Chapter 10: Regression with Panel Data

• In this chapter we will focus on state traffic fatalities (\( n = 48 \) and \( T = 7 \))
• An observed variable is denoted as \( Y_{it} \)

• There are approximately 40,000 highway traffic fatalities each year in the US.
• Approximately a quarter of fatal crashes involve a driver who was drinking.
• It is estimated that 25% of drivers on the road between 1am and 3am have been drinking.
• A driver who is legally drunk is at least 13 times as likely to cause a fatal crash as a driver who has not been drinking.

In this chapter, we want to study government policies that can be used to discourage drunk driving.

load("/usr/data/ec414/fatality.RData")

fatality.data$year <- factor(fatality.data$year)
fatality.data$mrall <- fatality.data$mrall * 10000  
fatal.82.88.data <- subset(fatality.data, year %in% c("1982", "1988"))

Consider regressing \( FatalityRate \) (annual traffic deaths per 10,000 people in a state) on \( BeerTax \) \[ \begin{alignat*}{3} \widehat{FatalityRate} = &~2.01 &{}+{} &0.15 BeerTax \qquad \text{(1982 data)} \\ &(0.15) & &(0.13) \\ \widehat{FatalityRate} = &~1.86 &{}+{} &0.44 BeerTax \qquad \text{(1988 data)} \\ &(0.11) & &(0.13) \end{alignat*} \]

• The previous regression very likely suffer from OVB (which omitted variables?)
• We can expand our specification to include these omitted variables, but some, like cultural attitudes towards drinking and driving in different states, would be difficult to control for.
• If such omitted variables are constant across time we control for them using panel data.
• Suppose we have observations for \( T = 2 \) periods for each of the \( n = 48 \) states.
• This approaches is based on comparing the “differences'' in the regression variables.

Consider having an omitted variable \( Z_i \) that is state specific but constant across time

\[ FatalityRate_{it} = \beta_0 + \beta_1 BeerTax_{it} + \beta_2 Z_i + u_{it} \]

Consider the model for the years 1982 and 1988

\[ \begin{align*} FatalityRate_{i,1982} &= \beta_0 + \beta_1 BeerTax_{i,1982} + \beta_2 Z_i + u_{i,1982} \\ FatalityRate_{i,1988} &= \beta_0 + \beta_1 BeerTax_{i,1988} + \beta_2 Z_i + u_{i,1988} \end{align*} \]

Subtract the previous models for the two observed years

\[ \begin{align*} FatalityRate_{i,1988}& - FatalityRate_{i,1982} = \\ &\beta_1 (BeerTax_{i,1988} - BeerTax_{i,1982}) \\ {} &+ (u_{i,1988} - u_{i,1982}) \end{align*} \]

Intuitively, by basing our regression on the change in fatalities we can ignore the time constant omitted variables.

If we regress the change in fatalities in states on the change in taxes we get \[ \begin{align*} &\widehat{FatalityRate_{1988} - FatalityRate_{1982}} = \\ &\qquad {} - 0.072 - 1.04 (BeerTax_{1988} - BeerTax_{1982}) \end{align*} \]

(with standard errors \( 0.064 \) and \( 0.26 \), respectively)

We reject the hypothesis that \( \beta_1 \) is zero at the 5% level. According to this estimate an increase of $1 tax would reduce traffic fatalities by 1.05 deaths/10,000 people (a reduction by 2 of current average fatality in the data).

We can modify our previous model

\[ Y_{it} = \beta_0 + \beta_1 X_{it} + \beta_2 Z_i + u_{it} \]

as

\[ Y_{it} = \beta_1 X_{it} + \alpha_i + u_{it} \]

where \( \alpha_i = \beta_0 + \beta_2Z_i \) are the \( n \) entity specific intercepts. These are also known as entity fixed effects, since the represent the constant effect of being in entity \( i \).

Let the dummy variable \( D1_i \) be binary variable that equal 1 when \( i = 1 \), 0 otherwise. We can likewise define \( n \) dummy variables \( Dj_i \) which are equal to 1 when \( j = i \). We can write our model as

\[ Y_{it} = \beta_0 + \beta_1 X_{it} + \underbrace{\gamma_2 D2_i + \gamma_3 D3_i + \cdots + \gamma_n Dn_i}_{n-1 \text{ dummy variables}} + u_{it} \]

We can easily extend both forms of the fixed effects model to include multiple regressors

• We now have \( k + n \) coefficients to estimate
• We are not interested in estimating the entity-specific effects so we need a way to subtract them from the regression. We use an “entity-demeaned” OLS algorithm
  1. We subtract the entity specific average from each variables
  2. We regress the demeaned dependent variable on the demeaned regressors

• Taking averages of the fixed effects model \[ \bar{Y}_i = \beta_1\bar{X}_i + \alpha_i + \bar{u}_i \] and subtract it from the original model \[ Y_{it} - \bar{Y}_i = \beta_1(X_{it} - \bar{X}_i) + (u_{it} - \bar{u}_i) \] or \[ \tilde{Y}_{it} = \beta_1 \tilde{X}_{it} + \tilde{u}_{it} \]
• We will discuss needed assumptions in order to carry out OLS regression and inference below.

library(plm)
regress.results.fe.only <- plm(mrall ~ beertax, 
                               data=fatality.data, 
                               index=c('state', 'year'), 
                               effect='individual', 
                               model='within')
robust.se <- vcovSCC(regress.results.fe.only)
coeftest(regress.results.fe.only, vcov = robust.se)

t test of coefficients:

        Estimate Std. Error t value Pr(>|t|)    
beertax   -0.656      0.106   -6.17  2.3e-09 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

• In the previous regression there is still a possibility of OVB due to effects are constant across states by different across time.

• We can modify our model to

  \[ Y_{it} = \beta_0 + \beta_1 X_{it} + \beta_2 Z_i + \beta_3 S_t + u_{it} \]

  where \( S_t \) is unobserved.

• Suppose that entity specific effects (\( Z_i \)) are not present in our model but time effects are.
• The same as with entity fixed effects we can model time effects as having a different intercept per time period \[ Y_{it} = \beta_1 X_{it} + \lambda_t + u_{it} \] where \( \lambda_1,\dots, \lambda_T \) are the time fixed effects.

• We can also use the dummy variable approach. Let \( Bs_t \) be equal to 1 if \( s = t \) and 0 otherwise.

  \[ Y_{it} = \beta_0 + \beta_1 X_{it} + \underbrace{\delta_2 B2_t + \cdots + \delta_T BT_t}_{T-1\text{ dummy variables}} + u_{it} \]

Now to control for both \( Z_i \) and \( S_t \) we have the combined entity and time fixed effects regression model

\[ Y_{it} = \beta_1 X_{it} + \alpha_i + \lambda_t + u_{it} \]

or

\[ \begin{align*} Y_{it} &= \beta_0 + \beta_1 X_{it} + \gamma_2 D2_i + \cdots + \gamma_n Dn_i \\ {}&+ \delta_2 B2_t + \cdots + \delta_T BT_t + u_{it} \end{align*} \]

regress.results.fe.time <- plm(mrall ~ beertax,
                               data = fatality.data, 
                               index = c('state', 'year'), 
                               effect = 'twoways', 
                               model = 'within')

robust.se <- vcovSCC(regress.results.fe.time)
coeftest(regress.results.fe.time, vcov = robust.se)

t test of coefficients:

        Estimate Std. Error t value Pr(>|t|)    
beertax   -0.640      0.154   -4.17  4.1e-05 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

\[ Y_{it} = \beta_1 X_{it} + \alpha_i + u_{it},~~i = 1, \dots,n,~~t = 1,\dots,T. \]

• FE.A.1: \( u_{it} \) has conditional mean zero: \( E[u_{it}|X_{i1}, \dots, X_{iT}, \alpha_i] = 0 \).
• FE.A.2: \( (X_{i1}, \dots, X_{iT}, u_{i1},\dots,u_{iT}), i = 1,\dots,n \) are i.i.d. draws from their joint distribution.
• FE.A.3: Large outliers are unlikely.
• FE.A.4: No perfect multicollinearity.

An important assumption for fixed effects regressions is FE.A.2

• The assumption requires variables to be independent across entities but not necessarily within entities.
• For example any \( X_{it} \) and \( X_{is} \) for \( t \ne s \) can be correlated: this is called autocorrelation or serial correlation.

• Autocorrelation is common in time series and panel data: what happens in one period is probably correlated with what happens in other periods for the same entity.
• The same can be said for the error terms \( u_{it} \)
• For that reason we need to use heteroskedasticity- and autocorrelation-consistent (HAC) standard errors.