# Exogeneity: Wu-Hausman and Sargan Tests in Python

Last Update: April 24, 2022

Exogeneity: Wu-Hausman and Sargan Tests in Python can be done using `linearmodels` package `IV2SLS` function, `wu_hausman` method and `woldridge_regression`, `sargan` attributes found within `linearmodels.iv.model` module for evaluating whether linear regression independent variables are not correlated with error term (exogenous) and whether instrumental variables are not correlated with second stage least squares linear regression error term (valid instruments). Main parameters within `IV2SLS` function are `dependent` with model dependent variable, `exog` with model exogenous independent variable, `endog` with model endogenous independent variable and `instruments` with model instrumental variables.

As example, we can do Wu-Hausman, Wu-Hausman (Wooldridge) and Sargan tests from original multiple linear regression of house price explained by its lot size and number of bedrooms with whether house has a driveway and number of garage places as instrumental variable using data included within `AER` R package `HousePrices` object [1].

First, we import packages `statsmodels` for data downloading and ordinary least squares original model fitting and `linearmodels` for two stage least squares model fitting, Wu-Hausman, Wu-Hausman (Wooldridge) and Sargan tests [2].

``````In [1]:
import statsmodels.api as sm
import statsmodels.formula.api as smf
import linearmodels.iv.model as lm
``````

Second, we create `houseprices` data object using `get_rdataset` function and display first five rows and first three columns together with sixth and eleventh columns of data using `print` function and `head` data frame method to view its structure.

``````In [2]:
houseprices = sm.datasets.get_rdataset(dataname="HousePrices", package="AER", cache=True).data
print(houseprices.iloc[:, list(range(3)) + [5] + [10]].head())
``````
``````Out [2]:
price  lotsize  bedrooms driveway  garage
0  42000.0     5850         3      yes       1
1  38500.0     4000         2      yes       0
2  49500.0     3060         3      yes       0
3  60500.0     6650         3      yes       0
4  61000.0     6360         2      yes       0
``````

Third, we fit original model with `ols` function using variables within `houseprices` data object and store outcome within `mlr1` object. Within `ols` function, parameter `formula = “price ~ lotsize + bedrooms”` fits model where house price is explained by its lot size and number of bedrooms.

``````In [3]:
mlr1 = smf.ols(formula="price ~ lotsize + bedrooms", data=houseprices).fit()
``````

Fourth, we create `mdatac` model data object and add a constant column using `add_constant` function. Within `add_constant` function, parameters `data=houseprices` includes `houseprices` data object and `prepend=False` includes logical value to add constant at last column of `mdatac` data object. Then, we fit two stage least squares model with `IV2SLS` function using variables within `mdatac` data object and store outcome within `mlr2` object. Within `IV2SLS` function, parameters `dependent=mdatac["price"]` includes model house price dependent variable, `exog=mdatac[["const", "bedrooms"]]` includes model number of bedrooms exogenous independent variable, `endog=mdatac["lotsize"]` includes model lot size endogenous independent variable, `instruments=mdatac[["driveway", "garage"]]` includes model whether house has a driveway and number of garage places instrumental variables, `cov_type="homoskedastic"` includes model homoskedastic variance covariance matrix estimation and `debiased=True` includes logical value to adjust model variance covariance matrix estimation for degrees of freedom. Notice that `IV2SLS` function parameters `cov_type="homoskedastic"` and `debiased=True` were only included as educational examples which can be modified according to your needs. Also, notice that doing stage by stage instead of simultaneous stages estimation of two stage least squares model with `ols` function would estimate correct coefficients but incorrect standard errors and F-statistic. Additionally, notice that two stage least squares `mlr2` model estimation assumes errors are homoskedastic.

``````In [4]:
mlr2 = lm.IV2SLS(dependent=mdatac["price"], exog=mdatac[["const", "bedrooms"]], endog=mdatac["lotsize"], instruments=mdatac[["driveway", "garage"]]).fit(cov_type="homoskedastic", debiased=True)
``````

Fifth, we can print Wu-Hausman test results using `mlr2` model `wu_hausman` method.

``````In [5]:
print(mlr2.wu_hausman())
``````
``````Out [5]:
Wu-Hausman test of exogeneity
H0: All endogenous variables are exogenous
Statistic: 50.9308
P-value: 0.0000
Distributed: F(1,542)
``````

Sixth, we can print Wu-Hausman (Wooldridge) test results using `mlr2` model `wooldridge_regression` attribute.

``````In [6]:
print(mlr2.wooldridge_regression)
``````
``````Out [6]:
Wooldridge's regression test of exogeneity
H0: Endogenous variables are exogenous
Statistic: 50.9046
P-value: 0.0000
Distributed: chi2(1)
``````

Seventh, we can print Sargan test results using `mlr2` model `sargan` attribute.

``````In [7]:
print(mlr2.sargan)
``````
``````Out [7]:
Sargan's test of overidentification
H0: The model is not overidentified.
Statistic: 0.0477
P-value: 0.8271
Distributed: chi2(1)
``````

Courses

My online courses are hosted at Teachable website.

For more details on this concept, you can view my Linear Regression in Python Course.

References

[1] Data Description: Sales prices of houses sold in the city of Windsor, Canada, during July, August and September, 1987.

Original Source: Anglin, P., and Gencay, R. (1996). Semiparametric Estimation of a Hedonic Price Function. Journal of Applied Econometrics, 11, 633–648.

[2] statsmodels Python package: Seabold, Skipper, and Josef Perktold. (2010). “statsmodels: Econometric and statistical modeling with python”. Proceedings of the 9th Python in Science Conference.

linearmodels Python package: Kevin Sheppard. (2021). “Linear (regression) models for Python. Extends statsmodels with Panel regression, instrumental variable estimators, system estimators and models for estimating asset prices”. Python package version 4.25.

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