# Fixed effects model

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In econometrics and statistics, a fixed effects model is a statistical model that represents the observed quantities in terms of explanatory variables that are treated as if the quantities were non-random. This is in contrast to random effects models and mixed models in which either all or some of the explanatory variables are treated as if they arise from random causes. Contrast this to the biostatistics definitions,[1][2][3][4] as biostatisticians use "fixed" and "random" effects to respectively refer to the population-average and subject-specific effects (and where the latter are generally assumed to be unknown, latent variables). Often the same structure of model, which is usually a linear regression model, can be treated as any of the three types depending on the analyst's viewpoint, although there may be a natural choice in any given situation.

In panel data analysis, the term fixed effects estimator (also known as the within estimator) is used to refer to an estimator for the coefficients in the regression model. If we assume fixed effects, we impose time independent effects for each entity that are possibly correlated with the regressors.

## Qualitative description

Such models assist in controlling for unobserved heterogeneity when this heterogeneity is constant over time and correlated with independent variables. This constant can be removed from the data through differencing, for example by taking a first difference which will remove any time invariant components of the model.

There are two common assumptions made about the individual specific effect, the random effects assumption and the fixed effects assumption. The random effects assumption (made in a random effects model) is that the individual specific effects are uncorrelated with the independent variables. The fixed effect assumption is that the individual specific effect is correlated with the independent variables. If the random effects assumption holds, the random effects model is more efficient than the fixed effects model. However, if this assumption does not hold (i.e., if the Durbin–Watson test fails), the random effects model is not consistent.

## Formal description

Consider the linear unobserved effects model for $N$ observations and $T$ time periods:

$y_{it} = X_{it}\mathbf{\beta}+\alpha_{i}+u_{it}$ for $t=1,..,T$ and $i=1,...,N$

where $y_{it}$ is the dependent variable observed for individual $i$ at time $t,$ $X_{it}$ is the time-variant $1\times k$ regressor matrix, $\alpha_{i}$ is the unobserved time-invariant individual effect and $u_{it}$ is the error term. Unlike $X_{it}$, $\alpha_{i}$ cannot be observed by the econometrician. Common examples for time-invariant effects $\alpha_{i}$ are innate ability for individuals or historical and institutional factors for countries.

Unlike the Random effects (RE) model where the unobserved $\alpha_{i}$ is independent of $x_{it}$ for all $t=1,...,T$, the FE model allows $\alpha_{i}$ to be correlated with the regressor matrix $x_{it}$. Strict exogeneity, however, is still required.

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

$y_{it}-\overline{y_{i}}=\left(X_{it}-\overline{X_{i}}\right) \beta+ \left( \alpha_{i} - \overline{\alpha_{i}} \right ) + \left( u_{it}-\overline{u_{i}}\right) \implies \ddot{y_{it}}=\ddot{X_{it}} \beta+\ddot{u_{it}}$

where $\overline{X_{i}}=\frac{1}{T}\sum\limits_{t=1}^{T}X_{it}$ and $\overline{u_{i}}=\frac{1}{T}\sum\limits_{t=1}^{T}u_{it}$. Since $\alpha_{i}$ is constant, $\overline{\alpha_{i}}=\alpha_{i}$ and hence the effect is eliminated. The FE estimator $\hat{\beta}_{FE}$ is then obtained by an OLS regression of $\ddot{y}$ on $\ddot{X}$.

Another alternative to the within transformation is to add a dummy variable for each individual $i$. This is numerically, but not computationally, equivalent to the fixed effect model and only works if $T,$ the number of time observations per individual, is much larger than the number of individuals in the panel.

## Equality of Fixed Effects (FE) and First Differences (FD) estimators when T=2

For the special two period case ($T=2$), the FE estimator and the 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: ${FE}_{T=2}= \left[ (x_{i1}-\bar x_{i}) (x_{i1}-\bar x_{i})' + (x_{i2}-\bar x_{i}) (x_{i2}-\bar x_{i})' \right]^{-1}\left[ (x_{i1}-\bar x_{i}) (y_{i1}-\bar y_{i}) + (x_{i2}-\bar x_{i}) (y_{i2}-\bar y_{i})\right]$

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

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

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

## Hausman–Taylor method

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

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

Estimating $\gamma$ via OLS on $\widehat{di}=Z_{i}\gamma+\varphi_{it}$ using $X_1$ and $Z_1$ as instruments yields a consistent estimate.

## Testing fixed effects (FE) vs. random effects (RE)

We can test whether a model is appropriate using a Hausman test.

$H_{0}$: $\alpha_{i}\perp X_{it},Z_{i}$
$H_{a}$: $\alpha_{i}\not \perp X_{it},Z_{i}$

If $H_{0}$ is true, both $\widehat{\beta}_{RE}$ and $\widehat{\beta}_{FE}$ are consistent, but only $\widehat{\beta}_{RE}$ is efficient. If $H_{a}$ is true, $\widehat{\beta}_{FE}$ is consistent and $\widehat{\beta}_{RE}$ is not.

$\widehat{Q}=$ $\widehat{\beta}_{RE}-\widehat{\beta}_{FE}$
$\widehat{HT}=T\widehat{Q}^{\prime}[Var(\widehat{\beta}_{FE})-Var(\widehat {\beta}_{RE})]^{-1}\widehat{Q}\sim\chi_{K}^{2}$ where $K=\dim(Q)$

The Hausman test is a specification test so a large test statistic might be indication that there might be Errors in Variables (EIV) or our model is misspecified. If the FE assumption is true, we should find that $\widehat {\beta}_{LD}\approx\widehat{\beta}_{FD}\approx\widehat{\beta}_{FE}$.

A simple heuristic is that if $\left\vert \widehat{\beta}_{LD}\right\vert >\left\vert \widehat{\beta}_{FE}\right\vert >\left\vert \widehat{\beta} _{FD}\right\vert$ there could be EIV.

## Steps in Fixed Effects Model for sample data

1. Calculate group and grand means
2. Calculate k=number of groups, n=number of observations per group, N=total number of observations (k x n)
3. Calculate SS-total (or total variance) as: (Each score - Grand mean)^2 then summed
4. Calculate SS-treat (or treatment effect) as: (Each group mean- Grand mean)^2 then summed x n
5. Calculate SS-error (or error effect) as (Each score - Its group mean)^2 then summed
6. Calculate df-total: N-1, df-treat: k-1 and df-error k(n-1)
7. Calculate Mean Square MS-treat: SS-treat/df-treat, then MS-error: SS-error/df-error
8. Calculate obtained f value: MS-treat/MS-error
9. Use F-table or probability function, to look up critical f value with a certain significance level
10. Conclude as to whether treatment effect significantly affects the variable of interest