# Endogeneity and Hausman’s Test we deal with the issue of endogeneity using Hausman’s test. In several tome series questions, , we have estimated the…

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: Endogeneity and Hausman’s Test

we deal with the issue of endogeneity using Hausman’s test. In several tome series questions, , we have estimated the model of interest rate determination. Economic theory shows that nominal interest rates depend on the nominal quantity of money and real income (real GDP):

NINTt = a + b1 Mt + b2 RGDPt +εt [Model 2]

where NINT = short-term nominal interest rate (measured in %), M = nominal quantity of money (M2, measured in billions of $), and RGDP = real GDP (measured in billions of $). Under the null of no endogeneity, the estimates in Model 2 are unbiased and efficient, but under the alternative hypothesis of endogeneity, the estimates are biased and inefficient

In Model 1, NINT and M are simultaneously determined within the model: money (M) affects the interest rate (NINT), and the interest rate in turn affects the amount of money (=

feedback effect

). We have seen that if there is a feedback relationship between the dependent variable and an explanatory variable, then the explanatory variable is endogenous. When an explanatory variable is endogenous, the explanatory variable is correlated with the error term, and the estimate is biased (= endogeneity bias). One common procedure to resolve the endogeneity bias problem is to use an instrumental variable as a proxy for the endogenous variable.

NINTt = a + b1IVt + b2 RGDPt +εt [Model 1]

The estimates in Model 1 are always consistent under the null and alternative hypotheses.

(1) First, estimate Model 2 using the same data as Computer Project 1. (In STATA, a model in which estimates are unbiased and efficient under the null hypothesis of no endogeneity, but inconsistent under the alternative hypothesis endogeneity is referred as ‘model 2’.)

(2) Next, estimate Model 1 using the past values of M and NINT (Mt-1 and NINTt-1) as IV variables. Are estimated coefficients significant at the 5% level of significance? (In STATA, a model in which estimates are consistent under the null and alternative hypotheses is referred as ‘model 1’.)

(3) Interpret the estimated coefficients of Model 2. Do they have expected signs?

(4) Using Hausman’s specification test, test whether there is an endogeneity bias in Model 2.

(5) Which estimates (Model 1 or Model 2) would you prefer to use? Explain.

STATA Command for Project #3

use c:econmacrovariables

reg nint m rgdp

estimates store model2 (*”Efficient under H0″, but “inefficient under Ha” should be model2.)

gen lm = L.m

gen lint = L.nint

ivreg nint rgdp (m = lm lint)

estimates store model1 (*”Always consistent under H0 and Ha” should be model1)

hausman model1 model2

STATA Command (General Description)

hausman performs Hausman’s specification test. To use hausman, perform the following steps:

(1) Obtain an estimator that is consistent whether or not the hypothesis is true (models with no errors in measurement or no endogeneity).

(2) Store the estimation results under name-consistent by using estimates store:

(3) Obtain an estimator that is efficient (and consistent) under the hypothesis that you are testing, but inconsistent otherwise (models with errors in measurement or endogeneity).

(4) Store the estimation results under name-efficient by using estimates store:

(5) Use hausman to perform the test.

hausman name-consistent name-efficient [, options]

The order of computing the two estimators may be reversed. Thus, you have to be careful to specify to hausman the models in the order “always consistent” first and “efficient under H0” second.

The following options are available:

constant includes estimated intercepts in comparison. The default is to exclude.

alleqs (eqlist) uses all equations to perform test; default is first equation only

eq specifies, by number, the pairs of equations that are to be compared. For instance, eq(1:1), eq(1:1, 2:2) or eq(1:2). eq(1:2) means that equation 1 of the always-consistent estimator is to be tested against equation 2 of the efficient estimator. eq(1:1, 2:2) means that equation 1 is to be tested against equation1 and that equation 2 is to be tested against equation2.

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