Chapter 14 Discovery Project

Home Sweet Home: Using Multiple Regression to Analyze and Predict Home Prices

An important problem in real estate is determining how to price homes to be sold. There are so many factors—size, age, and style of the home; number of bedrooms and bathrooms; size of the lot; and so on—which makes setting a price a challenging task. In this project, we will try to help realtors in this task by determining how different characteristics of homes relate to home prices, identifying the key variables in pricing, and building multiple-variable regression models to predict prices based on property characteristics.

Our analysis will be based on the Mount Pleasant Real Estate Data. This data set includes information about $245$ properties for sale in three communities in the suburban town of Mount Pleasant, South Carolina, in 2017.

Phase 1: Data Preparation

1. Download the Mount Pleasant Real Estate Data from stat.hawkeslearning.com and open it with Microsoft Excel.

2. Determine the mean, mode, median, maximum, minimum, standard deviation, and the coefficient of variation of the following variables: price, number of bedrooms, number of bathrooms, number of stories, and square footage, and briefly discuss the results. (Hint: these values can be quickly calculated using the Data Analysis Add-in: Descriptive Statistics in Excel.)

3. Fully summarize the qualitative variables (i.e., What percent of the sample has a pool?) and briefly discuss your findings.

4. To ensure the data contain comparable properties, eliminate duplexes and properties whose prices are outliers. What limitations does this impose on our analysis? How did you determine which prices were outliers?

   Consider the following variables associated with each property:

   $x_1=\text{number of bedrooms}$

   $x_2=\text{number of bathrooms}$

   $x_3=\text{number of stories}$

   $x_4=\text{subdivision}$

   $x_5=\text{square footage}$

   $x_6=\text{age (based on year built)}$

   $x_7=\text{acreage}$

   $x_8=\text{new owned}$

5. For the qualitative variables, adjust this data in a reasonable, quantitative way for use in a regression analysis.

6. Use the following correlation matrix and describe any issues with multicollinearity.

   

   	A	B	C	D	E	F	G	H	I	J	K	L	M	N	O	P	Q
   1		Bedrooms	Baths - Total	Baths - Full	Baths - Half	Stories	Subdivision	Square Footage	Age	Acreage	New Owned?	House Style	Covered Parking Spots	Fenced Yard	Screened Porch?	Golf Course?	Fireplace?
   2	Bedrooms	1
   3	Baths - Total	0.70	1.00
   4	Baths - Full	0.67	0.95	1.00
   5	Baths - Half	0.06	0.13	-0.18	1.00
   6	Stories	0.43	0.50	0.42	0.24	1.00
   7	Subdivision	0.08	-0.05	-0.04	-0.01	0.06	1.00
   8	Square Footage	0.71	0.74	0.68	0.19	0.44	0.11	1.00
   9	Age	0.05	-0.07	-0.12	0.14	-0.07	0.28	0.18	1.00
   10	Acreage	0.12	0.19	0.16	0.11	0.00	-0.03	0.35	0.33	1.00
   11	New Owned?	-0.14	0.00	0.01	-0.05	0.04	-0.29	-0.22	-0.78	-0.15	1.00
   12	House Style	-0.22	-0.22	-0.22	0.03	-0.15	0.07	-0.22	-0.06	-0.12	0.11	1.00
   13	Covered Parking Spots	0.28	0.35	0.31	0.13	0.16	0.03	0.47	0.14	0.24	-0.15	0.07	1.00
   14	Fenced Yard	0.04	-0.12	-0.13	0.04	-0.11	0.01	-0.03	0.29	-0.10	-0.41	-0.01	-0.04	1.00
   15	Screened Porch	0.17	0.24	0.23	0.02	0.14	-0.25	0.14	-0.04	0.03	-0.05	-0.18	-0.03	0.12	1.00
   16	Golf Course?	0.30	0.26	0.27	-0.02	0.19	-0.14	0.37	0.34	0.48	-0.19	-0.08	0.23	0.12	0.13	1.00
   17	Fireplace?	0.18	0.18	0.20	-0.08	0.06	0.04	0.22	0.14	0.09	-0.13	-0.17	0.05	0.11	0.30	0.22	1.00
   18	Number of Fireplaces	0.16	0.21	0.22	-0.03	0.04	0.11	0.35	0.24	0.14	-0.20	-0.13	0.15	0.13	0.28	0.17	0.80

Phase 2: Constructing Predictive Models

7. Construct the multiple regression model with input variables $x_1$, $x_2$, $x_3$, and $x_4$.

8. Examine the impact of adding additional variables to the model.

   1. Add $x_5$ to the model. Is the addition of $x_5$ to the model significant? How was the adjusted $R^2$ impacted? What is the P-value for $x_5$?

   2. Add $x_6$ to the model. Is the addition of $x_6$ to the model significant? How was the adjusted $R^2$ impacted? What is the P-value for $x_6$?

   3. Add $x_7$ to the model. Is the addition of $x_7$ to the model significant? How was the adjusted $R^2$ impacted? What is the P-value for $x_7$?

   4. Add $x_8$ to the model. Is the addition of $x_8$ to the model significant? How was the adjusted $R^2$ impacted? What is the P-value for $x_8$?

9. Perform a hypothesis test to determine if the model is useful for predicting home values at a significance level of $\alpha =0.05$. State the P-value and interpret its meaning.

10. Are any variables not useful predictors of home price at a significance level of $\alpha =0.05$? State the P-values of these variables. Intuitively, what does this mean with respect to pricing properties?

11. State the best model for the data and justify your answer.

Phase 3: Applying and Interpreting the Model

12. Suppose you own a $2000$ square foot $2$-story house in one of the communities in the data set with $3$ bedrooms, $2.5$ baths, a pool, and it is located on a golf course, but has no dock or fenced yard. What does the model predict the price of your house to be?

13. A common term in real estate is "comparables," or "comps" for short, which are properties that have similar characteristics. It is common for realtors to look up "comps" for a certain property to get an idea of how to price it. Locate the "comps" for your home in the data set. Create a box plot of the "comps" and estimate a price range for your house on this basis.

14. What advantages and disadvantages does this approach have to the multiple regression model above?