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10.3 Fixed Effects Regression

Consider the panel regression model

\[Y_{it} = \beta_0 + \beta_1 X_{it} + \beta_2 Z_i + u_{it}\] where the \(Z_i\) are unobserved time-invariant heterogeneities across the entities \(i=1,\dots,n\). We aim to estimate \(\beta_1\), the effect on \(Y_i\) of a change in \(X_i\) holding constant \(Z_i\). Letting \(\alpha_i = \beta_0 + \beta_2 Z_i\) we obtain the model \[\begin{align} Y_{it} = \alpha_i + \beta_1 X_{it} + u_{it} \tag{10.1}. \end{align}\] Having individual specific intercepts \(\alpha_i\), \(i=1,\dots,n\), where each of these can be understood as the fixed effect of entity \(i\), this model is called the fixed effects model. The variation in the \(\alpha_i\), \(i=1,\dots,n\) comes from the \(Z_i\). (10.1) can be rewritten as a regression model containing \(n-1\) dummy regressors and a constant: \[\begin{align} Y_{it} = \beta_0 + \beta_1 X_{it} + \gamma_2 D2_i + \gamma_3 D3_i + \cdots + \gamma_n Dn_i + u_{it} \tag{10.2}. \end{align}\] Model (10.2) has \(n\) different intercepts — one for every entity. (10.1) and (10.2) are equivalent representations of the fixed effects model.

The fixed effects model can be generalized to contain more than just one determinant of \(Y\) that is correlated with \(X\) and changes over time. Key Concept 10.2 presents the generalized fixed effects regression model.

Key Concept 10.2

The Fixed Effects Regression Model

The fixed effects regression model is

\[\begin{align} Y_{it} = \beta_1 X_{1,it} + \cdots + \beta_k X_{k,it} + \alpha_i + u_{it} \tag{10.3} \end{align}\]

with \(i=1,\dots,n\) and \(t=1,\dots,T\). The \(\alpha_i\) are entity-specific intercepts that capture heterogeneities across entities. An equivalent representation of this model is given by

\[\begin{align} Y_{it} = \beta_0 + \beta_1 X_{1,it} + \cdots + \beta_k X_{k,it} + \gamma_2 D2_i + \gamma_3 D3_i + \cdots + \gamma_n Dn_i + u_{it} \tag{10.4} \end{align}\]

where the \(D2_i,D3_i,\dots,Dn_i\) are dummy variables.

Estimation and Inference

Software packages use a so-called “entity-demeaned” OLS algorithm which is computationally more efficient than estimating regression models with \(k+n\) regressors as needed for models (10.3) and (10.4).

Taking averages on both sides of (10.1) we obtain \[\begin{align*} \frac{1}{n} \sum_{i=1}^n Y_{it} =& \, \beta_1 \frac{1}{n} \sum_{i=1}^n X_{it} + \frac{1}{n} \sum_{i=1}^n a_i + \frac{1}{n} \sum_{i=1}^n u_{it} \\ \overline{Y} =& \, \beta_1 \overline{X}_i + \alpha_i + \overline{u}_i. \end{align*}\] Subtraction from (10.1) yields \[\begin{align} \begin{split} Y_{it} - \overline{Y}_i =& \, \beta_1(X_{it}-\overline{X}_i) + (u_{it} - \overline{u}_i) \\ \overset{\sim}{Y}_{it} =& \, \beta_1 \overset{\sim}{X}_{it} + \overset{\sim}{u}_{it}. \end{split} \tag{10.5} \end{align}\] In this model, the OLS estimate of the parameter of interest \(\beta_1\) is equal to the estimate obtained using (10.2) — without the need to estimate \(n-1\) dummies and an intercept.

We conclude that there are two ways of estimating \(\beta_1\) in the fixed effects regression:

  1. OLS of the dummy regression model as shown in (10.2)

  2. OLS using the entity demeaned data as in (10.5)

Provided the fixed effects regression assumptions stated in Key Concept 10.3 hold, the sampling distribution of the OLS estimator in the fixed effects regression model is normal in large samples. The variance of the estimates can be estimated and we can compute standard errors, \(t\)-statistics and confidence intervals for coefficients. In the next section, we see how to estimate a fixed effects model using R and how to obtain a model summary that reports heteroskedasticity-robust standard errors. We leave aside complicated formulas of the estimators. See Chapter 10.5 and Appendix 10.2 of the book for a discussion of theoretical aspects.

Application to Traffic Deaths

Following Key Concept 10.2, the simple fixed effects model for estimation of the relation between traffic fatality rates and the beer taxes is \[\begin{align} FatalityRate_{it} = \beta_1 BeerTax_{it} + StateFixedEffects + u_{it}, \tag{10.6} \end{align}\] a regression of the traffic fatality rate on beer tax and 48 binary regressors — one for each federal state.

We can simply use the function lm() to obtain an estimate of \(\beta_1\).

fatal_fe_lm_mod <- lm(fatal_rate ~ beertax + state - 1, data = Fatalities)
#> Call:
#> lm(formula = fatal_rate ~ beertax + state - 1, data = Fatalities)
#> Coefficients:
#> beertax  stateal  stateaz  statear  stateca  stateco  statect  statede  statefl  statega  stateid  stateil  statein  stateia  
#> -0.6559   3.4776   2.9099   2.8227   1.9682   1.9933   1.6154   2.1700   3.2095   4.0022   2.8086   1.5160   2.0161   1.9337  
#> stateks  stateky  statela  stateme  statemd  statema  statemi  statemn  statems  statemo  statemt  statene  statenv  statenh  
#>  2.2544   2.2601   2.6305   2.3697   1.7712   1.3679   1.9931   1.5804   3.4486   2.1814   3.1172   1.9555   2.8769   2.2232  
#> statenj  statenm  stateny  statenc  statend  stateoh  stateok  stateor  statepa  stateri  statesc  statesd  statetn  statetx  
#>  1.3719   3.9040   1.2910   3.1872   1.8542   1.8032   2.9326   2.3096   1.7102   1.2126   4.0348   2.4739   2.6020   2.5602  
#> stateut  statevt  stateva  statewa  statewv  statewi  statewy  
#>  2.3137   2.5116   2.1874   1.8181   2.5809   1.7184   3.2491

As discussed in the previous section, it is also possible to estimate \(\beta_1\) by applying OLS to the demeaned data, that is, to run the regression

\[\overset{\sim}{FatalityRate} = \beta_1 \overset{\sim}{BeerTax}_{it} + u_{it}. \]

# obtain demeaned data
Fatalities_demeaned <- with(Fatalities,
            data.frame(fatal_rate = fatal_rate - ave(fatal_rate, state),
            beertax = beertax - ave(beertax, state)))

# estimate the regression
summary(lm(fatal_rate ~ beertax - 1, data = Fatalities_demeaned))

The function ave is convenient for computing group averages. We use it to obtain state specific averages of the fatality rate and the beer tax.

Alternatively one may use plm() from the package with the same name.

# install and load the 'plm' package
## install.packages("plm")

As for lm() we have to specify the regression formula and the data to be used in our call of plm(). Additionally, it is required to pass a vector of names of entity and time ID variables to the argument index. For Fatalities, the ID variable for entities is named state and the time id variable is year. Since the fixed effects estimator is also called the within estimator, we set model = “within”. Finally, the function coeftest() allows to obtain inference based on robust standard errors.

# estimate the fixed effects regression with plm()
fatal_fe_mod <- plm(fatal_rate ~ beertax, 
                    data = Fatalities,
                    index = c("state", "year"), 
                    model = "within")

# print summary using robust standard errors
coeftest(fatal_fe_mod, vcov. = vcovHC, type = "HC1")
#> t test of coefficients:
#> beertax -0.65587    0.28880  -2.271  0.02388 *
#> ---

The estimated coefficient is again \(-0.6559\). Note that plm() uses the entity-demeaned OLS algorithm and thus does not report dummy coefficients. The estimated regression function is

\[\begin{align} \widehat{FatalityRate} = -\underset{(0.29)}{0.66} \times BeerTax + StateFixedEffects. \tag{10.7} \end{align}\]

The coefficient on \(BeerTax\) is negative and significant. The interpretation is that the estimated reduction in traffic fatalities due to an increase in the real beer tax by \(\$1\) is \(0.66\) per \(10000\) people, which is still pretty high. Although including state fixed effects eliminates the risk of a bias due to omitted factors that vary across states but not over time, we suspect that there are other omitted variables that vary over time and thus cause a bias.