Specification Testing for Functional Forms in Dynamic Panel Data Models

The most popular econometric models in the panel data literature are the class of linear panel data models with unobserved individualand/or time-specific effects. The consistency of parameter estimators and the validity of their economic interpretations as marginal effects crucially depend on the correct functional form specification of the linear panel data model. Based on an individual-specific generalized spectral derivative approach, a new class of residual-based tests is proposed for functional form specification of dynamic panel data models with both large cross-sectional units and time series dimensions. Panel data models can have one-way or two-way error components, individual and time effects can be fixed or random, and panel data can be balanced or unbalanced. The tests can detect a wide range of model misspecifications in the conditional mean of a dynamic panel data model, including functional form and lag misspecification. No common alternative is assumed, thus allowing heterogeneity in the degrees and directions of functional form misspecification across individuals. Moreover, the proposed tests are robust to conditional heteroskedasticity and higher order time-varying conditional moments of unknown form. They check a large number of lags so that they can capture misspecification at any lag order asymptotically. The large number of lag orders does not cause loss of degrees of freedom because the proposed tests naturally discount higher order lags, which is consistent with the stylized fact that economic behaviors are more affected by recent past events than by remote past events. No specific estimation method for dynamic panel data models is required, and the tests have an appealing “nuisance parameter free” property that parameter estimation uncertainty has no impact on the asymptotic distribution of the test statistics. The asymptotic theory developed is the joint asymptotics for degenerate U -statistics, which provides an alternative approach to the existing joint limit approaches in the panel data literature. Thanks to the use of panel data with both large N and T, the proposed nonparametric tests have an asymptotic normal distribution under the null hypothesis without requiring the smoothing parameters to grow with the sample sizes, which suggests better nonparametric asymptotic approximation for panel data than for time series or cross sectional data. This is confirmed in a simulation study. The proposed tests have omnibus power against a variety of alternatives.