Report on the Factors influencing house prices in Ames, Iowa.

Name: Michelle O’ Regan
Student number: 114462288
Degree: BSc Finance. Second Year
Word Count: 1822 (not including appendix)
Submission Date: 14th April, 2016

Introduction
Throughout this report I endeavour to present a clear, concise documentation of the factors that influence house prices in Ames, Iowa. I will initiate this report with my estimate of the possible regression based on my economic theory, create a dummy variable in respect to the absence/presence of a garage, followed by a comprehensive description of continuous and discrete variables. Preceding this I aim to report an extensive description of my estimated regression, test said regression for multicollinearity and heteroscedasticity, predict possible solutions to these problems and re run the regression taking into consideration the possible solutions.
Main Body
Part (a)
From my study of econometrics and my knowledge of house prices, the following equation is my estimate of the factors that influence the price of houses PR= f (SI, YD, GA, lnAGE) + + + -
(see appendix 1.1 for variable details)
My reasoning for the inclusion of the above variables and their predicted signs are as follows:
SI: Generally speaking, the larger the home the more you pay as house buyers like to buy houses with as much space as possible. I believe there would be a positive relationship between the price and the size of a house in square feet.
YD: I believe that yard size is positively related to the cost of a house. Homeowners view yard size as an important aspect when choosing which house to purchase, especially those with/planning to have children.
GA: a garage is a pivotal aspect of a houses price as regarded by many people when looking to buy a house. Having a garage leaves your car less open to theft and may even lead to lower insurance premiums. I predicted the sign of this coefficient would be positive with the idea that as the number of cars you can occupy in the garage increases, so does the associated price. lAGE: I included the log of the variable age as I believe the older the house, the less the price of the house will be, but I believe the price decreases at a diminishing rate. I predicted a negative sign as theory and research would leave me to believe that the newer a house is, the more it will cost.
My reasoning for not including COND as part of my regression is that I believe AGE and COND are very highly correlated ie. The older the house the worse the condition, and therefore the results would not be as accurate as I would like.

Part (B)
“A dummy variable is a variable that takes on the values 1 and 0; 1 means something is true” (stata.com). * generate garage=0 * replace garage=1 if GA>=1

The above diagram is a scatter plot from Stata, in which I plotted the dummy variable “garage” against the dependant variable “Price”. This is a visual representation of the dummy variable being correctly made as the only values this variable can take on is zero and one. (See appendix 3.0)
Part ( C )
To describe the continuous variables I entered the command “summarize” into Stata followed by each of the variables separately and received the below results;
Size Variable Variable | Mean | Std.Dev | Min | Max | SI | 1509.331 | 511.9133 | 334 | 4476 |

Variable | Mean | Std.Dev | Min | Max | AGE | 36.62528 | 30.25181 | 0 | 136 |

Variable | Mean | Std.Dev | Min | max | YD | 10064.84 | 8526.558 | 1526 | 215245 |
For size the standard deviation is quite high and the mean vale of the 878 observations is large in respect of the minimum and maximum values illustrating that there is a large variance among this data set.
For the age variable the mean and standard deviations are much lower as the minimum and maximum values have much less variance, in contrast to size. This is quite realistic as the size of a house is much more likely to have a wide distribution, in contrast to age.
As regards to the yard size, the standard deviation and mean are extremely large, showing a greater variance among the data. This, again, makes sense theoretically as many houses many have an extremely small yard and others may have a very large garden- depending on their location.

For the discrete variable in my regression, GA, I entered the command “tab GA” into Stata and received the following output.

Garage Capacity (Number of Cars) | Frequency | Percent (%) | Cumulative Percent | 0 | 45 | 5.13 | 5.13 | 1 | 230 | 26.20 | 31.32 | 2 | 499 | 56.83 | 88.15 | 3 | 100 | 11.39 | 99.54 | 4 | 4 | 0.46 | 100.00 | Total | 878 | 100.00 | |
The output data and the histogram generated below show how the data is disturbed among the 878 observations. As can be seen from the table and graph the most prominent number of cars i.e. garage capacity, is 2 cars. A garage that can hold 2 cars holds a 56.83% of all the observations in the data set, in comparison to the least reoccurring number of 4 which only has a 0.46% of all observations. (See appendix 2.0&2.1)

Part (D) generate lnAGE=ln(AGE) regress PR SI YD GA lnAGE (See Appendix 4.0) Results: PR= 66.51363+.0899021SI+.0010049YD+16.82128GA-20.49697lnAGE (.0033592) (.0001824) (2.672804) (1.36098) t= 26.76 5.51 6.29 -15.06 N=878 Adjusted R2 =. 7166 All my chosen variables were statistically significant, as can be seen by all the t-scores above an absolute value of 2. All of the signs on the coefficients were as expected, with the lnAGE being the only negative. I was slightly surprised by the coefficients on both the YD and GA (a one unit increase in YD and GA cause an increase in price by $1 and $16000 respectively-holding constant all other variables). In my opinion YD is more theoretically important than the amount of cars you can fit in your garage when computing price, so I will attempt to remedy this later in my report. (See appendix 4.1) Part (e ) Test for Multicollinearity One informal method for checking for severe multicollinearity is investigating whether the regression has a very high R2 and low t-scores. In my regression, Stata generated a R2 value of .7179 and t-scores that range from 5.51 to 26.76. Although 0.7179 is a relatively high value for R2, all of the aforementioned t-scores can be categorized as relatively high figures, indicating informally that severe multicollinearity is not a problem in this equation. VIF test: * Quietly regress PR SI YD GA lnAGE * Display 1/(1-e(r2)) Result: 3.545087. This value illustrates that some degree of multicollinearity is present but as the value is under 5, it is not severe multicollinearity. (See appendix 5.1) * Regress PR SI YD GA lnAGE * Vif Results: Variable | VIF | 1/VIF | GA | 1.69 | 0.591629 | lnAGE | 1.44 | 0.696774 | SI | 1.32 | 0.754880 | YD | 1.11 | 0.903975 | Mean VIF: 1.39 As can be seen from above, none of the variables illustrate any signs of such a problem and suggest that multicollinearity is not a problem for this regression. (See appendix 5.2) Test for heteroscedasticity An informal method for testing for the aforementioned problem is by observing the pattern of the data on a scatter plot. The observations were as follows: The pattern of the data clearly illustrates an absence of homoscedasticity and the presence of heteroscedasticity as the variance of the data is not constant ie. As the values increases so does the variance. (See appendix 6.0) Breusch-Pagan / Cook-Weisberg test for heteroskedasticity
A formal method of testing for non-constant variance in a regression is by using the Breusch-Pagan test. Ho: Constant variance Variables: fitted values of PR chi2(1) = 714.62 Prob > chi2 = 0.0000
As can be seen from the above results, the null hypotheses of a constant variance must be rejected as the probability value does not exceed 5%- illustrating that heteroscedasticity is present in the regression. (See appendix 6.1)
White’s test for heteroscedasticity
In contrast to the Breusch-Pagan test which checks for constant variance through a hypothesis test, the White test checks for homoscedasticity as the null hypothesis. Ho: homoscedasticity against Ha: unrestricted heteroscedasticity chi2(14) = 322.87 Prob > chi2 = 0.0000
The above hypothesis test clearly shows the absence of homoscedasticity, and hence constant variance as the null hypothesis must be rejected as the probability is under 5%. (See appendix 6.2)
Part (F)
As heteroscedasticity is present in my regression I would recommend using the “robust” option while regressing in Stata. Heteroscedasticity influences the standard errors of an equation and not the coefficients so it is logical to use the “robust” command as it attempts to correct the errors caused by heteroscedasticity.
Part (G)
Through the application of the remedy for heteroscedasticity, I predict higher standard errors and lower t-scores (as the presence of heteroscedasticity underestimates standard errors and over estimates t-scores). As the coefficient on the YD variable is so low it would lead me to wonder whether I should log said variable, however I would not be able to log both AGE and YD as a double log can only be applied if both the coefficients have positive signs. As I believe yard size is a much more important aspect in determining the price than age and most certainly could have a diminishing effect on price , I suggest dropping the log on AGE and applying a log to YD . I predict that this would increase the coefficient on the variable YD while also decreasing it’s standard error.
Part (H)
I believe that the regression with the modifications specified in part H would deliver a more promising regression as it would hopefully help reduce the presence of heteroscedasticity, while almost increasing the coefficient on the variable YD- hence I chose to use this regression rather than my initial.
Part (I) Firstly, from applying the “robust” command on Stata and generating an updated rvfplot, a reduction in heteroscedasticity can be seen. (See appendix 8.0)

Rvfplot before applying robust command Rvfplot after robust command. As can be seen from comparing the two above graphs, there is a reduction in the variance. Also at a 95% confidence interval, all the standard errors increased bar that of GA and all the t-scores decreased bar tat again of GA which shows that some of the heteroscedasticity was removed from the regression- it would also lead me to believe that GA is the main cause of heteroscedasticity in the regression. (See appendix 9.0) I then decided it would appropriate to drop the log on AGE and apply a log to YD instead. The results of said regression are as follows: PR=110.4336+.0913386SI+.0011024lnYD+17.33099GA-.9180941AGE (.0069104) (3.370892) (2.673108) (.0538493) t= 13.22 5.13 6.48 -17.05 N= 878 R2= 0.7183
All of the above variables may be statistically significant but contrary to my predictions logging yard size only decreased the standard error and only made the slightest of changes to the coefficient. Also, dropping the log on AGE decreased the coefficient hugely (a one year increase in how old the house is only decreases the price by 918$ which would lead to believe my first regression was more accurate. The R2 values of both regressions are very similar, as are the standard errors and t-scores.
An additional squared foot of the size of the house increases the price by approximately $91- holding other variables constant; an additional 1000square foot of yard size outside the house would cause house prices to rise by $1.1, all other variables constant and for every extra car you can fit in the garage, holding all else constant, price increases by approximately $17330, according to this regression. Aside for the SI variable, these assumptions may be unrealistic.

APPENDIX
1.0 PR = the price (in thousands of dollars)
SI = size of the house (in square feet)
YD = size of the yard around the house (in 1000s of square feet)
GA = capacity of the garage (measured in the number of cars the garage can accommodate) lAGE = the log of the age of the house (in years)

1.1. version 11.0 capture log close set more off clear cd C:\Users\114462288\Desktop\ec2206
C:\Users\114462288\Desktop\ec2206
log using lab2.log, replace
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(note: file C:\Users\114462288\Desktop\ec2206\lab2.log not found) name: <unnamed> log: C:\Users\114462288\Desktop\ec2206\lab2.log log type: text opened on: 12 Apr 2016, 08:45:51 use housedata63

2.0. summarize SI AGE YD Variable | Obs Mean Std. Dev. Min Max
-------------+--------------------------------------------------------
SI | 878 1509.331 511.9133 334 4476 AGE | 878 36.62528 30.25181 0 136 YD | 878 10064.84 8526.558 1526 215245
2.1 tab GA garage | capacity | (number of | cars) | Freq. Percent Cum.
------------+-----------------------------------
0 | 45 5.13 5.13 1 | 230 26.20 31.32 2 | 499 56.83 88.15 3 | 100 11.39 99.54 4 | 4 0.46 100.00
------------+-----------------------------------
Total | 878 100.00
Hist GA
3.0 generate garage =0 replace garage=1 if GA>=1
(833 real changes made) generate lnAGE=ln(AGE)
(34 missing values generated)
4.0 label variable lnAGE "Natural Log of AGE"
4.1 regress PR SI YD GA lnAGE
Source | SS df MS Number of obs = 844
-------------+------------------------------ F( 4, 839) = 533.83 Model | 4070448.2 4 1017612.05 Prob > F = 0.0000 Residual | 1599335.58 839 1906.24026 R-squared = 0.7179
-------------+------------------------------ Adj R-squared = 0.7166 Total | 5669783.79 843 6725.72217 Root MSE = 43.661
------------------------------------------------------------------------------
PR | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
SI | .0899021 .0033592 26.76 0.000 .0833087 .0964956 YD | .0010049 .0001824 5.51 0.000 .000647 .0013629 GA | 16.82128 2.672804 6.29 0.000 11.57511 22.06745 lnAGE | -20.49697 1.36098 -15.06 0.000 -23.16829 -17.82564 _cons | 66.51363 8.108385 8.20 0.000 50.59853 82.42874
------------------------------------------------------------------------------
5.0 correlate PR SI YD GA lnAGE
(obs=844)
| PR SI YD GA lnAGE
-------------+---------------------------------------------
PR | 1.0000 SI | 0.7406 1.0000 YD | 0.2784 0.2741 1.0000 GA | 0.6012 0.4521 0.1690 1.0000 lnAGE | -0.5436 -0.2384 0.0244 -0.5377 1.0000
5.1 quietly regress PR SI YD GA lnAGE display 1/(1-e(r2))
3.545087
5.2 quietly regress PR SI YD GA lnAGE vif Variable | VIF 1/VIF
-------------+----------------------
GA | 1.69 0.591629 lnAGE | 1.44 0.696774 SI | 1.32 0.754880 YD | 1.11 0.903975
-------------+----------------------
Mean VIF | 1.39
5.3 regress PR SI YD GA lnAGE
Source | SS df MS Number of obs = 844
-------------+------------------------------ F( 4, 839) = 533.83 Model | 4070448.2 4 1017612.05 Prob > F = 0.0000 Residual | 1599335.58 839 1906.24026 R-squared = 0.7179
-------------+------------------------------ Adj R-squared = 0.7166 Total | 5669783.79 843 6725.72217 Root MSE = 43.661
------------------------------------------------------------------------------
PR | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
SI | .0899021 .0033592 26.76 0.000 .0833087 .0964956 YD | .0010049 .0001824 5.51 0.000 .000647 .0013629 GA | 16.82128 2.672804 6.29 0.000 11.57511 22.06745 lnAGE | -20.49697 1.36098 -15.06 0.000 -23.16829 -17.82564 _cons | 66.51363 8.108385 8.20 0.000 50.59853 82.4287
5.4 vif
Variable | VIF 1/VIF GA | 1.69 0.591629 lnAGE | 1.44 0.696774 SI | 1.32 0.754880 YD | 1.11 0.903975
-------------+----------------------
Mean VIF | 1.39 6.0 rvfplot
6.1 quietly regress PR SI YD GA lnAGE
. hettest
Breusch-Pagan / Cook-Weisberg test for heteroskedasticity Ho: Constant variance Variables: fitted values of PR chi2(1) = 714.62 Prob > chi2 = 0.0000
6.2 estat imtest,white
White's test for Ho: homoskedasticity against Ha: unrestricted heteroscedasticity chi2(14) = 322.87 Prob > chi2 = 0.0000
Cameron & Trivedi's decomposition of IM-test
---------------------------------------------------
Source | chi2 df p
---------------------+-----------------------------
Heteroskedasticity | 322.87 14 0.0000 Skewness | 91.11 4 0.0000 Kurtosis | 6.58 1 0.0103
---------------------+-----------------------------
Total | 420.56 19 0.0000
6.3 ovtest
Ramsey RESET test using powers of the fitted values of PR Ho: model has no omitted variables F(3, 836) = 90.04 Prob > F = 0.0000
7.0 regress PR SI GA YD lnAGE, robust
Linear regression Number of obs = 844 F( 4, 839) = 204.87 Prob > F = 0.0000 R-squared = 0.7179 Root MSE = 43.661 | Robust PR | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
SI | .0899021 .0072013 12.48 0.000 .0757675 .1040367 GA | 16.82128 2.540813 6.62 0.000 11.83418 21.80838 YD | .0010049 .0002257 4.45 0.000 .000562 .0014479 lnAGE | -20.49697 1.531049 -13.39 0.000 -23.50211 -17.49183 _cons | 66.51363 12.30974 5.40 0.000 42.35214 90.67513
8.0 rvfplot
8.1 estat imtest, white
White's test for Ho: homoskedasticity against Ha: unrestricted heteroscedasticity chi2(14) = 322.87 Prob > chi2 = 0.0000
Cameron & Trivedi's decomposition of IM-test Source | chi2 df p
---------------------+-----------------------------
Heteroskedasticity | 322.87 14 0.0000 Skewness | 91.11 4 0.0000 Kurtosis | 6.58 1 0.0103
---------------------+-----------------------------
Total | 420.56 19 0.0000
9.0 regress PR SI YD garage lnAGE
Source | SS df MS Number of obs = 844
-------------+------------------------------ F( 4, 839) = 510.57 Model | 4018810.43 4 1004702.61 Prob > F = 0.0000 Residual | 1650973.35 839 1967.78707 R-squared = 0.7088
-------------+------------------------------ Adj R-squared = 0.7074 Total | 5669783.79 843 6725.72217 Root MSE = 44.36 PR | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
SI | .096513 .0031999 30.16 0.000 .0902323 .1027937 YD | .0011024 .0001843 5.98 0.000 .0007405 .0014642 garage | 24.18087 6.943569 3.48 0.001 10.55207 37.80968 lnAGE | -24.20176 1.207036 -20.05 0.000 -26.57092 -21.83259
-------------------------------------------------
_cons | 73.25884 9.407592 7.79 0.000 54.79366 91.72401
.
10.0 generate lnYD=ln(YD) label variable lnYD "Natural Log of YD"

10.1 regress PR SI lnYD GA AGE, robust

Linear regression Number of obs = 878 F( 4, 873) = 236.10 Prob > F = 0.0000 R-squared = 0.7183 Root MSE = 44.835

------------------------------------------------------------------------------ | Robust PR | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
SI | .0913386 .0069104 13.22 0.000 .0777758 .1049015 lnYD | 17.27924 3.370892 5.13 0.000 10.66324 23.89524 GA | 17.33099 2.673108 6.48 0.000 12.08452 22.57745 AGE | -.9180941 .0538493 -17.05 0.000 -1.023783 -.812405 _cons | 110.4336 28.85213 3.83 0.000 167.0612 -53.80593

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Reference List
Internet
Stata.com: “FAQ creating a dummy variable”: http://www.stata.com/support/faqs/data-management/creating-dummy-variables/ .