2.2 Panel data models  (Page 2/10)

Estimation issues

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 $\ln q=\ln {\alpha }_0+{\beta }_1\ln I_1+\cdots +{\beta }_k\ln I_k.$ We can re-write this equation (adding an error term, as well as farm and year subscripts) giving:

where $y_{it}=\ln q_{it},$ , ${\beta }_0=\ln {\alpha }_0,$ $x_{jit}=\ln I_{jit},$ 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).

Fixed-effects models

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.