HW13

13.3

It makes no sense to look for pairs to difference since there is no fixed effect.

13.7

i)

It is not surprising that the coefficient on the interaction term changes little when \(afchnge\) is dropped from the equation, because the coefficient on \(afchnge\) in (13.12) is only 0.0077 (and very insignificant, means close to 0). The increase from 0.191 to 0.198 can be explained by sampling error.

ii)

We see that after \(highearn\) is dropped from the new equation, that means before policy change, there is no difference in average duration between high earners and low earners. But the very large and significant (0.256) coefficient on \(highearn\) in 13.12 contradicts this presumption.

Before the policy change, the high earning group spent about exp(0.256) − 1 = 29.2% higher on duration than the low earning group. After dropping \(highearn\) from the regression, much of the policy change effect are attributed by the intersection terms.

C5

i)

data <- wooldridge::rental
lm1 <- lm(data$lrent ~  data$y90 + data$lpop + data$lavginc+ data$pctstu)
summary(lm1)

## 
## Call:
## lm(formula = data$lrent ~ data$y90 + data$lpop + data$lavginc + 
##     data$pctstu)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.24233 -0.07824 -0.01642  0.04389  0.48082 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  -0.568807   0.534881  -1.063   0.2897    
## data$y90      0.262227   0.034763   7.543 8.78e-12 ***
## data$lpop     0.040686   0.022515   1.807   0.0732 .  
## data$lavginc  0.571446   0.053098  10.762  < 2e-16 ***
## data$pctstu   0.005044   0.001019   4.949 2.40e-06 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.1259 on 123 degrees of freedom
## Multiple R-squared:  0.8613, Adjusted R-squared:  0.8568 
## F-statistic: 190.9 on 4 and 123 DF,  p-value: < 2.2e-16

The positive and very significant coefficient on \(d90\) implies that, keep other things same, nominal rents grew by over 26% over the 10 year period. The coefficient on \(pctstu\) means that a 1% increase in \(pctstu\) is associated with a 0.5% increment in rent. Also, \(pctstu\) is very statistically significant.

ii)

The standard errors from (i) are not valid due to the existence of \(a_i\). If ai, the fixed effect, is in the error term, the errors across the two time periods for each city are positively correlated. But, this invalidates the assumption (independence of error terms) of usual OLS standard errors and t statistic.

iii)

lm2 <- lm(data$clrent  ~ data$clpop + data$clavginc + data$cpctstu)
summary(lm2)

## 
## Call:
## lm(formula = data$clrent ~ data$clpop + data$clavginc + data$cpctstu)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.18697 -0.06216 -0.01438  0.05518  0.23783 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   0.385521   0.036824  10.469 3.66e-15 ***
## data$clpop    0.072246   0.088343   0.818  0.41671    
## data$clavginc 0.309961   0.066477   4.663 1.79e-05 ***
## data$cpctstu  0.011203   0.004132   2.711  0.00873 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.09013 on 60 degrees of freedom
##   (64 observations deleted due to missingness)
## Multiple R-squared:  0.3223, Adjusted R-squared:  0.2884 
## F-statistic:  9.51 on 3 and 60 DF,  p-value: 3.136e-05

The effect of \(pctstu\) is over twice that in the pooled OLS equation. Now, a 1% increase in \(pctstu\) is estimated to increase rental rates by about 1.1%. Not surprisingly, we obtain a much less precise estimate when we difference (although the OLS standard errors from part (i) are likely to be much too small because of the positive serial correlation in the errors within each city). Even we have differenced away \(a_i\), there may be other unobservable variables change over time and are correlated with \(\Delta pctstu\).

iv)

library("lmtest")

## Loading required package: zoo

## 
## Attaching package: 'zoo'

## The following objects are masked from 'package:base':
## 
##     as.Date, as.Date.numeric

library("sandwich")

# Robust t test
coeftest(lm2, vcov = vcovHC(lm2, type = "HC0"))

## 
## t test of coefficients:
## 
##                Estimate Std. Error t value  Pr(>|t|)    
## (Intercept)   0.3855214  0.0471716  8.1727 2.487e-11 ***
## data$clpop    0.0722456  0.0674670  1.0708 0.2885354    
## data$clavginc 0.3099605  0.0864740  3.5844 0.0006786 ***
## data$cpctstu  0.0112033  0.0028428  3.9410 0.0002145 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

The heteroskedasticity-robust standard error on \(\Delta pctstu\) is about 0.00284, which is much smaller than the usual OLS standard error and \(pctstu\) becomes even more significant (3.94 > 2.71).