Last Update: March 24, 2022
Exogeneity: Wu-Hausman and Sargan Tests in R can be done using AER
package ivreg
, summary
for ivreg
functions 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 ivreg
function are formula
with y ~ x1 + x2 | x2 + z1 + z2
original model with x1
endogenous independent variable and x2
exogenous independent variable followed by first stage least squares model with x2
exogenous independent variable, z1
and z2
instrumental variables description and data
with data.frame
object including models variables. Main parameters within summary
for ivreg
function are object
with ivreg
function instrumental variables and two stage least squares estimation and diagnostics
with logical value to print Wu-Hausman (Wooldridge) and Sargan tests results.
As example, we can do 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 variables using data included within AER
package HousePrices
object [1].
First, we load package AER
for data, two stage least squares estimation, Wu-Hausman (Wooldridge) and Sargan tests [2].
In [1]:
library(AER)
Second, we create HousePrices
data object from AER
package using data
function and print first six rows, first three columns together with sixth and eleventh columns of data using head
function to view data.frame
structure.
In [2]:
data(HousePrices)
head(HousePrices[, c(1:3, 6, 11)])
Out [2]:
price lotsize bedrooms driveway garage
1 42000 5850 3 yes 1
2 38500 4000 2 yes 0
3 49500 3060 3 yes 0
4 60500 6650 3 yes 0
5 61000 6360 2 yes 0
6 66000 4160 3 yes 0
Third, we fit original model with lm
function using variables within HousePrices
data object and store outcome within mlr1
object. Within lm
function, parameter formula = price ~ lotsize + bedrooms
fits original model where house price is explained by its lot size and number of bedrooms.
In [3]:
mlr1 <- lm(formula = price ~ lotsize + bedrooms, data = HousePrices)
Fourth, we fit two stage least squares model with ivreg
function using variables within HousePrices
data object and store outcome within mlr2
object. Within ivreg
function, parameter formula = price ~ lotsize + bedrooms | bedrooms + driveway + garage
fits original model where house price is explained by its lot size endogenous independent variable and number of bedrooms exogenous independent variable followed by first stage least squares model number of bedrooms exogenous independent variable, whether house has a driveway and number of garage places instrumental variables. Notice that doing stage by stage instead of simultaneous stages estimation of two stage least squares model with lm
function would estimate correct coefficients but incorrect standard errors and F-statistic.
In [4]:
mlr2 <- ivreg(formula = price ~ lotsize + bedrooms | bedrooms + driveway + garage, data = HousePrices)
Fifth, we do Wu-Hausman (Wooldridge) and Sargan tests using summary
for ivreg
function. Within summary
for ivreg
function, parameters object = mlr2
includes mlr2
model results and diagnostics = TRUE
includes logical value to print Wu-Hausman (Wooldridge) and Sargan tests results. Notice that two stage least squares mlr2
model estimation assumes errors are homoskedastic unless heteroskedasticity consistent variance covariance matrix estimation is used within summary
for ivreg
function.
In [5]:
summary(object = mlr2, diagnostics = TRUE)
Out [5]:
Call:
ivreg(formula = price ~ lotsize + bedrooms | bedrooms + driveway +
garage, data = HousePrices)
Residuals:
Min 1Q Median 3Q Max
-115962 -11520 2287 14482 85515
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -19130.16 6540.67 -2.925 0.00359 **
lotsize 12.52 1.24 10.096 < 2e-16 ***
bedrooms 7680.13 1574.09 4.879 1.4e-06 ***
Diagnostic tests:
df1 df2 statistic p-value
Weak instruments 2 542 54.403 < 2e-16 ***
Wu-Hausman 1 542 50.905 3.12e-12 ***
Sargan 1 NA 0.048 0.827
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 25370 on 543 degrees of freedom
Multiple R-Squared: 0.1009, Adjusted R-squared: 0.09763
Wald test: 91.52 on 2 and 543 DF, p-value: < 2.2e-16
Courses
My online courses are hosted at Teachable website.
For more details on this concept, you can view my Linear Regression in R 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] AER R Package. Christian Kleiber and Achim Zeileis. (2008). Applied Econometrics with R. Springer-Verlag, New York.