Fixed effects model

Since ${\displaystyle \alpha _{i}}$ is not observable, it cannot be directly controlled for. The FE model eliminates ${\displaystyle \alpha _{i}}$ by demeaning the variables using the within transformation:

${\displaystyle y_{it}-{\overline {y}}_{i}=\left(X_{it}-{\overline {X}}_{i}\right)\beta +\left(\alpha _{i}-{\overline {\alpha }}_{i}\right)+\left(u_{it}-{\overline {u}}_{i}\right)\implies {\ddot {y}}_{it}={\ddot {X}}_{it}\beta +{\ddot {u}}_{it}}$

where ${\displaystyle {\overline {y}}_{i}={\frac {1}{T}}\sum \limits _{t=1}^{T}y_{it}}$, ${\displaystyle {\overline {X}}_{i}={\frac {1}{T}}\sum \limits _{t=1}^{T}X_{it}}$, and ${\displaystyle {\overline {u}}_{i}={\frac {1}{T}}\sum \limits _{t=1}^{T}u_{it}}$.

Since ${\displaystyle \alpha _{i}}$ is constant, ${\displaystyle {\overline {\alpha _{i}}}=\alpha _{i}}$ and hence the effect is eliminated. The FE estimator ${\displaystyle {\hat {\beta }}_{FE}}$ is then obtained by an OLS regression of ${\displaystyle {\ddot {y}}}$ on ${\displaystyle {\ddot {X}}}$.

At least three alternatives to the within transformation exist with variations.

One is to add a dummy variable for each individual ${\displaystyle i>1}$ (omitting the first individual because of multicollinearity). This is numerically, but not computationally, equivalent to the fixed effect model and only works if the sum of the number of series and the number of global parameters is smaller than the number of observations.[9] The dummy variable approach is particularly demanding with respect to computer memory usage and it is not recommended for problems larger than the available RAM, and the applied program compilation, can accommodate.

Second alternative is to use consecutive reiterations approach to local and global estimations.[10] This approach is very suitable for low memory systems on which it is much more computationally efficient than the dummy variable approach.

The third approach is a nested estimation whereby the local estimation for individual series is programmed in as a part of the model definition.[11] This approach is the most computationally and memory efficient, but it requires proficient programming skills and access to the model programming code; although, it can be programmed even in SAS.[12][13]

Finally, each of the above alternatives can be improved if the series-specific estimation is linear (within a nonlinear model), in which case the direct linear solution for individual series can be programmed in as part of the nonlinear model definition.[14]

An alternative to the within transformation is the first difference transformation, which produces a different estimator. For ${\displaystyle t=2,\dots ,T}$:

${\displaystyle y_{it}-y_{i,t-1}=\left(X_{it}-X_{i,t-1}\right)\beta +\left(\alpha _{i}-\alpha _{i}\right)+\left(u_{it}-u_{i,t-1}\right)\implies \Delta y_{it}=\Delta X_{it}\beta +\Delta u_{it}.}$

The FD estimator ${\displaystyle {\hat {\beta }}_{FD}}$ is then obtained by an OLS regression of ${\displaystyle \Delta y_{it}}$ on ${\displaystyle \Delta X_{it}}$.

When ${\displaystyle T=2}$, the first difference and fixed effects estimators are numerically equivalent. For ${\displaystyle T>2}$, they are not. If the error terms ${\displaystyle u_{it}}$ are homoskedastic with no serial correlation, the fixed effects estimator is more efficient than the first difference estimator. If ${\displaystyle u_{it}}$ follows a random walk, however, the first difference estimator is more efficient.[15]

For the special two period case (${\displaystyle T=2}$), the fixed effects (FE) estimator and the first difference (FD) estimator are numerically equivalent. This is because the FE estimator effectively "doubles the data set" used in the FD estimator. To see this, establish that the fixed effects estimator is: ${\displaystyle {FE}_{T=2}=\left[(x_{i1}-{\bar {x}}_{i})(x_{i1}-{\bar {x}}_{i})'+(x_{i2}-{\bar {x}}_{i})(x_{i2}-{\bar {x}}_{i})'\right]^{-1}\left[(x_{i1}-{\bar {x}}_{i})(y_{i1}-{\bar {y}}_{i})+(x_{i2}-{\bar {x}}_{i})(y_{i2}-{\bar {y}}_{i})\right]}$

Since each ${\displaystyle (x_{i1}-{\bar {x}}_{i})}$ can be re-written as ${\displaystyle (x_{i1}-{\dfrac {x_{i1}+x_{i2}}{2}})={\dfrac {x_{i1}-x_{i2}}{2}}}$, we'll re-write the line as:

${\displaystyle {FE}_{T=2}=\left[\sum _{i=1}^{N}{\dfrac {x_{i1}-x_{i2}}{2}}{\dfrac {x_{i1}-x_{i2}}{2}}'+{\dfrac {x_{i2}-x_{i1}}{2}}{\dfrac {x_{i2}-x_{i1}}{2}}'\right]^{-1}\left[\sum _{i=1}^{N}{\dfrac {x_{i1}-x_{i2}}{2}}{\dfrac {y_{i1}-y_{i2}}{2}}+{\dfrac {x_{i2}-x_{i1}}{2}}{\dfrac {y_{i2}-y_{i1}}{2}}\right]}$

${\displaystyle =\left[\sum _{i=1}^{N}2{\dfrac {x_{i2}-x_{i1}}{2}}{\dfrac {x_{i2}-x_{i1}}{2}}'\right]^{-1}\left[\sum _{i=1}^{N}2{\dfrac {x_{i2}-x_{i1}}{2}}{\dfrac {y_{i2}-y_{i1}}{2}}\right]}$

${\displaystyle =2\left[\sum _{i=1}^{N}(x_{i2}-x_{i1})(x_{i2}-x_{i1})'\right]^{-1}\left[\sum _{i=1}^{N}{\frac {1}{2}}(x_{i2}-x_{i1})(y_{i2}-y_{i1})\right]}$

${\displaystyle =\left[\sum _{i=1}^{N}(x_{i2}-x_{i1})(x_{i2}-x_{i1})'\right]^{-1}\sum _{i=1}^{N}(x_{i2}-x_{i1})(y_{i2}-y_{i1})={FD}_{T=2}}$

Gary Chamberlain's method, a generalization of the within estimator, replaces ${\displaystyle \alpha _{i}}$ with its linear projection onto the explanatory variables. Writing the linear projection as:

${\displaystyle \alpha _{i}=\lambda _{0}+X_{i1}\lambda _{1}+X_{i2}\lambda _{2}+\dots +X_{iT}\lambda _{T}+e_{i}}$

this results in the following equation:

${\displaystyle y_{it}=\lambda _{0}+X_{i1}\lambda _{1}+X_{i2}\lambda _{2}+\dots +X_{it}(\lambda _{t}+\mathbf {\beta } )+\dots +X_{iT}\lambda _{T}+e_{i}+u_{it}}$

which can be estimated by minimum distance estimation.[16]

Need to have more than one time-variant regressor (${\displaystyle X}$) and time-invariant regressor (${\displaystyle Z}$) and at least one ${\displaystyle X}$ and one ${\displaystyle Z}$ that are uncorrelated with ${\displaystyle \alpha _{i}}$.

Partition the ${\displaystyle X}$ and ${\displaystyle Z}$ variables such that ${\displaystyle {\begin{array}{c}X=[{\underset {TN\times K1}{X_{1it}}}\vdots {\underset {TN\times K2}{X_{2it}}}]\\Z=[{\underset {TN\times G1}{Z_{1it}}}\vdots {\underset {TN\times G2}{Z_{2it}}}]\end{array}}}$ where ${\displaystyle X_{1}}$ and ${\displaystyle Z_{1}}$ are uncorrelated with ${\displaystyle \alpha _{i}}$. Need ${\displaystyle K1>G2}$.

Estimating ${\displaystyle \gamma }$ via OLS on ${\displaystyle {\widehat {di}}=Z_{i}\gamma +\varphi _{it}}$ using ${\displaystyle X_{1}}$ and ${\displaystyle Z_{1}}$ as instruments yields a consistent estimate.

When there is input uncertainty for the ${\displaystyle y}$ data, ${\displaystyle \delta y}$, then the ${\displaystyle \chi ^{2}}$ value, rather than the sum of squared residuals, should be minimized.[17] This can be directly achieved from substitution rules:

${\displaystyle {\frac {y_{it}}{\delta y_{it}}}=\mathbf {\beta } {\frac {X_{it}}{\delta y_{it}}}+\alpha _{i}{\frac {1}{\delta y_{it}}}+{\frac {u_{it}}{\delta y_{it}}}}$,

then the values and standard deviations for ${\displaystyle \mathbf {\beta } }$ and ${\displaystyle \alpha _{i}}$ can be determined via classical ordinary least squares analysis and variance-covariance matrix.