What Affects Property Rent at San Jose?

Title: What Affects Property Rent in San Jose?

Author(s): Yang Jiang, Ke-Ying Mao, Kuan-Hao Su, Hsuan-Cheng Lee

Class: 2nd year o f SJ SU -FCU 2+2 Bachelors Program i n Business Anal yti cs

Student IDs: D0571973, D0572056, D0565664, D0565677

Course: Introduction to Data Analytics

Instructor: Dr. Cathy W. S. Chen

Department: SJ SU-FC U 2+2 Bachel ors Program i n Business Anal yti cs Academic Year: January 12, 2017-2018

Abstract

This paper examines the various factors influencing property rent in San Jose, by using multiple regression to investigate ten variables. We then utilize forward selection, backward elimination, stepwise regression, Mallow’s Cp, and Adjust-R² to select the best model. Our findings suggest that an increase in the internal conditions of property, such as number of bedrooms and baths, has positive effects on the rent. Geographical factors also have various influences, as evidence reveals that rent declines when the property is closer to San Jose State University, whereas the overall rent for San Jose decreases from South to North. Furthermore, rent for houses located in a community is higher than that for non-community houses.

Keywords: multicollinearity; model selection; outlier detection; regression analysis; residual

Table of Contents ABSTRACT ... 1 1. INTRODUCTION ... 3 2. METHOD... 3 2.1DATA DESCRIPTION ... 3 2.2BASIC STATISTICS ...5 2.3PLOTS ...5 2.4CORRELATION MATRIX ... 11 2.5FULL MODEL ... 12

2.6OUTLIERS AND INFLUENTIAL POINT DETECTION ... 13

2.7MODEL SELECTION ... 13

2.8ADDITIONAL TREATMENT ... 15

2.9VERIFY FOUR ASSUMPTIONS ... 16

3. RESULTS ... 18

4. DISCUSSION ... 19

4.1STUDENT SAFETY ... 19

4.2AMOUNT SAVED BY LIVING OFF-CAMPUS ... 19

5. CONCLUSION ... 20

1. Introduction

Housing is one of the most important issues for students travelling abroad for studies. Although with the progress in information technology, students are able to easily obtain data from the Internet, it is still a time-consuming process for them to completely understand a foreign rent market and estimate expenditure costs. Furthermore, students in the Department of Business Data Analytics are planning to attend San Jose State University after completing their two-year study at Feng Chia University in Taiwan, and thus it would be beneficial to know where the best place is to stay under various characteristics. Only 14% of students live on-campus at San Jose State University (U.S. News & World Report, n.d.), and on-campus rooms and living areas are small in comparison to the living room, resulting in it being uneconomical for students. As a result, students may prefer to live off-campus, but may also be subject to difficulties in choosing suitable residential areas.

We acquire and collect data for this study from three different sources using web crawler (A web robot that can get data from websites). The three sources used are (i) Trulia, an online residential real estate website; (ii) Google Maps Distance Matrix Application Programming Interface (API) – an API that provides routes and distances between housing and San Jose State University; and (iii) CrimReport - a website providing the latest cases reported by the police department. Details on the collection of data are available on the GitHub (Basically a website that store code). This study thus looks to understand the factors influencing rent values of property in San Jose and highlight the most suitable off-campus areas available to rent for students attending San Jose State University.

2. Method

We first employ summary statistics and scatter plots to provide a basic understanding of the dataset. Second, we use multiple linear regression to build the model. Third, we utilize studentized residual to detect outliers and Cook’s distance to identify influential points. Lastly, we apply five different model selection methods to pick out the best model.

2.1 Data Description

Table 1. Definition of variables

CrimeReports, and Google Maps Distance Matrix API, comprising 100 observations selected from full dataset (contains 846 observations) using simple random sampling. Table 1 lists the details of the variables.

We then classify predictors into five factors. The structural factor focuses on internal conditions of the property; geographical factors reflect the housing location; attitude factors deal with permission granted by the owner to have pets and efforts made to rent out the house (efforts are observed through the variable “photoCount”; the more photos posted by the owners onto the webpage, the greater the effort is to rent out the housing); and the safety factor accounts for the level of safety of the property.

Variable Definition Source

price Price of housing in dollars Trulia (December 2017)

sqft Size of housing in square feet Trulia (December 2017)

beds Number of bedroom in the housing Trulia (December 2017)

baths Number of bathrooms in the housing Trulia (December 2017)

lat Latitude of the housing Trulia (December 2017)

lng Longitude of the housing Trulia (December 2017)

dis Walking distance between the housing and San

Jose State University

Google Maps Distance Matrix API (December 2017)

photoCount Number of photos uploaded by the owner in

the description page Trulia (December 2017)

pet Pet dummy: = 1 if renter can have pets in the

house; = 0 otherwise Trulia (December 2017)

isComm Community dummy: = 1 if housing in a

community; = 0 if otherwise Trulia (December 2017)

crim Number of crime cases around the housing

within 1.45 km²

CrimReport (November 2017 - December 2017)

discrim dis divided by crim —

Response variable

Structural

Geographical

Safety Attitude

2.2 Basic Statistics

In order to provide a basic idea of variables influencing rent in San Jose, Table 2 provides their summary statistics. The statistics include 𝜇 (mean), 𝜎 (standard deviation), minimum, 1Q (lower quantile), median, 3Q (upper quantile), and maximum. The statistics in the table show that rent per month per person in San Jose is approximately US$1,207. Furthermore, on average, 77 crimes take place around housing on a monthly basis, which is a rather high number. We calculate the square root of crim, nature log of price, disCrim, and sqft due to their large range.

Variable 𝜇 𝜎 Minimum 1Q Median 3Q Maximum

price 3140.250 1184.49 975.000 2440.000 3125.000 3647.500 9200.000 sqft 1373.960 932.465 175.000 906.000 1214.500 1531.500 9000.000 beds 2.610 1.207 1.000 2.000 3.000 3.000 9.000 baths 1.915 0.817 1.000 1.000 2.000 2.000 5.000 photoCount 13.150 9.441 0.000 7.000 12.000 20.000 58.000 lat 37.308 0.044 37.209 37.288 37.314 37.336 37.414 lng -121.886 0.061 -122.031 -121.922 -121.890 -121.844 -121.740 dis 6.265 3.179 0.300 3.850 6.850 8.750 11.900 isComm 0.080 0.273 0.000 0.000 0.000 0.000 1.000 pet 0.230 0.423 0.000 0.000 0.000 0.000 1.000 disCrim 43.126 126.925 0.093 2.916 10.252 27.462 1007.500

Table 2. Summary statistics of the variables.

2.3 Plots

Figure 1 demonstrates a positive correlation between the size of housing and price, thereby indicating that an increase in this leads to higher rent. In Figure 2, we directly remove an outlier

(marked in black circle), and Figure 3 is the scatter plot after removal of this observation. The results indicate that property rent rises as the number of bedrooms increases.

Figure 1. Scatter plot - sqft versus price.

Figure 3. Scatter plot – No. of bedrooms versus price after removing the outlier.

Figure 4 exhibits a positive correlation between the number of bathrooms and price, which indicates that a greater number of baths lead to higher rent.

Figure 5. Scatter plot - photoCount versus price.

Figure 5 shows a positive correlation between photoCount and price, indicating that photos on the description page lead to a higher price.

Figure 6. Scatter plot lat versus price. Figure 7. Scatter plot lng versus price.

Figures 6 and 7 are scatter plots demonstrating latitude versus price and longitude versus price, respectively. The data indicate no significant correlation between these two variables and price.

Figure 8 shows the relationship between distance to San Jose State University and price. The result is that being closer to campus leads to lower rent.

Figure 8. Scatter plot - distance versus price.

Figures 9 and 10 are scatter plots of two dummy variables: isComm and pet. The results indicate that housing located in a community and pet-friendly housing do not have an impact on rent.

Figure 11. Scatter plot - crim versus price.

Figure 11 demonstrate a negative correlation between the number of crimes around housing and rent. A higher number of crimes leads to lower rent.

Figure 12 shows a negative correlation between disCrim and price.

2.4 Correlation Matrix

Figure 13. Correlation matrix of variables. The red block indicates a positive correlation, while the

blue block indicates a negative correlation.

Correlation quantifies the “strength of the linear association between two variables” (Barron & Kim, 1997), and a correlation matrix demonstrates correlations between a set of variables. The correlation matrix in Figure 13 reflects that the size of the housing, the number of bedrooms, the number of bathrooms, and distance to San Jose State University have a significantly positive correlation with price, thus supporting that an increase in these components lead to higher rent. In contrast, the number of crimes around housing has a strongly negative correlation with price, thereby

large negative correlation between distance to San Jose State University and number of crimes around housing, indicating that more crimes take place around San Jose State University.

2.5 Full Model Variable Parameter estimated Standard error t Value Pr > |t| Variance inflation Intercept 5.787 0.501 11.560 <.0001 0.000 sqft 0.200 0.069 2.900 0.005 2.251 beds 0.191 0.031 6.050 <.0001 2.584 baths 0.033 0.038 0.870 0.389 2.102 photoCount 0.005 0.002 1.930 0.057 1.304 lat -0.693 0.582 -1.190 0.237 1.603 lng -0.984 0.359 -2.740 0.008 1.174 dis 0.011 0.019 0.590 0.560 8.610 isComm 0.195 0.084 2.340 0.022 1.303 pet 0.037 0.049 0.760 0.449 1.070 disCrim -0.019 0.070 -0.270 0.789 32.395 crim 0.015 0.022 0.670 0.505 16.280

Table 3. Parameter estimates for the full model.

Using the multiple linear regression, the full model is:

𝑙𝑛(𝑝𝑟𝑖𝑐𝑒̂ ) = 0.200𝑙𝑛(𝑠𝑞𝑓𝑡) + 0.191𝑏𝑒𝑑𝑠 + 0.033𝑏𝑎𝑡ℎ𝑠 + 0.05𝑝ℎ𝑜𝑡𝑜𝐶𝑜𝑢𝑛𝑡

−0.693𝑙𝑎𝑡 − 0.984𝑙𝑛𝑔 + 0.011𝑙𝑛(𝑑𝑖𝑠) + 0.195𝑖𝑠𝐶𝑜𝑚𝑚 + 0.037𝑝𝑒𝑡

−0.019𝑙𝑛(𝑑𝑖𝑠𝐶𝑟𝑖𝑚) + 0.015√𝑐𝑟𝑖𝑚, 𝜎̂ = 0.040. 2

In order to use multiple linear regression, the assumption is that the distribution of the error term is

normal and with zero mean; furthermore, the variance 𝜎2 cannot be volatile (The Pennsylvania State

University, 2018a). In this model, the value for Adjust - R² is 0.655, which means that the model can explain 65.5% of the variance. We use Adjust - R² instead of R², because it is an “unbiased estimator that corrects for the sample size and numbers of coefficients estimated” (Nau, 2017). For this model,

𝜎̂ is 0.040, and it is the estimate of 𝜎2 2_{, which “quantifies how much the responses vary around the}

mean population regression line” (The Pennsylvania State University, 2018b).

Variance inflation (VIF) can detect the existence of multicollinearity; i.e. how strong a correlation is between predictors. Multicollinearity exists if VIF is greater than ten and can lead to insignificance of the individual Bata even if R² is large (Chen, 2017a). As demonstrated in Table 3, VIFs for the variables of disCrim and crim are greater than ten, which means that multicollinearity exists in the full model. We later use variable selections to resolve this problem by selecting a suitable subset of predictors.

2.6 Outliers and Influential Point Detection

Outliers are “extreme observations in the response variables” and generally have extreme values in the response variable. In contrast, leverage points have extreme values in the independent variable. Furthermore, outliers can substantially influence regression estimates, and if the outlier values have high leverage, then they denote ‘influential points’ (Refer Freund, Wilson, & Sa, 2006). We utilized the student residual (𝑟_𝑖) to detect outliers, with an observation considered as an outlier if |𝑟𝑖| > 3. There are three outliers in the dataset, and the locations of these outliers are 55, 60,

and 78, respectively. We delete all outliers immediately upon detection and locate influential points using Cook’s distance (Cook’s D). We then define an observation as a possible influential point if Cook’s D is greater than 0.5. In the dataset, the Cook’s D for each observation is less than 0.5 and therefore indicates that there are no influential points in the dataset. While influential points do not appear in the dataset, outliers still have a significant influence on estimation. For instance, the scatter plot on studentized residuals versus fit plot shows a significant pattern prior to deleting the outlier, indicating that the residual’s variance is not constant due to the existence of outliers.

2.7 Model Selection

This research uses five different methods to select the best model: backward elimination, forward selection, stepwise regression, adjust-R² selection, and Cp selection.

Backward elimination begins with the full model, and thereafter we calculated the p-value for every predictor in the model. We remove from the model any predictor with a p-value larger than the significant level (𝛼) and the largest from this set, halting elimination only after every in-model independent variable’s p-value is smaller than 𝛼. Forward selection starts with an empty model and then computes the p-value for every predictor that is not in the model. Only the predictor with the smallest p-value and the p-value fulfilling the criterion 𝛼 can enter the model. The selection continues until there are no external independent variable p-values less than 𝛼 . Stepwise regression is a combination of backward elimination and forward selection and has two criteria: 𝛼𝑖𝑛 and 𝛼𝑜𝑢𝑡. It

starts with an empty model and uses the same procedures as in forward selection. However, after the independent variable joins the model, it re-evaluates all independent variables in the model using the backward elimination process. Stepwise regression stops when the p-values of the model’s independent variables meet both criteria (𝛼_𝑖𝑛 and 𝛼_𝑜𝑢𝑡). Finally, the Adjust - R² selection uses the model with the highest Adjust - R², while the Cp selection uses the model with the lowest Cp (Chen, 2017b).

Variable Backword Forward Stepwise Adjust - R² Cp

price sqft ✓ ✓ ✓ ✓ ✓ beds ✓ ✓ ✓ ✓ ✓ baths ✓ photoCount ✓ ✓ ✓ ✓ ✓ lat lng ✓ ✓ ✓ ✓ ✓ dis ✓ ✓ ✓ ✓ ✓ isComm ✓ ✓ ✓ ✓ ✓ pet ✓ ✓ ✓ ✓ ✓ disCrim crim

Table 4. Models based on different selection methods. The significant level for backward elimination,

forward selection, and stepwise regression is 0.15. “✓” indicates variables selected by the relevant method.

Table 4 demonstrates backward elimination, forward selection, stepwise regression, and Cp selection methods using the same model. We use the model selected by these four methods. The model is:

𝑙𝑛(𝑝𝑟𝑖𝑐𝑒̂ ) = 5.190 + 0.334𝑙𝑛(𝑠𝑞𝑓𝑡) + 0.117𝑏𝑒𝑑𝑠 + 0.003𝑝ℎ𝑜𝑡𝑜𝐶𝑜𝑢𝑛𝑡 − 0.663𝑙𝑛𝑔

+0.013𝑑𝑖𝑠 + 0.162𝑖𝑠𝐶𝑜𝑚𝑚 + 0.061𝑝𝑒𝑡, 𝜎̂ = 0.017. 2

Adjust - R² for this model is 0.790, and 𝜎̂ is 0.017. In Table 5, every variable’s VIF in this model is 2

less than 10. Therefore, multicollinearity does not exist in the selected model.

Table 5. Parameter estimates for the selected model.

2.8 Additional Treatment

The performance of the model increases significantly after the model selection, but a new outlier appears. The location of this outlier is 98 with a student residual equal to 5.045. Upon detection of this outlier, we delete it. Furthermore, we repeat model selection and use the model nominated through stepwise selection, which is:

ln(𝑝𝑟𝑖𝑐𝑒̂ ) = 4.923 + 0.389𝑙𝑛(𝑠𝑞𝑓𝑡) + 0.089𝑏𝑒𝑑𝑠 − 0.578𝑙𝑛𝑔 + 0.011𝑑𝑖𝑠 ̂ = 0.012. Variable Parameter estimated Standard error t Value Pr > |t| Variance inflation Intercept 5.190 0.286 18.160 <.0001 0.000 sqft 0.334 0.045 7.340 <.0001 2.229 beds 0.117 0.020 5.940 <.0001 2.247 photoCount 0.003 0.002 1.680 0.098 1.189 lng -0.663 0.230 -2.890 0.005 1.092 dis 0.013 0.005 2.850 0.005 1.226 isComm 0.162 0.054 3.000 0.004 1.288 pet 0.061 0.032 1.920 0.059 1.038

Variable Parameter estimated Standard error t Value Pr > |t| Variance inflation Intercept 4.923 0.245 20.120 <.0001 0.000 sqft 0.389 0.039 9.990 <.0001 2.282 beds 0.089 0.017 5.170 <.0001 2.323 lng -0.578 0.195 -2.970 0.004 1.086 dis 0.011 0.004 2.850 0.005 1.219 isComm 0.180 0.043 4.240 <.0001 1.109 pet 0.046 0.027 1.680 0.096 1.039

Table 6. Parameter estimates for the selected model after deleting the outlier.

Adjust - R² for this model is 0.832, and 𝜎̂ is 0.012. As Table 6 demonstrates, every variable’s 2

VIF for this model is still less than 10. Thus, we use the same model throughout the rest of the study.

2.9 Verify Four Assumptions

There are four assumptions for error terms: zero mean, equality of variance, independence, and normality (Bansal, n.d.).

Mean of residuals equals zero, 𝑬(𝝐𝒊) = 𝟎. The hypothesis to test this assumption is:

𝐸(𝜖_𝑖) = 0 𝐸(𝜖_𝑖) ≠ 0.

Based on the student’s t test, sign test, and signed rank test in Table 7, the p-values for these three tests are all greater than 0.05, therefore failing to reject 𝐻0. The mean of the residuals is hence equal

to zero.

Test Statistics p Value

Student’s t t = 0.851 0.397

Sign M = -0.5 1.000

Signed rank S = 123 0.650

Table 7. Test for location: 𝜇 = 0

Variance of residuals is constant, 𝑽𝒂𝒓(𝝐𝒊) = 𝝈𝟐. Figure 14 exhibits the residual plot

locate randomly around zero and do not have significant patterns. Thus, the conclusion drawn is that the variance of residuals is constant.

Figure 14. Studentized residuals versus fit plot

Order DW Pr < DW Pr > DW

1 1.744 0.088 0.913

Table 8. Durbin-Watson statistics.

Independence of residuals, 𝑪𝒐𝒗(𝝐𝒊, 𝝐𝒋) = 𝟎 ∀𝒊 ≠ 𝒋. The hypotheses for this assumption

are: 𝐻0: 𝜌 = 0 𝐻𝑎: 𝜌 > 0, and 𝐻0: 𝜌 = 0 𝐻𝑎: 𝜌 < 0

The first hypothesis is a test for positive autocorrelation, while the second one is a test for negative autocorrelation. This study utilizes the Durbin-Watson statistic (DW) to check for the existence of

autocorrelation. Here, DW is between zero and four. A value equal to two means there is no autocorrelation, while values leaning toward zero reflect negative autocorrelation and values closer to four represent positive autocorrelation. In Table 8, although DW is not equal to two, the p-value for testing positive autocorrelation (Pr < DW) is 0.088, which is greater than 0.05. Thus, it fails to reject 𝐻₀ in the first hypothesis, and the failure results in no positive autocorrelation for residuals. The p-value for testing negative autocorrelation is 0.913, which is also greater than 0.05 and thus a failure to reject 𝐻₀ in the second hypothesis results in no negative autocorrelation for residuals.

Test Statistics p Value

Shapiro-Wilk W = 0.983 0.244

Kolmogorov-Smirnov D = 0.067 >0.1500

Cramer-von Mises W-Sq = 0.085 0.181

Anderson-Darling A - Sq = 0.556 0.151

Table 9. Test for normality.

Normal distribution of residuals, 𝝐_𝒊∼ 𝑵(𝟎, 𝝈𝟐_{). The hypothesis to test this assumption is:}

𝐻₀: 𝜖_𝑖 ∼ 𝑁(0, 𝜎2₎

𝐻𝑎: 𝜖𝑖 ≁ 𝑁(0, 𝜎2)

The four tests to calculate the p-value demonstrated in Table 9 reveal values that are all greater than 0.05 and therefore do not reject 𝐻₀, indicating that the distribution of the residual is normal.

3. Results

The dataset contains five outliers and has no influential point. Variables in the final model are all significant at the 0.05 level, except for the variable dealing with pet-friendly housing.

The results suggest that, in addition to structural factors, geographical and safety factors also influence a property’s rent. Houses within a community have a higher rent, and an increase in the following variables can also increase the rent: size of the house; number of bedrooms in the house; and distance between the housing to San Jose State University. However, in contrast, an increase in the longitude can decrease the rent value for the housing.

4. Discussion 4.1 Student Safety

Although the variable ‘crime’ is not selected for the final model, it is still an important factor that students need to be concerned about. Figure 14 is the scatter plot demonstrating ‘distance versus crimes’ for a total of 846 observations. As the plot shows, crime rates are fairly high in the areas located between 0 to 2.5 miles away from San Jose State University. Therefore, students may want to find rental property located 2.5 miles away from the campus in order to have a safer environment.

Figure 15. Distance versus crime plots.

4.2 Amount Saved by Living Off-campus

There are four dormitories in San Jose State University: building A, building B, building C, and building 2 (San Jose State University, 2017). In Figure 16, the y-axis is the average savings per month for students who live off-campus. These values are calculated as the average cost of living in a dormitory (without a meal plan) per month minus the average cost of living off-campus per month. On average, a student can save US$1,000 by living off campus for seven months. However, for those students who really want to save money, building 2 is the perfect option, as living in this building is more economical than living off-campus.

It is necessary to highlight that the rental housing market changes over time, and these estimates can be different in the future. Furthermore, the number of crimes around housing may fluctuate across different seasons.

Figure 16. Average savings per month by living off-campus.

5. Conclusion

Various factors influence rental property in San Jose, including those that are structural, geographical, attitudinal, and safety. The fact that the number of crimes is not accounted for in the final model is a surprising observation, but due to the rapid nature of changes in San Jose, it does not significantly affect rent. The following conclusion offers certain suggestions for landlords and students.

In the case of property owners in San Jose, there is a high opportunity cost associated with placing that focuses on uploading images of housing to the relevant description page, because this action does not have a significant influence on rent. Since it is almost impossible for owners to change the geographic factor of the property, one recommendation is for them to add a reasonable number of bedrooms to increase rent. Moreover, allowing tenants to have pets in the house can also facilitate higher rents for landlords.

-93.75 0 93.75 187.5 281.25

Building A Building B Building C Building 2

A ve ra ge s avi ngs pe r m ont h

In the case of students, in most situations it is cheaper to live off-campus than to live on-campus. Rent is lower on the north side of San Jose State University in comparison to properties to the south of campus. Furthermore, the number of crimes holds at a steady rate in areas located 2.5 miles away from campus. As Figure 17 demonstrates, one recommendation for students who want to live off-campus is to live north of the university and at least 2.5 miles away from it, as the rent is relatively cheap and the surroundings are safe.

Figure 17. Recommended rent area (blue area). Red area indicates high crime area, while yellow area

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