Regression topics
Contents
11. Regression topics#
This section will go into more detail on running regressions in Python. We already saw an example using factor models, like the CAPM and Fama-French 3-factor models.
We could spend an entire semester going over linear regression, how to put together models, how to interpret models, and all of the adjustments that we can make. In fact, this is basically what a first-semester Econometrics class is!
I will be following code examples from Coding for Economists, which has just about everything you need to know to do basic linear regression (OLS) in Python. I recommend giving it a read, especially if you’ve taken econometrics and have already seen the general ideas.
The Effect great book for getting starting with econometrics, regression, and how to add meaning to the regressions that we’re running. Chapter 13 of that book covers regression (with code in R).
You can read more about statsmodels on their help page.
I’ll be using our Zillow pricing error data in this example.
import numpy as np
import scipy.stats as st
import matplotlib.pyplot as plt
import pandas as pd
import seaborn as sns
import statsmodels.api as sm
import statsmodels.formula.api as smf
# Include this to have plots show up in your Jupyter notebook.
%matplotlib inline
pd.options.display.max_columns = None
housing = pd.read_csv('https://raw.githubusercontent.com/aaiken1/fin-data-analysis-python/main/data/properties_2016_sample10_1.csv')
pricing = pd.read_csv('https://raw.githubusercontent.com/aaiken1/fin-data-analysis-python/main/data/train_2016_v2.csv')
zillow_data = pd.merge(housing, pricing, how='inner', on='parcelid')
zillow_data['transactiondate'] = pd.to_datetime(zillow_data['transactiondate'], format='%Y-%m-%d')
/var/folders/kx/y8vj3n6n5kq_d74vj24jsnh40000gn/T/ipykernel_41859/4278326240.py:1: DtypeWarning: Columns (49) have mixed types. Specify dtype option on import or set low_memory=False.
housing = pd.read_csv('https://raw.githubusercontent.com/aaiken1/fin-data-analysis-python/main/data/properties_2016_sample10_1.csv')
zillow_data.describe()
| parcelid | airconditioningtypeid | architecturalstyletypeid | basementsqft | bathroomcnt | bedroomcnt | buildingclasstypeid | buildingqualitytypeid | calculatedbathnbr | decktypeid | finishedfloor1squarefeet | calculatedfinishedsquarefeet | finishedsquarefeet12 | finishedsquarefeet13 | finishedsquarefeet15 | finishedsquarefeet50 | finishedsquarefeet6 | fips | fireplacecnt | fullbathcnt | garagecarcnt | garagetotalsqft | heatingorsystemtypeid | latitude | longitude | lotsizesquarefeet | poolcnt | poolsizesum | pooltypeid7 | propertylandusetypeid | rawcensustractandblock | regionidcity | regionidcounty | regionidneighborhood | regionidzip | roomcnt | storytypeid | threequarterbathnbr | typeconstructiontypeid | unitcnt | yardbuildingsqft17 | yardbuildingsqft26 | yearbuilt | numberofstories | structuretaxvaluedollarcnt | taxvaluedollarcnt | assessmentyear | landtaxvaluedollarcnt | taxamount | taxdelinquencyyear | censustractandblock | logerror | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| count | 9.071000e+03 | 2871.000000 | 0.0 | 5.00000 | 9071.000000 | 9071.000000 | 3.0 | 5694.000000 | 8948.000000 | 64.0 | 695.000000 | 9001.000000 | 8612.000000 | 3.000000 | 337.000000 | 695.000000 | 49.000000 | 9071.000000 | 993.000000 | 8948.000000 | 3076.000000 | 3076.000000 | 5574.000000 | 9.071000e+03 | 9.071000e+03 | 8.020000e+03 | 1810.0 | 99.000000 | 1685.0 | 9071.000000 | 9.071000e+03 | 8912.000000 | 9071.000000 | 3601.000000 | 9070.000000 | 9071.000000 | 5.0 | 1208.000000 | 0.0 | 5794.000000 | 280.000000 | 7.000000 | 8991.000000 | 2138.000000 | 9.022000e+03 | 9.071000e+03 | 9071.0 | 9.071000e+03 | 9071.000000 | 168.000000 | 9.009000e+03 | 9071.000000 |
| mean | 1.298764e+07 | 1.838036 | NaN | 516.00000 | 2.266233 | 3.013670 | 4.0 | 5.572708 | 2.296826 | 66.0 | 1348.981295 | 1767.239307 | 1740.108918 | 1408.000000 | 2393.350148 | 1368.942446 | 2251.428571 | 6049.128982 | 1.197382 | 2.228990 | 1.800715 | 342.415475 | 3.909760 | 3.400230e+07 | -1.181977e+08 | 3.150909e+04 | 1.0 | 520.424242 | 1.0 | 261.835520 | 6.049436e+07 | 33944.006845 | 2511.879727 | 193520.398223 | 96547.689195 | 1.531364 | 7.0 | 1.004967 | NaN | 1.104764 | 290.335714 | 496.714286 | 1968.380047 | 1.428438 | 1.768673e+05 | 4.523049e+05 | 2015.0 | 2.763930e+05 | 5906.696988 | 13.327381 | 6.049368e+13 | 0.010703 |
| std | 1.757451e+06 | 3.001723 | NaN | 233.49197 | 0.989863 | 1.118468 | 0.0 | 1.908379 | 0.960557 | 0.0 | 664.508053 | 918.999586 | 880.213401 | 55.425626 | 1434.457485 | 709.622839 | 1352.034747 | 20.794593 | 0.480794 | 0.951007 | 0.598328 | 263.642761 | 3.678727 | 2.654493e+05 | 3.631575e+05 | 1.824345e+05 | 0.0 | 146.537109 | 0.0 | 5.781663 | 2.063550e+05 | 47178.373342 | 810.417898 | 169701.596819 | 412.732130 | 2.856603 | 0.0 | 0.070330 | NaN | 0.459551 | 172.987812 | 506.445033 | 23.469997 | 0.536698 | 1.909207e+05 | 5.229433e+05 | 0.0 | 3.901131e+05 | 6388.966672 | 1.796527 | 2.053649e+11 | 0.158364 |
| min | 1.071186e+07 | 1.000000 | NaN | 162.00000 | 0.000000 | 0.000000 | 4.0 | 1.000000 | 1.000000 | 66.0 | 49.000000 | 214.000000 | 214.000000 | 1344.000000 | 716.000000 | 49.000000 | 438.000000 | 6037.000000 | 1.000000 | 1.000000 | 0.000000 | 0.000000 | 1.000000 | 3.334420e+07 | -1.194143e+08 | 4.350000e+02 | 1.0 | 207.000000 | 1.0 | 31.000000 | 6.037101e+07 | 3491.000000 | 1286.000000 | 6952.000000 | 95982.000000 | 0.000000 | 7.0 | 1.000000 | NaN | 1.000000 | 41.000000 | 37.000000 | 1885.000000 | 1.000000 | 1.516000e+03 | 7.837000e+03 | 2015.0 | 2.178000e+03 | 96.740000 | 7.000000 | 6.037101e+13 | -2.365000 |
| 25% | 1.157119e+07 | 1.000000 | NaN | 485.00000 | 2.000000 | 2.000000 | 4.0 | 4.000000 | 2.000000 | 66.0 | 938.000000 | 1187.000000 | 1173.000000 | 1392.000000 | 1668.000000 | 938.000000 | 1009.000000 | 6037.000000 | 1.000000 | 2.000000 | 2.000000 | 0.000000 | 2.000000 | 3.380545e+07 | -1.184080e+08 | 5.746500e+03 | 1.0 | 435.000000 | 1.0 | 261.000000 | 6.037400e+07 | 12447.000000 | 1286.000000 | 46736.000000 | 96193.000000 | 0.000000 | 7.0 | 1.000000 | NaN | 1.000000 | 175.750000 | 110.500000 | 1953.000000 | 1.000000 | 8.028525e+04 | 1.926595e+05 | 2015.0 | 8.060700e+04 | 2828.645000 | 13.000000 | 6.037400e+13 | -0.025300 |
| 50% | 1.259048e+07 | 1.000000 | NaN | 515.00000 | 2.000000 | 3.000000 | 4.0 | 7.000000 | 2.000000 | 66.0 | 1249.000000 | 1539.000000 | 1513.000000 | 1440.000000 | 2157.000000 | 1257.000000 | 1835.000000 | 6037.000000 | 1.000000 | 2.000000 | 2.000000 | 430.000000 | 2.000000 | 3.401408e+07 | -1.181670e+08 | 7.200000e+03 | 1.0 | 504.000000 | 1.0 | 261.000000 | 6.037621e+07 | 25218.000000 | 3101.000000 | 118887.000000 | 96401.000000 | 0.000000 | 7.0 | 1.000000 | NaN | 1.000000 | 248.500000 | 268.000000 | 1969.000000 | 1.000000 | 1.315530e+05 | 3.416920e+05 | 2015.0 | 1.910000e+05 | 4521.580000 | 14.000000 | 6.037621e+13 | 0.007000 |
| 75% | 1.423676e+07 | 1.000000 | NaN | 616.00000 | 3.000000 | 4.000000 | 4.0 | 7.000000 | 3.000000 | 66.0 | 1612.000000 | 2090.000000 | 2055.000000 | 1440.000000 | 2806.000000 | 1617.500000 | 3732.000000 | 6059.000000 | 1.000000 | 3.000000 | 2.000000 | 484.000000 | 7.000000 | 3.417153e+07 | -1.179195e+08 | 1.161675e+04 | 1.0 | 600.000000 | 1.0 | 266.000000 | 6.059052e+07 | 45457.000000 | 3101.000000 | 274815.000000 | 96987.000000 | 0.000000 | 7.0 | 1.000000 | NaN | 1.000000 | 360.000000 | 792.500000 | 1986.000000 | 2.000000 | 2.076458e+05 | 5.361120e+05 | 2015.0 | 3.428715e+05 | 6865.565000 | 15.000000 | 6.059052e+13 | 0.040200 |
| max | 1.730050e+07 | 13.000000 | NaN | 802.00000 | 12.000000 | 12.000000 | 4.0 | 12.000000 | 12.000000 | 66.0 | 5416.000000 | 22741.000000 | 10680.000000 | 1440.000000 | 22741.000000 | 6906.000000 | 5229.000000 | 6111.000000 | 3.000000 | 12.000000 | 9.000000 | 2685.000000 | 24.000000 | 3.477509e+07 | -1.175604e+08 | 6.971010e+06 | 1.0 | 1052.000000 | 1.0 | 275.000000 | 6.111009e+07 | 396556.000000 | 3101.000000 | 764166.000000 | 97344.000000 | 13.000000 | 7.0 | 2.000000 | NaN | 9.000000 | 1018.000000 | 1366.000000 | 2015.000000 | 3.000000 | 4.588745e+06 | 1.275000e+07 | 2015.0 | 1.200000e+07 | 152152.220000 | 15.000000 | 6.111009e+13 | 2.953000 |
I’ll print a list of the columns, just to see what our variables are. There’s a lot in this data set.
zillow_data.columns
Index(['parcelid', 'airconditioningtypeid', 'architecturalstyletypeid',
'basementsqft', 'bathroomcnt', 'bedroomcnt', 'buildingclasstypeid',
'buildingqualitytypeid', 'calculatedbathnbr', 'decktypeid',
'finishedfloor1squarefeet', 'calculatedfinishedsquarefeet',
'finishedsquarefeet12', 'finishedsquarefeet13', 'finishedsquarefeet15',
'finishedsquarefeet50', 'finishedsquarefeet6', 'fips', 'fireplacecnt',
'fullbathcnt', 'garagecarcnt', 'garagetotalsqft', 'hashottuborspa',
'heatingorsystemtypeid', 'latitude', 'longitude', 'lotsizesquarefeet',
'poolcnt', 'poolsizesum', 'pooltypeid10', 'pooltypeid2', 'pooltypeid7',
'propertycountylandusecode', 'propertylandusetypeid',
'propertyzoningdesc', 'rawcensustractandblock', 'regionidcity',
'regionidcounty', 'regionidneighborhood', 'regionidzip', 'roomcnt',
'storytypeid', 'threequarterbathnbr', 'typeconstructiontypeid',
'unitcnt', 'yardbuildingsqft17', 'yardbuildingsqft26', 'yearbuilt',
'numberofstories', 'fireplaceflag', 'structuretaxvaluedollarcnt',
'taxvaluedollarcnt', 'assessmentyear', 'landtaxvaluedollarcnt',
'taxamount', 'taxdelinquencyflag', 'taxdelinquencyyear',
'censustractandblock', 'logerror', 'transactiondate'],
dtype='object')
Let’s run a really simple regression. Can we explain pricing errors using the size of the house? I’ll take the natural log of calculatedfinishedsquarefeet and use that as my independent (X) variable. My dependent (Y) variable will be logerror. I’m taking the natural log of the square footage, in order to have what’s called a “log-log” model.
zillow_data['ln_calculatedfinishedsquarefeet'] = np.log(zillow_data['calculatedfinishedsquarefeet'])
results = smf.ols("logerror ~ ln_calculatedfinishedsquarefeet", data=zillow_data).fit()
print(results.summary())
OLS Regression Results
==============================================================================
Dep. Variable: logerror R-squared: 0.001
Model: OLS Adj. R-squared: 0.001
Method: Least Squares F-statistic: 13.30
Date: Tue, 23 Jan 2024 Prob (F-statistic): 0.000267
Time: 12:40:20 Log-Likelihood: 3831.8
No. Observations: 9001 AIC: -7660.
Df Residuals: 8999 BIC: -7645.
Df Model: 1
Covariance Type: nonrobust
===================================================================================================
coef std err t P>|t| [0.025 0.975]
---------------------------------------------------------------------------------------------------
Intercept -0.0911 0.028 -3.244 0.001 -0.146 -0.036
ln_calculatedfinishedsquarefeet 0.0139 0.004 3.647 0.000 0.006 0.021
==============================================================================
Omnibus: 4055.877 Durbin-Watson: 2.005
Prob(Omnibus): 0.000 Jarque-Bera (JB): 2595715.665
Skew: 0.737 Prob(JB): 0.00
Kurtosis: 86.180 Cond. No. 127.
==============================================================================
Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
That’s the full summary of the regression. This is a “log-log” model, so we can say that a 1% change in square footage leads to a 1.39% increase in pricing error. The coefficient is positive and statistically significant at conventional levels (e.g. 1%).
We can pull out just a piece of this full result if we like.
results.summary().tables[1]
| coef | std err | t | P>|t| | [0.025 | 0.975] | |
|---|---|---|---|---|---|---|
| Intercept | -0.0911 | 0.028 | -3.244 | 0.001 | -0.146 | -0.036 |
| ln_calculatedfinishedsquarefeet | 0.0139 | 0.004 | 3.647 | 0.000 | 0.006 | 0.021 |
We can, of course, include multiple X variables in a regression. I’ll add bathroom and bedroom counts to the regression model.
results = smf.ols("logerror ~ ln_calculatedfinishedsquarefeet + bathroomcnt + bedroomcnt", data=zillow_data).fit()
print(results.summary())
OLS Regression Results
==============================================================================
Dep. Variable: logerror R-squared: 0.002
Model: OLS Adj. R-squared: 0.002
Method: Least Squares F-statistic: 6.718
Date: Tue, 23 Jan 2024 Prob (F-statistic): 0.000159
Time: 12:40:20 Log-Likelihood: 3835.2
No. Observations: 9001 AIC: -7662.
Df Residuals: 8997 BIC: -7634.
Df Model: 3
Covariance Type: nonrobust
===================================================================================================
coef std err t P>|t| [0.025 0.975]
---------------------------------------------------------------------------------------------------
Intercept -0.0140 0.041 -0.339 0.735 -0.095 0.067
ln_calculatedfinishedsquarefeet 0.0006 0.006 0.095 0.925 -0.012 0.013
bathroomcnt 0.0040 0.003 1.493 0.135 -0.001 0.009
bedroomcnt 0.0038 0.002 1.740 0.082 -0.000 0.008
==============================================================================
Omnibus: 4050.508 Durbin-Watson: 2.005
Prob(Omnibus): 0.000 Jarque-Bera (JB): 2598102.584
Skew: 0.733 Prob(JB): 0.00
Kurtosis: 86.219 Cond. No. 211.
==============================================================================
Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
Hey, all of my significance went away! Welcome to the world of multicollinearity. All of these variables are very correlated, so the coefficient estimates become difficult to interpret.
Watch what happens when I just run the model with the bedroom count. The \(t\)-statistic is quite large again.
results = smf.ols("logerror ~ bedroomcnt", data=zillow_data).fit()
print(results.summary())
OLS Regression Results
==============================================================================
Dep. Variable: logerror R-squared: 0.002
Model: OLS Adj. R-squared: 0.002
Method: Least Squares F-statistic: 21.69
Date: Tue, 23 Jan 2024 Prob (F-statistic): 3.24e-06
Time: 12:40:20 Log-Likelihood: 3856.7
No. Observations: 9071 AIC: -7709.
Df Residuals: 9069 BIC: -7695.
Df Model: 1
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [0.025 0.975]
------------------------------------------------------------------------------
Intercept -0.0101 0.005 -2.125 0.034 -0.019 -0.001
bedroomcnt 0.0069 0.001 4.658 0.000 0.004 0.010
==============================================================================
Omnibus: 4021.076 Durbin-Watson: 2.006
Prob(Omnibus): 0.000 Jarque-Bera (JB): 2560800.149
Skew: 0.697 Prob(JB): 0.00
Kurtosis: 85.301 Cond. No. 10.0
==============================================================================
Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
11.1. Indicators and categorical variables#
The variables used above are measured numerically. Some are continuous, like square footage, while others are counts, like the number of bedrooms. Sometimes, though, we want to include an indicator for something? For example, does this house have a pool or not?
There is a variable in the data called poolcnt. It seems to be either missing (NaN) or set equal to 1. I believe that a value of 1 means that the house has a pool and that NaN means that it does not. This is bit of a tricky assumption, because NaN could mean no pool or that we don’t know either way. But, I’ll make that assumption for illustrative purposes.
zillow_data['poolcnt'].describe()
count 1810.0
mean 1.0
std 0.0
min 1.0
25% 1.0
50% 1.0
75% 1.0
max 1.0
Name: poolcnt, dtype: float64
I am going to create a new variable, pool_d, that is set equal to 1 if poolcnt >= 1 and 0 otherwise. This type of 1/0 categorical variable is sometimes called an indicator, or dummy variable.
zillow_data['pool_d'] = np.where(zillow_data.poolcnt.isnull(), 0, zillow_data.poolcnt >= 1)
zillow_data['pool_d'].describe()
count 9071.000000
mean 0.199537
std 0.399674
min 0.000000
25% 0.000000
50% 0.000000
75% 0.000000
max 1.000000
Name: pool_d, dtype: float64
I can then use this 1/0 variable in my regression.
results = smf.ols("logerror ~ ln_calculatedfinishedsquarefeet + pool_d", data=zillow_data).fit()
print(results.summary())
OLS Regression Results
==============================================================================
Dep. Variable: logerror R-squared: 0.001
Model: OLS Adj. R-squared: 0.001
Method: Least Squares F-statistic: 6.684
Date: Tue, 23 Jan 2024 Prob (F-statistic): 0.00126
Time: 12:40:20 Log-Likelihood: 3831.8
No. Observations: 9001 AIC: -7658.
Df Residuals: 8998 BIC: -7636.
Df Model: 2
Covariance Type: nonrobust
===================================================================================================
coef std err t P>|t| [0.025 0.975]
---------------------------------------------------------------------------------------------------
Intercept -0.0898 0.029 -3.150 0.002 -0.146 -0.034
ln_calculatedfinishedsquarefeet 0.0137 0.004 3.519 0.000 0.006 0.021
pool_d 0.0011 0.004 0.262 0.794 -0.007 0.009
==============================================================================
Omnibus: 4055.061 Durbin-Watson: 2.006
Prob(Omnibus): 0.000 Jarque-Bera (JB): 2593138.691
Skew: 0.737 Prob(JB): 0.00
Kurtosis: 86.139 Cond. No. 129.
==============================================================================
Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
Pools don’t seem to influence pricing errors.
We can also create more general categorical variables. For example, instead of treating bedrooms like a count, we can create new categories for each number of bedrooms. This type of model is helpful when dealing states or regions. For example, you could turn a zip code into a categorical variable. This would allow zip codes, or a location, to explain the pricing errors.
In Python, you can turn something into a categorical variable by using C() in the regression formula.
I’ll try the count of bedrooms first.
results = smf.ols("logerror ~ ln_calculatedfinishedsquarefeet + C(bedroomcnt)", data=zillow_data).fit()
print(results.summary())
OLS Regression Results
==============================================================================
Dep. Variable: logerror R-squared: 0.004
Model: OLS Adj. R-squared: 0.003
Method: Least Squares F-statistic: 3.118
Date: Tue, 23 Jan 2024 Prob (F-statistic): 0.000196
Time: 12:40:20 Log-Likelihood: 3843.8
No. Observations: 9001 AIC: -7662.
Df Residuals: 8988 BIC: -7569.
Df Model: 12
Covariance Type: nonrobust
===================================================================================================
coef std err t P>|t| [0.025 0.975]
---------------------------------------------------------------------------------------------------
Intercept -0.0680 0.045 -1.523 0.128 -0.155 0.020
C(bedroomcnt)[T.1.0] 0.0370 0.021 1.756 0.079 -0.004 0.078
C(bedroomcnt)[T.2.0] 0.0279 0.020 1.428 0.153 -0.010 0.066
C(bedroomcnt)[T.3.0] 0.0319 0.019 1.648 0.099 -0.006 0.070
C(bedroomcnt)[T.4.0] 0.0357 0.020 1.825 0.068 -0.003 0.074
C(bedroomcnt)[T.5.0] 0.0580 0.021 2.799 0.005 0.017 0.099
C(bedroomcnt)[T.6.0] 0.0491 0.024 2.007 0.045 0.001 0.097
C(bedroomcnt)[T.7.0] 0.0903 0.040 2.266 0.023 0.012 0.168
C(bedroomcnt)[T.8.0] -0.0165 0.043 -0.383 0.702 -0.101 0.068
C(bedroomcnt)[T.9.0] -0.1190 0.081 -1.461 0.144 -0.279 0.041
C(bedroomcnt)[T.10.0] 0.0312 0.159 0.196 0.845 -0.281 0.343
C(bedroomcnt)[T.12.0] 0.0399 0.114 0.351 0.725 -0.183 0.262
ln_calculatedfinishedsquarefeet 0.0062 0.005 1.134 0.257 -0.005 0.017
==============================================================================
Omnibus: 4046.896 Durbin-Watson: 2.006
Prob(Omnibus): 0.000 Jarque-Bera (JB): 2594171.124
Skew: 0.731 Prob(JB): 0.00
Kurtosis: 86.156 Cond. No. 716.
==============================================================================
Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
And here are zip codes as a categorical variable. This is saying: Is the house in this zip code or no? If it is, the indicator for that zip code gets a 1, and a 0 otherwise. If we didn’t do this, then the zip code would get treated like a numerical variable in the regression, like square footage, which makes no sense!
results = smf.ols("logerror ~ ln_calculatedfinishedsquarefeet + C(regionidzip)", data=zillow_data).fit()
print(results.summary())
OLS Regression Results
==============================================================================
Dep. Variable: logerror R-squared: 0.054
Model: OLS Adj. R-squared: 0.012
Method: Least Squares F-statistic: 1.300
Date: Tue, 23 Jan 2024 Prob (F-statistic): 0.000104
Time: 12:40:20 Log-Likelihood: 4075.3
No. Observations: 9001 AIC: -7391.
Df Residuals: 8621 BIC: -4691.
Df Model: 379
Covariance Type: nonrobust
===================================================================================================
coef std err t P>|t| [0.025 0.975]
---------------------------------------------------------------------------------------------------
Intercept -0.1624 0.050 -3.261 0.001 -0.260 -0.065
C(regionidzip)[T.95983.0] 0.0855 0.053 1.621 0.105 -0.018 0.189
C(regionidzip)[T.95984.0] -0.0735 0.050 -1.481 0.139 -0.171 0.024
C(regionidzip)[T.95985.0] 0.0679 0.051 1.326 0.185 -0.032 0.168
C(regionidzip)[T.95986.0] 0.0356 0.060 0.593 0.553 -0.082 0.153
C(regionidzip)[T.95987.0] 0.1185 0.066 1.807 0.071 -0.010 0.247
C(regionidzip)[T.95988.0] 0.1128 0.088 1.283 0.199 -0.060 0.285
C(regionidzip)[T.95989.0] 0.0356 0.058 0.618 0.537 -0.077 0.148
C(regionidzip)[T.95991.0] 0.0894 0.162 0.551 0.581 -0.228 0.407
C(regionidzip)[T.95992.0] -0.0856 0.052 -1.641 0.101 -0.188 0.017
C(regionidzip)[T.95993.0] 0.0710 0.099 0.718 0.473 -0.123 0.265
C(regionidzip)[T.95994.0] 0.1135 0.088 1.291 0.197 -0.059 0.286
C(regionidzip)[T.95996.0] 0.1052 0.071 1.476 0.140 -0.034 0.245
C(regionidzip)[T.95997.0] 0.0820 0.049 1.674 0.094 -0.014 0.178
C(regionidzip)[T.95998.0] 0.0517 0.088 0.589 0.556 -0.121 0.224
C(regionidzip)[T.95999.0] 0.1369 0.057 2.423 0.015 0.026 0.248
C(regionidzip)[T.96000.0] 0.1968 0.050 3.937 0.000 0.099 0.295
C(regionidzip)[T.96001.0] 0.1037 0.063 1.636 0.102 -0.021 0.228
C(regionidzip)[T.96003.0] 0.0724 0.060 1.205 0.228 -0.045 0.190
C(regionidzip)[T.96004.0] 0.0467 0.088 0.531 0.596 -0.126 0.219
C(regionidzip)[T.96005.0] 0.1988 0.048 4.106 0.000 0.104 0.294
C(regionidzip)[T.96006.0] 0.0866 0.049 1.770 0.077 -0.009 0.183
C(regionidzip)[T.96007.0] 0.0738 0.051 1.454 0.146 -0.026 0.173
C(regionidzip)[T.96008.0] 0.0746 0.052 1.430 0.153 -0.028 0.177
C(regionidzip)[T.96009.0] 0.0495 0.088 0.564 0.573 -0.123 0.222
C(regionidzip)[T.96010.0] 0.6989 0.162 4.312 0.000 0.381 1.017
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C(regionidzip)[T.96496.0] 0.0351 0.056 0.632 0.528 -0.074 0.144
C(regionidzip)[T.96497.0] 0.0483 0.062 0.784 0.433 -0.072 0.169
C(regionidzip)[T.96505.0] 0.0959 0.046 2.108 0.035 0.007 0.185
C(regionidzip)[T.96506.0] 0.0771 0.048 1.610 0.107 -0.017 0.171
C(regionidzip)[T.96507.0] 0.0997 0.050 1.997 0.046 0.002 0.198
C(regionidzip)[T.96508.0] 0.1185 0.060 1.973 0.048 0.001 0.236
C(regionidzip)[T.96510.0] 0.0638 0.053 1.210 0.226 -0.040 0.167
C(regionidzip)[T.96513.0] 0.0712 0.050 1.415 0.157 -0.027 0.170
C(regionidzip)[T.96514.0] 0.0766 0.062 1.243 0.214 -0.044 0.197
C(regionidzip)[T.96515.0] 0.1186 0.060 1.975 0.048 0.001 0.236
C(regionidzip)[T.96517.0] 0.1010 0.051 1.991 0.046 0.002 0.201
C(regionidzip)[T.96522.0] 0.0969 0.047 2.068 0.039 0.005 0.189
C(regionidzip)[T.96523.0] 0.0107 0.048 0.223 0.824 -0.083 0.105
C(regionidzip)[T.96524.0] 0.0808 0.050 1.629 0.103 -0.016 0.178
C(regionidzip)[T.96525.0] 0.0685 0.059 1.166 0.244 -0.047 0.184
C(regionidzip)[T.96531.0] 0.0708 0.057 1.253 0.210 -0.040 0.182
C(regionidzip)[T.96533.0] 0.0839 0.056 1.509 0.131 -0.025 0.193
C(regionidzip)[T.96939.0] 0.0675 0.055 1.233 0.218 -0.040 0.175
C(regionidzip)[T.96940.0] 0.0961 0.048 1.992 0.046 0.002 0.191
C(regionidzip)[T.96941.0] 0.0814 0.049 1.651 0.099 -0.015 0.178
C(regionidzip)[T.96943.0] 0.0663 0.056 1.192 0.233 -0.043 0.175
C(regionidzip)[T.96946.0] 0.0726 0.071 1.019 0.308 -0.067 0.212
C(regionidzip)[T.96947.0] 0.0793 0.051 1.550 0.121 -0.021 0.180
C(regionidzip)[T.96948.0] 0.0803 0.052 1.555 0.120 -0.021 0.182
C(regionidzip)[T.96951.0] 0.3394 0.088 3.862 0.000 0.167 0.512
C(regionidzip)[T.96952.0] 0.0872 0.053 1.654 0.098 -0.016 0.191
C(regionidzip)[T.96954.0] 0.0831 0.044 1.870 0.061 -0.004 0.170
C(regionidzip)[T.96956.0] 0.0063 0.057 0.112 0.911 -0.104 0.117
C(regionidzip)[T.96957.0] 0.0914 0.053 1.712 0.087 -0.013 0.196
C(regionidzip)[T.96958.0] 0.0816 0.048 1.685 0.092 -0.013 0.176
C(regionidzip)[T.96959.0] 0.0840 0.048 1.745 0.081 -0.010 0.178
C(regionidzip)[T.96961.0] 0.0818 0.045 1.802 0.072 -0.007 0.171
C(regionidzip)[T.96962.0] 0.0672 0.045 1.494 0.135 -0.021 0.155
C(regionidzip)[T.96963.0] 0.0014 0.046 0.030 0.976 -0.089 0.092
C(regionidzip)[T.96964.0] 0.0961 0.044 2.173 0.030 0.009 0.183
C(regionidzip)[T.96965.0] 0.1011 0.048 2.111 0.035 0.007 0.195
C(regionidzip)[T.96966.0] 0.0641 0.046 1.383 0.167 -0.027 0.155
C(regionidzip)[T.96967.0] 0.0810 0.047 1.721 0.085 -0.011 0.173
C(regionidzip)[T.96969.0] 0.1331 0.050 2.645 0.008 0.034 0.232
C(regionidzip)[T.96971.0] 0.0707 0.049 1.453 0.146 -0.025 0.166
C(regionidzip)[T.96973.0] -0.0640 0.099 -0.647 0.517 -0.258 0.130
C(regionidzip)[T.96974.0] 0.0952 0.042 2.242 0.025 0.012 0.178
C(regionidzip)[T.96975.0] 0.1186 0.063 1.870 0.062 -0.006 0.243
C(regionidzip)[T.96978.0] 0.0983 0.045 2.190 0.029 0.010 0.186
C(regionidzip)[T.96979.0] 0.0752 0.088 0.855 0.392 -0.097 0.247
C(regionidzip)[T.96980.0] 0.0799 0.075 1.061 0.289 -0.068 0.227
C(regionidzip)[T.96981.0] 0.0486 0.053 0.910 0.363 -0.056 0.153
C(regionidzip)[T.96982.0] 0.0956 0.048 1.996 0.046 0.002 0.190
C(regionidzip)[T.96983.0] 0.0658 0.047 1.402 0.161 -0.026 0.158
C(regionidzip)[T.96985.0] 0.0929 0.047 1.980 0.048 0.001 0.185
C(regionidzip)[T.96986.0] 0.1024 0.162 0.632 0.528 -0.215 0.420
C(regionidzip)[T.96987.0] 0.0943 0.043 2.203 0.028 0.010 0.178
C(regionidzip)[T.96989.0] 0.0724 0.045 1.593 0.111 -0.017 0.161
C(regionidzip)[T.96990.0] 0.0590 0.046 1.272 0.203 -0.032 0.150
C(regionidzip)[T.96993.0] 0.0530 0.043 1.227 0.220 -0.032 0.138
C(regionidzip)[T.96995.0] 0.0861 0.044 1.940 0.052 -0.001 0.173
C(regionidzip)[T.96996.0] 0.0688 0.044 1.575 0.115 -0.017 0.154
C(regionidzip)[T.96998.0] 0.0960 0.045 2.129 0.033 0.008 0.184
C(regionidzip)[T.97001.0] 0.1768 0.059 3.012 0.003 0.062 0.292
C(regionidzip)[T.97003.0] 0.1598 0.058 2.777 0.006 0.047 0.273
C(regionidzip)[T.97004.0] 0.1275 0.048 2.674 0.008 0.034 0.221
C(regionidzip)[T.97005.0] 0.0989 0.047 2.101 0.036 0.007 0.191
C(regionidzip)[T.97006.0] 0.0975 0.056 1.754 0.079 -0.011 0.206
C(regionidzip)[T.97007.0] 0.0706 0.048 1.458 0.145 -0.024 0.165
C(regionidzip)[T.97008.0] 0.0578 0.047 1.229 0.219 -0.034 0.150
C(regionidzip)[T.97016.0] 0.0651 0.045 1.433 0.152 -0.024 0.154
C(regionidzip)[T.97018.0] 0.0676 0.050 1.362 0.173 -0.030 0.165
C(regionidzip)[T.97020.0] 0.0807 0.053 1.513 0.130 -0.024 0.185
C(regionidzip)[T.97021.0] 0.0869 0.053 1.629 0.103 -0.018 0.191
C(regionidzip)[T.97023.0] 0.0114 0.048 0.239 0.811 -0.082 0.105
C(regionidzip)[T.97024.0] 0.0958 0.047 2.018 0.044 0.003 0.189
C(regionidzip)[T.97025.0] 0.0797 0.063 1.258 0.208 -0.045 0.204
C(regionidzip)[T.97026.0] 0.0890 0.046 1.926 0.054 -0.002 0.180
C(regionidzip)[T.97027.0] 0.0911 0.051 1.794 0.073 -0.008 0.191
C(regionidzip)[T.97035.0] 0.0835 0.049 1.705 0.088 -0.012 0.180
C(regionidzip)[T.97037.0] 0.0635 0.081 0.788 0.431 -0.095 0.221
C(regionidzip)[T.97039.0] 0.0504 0.049 1.030 0.303 -0.046 0.146
C(regionidzip)[T.97040.0] 0.0443 0.060 0.738 0.461 -0.073 0.162
C(regionidzip)[T.97041.0] 0.0678 0.046 1.472 0.141 -0.022 0.158
C(regionidzip)[T.97043.0] 0.0745 0.050 1.490 0.136 -0.024 0.173
C(regionidzip)[T.97047.0] 0.0961 0.047 2.034 0.042 0.003 0.189
C(regionidzip)[T.97048.0] 0.0798 0.051 1.559 0.119 -0.021 0.180
C(regionidzip)[T.97050.0] 0.0749 0.051 1.464 0.143 -0.025 0.175
C(regionidzip)[T.97051.0] 0.1045 0.058 1.817 0.069 -0.008 0.217
C(regionidzip)[T.97052.0] 0.0708 0.056 1.273 0.203 -0.038 0.180
C(regionidzip)[T.97059.0] 0.0552 0.075 0.732 0.464 -0.093 0.203
C(regionidzip)[T.97063.0] 0.1158 0.053 2.170 0.030 0.011 0.220
C(regionidzip)[T.97064.0] 0.0613 0.059 1.044 0.296 -0.054 0.176
C(regionidzip)[T.97065.0] 0.0695 0.049 1.409 0.159 -0.027 0.166
C(regionidzip)[T.97066.0] 0.0919 0.068 1.350 0.177 -0.042 0.225
C(regionidzip)[T.97067.0] 0.1008 0.046 2.187 0.029 0.010 0.191
C(regionidzip)[T.97068.0] 0.0863 0.047 1.834 0.067 -0.006 0.179
C(regionidzip)[T.97078.0] 0.0657 0.047 1.406 0.160 -0.026 0.157
C(regionidzip)[T.97079.0] 0.0955 0.050 1.894 0.058 -0.003 0.194
C(regionidzip)[T.97081.0] 0.0527 0.050 1.046 0.295 -0.046 0.151
C(regionidzip)[T.97083.0] 0.0747 0.045 1.645 0.100 -0.014 0.164
C(regionidzip)[T.97084.0] 0.0804 0.049 1.632 0.103 -0.016 0.177
C(regionidzip)[T.97089.0] 0.0935 0.046 2.031 0.042 0.003 0.184
C(regionidzip)[T.97091.0] 0.0889 0.045 1.977 0.048 0.001 0.177
C(regionidzip)[T.97094.0] 0.0986 0.071 1.384 0.166 -0.041 0.238
C(regionidzip)[T.97097.0] 0.0831 0.048 1.726 0.084 -0.011 0.178
C(regionidzip)[T.97098.0] 0.1622 0.066 2.476 0.013 0.034 0.291
C(regionidzip)[T.97099.0] 0.0472 0.050 0.945 0.345 -0.051 0.145
C(regionidzip)[T.97101.0] 0.0767 0.050 1.547 0.122 -0.021 0.174
C(regionidzip)[T.97104.0] 0.1146 0.053 2.148 0.032 0.010 0.219
C(regionidzip)[T.97106.0] 0.0840 0.047 1.771 0.077 -0.009 0.177
C(regionidzip)[T.97107.0] 0.0685 0.051 1.350 0.177 -0.031 0.168
C(regionidzip)[T.97108.0] -0.2492 0.162 -1.537 0.124 -0.567 0.069
C(regionidzip)[T.97109.0] 0.1068 0.051 2.105 0.035 0.007 0.206
C(regionidzip)[T.97111.0] 0.0898 0.162 0.554 0.579 -0.228 0.407
C(regionidzip)[T.97113.0] 0.0906 0.056 1.630 0.103 -0.018 0.200
C(regionidzip)[T.97116.0] 0.1033 0.045 2.288 0.022 0.015 0.192
C(regionidzip)[T.97118.0] 0.0796 0.044 1.795 0.073 -0.007 0.166
C(regionidzip)[T.97298.0] 0.1319 0.060 2.196 0.028 0.014 0.250
C(regionidzip)[T.97316.0] 0.1940 0.118 1.645 0.100 -0.037 0.425
C(regionidzip)[T.97317.0] -0.0006 0.046 -0.013 0.989 -0.090 0.089
C(regionidzip)[T.97318.0] 0.0454 0.044 1.036 0.300 -0.040 0.131
C(regionidzip)[T.97319.0] 0.0748 0.043 1.749 0.080 -0.009 0.159
C(regionidzip)[T.97323.0] 0.1476 0.059 2.514 0.012 0.033 0.263
C(regionidzip)[T.97324.0] 0.1182 0.118 1.002 0.316 -0.113 0.349
C(regionidzip)[T.97328.0] 0.0797 0.043 1.853 0.064 -0.005 0.164
C(regionidzip)[T.97329.0] 0.0782 0.044 1.774 0.076 -0.008 0.165
C(regionidzip)[T.97330.0] 0.0625 0.046 1.347 0.178 -0.028 0.153
C(regionidzip)[T.97331.0] 0.1023 0.099 1.034 0.301 -0.092 0.296
C(regionidzip)[T.97344.0] 0.1660 0.071 2.329 0.020 0.026 0.306
ln_calculatedfinishedsquarefeet 0.0126 0.004 2.969 0.003 0.004 0.021
==============================================================================
Omnibus: 4118.501 Durbin-Watson: 2.002
Prob(Omnibus): 0.000 Jarque-Bera (JB): 2425694.947
Skew: 0.796 Prob(JB): 0.00
Kurtosis: 83.407 Cond. No. 3.45e+03
==============================================================================
Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
[2] The condition number is large, 3.45e+03. This might indicate that there are
strong multicollinearity or other numerical problems.