# 2.2 Panel data models  (Page 2/10)

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## 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:

$q={\alpha }_{0}{I}_{1}^{{\beta }_{1}}\cdots {I}_{k}^{{\beta }_{k}},$

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:

${y}_{it}={\beta }_{0}+{\beta }_{1}{x}_{1it}+\cdots +{\beta }_{k}{x}_{kit}+{\epsilon }_{it,}$

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:

${\epsilon }_{it}=\lambda {F}_{i}+\eta {P}_{t}+{\upsilon }_{it},$

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: ${y}_{it}=\left({\beta }_{0}+\lambda {F}_{i}+\eta {P}_{t}\right)+\sum _{j=1}^{k}{\beta }_{j}{x}_{jit}+{\upsilon }_{it}$ or

${y}_{it}={\alpha }_{it}+\sum _{j=1}^{k}{\beta }_{j}{x}_{jit}+{\upsilon }_{it},$

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

${y}_{it}={\alpha }_{i}+\sum _{j=1}^{k}{\beta }_{j}{x}_{jit}+{\upsilon }_{it},$

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:

${\alpha }_{i}={\alpha }_{1}+{\alpha }_{2}{D}_{2}+\cdots +{\alpha }_{m}{D}_{m}={\alpha }_{1}+\sum _{j=2}^{m}{\alpha }_{j}{D}_{j},$

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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no can't
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William
currently
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nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
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learn
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Preparation and Applications of Nanomaterial for Drug Delivery
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Application of nanotechnology in medicine
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I think
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The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
how did you get the value of 2000N.What calculations are needed to arrive at it
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