ECON 407: Introduction to Panel Data

To formalize this, consider a model of the dependent variable \(y\) that is a linear function of \(K\) independent variables (and parameters) and an individual specific unobservable factors \(c_i\). Write this as

\begin{equation} y_{it}=x_{it}\beta+c_i+\epsilon_{it} \end{equation}

where \(\beta\) is of dimension \(K\times 1\). If \(c_i\) is uncorrelated with each \(x_{it}\) then we can proceed using OLS and allow the effect of \(c_i\) to be soaked up into the error term. If on the other hand \(c_i\) is correlated with \(x_{it}\) (\(cov(x_i,c_i) \ne 0\)), then putting the effect of \(c_i\) in the error term can lead to biased estimated of \(\beta\) by way of missing variable bias.

Note that we are assuming that the unobserved individual specific effects (\(c_i\)) are constant across time. What is the economic meaning of \(c_i\)? If we are estimating the labor market decisions of women in their twenties and we observed these women in each year, then this time invariant unobservable information might be capturing cognitive ability, drive and determination, or family upbringing that may well influence outcomes for an individual but not vary over time. If our unit of analysis is a firm in an analysis of productivity, \(c_i\) might be capturing information about the managerial ability or organization that is roughly constant through time. Ultimately, these unobserved factors are not of direct interest in these models, we need to worry about them to avoid biased parameters of the marginal effects

\begin{equation} \beta=\frac{\partial E[y_{it}|x_{it}]}{\partial x_{it}} \end{equation}

To estimate these marginal effects, we need to invoke some assumptions. As in the OLS case, we need exogeneity

\begin{equation} E(\epsilon_{t}|\mathbf{x}_{t},\mathbf{c})=0 \end{equation}

Which gives us the familiar condition that

\begin{equation} E(\mathbf{x}_{t}'\epsilon_{t})=0 \label{eq:ols_orthog_assumption} \end{equation}

for \(t=1,2\). If we assume that \(E(\mathbf{x}_{t}'c)=0\) then we could pool the data and estimate an OLS model. If this condition does not hold, however, an OLS model will lead to biased and inconsistent parameter estimates.

Suppose that for each individual, we have a panel of two periods (\(t=1,2\)). Perhaps we could apply a differencing approach for each individual i to rid the model of \(c_i\), since

\begin{aligned} \Delta y_{i} = & \beta \left (x_{i2}-x_{i1} \right ) + \left (c_i-c_i \right) + \left (\epsilon_{i2}-\epsilon_{i1} \right) \\ = & \Delta x_{i} \beta + \Delta \epsilon_{i} \end{aligned}

since the individual-specific effect is time invariant, these constants drop from the model, and we are still left with our parameters of interest (\(\beta\)). So in a panel with two time periods, we are left with one standard cross section equation consisting of individual level data denoted by

\[\Delta y = \Delta x \beta + \Delta \epsilon\]

where

\[ \Delta y = \begin{bmatrix} \Delta y_1 \\ \vdots \\ \Delta y_i \\ \vdots \\ \Delta y_N \end{bmatrix} \]

and \(\Delta x\) is defined in a similar fashion. Will a simple OLS estimate of this cross section be unbiased and consistent?

The orthogonality condition is the first place to start with answering this question. Recall that in a standard regression model of the previous chapter, the proof of unbiasedness rests with Equation \(\eqref{eq:ols_orthog_assumption}\), and invoking the fact that \(E(x'\epsilon)=0\) allows us to show that \(E(b)=\beta\). For OLS to be unbiased, we need a similar condition here. Note that this means

\begin{equation} E(\Delta x' \Delta \epsilon)=0 \label{eq:first_diff_assumption1} \end{equation}

and the following inverse must exist

\begin{equation} \label{eq:first_diff_assumption2} (\Delta x' \Delta x)^{-1} \end{equation}

requiring that \(\Delta x\) must be of full rank.

What are the implications of these assumptions? First consider Equation \(\eqref{eq:first_diff_assumption1}\). This condition can be rewritten as

\begin{aligned} & E \left [(x_{2}-x_{1})'(\epsilon_{2}-\epsilon_{1}) \right]=0 \\ & E \left [(x_{2}'\epsilon_{2}+x_{1}'\epsilon_{1}-x_{1}'\epsilon_{2}-x_{2}'\epsilon_{1}) \right]=0 \end{aligned}

The first two terms are zero by definition given assumption \(\eqref{eq:ols_orthog_assumption}\). Note however, that in this framework, we also need to assume

\begin{aligned} E(x_{1}'\epsilon_{2})&=0 \\ E(x_{2}'\epsilon_{1})&=0 \end{aligned}

these highlight the even more restrictive orthogonality conditions we need to assume to implement this type of approach.

It is also useful to think carefully about equation \(\eqref{eq:first_diff_assumption2}\). Since the matrix \(\Delta x\) is comprised of elements differenced across two periods, x may not contain any variable that is constant across time for every person in the sample. For example, if the Kth element of x were to contain a dummy variable equal 1 if a person's sex is female and 0 if male, then

\[ \Delta x_K= \begin{bmatrix} \color{blue}{1-1} \\ \color{blue}{\vdots} \\ \color{blue}{1-1} \\ \vdots \\ \color{red}{0-0} \\ \color{red}{\vdots} \\ \color{red}{0-0} \\ \vdots \\ \color{green}{1-1} \\ \color{green}{\vdots} \\ \color{green}{1-1} \\ \end{bmatrix} \]

so the matrix \(\Delta x\) would not be of full rank since every element in column K is equal to zero. For some of the panel data models we will consider (i.e. the Fixed Effects Model), your independent variables must vary across periods in your panel. This is very important to remember.

Following from our framework above, the unobserved effects model can be written

\[y_{it}=x_{it}\beta+c_i+\epsilon_{it}\]

where \(x_{it}\) is \(1 \times K\). The literature has implemented this type of model in a number of ways, but the most prevalent are termed (1) random effects and (2) fixed effects. These terms refer to assumptions about the individual-specific constant terms \(c_i\).

The labels fixed and random effects are misleading. What matters here is the nature of the unobserved, time invariant, and individual-specific quantities. If there is a dependence between these unobserved factors and the observed independent variables we employ the fixed effects approach. If on the other hand, these effects are independent of the observed independent variables, then we use the random effects estimator. We will explore the difference in the models, compare these approaches to Pooled OLS, and develop some test statistics for model selection.

The pooled model simply applies an OLS estimate to the pooled data set (where each individual i's data is ordered from \(t=1,\ldots,T\), and then vertically stacked.). For a data set of N individuals across T periods, the vector \(y\) and the matrix \(x\) will look like

\[ y=\begin{bmatrix} \color{blue}{y_{11}} \\ \color{blue}{\vdots} \\ \color{blue}{y_{1T}} \\ \vdots \\ \color{red}{y_{i1}} \\ \color{red}{\vdots} \\ \color{red}{y_{iT}} \\ \vdots \\ \color{green}{y_{N1}} \\ \color{green}{\vdots} \\ \color{green}{y_{NT}} \\ \end{bmatrix} x=\begin{bmatrix} \color{blue}{x_{111} \ldots x_{11k} \ldots x_{11K}}\\ \color{blue}{\ddots} \\ \color{blue}{x_{1T} \ldots x_{1T} \ldots x_{1T}}\\ \vdots \\ \color{red}{x_{i1} \ldots x_{i1} \ldots x_{i1}}\\ \color{red}{\ddots} \\ \color{red}{x_{iT} \ldots x_{iT} \ldots x_{iT}}\\ \vdots \\ \color{green}{x_{N1} \ldots x_{N1} \ldots x_{N1}}\\ \color{green}{\ddots} \\ \color{green}{x_{NT} \ldots x_{NT} \ldots x_{NT}} \\ \end{bmatrix} \]

For pooled OLS to be the appropriate estimator, we need to assume

Assumption 1: Strict Endogeneity

\[E[\epsilon_{it}|x_{i1},\ldots,x_{iT}]=0\]

Assumption 2: Uncorrelated individual-specific effects

\[E[c_i|x_{i1},\ldots,x_{iT}]=0\]

Defining \(v_{it}=c_i+\epsilon_{it}\) now completely uncorrelated with the covariates, we have an error structure with orthogonality just as in the previous chapter and our analysis of the OLS model. So we can recover the parameter using

\[\mathbf{b_{ols}}=\mathbf{(x'x)}^{-1}\mathbf{x'y}\]

The usual estimate for the variance covariance matrix, and associated standard errors is

\[Var[\mathbf{b_{ols}}]=\mathbf{\sigma^2E_x[(x'x)^{-1}]}\]

Taking note of the error structure inherent in an OLS model (where \(\epsilon_{i}\sim N(0,\sigma I)\)) compared to this case where the error contain the usual \(\epsilon_{it}\sim N(0,\sigma)\) plus \(c_i\) which is shared by each cross section unit (e.g. each individual or firm). This may well introduce a systematic relationship between the composite error (\(v_{it}\)) among cross section units that the OLS variance-covariance estimate ignores. To fix, this can apply the robust standard errors using the technique we discussed in the previous chapter.

Following the recovery of \(\mathbf{b_{ols}}\), calculate the estimated residuals

\[\mathbf{\hat{v}=y-x b_{ols}}\]

Using the notation adopted in the OLS Chapter for correcting for heteroskedasticity, the expected variance-covariance matrix for the error is

\[\label{eq:het_correction_pooled_OLS} E[var(\mathbf{vv'})]=\hat{\mathbf{V}}=diagonal\left(\hat{\mathbf{v}}\hat{\mathbf{v}}' \right )_{N \times N}\]

let's us write the robust pooled variance covariance matrix of the parameters as

\[Var^{robust}=\mathbf{(x'x)^{-1}(x'\hat{V}x)(x'x)^{-1}}\]

Notice that this equation is exactly equivalent to the definition of robust standard errors in OLS and highlites the strong assumptions we are making when we estimate panel data with a pooled OLS model.