fixed effects model

Such models assist in controlling for omitted variable bias due to unobserved heterogeneity when this heterogeneity is constant over time. This heterogeneity can be removed from the data through differencing, for example by subtracting the group-level average over time, or 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 is that the individual-specific effects are uncorrelated with the independent variables. The fixed effect assumption is that the individual-specific effects are correlated with the independent variables. If the random effects assumption holds, the random effects estimator is more efficient than the fixed effects estimator. However, if this assumption does not hold, the random effects estimator is not consistent. The Durbin–Wu–Hausman test is often used to discriminate between the fixed and the random effects models.{{cite book |first1=A. Colin |last1=Cameron |first2=Pravin K. |last2=Trivedi |title=Microeconometrics: Methods and Applications |publisher=Cambridge University Press |year=2005 |pages=717–19 |isbn=9780521848053 |url=https://books.google.com/books?id=Zf0gCwxC9ocC&pg=PA717 }}{{cite book |first=Marc |last=Nerlove |author-link=Marc Nerlove |title=Essays in Panel Data Econometrics |publisher=Cambridge University Press |year=2005 |pages=36–39 |isbn=9780521022460 |url=https://books.google.com/books?id=2eZpoAZnu9UC&pg=PA36 }}

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

:$y\_\{it\}\; =\; X\_\{it\}\backslash mathbf\{\backslash beta\}+\backslash alpha\_\{i\}+u\_\{it\}$ for $t=1,\backslash dots,T$ and $i=1,\backslash dots,N$

Where:

• $y\_\{it\}$ is the dependent variable observed for individual $i$ at time $t$.
• $X\_\{it\}$ is the time-variant $1\backslash times\; k$ (the number of independent variables) regressor vector.
• $\backslash beta$ is the $k\backslash times\; 1$ matrix of parameters.
• $\backslash alpha\_\{i\}$ is the unobserved time-invariant individual effect. For example, the innate ability for individuals or historical and institutional factors for countries.
• $u\_\{it\}$ is the error term.

Unlike $X\_\{it\}$, $\backslash alpha\_\{i\}$ cannot be directly observed.

Unlike the random effects model where the unobserved $\backslash alpha\_\{i\}$ is independent of $X\_\{it\}$ for all $t=1,...,T$, the fixed effects (FE) model allows $\backslash alpha\_\{i\}$ to be correlated with the regressor matrix $X\_\{it\}$. Strict exogeneity with respect to the idiosyncratic error term $u\_\{it\}$ is still required.

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

:$y\_\{it\}-\backslash overline\{y\}\_\{i\}=\backslash left(X\_\{it\}-\backslash overline\{X\}\_\{i\}\backslash right)\; \backslash beta+\; \backslash left(\; \backslash alpha\_\{i\}\; -\; \backslash overline\{\backslash alpha\}\_\{i\}\; \backslash right\; )\; +\; \backslash left(\; u\_\{it\}-\backslash overline\{u\}\_\{i\}\backslash right)\; \backslash implies\; \backslash ddot\{y\}\_\{it\}=\backslash ddot\{X\}\_\{it\}\; \backslash beta+\backslash ddot\{u\}\_\{it\}$

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

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

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

One is to add a dummy variable for each individual $i>1$ (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.{{cite journal |first=Oscar. |last=Garcia |year=1983 |title=A stochastic differential equation model for the height growth of forest stands |journal=Biometrics |volume=39 |issue=4 | pages=1059–1072 |doi=10.2307/2531339 |jstor=2531339 }} 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.{{cite journal |first1=David |last1=Tait |first2=Chris J. |last2=Cieszewski |first3=Imre E. |last3=Bella |year=1986 |title=The stand dynamics of lodgepole pine |journal=Can. J. For. Res. |volume=18 |issue=10 | pages=1255–1260 |doi=10.1139/x88-193 }} 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.{{cite journal |first1=Mike |last1=Strub |first2=Chris J. |last2=Cieszewski |year=2006 |title=Base–age invariance properties of two techniques for estimating the parameters of site index models |journal=Forest Science |volume=52 |issue=2 |pages=182–186 |doi=10.1093/forestscience/52.2.182 }} 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 including in SAS.{{cite conference|first1=Mike |last1=Strub |first2=Chris J. |last2=Cieszewski |year=2003 |title= Fitting global site index parameters when plot or tree site index is treated as a local nuisance parameter|editor=Burkhart, HA|conference=Proceedings of the Symposium on Statistics and Information Technology in Forestry; 2002 September 8–12; Blacksburg, Virginia: Virginia Polytechnic Institute and State University | pages=97–107 }}{{cite journal |first1=Chris J. |last1=Cieszewski |first2=Mike |last2=Harrison |first3=Stacey W. |last3=Martin |year=2000 |title=Practical methods for estimating non-biased parameters in self-referencing growth and yield models |journal=PMRC Technical Report |volume=2000 |issue=7 |pages=12|url=http://pmrc.uga.edu/TR2000-7.pdf}}

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.{{cite journal |first1=Jon |last1=Schnute |first2=Skip |last2=McKinnell |year=1984 |title= A biologically meaningful approach to response surface analysis |journal=Can. J. Fish. Aquat. Sci. |volume=41 |issue=6 |pages=936–953|doi=10.1139/f84-108 }}

{{main|First-difference estimator}}

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

:$y\_\{it\}-y\_\{i,t-1\}=\backslash left(X\_\{it\}-X\_\{i,t-1\}\backslash right)\; \backslash beta+\; \backslash left(\; \backslash alpha\_\{i\}\; -\; \backslash alpha\_\{i\}\; \backslash right\; )\; +\; \backslash left(\; u\_\{it\}-u\_\{i,t-1\}\backslash right)\; \backslash implies\; \backslash Delta\; y\_\{it\}=\backslash Delta\; X\_\{it\}\; \backslash beta+\; \backslash Delta\; u\_\{it\}.$

The FD estimator $\backslash hat\backslash beta\_\{FD\}$ is then obtained by an OLS regression of $\backslash Delta\; y\_\{it\}$ on $\backslash Delta\; X\_\{it\}$.

When $T=2$, the first difference and fixed effects estimators are numerically equivalent. For $T>2$, they are not. If the error terms $u\_\{it\}$ are homoskedastic with no serial correlation, the fixed effects estimator is more efficient than the first difference estimator. If $u\_\{it\}$ follows a random walk, however, the first difference estimator is more efficient.{{cite book |first=Jeffrey M. |last=Wooldridge |title=Econometric Analysis of Cross Section and Panel Data |url=https://archive.org/details/econometricanaly0000wool |url-access=registration |year=2001 |publisher=MIT Press |isbn=978-0-262-23219-7 |pages=[https://archive.org/details/econometricanaly0000wool/page/279 279]–291}}

For the special two period case ($T=2$), 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:

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

$\{FE\}\_\{T=2\}=\; \backslash left[\backslash sum\_\{i=1\}^\{N\}\; \backslash dfrac\{x\_\{i1\}-x\_\{i2\}\}\{2\}\; \backslash dfrac\{x\_\{i1\}-x\_\{i2\}\}\{2\}\; \text{'}\; +\; \backslash dfrac\{x\_\{i2\}-x\_\{i1\}\}\{2\}\; \backslash dfrac\{x\_\{i2\}-x\_\{i1\}\}\{2\}\; \text{'}\; \backslash right]^\{-1\}\; \backslash left[\backslash sum\_\{i=1\}^\{N\}\; \backslash dfrac\{x\_\{i1\}-x\_\{i2\}\}\{2\}\; \backslash dfrac\{y\_\{i1\}-y\_\{i2\}\}\{2\}\; +\; \backslash dfrac\{x\_\{i2\}-x\_\{i1\}\}\{2\}\; \backslash dfrac\{y\_\{i2\}-y\_\{i1\}\}\{2\}\; \backslash right]$

:$=\; \backslash left[\backslash sum\_\{i=1\}^\{N\}\; 2\; \backslash dfrac\{x\_\{i2\}-x\_\{i1\}\}\{2\}\; \backslash dfrac\{x\_\{i2\}-x\_\{i1\}\}\{2\}\; \text{'}\; \backslash right]^\{-1\}\; \backslash left[\backslash sum\_\{i=1\}^\{N\}\; 2\; \backslash dfrac\{x\_\{i2\}-x\_\{i1\}\}\{2\}\; \backslash dfrac\{y\_\{i2\}-y\_\{i1\}\}\{2\}\; \backslash right]$

:$=\; 2\backslash left[\backslash sum\_\{i=1\}^\{N\}\; (x\_\{i2\}-x\_\{i1\})(x\_\{i2\}-x\_\{i1\})\text{'}\; \backslash right]^\{-1\}\; \backslash left[\backslash sum\_\{i=1\}^\{N\}\; \backslash frac\{1\}\{2\}\; (x\_\{i2\}-x\_\{i1\})(y\_\{i2\}-y\_\{i1\})\; \backslash right]$

:$=\; \backslash left[\backslash sum\_\{i=1\}^\{N\}\; (x\_\{i2\}-x\_\{i1\})(x\_\{i2\}-x\_\{i1\})\text{'}\; \backslash right]^\{-1\}\; \backslash sum\_\{i=1\}^\{N\}\; (x\_\{i2\}-x\_\{i1\})(y\_\{i2\}-y\_\{i1\})\; =\{FD\}\_\{T=2\}$

{{main|Chamberlain's approach to unobserved effects models}}

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

:$\backslash alpha\_\{i\}\; =\; \backslash lambda\_0\; +\; X\_\{i1\}\; \backslash lambda\_1\; +\; X\_\{i2\}\; \backslash lambda\_2\; +\; \backslash dots\; +\; X\_\{iT\}\; \backslash lambda\_T\; +\; e\_i$

this results in the following equation:

:$y\_\{it\}\; =\; \backslash lambda\_0\; +\; X\_\{i1\}\; \backslash lambda\_1\; +\; X\_\{i2\}\; \backslash lambda\_2\; +\; \backslash dots\; +\; X\_\{it\}(\backslash lambda\_t\; +\; \backslash mathbf\{\backslash beta\})\; +\; \backslash dots\; +\; X\_\{iT\}\; \backslash lambda\_T\; +\; e\_i\; +\; u\_\{it\}$

which can be estimated by minimum distance estimation.{{cite book|last1=Chamberlain|first1=Gary|title=Chapter 22 Panel data|series=Handbook of Econometrics |volume=2|year=1984|pages=1247–1318|issn=1573-4412|doi=10.1016/S1573-4412(84)02014-6|isbn=9780444861863 }}

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

$\backslash alpha\_\{i\}$.

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 $\backslash alpha\_\{i\}$. Need $K1>G2$.

Estimating $\backslash gamma$ via OLS on $\backslash widehat\{di\}=Z\_\{i\}\backslash gamma+\backslash varphi\_\{it\}$ using $X\_1$ and $Z\_1$ as instruments yields a consistent estimate.

When there is input uncertainty for the $y$ data, $\backslash delta\; y$, then the $\backslash chi^2$ value, rather than the sum of squared residuals, should be minimized.{{Cite journal|arxiv=1803.06776|last1= Ren|first1= Bin |title= A Decade of MWC 758 Disk Images: Where Are the Spiral-Arm-Driving Planets?|journal= The Astrophysical Journal Letters|volume= 857|pages= L9|last2= Dong|first2= Ruobing|last3= Esposito | first3 = Thomas M.|last4=Pueyo|first4= Laurent|last5= Debes|first5= John H. |last6= Poteet|first6= Charles A. |last7= Choquet|first7= Élodie|last8=Benisty|first8= Myriam|last9=Chiang|first9= Eugene |last10=Grady|first10= Carol A.|last11=Hines|first11= Dean C.|last12=Schneider|first12= Glenn|last13=Soummer|first13= Rémi|year= 2018|issue= 1|doi=10.3847/2041-8213/aab7f5 |bibcode = 2018ApJ...857L...9R |s2cid= 59427417|doi-access= free}} This can be directly achieved from substitution rules:

:$\backslash frac\{y\_\{it\}\}\{\backslash delta\; y\_\{it\}\}\; =\; \backslash mathbf\{\backslash beta\}\backslash frac\{X\_\{it\}\}\{\backslash delta\; y\_\{it\}\}+\backslash alpha\_\{i\}\backslash frac\{1\}\{\backslash delta\; y\_\{it\}\}+\backslash frac\{u\_\{it\}\}\{\backslash delta\; y\_\{it\}\}$,

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

Random effects estimators may be inconsistent sometimes in the long time series limit, if the random effects are misspecified (i.e. the model chosen for the random effects is incorrect). However, the fixed effects model may still be consistent in some situations. For example, if the time series being modeled is not stationary, random effects models assuming stationarity may not be consistent in the long-series limit. One example of this is if the time series has an upward trend. Then, as the series becomes longer, the model revises estimates for the mean of earlier periods upwards, giving increasingly biased predictions of coefficients. However, a model with fixed time effects does not pool information across time, and as a result earlier estimates will not be affected.

In situations like these where the fixed effects model is known to be consistent, the Durbin-Wu-Hausman test can be used to test whether the random effects model chosen is consistent. If $H\_\{0\}$ is true, both $\backslash widehat\{\backslash beta\}\_\{RE\}$ and $\backslash widehat\{\backslash beta\}\_\{FE\}$ are consistent, but only $\backslash widehat\{\backslash beta\}\_\{RE\}$ is efficient. If $H\_\{a\}$ is true the consistency of $\backslash widehat\{\backslash beta\}\_\{RE\}$ cannot be guaranteed.

• Random effects model
• Mixed model
• Dynamic unobserved effects model
• Fixed-effect Poisson model
• Panel analysis
• First-difference estimator

{{reflist}}

• {{cite book |title=Plane Answers to Complex Questions: The Theory of Linear Models|last=Christensen|first=Ronald|location=New York|publisher=Springer|year=2002| edition=Third|isbn=0-387-95361-2}}
• {{cite book |last1=Gujarati |first1=Damodar N. |last2=Porter |first2=Dawn C.|author2-link=Dawn C. Porter |chapter=Panel Data Regression Models |title=Basic Econometrics |location=Boston |publisher=McGraw-Hill |year=2009 |edition=Fifth international |isbn=978-007-127625-2 |pages=591–616 }}
• {{cite book |last=Hsiao |first=Cheng |chapter=Fixed-effects models |title=Analysis of Panel Data |location=New York |publisher=Cambridge University Press |edition=2nd |year=2003 |isbn=0-521-52271-4 |pages=95–103 |chapter-url=https://books.google.com/books?id=i9iPG7C3EP4C&pg=PA95 }}
• {{cite book |last=Wooldridge |first=Jeffrey M. |year=2013 |chapter=Fixed Effects Estimation |pages=466–474 |title=Introductory Econometrics: A Modern Approach |location=Mason, OH |publisher=South-Western |edition=Fifth international |isbn=978-1-111-53439-4 }}

• [http://teaching.sociology.ul.ie/DCW/confront/node45.html Fixed and random effects models]
• [https://www.southampton.ac.uk/~cpd/anovas/datasets/index.htm Examples of all ANOVA and ANCOVA models with up to three treatment factors, including randomized block, split plot, repeated measures, and Latin squares, and their analysis in R]

Category:Analysis of variance

Category:Regression models