Question B1: Hedonic model. A dataset comprising 88 houses and their characteristics are collected by a real estate agent. The house features are: assess = the assessed value of the house ($'000) bdrms = the number of bedrooms colonial = a dummy which equals 1 if the house has a colonial style price = the price of the house ($'000) sqrft = the size of the house in square feet lotsize = the size of the lot (the area where the house resides) in square feet The table below shows the correlation between the variables of interest. LOTSIZE SQRFT LOTSIZE 1.000000 0.183842 SQRFT 0.183842 1.000000 PRICE 0.347124 0.787907 ASSESS 0.328146 0.865634 BDRMS 0.136326 0.531474 COLONIAL 0.014019 0.065421 PRICE 0.347124 0.787907 1.000000 0.905279 0.508084 0.137946 ASSESS 0.328146 0.865634 0.905279 1.000000 0.482474 0.082936 BDRMS 0.136326 0.531474 0.508084 0.482474 1.000000 0.304575 COLONIAL 0.014019 0.065421 0.137946 0.082936 0.304575 1.000000 (a) The agent wanted to study the relationship between house prices and their assessed values. He plotted the two series together. GRAPH 1 shows the plot for assess and price" across the 88 houses. GRAPH 2 shows the scatter plot of the two series with each series represented on one of the axes. What can you infer about the relationship of the two series? (2 marks) GRAPH 1 GRAPH 2 800 800 700 700 600 600 500 500 PRICE 400 400 300 300 200 200 100 10 20 30 40 50 60 70 80 100 100 200 300 400 500 600 700 800 ASSESS - PRICE ASSESS (6) The agent believes that the assessed value of the house has predictive power over the settlement price of the house. He ran a regression of PRICE' on 'ASSESS' and the results are shown below: Dependent Variable: PRICE Method: Least Squares Sample: 1 88 Included observations: 88 Variable Coefficient Std. Error t-Statistic Prob. ASSESS 0.933507 0.014175 65.85659 0.0000 R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat 0.817871 Mean dependent var 0.817871 S.D. dependent var 43.83456 Akaike info criterion 167167.8 Schwarz criterion -457.0409 Hannan-Quinn criterion. 1.943617 293.5460 102.7134 10.41002 10.43817 10.42136 What can be inferred from this regression result about the average price of houses based on their assessed values? (2 marks) (c) The agent obtained the descriptive statistic of the resulting residuals from the above regression in (b). Based on the descriptive statistics, it can be seen that one of the assumptions of the classical linear regression model is violated. State the assumption that is violated. Explain your answer. How would you change the regression model to ensure that the assumption is not violated? (2 marks, 2 marks) 20 Series: RESID1 Sample 1 88 Observations 88 16 12 Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis -1.196085 -5.873754 189.2895 -100.8495 43.81805 0.948869 6.251329 8 4 Jarque-Bera Probability 51.96600 0.000000 -80 -40 40 80 120 160 (d) The agent seeks to understand the determinants of house prices. He ran a regression of house price on all the features of the house. The regression results are shown below: Dependent Variable: PRICE Method: Least Squares Sample: 1 88 Included observations: 88 Variable Coefficient Std. Emor t-Statistic Prob. 40.44766 21.59420 -1.873080 0.0646 LOTSIZE 0.000599 0.000497 1.205584 0.2314 SQRFT 0.001071 0.017197 0.062301 0.9505 BDRMS 9.630256 6.916290 1.392402 0.1676 COLONIAL 9.547571 10.64735 0.896709 0.3725 ASSESS 0.904078 0.104268 8.670721 0.0000 R-squared 0.830864 Mean dependent var 293.5460 Adjusted R-squared 0.820551 S.D. dependent var 102.7134 S.E. of regression 43.51092 Akaike info criterion 10.44965 Sum squared resid 155242.4 Schwarz criterion 10.61856 Log likelihood 453.7845 Hannan-Quinn criter. 10.51770 F-statistic 80.56328 Durbin-Watson stat 2.118382 Prob(F-statistic) 0.000000 The agent showed the results to you given your knowledge in financial modelling from ECON339. You believe there is something odd about the results. What is uncanny (or strange) about the results? (Hint: Look at the R-squared and the statistical significance of most of the variables). What do you suspect may be the problem with this regression? (2 marks, 2 marks) (e) You advised the real estate agent to drop the variable "ASSESS from the regression and he re- estimated the model to produce a set of new results: Dependent Variable: PRICE Method: Least Squares Sample: 1 88 Included observations: 88 Variable Coefficient Std. Error t-Statistic Prob -24.12653 29.60345 -0.814990 0.4174 LOTSIZE 0.002076 0.000643 3.230108 0.0018 SQRFT 0.124237 9.314370 0.0000 BDRMS 11.00429 9.515260 1.156489 0.2508 COLONIAL 13.71554 14.63727 0.937029 0.3515 R-squared 0.675792 Mean dependent var 293.5460 Adjusted R-squared 0.660167 S.D. dependent var 102.7134 S.E. of regression 59.87697 Akaike info criterion 11.07760 Sum squared resid 297575.9 Schwarz criterion 11.21836 Log likelihood -482.4144 Hannan-Quinn criter. 11.13431 Looking at the statistical significance of the variables, what is so different about this set of results from the regression results in part (d)? What do you believe is the cause for the differences in results? [Hint: Look at the table of correlations of the variables of interest.) (2 marks, 2 marks) 0.013338 (f) Suppose you run a regression of PRICE on LOTSIZE and SQRFT including an intercept. What would happen to the R of the new regression if you were to include BDRMS as a regressor in the regression? How would you determine whether the new regression which includes BDRMS is a better fit of the regression compared to the regression excluding BDRMS in characterising the data? Explain. (1 mark, 3 marks) Question B1: Hedonic model. A dataset comprising 88 houses and their characteristics are collected by a real estate agent. The house features are: assess = the assessed value of the house ($'000) bdrms = the number of bedrooms colonial = a dummy which equals 1 if the house has a colonial style price = the price of the house ($'000) sqrft = the size of the house in square feet lotsize = the size of the lot (the area where the house resides) in square feet The table below shows the correlation between the variables of interest. LOTSIZE SQRFT LOTSIZE 1.000000 0.183842 SQRFT 0.183842 1.000000 PRICE 0.347124 0.787907 ASSESS 0.328146 0.865634 BDRMS 0.136326 0.531474 COLONIAL 0.014019 0.065421 PRICE 0.347124 0.787907 1.000000 0.905279 0.508084 0.137946 ASSESS 0.328146 0.865634 0.905279 1.000000 0.482474 0.082936 BDRMS 0.136326 0.531474 0.508084 0.482474 1.000000 0.304575 COLONIAL 0.014019 0.065421 0.137946 0.082936 0.304575 1.000000 (a) The agent wanted to study the relationship between house prices and their assessed values. He plotted the two series together. GRAPH 1 shows the plot for assess and price" across the 88 houses. GRAPH 2 shows the scatter plot of the two series with each series represented on one of the axes. What can you infer about the relationship of the two series? (2 marks) GRAPH 1 GRAPH 2 800 800 700 700 600 600 500 500 PRICE 400 400 300 300 200 200 100 10 20 30 40 50 60 70 80 100 100 200 300 400 500 600 700 800 ASSESS - PRICE ASSESS (6) The agent believes that the assessed value of the house has predictive power over the settlement price of the house. He ran a regression of PRICE' on 'ASSESS' and the results are shown below: Dependent Variable: PRICE Method: Least Squares Sample: 1 88 Included observations: 88 Variable Coefficient Std. Error t-Statistic Prob. ASSESS 0.933507 0.014175 65.85659 0.0000 R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat 0.817871 Mean dependent var 0.817871 S.D. dependent var 43.83456 Akaike info criterion 167167.8 Schwarz criterion -457.0409 Hannan-Quinn criterion. 1.943617 293.5460 102.7134 10.41002 10.43817 10.42136 What can be inferred from this regression result about the average price of houses based on their assessed values? (2 marks) (c) The agent obtained the descriptive statistic of the resulting residuals from the above regression in (b). Based on the descriptive statistics, it can be seen that one of the assumptions of the classical linear regression model is violated. State the assumption that is violated. Explain your answer. How would you change the regression model to ensure that the assumption is not violated? (2 marks, 2 marks) 20 Series: RESID1 Sample 1 88 Observations 88 16 12 Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis -1.196085 -5.873754 189.2895 -100.8495 43.81805 0.948869 6.251329 8 4 Jarque-Bera Probability 51.96600 0.000000 -80 -40 40 80 120 160 (d) The agent seeks to understand the determinants of house prices. He ran a regression of house price on all the features of the house. The regression results are shown below: Dependent Variable: PRICE Method: Least Squares Sample: 1 88 Included observations: 88 Variable Coefficient Std. Emor t-Statistic Prob. 40.44766 21.59420 -1.873080 0.0646 LOTSIZE 0.000599 0.000497 1.205584 0.2314 SQRFT 0.001071 0.017197 0.062301 0.9505 BDRMS 9.630256 6.916290 1.392402 0.1676 COLONIAL 9.547571 10.64735 0.896709 0.3725 ASSESS 0.904078 0.104268 8.670721 0.0000 R-squared 0.830864 Mean dependent var 293.5460 Adjusted R-squared 0.820551 S.D. dependent var 102.7134 S.E. of regression 43.51092 Akaike info criterion 10.44965 Sum squared resid 155242.4 Schwarz criterion 10.61856 Log likelihood 453.7845 Hannan-Quinn criter. 10.51770 F-statistic 80.56328 Durbin-Watson stat 2.118382 Prob(F-statistic) 0.000000 The agent showed the results to you given your knowledge in financial modelling from ECON339. You believe there is something odd about the results. What is uncanny (or strange) about the results? (Hint: Look at the R-squared and the statistical significance of most of the variables). What do you suspect may be the problem with this regression? (2 marks, 2 marks) (e) You advised the real estate agent to drop the variable "ASSESS from the regression and he re- estimated the model to produce a set of new results: Dependent Variable: PRICE Method: Least Squares Sample: 1 88 Included observations: 88 Variable Coefficient Std. Error t-Statistic Prob -24.12653 29.60345 -0.814990 0.4174 LOTSIZE 0.002076 0.000643 3.230108 0.0018 SQRFT 0.124237 9.314370 0.0000 BDRMS 11.00429 9.515260 1.156489 0.2508 COLONIAL 13.71554 14.63727 0.937029 0.3515 R-squared 0.675792 Mean dependent var 293.5460 Adjusted R-squared 0.660167 S.D. dependent var 102.7134 S.E. of regression 59.87697 Akaike info criterion 11.07760 Sum squared resid 297575.9 Schwarz criterion 11.21836 Log likelihood -482.4144 Hannan-Quinn criter. 11.13431 Looking at the statistical significance of the variables, what is so different about this set of results from the regression results in part (d)? What do you believe is the cause for the differences in results? [Hint: Look at the table of correlations of the variables of interest.) (2 marks, 2 marks) 0.013338 (f) Suppose you run a regression of PRICE on LOTSIZE and SQRFT including an intercept. What would happen to the R of the new regression if you were to include BDRMS as a regressor in the regression? How would you determine whether the new regression which includes BDRMS is a better fit of the regression compared to the regression excluding BDRMS in characterising the data? Explain. (1 mark, 3 marks)