A penalized spline estimator for fixed effects panel data models

Abstract

Estimating nonlinear effects of continuous covariates by penalized splines is well established for regressions with cross-sectional data as well as for panel data regressions with random effects. Penalized splines are particularly advantageous since they enable both the estimation of unknown nonlinear covariate effects and inferential statements about these effects. The latter are based, for example, on simultaneous confidence bands that provide a simultaneous uncertainty assessment for the whole estimated functions. In this paper, we consider fixed effects panel data models instead of random effects specifications and develop a first-difference approach for the inclusion of penalized splines in this case. We take the resulting dependence structure into account and adapt the construction of simultaneous confidence bands accordingly. In addition, the penalized spline estimates as well as the confidence bands are also made available for derivatives of the estimated effects which are of considerable interest in many application areas. As an empirical illustration, we analyze the dynamics of life satisfaction over the life span based on data from the German Socio-Economic Panel. An open-source software implementation of our methods is available in the R package pamfe.

Appendix: Serial correlation in the first-difference errors

Consider Eq. (7): If the error terms $u_{it},\; i=1,\ldots ,N,\, t=1,\ldots ,T_{i}$ are homoscedastic and independent with expectation zero, then $\text {E}(u_{it}u_{i,t-1})=0 \text { and }\text {E}(u_{it}u_{it})=\sigma _{u}^{2}.$ It follows for the errors $\triangle u_{it}=u_{it}-u_{i,t-1}$ in Eq. (9):

$$\begin{aligned} \text {E}(\Delta {u_{it}})=\text {E}(u_{it}-u_{i,t-1})=0 \end{aligned}$$

Fig. 4

Nonparametrically estimated relationship between household income (in 1000 €) and life satisfaction with confidence bands

Table 2 Estimation results for strictly parametric components

and

$$\begin{aligned} \text {Var}(\Delta {u_{it}})=\text {Var}(u_{it}-u_{i,t-1}) =\text {Var}(u_{it})+\text {Var}(u_{i,t-1})=2\sigma _{u}^{2}. \end{aligned}$$

The correlation of two consecutive error terms for the same individual after applying first differences is then given by

$$\begin{aligned} \text {Cor}(\Delta {u_{it}},\Delta {u_{i,t-1}})= & {} \frac{\text {E}\left[ (\Delta {u_{it}}) (\Delta {u_{i,t-1}})\right] }{\sqrt{\text {Var}(\Delta {u_{it}})\text {Var}(\Delta {u_{i,t-1}})}}\\= & {} \frac{\text {E}\left[ (u_{it}-u_{i,t-1})(u_{i,t-1}-u_{i,t-2})\right] }{\sqrt{2\sigma _{u}^{2}2\sigma _{u}^{2}}}\\= & {} \frac{\text {E}\left( -u_{i,t-1}^{2}\right) }{2\sigma _{u}^{2}}=\frac{-\sigma _{u}^{2}}{2 \sigma _{u}^{2}}=-0.5. \end{aligned}$$