# 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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Joseph
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Lohitha
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nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
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da
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Bhagvanji
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Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
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da
Application of nanotechnology in medicine
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Kamaluddeen
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Bhagvanji
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what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
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Professor
I think
Professor
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Damian
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scanning tunneling microscope
Sahil
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Rafiq
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Mahi
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Hafiz
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write examples of Nano molecule?
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The nanotechnology is as new science, to scale nanometric
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nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
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