Best The Model for Predicting Home
Prices
-A Multi-regression Analysis of homes at Hemlock Farms
Problem Set #6
BADM 5140
Dr. D
Brandon Kear
Harold Riddle
Hui Li
Nicolas Bilweis
Jed Dahl
I. Executive Summary
We were recently hired as a market analysis group by a private investor to conduct research on
what causes house prices to increase or decrease in the hemlock farms area. We have been
provided by Hemlock Farms with the following variables: list price, hot tub, lake view,
bathrooms, bedrooms, loft/den, finished basement, and acres. We examine the data using several
multiple regression methods (VIF, Best subset and stepwise methods). The best model will keep
the best independent variables predicting asking price for houses and dismiss the others. After
we evaluated all the possible combinations of independent variables, we found the most
appropriate multiple regression model to predict the asking price of house. The best model
consists of the following independent variables: number of rooms, whether or not the house has a
hot tub or not, and whether or not the house has a Lakeview or not. Therefore, using the best
model, a house with 6 rooms, a hot tub, and a lake view, will have a predictive value of
$574,514.30. We recommend the investor to refrain from extrapolation. For future research we
also recommend including other variables in the regression analysis, including personal
disposable income, unemployment rate, consumer sentiment, and the change in CPI, etc.
II. Statement of the problem
A private investor, whom requests that his /her identity remains anonymous, hired us as a market
analysis group that specializes in predicting housing prices. After providing us with data on 61
homes our investor wants us to be able to predict which variables change house prices in the
Hemlock Farms area and to what extent each variable influences the market price. We will need
to test which independent variables are significant in predicting the dependent variable using
VIF, Best subset and stepwise methods. Our goal: "to find statistically significant evidence
clarifying which variables, if any, are directly impacting market price." Once we have completed
our statistical research, we built the most appropriate multiple regression models to predict the
asking price of house. It will be given to our investor to evaluate and use as he/she deems fit.
III. Statement of research objective
The Hemlock farms have provided us information on 61 homes that were on sale for July,
2010. The single dependent variable is Home Sales Price; and the eight independent variables
include: List Price, Hot Tub, Lake View, Bathrooms, Bedrooms, Loft/Den, Finished Basement,
and Acres. Our job is to create a regression model to predict the price of home sales in Hemlock
Farms.
IV. Assumptions
We first assume the data provided by the Hemlock farms is accurate and free from
misstatements. We assume the eight variables are independent variables. We assume a linear
relationship between the dependent variable List Price and the independent variables. We also
assume normal distribution for all the independent variables. As for the errors, we assume: The
errors are independent; the errors are normally distributed; and Errors have an equal variance.
V. Methodology
VIF
We will regress all independent variables against home sales and compute the VIF for each
variable. If any of the independent variables have a VIF greater than 5, we will dismiss those
variables. Then we regress the remaining independent variables against home sales. We will
continue this process until we have eliminated all non-statistically significant independent
variables, which is found when the VIF is less than 5.
Best Subset Regression model
We will run the subset for all independent variables with a statistically significant VIF against
home sales. We will then employ the equations that have a Cp close to or less than K+1. If any
potential models have a Cp less than K+1, we will dismiss them as significant models.
Stepwise
We will then run an automated iteration against all possible regression models which will print
out the F-statistical test for each possible Regression model. We will then choose the regression
model that has the highest F-statistic as that model will be the most statistically significant
model, which is listed below:
F-stat
Based on the culmination of the three previous methods, we will determine the most statistically
significant regression model. Then we need to determine if there is a relationship between the
independent variables and the dependent variable. While the Stepwise test will employ F-stat to
determine the best model, we will look at the F-stat closer to guarantee that regression model is
statistically significant.
H0:
HA:
The overall F-stat will determine if the Beta coefficients are statistically different from 0, zero
meaning that the equation is not statistically significant. Thus we wish to reject the null and see
that at least one variable within the entire equation is statistically significant.
Adjusted R2
The Adjusted R2 explains what effect the independent variables have on the change in the
dependent variable. An R2 of .5 means that 50% of the change in the dependent variable is
caused by the culminated effects of the independent variables. That would also indicate that
50% of the change is effected by other variables not included in the model.
H0:
HA:
The null hypothesis states that the R2 is 0. If we reject the null hypothesis, it means that the R2 is
greater than zero, thus the independent variables do have an effect on the change of dependent
variable.
Standard Deviation v. Standard Error
This will determine which one of the standard error of the predicted value or the dependent
variable has the lowest variance.
H0:
HA:
The null hypothesis states that the standard deviation of the dependent variable (Home price) is
greater than the standard error of the predicted value. If this is the case, the model is useless and
we should employ a mean-variance determination for home price or need to select another
model. If we reject the null, then the standard deviation of home prices, the dependent variable,
is greater than the standard error of the model, and therefore our model will produce better
results with less risk.
Beta-Coefficients (T-tests)
We will then run a t-test against the beta-coefficients to determine that they have a linear
statistically significant relationship.
H0:
HA:
The null hypothesis would indicate that there is no linear relationship between the dependent
variable and each of the dependent variables. If we reject the null hypothesis, the betacoefficients are different from zero, meaning they may have a linear relationship. We determine
if we can reject the null hypothesis by looking at the t-values to see if they have an absolute
value greater 2.0 than and looking at the p-values to see if they are below alpha (alpha = 0.05).
We will also look at the Upper and Lower Critical Range to determine that the independent
variables do not cross zero on the y-axis.
Residuals
Finally, we will look at the residuals and determine if the residuals are normally distributed. We
will run Jarque-Bera test to determine if the residuals are normally distributed.
H0: residuals are normally distributed
HA: residuals are not normally distributed.
If we Fail to Reject the null, then the residuals are not normally distributed. If we do reject the
null, then they are normally distributed.
VI. Data Analysis Results and Conclusions
VIF
All independent variables had a VIF below 5. Thus, we could not reject any of the independent
variables with the use of VIF.
Best Subset Regression model
When we ran the best subset test, none of the CPs were less than corresponding K+1, thus we
could not reject any of the independent variables.
Stepwise
When we ran the iteration against all possible regression models, we found the following
equation was the most statistically significant, allowing us to dismiss variables: bedrooms,
bathrooms, a finished basement, acres, loft/den. Below is the designation of the beta
coefficients:
X1: Number of Rooms
X2: If the house has a hot tub
X3: If the house has a lake view.
F-stat
H0:
HA:
Because the p-value is 0.0000001, we can Reject the null with nearly 100% (99.99999%)
confidence, and conclude that this equation is statistically significant.
Adjusted R2
H0:
HA:
Because we were able to reject the null on the F-statistic, we can reject the null on the Adjusted
R2 and conclude that with 99.9999% confidence that this equation does have an effect on the
change in the dependent variable, home prices. The value of the adjusted R2 is 0.514432, which
means that 51.4432% of the change in the home prices of is caused by the combined changes in
the number rooms, whether a house as a hot tub or not, and whether it has a lake view or not.
This also means that there is a 48.5568% change in home prices that is caused by other variables
that are not included in this regression model.
Standard Deviation v. Standard Error
H0:
HA:
We found that the sample standard deviation of home prices (SY) is 133.0704 while the sample
Standard Error of our equation,
is 92.72708.
133.0704 > 92.72708, thus we reject the null hypothesis and conclude that the prediction model
will produce more accurate results with less variance than a mean-variance of home prices alone.
Beta-Coefficients (T-tests)
H0:
HA:
1 = Rooms has a coefficient of 42.6382 and has a p-value of 0.0000, thus with nearly 100%
(99.9999%) confidence, we can Reject the null and state that the coefficient of the number of
rooms is significantly different from zero.
2 = Hot tub has a beta coefficient of 90.3061 with a p-value of 0.0147, thus we Reject the null
with 98.53%, and can conclude that whether a house has a hot tub or not is a statistically
significant variable.
3 = Lake view has a coefficient of 194.6314 with a p-value of 0.0000, thus with nearly 100%
(99.9999%) confidence, we can Reject the null and state that whether a house has a lake view or
not is a statistically significant variable.
Residuals
H0: residuals are normally distributed
HA: residuals are not normally distributed.
The Jarque-Bera test is 105.0615 and has a p-value of 0.0000000, thus we Reject the null
hypothesis with nearly 100% (99.999999%) confidence, and can conclude that the residuals are
normally distributed.
Predictive Model
Based upon the results of our statistical test from above, we have determined that the following
equation is a predictive model for home sales prices.
Because the raw data is stored in thousands, we have to calculate the results of the model by
1000, hence
. The constant is 33.7475, which is the point where the
line crosses the x-axis. If a house had 0 rooms, no hot tubs, and no lake view, the base house
price would be 33.7475*1000, which is $33,747.50. 1 is $42.6382 which is multiplied by the
number of rooms in the house. Thus a house with 6 rooms will increase the value of a house by
$255,829.29. 2 is $90.3061, thus if a house has a hot tub, the value of the house will increase by
$90,306.10. 3 is $194.6314, thus if a house has a lake view, its value will increase by
$194,631.40. Thus a house with 6 rooms, a hot tub, and a lake view, will have a predicted value
of $574,514.30.
VII. Recommendations
The first recommendation we give regarding this data is to make sure that the predictive model is
not applied to houses that are not indicative of the sample houses. In addition, this model should
not be employed for houses that have less than 3.5 rooms or have more than 10.5 rooms.
In addition, because this model only explains 51.4432%, more analysis should be done to
attempt to build a model that has at least a 70% correlation. According to the Standards and
Poors Case-Shiller index researchers, there appears to be a statistical relationship between
personal disposable income and the change in home prices. Therefore, personal disposable
income could be included as an independent variable in future research. In addition, the model
should employ lagged data from the change in home prices from previous months. Some studies
have shown that current prices are responsive to the change in home prices from 2 months ago
and 6 months ago. In addition, it may also be useful to include the unemployment rate,
consumer sentiment, and the change in CPI to find the effect that these macro-economic
variables have on home prices.
Appendix
Part I-Terminology
H0 Null hypothesis
HA Alternative hypothesis
(alpha) significance level
n sample size
Proportion of population
p Proportion of Sample
VIF Variance Inflation Factor
k number of independent variable
Cp Process capability
1: (Number of Rooms) slopes if other independent variables held constant, represent the change
in the dependent variable y by 1 with 1 unit of change in x1
2: (If the house has a hot tub) slope if other independent variables held constant, represent the
change in the dependent variable y by 2 with 1 unit of change in x2
3: (If the house has a lake view) slope if other independent variables held constant, represent
the change in the dependent variable y by 3 with 1 unit of change in x3
X1: Number of Rooms
X2: If the house has a hot tub
X3: If the house has a lake view.
R2- Coefficient of determination: measures the proportion of variation in the dependent variable
due to the independent variables
sample standard deviation of the dependent variable (Home price)
- sample Standard Error of our equation
Part II: Equition produced by the best model:
Home Sales Price*1,000=$33.7475+$42.6382(#of Rooms) +$90.3061(Has Hot tub) +
$194.6314(Has Lakeview)
Part III: Excel Data Analysis Results