Model: log(price) = Bo + B1*log(dist) + u
# Subset the data for the year 1981
data_1981 <- subset(data, year == 1981)
# Perform simple regression
model1 <- lm(log(price) ~ log(dist), data = data_1981)
summary(model1)##
## Call:
## lm(formula = log(price) ~ log(dist), data = data_1981)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.87318 -0.22657 -0.01985 0.25687 0.95045
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 8.04716 0.64624 12.452 < 2e-16 ***
## log(dist) 0.36488 0.06576 5.548 1.39e-07 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.3543 on 140 degrees of freedom
## Multiple R-squared: 0.1803, Adjusted R-squared: 0.1744
## F-statistic: 30.79 on 1 and 140 DF, p-value: 1.395e-07In the event that B1 is statistically significant and negative, it implies that the incinerator’s existence lowers home values as one gets farther away from it.
Model: log(price) = Bo + B1log(dist) + B2log(intst) + B3log(area) + B4log(land) + B5rooms + B6baths + B7*age + u
model2 <- lm(log(price) ~ log(dist) + log(intst) + log(area) + log(land) + rooms + baths + age, data = data_1981)
summary(model2)##
## Call:
## lm(formula = log(price) ~ log(dist) + log(intst) + log(area) +
## log(land) + rooms + baths + age, data = data_1981)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.74072 -0.10669 0.00932 0.11817 0.61387
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 7.592332 0.641711 11.831 < 2e-16 ***
## log(dist) 0.055389 0.057621 0.961 0.338153
## log(intst) -0.039032 0.051662 -0.756 0.451261
## log(area) 0.319294 0.076418 4.178 5.27e-05 ***
## log(land) 0.076824 0.039505 1.945 0.053908 .
## rooms 0.042528 0.028251 1.505 0.134588
## baths 0.166923 0.041944 3.980 0.000113 ***
## age -0.003567 0.001059 -3.369 0.000985 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.201 on 134 degrees of freedom
## Multiple R-squared: 0.7475, Adjusted R-squared: 0.7344
## F-statistic: 56.68 on 7 and 134 DF, p-value: < 2.2e-16After adjusting for other factors, it indicates that the incinerator’s existence still has an impact on house prices if it is statistically significant and stays negative. The extra control variables in model (ii) could be the cause of differences in the outcomes between (i) and (ii).
Model: log(price) = Bo + B1log(dist) + B2log(intst) + B3log(area) + B4log(land) + B5rooms + B6baths + B7*age + u The significance of the functional form was assessed by comparing the results with model (ii).
model3 <- lm(log(price) ~ log(dist) + log(intst), data = data_1981)
summary(model3)##
## Call:
## lm(formula = log(price) ~ log(dist) + log(intst), data = data_1981)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.92562 -0.19871 -0.02897 0.22916 0.88075
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 8.65979 0.62227 13.917 < 2e-16 ***
## log(dist) 0.04682 0.09449 0.495 0.621
## log(intst) 0.26555 0.05971 4.447 1.77e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.3327 on 139 degrees of freedom
## Multiple R-squared: 0.2824, Adjusted R-squared: 0.272
## F-statistic: 27.35 on 2 and 139 DF, p-value: 9.664e-11A significant coefficient indicates that the relationship between distance and housing prices is not linear.
data_1981$dist_squared <- (log(data_1981$dist))^2
model4 <- lm(log(price) ~ log(dist) + log(intst) + log(area) + log(land) + rooms + baths + age + dist_squared, data = data_1981)
summary(model4)##
## Call:
## lm(formula = log(price) ~ log(dist) + log(intst) + log(area) +
## log(land) + rooms + baths + age + dist_squared, data = data_1981)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.75771 -0.10073 0.00257 0.11296 0.56269
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -15.758968 7.768580 -2.029 0.04450 *
## log(dist) 4.844616 1.589145 3.049 0.00277 **
## log(intst) 0.052321 0.058606 0.893 0.37360
## log(area) 0.311436 0.074255 4.194 4.97e-05 ***
## log(land) 0.059774 0.038777 1.541 0.12558
## rooms 0.039987 0.027448 1.457 0.14752
## baths 0.166285 0.040733 4.082 7.65e-05 ***
## age -0.002855 0.001055 -2.706 0.00770 **
## dist_squared -0.251447 0.083383 -3.016 0.00307 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1952 on 133 degrees of freedom
## Multiple R-squared: 0.7637, Adjusted R-squared: 0.7495
## F-statistic: 53.73 on 8 and 133 DF, p-value: < 2.2e-16library(wooldridge)
data("wage1")# Perform OLS estimation
model <- lm(log(wage) ~ educ + exper + I(exper^2), data = wage1)
summary(model)##
## Call:
## lm(formula = log(wage) ~ educ + exper + I(exper^2), data = wage1)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.96387 -0.29375 -0.04009 0.29497 1.30216
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.1279975 0.1059323 1.208 0.227
## educ 0.0903658 0.0074680 12.100 < 2e-16 ***
## exper 0.0410089 0.0051965 7.892 1.77e-14 ***
## I(exper^2) -0.0007136 0.0001158 -6.164 1.42e-09 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.4459 on 522 degrees of freedom
## Multiple R-squared: 0.3003, Adjusted R-squared: 0.2963
## F-statistic: 74.67 on 3 and 522 DF, p-value: < 2.2e-16No need to access the “t value” for this purpose.
# Test the significance of exper
exper_summary <- summary(model)
exper_p_value <- coef(exper_summary)["exper", "Pr(>|t|)"]
exper_p_value## [1] 1.765073e-14Using the given formula, approximate returns for fifth and twentieth years of experience were found.
# Calculate approximate returns
B_exper <- coef(model)["exper"]
B_exper_squared <- coef(model)["I(exper^2)"]
approx_return_5_years <- 100 * (B_exper + 2 * B_exper_squared * 5)
approx_return_20_years <- 100 * (B_exper + 2 * B_exper_squared * 20)
approx_return_5_years## exper
## 3.387329approx_return_20_years## exper
## 1.246655