Introduction

• Real estate investments are notoriously risky.
• However many investors are still willing to take the chance in hopes of a lurcative profit.
• In this presentation, we analyse the various factors that decides the price value for residential properties.
• We aim to find the feature that proves to be the deciding factor to make a profitable investment.

Problem Statement

In order to find the deciding factor for a profitable investment on residential properties, We determined the relationship between the price of a house and other factors using linear regression

Data

To guide our investigation, We took the “Real estate valuation” data set from UCI repository. The data set consists of real estate valuation collected from Sindian District, New Taipei City, Taiwan during 2012 to 2013.

The available features in the data set are

• Transaction date (example: 2013.500= 2013 June)
• House age (unit: year)
• the distance to the nearest MRT station (unit: meter)
• the number of convenience stores in the living circle on foot (integer)
• Geographic coordinate, latitude. (unit: degree)
• Geographic coordinate, longitude. (unit: degree)
• House price of unit area (10000 New Taiwan Dollar/Ping, where Ping is a local unit, 1 Ping = 3.3 meter squared)

During data preparation, we

• Renamed the column names the appropriately
• Checked for duplicates and null values
• Removed ‘Transaction date’ column since, all dates fall between year 2012 to 2013 without any comparison between prices on different periods.

Visualization and Decsriptive Statistics

Price distribution

Below plots represents the price distribution on houses and check for outliers. According to the distribution most houses were priced between 35 to 45

RealEstate$House_price_of_unit_area %>% hist(main="Price distribution of houses", col= "blue", breaks = 20)
grid()

boxplot(RealEstate$House_price_of_unit_area,main = "Box plot of House prices", ylab="Prices")
grid()

House age distribution

Below plots represents the age distribution on houses and check for outliers. The maximum age of the house from the data set is 40 and most number of houses are aged between 10 and 20

RealEstate$House_age %>% hist(main="Age distribution of houses", col= "yellow", breaks = 20)
grid()

boxplot(RealEstate$House_age,main = "Box plot of age of houses", ylab="age")
grid()

Distribution of the distance to the nearest MRT station

Below plots represents the distance to the nearest MRT station distribution on houses and check for outliers.

• The maximum amount of house properties were situated closer to the MRT station.
• Few outliers were observed. We chose to retain the outliers for data preservation.

RealEstate$Distance_to_the_nearest_MRT_station %>% hist(main="Distribution of the distance to the nearest MRT", col= "green", breaks = 20)
grid()

boxplot(RealEstate$Distance_to_the_nearest_MRT_station ,main = "Box plot of distance to the nearest MRT", ylab="Distance to the nearest MRT")
grid()

Distribution of the number of convienience stores

Below plots represents the number of convienience store closer to houses and check for outliers.

RealEstate$Number_of_convenience_stores %>% hist(main="Distribution of the number of convienience stores", col= "red", breaks = 10)
grid()

boxplot(RealEstate$Number_of_convenience_stores,main = "Box plot of number of convenient stores", ylab="No.of convenient stores")
grid()

Comparison 1 continued

# Fitting a Linear regression model
PriceOnAge <- lm(House_price_of_unit_area ~ House_age, data = RealEstate)
PriceOnAge %>% summary()

## 
## Call:
## lm(formula = House_price_of_unit_area ~ House_age, data = RealEstate)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -30.895 -10.502   1.837   8.227  45.213 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 42.06794    1.16255  36.186  < 2e-16 ***
## House_age   -0.24142    0.05517  -4.376 1.54e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 12.77 on 411 degrees of freedom
## Multiple R-squared:  0.04451,    Adjusted R-squared:  0.04219 
## F-statistic: 19.15 on 1 and 411 DF,  p-value: 1.536e-05

PriceOnAge %>% confint()

##                  2.5 %     97.5 %
## (Intercept) 39.7826466 44.3532305
## House_age   -0.3498796 -0.1329695

Comparison 1 continued

# Validating the assumptions
# 1.Homoscedastisity by Residuals vs. Fitted plot
PriceOnAge %>% plot(which = 1, main="Homoscedastisity by Residuals vs. Fitted plot")
grid()

# 2.Normality distribution of Residuals by Q-Q plot
PriceOnAge %>% plot(which = 2, main="Normality distribution of Residuals by Q-Q plot")
grid()

Comparison 1 : Summary

• The below regression model cannot be declared as a best fitted model due to the low R2 value.
• Only 4.2% of the variability in price of house can be explained by a linear relationship with its age.
• However the overall regression model and its coefficients are still statistically significant according to the p value and the confident intervals.
• The model slightly supports the popular fact that modern houses are expensive when compared to old houses.
• The Residuals vs Fitted plot shows no pattern in variability, and hence Homoscedasticity can be assured. Also the trend line is almost flat, thus there is no sign of non-linearity.
• According to the Q-Q plot the residuals fall close to the line. No obvious departures assures normality of the residuals.

Comparison 2 : Price with distance to the nearest MRT station

Hypothesis:

Null Hypothesis ( \(H_0\)): The data does not fit the linear regression model ; Alternative Hypothesis (\(H_1\)): The data fits the linear regression model

plot (House_price_of_unit_area ~ Distance_to_the_nearest_MRT_station, data = RealEstate, main="Comparison of price with distance to the nearest MRT station")

The scatter plot of house prices vs distance to the nearest MRT shows an exponential relationship. Hence both of the attributes were converted in to their log values before constructing the linear regression model.

Comparison 2 continued

Scatter plot for the log convertion of house price and Distance to the nearest MRT is approximately linear and it shows a negative relationship among the two variables.

par(mfrow=c(2,2))
RealEstate$House_price_of_unit_area %>% hist(main = "Price")
log(RealEstate$House_price_of_unit_area) %>% hist(main = "log(Price)")
RealEstate$Distance_to_the_nearest_MRT_station %>% hist(main = "Distance to MRT")
log(RealEstate$Distance_to_the_nearest_MRT_station) %>% hist(main = "log(Distance to MRT)")

plot (log(House_price_of_unit_area) ~ log(Distance_to_the_nearest_MRT_station), data = RealEstate)

Comparison 2 continued

# Fitting a Linear regression model
PriceOnDistance <- lm(log(House_price_of_unit_area) ~ log(Distance_to_the_nearest_MRT_station), data = RealEstate)
PriceOnDistance %>% summary()

## 
## Call:
## lm(formula = log(House_price_of_unit_area) ~ log(Distance_to_the_nearest_MRT_station), 
##     data = RealEstate)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1.64679 -0.12144  0.01408  0.14479  0.73385 
## 
## Coefficients:
##                                          Estimate Std. Error t value
## (Intercept)                               5.25862    0.07158   73.47
## log(Distance_to_the_nearest_MRT_station) -0.26507    0.01103  -24.04
##                                          Pr(>|t|)    
## (Intercept)                                <2e-16 ***
## log(Distance_to_the_nearest_MRT_station)   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.2508 on 411 degrees of freedom
## Multiple R-squared:  0.5843, Adjusted R-squared:  0.5833 
## F-statistic: 577.8 on 1 and 411 DF,  p-value: < 2.2e-16

PriceOnDistance %>% confint()

##                                              2.5 %     97.5 %
## (Intercept)                               5.117916  5.3993297
## log(Distance_to_the_nearest_MRT_station) -0.286752 -0.2433974

Comparison 3 : Price with number of convenience stores

Null Hypothesis ( \(H_0\)): The data does not fit the linear regression model ; Alternative Hypothesis (\(H_1\)): The data fits the linear regression model

plot (House_price_of_unit_area ~ Number_of_convenience_stores, data = RealEstate, main="comparison of price with number of convenience stores")

# Fitting a Linear regression model
PriceOnNoOfStores <- lm(House_price_of_unit_area ~ Number_of_convenience_stores, data = RealEstate)

Comparison 3: continued

PriceOnNoOfStores %>% summary()

## 
## Call:
## lm(formula = House_price_of_unit_area ~ Number_of_convenience_stores, 
##     data = RealEstate)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -35.339  -7.098  -1.398   6.002  30.661 
## 
## Coefficients:
##                              Estimate Std. Error t value Pr(>|t|)    
## (Intercept)                   26.6567     0.8717   30.58   <2e-16 ***
## Number_of_convenience_stores   2.7138     0.1727   15.71   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 10.32 on 411 degrees of freedom
## Multiple R-squared:  0.3753, Adjusted R-squared:  0.3738 
## F-statistic: 246.9 on 1 and 411 DF,  p-value: < 2.2e-16

PriceOnNoOfStores %>% confint()

##                                  2.5 %    97.5 %
## (Intercept)                  24.943184 28.370141
## Number_of_convenience_stores  2.374281  3.053226

Comparison 3: continued..

assumption homoscedastisity. Thus this model should not come in to the consideration even though it is statistically significant.

# Validating the assumptions
# 1.Homoscedastisity by Residuals vs. Fitted plot
PriceOnNoOfStores %>% plot(which = 1, main="Homoscedastisity by Residuals vs. Fitted plot")

# 2.Normality distribution of Residuals by Q-Q plot
PriceOnNoOfStores %>% plot(which = 2, main="Normality distribution of Residuals by Q-Q plot")

Discussion

Based on liner regession it can be concluded that

• Only 4.2% of the variability in price of house can be explained by a linear relationship with its age.
• 58% of the variability due to the nearest MRT station distance.
• A successfull Simple linear regression model could not be generated to explain the price of house with the predictor, number of convinience stores surrounding the house due to the presence of heteroscedasticity.

Hence to buy a profitable house in Sindian District, New Taipei City, Taiwan, the house must be situated closer to the MRT station.

References

• [1] Data Source: https://archive.ics.uci.edu/ml/datasets/Real+estate+valuation+data+set

• [2] Yeh, I. C., & Hsu, T. K. (2018). Building real estate valuation models with comparative approach through case-based reasoning. Applied Soft Computing, 65, 260-271.