10.3 DETERMINING THE NUMBER OF COMMON FACTORS

10.3 DETERMINING THE NUMBER OF COMMON FACTORS

A key issue when forecasting with common factors is how many factors to include in the model. Two separate issues must be considered in constructing a forecasting model: first, how many factors are required to characterize the variation in the xvariables through the dynamics in (10.7) and (10.12), and second, how many factors should be included in a forecasting model such as (10.13).

Consider the first of these problems—how many factors should be chosen to replace the large-dimensional $X_t$ with a smaller set of factors. Let the number of factors be r and consider the static factor representation

$$X_{it}={\lambda }_i^{\prime }{F}_t^{\hat{P}}+{\epsilon }_{xit},i=1,\dots ,N.\text{\hspace{1em}(10.23)}$$

Inspection of the eigenvalues of the sample correlation matrix, ${\hat{\Sigma }}_{XX}=$ $\begin{array}{r}{T}^{-1}{\sum }_{t=1}^T{X}_t{X}_t^{\prime },\end{array}$ , provides a first informal way to select the number of factors, r . Letting ${\mu }_i$ be the i th eigenvalue of the covariance matrix arranged in descending order, the fraction of the total variance of X explained by the first r common factors is equal to ${\sum }_{i=1}^r{\mu }_i/N$ . Hence, factors with an eigenvalue exceeding 1 contribute more than the average factor. Eigenvalues close to 0 suggest that very little variation is explained by the accompanying factor. In any sample, the ordered eigenvalues will typically tail off towards 0 with no obvious “separation” between those that are nonzero in the limit and those that might be 0. This is entirely analogous to the usual model selection problem where coefficients that are truly nonzero and those that are truly 0 do not reveal themselves in a sample. There may not be an obvious threshold or cutoff point for determining the optimal value of r in any sample. Mirroring the general model selection problem, formal methods have been developed to address this point.

For consistent estimation of $r,$ formal methods require assumptions on the eigenvalues of the variance–covariance matrix of $X_t$ . What is required for the r factors to continue to account for the majority of the variation in $X_t$ as N becomes large is that the r th largest eigenvalue diverges to infinity. This ensures that the r th largest eigenvalue becomes separated from the remaining eigenvalues (asymptotically in N). Thus methods that consistently select the correct number of factors become available. Again, notice that this is analogous to the usual consistent model selection procedures outlined in chapter 6 where estimates of nonzero parameters and zero parameters asymptotically become far apart, enabling consistent model selection procedures such as BIC to pick the right model.

Differences across estimation procedures reflect the assumptions made on the speed at which the r th largest eigenvalue diverges. Additional differences center on the precise correlation structure of the idiosyncratic component ${\epsilon }_{xit}$ , particularly the allowance for temporal and cross-sectional dependence.

Bai and Ng (2002) develop methods for consistently selecting the number of factors in the context of approximate factor models. Their analysis allows for time series and cross-sectional heteroskedasticity as well as weak dependencies in ${\epsilon }_{xit}$ . Bai and Ng (2002) characterize the conditions under which a class of selection criteria consistently identify the correct number of factors, r . Let ${\hat{F}}_k^P$ be a matrix of k factors obtained through principal components so the sum of squared residuals from the associated model is given by

$$V(k,{\hat{F}}_k^P)=\frac{1}{NT}\sum _{i=1}^N\sum _{t=1}^T(X_{it}-{\lambda }_i^{k\prime }{\hat{F}}_{kt}^P{)}^2.$$

The three information criteria proposed by Bai and Ng take the following form:

$$\begin{array}{rl}& {\mathrm{IC}}_{p1}(k)=\ln (V(k,{\hat{F}}_k^p))+k\left(\frac{N+T}{NT}\right)\ln \left(\frac{NT}{N+T}\right),\\ & {\mathrm{IC}}_{p2}(k)=\ln (V(k,{\hat{F}}_k^p))+k\left(\frac{N+T}{NT}\right)\ln \left(\min (N,T)\right),\\ & {\mathrm{IC}}_{p3}(k)=\ln (V(k,{\hat{F}}_k^p))+k\frac{\ln \left(\min (N,T)\right)}{\min (N,T)}.\end{array}\text{\hspace{1em}(10.24)}$$

In each case the first term is a measure of fit while the second is a penalty term that increases as a function of min(N, T ) and the number of factors, k. Different values of $k=0,\dots ,\bar{k}$ can be tried and the value of k that minimizes (10.24) gives the number of selected factors. The penalty terms in (10.24) depend on both N and T. Conventional criteria that depend only on T or N generally choose too many factors. Bai and Ng establish conditions under which these criteria are consistent in the sense that when N and T both go to infinity, the selected number of factors, $\hat{k},$ tends to the true number of factors, r . For small values of N or T , Bai and Ng report that the criteria proposed in (10.24) fail to select the right number of factors, while for min $(N,T)\ge 40$ , the criteria are found to work well in a set of Monte Carlo simulations.

Results are also available under weaker assumptions on the rate of divergence of the r th largest eigenvalue. Under weaker assumptions, the separation is not as clear, even asymptotically; the smaller eigenvalues converge to a distribution and so the separation between the r th largest eigenvalue and the remaining ones is less clear. This analysis is intended to provide a better asymptotic approximation to the empirical sample distribution of the eigenvalues, which as noted above typically do not show such a large separation. The weaker assumption in this direction is often accompanied by stronger assumptions on the correlation structure of ${\epsilon }_{xit:}$ , although Monte Carlo evidence suggests that the effect of misspecifying the assumptions on the errors is small for reasonable models.

Onatski (2010) suggests a method based on these weaker assumptions on the rate of divergence of the eigenvalues. Under the weaker assumptions, the distribution of the eigenvalues smaller than the r th largest eigenvalue can be determined and the maximum point of this distribution can be employed as a cutoff value. The method involves estimating this cutoff value and then estimating the number of factors as the number of eigenvalues above the cutoff. The steps in the method of Onatski (2010) to estimate this cutoff are as follows in our notation:

Step 1: Compute the N eigenvalues of ${\hat{\Sigma }}_{XX}$ and denote them by ${\mu }_i,i=$ $\{1,2,\dots ,N\}$ , listed in descending order. Let $r_{\max }$ be the largest number of factors to be considered, and set $j=r_{\max }+1$

Step 2: Run a regression of $\{{\mu }_j,\dots ,{\mu }_{j+4}\}$ on $\{(j-1{)}^{2/3},(j{)}^{2/3},\dots ,(j+3{)}^{2/3}\}$ and set δ equal to 2 times the absolute value of the coefficient from this regression.

Step 3: Compute $\hat{r}(\delta )=\max \{i\le r_{\max }:{\mu }_i-{\mu }_{i+1}\ge \delta \}$ . If ${\mu }_i-{\mu }_{i+1}<\delta$ , set $\hat{r}(\delta )=0$

Step 4: Set $j=\hat{r}(\delta )+1$ and repeat steps (2) and (3) until convergence.

Here $r_{\max }$ is set to the largest number of factors to be considered.

Technically, $r_{\max }=r_{\max }^N$ should be chosen to be a sequence in N such that $r_{\max }^N/N\to 0$ as $N\to \infty ,$ , although rate conditions offer no specific insights for particular applications. This is standard when relying on as weak a concept as consistency and is similar to consistent model selection for information criteria discussed in chapter 6.

As always, there is a trade-off between parsimony and model fit. If too many factors are included in the forecast equation, this is likely to adversely affect the precision of the parameter estimates. Conversely, including too few factors means that potentially relevant information is not being used and leads to a loss of efficiency.