Fixed Effect Model

• A panel dataset contains observations on multiple entities $i$ at two or more points in time $t$
• Whether some observations are missing: balanced and unbalanced
• Fixed effects is a method of controlling for all variables, whether they’re observed or not, as long as they stay constant within some larger category
• The idea of fixed effects is that by sweeping away all that variation between individuals
• In panel data, when $corr(\alpha_i, x_{it} \neq 0)$, we should use fixed effect model, otherwise we use random effect model
• The assumption of fixed effect is everyone has a different intercept, and the individual’s impacts is the same within the group; while the random effect assume everyone has the same intercept representing the random differences between individuals

Estimators

• A standard regression equation is:
  • $y_{it} = \beta_i + \beta_1 X_{it} + \epsilon_{it}$
  • fixed effects: relative to others represented by intercept
  • The equation can be written equivalently as: $Y_{it} = \beta_0 + \beta_1 X_{it} + \gamma_2 D2i + \gamma_3 D3_i + … + \gamma_n Dn_i + \mu{it}$
• To estimate a regression with individual intercepts:
  • $Y_{it} - \overline{Y_i} = \beta_0 + \beta_1 (X_{it} - \overline{X_i})+ \epsilon_{it}$
• Two-way fixed effects model
  • $Y_{it} = \beta_i + \beta_t + \beta_1 X_{it} + \epsilon_{it}$
  • We are looking at variation within individual as well as within year at this point

Test

• Hausman

  xtreg invest mvalue kstock,fe
  est store fe_result
  xtreg invest mvalue kstock,re
  est store re_result
  hausman fe_result re_result
  

• Serial correlation test and Heteroskedasticity test ```Stata xtserial invest mvalue kstock

xtreg invest mvalue kstock,fe xttest3

- Robust Hausman
```Stata
webuse grunfeld, clear
xtset company year
quiet xtreg invest mvalue kstock,re
scalar theta = e(theta)
global xlist2 invest mvalue kstock
sort company
foreach x of varlist $xlist2 {
     by company: egen mean`x' = mean(`x')
     generate md`x' = `x' - mean`x'
     generate red`x' = `x' - theta*mean`x'
      }
quiet reg redinvest redmvalue redkstock mdmvalue mdkstock, vce(cluster company)
test mdmvalue mdkstock

• Corrected Hausman

  xtreg invest mvalue kstock,fe
  est store fe_result
  xtreg invest mvalue kstock,re
  est store re_result
  hausman fe_result re_result,sigmamore
  

• Over-estimation Wald

  xtreg invest mvalue kstock, re cluster(company)
   xtoverid
  

• Mundlak ```Stata local xlist “mvalue kstock” foreach f of local xlist{ bysort company: egen mean_f' = mean(f’) } xtreg invest mvalue kstock mean_mvalue mean_kstock, re vce(robust) est store Mundlak_result

test mean_mvalue mean_kstock

- Bootstrap Hausman
 ```Stata
xtreg invest mvalue kstock,fe
est store fe_result
xtreg invest mvalue kstock,re
est store re_result
rhausman fe_result re_result,reps(200) cluster

• Interface related inspection ```Stata qui xtreg invest mvalue kstock, fe xttest2

qui xtreg invest mvalue kstock, re xtcsd, pesaran ```