Panel Data Models

Model Framework

The basic framework of a regression model for panel data is

$${𝑦}_{𝑖𝑡}={𝑥}_{𝑖𝑡}^{\prime }\beta +{𝑧}_{𝑖}^{\prime }\alpha +{\epsilon }_{𝑖𝑡}$$

• Pooled Regression: If ${𝑧}_{𝑖}$ contains only a constant term.
• Fixed Effects: If ${𝑧}_{𝑖}$ is unobserved, but correlated with ${𝑥}_{𝑖𝑡}$, the model becomes ${𝑦}_{𝑖𝑡}={𝑥}_{𝑖𝑡}^{\prime }\beta +{\alpha }_{𝑖}+{\epsilon }_{𝑖𝑡}$.
• Random Effects: If ${𝑧}_{𝑖}$ is unobserved and uncorrelated with ${𝑥}_{𝑖𝑡}$, the model becomes ${𝑦}_{𝑖𝑡}={𝑥}_{𝑖𝑡}^{\prime }\beta +\alpha +{𝑢}_{𝑖}+{\epsilon }_{𝑖𝑡}$.
• Random Parameters: Coefficients vary randomly across individuals, i.e., ${𝑦}_{𝑖𝑡}={𝑥}_{𝑖𝑡}^{\prime }(\beta +{ℎ}_{𝑖})+(\alpha +{𝑢}_{𝑖})+{\epsilon }_{𝑖𝑡}$.

Assumptions

1. Full rank: $X$ has full column rank.
2. Exogeneity of the independent variables: $E[{\epsilon }_{𝑖}|{𝑥}_{𝑗1},{𝑥}_{𝑗2},\dots ,{𝑥}_{𝑗𝐾}]=0,i,j=1,...,n$.
3. Homoscedasticity and nonautocorrelation.

For consistency of $b$, we need $\mathrm{plim}\frac{1}{n}{𝑋}^{\prime }𝑋=\mathrm{plim}{\overline{Q}}_n=Q$, a positive definite matrix, and

$$\mathrm{plim}\frac{1}{n}{𝑋}^{\prime }\epsilon =\mathrm{plim}{\overline{w}}_n=E[{\overline{w}}_n]=0.$$

The Pooled Regression Model

Basic Pooled Regression Model

Assumptions.

$${𝑦}_{𝑖𝑡}=\alpha +{𝑥}_{𝑖𝑡}^{\prime }\beta +{\epsilon }_{𝑖𝑡},i=1,\dots ,n,t=1,\dots ,{𝑇}_{𝑖}$$$$𝐸[{\epsilon }_{𝑖𝑡}|{𝑥}_{1𝑡},{𝑥}_{2𝑡},\dots ,{𝑥}_{𝑖{𝑇}_{𝑖}}]=0$$$$\mathrm{Var}[{\epsilon }_{𝑖𝑡}|{𝑥}_{1𝑡},{𝑥}_{2𝑡},\dots ,{𝑥}_{𝑖{𝑇}_{𝑖}}]={\sigma }_{\epsilon }^2$$$$Cov[{\epsilon }_{i𝑡},x_{𝑗𝑠}|{𝑥}_{1𝑡},{𝑥}_{2𝑡},\dots ,{𝑥}_{𝑖{𝑇}_{𝑖}}]=0,\text{ if }i\ne j\text{ or }t\ne s.$$

OLS estimator is unbiased, consistent, and efficient.

Heterogeneity in Pooled Regression Model

A general model $y_i=X_i\beta +w_i$ and $\mathrm{Var}[w_i|X_i]={\sigma }_{\epsilon }^2{I}_{T_i}+{\mathrm{\Sigma }}_i={\mathrm{\Omega }}_i$.

$$\text{Asy.Var}[b]=\frac{1}{n}\mathrm{plim}{\left[\frac{1}{n}\sum _{i=1}^n{X}_i^{\prime }{X}_i\right]}^{-1}\mathrm{plim}\left[\frac{1}{n}\sum _{i=1}^n{X}_i^{\prime }{\mathrm{\Omega }}_i{X}_i\right]\mathrm{plim}{\left[\frac{1}{n}\sum _{i=1}^n{X}_i^{\prime }{X}_i\right]}^{-1}$$$$\boxed{{\displaystyle \text{No }\text{ Est. Asy. }\mathrm{Var}[\mathrm{b}]=\frac{1}{n}{\left(\frac{X^{\prime }X}{n}\right)}^{-1}\left(\frac{1}{n}\sum _{i=1}^n{e}_i^2{x}_i{x}_i^{\prime }\right){\left(\frac{X^{\prime }X}{n}\right)}^{-1}}}$$

White heteroscedasticity consistent estimator is not appropriate, since the problem is the correlation across observations, not heteroscedasticity.

Robust Estimation using Group Means

The pooled regression model can be estimated using the sample means of the data,

$$\frac{1}{T}{i}^{\prime }{y}_i=\frac{1}{T}{i}^{\prime }{X}_i\beta +\frac{1}{T}{i}^{\prime }{w}_i,$$

or

$${\overline{y}}_i={\overline{x}}_i^{\prime }\beta +{\overline{w}}_i,$$

where $i^{\prime }$ is a row vector of ones.

Estimation with First Differences

The first difference model is,

$$\begin{array}{rl}\mathrm{\Delta }{y}_{it}& =\mathrm{\Delta }\alpha +{\left(\mathrm{\Delta }{x}_{it}\right)}^{\prime }\beta +\mathrm{\Delta }{\epsilon }_{it}\\ & =\mathrm{\Delta }\alpha +{\left(\mathrm{\Delta }{x}_{it}\right)}^{\prime }\beta +{\epsilon }_{it}-{\epsilon }_{i,t-1}\\ & ={\left(\mathrm{\Delta }{x}_{it}\right)}^{\prime }\beta +u_{it}.\end{array}$$

The Within and Between Groups Estimation

General regression:

$$y_{it}=\alpha +x_{it}^{\prime }\beta +{\epsilon }_{it}.$$

Group means:

$${\overline{y}}_i=\alpha +{\overline{x}}_i^{\prime }\beta +{\overline{\epsilon }}_i.$$

Deviation from the group means:

$$y_{it}-{\overline{y}}_i=(x_{it}-{\overline{x}}_i{)}^{\prime }\beta +({\epsilon }_{it}-{\overline{\epsilon }}_i).$$

Define

$$S_{xx}^{\text{total }}=\sum _{i=1}^n\sum _{t=1}^T(x_{it}-\bar{\overline{x}}){(x_{it}-\bar{\overline{x}})}^{\prime }\text{ and }{S}_{xy}^{\text{total }}=\sum _{i=1}^n\sum _{t=1}^T(x_{it}-\bar{\overline{x}}){(y_{it}-\bar{\overline{y}})}^{\prime },$$$$S_{xx}^{within}=\sum _{i=1}^n\sum _{t=1}^T(x_{it}-{\overline{x}}_i){(x_{it}-{\overline{x}}_i)}^{\prime }\text{ and }{S}_{xy}^{within}=\sum _{i=1}^n\sum _{t=1}^T(x_{it}-{\overline{x}}_i){(y_{it}-{\overline{y}}_i)}^{\prime },$$

and

$$S_{xx}^{\text{between }}=\sum _{i=1}^n\sum _{t=1}^T({\overline{x}}_i-\bar{\overline{x}}){({\overline{x}}_i-\bar{\overline{x}})}^{\prime }\text{ and }{S}_{xy}^{\text{between }}=\sum _{i=1}^n\sum _{t=1}^T({\overline{x}}_i-\bar{\overline{x}}){({\overline{y}}_i-\bar{\overline{y}})}^{\prime },$$

then

$$b^{\text{total }}=F^{\text{within }}{b}^{\text{within }}+F^{\text{betweeen }}{b}^{\text{between }},$$

where

$$F^{\text{within }}={\left[S_{xx}^{\text{total }}\right]}^{-1}{S}_{xx}^{\text{within }}=\mathrm{I}-F^{\text{betweeen }}.$$$$\boxed{{\displaystyle S_{xx}^{\text{total }}=S_{xx}^{\text{within}}+S_{xx}^{\text{between }}.}}$$

The Fixed Effects Model

The fixed effects model has the form

$${𝑦}_{𝑖𝑡}={𝑥}_{𝑖𝑡}^{\prime }\beta +{\alpha }_{𝑖}+{\epsilon }_{𝑖𝑡}$$

• ${\alpha }_{𝑖}$ is allowed to be correlated with ${𝑥}_{𝑖𝑡}$, and each ${\alpha }_{𝑖}$ is treated as an unknown parameter to be estimated.
• The coefficients on the time invariant variables cannot be estimated.

In each group, we have

$$y_i=X_i\beta +i{\alpha }_i+{\epsilon }_i$$

Least Squares Estimation

Least squares dummy variables (LSDV) model:

$$𝑦=𝑋\beta +𝐷\alpha +\epsilon .$$

Estimator of $\beta$:

$$𝑏=[{𝑋}^{\prime }{𝑀}_{𝐷}𝑋{]}^{-1}[{𝑋}^{\prime }{𝑀}_{𝐷}𝑌]={𝑏}^{\text{within}},$$

where

$${𝑀}_{𝐷}=I-𝐷({𝐷}^{\prime }𝐷{)}^{-1}{𝐷}^{\prime }.$$

Estimator of ${\alpha }_{𝑖}$:

$$a=[{𝐷}^{\prime }𝐷{]}^{-1}{𝐷}^{\prime }(𝑦-𝑋𝑏),$$

so we have

$${\alpha }_{𝑖}={\overline{𝑦}}_{𝑖}-{\overline{𝑥}}_{𝑖}^{\prime }𝑏.$$

Covariance matrix for $b$ is

$$\mathrm{Est}.\mathrm{Asy}.\mathrm{Var}[b]=s^2{\left[X^{\prime }{M}_{D}X\right]}^{-1}=s^2{\left[S_{XX}^{\text{within }}\right]}^{-1},$$

and

$$s^2=\frac{\sum _{i=1}^n\sum _{t=1}^T{(y_{it}-x_{it}^{\prime }b-a_i)}^2}{nT-n-K}=\frac{{(M_{D}y-M_{D}Xb)}^{\prime }(M_{D}y-M_{D}Xb)}{nT-n-K}.$$

Significance of the Group Effects

Test:

$$\begin{array}{ccc}{H}_0& :& \text{All }{\alpha }_i\text{ are equal};\\ H_1& :& \text{Otherwise}.\end{array}$$

Statistic:

$$F(n-1,nT-n-K)=\frac{(R_{LSDV}^2-R_{Pooled}^2)/(n-1)}{(1-R_{LSDV}^2)/(nT-n-K)}.$$

The Random Effects Model

Assumptions

The random effect model is

$$y_{it}=\alpha +x_{it}^{\prime }\beta +u_i+{\epsilon }_{it}=\alpha +x_{it}^{\prime }\beta +{\eta }_{it},$$

with assumptions

$$\begin{array}{rl}E[{\epsilon }_{it}\mid X]=E[u_i\mid X]& =0,\\ E[{\epsilon }_{it}^2\mid X]& ={\sigma }_{\epsilon }^2,\\ E[u_{it}^2\mid X]& ={\sigma }_u^2,\\ E[{\epsilon }_{it}{u}_j\mid X]& =0\text{ for all i, t, and j, }\\ E[{\epsilon }_{it}{\epsilon }_{js}\mid X]& =0\text{ for }\mathrm{t}\ne \mathrm{s}\text{ or }\mathrm{i}\ne \mathrm{j},\\ E[u_i{u}_j\mid X]& =0\text{ if }\mathrm{i}\ne \mathrm{j}.\end{array}$$

Least Squares Estimation

Three models,

$$\begin{array}{crl}(1)& y_{it}& =\alpha +x_{it}^{\prime }\beta +u_i+{\epsilon }_{it},\\ (2)& y_{it}-{\overline{y}}_i& =(x_{it}-\overline{x}{)}^{\prime }\beta +({\epsilon }_{it}-{\overline{\epsilon }}_i),\\ (3)& {\overline{y}}_i& =\alpha +{\overline{x}}_i^{\prime }\beta +u_i+{\overline{\epsilon }}_i.\end{array}$$

Variance estimation,

$$\begin{array}{l}\text{ (Pooled) }\mathrm{plim}[e_{\text{pooled }}^{\prime }{e}_{\text{pooled }}/(nT)]={\sigma }_{\epsilon }^2+{\sigma }_u^2,\\ \text{(LSDV)}\mathrm{plim}[e_{LSDV}^{\prime }{e}_{LSDV}/(nT)]={\sigma }_{\epsilon }^2[1-1/\mathrm{T}],\\ \text{ (Means) }\mathrm{plim}[e^{\prime }{}_{\text{means }}{e}_{\text{means }}/(nT)]={\sigma }_{\epsilon }^2/T+{\sigma }_u^2\end{array}$$

Generalized Least Squares

The GLS estimator is

$$b^{\text{total }}={\hat{F}}^{\text{within }}{b}^{\text{within }}+(\mathrm{I}-{\hat{F}}^{\text{within }})b^{\text{between }},$$

where

$${\hat{F}}^{\text{within}}=[S_{xx}^{within}+\lambda S_{xx}^{\text{between}}{]}^{-1}{S}_{xx}^{\text{within}},$$

and

$$\lambda =\frac{{\sigma }_{\epsilon }^2}{{\sigma }_{\epsilon }^2+T{\sigma }_u^2}=(1-\theta {)}^2.$$

FGLS

Estimate variances,

$${\sigma }_{\epsilon }^2=s_{\text{LSDV}}^2=\frac{\sum _{i=1}^n\sum _{t=1}^T(e_{it}-{\overline{e}}_i{)}^2}{nT-n-K},$$

and

$${\hat{\sigma }}_u^2=s_{\text{Pooled}}^2-s_{\text{LSDV}}^2.$$

Testing for Random Effects

Lanrange multiplier test,

$$H_0:{\sigma }_u^2=0\text{ (or }Corr[{\eta }_{it},{\eta }_{is}]=0);$$$$H_1:{\sigma }^2\ne 0.$$

The test statistic is

$$LM=\frac{nT}{2(T-1)}{[\frac{\sum _{i=1}^n[\sum _{t=1}^T{e}_{it}{]}^2}{\sum _{i=1}^n\sum _{t=1}^T{e}_{it}^2}-1]}^2\stackrel{d}{\sim }{\chi }^2(1).$$

Hausman’s Specification Test

Hausman’s test is used to decide which model, fixed effects or random effects, to use.

The test statistic is

$$(b-\hat{\beta }{)}^{\prime }[\mathrm{Var}(b)-\mathrm{Var}(\hat{\beta }){]}^{-1}(b-\hat{\beta })\sim {\chi }^2(K-1).$$