Accurate Confidence Regions for Principal Components Factors*

Abstract

In dynamic factor models, factors are often extracted using principal components with their asymptotic confidence regions having empirical coverages below the nominal ones when the temporal dimension is small. We propose a subsampling procedure to compute the factor loadings uncertainty and correct the asymptotic covariance matrix of the extracted factors. We show that the empirical coverages of the modified confidence regions are closer to the nominal ones than those of asymptotic regions and asymptotically valid bootstrap regions. The results are empirically illustrated obtaining confidence intervals of the underlying factor in a system of Spanish macroeconomic variables.

Ignoring Cross-Correlated Idiosyncratic Components When Extracting Factors in Dynamic Factor Models1

“…Figure 1 plots averages of the empirical coverages obtained through the replicates for dierent values of the cross-sectional dependence, τ . We can observe that, when T is small, the coverages are well below the 95% nominal level regardless of the cross-sectional dimension even if there is not idiosyncratic cross-sectional dependence; see Maldonado and Ruiz (2021), who show that, in the case of PC, parameter uncertainty is not included in the asymptotic MSE and, consequently, the coverage is below nominal. The coverages are close to the nominal level when T is large and τ = 0.…”

“…However, if T = 500, the percentages are 23% and 21%, respectively. It is important to note that, as far as we are concerned, this underestimation of the MSE of the factors cannot be corrected by using available resampling procedures; see, for example, the results in Maldonado and Ruiz (2021), who use subsampling to incorporate parameter uncertainty but still have large biases in the MSEs as they do not consider the misspecication of Σ ε . Constructing prediction intervals for the factors with coverages close to the nominal is still an open challenge in the presence of idiosyncratic cross-sectional dependence.…”

“…Table 1 reports M SE T t and M SE M t , together with the three terms in the decomposition in (14), denoted as P , M and E, respectively, when τ = 0 and 0.5. First, comparing M SE T t and M SE M t , we can observe that, regardless, of N , T and τ , and of the procedure used to extract the factors, the former MSE is clearly larger than the latter, explaining the large undercoverage of the prediction intervals for the factors observed in Figure 1.…”

“…In this case, the dierence between M SE T t and M SE M t is mainly due to the parameter estimation uncertainty not being considered when computing M SE M t ; observe that M is always close to zero. The parameter estimation uncertainty decreases with T and can be incorporated into the MSE by using resampling procedures as those proposed by Maldonado and Ruiz (2021) and Rodríguez and Ruiz (2012) for PC-based and KFS procedures, respectively. However, when τ = 0.5, the percentage of the dierence between M SE T t and M SE M t due to misspecication, M , is dierent from zero and its magnitude decreases with the cross-sectional dimension, N , but increases with T .…”