Salt Lake City Real Estate
We will be analyzing SLC real estate data. The dataset contains multiple listing service (MLS) real estate transactions for houses sold in 2015-16 in zip code 84103 (SLC avenues neighborhood). We are primarily interested in regressing the SoldPrice on the house attributes (property size, house size, number of bedrooms, etc...).
Data Overview
There are 206 different variables associated with each of the 348 houses in this dataset. There is some information about real estate data on this website to get a better sense of the meanings of the column headers in the dataset.
We are only keeping houses with a list price between 200,000 and 1,000,000 dollars. This is an arbitrary choice, and we realize that some people are high rollers, but for our purposes, we will consider the others as outliers.
We will be keeping these variables:
['Acres', 'Deck', 'GaragCap', 'Latitude', 'Longitude', 'LstPrice', 'Patio', 'PkgSpacs']['PropType', 'SoldPrice', 'Taxes', 'TotBed', 'TotBth', 'TotSqf', 'YearBlt']We will check the datatypes and convert any numbers that were read as strings to numerical values. In particular, we converted
TotSqfto an integer and add a column titledProp_Type_num(Prop) that is:
$$ Prop_i = \begin{cases} 0, & \text{condo or townhouse} \\ 1, & \text{single family house} \end{cases} $$
- We removed the listings with erroneous
Longitude(one has Longitude = 0) andTaxesvalues (two have unreasonably large values).
Exploratory Data Analysis
Python Code:
import seaborn as sns
rel = sns.pairplot(data = train_subset, hue="PropType", diag_kws={'bw': 0.2})
In our exploratory analysis, there are 260 observations and 18 variables in the training dataset. Based on the correlation scatterplot, the columns with the strongest correlation are the LstPrice and the SoldPrice. The plot is almost a linear line. Additionally, there are more than own a Single Family than any other Property Type.
Geospatial Analysis
Geospatial data defines specific geographical locations, either in the form of latitude and longitude coordinates or text fields with names of geographical areas, such as countries or states. Geospatial charts combine geospatial data with other forms of data to create map-based charts.
Two of the variables are the latitude and longitude of each listing. Salt Lake City is on a east-west, north-south grid, so even a simple plot of latitude and longitude makes sense. Thus, we will be creating a scatterplot of these two variables.
Python Code:
rel2 = sns.scatterplot(train['Latitude'], train['Longitude'], hue = train['LstPrice'], palette = 'hot')
As we can see from the chart, the bigger the longitude and latitude, the higher the price is for the house. Houses closer to (-111.91, 40.770) are cheaper in price.
Linear Regression
Ordinary Least Squares Assumptions:
- Standard Errors assume that the covariance matrix of the errors is correctly specified.
- The condition number is large, 1.22e+03. This might indicate that there are strong multicollinearity or other numerical problems.
- The linear regression model is “linear in parameters.”
- There is a random sampling of observations.
- There is homoscedasticity and no autocorrelation.
We will now create a linear regression model to regress the Sold price on some of the other variables.
$$ SoldPrice = β_0 + β_1x $$where x is one of the other variables.
After we run the regression, we will make a scatterplot of list price vs sold price and overlay the prediction with the regression model.
Python Code:
# Regression Model
uni_mod = sm.ols(formula="SoldPrice ~ LstPrice", data = train)
uni_result = uni_mod.fit()
print(uni_result.summary())
Output:
OLS Regression Results
==============================================================================
Dep. Variable: SoldPrice R-squared: 0.990
Model: OLS Adj. R-squared: 0.990
Method: Least Squares F-statistic: 2.611e+04
Date: Thu, 21 Jul 2022 Prob (F-statistic): 2.97e-261
Time: 13:50:46 Log-Likelihood: -2927.2
No. Observations: 260 AIC: 5858.
Df Residuals: 258 BIC: 5866.
Df Model: 1
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [0.025 0.975]
------------------------------------------------------------------------------
Intercept 1.317e+04 2948.953 4.465 0.000 7361.378 1.9e+04
LstPrice 0.9441 0.006 161.593 0.000 0.933 0.956
==============================================================================
Omnibus: 123.259 Durbin-Watson: 2.015
Prob(Omnibus): 0.000 Jarque-Bera (JB): 881.446
Skew: -1.750 Prob(JB): 3.95e-192
Kurtosis: 11.313 Cond. No. 1.27e+06
==============================================================================
Based on the OLS Regression, the R-squared is 0.988. This is saying that 98.8% of the variation is explained in the model. The Beta one of LstPrice is 0.9019. In other words, as LstPrice goes up by 1, SoldPrice increases by 0.9019 points.
Python Code:
plt.scatter(train['LstPrice'], train['SoldPrice'], color='red')
Multilinear Regression
We are now developing a multilinear regression model for house prices in this neighborhood. We could use this to come up with a list price for houses coming on the market, so will not be including the list price in our model and also ignore the categorical variable Prop_Type. Our model is now in the form:
$$ SoldPrice = β_0 + β_1x_1 + \cdots + β_nx_n $$where the x's are predictive variables.
Next we look at the difference between list price and sold price explicitly and calculate two new columns for our dataset. DiffPriceAbsolute and DiffPriceRelative.
DiffPriceAbsolute- This is difference between sold price and list price. If it is positive, that means the house sold for more than it was listed at.DiffPriceRelative- This is the relative difference between sold price and list price. A value of 1.1 here means that the house sold for 110% of the asking price, and 0.9 means the house sold for 90% of the asking price.
After that, we created two new models. One to predict DiffPriceAbsolute, and one to predict DiffPriceRelative and use the same predictive variables as in the last model. From there, we compare which of the two new models makes better predictions.
From the two models, we will train two models to predict DiffPriceAbsolute and DiffPriceRelative based on just SoldPrice and create a scatterplot for each model.
SoldPrice Regression
Python Code:
uni_mod0 = sm.ols(formula="SoldPrice ~ Acres + Taxes + TotBed + TotBth + TotSqf + YearBlt", data = train)
uni_result0 = uni_mod0.fit()
print(uni_result0.summary())
Output:
OLS Regression Results
==============================================================================
Dep. Variable: SoldPrice R-squared: 0.838
Model: OLS Adj. R-squared: 0.835
Method: Least Squares F-statistic: 218.7
Date: Thu, 21 Jul 2022 Prob (F-statistic): 4.45e-97
Time: 13:51:02 Log-Likelihood: -3291.9
No. Observations: 260 AIC: 6598.
Df Residuals: 253 BIC: 6623.
Df Model: 6
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [0.025 0.975]
------------------------------------------------------------------------------
Intercept 2.765e+05 2.94e+05 0.942 0.347 -3.02e+05 8.55e+05
Acres 1.169e+05 6.15e+04 1.900 0.059 -4276.513 2.38e+05
Taxes 58.6029 5.466 10.721 0.000 47.838 69.368
TotBed 1.286e+04 6045.229 2.127 0.034 951.492 2.48e+04
TotBth 6050.4777 8140.361 0.743 0.458 -9981.025 2.21e+04
TotSqf 45.5661 9.878 4.613 0.000 26.113 65.019
YearBlt -96.9146 151.794 -0.638 0.524 -395.855 202.025
==============================================================================
Omnibus: 30.768 Durbin-Watson: 2.174
Prob(Omnibus): 0.000 Jarque-Bera (JB): 43.356
Skew: 0.759 Prob(JB): 3.85e-10
Kurtosis: 4.302 Cond. No. 2.90e+05
==============================================================================
DiffPriceAbsolute Regression
Python Code:
uni_mod1 = sm.ols(formula="DiffPriceAbsolute ~ Acres + Taxes + TotBed + TotBth + TotSqf + YearBlt", data = train)
uni_result1 = uni_mod1.fit()
print(uni_result1.summary())
Output:
OLS Regression Results
==============================================================================
Dep. Variable: DiffPriceAbsolute R-squared: 0.261
Model: OLS Adj. R-squared: 0.244
Method: Least Squares F-statistic: 14.92
Date: Thu, 21 Jul 2022 Prob (F-statistic): 1.32e-14
Time: 13:51:02 Log-Likelihood: -2927.3
No. Observations: 260 AIC: 5869.
Df Residuals: 253 BIC: 5894.
Df Model: 6
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [0.025 0.975]
------------------------------------------------------------------------------
Intercept 2.867e+04 7.22e+04 0.397 0.692 -1.14e+05 1.71e+05
Acres -1.235e+04 1.51e+04 -0.815 0.416 -4.22e+04 1.75e+04
Taxes -1.1223 1.345 -0.834 0.405 -3.771 1.527
TotBed 528.3073 1487.544 0.355 0.723 -2401.240 3457.854
TotBth -2969.8850 2003.091 -1.483 0.139 -6914.743 974.973
TotSqf -4.9697 2.431 -2.045 0.042 -9.756 -0.183
YearBlt -9.4079 37.352 -0.252 0.801 -82.968 64.152
==============================================================================
Omnibus: 98.703 Durbin-Watson: 2.026
Prob(Omnibus): 0.000 Jarque-Bera (JB): 618.218
Skew: -1.371 Prob(JB): 5.70e-135
Kurtosis: 10.039 Cond. No. 2.90e+05
==============================================================================
DiffPriceRelative Regression
Python Code:
uni_mod2 = sm.ols(formula="DiffPriceRelative ~ Acres + Taxes + TotBed + TotBth + TotSqf + YearBlt", data = train)
uni_result2 = uni_mod2.fit()
print(uni_result2.summary())
Output:
OLS Regression Results
==============================================================================
Dep. Variable: DiffPriceRelative R-squared: 0.079
Model: OLS Adj. R-squared: 0.057
Method: Least Squares F-statistic: 3.618
Date: Thu, 21 Jul 2022 Prob (F-statistic): 0.00184
Time: 13:51:02 Log-Likelihood: 486.61
No. Observations: 260 AIC: -959.2
Df Residuals: 253 BIC: -934.3
Df Model: 6
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [0.025 0.975]
------------------------------------------------------------------------------
Intercept 1.0221 0.143 7.131 0.000 0.740 1.304
Acres -0.0145 0.030 -0.483 0.629 -0.074 0.045
Taxes 9.371e-07 2.67e-06 0.351 0.726 -4.32e-06 6.19e-06
TotBed 0.0025 0.003 0.838 0.403 -0.003 0.008
TotBth -0.0080 0.004 -2.022 0.044 -0.016 -0.000
TotSqf -3.941e-06 4.82e-06 -0.817 0.415 -1.34e-05 5.56e-06
YearBlt -1.261e-05 7.41e-05 -0.170 0.865 -0.000 0.000
==============================================================================
Omnibus: 119.899 Durbin-Watson: 2.053
Prob(Omnibus): 0.000 Jarque-Bera (JB): 840.861
Skew: -1.697 Prob(JB): 2.57e-183
Kurtosis: 11.130 Cond. No. 2.90e+05
==============================================================================
The multilinear regression model has a lower R squared than the simple linear regression model including the List Price. Therefore, there is more variation explained by the simple regression model. Additionally, based on the two models, the model that predicts the DiffPriceAbsolute is better than the model that predicts the DiffPriceRelative because the R squared is greater. Furthermore, from the different trained plots, the plot for the DiffPriceAbsolute has more plots closer to the linear line, thus showing why more of the variation is explained. Although DiffPriceAbsolute is better, it still doesn't have a high R squared.
Python Code:
from sklearn.model_selection import train_test_split
from sklearn import linear_model
lr = linear_model.LinearRegression()
# Generate Values
X = train[['SoldPrice']].values
y = train['DiffPriceAbsolute'].values
# Split data into training data and test data sets
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.9, random_state=1)
print(X_train.shape, y_train.shape)
print(X_test.shape, y_test.shape)
for m in np.arange(1,2):
lr.fit(X=X_train[:,:m], y=y_train)
print('m=', m, ', train: ', lr.score(X_train[:,:m], y_train), ' test: ', lr.score(X_test[:,:m], y_test))
Python Code:
from sklearn.model_selection import train_test_split
from sklearn import linear_model
lr = linear_model.LinearRegression()
# Generate Values
X = train[['SoldPrice']].values
y = train['DiffPriceRelative'].values
# Split data into training data and test data sets
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.9, random_state=1)
print(X_train.shape, y_train.shape)
print(X_test.shape, y_test.shape)
for m in np.arange(1,2):
lr.fit(X=X_train[:,:m], y=y_train)
print('m=', m, ', train: ', lr.score(X_train[:,:m], y_train), ' test: ', lr.score(X_test[:,:m], y_test))
Incorporating a Categorical Variable
Above, we considered houses, townhouses, and condos together. Now we will now distinguish the difference between houses, townhouses, and condos. Consider the two regression models:
$$ SoldPrice = β_0 + β_1 (Prop) $$ $$ SoldPrice = β_0 + β_1 (Prop) + β_2 (TotSqf) $$From the first model, property type is significant in predicting the sold price. On the other hand, the second model indicates that when we consider total square footage, property type is no longer predictive.
We will also make a scatterplot of TotSqf vs. SoldPrice where the house types are colored differently to illustrate your explanation.
Python Code:
rel3 = sns.scatterplot(train['TotSqf'], train['SoldPrice'], hue = train['PropType'])
