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Hsiao (2003: 27-30) discusses a convenient example of a panel data model that illustrates many of the important issues that arise with panel data. We make use of this example in what follows. Assume that we want to estimate a production function for farm production in order to determine if the farm industry exhibits increasing returns to scale. Assume the sample consists of observations for N farms over T years, giving a total sample size of $NT.$ For simplicity, we assume that the Cobb-Douglas production is an adequate description of the production process. The general form of the Cobb-Douglas production function is:
where q is output and ${I}_{j}$ is the quantity of the j-th input (for example, land, machinery, labor, feed, and fertilizer). The parameter, ${\beta}_{j},$ is the output elasticity of the j-th input; the farms exhibit constant returns to scale if the output elasticities sum to one and either increasing or decreasing returns to scale if they sum to a value greater than or less than one, respectively. is the quantity of the j -th input (for example, land, machinery, labor, feed, and fertilizer). The parameter, is the output elasticity of the j -th input; the farms exhibit constant returns to scale if the output elasticities sum to one and either increasing or decreasing returns to scale if they sum to a value greater than or less than one, respectively.
Taking the natural logarithm of (5) gives $\mathrm{ln}q=\mathrm{ln}{\alpha}_{0}+{\beta}_{1}\mathrm{ln}{I}_{1}+\cdots +{\beta}_{k}\mathrm{ln}{I}_{k}.$ We can re-write this equation (adding an error term, as well as farm and year subscripts) giving:
where ${y}_{it}=\mathrm{ln}{q}_{it},$ , ${\beta}_{0}=\mathrm{ln}{\alpha}_{0},$ ${x}_{jit}=\mathrm{ln}{I}_{jit},$ for $j=1,\dots ,k$ and ${\epsilon}_{it}$ is an error term. One way to account for year and time effects is to assume:
where F _{i} is a measure of the unobserved farm specific effects on productivity and P _{t} is a measure of the unobserved changes in productivity that are the same for all farms but vary annually. Substitution of (7) into (6) gives: ${y}_{it}=\left({\beta}_{0}+\lambda {F}_{i}+\eta {P}_{t}\right)+{\displaystyle \sum _{j=1}^{k}{\beta}_{j}{x}_{jit}}+{\upsilon}_{it}$ or
where ${\alpha}_{it}={\beta}_{0}+\lambda {F}_{i}+\eta {P}_{t}.$ Thus, (8) is equivalent to (2). Moreover, if we assume that $\eta =0,$ we get
where ${\alpha}_{i}={\beta}_{0}+\lambda {F}_{i}.$ Thus, (9) is equivalent to (1).
A natural way to make (9) operational is to introduce a dummy variable, D _{i} , for each farm so that the intercept term becomes:
where ${D}_{j}=1$ if $j=i$ and 0 otherwise. This substitution is equivalent to replacing the intercept term with a dummy variable for each farm and letting the farm dummy variable “sweep out” the farm-specific effects. In this specification the slope terms are the same for every farm while the intercept term is given for farm j by ${\alpha}_{1}+{\alpha}_{j}.$ Clearly, the intercept term for the first farm is equal to just ${\alpha}_{1}.$ This specification is known as the fixed effect model and is estimated using ordinary least squared (OLS). We can extend the fixed-effects model to fit (8) by including a dummy variable for each time period except one.
In sum, fixed-effects models assume either (or both) that the omitted effects that are specific to cross-sectional units are constant over time or that the effects specific to time are constant over the cross-sectional units. This method is equivalent to including a dummy variable for all but one of the cross-sectional units and/or a dummy variable for all but one of the time periods.
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