6 Fixed Effects
6.1 Preliminaries
These notes are based on Nick Huntington-Klein’s lecture on Within Variation and Fixed Effects. Check his video lecture here. In this exercise, we use data from Cornwell and Trumbull (1994). This dataset download here comprehends 90 North Carolina counties during the period from 1981 to 1987. The table below describes the variables:
| Variable | Definition |
|---|---|
| county | County id |
| year | Time period (year) |
| crmrte | Crime rate (per person) |
| prbarr | Probability of arrest (arrests/offenses) |
| prbconv | Probability of conviction (conviction/arrests) |
| prbpris | Probability of prison (prison/convictions) |
| avgsen | Average sentence lenght |
| polpc | Police per capita |
| density | Population density |
| taxpc | Per capita tax revenue |
| west | Western counties dummy |
| central | Central counties dummy |
| urban | Urban counties dummy |
| pctmin80 | Percentage of minority 1980 |
| pctymle | Percentage of young male |
| w… |
All other variables starting with w represent wages in different sectors/industries
|
When a column’s name starts with l, that means the logarithm transformation was applied to the variable.
6.2 Between variation
Reading the CT1993.RDS file and filtering by counties 1, 3, 23 and 91:
library(tidyverse)
setwd("C:/Users/User/Desktop/474-Rlab/datasets")
ct93<-readRDS("CT1993.RDS")
ct93_sub<-ct93%>%
select(county, year, crmrte, prbarr, polpc, density, urban)%>%
filter(county %in% c(1,3,23,91))Exploring the between variation with a pooled regression, one can see the positive relationship between crime rate and probability of arrest:
library(fixest)
pooled<-feols(crmrte~prbarr, se="hetero", data=ct93_sub)
summary(pooled)## OLS estimation, Dep. Var.: crmrte
## Observations: 28
## Standard-errors: Heteroskedasticity-robust
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.000776 0.004050 0.191625 8.49524e-01
## prbarr 0.095990 0.012966 7.403400 7.33000e-08 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## RMSE: 0.004859 Adj. R2: 0.6405636.3 Within variation
There are three ways to get estimates using the within variation. Let’s explore each one in more detail:
6.3.1 Regress the demeaned values of \(Y_{it}\) on \(D_{it}\)
The first step is to get the group means (i.e., average crime rate and the average probability of arrest for each county during 1981-1987). Then, take the difference of each observation and its group mean:
within<-ct93_sub%>%
group_by(county)%>%
mutate(mean_crime=mean(crmrte),
mean_prbarr=mean(prbarr))%>%
mutate(within_crime=crmrte-mean_crime,
within_prbarr=prbarr-mean_prbarr)To get the estimates using the within variation, regress within_crime on within_prbarr:
within_demean<-lm(within_crime~within_prbarr, data=within)
summary(within_demean)##
## Call:
## lm(formula = within_crime ~ within_prbarr, data = within)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.0058380 -0.0015729 -0.0000249 0.0013558 0.0068621
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.130e-18 5.242e-04 0.00 1.000
## within_prbarr -1.392e-02 1.809e-02 -0.77 0.448
##
## Residual standard error: 0.002774 on 26 degrees of freedom
## Multiple R-squared: 0.02227, Adjusted R-squared: -0.01533
## F-statistic: 0.5923 on 1 and 26 DF, p-value: 0.4485Differently from the between regression, the coefficient on the probability of arrest is negative (but not statistically significant).
6.3.2 Regress \(Y_{it}\) on \(D_{it}\) and county unit dummies
The demeaning process might be exhaustive because you need to do it not only for the dependent variable but also for all the covariates you are using. One easy way to get the same result is to allow different intercepts for each county:
lsdv<-lm(crmrte~-1+prbarr+factor(county), data=ct93_sub)
summary(lsdv)##
## Call:
## lm(formula = crmrte ~ -1 + prbarr + factor(county), data = ct93_sub)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.0058380 -0.0015729 -0.0000249 0.0013558 0.0068621
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## prbarr -0.013923 0.019235 -0.724 0.476
## factor(county)1 0.040257 0.006338 6.352 1.76e-06 ***
## factor(county)3 0.017396 0.003576 4.864 6.54e-05 ***
## factor(county)23 0.032365 0.005546 5.835 6.03e-06 ***
## factor(county)91 0.036341 0.006594 5.511 1.33e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.002949 on 23 degrees of freedom
## Multiple R-squared: 0.9914, Adjusted R-squared: 0.9896
## F-statistic: 532.8 on 5 and 23 DF, p-value: < 2.2e-16This is what we call least squares dummy variables (LSDV) regression.
6.3.3 Regress \(Y_{it}\) on \(D_{it}\) with canned fixed effects routine
When you have many fixed effects (sometimes thousands, sometimes millions), LSDV regression can be time-consuming and computationally inefficient. -users have Fixest as an option to perform estimations with multiple fixed effects at a breakneck pace:
fe<-feols(crmrte~prbarr|county, data=ct93_sub)
summary(fe)## OLS estimation, Dep. Var.: crmrte
## Observations: 28
## Fixed-effects: county: 4
## Standard-errors: Clustered (county)
## Estimate Std. Error t value Pr(>|t|)
## prbarr -0.013923 0.026278 -0.529824 0.632941
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## RMSE: 0.002673 Adj. R2: 0.877019
## Within R2: 0.022272The estimated coefficient is the same \(-0.013923\) in all the regressions above - what is different is the standard error from the demeaned and the last two regressions.
6.4 Two-way fixed effects
A common approach is to add a fixed effect for the panel unit and a time-varying intercept to capture common time trends across units, i.e., to control for something that varies over time in the same way for all panel units. To do that, add time to the panel unit on the right side of the vertical bar |.
twfe<-feols(crmrte~prbarr|county+year, data=ct93_sub)
summary(twfe)## OLS estimation, Dep. Var.: crmrte
## Observations: 28
## Fixed-effects: county: 4, year: 7
## Standard-errors: Clustered (county)
## Estimate Std. Error t value Pr(>|t|)
## prbarr 0.005353 0.025778 0.207665 0.848789
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## RMSE: 0.001947 Adj. R2: 0.911743
## Within R2: 0.0044856.5 Panel data alone can’t deal with simultaneity
A two-way fixed effects regression between crime rate and police per capita deals with common time trends and time-invariant unobserved heterogeneity. However, TWFE does not address simultaneity:
pol_crime<-feols(crmrte~polpc|county+year, data=ct93)
summary(pol_crime)## OLS estimation, Dep. Var.: crmrte
## Observations: 630
## Fixed-effects: county: 90, year: 7
## Standard-errors: Clustered (county)
## Estimate Std. Error t value Pr(>|t|)
## polpc 1.7938 0.339849 5.2783 9.12e-07 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## RMSE: 0.005577 Adj. R2: 0.88804
## Within R2: 0.226753One can see that the coefficient on police per capita is positive and statistically significant. That positive association is most likely driven by counties with higher crime rates hiring more police officers per capita.
6.6 Intepreting Within Relationships
This regression replicates columns between and within in Table 3 in Cornwell and Trumbuil (1993). All the variables are in log, so the results are interpretable as elasticities.
full_between<-feols(lcrmrte~lprbarr+lprbconv+lprbpris+lavgsen+lpolpc+
ldensity+lpctymle+lwtuc+lwtrd+lwfir+lwser+lwmfg+
lwfed+lwsta+lwloc+central+urban+pctmin80,
se="standard", data=ct93)
full_within<-feols(lcrmrte~lprbarr+lprbconv+lprbpris+lavgsen+lpolpc+
ldensity+lpctymle+lwtuc+lwtrd+lwfir+lwser+lwmfg+
lwfed+lwsta+lwloc|county+year,
se="standard", data=ct93)
etable(full_between, full_within, signifCode = c("***"=0.01, "**"=0.05, "*"=0.10))## full_between full_within
## Dependent Var.: lcrmrte lcrmrte
##
## (Intercept) -1.711*** (0.5940)
## lprbarr -0.6426*** (0.0312) -0.3556*** (0.0322)
## lprbconv -0.4750*** (0.0223) -0.2815*** (0.0211)
## lprbpris -0.1678*** (0.0502) -0.1710*** (0.0322)
## lavgsen -0.0626 (0.0395) -0.0034 (0.0261)
## lpolpc 0.3543*** (0.0230) 0.4136*** (0.0266)
## ldensity 0.3633*** (0.0289) 0.4168 (0.2825)
## lpctymle -0.1962*** (0.0658) 0.6086* (0.3631)
## lwtuc 0.0148 (0.0300) 0.0464** (0.0190)
## lwtrd 0.0319 (0.0641) -0.0195 (0.0405)
## lwfir -0.0024 (0.0463) -0.0040 (0.0283)
## lwser -0.0538* (0.0320) 0.0083 (0.0191)
## lwmfg -0.1029* (0.0543) -0.3601*** (0.1118)
## lwfed -0.0337 (0.1176) -0.3074* (0.1761)
## lwsta -0.3136*** (0.0839) 0.0592 (0.1133)
## lwloc 0.1870* (0.1117) 0.1821 (0.1176)
## central -0.0483* (0.0248)
## urban -0.2132*** (0.0546)
## pctmin80 0.0147*** (0.0007)
## Fixed-Effects: ------------------- -------------------
## county No Yes
## year No Yes
## _______________ ___________________ ___________________
## S.E. type Standard Standard
## Observations 630 630
## R2 0.79498 0.95314
## Within R2 -- 0.40013Since we are using within variation, the interpretation must be within-county. Hence, raising the probability of arrest by 10% in a given county reduces the number of crimes per person in that county by 3.556%, on average.
Finally, we know that standard errors are hardly homoskedastic, and it is better to estimate the robust standard errors. However, when using fixed effects, we should cluster the standard errors at the main fixed effect level: in this way, we allow the error term to be correlated within units. Estimates are less precise but still significant at the 1% level.
full_between2<-feols(lcrmrte~lprbarr+lprbconv+lprbpris+lavgsen+lpolpc+
ldensity+lpctymle+lwtuc+lwtrd+lwfir+lwser+lwmfg+
lwfed+lwsta+lwloc+central+urban+pctmin80,
cluster=~county, data=ct93)
full_within2<-feols(lcrmrte~lprbarr+lprbconv+lprbpris+lavgsen+lpolpc+
ldensity+lpctymle+lwtuc+lwtrd+lwfir+lwser+lwmfg+
lwfed+lwsta+lwloc|county+year,
cluster=~county, data=ct93)
etable(full_between2, full_within2, signifCode = c("***"=0.01, "**"=0.05, "*"=0.10))## full_between2 full_within2
## Dependent Var.: lcrmrte lcrmrte
##
## (Intercept) -1.711 (1.130)
## lprbarr -0.6426*** (0.0951) -0.3556*** (0.0600)
## lprbconv -0.4750*** (0.0644) -0.2815*** (0.0494)
## lprbpris -0.1678** (0.0678) -0.1710*** (0.0456)
## lavgsen -0.0626 (0.0600) -0.0034 (0.0322)
## lpolpc 0.3543*** (0.1079) 0.4136*** (0.0839)
## ldensity 0.3633*** (0.0620) 0.4168 (0.3690)
## lpctymle -0.1962 (0.1196) 0.6086 (0.5625)
## lwtuc 0.0148 (0.0248) 0.0464*** (0.0167)
## lwtrd 0.0319 (0.1006) -0.0195 (0.0313)
## lwfir -0.0024 (0.0404) -0.0040 (0.0124)
## lwser -0.0538 (0.0392) 0.0083 (0.0220)
## lwmfg -0.1029 (0.1220) -0.3601*** (0.1132)
## lwfed -0.0337 (0.2434) -0.3074 (0.2224)
## lwsta -0.3136** (0.1385) 0.0592 (0.1133)
## lwloc 0.1870 (0.2341) 0.1821 (0.1624)
## central -0.0483 (0.0454)
## urban -0.2132** (0.0837)
## pctmin80 0.0147*** (0.0016)
## Fixed-Effects: ------------------- -------------------
## county No Yes
## year No Yes
## _______________ ___________________ ___________________
## S.E.: Clustered by: county by: county
## Observations 630 630
## R2 0.79498 0.95314
## Within R2 -- 0.40013