CHAPTER 19
STOCHASTIC REGRESSORS AND THE
METHOD OF INSTRUMENTAL
VARIABLES
Damodar Gujarati
Econometrics by Example
STOCHASTIC REGRESSORS
 One of the critical assumptions of the classical linear
regression model is that the error term and regressor(s) are
uncorrelated.
 If they are correlated, we call such regressor(s) stochastic
or endogenous regressors.
 In this situation the OLS estimators are biased and the bias
does not disappear even if the sample size increases
indefinitely.
 Therefore, the OLS estimators are not even consistent.
 As a result, tests of significance and hypothesis testing becomes
suspect.
Damodar Gujarati
Econometrics by Example
ASYMPTOTIC BIAS
 It can be shown that the difference between b2, the
estimate of B2 (the true value of the coefficient on the
stochastic variable) is equal to:
cov( X i , ui )
var( X i )
 If the covariance between the regressor and the error term is
positive, b2 will overestimate the true B2, a positive bias.
 If the covariance term is negative, b2 will underestimate B2, a
negative bias.
Damodar Gujarati
Econometrics by Example
REASONS FOR CORRELATION BETWEEN
REGRESSORS AND THE ERROR TERM
 1. Measurement errors in the regressor(s)
 2. Omitted Variable Bias
 3. Simultaneous Equation Bias
 4. Dynamic regression model with serial correlation in
the error term
Damodar Gujarati
Econometrics by Example
INSTRUMENTAL VARIABLES (IV)
 Proxy variables can give us consistent estimates of the
coefficients of the suspected stochastic regressors
 They must be:
 (1) Uncorrelated with the error term
 (2) Correlated with the stochastic regressors
 (3) Not regressors in their own right
 Such variables are called instrumental variables or
instruments.
Damodar Gujarati
Econometrics by Example
IV ESTIMATORS
 In large samples IV estimators are normally
distributed with mean equal to the true population
value of the regressor under stress and the variance that
involves the population correlation coefficient of the
instrument with the suspect stochastic regressor.
 But in small, or finite, samples, IV estimators are
biased and their variances are less efficient than the
OLS estimators.
Damodar Gujarati
Econometrics by Example
SUCCESS OF IV
 The success of IV depends on how strong they are.
 Instrumental variables must be strongly correlated with
the stochastic regressor.
 If this correlation is high, we say such IVs are strong,
but if it is low, we call them weak instruments.
 If the instruments are weak, IV estimators may not be
normally distributed even in large samples.
 A rule of thumb says that an F- statistic in the first
step of the Hausman test less than 10 suggests that the
chosen instrument is weak.
Damodar Gujarati
Econometrics by Example
IV ESTIMATION
 Stage 1: Regress the suspected endogenous variable (S) on the chosen instrument
(Z) and the other regressors in the original model and obtain the estimated value
of S from this regression, S-hat.
 Step 2: Run the main regression on the regressors included in the original model
but replace the potentially endogenous variable by its value estimated from the
Step 1 regression.
 Note that the standard errors will need to be altered.
 This method of estimating the parameters of the model of interest is appropriately
called the method of two-stage least squares (2SLS), for we apply OLS twice.
 The value of the coefficient on S-hat will be equal to:
IV
2
b
 zi yi

 zi xi
where z, y, and x represent deviations from the mean values of Z, Y, and X, respectively.
Damodar Gujarati
Econometrics by Example
TEST OF ENDOGENEITY OF A REGRESSOR
 Hausman Test
 Step 1: Regress the endogenous S on all the (non-stochastic)
regressors in the original equation plus the instrumental
variable(s) and obtain residuals from this regression.
 Step 2: Regress the dependent variable (Y) on all the regressors,
including the (stochastic) S and the residuals from Step 1.
 If in this regression the t value of the residuals variable is
statistically significant, we conclude that S is endogenous or
stochastic.
 If it is not, then there is no need for IV estimation.
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Econometrics by Example
IDENTIFICATION
 1. If the number of instruments equals the number of
endogenous regressors, the regression coefficients are exactly
identified, and we can obtain unique estimates of them.
 2. If the number of instruments exceeds the number of
regressors, the regression coefficients are overidentified, in
which case we may obtain more than one estimate of one or
more of the regressors.
 3. If the number of instruments is less than the number of
endogenous regressors, the regression coefficients are
underidentified, and we cannot obtain unique values of the
regression coefficients.
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Econometrics by Example
TESTING THE VALIDITY OF SURPLUS INSTRUMENT
 TEST OF OVERIDENTIFYING RESTRICTIONS
 1. Obtain the IV estimates of the regression coefficients for the
main regression.
 2. Obtain residuals from this regression, e.
 3. Regress e on all the original regressors, including the
instruments, and obtain the R2 value from this regression.
 4. Multiply R2 value obtained in (3) by the sample size n.
 It can be shown that nR2 follows the chi-square distribution with m df,
where m is the number of surplus instruments.
 5. If the estimated chi-square value exceeds the critical chi-square
value, we conclude that at one surplus instrument is not valid.
Damodar Gujarati
Econometrics by Example