10.2.3 Consistency and Efficiency

10.2.3 Consistency and Efficiency

Stock and Watson (2002a) provide conditions under which the unobserved common factors can be consistently estimated and principal components analysis consistently estimates the space spanned by the common factors. Equally important, they establish when the feasible forecasts based on factor estimates and estimated coefficients of the forecasting relation (10.13) are asymptotically efficient in the sense that, at least up to first order, nothing is lost by using the extracted as opposed to the true factors. Their conditions are quite general and allow for serial correlation in the errors as well as “weak” temporal instability.

Specifically, Stock and Watson (2002a) assume that the factors and factor loadings in representations such as (10.12) satisfy (a) $\left({\Lambda }^{\prime }\Lambda /N\right)\to I_r;\left(\mathbf{b}\right)E\left[F_t^{\mathcal{P}}{F}_t^{\mathcal{P}\prime }\right]={\Sigma }_{FF}$ with ${\Sigma }_{FF}$ being a diagonal matrix for which ${\sigma }_{ii}>{\sigma }_{jj}$ for $i<j;\left(c\right)\left|{\lambda }_{i,m}\right|\le \bar{\lambda }<\infty ;$ and (d) $T^{-1}{\sum }_t{F}_t^{\mathcal{P}}{\check{F}}_t^{{\mathcal{P}}^{\prime }}{}^{\mathcal{P}}{\Sigma }_{FF}$ , where ${\lambda }_{i,m}$ are the elements of the factor loadings, -.

Assumptions (a) and (d) ensure that each of the factors contributes a nonvanishing amount to the variation in each of the observed variables ($X_t$). The requirement that the matrix $N^{-1}{\Lambda }^{\prime }\Lambda$ converges to a positive-definite limit means that, on average, factors affect all variables in a similar way. For example, some factors cannot affect only the first few variables but have a zero impact (loading) on a large set of additional variables.

Assumptions (a) and (b) together identify the factors up to a diagonal matrix with elements ±1. The assumptions do not rule out serial correlation in the factors, nor do they rule out conditional dynamics in the second moments of the factors. Temporal variations such as trends or structural breaks in the variance of the factors are ruled out, however.

Stock and Watson allow the innovations in the factor model ${\epsilon }_{xt}$ to be serially correlated and weakly cross-sectionally correlated. Although no specific distributional assumptions such as normality are made, their assumption on the fourth moments of the innovations rules out some error distributions.

These assumptions are related to the distinction between strict and approximate factor models. Strict factor models assume that the elements of the vector of innovations ${\epsilon }_{xt}$ are mutually uncorrelated so that $E\left[{\epsilon }_{xt}\right]=0,E\left[{\epsilon }_{xt}{\epsilon }_{xt}^{\prime }\right]=$ $\mathrm{diag}({\sigma }_1^2,\dots ,{\sigma }_N^2)]$ , and $E[F_t{\epsilon }_{xt}^{\prime }]=0$ . Approximate factor models relax these strong assumptions in two ways. First, they allow for weak serial correlation in the idiosyncratic errors. It turns out that as long as the idiosyncratic errors follow stationary ARMA processes, the principal component estimator will still be consistent as N tends to infinity. Second, the idiosyncratic errors can be weakly cross-sectionally correlated and heteroskedastic. This may be relevant if certain variables form clusters. The idiosyncratic errors and factors can also be weakly correlated.4

Under these assumptions, Stock and Watson (2002a) show that the true factors can be consistently estimated and are identified up to a sign transformation. Specifically, letting $N,T\to \infty$ for a model with k estimated factors, a set of indicator variables $S_i\in \{-1,+1\}$ can be selected such that, from Stock and Watson (2002a, Theorem 1),

$$T^{-1}\sum _{t=1}^T(S_i{\hat{F}}_{it}^p-F_{it}^p{)}^2{\to }^{p}0\mathrm{for}i=1,\dots ,r,$$

S _ { i } \hat { F } _ { i t } ^ {p} ^ {p} F _ { i t } ^ {p} \quad \mathrm { f o r} i = 1, . . . , r ,

$$T^{-1}\sum _{t=1}^T({\hat{F}}_{it}^{\hat{P}}{)}^2{\to }^{\hat{P}}0\mathrm{for}i=r+1,\dots ,k.$$

Returning to the general forecasting equation based on the static factor representation,

$$y_{t+1}={\beta }_F^{\prime }{F}_t^{\mathring{P}}+{\beta }_w^{\prime }{w}_t+{\epsilon }_{yt+1},\text{\hspace{1em}(10.19)}$$

where $w_t$ contains observable predictor variables, including lagged values of $y.$ Perhaps the most important result proved by Stock and Watson is that the feasible forecast (which is based on extracted factors and estimates of the parameters of the forecasting model) converges asymptotically to the optimal infeasible forecast which assumes that $F_t^P$ and $({\beta }_F^{\prime }{\bar{\beta }}_w^{\prime }{)}^{\prime }$ are known, thus ensuring that the feasible coefficient estimates are consistent. To prove this, Stock and Watson use standard assumptions on the forecasting equation that ensure consistency of the parameter estimates, but add extra assumptions to deal with the fact that an estimate, ${\hat{F}}_t^p$ , is used in place of $F_t^P$

Defining $z_t=(F_t^{p\prime }{w}_t^{\prime }{)}^{\prime }$ and $\beta =({\beta }_F^{\prime }{\beta }_{\omega }^{\prime }{)}^{\prime }$ with $|\beta |<\infty$ , Stock and Watson (2002a) assume that $E\left[z_t{z}_t^{\prime }\right]={\Sigma }_{zz}$ is a positive-definite matrix. Moreover,

$$T^{-1}\sum _t{z}_t{z}_t^{\prime }\to P_{\tau z},$$

$$T^{-1}\sum _t{z}_t{\epsilon }_{yt+1}{\to }^{p}0,$$

$$T^{-1}\sum _t{\epsilon }_{yt+1}^2{\to }^p{\sigma }^2.$$

Using these assumptions along with the earlier factor and error moment assumptions, Stock and Watson (2002a) establish results for the OLS estimates $({\hat{\mathbf{\beta }}}_F^{\prime }{\hat{\mathbf{\beta }}}_{x^{\ast }}^{\prime }{)}^{\prime }$ obtained from regressions of $\{y_{t+1}{\}}_{t=1}^{T-1}$ on $\{{\hat{F}}_t^p,x_t^{\ast }{\}}_{t=1}^{T-1}$ . Specifically, they show that

$$({\hat{\beta }}_F^{\prime }{\hat{F}}_T^{\mathcal{P}}+{\hat{\beta }}_w^{\prime }{w}_T)-({\beta }_F^{\prime }{F}_T^{\mathcal{P}}+{\beta }_w^{\prime }{w}_T){\to }^{\mathcal{P}}0.$$

Moreover, ${\hat{\beta }}_{\omega }-{\beta }_{\omega }\to P0$ and there exists an $S_i\in \{-1,+1\}$ such that $S_i{\hat{\beta }}_{iF}-$ ${\beta }_{iF}^{P}0$ for $i=1,\dots ,r{.}^5$

To summarize, even under an approximate factor structure entailing weak serial and cross-sectional correlation in the idiosyncratic shocks, common factors can be consistently estimated under fairly mild rate and moment conditions. In turn, such factor estimates can be treated as if they were the true factors in the forecast model that projects future realizations on the factors in the sense that the asymptotic properties of the regression coefficients will not be affected by estimation error in the factors.

Despite the elegance and generality of these theoretical results, one should also bear in mind that, as emphasized by Ng (2013) as well as our discussion in chapter 6, there is an inevitable tension between selecting the true model (consistent model selection) versus generating accurate forecasts. Hence, it is not necessarily the case that methods that succeed in (asymptotically) identifying the correct factor structure and consistently estimate the model parameters lead to the best forecasts.