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Dimension Reduction for High Dimensional Vector Autoregressive Models
For decades, the vector autoregressive (VAR) model is de facto a standard tool for investigating multivariate time series data.\footnote{It is usual and correct to quote Sims (1980) for his fundamental contribution to the VAR literature. An earlier reference is Quenouille (1957).} In macroeconometrics, VARs are routinely used for forecasting, for extracting co-movements such as the presence of cointegration, to test for Granger causality as well as to perform structural analyses. However, the number of VAR parameters to be estimated increases quadratically with the number of variables and linearly with the number of lags. This quickly compromises the estimation results outside the case of small scale models. There is currently a growing interest for jointly modeling many variables and consequently to look at the feasibility to work with large dimensional VAR models. Indeed, the increase of data availability in economics and finance\footnote{Both in terms of the number of series that are easily available and in the use of different sampling frequencies (e.g. mixed frequency VARs, see Goetz et al., 2016).} is associated with the common belief that using more information with high dimensional econometric and statistical models will improve our understanding of the macroeconomy as well as forecast accuracy (see Boivin and Ng, 2006 for counter examples). Consequently, as the increase of the number of time series jointly considered in VARs cannot be too large compared to a given number of observations, different attempts have been proposed in the literature to curb the curse of dimensionality problem. These methods can be gathered in two categories: dimension reduction approaches on the one hand and regularization techniques on the other hand. We include in the latter group both Bayesian methods (surveyed in Karlsson, 2013, Koop, 2018), although Bayesian techniques are also used to estimate large reduced-rank VARs (see Carriero et al., 2011), and the more recent booming contributions on penalized estimation of sparse VARs (Wilms and Croux, 2016, Hsu et al., 2008, Nicholson et al., 2018, Davis et al., 2016, Smeekes and Wijler, 2018, Kock and Callot, 2015, Hecq \textit{et al.}, 2021). The former group of methods, to which our paper wishes to contribute, includes reduced rank techniques (Reinsel, 1983, Ahn and Reinsel, 1988, Carriero \textit{et al}., 2011, Cubadda and Hecq, 2011, Cubadda and Hecq, 2021, Bernardini and Cubadda, 2015) and the huge literature on factor models (surveyed in Stock and Watson, 2016, and Lippi, 2018, 2019).
Differently from those contributions on system dimension reductions, we provide a framework where the whole dynamics of the system is due to an underlying small scale VAR model. Following Lam et al. (2011) and Lam and Yao (2012), we first decompose the large multivariate time series $Y_{t}$ into two parts: a linear function of a small scale dynamic component $x_{t}$ and a static component $\varepsilon_{t}$ that, and this is the key point that makes our approach different from the usual dynamic factor model, is unpredictable from the past. The capital $Y_{t}$ stresses that we start from a potentially high-dimensional time series process whereas the small $x_{t}$ stresses that a small number of factors are responsible for the entire dynamics of the system. Then we provide the conditions under which such a dynamic component $x_{t}$ is generated by a small scale VAR model. Particularly, we show that it is required that the large VAR model of series $Y_{t}$ is endowed with both the serial correlation common feature (Engle and Kozicki, 1993) and an index structure (Reinsel, 1983) in order to ensure that the dynamic component $x_{t}$ follows a VAR model. Hence, we provide a link between the factor modeling for high-dimensional time series and the reduced-rank VAR approach, which, to the best of our knowledge, was not noted before. This bridge allows us to unravel common cyclical features and to impose their presence in large VARs.
Obviously, the decomposition that we consider might not exist. Based on the eigenanalysis proposed in Lam et al. (2011) and Lam and Yao (2012), we provide statistical tools to verify whether there exists in series $Y_{t}$ a dynamic component $x_{t}$ that is generated by such a small scale VAR model and to estimate the associated parameters. If this is the case, the forecasts of $x_{t}$ can be used to predict the future realizations of the large dimensional system $Y_{t}$ and structural shocks may be recovered from the reduced form errors of the dynamic component only.
Our contribution can be related to the early literature on the analysis of linear transformations of vector autoregressive moving average (VARMA) processes, see, inter alia, Kohn (1982), and L\"{u}tkepohl (1984a, 1984b). These previous contributions established that non-singular linear transformations of a VARMA process still have a VARMA structure, and if some restrictions on the relationships among variables apply, the VARMA model for the linear transformation can be parsimonious enough to be used in empirical applications. However, at the representation theory level the aim of our analysis is rather to find the conditions that allow to model a large dimensional VAR process through a small scale VAR of the same order without any information loss. To the best of our knowledge, such issue has not been tackled so far. So doing we also provide the basis for further extending the link between the final equation representation and VAR models with reduced rank restrictions (Cubadda et al., 2009).
The rest of the paper is organized as follows. Section 2 presents the main results on the model representation as well as the restrictions that our new modelling entails. We observe that, so far, the reduced-rank VAR and multivariate autoregressive index model have been considered separately in the literature. We find out that the combination of the two models allows for the important dimension reduction in large VARs that we seek. We show that information criteria can be used to determine the dimension of a small VAR within a large dimensional system and we provide estimators for its parameters. Section 3 conducts an extensive Monte Carlo study to evaluate the finite sample properties of the proposed tools. Section 4 provides an empirical application a large set of macroeconomic and financial time series to illustrate the practical value of our approach. Based on our methodology, we propose a novel approach to identify the shock that is responsible for most of the common volatility at the business cycle frequency band. Finally, Section 5 concludes.
This section starts by presenting the derivation of the proposed modelling, then it discusses the statistical inference in a large dimensional framework.
Let us assume that the $n$-vector time series $Y_{t}$ is generated by the following second-order stationary VAR($p$) model
where $t=1,...,T$, $\Phi_{j}$ is an $n\times n$ matrix for $j=1,...,p$ with $\Phi_{p}\neq0$ such that the roots of $\det\left( I_{n}-\sum_{j=1}^{p} \Phi_{j}z^{j}\right) $ lie outside the unit circle; ${\small u}_{t}$ is an $n$-vector of errors such that $\mathrm{E}({\small u}_{t})=0$, $\mathrm{E} ({\small u}_{t}{\small u}_{t}^{\prime})=\Sigma_{{\small u}}$ is a finite and positive definite matrix, $\mathrm{E}({\small u}_{t}|\digamma_{t-1})=0$ and $\digamma_{t}$ is the natural filtration of the process $Y_{t}$. For the sake of simplicity, we assume that deterministic elements are absent (or that the series have been demeaned or detrended first).
Assumption 1 is popularly known in time series econometrics as the serial correlation common feature (Engle and Kozicki, 1993). It was extensively studied in connection with cointegration (see, inter alia, Vahid and Engle, 1993, Ahn, 1997, Cubadda and Hecq, 2001, Hecq et al. 2006, Cubadda, 2007, and Athanasopoulos et al., 2011). Moreover, it implies that the marginal processes of series $Y_{t}$\ follow parsimonious univariate models, thus solving the so-called autoregressivity paradox (Cubadda et al., 2009). See Centoni and Cubadda (2015) and Cubadda and Hecq (2021) for recent surveys. In the analysis that follows, we focus on the case where $n$ is large, virtually with a similar magnitude as the sample size $T$, whereas $r$ is small compared to $T$.
We start by noting that under Assumption 1 we can use the identity
in order to decompose series $Y_{t}$ as
where $x_{t}=A^{\prime}Y_{t}$, $\varepsilon_{t}=A_{\bot}A_{\bot}^{\prime }{\small u}_{t}$ and $A_{\bot}$ is a full-rank $n\times(n-r)$ matrix such that $A_{\bot}^{\prime}A=0$ and $A_{\bot}^{\prime}A_{\bot}=I_{n-r}$.
Following Lam et al. (2011) and Lam and Yao (2012), we call $x_{t}$ and $\varepsilon_{t}$ respectively\ the dynamic and the static component of series $Y_{t}$. Indeed, we have for the disturbances $\varepsilon_{t}$ \[ \mathrm{E}(\varepsilon_{t}|\digamma_{t-1})=A_{\bot}^{\prime}A_{\bot} \mathrm{E}({\small u}_{t}|\digamma_{t-1})=0 \] from which it follows
where $k$, the forecast horizon, is any positive integer.
Lam et al. (2011) and Lam and Yao (2012) showed how to consistently estimate both $r$ and $A$ (or, more formally, an $n\times r$\ matrix that lies in the space generated by the columns of $A$) under assumptions that are compatible with those in Assumption 1 even when the dimension $n$ diverges.\footnote{Notice that the aforementioned contributors assume that $\mathrm{E}(\varepsilon_{t}x_{t}^{\prime})=0$. We tackle with this issue in Remark 4.} Their method is simple to perform since it is based on the eigenanalysis of the sum of the squared autocovariance matrices of series $Y_{t}$. Having obtained an estimator, say $\hat{A}$, it is then possible to get the dynamic and static components in ((ref)) respectively as \[ \hat{x}_{t}=\hat{A}^{\prime}Y_{t} \] and \[ \hat{\varepsilon}_{t}=(I-\hat{A}\hat{A}^{\prime})Y_{t}. \] \qquad
However, in order to forecast series $Y_{t},$ as well as to perform structural analysis, estimating the loading matrix $A$ is not enough. In a dimension reduction perspective, we further look at the conditions under which the dynamic component $x_{t}$\ is generated by a small-scale VAR($p$) model. To the best of our knowledge, such goal has not been pursued so far.
We start by noting that under Assumption 1 we can rewrite model ((ref)) as follows
which is popularly known as the reduced-rank VAR model (RRVAR) and it was extensively analyzed, inter alia, by Velu et al. (1986) and Ahn and Reinsel (1988). We have already emphasized that restrictions such as ((ref)) are at the heart of the serial correlation common feature literature.
Second, pre-multiplying both sides of Equation ((ref)) by $A^{\prime}$ we get
where $\xi_{t}=A^{\prime}{\small u}_{t}$. However, it is important to see that Equation ((ref)) does not yet provide a small-scale model for series $Y_{t}$. Indeed, $\Omega$ being a $np\times r$ matrix with elements $\omega$ in ((ref)), the number of parameters still grows proportionally with $n$, although not with $n^{2}$ as in the unrestricted VAR.
Finally, we insert Equation ((ref)) into ((ref)) such that we obtain \[ x_{t}=\sum_{j=1}^{p}\omega_{j}^{\prime}Ax_{t-j}+\sum_{j=1}^{p}\omega _{j}^{\prime}\varepsilon_{t-j}+\xi_{t}, \] which allows us to derive the condition under which the dynamic component $x_{t}$ is generated by a VAR($p$) process as follows:
Indeed, under Assumption 2 we have that $\omega_{j}^{\prime}Ax_{t-j} =\alpha_{j}x_{t-j}$ and $\omega_{j}^{\prime}\varepsilon_{t-j}=0$ for $j=1,...,p$, hence the data generating process of the dynamic component $x_{t}$ boils down to
The intuition behind the algebra is that Assumption 2 requires that the lags of the same linear combinations of $Y_{t}$ that are unpredictable from the past are also irrelevant predictors of the dynamic component $x_{t}$.
Remarkably, the RRVAR model ((ref)) of series $Y_{t}$\ can be rewritten as
Model ((ref)) is interesting since it combines the features of the RRVAR model with those of the multivariate autoregressive index (MAI) model proposed by Reinsel (1983). Recently, there has been a renewed interest in the MAI because it allows to rewrite the VAR in a similar way as the popular dynamic factor model, see inter alia Carriero et al. (2016), Cubadda et al. (2017), and Cubadda and Guardabascio (2019). So far, the RRVAR and the MAI have been considered separately in the literature whereas Assumption 2 reveals that the combination of the two models allows for an important dimension reduction in large VARs. In what follows, we call model ((ref)) as the dimension-reducible VAR model (DRVAR).
In order to perform structural analysis through the DRVAR, one way to go is inverting the polynomial VAR coefficient matrix in Equation ((ref)) to obtain the Wold representation of series $Y_{t}$. Here we offer an alternative route. If we invert the polynomial coefficient matrix in Equation ((ref)) and insert the Wold representation of the dynamic components $x_{t}$\ in decomposition ((ref)) we get
where $\gamma(L)^{-1}=I_{n}-\sum_{j=1}^{p}\alpha_{j}L^{j}$. Finally, by linearly projecting $\varepsilon_{t}$ on $\xi_{t}$, we can decompose the static component as $\varepsilon_{t}=\rho\xi_{t}+\nu_{t}$, where $\rho =A_{\bot}A_{\bot}^{\prime}\Sigma_{{\small u}}A(A^{\prime}\Sigma_{{\small u} }A)^{-1}$, and then rewrite Equation ((ref)) as
where $C_{0}=A+\rho$\ and $C_{j}=A\gamma_{j}$ for $j>0$.
Representation ((ref)) highlights that the system dynamics are entirely generated by errors $\xi_{t}$. Hence, we label $\chi_{t}$ as the common component of $Y_{t}$ and $\nu_{t}$\ as the ignorable errors, as we assume the latter are not endowed with a structural interpretation. Since the errors $\xi_{t}$\ and $\nu_{t}$ are uncorrelated at any lead and lag, it is then possible to recover the structural shocks solely from the reduced form errors $\xi_{t}$ of the common component $\chi_{t}$ using any of the procedures that are commonly employed in structural VAR analysis.
For instance, one may obtain the structural shocks as $u_{t}=C^{-1}D\xi_{t}$ and the impulse response functions from $\Psi(L)=C(L)D^{-1}C$, where $D$ is the matrix formed by the first $r$ rows of $C_{0}$ and $C$ is a lower triangular matrix such that $CC^{\prime}=DA^{\prime}\Sigma_{{\small u} }AD^{\prime}$. Since the first $r$ rows of $\Psi(0)$, being equal to $C$, form a lower triangular matrix, the usual interpretation of the structural shocks $u_{t}$ applies as long as the $s$ $(s\leq r)$ variables of interest are placed and properly ordered in the first $s$ elements of $Y_{t}$. Notice that such identification strategy is based on a unique rotation of the reduced form common shocks $\xi_{t}$, and hence it does not require to endow the dynamic component $x_{t}$ with an economic interpretation.
Clearly, it is always possible to identify the structural shocks directly from the reduced form errors ${\small u}_{t}$ of the large VAR. However, the advantage of the approach based on representation ((ref)) is that it requires to identify $r$ shocks only instead of $n$ of them. Hence, in the structural DRVAR analysis based on ((ref)) we have a number of structural shocks that is smaller than the number of variables, as it is typical in both structural factor models (see e.g. Forni et al, 2009) and dynamic stochastic general equilibrium (DSGE) models (see e.g. Fern\'{a}ndez-Villaverde et al., 2016).
In order to consistently estimate the matrix $A$, we start by relying on the approach suggested by Lam et al. (2011), and Lam and Yao (2012). This approach has recently been extended in various directions, such as cointegration (Zhang et al, 2019), principal component analysis for stationary time series (Chang et al, 2018), and multivariate volatilities modelling (Tao et al. 2011; Li et al., 2016).
Let us denote the autocovariance matrix of series $Y_{t}$ at lag $j$ as $\Sigma_{y}(j)=\mathrm{E}(Y_{t}Y_{t-j}^{\prime})$. In view of Equation ((ref)), we see that \[ A_{\bot}^{\prime}\Sigma_{y}(j)=\mathrm{E}(A_{\bot}^{\prime}\varepsilon _{t}Y_{t-j}^{\prime})=0\text{ \ \ }\forall j>0. \] Hence, the matrix $A$ lies in the space generated by the eigenvectors associated with the $r$ non-zero eigenvalues of the symmetric and semi-positive definite matrix \[ M=\sum_{j=1}^{p_{0}}\Sigma_{y}(j)\Sigma_{y}(j)^{\prime}, \] where $p_{0}$ is a positive integer. Given the assumption that series $Y_{t}$ follow a finite order VAR($p$) model, one would ideally fix $p_{0}=p$.
Let us indicate with $\hat{V}_{q}$ the matrix formed by the eigenvectors associated with the $q$ $(\leq n)$ largest eigenvalues of the matrix \[ \hat{M}=\sum_{j=1}^{p_{0}}\hat{\Sigma}_{y}(j)\hat{\Sigma}_{y}(j)^{\prime}, \] where $\hat{\Sigma}_{y}(j)$ denotes the sample autocovariance matrix of $Y_{t}$ at lag $j$.
Under regularity conditions that are compatible with our assumptions, $\hat {V}_{r}$\ estimates $A$\ (up to an orthonormal transformation) with a rate equal to $n^{-\delta}T^{1/2}$ when $r$ is fixed, $n,T\rightarrow\infty$, and $\bar{a}_{i}^{\prime}\bar{a}_{i}\asymp n^{1-\delta}$ for $i=1,...,r$, where $\bar{A}=[\bar{a}_{1},...,\bar{a}_{r}]$ and $\delta\in\lbrack0,1]$. Notice that $\delta$\ can be interpreted as an inverse measure of the strength of the factors: when $\delta=0$\ the factors are strong since the common component is shared by most of the $n$\ time series, whereas the factors are weak when $\delta\in(0,1]$ (see Theorem 1 of Lam et al., 2011). Moreover, Lam and Yao (2012) proved that a consistent estimator of $r$ is provided by
where $R$ is a constant such that $r<R<n$ and $\hat{\lambda}_{i}$ is the $i-$th largest eigenvalue of matrix $\hat{M}$.\footnote{Li et al. (2017) proposed an improved estimator of $r$ that is consistent even when not all the factors have the same strength. However, they take the assumption of independence between $x_{t}$ and $\varepsilon_{t}$, which clearly does not hold here.}
Notice that $\hat{r}$ in ((ref)) consistently estimates the rank of the matrix $M$ when Assumption 1 only applies, whereas we need an estimator of $r$ that is subject to Assumption 2 as well. Let us first consider the problem of estimating the parameters of model ((ref)) assuming that $r$ is known and having fixed $A$ equal to $\hat{V}_{r}$. In order to accomplish this goal, it is convenient to rewrite model ((ref)) in its matrix form
where $Y=\left[ y_{p+1},...,y_{T}\right] ^{\prime}$, ${\small u}=\left[ {\small u}_{p+1},...,{\small u}_{T}\right] ^{\prime}$, $z_{t}=\left[ x_{t}^{\prime},...,x_{t-p+1}^{\prime}\right] ^{\prime}$, and $Z=\left[ z_{p},...,z_{T-1}\right] ^{\prime}$. Then apply the $\mathrm{Vec}$ operator to both the sides of Equation ((ref)) and use the property $\mathrm{Vec} (ABC)=(C^{\prime}\otimes A)\mathrm{Vec}(B)$ to get
from which it is easy to see that the ordinary least squares (OLS) estimator of $\mathrm{Vec}(\alpha)$\ in Equation ((ref)) takes the following form:
The main theoretical justification for considering the estimator ((ref)) is that it is equivalent to applying OLS on ((ref)),\footnote{This result is obtained by post-multiplying with $A$ both sides of Equation ((ref)) and then applying the Vec operator to get \[ \mathrm{Vec}(YA)=(I_{r}\otimes Z)\mathrm{Vec}(\alpha)+\mathrm{Vec}(\epsilon A) \] It is easy to see that the OLS estimator of $\mathrm{Vec}(\alpha)$ in the model above is the same as ((ref)).} which turns out to be the quasi maximum likelihood (QML) estimator of parameters $\alpha$ in the small-scale VAR model of the factor $x_{t}$ under the assumption that $A$\ is known.
An alternative estimator of parameters $\alpha$ can be obtained by applying the generalized least squares (GLS) on Equation ((ref)). In particular, pre-multiply both the sides of Equation ((ref)) by $\Sigma_{{\small u} }^{-1/2}\otimes I_{T-p}$ to get
Tedious but simple algebra reveals that the OLS estimator of $\mathrm{Vec} (\alpha)$\ in Equation ((ref)) takes the following form:
In view of Equation ((ref)), it is easy to see that the GLS estimator ((ref)) is the QML estimator of parameters $\alpha$ in model ((ref)) under the assumption that $A$ and $\Sigma_{{\small u}}$ are known.
The relation, in terms of efficiency, between the estimators ((ref)) and ((ref)) is provided in the following theorem.
In order to derive a feasible GLS (FGLS) estimator, we suggest the following switching algorithm, which has the property to increase the Gaussian likelihood conditional to $A$ in each step.
In order to speed up the numerical convergence of that algorithm, it is important to choose the initial values for the coefficient matrix $\alpha$ correctly. An obvious choice is resorting to $\hat{\alpha}$, which provides a consistent estimate of $\alpha$ as $T$ increases.
A practical problem that arises when the sample size $T$ and the dimension $n$ are of similar magnitude is that the estimate of matrix $\Sigma_{{\small u}} $\ is singular or nearly singular. We propose to solve this problem by ignoring the error cross-correlations in the estimation method. In particular, we suggest to use a diagonal matrix $\Delta_{{\small u}}$ with the same diagonal as $\Sigma_{{\small u}}$ in place of $\Sigma_{{\small u}}$ itself in the FGLS procedure. This solution has two main motivations. First, it makes the objective function of the switching algorithm to become $\mathrm{trace} (\ln(\Delta_{{\small u}}))$, which is a common approximation of $\ln (\det(\Sigma_{{\small u}}))$ in high-dimensional settings, see Hu et al. (2017) and the references therein. Second, it is reasonable to presume that the fraction of unanticipated co-movements among variables is small when the conditioning information set is large.
Finally, in order to identify the dimension the dynamic component $r,$ we suggest the following strategy. For $q=1,...,R$ estimate either by OLS or FGLS the models \[ Y_{t}=\sum_{j=1}^{p}\hat{V}_{q}\alpha_{j,q}\hat{V}_{q}^{\prime}Y_{t-j} +{\small u}_{t}(q), \] where $\alpha_{j,q}$ is a $q\times q$ matrix for $j=1,...,p$, and estimate $r$ as the index $\widehat{r}$ that minimizes an information criterion such as
where $\hat{\sigma}_{i}^{2}(q)=(T-p)^{-1} {\textstyle\sum\limits_{t=p+1}^{T}} {\small u}_{t,i}^{2}(q)$, ${\small u}_{t}(q)=[{\small u}_{t,1} (q),...,{\small u}_{t,n}(q)]^{\prime}$, $k=nq+(p-1)q^{2}\mathtt{,}$ $c_{T} $\ is a penalty term such that $c_{T}=2$ for the Akaike information criterion (AIC), $c_{T}=2\ln(\ln(T))$ for the Hannan-Quinn information criterion (HQIC), and $c_{T}=\ln(T)$ for the Bayes information criterion (BIC). Notice that the measure of fit to be used is $\mathrm{trace}(\ln(\Delta_{{\small u}}))$, given the assumption that $\Sigma_{{\small u}}=\Delta_{{\small u}}$, and the overall number of parameters is $k=nq+(p-1)q^{2}$, given that the number of free parameters in a base of the space spanned by $A$ is equal to $nq-q^{2}$ and each of the $p$ $\alpha_{j,q}$ matrices has $q^{2}$ coefficients$.$
The asymptotic behavior of the information criteria ((ref)) is given in the following proposition.
In this section we perform a Monte Carlo study to evaluate the finite sample performances of the OLS and FGLS estimators of model ((ref)) parameters having estimated the matrix $A$ according to Lam et al. (2011) in both cases. We consider the following $n$-dimensional stationary VAR$(2)$ process
where $\bar{A}$ is a $n\times r$ matrix such that its columns are generated by $r$ i.i.d. $\mathrm{N}_{n}(0,I_{n})$, $\bar{A}^{+}=(\bar{A}^{\prime}\bar {A})^{-1}\bar{A}^{\prime}$ is the Moore--Penrose pseudo inverse of the matrix $\bar{A}$, $\delta_{1}=2\mathrm{diag}(m)\cos(\omega)$, $m$\ is a $r-$vector drawn from a $\mathrm{U}_{r}[0.3,0.9]$, $\omega$\ is a $r-$vector drawn from a $\mathrm{U}_{r}[0,\pi]$, $\delta_{2}=-m^{2}$, and ${\small u}_{t}$ are i.i.d. $\mathrm{N}_{n}(0,\Sigma_{{\small u}})$.\footnote{Notice that the squared Euclidean norms of the columns of $\bar{A}$ are proportional to $n$ on average over the replications. However, the randomness of the matrix $\bar{A}$ ensures for each replication some variability of the degree of strength over the $r$ factors.} Notice that Equation ((ref)) is equivalent to \[ Y_{t}=A\alpha_{1}A^{\prime}Y_{t-1}+A\alpha_{2}A^{\prime}Y_{t-2}+{\small u} _{t}, \] where $A=\bar{A}S^{-1}$, $S=(\bar{A}^{\prime}\bar{A})^{1/2}$, and $\alpha _{i}=S\mathrm{diag}(\delta_{i})S^{-1}$ for $i=1,2$ and consequently imposes Assumptions 1 and 2 of our specification.
In order to simulate series $Y_{t}$, we first generate the diagonal VAR$(2)$ process \[ \bar{x}_{t}=\mathrm{diag}(\delta_{1})\bar{x}_{t-1}+\mathrm{diag}(\delta _{2})\bar{x}_{t-2}+\bar{A}^{+}{\small u}_{t} \] and then the static component $\varepsilon_{t}=\bar{A}_{\bot}\bar{A}_{\bot }^{+}{\small u}_{t}$ with $\eta_{t}=[(\bar{A}^{+}{\small u}_{t})^{^{\prime} },(\bar{A}_{\bot}^{+}{\small u}_{t})^{\prime}]^{\prime}$ that are i.i.d. $\mathrm{N}_{n}(0,\Sigma_{\eta})$. We can finally obtain
where $x_{t}=S\bar{x}_{t}$ is the dynamic component.
An important role in the data generating process is played by the covariance matrix $\Sigma_{\eta}$, which has the following Toeplitz structure \[ \Sigma_{\eta}=\left[
\right] , \] where $\tau$ is a scalar drawn from a $\mathrm{U}[-0.5,0.5]$. Notice that since $\Sigma_{\eta}$ is not diagonal by implication the covariance matrix of the VAR errors $\Sigma_{{\small u}}$ is not diagonal as well. This allows us to evaluate the performances of both the FGLS estimator and the information criteria when $\ln(\det(\Sigma_{{\small u}}))$ is not equal to $\mathrm{trace} (\ln(\Delta_{{\small u}}))$.
From ((ref)) we generate systems of successively$\ n=150,300,600,1200$ variables. We consider $r=3,9$ and $n$ dynamic components; $r=3$ is indeed often assumed in financial applications (the Fama-French factors) whereas several studies find that there exist from around 8 up to 10 factors in large macroeconomic datasets. The case $r=n$ is considered to evaluate the performances of the various estimators of $r$ when Assumptions 1 and 2 are not valid.\footnote{Notice that when $r=n$ the DGP((ref)) boils down to $Y_{t}=\bar{A}\bar{x}_{t}$, where $\bar{A}$ is a $n\times n$ matrix such that its columns are generated by $n$ i.i.d. $\mathrm{N}_{n}(0,I_{n})$ and $\bar {x}_{t}$ is generated by a $n-$dimensional diagonal VAR(2) process.}
The number of observations is successively $T=\frac{1}{2}n,n,1.5n$. We consequently evaluate the performance of our approach when the number of variables is respectively less, equal or larger than the sample size. We simulate $T+50$ observations and the first $50$ points are used as a burn-in period, the remaining ones for estimations. Results are based on 1000 replications.
The proposed methods are evaluated by means of two statistics. We first compute the percentage with which the number of dynamic components $r$ is either correctly identified ($\%\widehat{r}=r$) when $r=3,9$ or hits the upper bound ($\%\widehat{r}=R=11$) when $r=n$ using both the estimator ((ref)) proposed by Lam and Yao (2012), hereafter denoted as LY, and the usual information criteria under the assumption that $\Sigma_{{\small u}}$ is a diagonal matrix. We also compute the average of $\hat{r}$ over the replications as well as the frequencies with which those procedures underestimate the correct number of dynamic components ($\%\widehat{r}<r$) when $r<n$. Second, when the DRVAR restrictions are valid, i.e. $r=3,9$, we compute the Frobenius distance between the estimates of $\Phi=[\Phi_{1} ,\Phi_{2}]^{\prime}$\ and the true ones relative to the Frobenius norm of $\Phi$ (RFD) as a measure of estimation precision. We only document the results for the LY procedure and the OLS estimator since those with FGLS are rather similar to the OLS ones and are then omitted for the sake of space.\footnote{In particular, the percentages of correct identification that are obtained by each criterion are almost identical irrespective of the estimation method, whereas FGLS generally exhibits slightly lower RFDs than OLS, although the differences are significant in about one third of the cases. Results are available upon request.}
We first examine the results for $r=3,9$, which are respectively reported in Tables (ref) and (ref). As expected, we notice that all methods perform better as the dimension of the system $n$ increases and, conditional to a given $n$, as the sample size $T$ gets larger. For what regards the estimation of the true number of dynamic components, we see that the information criteria outperform the LY procedure by a clear margin. This finding is hardly surprising given the parametric nature of the information criteria. In particular, HQIC identifies the correct model better than the competitors but in 3 cases where BIC performs best. In contrast, LY [AIC] systematically underestimates [overestimates] the true $r$. With respect to the RFD, we observe that the models identified by the BIC [AIC] provide estimates of $\Phi$ that are more [less] accurate than those obtained by the other criteria over all the settings. Such outcomes are likely due the fact that the BIC is downward biased, thus implicitly shrinking to zero the dynamic components that are only mildly autocorrelated, whereas the AIC systematically overfits the model, thus inflating the estimation error. This finding discourages the use of the AIC in empirical applications. Overall, BIC and HQIC tend to perform similarly in both identification and estimation precision as both $n$ and $T$ get large.
Table (ref) reports the results when the DRVAR restrictions are not valid, i.e. for $r=n$. We see that the LY procedure, which is designed to estimate $r$ when Assumption 1 is valid, spuriously suggests the presence of about two dynamic components on average irrespective of the sample size and the system dimension. In contrast, the usual information criteria, especially HQIC and AIC, provide estimates of $r$ that get close or hit the upper bound starting from $n,T=300,300$ up to larger values of both $n$ and $T$. These findings indicate that, when there exist no common dynamic components in the data, the information criteria ((ref)) correctly provide estimates of $r$ that tend to become larger as $n,T$ increase.
Finally, as suggested by a referee we build an alternative Monte Carlo design that is based on the features of the data that we use in Section 4. Our favorite DRVAR empirical specification is a model with $r=8$ and $p=2$ for a dataset of $211$ aggregate US time series and $242$ observations. Obviously, $n$ cannot be increased beyond $211$ in this framework but we can evaluate the effects of doubling the sample size $T$ on the proposed methods. Hence, data are generated by a DRVAR having Gaussian errors and the same parameters as the model that we estimate in Section 4. The results, reported in (ref) , confirm most of the main findings of the previous experiments, with the exceptions that AIC outperforms BIC and that LY behaves even more poorly than it does with the artificial DGP.\footnote{Again, the RFDs of FGLS are about 1% lower than than those of OLS but the differences are significant in all cases with the data-based DGP.}
Remarkably, the general outcome that (consistent) information criteria are particularly useful in selection of VAR models is fully in line with previous contributions in the literature (see e.g. Cavaliere et al., 2015, 2018; Kapetanios, 2004; Nielsen, 2006). Regarding the poor performances of the LY procedure, we remark that, differently from our DGPs, the dynamic component and the white noise are generated independently in the Monte Carlo study by Lam in Yao (2012). As noted by Lam et al. (2011), the cross correlation between the dynamic and the static component is beneficial to the estimation of $A$, resulting even in a faster convergence rate as $n^{-\delta/2}T^{1/2}$ when such cross correlation is strong, whereas it creates difficulties in the estimation of $r$ by the LY procedure, see the asymptotic analysis in Lam in Yao (2012).
This section illustrates the feasibility and the practical value of our dimension reduction approach to VAR modelling. We first search for the presence of co-movements among 211 US quarterly economic and financial time series. Based on our methodology, we then propose a novel approach to identify the shock that is responsible for most of the variability of the common components at the business cycle frequencies.
We start by investigating a dimension reduction for a high-dimensional VAR of the US economy. The data are obtained from the Federal Reserve Economic Quaterely Database (FRED-QD henceforth), to which we added the total factor productivity time series corrected for utilization produced by Fernald (2012).\footnote{This variable is relevant for the structural analysis that we conduct in the next subsection.} FRED-QD is regularly updated and different releases of all the series are available online. A detailed description of the variables and the proposed transformations used to achieve stationarity of each series is provided by McCracken and Ng (2020).
After some necessary cleaning of the dataset and various stationarity transformations of the series, we have at disposal $n=211$ variables with $T=242$ quarterly observations from 1959Q3 up to 2019Q4.\footnote{We did not include the variables that were not observed from 1959Q1 up to 2019Q4.} For comparison convenience, we use the transformations proposed in FRED-QD to make the time series stationary, although the results of unit root tests might suggest alternative transformations in some cases. Moreover, series are demeaned and standardized to have a unit variance after having corrected them for outliers.\footnote{In particular, we removed 20 outliers from 16 series. Our dataset is available upon request.}
We start the analysis by fixing $p_{0}=5$, a rather typical lag length of a VAR model for quarterly data, and $R=14$ as the upper bound of the dimension of the dynamic component.\footnote{The results that will be reported later are robust to alternative choices as $p_{0}=3,...,9$ and $R=12,...,15.$} The LY procedure detects $r=2$ dynamic components. This finding is rather dubious given the huge heterogeneity in the series we work with. In order to determine the largest VAR order, we use the traditional information criteria to estimate the lag length in a VAR model for series $\hat{V}_{R}^{\prime}Y_{t} $.\footnote{Notice that, if series $Y_{t}$ are generated by a DVAR model, then the linear combinations $V_{R}^{\prime}Y_{t}$ follow a VAR model of the same order as the dynamic component $x_{t}$.} We get $p=1$ according to the BIC, $p=2$ according to the HQIC, and $p=4$ according to the AIC. Consequently, we consider successively $p=1,...,4$ lags when estimating $r$ through the information criteria ((ref)) using OLS and FGLS in estimation. As in the Monte Carlo study, the two estimation methods provide identical estimates of $r$: BIC [AIC] systematically indicates $r=7$ [$r=14$], whereas HQIC indicates either $r=8$ or $r=12$ according to the VAR lag length. After a careful comparison of the various specifications, we opt for $r=8$ and $p=2$, with the latter being the indication coming from the HQIC when it is used to determine the lag length having fixed $r=8$.\footnote{Choosing either $r=12$ or $p=1$ leads to empirical results of the subsequent analysis that are qualitatively very similar to the ones that will be documented later.}\ We remark that the presence of eight common components is a rather typical finding in the empirical literature on factor models using similar data as ours.
Next, we compute two statistics in order to evaluate how the model fits to the data. First, we consider the coefficients of determination of each element of $Y_{t}$ as obtained by model ((ref)). Second, we compute the squared correlation coefficients between each element of $Y_{t}$ and its counterpart in the common component $\chi_{t}$ of representation ((ref)). We denote the former statistic as $R_{Y,Z}^{2}$\ and the latter as $R_{Y,\Xi} ^{2}$. It is easy to see that $R_{Y,\Xi}^{2}\geq$ $R_{Y,Z}^{2}$.
Whereas $R_{Y,Z}^{2}$ has the usual interpretation in terms of measure of the degree of predictability, $R_{Y,\Xi}^{2}$ indicates the fraction of the variability of each element of $Y_{t}$ that is explained by a linear projection on the present and past values of the dynamic errors $\xi_{t}$. Hence, $R_{Y,\Xi}^{2}$\ measures the importance of the common component $\chi_{t}$ in the variability of each series.\footnote{Notice that $R_{Y,\Xi }^{2}$ is actually computed as the sample analogous of the squared correlation coefficient between each element of $Y_{t}$ and the corresponding element in $Y_{t}-\nu_{t}$.}
Based on the FGLS estimates of the coefficients of the DRVAR model, we report in Table (ref) the averages as well as the quartiles of the empirical distributions of both $R_{Y,Z}^{2}$\ and $R_{Y,\Xi}^{2}$.
Moreover, in Table (ref) we report the estimates of both $R_{Y,Z}^{2}$\ and $R_{Y,\Xi}^{2}$ for nine macroeconomic variables that we are going to analyze in the subsequent subsection: Output (GDP), Consumption (Con), Investment (Inv), Unemployment Rate, (UR), Worked Hours (Hours), Inflation Rate (Inf), Interest Rate (IR), Labor Productivity (LP), and Total Factor Productivity (TFP). The exact denominations of the variables along with their stationarity transformations are reported in the appendix.
We see that, as expected, the estimates of $R_{Y,\Xi}^{2}$ are considerably larger than those of $R_{Y,Z}^{2}$ over all the series and the nine key variables as well. Remarkably, the role of the common component of TFP is smaller than the one of the other key variables. This finding may reflect the partial exogenous nature of TFP, as well as the possible presence of large estimation errors in a variable that it is not directly observable.
In light of Remark 4, it is of interest to check wether a FAVAR or an (unrestricted)\ DRVAR fits better to the data. The first step of such comparison requires to estimate a FAVAR model on our dataset. We use the nine key variables as the observed factors and the remaining variables to construct the unobserved factors, which are estimated by the principal components of the remaining $202$ variables.
In order to fix the number of unobserved factors, rather that relying on criteria that take into account the internal variability of the predictors only, we follow Pesaran et al. (2011) and use the traditional information criteria in a predictive model where the target variables are the nine key series and the predictors are two lags of both the targets and the estimates of the unobserved factors The BIC and HQIC respectively suggest the presence of $1$ and $3$ unobserved factors, whereas the AIC hits the upper bound. As in the case of the DRVAR, we follow the indication coming from the HQIC.
In light of Proposition 1, it is not obvious how to rank the two empirical models at the system level. A possible solution is contrasting the alternative specifications of the partial model of the key series only (i.e., the equations of the large VAR corresponding to the nine key variables). Given the different number of parameters in the DRVAR and FAVAR specifications of the considered partial model, a simple comparison of the respective log-likelihoods would be misleading. Hence, Table (ref) reports the results of the application of traditional information criteria to the competing specifications of the partial model for the key series, where the estimated factors are treated as observable in the computation of the penalty terms.\footnote{Since in both models the unobservable factors are estimated by the eigenvectors of large covariance matrices, the sample variability of the two estimators is expected to be similar.}
We see that BIC clearly supports the DRVAR specification, whereas AIC [HQIC] favors the FAVAR [DRVAR]. Hence, notwithstanding the specification of the FAVAR is tailored for the key variables, the two empirical models fit similarly to the key series.
A relevant issue in macroeconomics is the identifications of the shocks that drive the macroeconomic fluctuations. The textbook approach consists in identifying a given structural shock according to the guidance of economic theory (e.g. a productivity shock or a monetary shock) and to evaluate its impact over the key macroeconomic variables at various time horizons by means of the impulse response function and the forecast error variance decomposition.
Recently, this strategy has been subject of some criticism on the ground that empirical findings are strictly conditional on the validity of the underlying economic assumption. For instance, it may be too restrictive to assume that technology is the only shock that can permanently affect productivity. Alternatively, several authors have resorted to identification schemes that are based on the max-share identification strategy, as originally proposed by Uhlig (2003). The max-share methodology is a kind of reverse engineering approach, through which a shocks is identified as the main driver of a macroeconomic variable at a given time horizon. Contributions along this line of research include Barsky and Sims (2011) and Francis et al. (2014). Lately, Angeletos et al. (2020) proposed a max-share approach to identify the main driver of the business cycle. In particular, they identify the main business cycle shock as the shock that maximizes the volatility at the business cycle frequency band of a target variable. Using a Bayesian VAR model for ten macroeconomic variables, they show that alternatively targeting unemployment, output, hours worked, consumption and investment, their approach provides very similar impulse responses functions for ten key macroeconomic variables.
The approach that we adopt here is similar as the one by Angeletos et al. (2020) but with some relevant differences. First, we rely on a much richer information set coming from the large dimensional VAR that we have previously estimated. Second, we recognize that the business cycle is inherently a multivariate phenomenon and we aim at disentangling a unique driver of the business cycle for the whole economy rather than targeting a specific variable. Third, we search for the shock of the direction that maximizes the variability at the business cycle frequencies of the common component $\chi_{t}$ in decomposition ((ref)). In this way, we are able to filter out the effect of the ignorable errors, which cannot generate cyclical fluctuations by construction but still contaminate the observed variables $Y_{t}$. Notice that the last goal could not be pursued simply by estimating a large VAR with some shrinkage method.
Formally, in view of Equation ((ref)), the spectral density matrix of the common component $\chi_{t}$ is
where $z=\exp(-i\varpi)$ and $\varpi\in\lbrack0,2\pi)$.
Given that $\operatorname{Re}F_{\varkappa}(\varpi)$ is proportional to the variance matrix of the $\varpi$-frequency component in the spectral representation of $\chi_{t}$\ (see, e.g., Subsections 4.6 and 7.1 in Brillinger, 2001), the matrix \[ \Theta(\varpi_{0},\varpi_{1})=\int_{\varpi_{0}}^{\varpi_{1}}\operatorname{Re} F_{\varkappa}(\varpi)\mathrm{d}\varpi \] measures the (co-)volatility of the common component $\chi_{t}$ at the frequency band $[\varpi_{0},\varpi_{1}]$, where $0<\varpi_{0}<\varpi_{1}<\pi$.
Let $Q$ be the matrix formed by the eigenvectors that are associated with the first $r$ non-increasing eigenvalues of the matrix $\Theta(\varpi_{0} ,\varpi_{1})$, then the linear combinations $Q^{\prime}\chi_{t}$\ represent the (static) principal components of $\chi_{t}$\ at the frequency band $[\varpi_{0},\varpi_{1}]$. The Wold representation and the structural vector moving average representation of $Q^{\prime}\chi_{t}$ are respectively given by \[ Q^{\prime}\chi_{t}=Q(L)D\xi_{t}=Q^{\prime}\Psi(L)u_{t}, \] where $D=Q^{\prime}C_{0}$, $Q(L)=Q^{\prime}C(L)D^{-1}$, $\Psi(L)=C(L)D^{-1}C$, $u_{t}=C^{-1}D\xi_{t}$, and $C$ is a lower triangular matrix such that $CC^{\prime}=DA^{\prime}\Sigma_{{\small u}}AD^{\prime}$.
When $[\varpi_{0},\varpi_{1}]$ is the typical business cycle frequency band, i.e. $[2\pi/32,2\pi/6]$ for quarterly data, we label the first element in $u_{t}$\ as the Main Business Cycle Common Shock (MBCCS). In words, the MBCCS is the (standardized) shock of the direction that maximizes the contemporaneous variability of the common component $\chi_{t}$ at frequencies corresponding to periods between $6$ and $32$ quarters.\footnote{Notice that, since we are considering the real part only of the spectral density matrix $F_{\varkappa}(\varpi)$, we are implicitly focusing on the waves of $\chi_{t}$ at the frequency band $[\pi/16,\pi/3]$ that move in phase.} It is easy to see that the MBCCS is \[ u_{1t}=c_{11}^{-1}D_{1\bullet}^{\prime}\xi_{t} \] where $c_{11}$ is the entry at the first row and first column of $C$ and $D_{1\bullet}^{\prime}$\ is the first row of $D$, whereas, in light of decomposition ((ref)), the associated impulse response function (IRF) for series $Y_{t}$ is \[ \Psi_{\bullet1}(L)=C(L)D^{-1}C_{\bullet1}, \] where $C_{\bullet1}$ is the first column of $C$.
Given the frequency domain nature of our identification scheme, we evaluate the effects of the MBCCS on the variable of interest over frequencies rather than over time horizons (Centoni and Cubadda, 2003; Angeletos et al., 2020). In particular, we look at the contribution of the MBCCS to the variability of the $i-$th series at the business cycle frequency band \[ \frac{\int_{\pi/16}^{\pi/3}e_{i}^{\prime}\Psi_{\bullet1}(z^{-1})\Psi _{\bullet1}(z)^{\prime}e_{i}\mathrm{d}\varpi}{2\pi\int_{\pi/16}^{\pi/3} e_{i}^{\prime}F_{Y}(\varpi)e_{i}\mathrm{d}\varpi}, \] and at the zero frequency \[ \frac{e_{i}^{\prime}\Psi_{\bullet1}(1)\Psi_{\bullet1}(1)^{\prime}e_{i}}{2\pi e_{i}^{\prime}F_{Y}(0)e_{i}}, \] where $e_{i}$ is an $n-$vector with unity as its $i-$th element and zeros elsewhere, and the spectral density matrix of series $Y_{t}$ is \[ F_{Y}(\varpi)=F_{\varkappa}(\varpi)+(2\pi)^{-1}\mathrm{E}(\nu_{t}\nu _{t}^{\prime}) \]
Regarding the computational aspects, we truncate the order of the estimated polynomial matrix $C(L)$ by setting $\hat{C}_{j}=0$ for $j>199$ and we approximate integrals over the business cycle frequency band with sums over $100$ evenly spaced frequencies from $\pi/16$ up to $\pi/3$.
We start the empirical analysis by computing the sample eigenvalues of matrix $\Theta(\pi/16,\pi/3)$. The largest eigenvalue, namely the one that is associated with the MBCCS, accounts for about $54.8\%$ of the variability of the common component at the business cycle frequency band. Figure ((ref)) shows the estimate of the linear combination of the common component that is associated with the MBCCS, i.e. $Q_{1\bullet}^{\prime}\chi_{t}$\ where $Q_{1\bullet}^{\prime}$ is the first row of $Q$, which we label as the Main Business Cycle Common Component (MBCCC). Shaded areas in the figure represent NBER-defined recessions. We see that the MBCCC accurately reproduces the main features of the US aggregate cycle. Moreover, its estimated spectrum exhibits a unique large peak corresponding to fluctuations with a period of about five years.
We report in Table (ref) the estimates of the contributions of the MBCCS to the variability of the 9 key series at periods 6-32 and $\infty$. This information helps to asses the role of the MBCCS both at the business cycle frequencies and in the long-run.
Finally, Figure ((ref)) reports the estimates of the cumulated IRF to the MBCCS of the 9 variables already considered in the previous subsection. The IRF are cumulated to provide readers with the dynamic effects of the MBCCS on the levels of variables, thus facilitating the comparison with results of previous studies.
We see in Figure ((ref)) that the MBCCS triggers a procyclical effect on GDP and Inv peaking with one quarter delay, on Con with no delay, as well as on UR [Hours and IR] with a peak at two [three] quarters. Moreover, in view of Table (ref), it explains a large fraction of the cyclical variability of Inv, Hours, UR and GDP.\footnote{In general, the contribution of the MBCCS to the cyclical variability of the various variables is smaller than those of the shocks that Angeletos et al. (2020) obtain by targeting UR or other procyclical variables. However, it should be reminded that the aforementioned authors make use of a VAR with 10 series, whereas in our empirical model each series is hit by 211 reduced form errors. This considerably reduces the chances that a single shock can generate a very large fraction of cyclical volatility.} These results corroborate the claim that the considered shock is the main driver of the business cycle fluctuations.
However, the MBCCS has a limited positive impact on Inf, which peaks at one quarter and slowly dies out, and it marginally affects both LP and, especially, TFP. Moreover, it explains a small portion of the cyclical movements of both Inf and TFP.
Regarding the long-run scenario, the MBCCS explains around 35% of the zero-frequency variability of UR, Hours, Inv, and GDP, whereas it has negligible explanatory power for the permanent variation in Inf and TFP. Surprisingly, the MBCCS is responsible for almost the same portion of variability of Con (around 14%) both in the long and short run.\footnote{With the important caveat in mind that the validity of bootstrap for the DRVAR needs to be formally proven, the contributions of the MBCCS to the variability of GDP, INV, UR, HOURS at periods 6-32 and $\infty$ are larger than twice their bootstrap standard errors. For the same variables and IR, some short horizons of their IRFs are significant at the the 68% level. Bootstrap was implemented by sampling with replacement from the DRVAR residuals.}
All in all, the findings above seem to preclude the interpretation of the MBCCS as either a productivity or a news shock on the one hand, and a traditional demand shock on the other hand. However, a meticulous interpretation of the empirical results of this application, in particular by means of a rigorous comparison with DSGE models, is beyond the scope of the present paper.
This paper provides a link between two related but different strands of the literature on dimension reduction of multivariate time series, namely dynamic factor modelling on the one hand and the common feature methodology on the other hand. The former approach has the advantage that a limited number of factors is often enough to summarize the variation of a large dataset in economic and financial applications. However, the common feature approach has the nice theoretical property that the related common components posses a given time series feature (autocorrelation, volatility, trends, etc.) whereas the uncommon components do not.
Building on Lam et al. (2011) and Lam and Yao (2012), we propose a dimensional reduction approach such that both a common right space and a common left null space are present in the coefficient matrices of a large VAR model. This specification allows to detect a small dimensional VAR that is responsible for the whole dynamics of the system. This approach has many potential applications such as forecasting big data from a small scale VAR without loosing relevant information, structural VAR analysis, realized covariance matrices modelling, etc.
Our Monte Carlo study shows that we should consider either the BIC or the HQIC to detect the number of the common dynamic components. We illustrate the feasibility of our framework on large dimensional macroeconomic times series relative to the US economy. Around 8 common components generate the entire dynamics of 211 aggregate economic variables. Moreover, we offer a novel approach to identify the shock that is responsible for most of the volatility of the common component at the business cycle frequency band. Such shock as a clear expansionary effect, both in the short and long run, on the labor market variables, output, and investments but it affects marginally inflation and total factor productivity.