1. Interpreting the regression results obtained by pooled OLS method
log (rentit) = β0 + β1*log (popit) + β2*log (avgincit) + β3*pctstuit + δ*y90t + αi +uit log = -1.500 + 0.407*log (pop) + 0.571*log (avginc) + 0.005*pctstu + 0.262*y90 se = (0.535) (0.023) (0.053) (0.001) (0.348) t = (-1.06) (1.81) (10.76) (4.95) (7.54) p = (0.073) (0.009) (0.000) (0.000) (0.000) r2 = 0.861
adj r2 = 0.857
df = 123 = 0.016
The regression table shows that 100% increase in city population will result in 4% increase in average rental prices, while holding all other variables constant. Similarly, 100% increase in average income will lead to 57% increase in dependent variable, holding other variables constant. 100% increase of student population in total population will result in 0.5% increase in rental prices, holding other variables constant. Since we have variable αi (which captures all the unobserved, time constant factors affecting rent prices) in our equation, we have fixed problem. In other words, αi is unobserved rent effect or rent price fixed effect. αi represents all factors affecting rent prices that don’t change over time. We can see that “y90” is dummy variable, which only counts observations for 1990 while in our regression model we have 128 observations, which means that observations for 1980 we also included in the model. 2. Estimating the first-difference model of equation (a) using OLS method xtset city year, delta(10)
panel variable: city (strongly balanced)
time variable: year, 80 to 90
delta: 10 units
log = 0.386 + 0.072*log (pop_d) + 0.310*log (avginc_d) + 0.011*pctstu_d se = (0.368) (0.088) (0.066) (0.004) t = (10.47) (0.82) (4.66) (2.71) p = (0.000) (0.417) (0.000) (0.009) r2 = 0.322
adj r2 = 0.288
df = 60 = 0.008
In order to eliminate fixed effect αi, we use first differencing estimation. The coefficient for “y90_d” could not be estimated because it is time-varying variable in our dataset. After taking first differences using OLS, we can see that there is a slight increase in coefficient percentage of log of student population in total population. In other words, doubling percentage of student population in total population will only increase the rent prices by 1.1%, while holding other variables constant.
3. Obtaining heteroscedasticity-robust standard errors
log = 0.386 + 0.072*log (pop_d) + 0.310*log (avginc_d) + 0.011*pctstu_d se = (0.049) (0.070) (0.089) (0.002) t = (7.91) (1.04) (3.47) (3.82) p = (0.000) (0.304) (0.000) (0.009) r2 = 0.322
After running robust regression, we can notice that there are certain changes in standard errors and significance values compared to the results in previous first differenced model regression. In particular, the size of standard errors for variables log of city population and log of student population in total population have decreased while the size of standard error for variable log of average income has increased. However, any variable that was statistically significant using the first differences models is still significant using the heteroscedasticity-robust t statistic. We can conclude, since the results of the robust standard error do not change significance levels, there are no signs of heteroscedasticity.
4. Estimating equation (a) by fixed effects
log = 1.409 + 0.072*log (pop) + 0.310*log (avginc) + 0.011*pctstu + 0.386*y90 se =...
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