbf L_{t}^*-\mathbf L_t$$ | $$\displaystyle =\widehat {\mathbf A}^*-\mathbf A-\sqrt n\left[\widehat {\mathbf A}^* \widehat {{\mathcal K}}_t^* \frac{\widehat {\Lambda}^*}{\sqrt n}-{\mathbf A} {{\mathcal K}}_t \frac{{\Lambda}}{\sqrt n}\right]= O_p(T^{-1/2}).$$ | (C97) |
Then, from (C92), because of (C67), (C77), (C90), (C96) and (C97), at $$ t=T-1$$ we have
| $$\displaystyle \mathbf r_{T-2}^*-\mathbf r_{T-2}=O_p(T^{-1/2}),\qquad \mathbf N_{T-2}^*-\mathbf N_{T-2}=O_p(T^{-1/2}).$$ | (C98) |
Therefore, from (C91) and (C93), because of (C90) and (C98), we have
| $$\displaystyle \mathbf F_{T-1\vert T}^*-\mathbf F_{T-1\vert T}= O_p(T^{-1/2}), \qquad \mathbf P_{T-1\vert T}^*-\mathbf P_{T-1\vert T}= O_p(T^{-1/2}).$$ | (C99) |
By repeating the same reasoning for $$ t=(T-2),\ldots ,1$$, we have
| $$\displaystyle \Vert\mathbf F_{t\vert T}^*-\mathbf F_{t\vert T}\Vert=O_p(T^{-1/2}), \qquad \Vert\mathbf P_{t\vert T}^*-\mathbf P_{t\vert T}\Vert=O_p(T^{-1/2}).$$ | (C100) |
Because of Lemma 5 and (C100), we have for $$ \bar t\le t \le T$$
| $$\displaystyle \Vert\mathbf F_{t\vert T}^*-\mathbf F_{t} \Vert \le \Vert\mathbf F_{t\vert T}^*-\mathbf F_{t\vert T} \Vert + \Vert\mathbf F_{t\vert T}-\mathbf F_{t} \Vert = O_p(T^{-1/2}) + O(n^{-1/2}),$$ | ||
| $$\displaystyle \Vert\mathbf P_{t\vert T}^* \Vert \le \Vert\mathbf P_{t\vert T}^*-\mathbf P_{t\vert T} \Vert + \Vert\mathbf P_{t\vert T} \Vert = O_p(T^{-1/2}) + O(n^{-1}),$$ |
which completes the proof.
| $$\displaystyle \min(\sqrt n,\sqrt T)\,\Vert\widehat {\lambda}_{i0}'-\lambda_i'\mathbf J^{-1}\Vert = O_p(1), \quad i=1,\ldots, n,$$ | ||
| $$\displaystyle \min(\sqrt n,\sqrt T)\,\Vert\widehat {\mathbf A}_0-\mathbf J\mathbf A\mathbf J^{-1}\Vert = O_p(1),$$ | ||
| $$\displaystyle \min(\sqrt n,\sqrt T)\,\Vert\widehat {\mathbf H}_0-\mathbf J\mathbf H\Vert = O_p(1),$$ | ||
| $$\displaystyle \min(\sqrt n,\sqrt T)\,\vert[\widehat {\mathbf R}]_{ii,0}-[\mathbf R]_{ii}\vert = O_p(1), \quad i=1,\ldots, n.$$ |
Moreover, under E1 and E2 we have $$ \mathbf J=\mathbf I_r$$.
Proof. When E2 holds the proof Lemmas 3 and 5 in Barigozzi et al. (2016b) where is shown that $$ \mathbf J$$ is a diagonal matrix with entries $$ \pm 1$$. If we impose also E1 the sign indeterminacy is fixed and $$ \mathbf J=\mathbf I_r$$.
| $$\displaystyle \sqrt T\,\Vert\widehat {\lambda}_{ik^*}-\lambda_i \Vert = O_p(1), \quad i=1,\ldots, n,$$ | ||
| $$\displaystyle \sqrt T\,\Vert\widehat {\mathbf A}_{k^*}-\mathbf A\Vert = O_p(1),$$ | ||
| $$\displaystyle \sqrt T\,\Vert\widehat {\mathbf H}_{k^*}-\mathbf H\Vert = O_p(1),$$ | ||
| $$\displaystyle \sqrt T\,\vert[\widehat {\mathbf R}]_{ii,k^*}-[\mathbf R]_{ii}\vert = O_p(1), \quad i=1,\ldots, n.$$ |
Proof. Define the $$ Q\times Q$$ matrices
| $$\displaystyle {\mathcal I}(\Theta)=-\nabla^2_{\Theta\Theta'}\ \ell({\mathcal X}_T;\Theta),$$ | ||
| $$\displaystyle {\mathcal I}_0(\Theta) = -\int_{\mathbb{R}^{r\times T}}\nabla^2_{\Theta\Theta'}\ \ell({\mathcal X}_T,{\mathcal F}_T;\Theta) f({\mathcal F}_T\vert{\mathcal X}_T;\Theta)\mathrm d{\mathcal F}_T = -\mathsf{E}_{\Theta}[\nabla^2_{\Theta\Theta'}\ \ell({\mathcal X}_T,{\mathcal F}_T;\Theta)\vert{\mathcal X}_T],$$ | ||
| $$\displaystyle {\mathcal I}_1(\Theta) = -\int_{\mathbb{R}^{r\times T}}\nabla^2_{\Theta\Theta'}\ \ell({\mathcal F}_T\vert{\mathcal X}_T,{\mathcal F}_T;\Theta) f({\mathcal F}_T\vert{\mathcal X}_T;\Theta)\mathrm d{\mathcal F}_T= -\mathsf{E}_{\Theta}[\nabla^2_{\Theta\Theta'}\ \ell({\mathcal F}_T\vert{\mathcal X}_T;\Theta)\vert{\mathcal X}_T],$$ |
then, since $$ {\mathcal I}(\Theta)$$ does not depend on $$ {\mathcal F}_T$$ from (B1) and (B11)
Moreover, at convergence of the EM algorithm (iteration $$ k^*$$) we have the Taylor approximation
| $$\displaystyle (\widehat {\Theta}_{k^*}-\widehat {\Theta}^*)$$ | $$\displaystyle = (\widehat {\Theta}_{k^*-1}-\widehat {\Theta}^*){\mathcal R}(\widehat {\Theta}^*) + O\left(\Vert\widehat {\Theta}_{k^*}-\widehat {\Theta}^*\Vert^2\right),$$ | (C102) |
where following Meng & Rubin (1994) we have (see also (C101))
| $$\displaystyle \Vert\widehat {\Theta}_{k^*}-\widehat {\Theta}^*\Vert$$ | $$\displaystyle \le \Vert \widehat {\Theta}_{0}-\widehat {\Theta}^*\Vert \;\Vert{\mathcal R}(\widehat {\Theta}^*)\Vert^{k^*}.$$ | (C103) |
Hereafter, denote $$ \zeta_{nT}:=\max(n^{-1/2},T^{-1/2})$$. Consider (C103) for the estimated loadings, for any $$ i=1,\ldots, n$$, and using Gaussianity (see D2 and D3), we have
| $$\displaystyle \Vert\widehat {\lambda}_{ik^*}-\widehat {\lambda}_i^*\Vert$$ | $$\displaystyle \le \Vert \widehat {\lambda}_{i0}-\widehat {\lambda}_i^*\Vert \;\Big\Vert\mathbf I_r- \Big(\sum_{t=1}^T\mathbf F_t\mathbf F_t'\Big)\Big(\sum_{t=1}^T\mathsf{E}_{\widehat {\Theta}^*}[\mathbf F_t\mathbf F_t'\vert{\mathcal X}_T]\Big)^{-1}\Big\Vert^{k^*}$$ | |
| $$\displaystyle =\Vert \widehat {\lambda}_{i0}-\widehat {\lambda}_i^*\Vert \;\Big\Vert\mathbf I_r- \Big(\frac 1 {T^2}\sum_{t=1}^T\mathbf F_t\mathbf F_t'\Big)\Big(\frac 1 {T^2}\sum_{t=1}^T\left(\mathbf F_{t\vert T}^*\mathbf F_{t\vert T}^{*'}+\mathbf P^*_{t\vert T}\right)\Big)^{-1}\Big\Vert^{k^*}.$$ | (C104) |
Therefore, by condition (15) in the text $$ {\bar t}=O(\log T)$$, because of Lemma 7, we have
since $$ \Vert\mathbf F_{t\vert T}^*\mathbf F_{t\vert T}^{*'}+\mathbf P^*_{t\vert T}\Vert = O_p(T)$$ and $$ \Vert\mathbf F_t\mathbf F_t'\Vert = O_p(T)$$, because $$ \mathbf F_t\sim I(1)$$ and $$ \mathbf F_{t\vert T}^*\sim I(1)$$, and
| $$\displaystyle \frac 1 {T^2} \sum_{t=1}^{\bar t-1}\left(\mathbf F_{t\vert T}^*\mathbf F_{t\vert T}^{*'}+\mathbf P^*_{t\vert T}\right)=O_p\left(\frac {\log T}{T}\right),\qquad \frac 1{T^2}\sum_{t=1}^{T}\mathbf F_{t}\mathbf F_{t}'- \frac 1 {T^2}\sum_{t=\bar t}^{T}\mathbf F_{t}\mathbf F_{t}'= O_p\left(\frac {\log T}{T}\right).$$ |
Moreover, because of Lemmas 8 and 6, we have
| $$\displaystyle \Vert \widehat {\lambda}_{i0}-\widehat {\lambda}_i^*\Vert$$ | $$\displaystyle \le \Vert \widehat {\lambda}_{i0}-{\lambda}_i\Vert +\Vert \widehat {\lambda}_i^*-{\lambda}_i\Vert = O_p(\zeta_{nT}) + O_p(T^{-1/2}).$$ | (C106) |
By substituting (C105) and (C106) into (C104), we have
| $$\displaystyle \Vert\widehat {\lambda}_{ik^*}-\widehat {\lambda}_i^*\Vert=o_p(\zeta_{nT}\,T^{-k^*/2}).$$ | (C107) |
Finally, because of Lemma 6 and (C107), we have
| $$\displaystyle \Vert \widehat {\lambda}_{ik^*}-{\lambda}_i\Vert$$ | $$\displaystyle \le \Vert \widehat {\lambda}_{ik^*}-\widehat {\lambda}_i^*\Vert + \Vert \widehat {\lambda}_{i}^*-{\lambda}_i\Vert =o_p(T^{-k^*/2})+O_p(T^{-1/2}).$$ |
The proof for the other parameters follows the same steps by taking the appropriate second derivatives and applying the results in Lemma 7.
C.4 Proof of Proposition 1
Consistency of the estimated loadings is proved in Lemma 9. Recalling that $$ \widehat {\Theta}:= \widehat {\Theta}_{k^*}$$ and also $$ \lambda_i = \mathbf K^{-1'} ( b_{i0}'\, b_{i1}')$$, because of (A12) we prove (16).
Consistency of the estimated static factors is then proved as in Lemma 7 but using the results of Lemma 9 for the estimated parameters. In particular, for all $$ \bar t\le t \le T$$
| $$\displaystyle \Vert{\mathbf F}_{t\vert T,k^*}-\mathbf F_{t} \Vert \le \Vert{\mathbf F}_{t\vert T,k^*}-\mathbf F_{t\vert t} \Vert + \Vert\mathbf F_{t\vert t}-\mathbf F_{t} \Vert = O_p(T^{-1/2}) + O(n^{-1/2}).$$ |
Recalling that $$ \widehat {\mathbf F}_t:= \mathbf F_{t\vert T,k^*}$$ and also $$ \mathbf F_t=\mathbf K ( f_t'\, f_{t-1}')'$$ because of (A11) we prove (17). A similar result can be proved for the KF estimator of the static factors, $$ \mathbf F_{t\vert t,k^*}$$ using again Lemmas 7 and 9.
Denote $$ \zeta_{nT}:=\max(n^{-1/2},T^{-1/2})$$. Then, recalling that $$ \widehat {\lambda}_i'\widehat {\mathbf F}_t:= \widehat {\lambda}_{ik^*}'{\mathbf F}_{t\vert T,k^*}$$, because of (16), (17) and Lemma 5, we have
| $$\displaystyle \vert \widehat {\chi}_{it} - \chi_{it}\vert$$ | $$\displaystyle = \vert \widehat {\lambda}_{i}'\widehat {\mathbf F}_{t}-\lambda_i'\mathbf F_t\vert \le \Vert(\widehat {\lambda}_i-\lambda_i)\mathbf F_t\Vert + \Vert\lambda_i\Vert\;\Vert\widehat {\mathbf F}_t-\mathbf F_t\Vert+ \Vert\widehat {\lambda}_i-\lambda_i\Vert\;\Vert\widehat {\mathbf F}_t-\mathbf F_t\Vert$$ | |
| $$\displaystyle =\Vert(\widehat {\lambda}_i-\lambda_i)\mathbf F_t\Vert + O_p(\zeta_{nT})$$ | ||
| $$\displaystyle =\Vert\widehat {\lambda}_i-\widehat {\lambda}_i^*\Vert \;\Vert\mathbf F_t\Vert + \Vert(\widehat {\lambda}_i^*-\lambda_i)\mathbf F_t\Vert + O_p(\zeta_{nT}).$$ | (C108) |
Now, from Lemma 9 (see in particular (C107)) and since $$ \Vert\mathbf F_t\Vert=O_p(T^{-1/2})$$, we have
| $$\displaystyle \Vert\widehat {\lambda}_i-\widehat {\lambda}_i^*\Vert \;\Vert\mathbf F_t\Vert= o_p(\zeta_{nT}\,T^{-(k^*-1)/2}).$$ | (C109) |
Moreover, from transformation (C57), defined in the proof of Lemma 6, $$ {\mathcal D}\mathbf F_t=(\mathbf Z_{1t}'\;\mathbf Z_{0t}')'$$, we have $$ \Vert\mathbf Z_{1t}\Vert=O_p(\sqrt T)$$ and $$ \Vert\mathbf Z_{0t}\Vert=O_p(1)$$. Then, since $$ {\mathcal D}'{\mathcal D}=\mathbf I_r$$, as a consequence of Lemma 6 (see in particular (C59)), we have
| $$\displaystyle (\widehat {\lambda}_{i}^*-\lambda_i)'{\mathcal D}'{\mathcal D}\mathbf F_t= O_p(T^{-1/2}).$$ | (C110) |
By substituting (C109) and (C110) into (C108) and since $$ k^*\ge 1$$ because we run the EM algorithm at least once after initialization, we prove (18).
Appendix D Data Description and Data Treatment
This Appendix present the dataset used in the analysis. All variables where downloaded from Haver on June 16th 2017. None of the variables where adjusted for outliers but variables 57, 83, 87, and 94. All variables are from the USECON database but variable 103 that is from the DAILY database. All monthly and daily series are transformed into quarterly observation by simple averages.
In order to choose whether or not to de-trend a variable we apply the following procedure: let mi be the sample mean of $$ \Delta y_{it}$$, $$ \gamma_{i}(j)$$ be the auto-covariance of order j of $$ \Delta y_{it}$$, and $$ \bar{\gamma}_i=\sqrt{\frac{1}{T}\sum_{j=1}^J \gamma_i(j)}$$, then if $$ \frac{\vert m_i\vert}{\bar{\gamma}_i}\ge 1.96$$ we estimate ai and bi from an OLS regression of yit on a constant and a time trend, whereas if $$ \frac{\vert m_i\vert}{\bar{\gamma}_i}< 1.96$$ we set $$ \widehat {a}_i=m_i$$ and $$ \widehat {b}_i=0$$.
LIST OF ABBREVIATIONS
Source:
BLS=U.S. Department of Labor: Bureau of Labor Statistics
BEA=U.S. Department of Commerce: Bureau of Economic Analysis
ISM = Institute for Supply Management
CB=U.S. Department of Commerce: Census Bureau
FRB=Board of Governors of the Federal Reserve System
EIA=Energy Information Administration
WSJ=Wall Street Journal
CBO=Congressional Budget Office
FRBPHIL=Federal Reserve Bank of Philadelphia
F = Frequency
Q = Quarterly
M = Monthly
D = Daily
T=Transformation
0 = None
1 = $$ \log$$
2 = $$ \Delta \log$$
SA
0 = no
1 = yes
$$ \xi$$=Idiosyncratic
0=$$ I(0)$$
1=$$ I(1)$$
D = Deterministic Component
0 = $$ \widehat {a}_i=\frac{1}{T}\sum_{t=1}^T\Delta y_{it}$$, $$ \widehat {b}_i=0$$
1 = OLS Detrending
U=Units
1000-P = Thousands of Persons
1000-U = Thousands of Units
BoC = Billions of Chained
$-B = Dollars per Barrel
| N | Series ID | Definition | Unit | F | S | SA | T | D | $$ \xi$$ |
|---|---|---|---|---|---|---|---|---|---|
| 1 | GDPH | Real Gross Domestic Product | BoC 2009$ | Q | BEA | 1 | 1 | 1 | 0 |
| 2 | GDYH | Real gross domestic income | BoC 2009$ | Q | BEA | 1 | 1 | 1 | 0 |
| 3 | FSH | Real Final Sales of Domestic Product | BoC 2009$ | Q | BEA | 1 | 1 | 1 | 1 |
| 4 | IH | Real Gross Private Domestic Investment | BoC 2009$ | Q | BEA | 1 | 1 | 1 | 1 |
| 5 | GSH | Real State & Local$$ ^\ast$$ | BoC 2009$ | Q | BEA | 1 | 1 | 1 | 1 |
| 6 | FRH | Real Private Residential Fixed Investment | BoC 2009$ | Q | BEA | 1 | 1 | 1 | 0 |
| 7 | FNH | Real Private Nonresidential Fixed Investment | BoC 2009$ | Q | BEA | 1 | 1 | 1 | 1 |
| 8 | MH | Real Imports of Goods & Services | BoC 2009$ | Q | BEA | 1 | 1 | 1 | 0 |
| 9 | GH | Real Government$$ ^\ast$$ | BoC 2009$ | Q | BEA | 1 | 1 | 1 | 1 |
| 10 | XH | Real Exports of Goods & Services | BoC 2009$ | Q | BEA | 1 | 1 | 1 | 0 |
| 14 | CH | Real Personal Consumption Expenditures (PCE) | BoC 2009$ | Q | BEA | 1 | 1 | 1 | 0 |
| 11 | CNH | Real PCE: Nondurable Goods | BoC 2009$ | Q | BEA | 1 | 1 | 1 | 1 |
| 12 | CSH | Real PCE: Services | BoC 2009$ | Q | BEA | 1 | 1 | 1 | 0 |
| 13 | CDH | Real PCE: Durable Goods | BoC 2009$ | Q | BEA | 1 | 1 | 1 | 0 |
| 15 | GFDIH | Real National Defense Gross Investment | BoC 2009$ | Q | BEA | 1 | 1 | 1 | 0 |
| 16 | GFNIH | Real Federal Nondefense Gross Investment | BoC 2009$ | Q | BEA | 1 | 1 | 1 | 0 |
| 17 | YPDH | Real Disposable Personal Income | BoC 2009$ | Q | BEA | 1 | 1 | 1 | 0 |
| 18 | JI | Gross Private Domestic Investment:$$ ^\star$$ | 2009=100 | Q | BEA | 1 | 2 | 0 | 0 |
| 19 | JGDP | Gross Domestic Product:$$ ^\star$$ | 2009=100 | Q | BEA | 1 | 2 | 0 | 1 |
| 20 | LXNFU | Unit Labor Cost$$ ^\dagger$$ | 2009=100 | Q | BLS | 1 | 1 | 1 | 1 |
| 21 | LXNFR | Real Compensation Per Hour$$ ^\dagger$$ | 2009=100 | Q | BLS | 1 | 1 | 1 | 1 |
| 22 | LXNFC | Compensation Per Hour$$ ^\dagger$$ | 2009=100 | Q | BLS | 1 | 1 | 1 | 1 |
| 23 | LXNFH | Hours of All Persons$$ ^\dagger$$ | 2009=100 | Q | BLS | 1 | 1 | 1 | 0 |
| 24 | LXNFA | Output Per Hour of All Persons$$ ^\dagger$$ | 2009=100 | Q | BLS | 1 | 1 | 1 | 0 |
| 25 | LXMU | Unit Labor Cost$$ ^\ddagger$$ | 2009=100 | Q | BLS | 1 | 1 | 1 | 1 |
| 26 | LXMR | Real Compensation Per Hour$$ ^\ddagger$$ | 2009=100 | Q | BLS | 1 | 1 | 1 | 1 |
| 27 | LXMC | Compensation Per Hour$$ ^\ddagger$$ | 2009=100 | Q | BLS | 1 | 1 | 1 | 0 |
| 28 | LXMH | Hours of All Persons$$ ^\ddagger$$ | 2009=100 | Q | BLS | 1 | 1 | 1 | 0 |
| 29 | LXMA | Output Per Hour of All Persons$$ ^\ddagger$$ | 2009=100 | Q | BLS | 1 | 1 | 1 | 1 |
| 30 | IP | Industrial Production (IP) Index | 2012=100 | M | FRB | 1 | 1 | 1 | 0 |
| 31 | IP521 | IP: Business Equipment | 2012=100 | M | FRB | 1 | 1 | 1 | 1 |
| 32 | IP511 | IP: Durable Consumer Goods | 2012=100 | M | FRB | 1 | 1 | 1 | 0 |
| 33 | IP531 | IP: Durable Materials | 2012=100 | M | FRB | 1 | 1 | 1 | 1 |
| 34 | IP512 | IP: Nondurable Consumer Goods | 2012=100 | M | FRB | 1 | 1 | 1 | 0 |
| 35 | IP532 | IP: nondurable Materials | 2012=100 | M | FRB | 1 | 1 | 1 | 0 |
$$ ^\ast$$ Consumption Expenditures & Gross Investment
$$ ^\star$$ Chain-type Price Index
$$ ^\dagger$$ Nonfarm Business Sector
$$ ^\ddagger$$ Manufacturing Sector
| N | Series ID | Definition | Unit | F | S | SA | T | D | $$ \xi$$ |
|---|---|---|---|---|---|---|---|---|---|
| 36 | PCU | CPI-U: All Items | 82-84=100 | M | BLS | 1 | 2 | 0 | 0 |
| 37 | PCUSE | CPI-U: Energy | 82-84=100 | M | BLS | 1 | 2 | 0 | 0 |
| 38 | PCUSLFE | CPI-U: All Items Less Food and Energy | 82-84=100 | M | BLS | 1 | 2 | 0 | 0 |
| 39 | PCUFO | CPI-U: Food | 82-84=100 | M | BLS | 1 | 2 | 0 | 0 |
| 40 | JCBM | PCE: Chain Price Index | 2009=100 | M | BEA | 1 | 2 | 0 | 0 |
| 41 | JCEBM | PCE: Energy Goods & Services-price index | 2009=100 | M | BEA | 1 | 2 | 0 | 0 |
| 42 | JCNFOM | PCE: Food & Beverages-price index$$ ^\ast$$ | 2009=100 | M | BEA | 1 | 2 | 0 | 0 |
| 43 | JCXFEBM | PCE less Food & Energy-price index | 2009=100 | M | BEA | 1 | 2 | 0 | 0 |
| 44 | JCSBM | PCE: Services-price index | 2009=100 | M | BEA | 1 | 2 | 0 | 0 |
| 45 | JCDBM | PCE: Durable Goods-price index | 2009=100 | M | BEA | 1 | 2 | 0 | 0 |
| 46 | JCNBM | PCE: Nondurable Goods-price index | 2009=100 | M | BEA | 1 | 2 | 0 | 0 |
| 47 | PC1 | PPI: Intermediate Demand Processed Goods | 1982=100 | M | BLS | 1 | 2 | 0 | 0 |
| 48 | P05 | PPI: Fuels and Related Products and Power | 1982=100 | M | BLS | 0 | 2 | 0 | 0 |
| 49 | SP3000 | PPI: Finished Goods | 1982=100 | M | BLS | 1 | 2 | 0 | 0 |
| 50 | PIN | PPI: Industrial Commodities | 1982=100 | M | BLS | 0 | 2 | 0 | 0 |
| 51 | PA | PPI: All Commodities | 1982=100 | M | BLS | 0 | 2 | 0 | 0 |
| 52 | PC1 | PPI: Intermediate Demand Processed Goods | 1982=100 | M | BLS | 1 | 2 | 0 | 0 |
| 53 | FMC | Money Stock: Currency | Bil. of $ | M | FRB | 1 | 2 | 0 | 0 |
| 54 | FM1 | Money Stock: M1 | Bil. of $ | M | FRB | 1 | 2 | 0 | 1 |
| 55 | FM2 | Money Stock: M2 | Bil. of $ | M | FRB | 1 | 2 | 0 | 0 |
| 56 | FABWC | C & I Loans in Bank Credit:$$ ^\dagger$$ | Bil. of $ | M | FRB | 1 | 1 | 1 | 1 |
| 57 | FABWQ | Consumer Loans in Bank Credit:$$ ^\dagger$$ | Bil. of $ | M | FRB | 1 | 1 | 1 | 1 |
| 58 | FAB | Bank Credit:$$ ^\dagger$$ | Bil. of $ | M | FRB | 1 | 1 | 1 | 1 |
| 59 | FABW | Loans & Leases in Bank Credit:$$ ^\dagger$$ | Bil. of $ | M | FRB | 1 | 1 | 1 | 1 |
| 60 | FABYO | Other Securities in Bank Credit:$$ ^\dagger$$ | Bil. of $ | M | FRB | 1 | 1 | 1 | 1 |
| 61 | FABWR | Real Estate Loans in Bank Credit:$$ ^\dagger$$ | Bil. of $ | M | FRB | 1 | 1 | 1 | 0 |
| 62 | FOT | Consumer Credit Outstanding | Bil. of $ | M | FRB | 1 | 1 | 1 | 0 |
| 63 | HSTMW | Housing Starts: Midwest | 1000-U | M | CB | 1 | 1 | 0 | 0 |
| 64 | HSTNE | Housing Starts: Northeast | 1000-U | M | CB | 1 | 1 | 0 | 0 |
| 65 | HSTS | Housing Starts: South | 1000-U | M | CB | 1 | 1 | 0 | 0 |
| 66 | HSTGW | Housing Starts: West | 1000-U | M | CB | 1 | 1 | 0 | 0 |
| 67 | HPT | Building Permit $$ ^\star$$ | 1000-U | M | CB | 1 | 1 | 0 | 0 |
| 68 | FBPR | Bank Prime Loan Rate | Percent | M | FRB | 0 | 0 | 0 | 0 |
| 69 | FFED | Federal Funds [effective] Rate | Percent | M | FRB | 0 | 0 | 0 | 0 |
| 70 | FCM1 | 1-Year Treasury Bill Yield$$ ^\ddagger$$ | Percent | M | FRB | 0 | 0 | 0 | 0 |
| 71 | FCM10 | 10-Year Treasury Note Yield$$ ^\ddagger$$ | Percent | M | FRB | 0 | 0 | 0 | 0 |
$$ ^\ast$$ Purchased for Off-Premises Consumption
$$ ^\dagger$$ All Commercial Banks
$$ ^\star$$ New Private Housing Units Authorized by
$$ ^\ddagger$$ at Constant Maturity
| N | Series ID | Definition | Unit | F | S | SA | T | D | $$ \xi$$ |
|---|---|---|---|---|---|---|---|---|---|
| 72 | LP | Civilian Participation Rate: 16 yr + | Percent | M | BLS | 0 | 0 | 0 | 1 |
| 73 | LQ | Civilian Employment/Population Ratio: 16 yr + | Percent | M | BLS | 0 | 0 | 0 | 1 |
| 74 | LE | Civilian Employment: Sixteen Years & Over | 1000-P | M | BLS | 0 | 1 | 1 | 0 |
| 75 | LR | Civilian Unemployment Rate: 16 yr + | Percent | M | BLS | 0 | 0 | 0 | 0 |
| 76 | LU0 | Civilians Unemployed for Less Than 5 Weeks | 1000-P | M | BLS | 0 | 1 | 0 | 0 |
| 77 | LU5 | Civilians Unemployed for 5-14 Weeks | 1000-P | M | BLS | 0 | 1 | 0 | 1 |
| 78 | LU15 | Civilians Unemployed for 15-26 Weeks | 1000-P | M | BLS | 0 | 1 | 0 | 1 |
| 79 | LUT27 | Civilians Unemployed for 27 Weeks and Over | 1000-P | M | BLS | 0 | 1 | 0 | 1 |
| 80 | LUAD | Average [Mean] Duration of Unemployment | Weeks | M | BLS | 0 | 1 | 0 | 0 |
| 81 | LANAGRA | All Employees: Total Nonfarm | 1000-P | M | BLS | 0 | 1 | 1 | 1 |
| 82 | LAPRIVA | All Employees: Total Private Industries | 1000-P | M | BLS | 0 | 1 | 1 | 0 |
| 83 | LANTRMA | All Employees: Mining and Logging | 1000-P | M | BLS | 0 | 1 | 1 | 1 |
| 84 | LACONSA | All Employees: Construction | 1000-P | M | BLS | 0 | 1 | 1 | 1 |
| 85 | LAMANUA | All Employees: Manufacturing | 1000-P | M | BLS | 0 | 1 | 1 | 0 |
| 86 | LATTULA | All Employees: Trade, Transportation & Utilities | 1000-P | M | BLS | 0 | 1 | 1 | 1 |
| 87 | LAINFOA | All Employees: Information Services | 1000-P | M | BLS | 0 | 1 | 1 | 1 |
| 88 | LAFIREA | All Employees: Financial Activities | 1000-P | M | BLS | 0 | 1 | 1 | 1 |
| 89 | LAPBSVA | All Employees: Professional & Business Services | 1000-P | M | BLS | 0 | 1 | 1 | 1 |
| 90 | LAEDUHA | All Employees: Education & Health Services | 1000-P | M | BLS | 0 | 1 | 1 | 1 |
| 91 | LALEIHA | All Employees: Leisure & Hospitality | 1000-P | M | BLS | 0 | 1 | 1 | 1 |
| 92 | LASRVOA | All Employees: Other Services | 1000-P | M | BLS | 0 | 1 | 1 | 1 |
| 93 | LAGOVTA | All Employees: Government | 1000-P | M | BLS | 0 | 1 | 1 | 0 |
| 94 | LAFGOVA | All Employees: Federal Government | 1000-P | M | BLS | 0 | 1 | 1 | 1 |
| 95 | LASGOVA | All Employees: State Government | 1000-P | M | BLS | 0 | 1 | 1 | 0 |
| 96 | LALGOVA | All Employees: Local Government | 1000-P | M | BLS | 0 | 1 | 1 | 0 |
| 97 | PETEXA | West Texas Intermediate Spot Price FOB$$ ^\ast$$ | $-B | M | EIA | 0 | 2 | 0 | 0 |
| 98 | NAPMNI | ISM Mfg: New Orders Index | Index | M | ISM | 1 | 0 | 0 | 1 |
| 99 | NAPMOI | ISM Mfg: Production Index | Index | M | ISM | 1 | 0 | 0 | 1 |
| 100 | NAPMEI | ISM Mfg: Employment Index | Index | M | ISM | 1 | 0 | 0 | 1 |
| 101 | NAPMVDI | ISM Mfg: Supplier Deliveries Index | Index | M | ISM | 1 | 0 | 0 | 0 |
| 102 | NAPMII | ISM Mfg: Inventories Index | Index | M | ISM | 1 | 0 | 0 | 0 |
| 103 | SP500 | Standard & Poor's 500 Stock Price Index | 41-43=10 | D | WSJ | 0 | 1 | 1 | 0 |
| Series ID | Definition | Unit | F | Source |
|---|---|---|---|---|
| GDPPOTHQ | Real Potential Gross Domestic Product | BoC 2009$ | Q | CBO |
| NAIRUQ | Natural Rate of Unemployment | percent | Q | CBO |
| GDPPLUS | US GDPplus | percent | Q | FRBPHIL |