CCE in fixed‐T panels

Abstract

Summary The presence of unobserved heterogeneity and its likely detrimental effect on inference has recently motivated the use of factor‐augmented panel regression models. The workhorse of this literature is based on first estimating the unknown factors using the cross‐section averages of the observables, and then applying ordinary least squares conditional on the first‐step factor estimates. This is the common correlated effects (CCE) approach, the existing asymptotic theory for which is based on the requirement that both the number of time series observations, T, and the number of cross‐section units, N, tend to infinity. The obvious implication of this theory for empirical work is that both N and T should be large, which means that CCE is impossible for the typical micro panel where only N is large. In the current paper, we put the existing CCE theory and its implications to a test. This is done by developing a new theory that enables T to be fixed. The results show that many of the previously derived large‐T results hold even if T is fixed. In particular, the pooled CCE estimator is still consistent and asymptotically normal, which means that CCE is more applicable than previously thought. In fact, not only do we allow T to be fixed, but the conditions placed on the time series properties of the factors and idiosyncratic errors are also much more general than those considered previously.

Information Equivalence among Transformations of Semi‐parametric Nonlinear Panel Data Models*

“…My work generalizes a result of Im et al (1999) on linear models with additive heterogeneity to a factor-augmented error structure as studied in Pesaran (2006), Ahn et al (2013), Westerlund et al (2019) and Brown (2022). I show that any T − p rank transformation of the data that eliminates the factors is information equivalent to the infeasible transformation that treats the factors as known.…”

“…(2013), Westerlund et al . (2019) and Brown (2022). I show that any $$$ T-p $$$ rank transformation of the data that eliminates the factors is information equivalent to the infeasible transformation that treats the factors as known.…”

“…where Westerlund, Petrova, and Norkut (2019) show that when T is fixed and N → ∞, M F p − → M F − P −p where P −p is a nonlinear function of the model's errors. When p = K + 1 and the number of cross-sectional averages equals the number of factors, P −p = 0, and so the CCE transformation is asymptotically equivalent to the infeasible factor residual-maker matrix M F .…”

“…Westerlund, Petrova, and Norkut (2019) show that when $$$ T $$$ is fixed and $$$ N\to \infty $$$, $$$ {\boldsymbol{M}}_{\hat{\boldsymbol{F}}}\overset{p}{\to }{\boldsymbol{M}}_{\boldsymbol{F}}-{\boldsymbol{P}}_{-p} $$$ where $$$ {\boldsymbol{P}}_{-p} $$$ is a nonlinear function of the model's errors. When $$$ p=K+1 $$$ and the number of cross‐sectional averages equals the number of factors, $$$ {\boldsymbol{P}}_{-p}=\mathbf{0} $$$, and so the CCE transformation is asymptotically equivalent to the infeasible factor residual‐maker matrix $$$ {\boldsymbol{M}}_{\boldsymbol{F}} $$$.…”

The efficacy of ability proxies for estimating the returns to schooling: A factor model‐based evaluation

“…The assumptions required for identification in the CCE approach are similar to those previously outlined for the IFE approach except that the CCE approach does not require all the factors to be strong (Chudik et al, 2011) or the time dimension to be large (Zhou & Zhang, 2016; Westerlund et al, 2019). However, both CCE estimators can be sensitive to a particular rank condition that requires that the total number of factors does not exceed the total number of observed variables.…”