Week 7 - Course Project
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(a) Consider a linear model where the sale price of a house is the dependent variable and the explanatory variables are the other variables given above. Perform a test for linearity. What do you conclude based on the test result?
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data <- read.table("./Week 7 - Test.txt", header = TRUE)
reg <- lm(data = data, sell ~ lot + bdms + fb + sty + drv + rec + ffin + ghw + ca + gar + reg)
summary(reg)##
## Call:
## lm(formula = sell ~ lot + bdms + fb + sty + drv + rec + ffin +
## ghw + ca + gar + reg, data = data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -41389 -9307 -591 7353 74875
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -4038.3504 3409.4713 -1.184 0.236762
## lot 3.5463 0.3503 10.124 < 2e-16 ***
## bdms 1832.0035 1047.0002 1.750 0.080733 .
## fb 14335.5585 1489.9209 9.622 < 2e-16 ***
## sty 6556.9457 925.2899 7.086 4.37e-12 ***
## drv 6687.7789 2045.2458 3.270 0.001145 **
## rec 4511.2838 1899.9577 2.374 0.017929 *
## ffin 5452.3855 1588.0239 3.433 0.000642 ***
## ghw 12831.4063 3217.5971 3.988 7.60e-05 ***
## ca 12632.8904 1555.0211 8.124 3.15e-15 ***
## gar 4244.8290 840.5442 5.050 6.07e-07 ***
## reg 9369.5132 1669.0907 5.614 3.19e-08 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 15420 on 534 degrees of freedom
## Multiple R-squared: 0.6731, Adjusted R-squared: 0.6664
## F-statistic: 99.97 on 11 and 534 DF, p-value: < 2.2e-16reset <- resettest(reg, power = 2, type = "fitted")
jarque <- jarque.bera.test(reg$residuals)
format(data.frame("Ramsey RESET" = c(reset$statistic, reset$p.value), "Jarque-Bera" = c(jarque$statistic, jarque$p.value), row.names = c("Statistic", "P-Value")), scientific= FALSE)## Ramsey.RESET Jarque.Bera
## Statistic 26.986035203525 247.6198
## P-Value 0.000000292211 0.0000\(~\)
With significant statistics and p-values close to 0, both tests reject the null hypothesis that the model has been correctly specified, meaning it is not a good fit for the data.
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(b) Now consider a linear model where the log of the sale price of the house is the dependent variable and the explanatory variables are as before. Perform again the test for linearity. What do you conclude now?
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reg2 <- lm(data = data, log(sell) ~ lot + bdms + fb + sty + drv + rec + ffin + ghw + ca + gar + reg)
summary(reg2)##
## Call:
## lm(formula = log(sell) ~ lot + bdms + fb + sty + drv + rec +
## ffin + ghw + ca + gar + reg, data = data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.67865 -0.12211 0.01666 0.12868 0.67737
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.003e+01 4.724e-02 212.210 < 2e-16 ***
## lot 5.057e-05 4.854e-06 10.418 < 2e-16 ***
## bdms 3.402e-02 1.451e-02 2.345 0.01939 *
## fb 1.678e-01 2.065e-02 8.126 3.10e-15 ***
## sty 9.227e-02 1.282e-02 7.197 2.10e-12 ***
## drv 1.307e-01 2.834e-02 4.610 5.04e-06 ***
## rec 7.352e-02 2.633e-02 2.792 0.00542 **
## ffin 9.940e-02 2.200e-02 4.517 7.72e-06 ***
## ghw 1.784e-01 4.458e-02 4.000 7.22e-05 ***
## ca 1.780e-01 2.155e-02 8.262 1.14e-15 ***
## gar 5.076e-02 1.165e-02 4.358 1.58e-05 ***
## reg 1.271e-01 2.313e-02 5.496 6.02e-08 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.2137 on 534 degrees of freedom
## Multiple R-squared: 0.6766, Adjusted R-squared: 0.6699
## F-statistic: 101.6 on 11 and 534 DF, p-value: < 2.2e-16reset2 <- resettest(reg2, power = 2, type = "fitted")
jarque2 <- jarque.bera.test(reg2$residuals)
format(data.frame("Ramsey RESET" = c(reset2$statistic, reset2$p.value), "Jarque-Bera" = c(jarque2$statistic, jarque2$p.value), row.names = c("Statistic", "P-Value")), scientific=FALSE)## Ramsey.RESET Jarque.Bera
## Statistic 0.2703143 8.44324122
## P-Value 0.6033369 0.01467484\(~\)
With a p-value above 0.05, the RESET Test fails to reject the null hypothesis, suggesting that the new model is not misspecified. The Jarque-Bera test, on the other hand, still rejects its null hypothesis, indicating that residuals are not normally distributed.
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(c) Continue with the linear model from question (b). Estimate a model that includes both the lot size variable and its logarithm, as well as all other explanatory variables without transformation. What is your conclusion, should we include lot size itself or its logarithm?
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reg3 <- lm(data = data, log(sell) ~ lot + log(lot) + bdms + fb + sty + drv + rec + ffin + ghw + ca + gar + reg)
summary(reg3)##
## Call:
## lm(formula = log(sell) ~ lot + log(lot) + bdms + fb + sty + drv +
## rec + ffin + ghw + ca + gar + reg, data = data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.68573 -0.12380 0.00785 0.12521 0.68112
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 7.150e+00 6.830e-01 10.469 < 2e-16 ***
## lot -1.490e-05 1.624e-05 -0.918 0.359086
## log(lot) 3.827e-01 9.070e-02 4.219 2.88e-05 ***
## bdms 3.489e-02 1.429e-02 2.442 0.014915 *
## fb 1.659e-01 2.033e-02 8.161 2.40e-15 ***
## sty 9.121e-02 1.263e-02 7.224 1.76e-12 ***
## drv 1.068e-01 2.847e-02 3.752 0.000195 ***
## rec 5.467e-02 2.630e-02 2.078 0.038156 *
## ffin 1.052e-01 2.171e-02 4.848 1.64e-06 ***
## ghw 1.791e-01 4.390e-02 4.079 5.20e-05 ***
## ca 1.643e-01 2.146e-02 7.657 9.01e-14 ***
## gar 4.826e-02 1.148e-02 4.203 3.09e-05 ***
## reg 1.344e-01 2.284e-02 5.884 7.10e-09 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.2104 on 533 degrees of freedom
## Multiple R-squared: 0.687, Adjusted R-squared: 0.68
## F-statistic: 97.51 on 12 and 533 DF, p-value: < 2.2e-16reset3 <- resettest(reg3, power = 2, type = "fitted")
jarque3 <- jarque.bera.test(reg3$residuals)
format(data.frame("Ramsey RESET" = c(reset3$statistic, reset3$p.value), "Jarque-Bera" = c(jarque3$statistic, jarque3$p.value), row.names = c("Statistic", "P-Value")), scientific=FALSE) ## Ramsey.RESET Jarque.Bera
## Statistic 0.067690 9.364331765
## P-Value 0.794831 0.009258938\(~\)
While the Jarque-bera test still rejects the null hypothesis, the values of the RESET test present even stronger evidence against the null hypothesis than on the last model, suggesting that including the logarithm of the lot size is beneficial for the model’s specification. In this model, the lot size itself is no longer statistically significant.
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(d) Consider now a model where the log of the sale price of the house is the dependent variable and the explanatory variables are the log transformation of lot size, with all other explanatory variables as before. We now consider interaction effects of the log lot size with the other variables. Construct these interaction variables. How many are individually significant?
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reg4 <- lm(data = data, log(sell) ~ log(lot) + bdms + fb + sty + drv + rec + ffin + ghw + ca + gar + reg + log(lot) * bdms + log(lot) * fb + log(lot) * sty + log(lot) * drv + log(lot) * rec + log(lot) * ffin + log(lot) * ghw + log(lot) * ca + log(lot) * gar + log(lot) * reg)
summary(reg4)##
## Call:
## lm(formula = log(sell) ~ log(lot) + bdms + fb + sty + drv + rec +
## ffin + ghw + ca + gar + reg + log(lot) * bdms + log(lot) *
## fb + log(lot) * sty + log(lot) * drv + log(lot) * rec + log(lot) *
## ffin + log(lot) * ghw + log(lot) * ca + log(lot) * gar +
## log(lot) * reg, data = data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.68306 -0.11612 0.00591 0.12486 0.65998
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 8.966499 1.070667 8.375 5.09e-16 ***
## log(lot) 0.152685 0.128294 1.190 0.2345
## bdms 0.019075 0.326700 0.058 0.9535
## fb -0.368234 0.429048 -0.858 0.3911
## sty 0.488885 0.309700 1.579 0.1150
## drv -1.463371 0.717225 -2.040 0.0418 *
## rec 1.673992 0.655919 2.552 0.0110 *
## ffin -0.031844 0.445543 -0.071 0.9430
## ghw -0.505889 0.902733 -0.560 0.5754
## ca -0.340276 0.496041 -0.686 0.4930
## gar 0.401941 0.258646 1.554 0.1208
## reg 0.118484 0.479856 0.247 0.8051
## log(lot):bdms 0.002070 0.038654 0.054 0.9573
## log(lot):fb 0.062037 0.050145 1.237 0.2166
## log(lot):sty -0.046361 0.035942 -1.290 0.1977
## log(lot):drv 0.191542 0.087361 2.193 0.0288 *
## log(lot):rec -0.188462 0.076373 -2.468 0.0139 *
## log(lot):ffin 0.015913 0.052851 0.301 0.7635
## log(lot):ghw 0.081135 0.106929 0.759 0.4483
## log(lot):ca 0.059549 0.058024 1.026 0.3052
## log(lot):gar -0.041359 0.030142 -1.372 0.1706
## log(lot):reg 0.001515 0.055990 0.027 0.9784
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.2095 on 524 degrees of freedom
## Multiple R-squared: 0.6951, Adjusted R-squared: 0.6829
## F-statistic: 56.89 on 21 and 524 DF, p-value: < 2.2e-16\(~\)
The individually significant interactions are:
- drv: Driveway dummy
- rev: Recreational room dummy
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(e) Perform an F-test for the joint significance of the interaction effects from question (d).
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reg5 <- lm(data = data, log(sell) ~ log(lot) + bdms + fb + sty + drv + rec + ffin + ghw + ca + gar + reg + log(lot) * drv + log(lot) * rec)
data.frame("All Interactions" = summary(reg4)$fstatistic[1], "Only DRV and REV" = summary(reg5)$fstatistic[1], row.names = "F-statistic")## All.Interactions Only.DRV.and.REV
## F-statistic 56.88817 91.78705\(~\)
With a greater f-statistic, the joint significance of the model including only the ‘drv’ and ‘rev’ interactions is higher than the one including interactions with all variables.
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(f) Now perform model specification on the interaction variables using the general-to-specific approach. (Only eliminate the interaction effects.)
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summary(reg4)##
## Call:
## lm(formula = log(sell) ~ log(lot) + bdms + fb + sty + drv + rec +
## ffin + ghw + ca + gar + reg + log(lot) * bdms + log(lot) *
## fb + log(lot) * sty + log(lot) * drv + log(lot) * rec + log(lot) *
## ffin + log(lot) * ghw + log(lot) * ca + log(lot) * gar +
## log(lot) * reg, data = data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.68306 -0.11612 0.00591 0.12486 0.65998
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 8.966499 1.070667 8.375 5.09e-16 ***
## log(lot) 0.152685 0.128294 1.190 0.2345
## bdms 0.019075 0.326700 0.058 0.9535
## fb -0.368234 0.429048 -0.858 0.3911
## sty 0.488885 0.309700 1.579 0.1150
## drv -1.463371 0.717225 -2.040 0.0418 *
## rec 1.673992 0.655919 2.552 0.0110 *
## ffin -0.031844 0.445543 -0.071 0.9430
## ghw -0.505889 0.902733 -0.560 0.5754
## ca -0.340276 0.496041 -0.686 0.4930
## gar 0.401941 0.258646 1.554 0.1208
## reg 0.118484 0.479856 0.247 0.8051
## log(lot):bdms 0.002070 0.038654 0.054 0.9573
## log(lot):fb 0.062037 0.050145 1.237 0.2166
## log(lot):sty -0.046361 0.035942 -1.290 0.1977
## log(lot):drv 0.191542 0.087361 2.193 0.0288 *
## log(lot):rec -0.188462 0.076373 -2.468 0.0139 *
## log(lot):ffin 0.015913 0.052851 0.301 0.7635
## log(lot):ghw 0.081135 0.106929 0.759 0.4483
## log(lot):ca 0.059549 0.058024 1.026 0.3052
## log(lot):gar -0.041359 0.030142 -1.372 0.1706
## log(lot):reg 0.001515 0.055990 0.027 0.9784
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.2095 on 524 degrees of freedom
## Multiple R-squared: 0.6951, Adjusted R-squared: 0.6829
## F-statistic: 56.89 on 21 and 524 DF, p-value: < 2.2e-16\(~\)
1st eliminated interaction: reg (t-value = 0.027)
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reg6 <- lm(data = data, log(sell) ~ log(lot) + bdms + fb + sty + drv + rec + ffin + ghw + ca + gar + reg + log(lot) * bdms + log(lot) * fb + log(lot) * sty + log(lot) * drv + log(lot) * rec + log(lot) * ffin + log(lot) * ghw + log(lot) * ca + log(lot) * gar)
summary(reg6)##
## Call:
## lm(formula = log(sell) ~ log(lot) + bdms + fb + sty + drv + rec +
## ffin + ghw + ca + gar + reg + log(lot) * bdms + log(lot) *
## fb + log(lot) * sty + log(lot) * drv + log(lot) * rec + log(lot) *
## ffin + log(lot) * ghw + log(lot) * ca + log(lot) * gar, data = data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.68292 -0.11619 0.00573 0.12491 0.65976
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 8.96795 1.06831 8.394 4.37e-16 ***
## log(lot) 0.15252 0.12802 1.191 0.2341
## bdms 0.01949 0.32603 0.060 0.9523
## fb -0.36774 0.42824 -0.859 0.3909
## sty 0.48721 0.30316 1.607 0.1086
## drv -1.46786 0.69713 -2.106 0.0357 *
## rec 1.67468 0.65480 2.558 0.0108 *
## ffin -0.03494 0.43021 -0.081 0.9353
## ghw -0.50427 0.89990 -0.560 0.5755
## ca -0.33954 0.49483 -0.686 0.4929
## gar 0.40234 0.25797 1.560 0.1194
## reg 0.13145 0.02304 5.705 1.94e-08 ***
## log(lot):bdms 0.00202 0.03857 0.052 0.9582
## log(lot):fb 0.06198 0.05005 1.238 0.2161
## log(lot):sty -0.04617 0.03518 -1.312 0.1900
## log(lot):drv 0.19207 0.08504 2.259 0.0243 *
## log(lot):rec -0.18855 0.07623 -2.473 0.0137 *
## log(lot):ffin 0.01629 0.05098 0.319 0.7495
## log(lot):ghw 0.08094 0.10658 0.759 0.4479
## log(lot):ca 0.05946 0.05788 1.027 0.3047
## log(lot):gar -0.04140 0.03007 -1.377 0.1691
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.2093 on 525 degrees of freedom
## Multiple R-squared: 0.6951, Adjusted R-squared: 0.6835
## F-statistic: 59.85 on 20 and 525 DF, p-value: < 2.2e-16\(~\)
2nd eliminated interaction: bdms (t-value = 0.052)
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reg7 <- lm(data = data, log(sell) ~ log(lot) + bdms + fb + sty + drv + rec + ffin + ghw + ca + gar + reg + log(lot) * fb + log(lot) * sty + log(lot) * drv + log(lot) * rec + log(lot) * ffin + log(lot) * ghw + log(lot) * ca + log(lot) * gar)
summary(reg7)##
## Call:
## lm(formula = log(sell) ~ log(lot) + bdms + fb + sty + drv + rec +
## ffin + ghw + ca + gar + reg + log(lot) * fb + log(lot) *
## sty + log(lot) * drv + log(lot) * rec + log(lot) * ffin +
## log(lot) * ghw + log(lot) * ca + log(lot) * gar, data = data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.68301 -0.11617 0.00574 0.12490 0.66020
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 8.93486 0.86069 10.381 < 2e-16 ***
## log(lot) 0.15647 0.10329 1.515 0.1304
## bdms 0.03655 0.01459 2.506 0.0125 *
## fb -0.37745 0.38563 -0.979 0.3281
## sty 0.48200 0.28614 1.685 0.0927 .
## drv -1.46253 0.68903 -2.123 0.0343 *
## rec 1.67592 0.65375 2.564 0.0106 *
## ffin -0.03743 0.42717 -0.088 0.9302
## ghw -0.50161 0.89761 -0.559 0.5765
## ca -0.33869 0.49409 -0.685 0.4933
## gar 0.40103 0.25652 1.563 0.1186
## reg 0.13144 0.02302 5.710 1.89e-08 ***
## log(lot):fb 0.06313 0.04494 1.405 0.1607
## log(lot):sty -0.04556 0.03321 -1.372 0.1706
## log(lot):drv 0.19143 0.08408 2.277 0.0232 *
## log(lot):rec -0.18868 0.07612 -2.479 0.0135 *
## log(lot):ffin 0.01658 0.05062 0.328 0.7434
## log(lot):ghw 0.08062 0.10631 0.758 0.4486
## log(lot):ca 0.05935 0.05779 1.027 0.3049
## log(lot):gar -0.04125 0.02989 -1.380 0.1682
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.2091 on 526 degrees of freedom
## Multiple R-squared: 0.6951, Adjusted R-squared: 0.6841
## F-statistic: 63.12 on 19 and 526 DF, p-value: < 2.2e-16\(~\)
3rd eliminated interaction: ffin (t-value = 0.328)
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reg8 <- lm(data = data, log(sell) ~ log(lot) + bdms + fb + sty + drv + rec + ffin + ghw + ca + gar + reg + log(lot) * fb + log(lot) * sty + log(lot) * drv + log(lot) * rec + log(lot) * ghw + log(lot) * ca + log(lot) * gar)
summary(reg8)##
## Call:
## lm(formula = log(sell) ~ log(lot) + bdms + fb + sty + drv + rec +
## ffin + ghw + ca + gar + reg + log(lot) * fb + log(lot) *
## sty + log(lot) * drv + log(lot) * rec + log(lot) * ghw +
## log(lot) * ca + log(lot) * gar, data = data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.68181 -0.11724 0.00567 0.12594 0.65662
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 8.87651 0.84134 10.550 < 2e-16 ***
## log(lot) 0.16359 0.10089 1.621 0.1055
## bdms 0.03655 0.01458 2.507 0.0125 *
## fb -0.38191 0.38506 -0.992 0.3217
## sty 0.48851 0.28520 1.713 0.0873 .
## drv -1.45022 0.68742 -2.110 0.0354 *
## rec 1.62140 0.63167 2.567 0.0105 *
## ffin 0.10232 0.02181 4.691 3.47e-06 ***
## ghw -0.51600 0.89578 -0.576 0.5648
## ca -0.35449 0.49131 -0.722 0.4709
## gar 0.40146 0.25629 1.566 0.1179
## reg 0.13227 0.02286 5.786 1.24e-08 ***
## log(lot):fb 0.06360 0.04487 1.417 0.1570
## log(lot):sty -0.04640 0.03308 -1.403 0.1613
## log(lot):drv 0.18991 0.08388 2.264 0.0240 *
## log(lot):rec -0.18218 0.07343 -2.481 0.0134 *
## log(lot):ghw 0.08250 0.10606 0.778 0.4370
## log(lot):ca 0.06123 0.05746 1.066 0.2871
## log(lot):gar -0.04129 0.02987 -1.383 0.1674
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.2089 on 527 degrees of freedom
## Multiple R-squared: 0.695, Adjusted R-squared: 0.6846
## F-statistic: 66.73 on 18 and 527 DF, p-value: < 2.2e-16\(~\)
4th eliminated interaction: ghw (t-value = 0.778)
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reg9 <- lm(data = data, log(sell) ~ log(lot) + bdms + fb + sty + drv + rec + ffin + ghw + ca + gar + reg + log(lot) * fb + log(lot) * sty + log(lot) * drv + log(lot) * rec + log(lot) * ca + log(lot) * gar)
summary(reg9)##
## Call:
## lm(formula = log(sell) ~ log(lot) + bdms + fb + sty + drv + rec +
## ffin + ghw + ca + gar + reg + log(lot) * fb + log(lot) *
## sty + log(lot) * drv + log(lot) * rec + log(lot) * ca + log(lot) *
## gar, data = data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.68122 -0.11898 0.00738 0.12611 0.65311
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 8.80857 0.83648 10.530 < 2e-16 ***
## log(lot) 0.17159 0.10033 1.710 0.0878 .
## bdms 0.03661 0.01457 2.513 0.0123 *
## fb -0.37636 0.38485 -0.978 0.3286
## sty 0.49092 0.28508 1.722 0.0857 .
## drv -1.43262 0.68679 -2.086 0.0375 *
## rec 1.63058 0.63133 2.583 0.0101 *
## ffin 0.10361 0.02174 4.766 2.44e-06 ***
## ghw 0.17991 0.04391 4.098 4.83e-05 ***
## ca -0.33972 0.49076 -0.692 0.4891
## gar 0.39730 0.25614 1.551 0.1215
## reg 0.13113 0.02281 5.750 1.51e-08 ***
## log(lot):fb 0.06302 0.04485 1.405 0.1606
## log(lot):sty -0.04669 0.03306 -1.412 0.1585
## log(lot):drv 0.18782 0.08380 2.241 0.0254 *
## log(lot):rec -0.18320 0.07339 -2.496 0.0129 *
## log(lot):ca 0.05932 0.05738 1.034 0.3017
## log(lot):gar -0.04085 0.02985 -1.368 0.1718
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.2088 on 528 degrees of freedom
## Multiple R-squared: 0.6947, Adjusted R-squared: 0.6849
## F-statistic: 70.67 on 17 and 528 DF, p-value: < 2.2e-16\(~\)
5th eliminated interaction: ca (t-value = 1.034)
\(~\)
reg10 <- lm(data = data, log(sell) ~ log(lot) + bdms + fb + sty + drv + rec + ffin + ghw + ca + gar + reg + log(lot) * fb + log(lot) * sty + log(lot) * drv + log(lot) * rec + log(lot) * gar)
summary(reg10)##
## Call:
## lm(formula = log(sell) ~ log(lot) + bdms + fb + sty + drv + rec +
## ffin + ghw + ca + gar + reg + log(lot) * fb + log(lot) *
## sty + log(lot) * drv + log(lot) * rec + log(lot) * gar, data = data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.67934 -0.12004 0.00644 0.12660 0.64601
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 8.78218 0.83615 10.503 < 2e-16 ***
## log(lot) 0.17484 0.10028 1.743 0.0818 .
## bdms 0.03523 0.01451 2.428 0.0155 *
## fb -0.38030 0.38485 -0.988 0.3235
## sty 0.47196 0.28451 1.659 0.0977 .
## drv -1.42861 0.68683 -2.080 0.0380 *
## rec 1.55669 0.62731 2.482 0.0134 *
## ffin 0.10341 0.02174 4.756 2.55e-06 ***
## ghw 0.17721 0.04383 4.043 6.06e-05 ***
## ca 0.16716 0.02121 7.880 1.87e-14 ***
## gar 0.33850 0.24976 1.355 0.1759
## reg 0.13298 0.02274 5.848 8.69e-09 ***
## log(lot):fb 0.06359 0.04485 1.418 0.1569
## log(lot):sty -0.04432 0.03299 -1.344 0.1797
## log(lot):drv 0.18733 0.08381 2.235 0.0258 *
## log(lot):rec -0.17463 0.07293 -2.395 0.0170 *
## log(lot):gar -0.03385 0.02908 -1.164 0.2448
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.2088 on 529 degrees of freedom
## Multiple R-squared: 0.6941, Adjusted R-squared: 0.6848
## F-statistic: 75.01 on 16 and 529 DF, p-value: < 2.2e-16\(~\)
6th eliminated interaction: gar (t-value = -1.164)
\(~\)
reg11 <- lm(data = data, log(sell) ~ log(lot) + bdms + fb + sty + drv + rec + ffin + ghw + ca + gar + reg + log(lot) * fb + log(lot) * sty + log(lot) * drv + log(lot) * rec)
summary(reg11)##
## Call:
## lm(formula = log(sell) ~ log(lot) + bdms + fb + sty + drv + rec +
## ffin + ghw + ca + gar + reg + log(lot) * fb + log(lot) *
## sty + log(lot) * drv + log(lot) * rec, data = data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.68420 -0.12071 0.00669 0.12322 0.64513
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 8.77393 0.83640 10.490 < 2e-16 ***
## log(lot) 0.17584 0.10031 1.753 0.0802 .
## bdms 0.03530 0.01451 2.432 0.0153 *
## fb -0.34021 0.38344 -0.887 0.3753
## sty 0.46819 0.28459 1.645 0.1005
## drv -1.23688 0.66702 -1.854 0.0642 .
## rec 1.51405 0.62645 2.417 0.0160 *
## ffin 0.10279 0.02174 4.727 2.92e-06 ***
## ghw 0.18002 0.04378 4.112 4.55e-05 ***
## ca 0.16697 0.02122 7.869 2.02e-14 ***
## gar 0.04802 0.01143 4.200 3.13e-05 ***
## reg 0.12990 0.02259 5.750 1.51e-08 ***
## log(lot):fb 0.05903 0.04469 1.321 0.1872
## log(lot):sty -0.04392 0.03300 -1.331 0.1837
## log(lot):drv 0.16448 0.08150 2.018 0.0441 *
## log(lot):rec -0.16943 0.07281 -2.327 0.0203 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.2089 on 530 degrees of freedom
## Multiple R-squared: 0.6933, Adjusted R-squared: 0.6846
## F-statistic: 79.87 on 15 and 530 DF, p-value: < 2.2e-16\(~\)
7th eliminated interaction: fb (t-value = 1.321)
\(~\)
reg12 <- lm(data = data, log(sell) ~ log(lot) + bdms + fb + sty + drv + rec + ffin + ghw + ca + gar + reg + log(lot) * sty + log(lot) * drv + log(lot) * rec)
summary(reg12)##
## Call:
## lm(formula = log(sell) ~ log(lot) + bdms + fb + sty + drv + rec +
## ffin + ghw + ca + gar + reg + log(lot) * sty + log(lot) *
## drv + log(lot) * rec, data = data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.68209 -0.11831 0.00758 0.12350 0.63856
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 8.29846 0.75549 10.984 < 2e-16 ***
## log(lot) 0.23088 0.09131 2.529 0.0117 *
## bdms 0.03623 0.01451 2.497 0.0128 *
## fb 0.16549 0.02061 8.030 6.30e-15 ***
## sty 0.38420 0.27758 1.384 0.1669
## drv -1.25462 0.66735 -1.880 0.0607 .
## rec 1.47254 0.62610 2.352 0.0190 *
## ffin 0.10042 0.02168 4.631 4.58e-06 ***
## ghw 0.18093 0.04381 4.130 4.21e-05 ***
## ca 0.16623 0.02123 7.831 2.64e-14 ***
## gar 0.04751 0.01143 4.155 3.79e-05 ***
## reg 0.13126 0.02258 5.812 1.06e-08 ***
## log(lot):sty -0.03402 0.03216 -1.058 0.2906
## log(lot):drv 0.16690 0.08154 2.047 0.0412 *
## log(lot):rec -0.16467 0.07278 -2.263 0.0241 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.2091 on 531 degrees of freedom
## Multiple R-squared: 0.6923, Adjusted R-squared: 0.6842
## F-statistic: 85.33 on 14 and 531 DF, p-value: < 2.2e-16\(~\)
8th eliminated interaction: sty (t-value = -1.058)
\(~\)
reg13 <- lm(data = data, log(sell) ~ log(lot) + bdms + fb + sty + drv + rec + ffin + ghw + ca + gar + reg + log(lot) * drv + log(lot) * rec)
summary(reg13)##
## Call:
## lm(formula = log(sell) ~ log(lot) + bdms + fb + sty + drv + rec +
## ffin + ghw + ca + gar + reg + log(lot) * drv + log(lot) *
## rec, data = data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.67934 -0.12225 0.00849 0.12259 0.65051
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 8.74189 0.62863 13.906 < 2e-16 ***
## log(lot) 0.17906 0.07707 2.323 0.02053 *
## bdms 0.03881 0.01430 2.714 0.00686 **
## fb 0.16145 0.02025 7.971 9.62e-15 ***
## sty 0.09083 0.01254 7.242 1.56e-12 ***
## drv -1.18996 0.66462 -1.790 0.07395 .
## rec 1.50253 0.62553 2.402 0.01665 *
## ffin 0.10276 0.02157 4.763 2.46e-06 ***
## ghw 0.18448 0.04368 4.223 2.83e-05 ***
## ca 0.16526 0.02121 7.792 3.48e-14 ***
## gar 0.04690 0.01142 4.107 4.65e-05 ***
## reg 0.13260 0.02255 5.880 7.24e-09 ***
## log(lot):drv 0.15943 0.08124 1.962 0.05024 .
## log(lot):rec -0.16826 0.07270 -2.314 0.02103 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.2091 on 532 degrees of freedom
## Multiple R-squared: 0.6916, Adjusted R-squared: 0.6841
## F-statistic: 91.79 on 13 and 532 DF, p-value: < 2.2e-16\(~\)
9th eliminated interaction: drv (t-value = -1.962)
\(~\)
reg14 <- lm(data = data, log(sell) ~ log(lot) + bdms + fb + sty + drv + rec + ffin + ghw + ca + gar + reg + log(lot) * rec)
summary(reg14)##
## Call:
## lm(formula = log(sell) ~ log(lot) + bdms + fb + sty + drv + rec +
## ffin + ghw + ca + gar + reg + log(lot) * rec, data = data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.68111 -0.12208 0.00593 0.12731 0.66275
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 7.59071 0.22656 33.505 < 2e-16 ***
## log(lot) 0.32024 0.02770 11.562 < 2e-16 ***
## bdms 0.03842 0.01434 2.680 0.0076 **
## fb 0.16318 0.02029 8.043 5.71e-15 ***
## sty 0.09080 0.01258 7.220 1.80e-12 ***
## drv 0.11312 0.02815 4.018 6.72e-05 ***
## rec 1.44313 0.62646 2.304 0.0216 *
## ffin 0.10450 0.02161 4.835 1.74e-06 ***
## ghw 0.18429 0.04380 4.208 3.03e-05 ***
## ca 0.16593 0.02126 7.804 3.19e-14 ***
## gar 0.04810 0.01144 4.206 3.05e-05 ***
## reg 0.13373 0.02260 5.917 5.89e-09 ***
## log(lot):rec -0.16112 0.07281 -2.213 0.0273 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.2096 on 533 degrees of freedom
## Multiple R-squared: 0.6894, Adjusted R-squared: 0.6824
## F-statistic: 98.59 on 12 and 533 DF, p-value: < 2.2e-16\(~\)
No interactions left to eliminate.
\(~\)
(g) One may argue that some of the explanatory variables are endogenous and that there may be omitted variables. For example, the ‘condition’ of the house in terms of how it is maintained is not a variable (and difficult to measure) but will affect the house price. It will also affect, or be reflected in, some of the other variables, such as whether the house has an air conditioning (which is mostly in newer houses). If the condition of the house is missing, will the effect of air conditioning on the (log of the) sale price be over- or underestimated? (For this question no computer calculations are required.)
\(~\)
Endogeneity causes overestimation when the omitted variable has the same effect on the dependent and independent variables, while underestimation happens when the effect is opposite.
The condition of a house tends to increase both sale prices and likelihood of air conditioning, meaning it affects the dependent and independent variables in similar ways. This would cause an overestimation of the effect of air conditioning on sale prices.
\(~\)
(h) Finally we analyze the predictive ability of the model. Consider again the model where the log of the sale price of the house is the dependent variable and the explanatory variables are the log transformation of lot size, with all other explanatory variables in their original form (and no interaction effects). Estimate the parameters of the model using the first 400 observations. Make predictions on the log of the price and calculate the MAE for the other 146 observations. How good is the predictive power of the model (relative to the variability in the log of the price)?
\(~\)
data_sub <- data[1:400,]
reg_sub <- lm(data = data_sub, log(sell) ~ log(lot) + bdms + fb + sty + drv + rec + ffin + ghw + ca + gar + reg)
summary(reg_sub)##
## Call:
## lm(formula = log(sell) ~ log(lot) + bdms + fb + sty + drv + rec +
## ffin + ghw + ca + gar + reg, data = data_sub)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.66582 -0.13906 0.00796 0.14694 0.67596
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 7.67309 0.29240 26.241 < 2e-16 ***
## log(lot) 0.31378 0.03615 8.680 < 2e-16 ***
## bdms 0.03787 0.01744 2.172 0.030469 *
## fb 0.15238 0.02469 6.170 1.71e-09 ***
## sty 0.08824 0.01819 4.850 1.79e-06 ***
## drv 0.08641 0.03141 2.751 0.006216 **
## rec 0.05465 0.03392 1.611 0.107975
## ffin 0.11471 0.02673 4.291 2.25e-05 ***
## ghw 0.19870 0.05301 3.748 0.000205 ***
## ca 0.17763 0.02724 6.521 2.17e-10 ***
## gar 0.05301 0.01480 3.583 0.000383 ***
## reg 0.15116 0.04215 3.586 0.000378 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.2238 on 388 degrees of freedom
## Multiple R-squared: 0.6705, Adjusted R-squared: 0.6611
## F-statistic: 71.77 on 11 and 388 DF, p-value: < 2.2e-16new <- data[401:546,]
new <- data.frame(new, "log.sell" = log(new$sell))
forecast <- predict.lm(reg_sub, newdata = new)
comparison <- data.frame("Observation" = new$obs, "Forecast" = forecast, "Actual Data" = log(new$sell))
data.frame("MAE" = mae(new$log.sell, forecast))## MAE
## 1 0.1278416ggplot(data = comparison, aes(x = Observation, y = Forecast))+
geom_line(aes(color = "red"))+
geom_line(y = comparison$Actual.Data, aes(color = "blue"))+
labs(x = "Observation", y = "Log(sale prices)")+
ggtitle("Forecast")+
scale_color_identity(name = "Legend", breaks = c("red", "blue"), labels =
c("Forecast", "Actual Values"), guide = "legend")
head(comparison)## Observation Forecast Actual.Data
## 401 401 11.51385 11.43496
## 402 402 11.47529 11.23849
## 403 403 11.38201 11.25803
## 404 404 11.18935 11.28978
## 405 405 11.32946 11.28978
## 406 406 11.36018 11.36210