10.1.1 Dynamic Factor Models

10.1.1 Dynamic Factor Models

As shown by Stock and Watson (2006), the forecasting model in (10.3) is implied by quite general procedures for the dynamics in x and $y.$ This section follows their analysis and lays out the framework leading to the forecasting model (10.3).

Suppose that a panel of data $\{x_{it}\},i=1,\dots ,N,t=1,\dots ,T$ is available and that the variation in each of the observed variables, $x_{it}$ , can be decomposed into the effects of past and current values of a common component, $F_t$ , and an idiosyncratic component, ${\epsilon }_{xit}:$

$$x_{it}={\lambda }_i(L{)}^{\prime }{F}_t+{\epsilon }_{xit},\text{\hspace{1em}(10.5)}$$

where $x_{it}$ and ${\epsilon }_{xit}$ are scalars, $F_t$ is a $(q\times 1)$ vector of unobserved common factors, while ${\lambda }_i(L)=({\lambda }_{i1}(L){\lambda }_{i2}(L)\dots {\lambda }_{iq}(L){)}^{\prime }$ is a vector lag polynomial containing the dynamic factor loadings. Moreover, the idiosyncratic shocks are usually assumed to be uncorrelated with the factors at all leads and lags, i.e., $E\left[F_t{\epsilon }_{xis}\right]=0$ for all $i,t,$ and s . Because current and past values of the factors can affect the current value of the x-variables, this model is clearly dynamic.2 Dynamics may also arise through serial correlation in the idiosyncratic shocks, ${\epsilon }_{xit}$

$$\begin{array}{r}{\delta }_i(L){\epsilon }_{xit}=v_{xit},\end{array}\text{\hspace{1em}(10.6)}$$

where $v_{xit}$ is unpredictable given past information, i.e., $E[v_{xit}|F_{t-1},x_{t-1}$ $F_{t-2},x_{t-2},...]=0$ and ${\mathbf{x}}_{t-1}=(x_{1t-1},\dots ,x_{Nt-1}{)}^{\prime }$ contains the complete set of variables.

Suppose dynamics in the common factors can be captured through an autoregressive process of the form

$$\Gamma (L)F_t=v_{ft},\text{\hspace{1em}(10.7)}$$

where $\Gamma (L)$ is a matrix lag polynomial and $v_{ft}$ is a vector of unpredictable innovations.

The common factor structure in (10.5) is assumed to also carry over to $y,$ the variable we are interested in forecasting:

$$y_t={\lambda }_y(L{)}^{\prime }{F}_t+{\epsilon }_{yt},\text{\hspace{1em}(10.8)}$$

where ${\epsilon }_y$ may be serially correlated as captured through an autoregressive process:

$$\begin{array}{r}{\delta }_y(L){\epsilon }_{yt}=v_{yt},\end{array}\text{\hspace{1em}(10.9)}$$

and $v_{yt}$ is assumed to be unpredictable given current information, $E\left[{\mathbf{v}}_{yt+1}\right|F_t,{\mathbf{x}}_{\mathbf{t}}.$ $y_t,F_{t-1},x_{t-1},y_{t-1},...]=0$

Stock and Watson (2006) show that (10.8) and (10.9) are consistent with the factor model in (10.3). Specifically, if we pre-multiply equation (10.8) by ${\delta }_y(L)$ and use (10.9), we have

$${\delta }_y(L)y_{t+1}={\delta }_y(L){\lambda }_y(L{)}^{\prime }{F}_{t+1}+v_{yt+1},$$

or, equivalently,

$$y_{t+1}={\delta }_y(L){\lambda }_y(L{)}^{\prime }{F}_{t+1}+L^{-1}(1-{\delta }_y(L))y_t+v_{yt+1}.\text{\hspace{1em}(10.10)}$$

Defining ${\beta }_f(L)F_t=E[{\delta }_y(L){\lambda }_y(L{)}^{\prime }{F}_{t+1}|F_t,F_{t-1},\dots ]$ and $\gamma (L)=L^{-1}(1-{\delta }_y(L))$ we can rewrite (10.10) as

$$y_{t+1}=\beta (L)F_t+\gamma (L)y_t+u_{yt+1},\text{\hspace{1em}(10.11)}$$

where $u_{yt+1}=v_{yt+1}+\{{\delta }_y(L){\lambda }_y(L{)}^{\prime }{F}_{t+1}-E\left[{\delta }_y(L){\lambda }_Y(L{)}^{\prime }{F}_{t+1}|F_t,F_{t-1},...\right]\}$ comprises both idiosyncratic shocks that are specific to ${\mathbf{y}}_{}({\mathbf{v}}_{\mathbf{y}t+1})$ as well as shocks to the common factors, $v_{ft+1}$ . This equation thus expresses the future value of $\mathbf{y}$ in terms of past values of the common factors, past values of $\mathbf{y}$ itself, and future shocks, $u_{yt+1}$ , whose mean is unpredictable given current information. This is consistent with (10.3), the only difference being the $x^{\ast }$ -variables which, for simplicity, we ignore here.

In cases where N is large, it is convenient to recast the dynamic factor model using the following static representation:

$$X_t=\Lambda F_t^p+{\epsilon }_{xt},\text{\hspace{1em}(10.12)}$$

where now $F_t^p=(F_t^{\prime },F_{t-1}^{\prime },\dots ,F_{t-p+1}^{\prime }{)}^{\prime }$ is an $(r\times 1)$ vector that stacks current and lagged values of the common components (with $r=p\times q)$ and ${\epsilon }_{xt}$ again represents the idiosyncratic components. Supplementing (10.12) with an autoregressive model for the factor dynamics gives a state-space equation with $F_t^P$ as the latent state and (10.12) as the observation equation.3

The factor representation in (10.12) gives rise to a forecasting model of the form

$$y_{t+1}={\beta }^{\prime }{F}_t^{\text{p}}+\gamma (L)y_t+{\epsilon }_{yt+1},\text{\hspace{1em}(10.13)}$$

where the effects of current and lagged values of the factors come through the $\beta -$ coefficients.

10.2 ESTIMATION OF FACTORS

So far we have assumed that a set of factor estimates are available. In practice the factors are almost always unobserved and so have to be estimated. The two main methods for extracting factors are the parametric approach which makes use of assumptions about the dynamics and distributions of innovations in the model and nonparametric methods such as principal components which make weaker assumptions to identify the factors.