Suppose that you want to estimate the effect of years of education on (log)wages. As yourealize that years of education is likely endogenous, youdecide to use the instrumental variables (IV) method.

a. Explain the conditions needed for an instrument to be valid.

b. State a proof that the IV estimator is a consistent estimator for a modelwhich includes just one explanatory variable (say x).

c. Give some examples of what could be a good instrument for years ofeducation.

2.

Consider a simple regression model in which the observed regressor (????????1)contains measurement error. The model is:Where the notation is an in the curriculum. ????????1∗ is the true value of the

regressor, whereas ????????1 is the measurement error. ????????1 is uncorrelated with u and

a. Suppose that the variance in ????????1

∗ is 3 and that the variance in ????????1 is 3.

Assume that the true value of ????????1 = 4. Using the result on theprobability limit on the OLS estimator of ????????1 from the curriculum,what estimated value of the parameter will be observedin largesamples?

b. What happens to the bias if the variance in ????????1 is 0?

c. What estimation method could be used to obtain a consistent estimate

of ????????1?

3.

a. Explain assumption MLR.5 made in chapter 4 in the curriculum.

b. Explain whether assumption MLR.5 affects the results that OLS providesan unbiased estimator of the parameters in a multiple linear regression.

4 (5)

c. Explain what you would do If MLR.5 does not hold and you wanted to testwhether a coefficient in a linear regression is different from zero.

4.

Consider the first stage equation in which you have regressed ”averageprotection against expropriation risk” on the instrument log settler mortality.

You get the following output:

a. What does the t-ratio tell us about the significance of log settlermortality (logem4)?

b. What assumption needed for valid IV estimation is being tested by

the above regression?

c. What is the 95 % confidence interval for the constant?_cons 9.514584 .5715876 16.65 0.000 8.373998 10.65517

logem4 -.6314299 .1243154 -5.08 0.000 -.8794974 -.3833624

avexpr Coef. Std. Err. t P>|t| [95% Conf. Interval]

Robust

Root MSE = 1.3041

R-squared = 0.3047

Prob > F = 0.0000

F(1, 68) = 25.80

Linear regression Number of obs = 70

5 (5)

5.

The following graph shows the general fertility rate (gfr) and the value of the

allowance for dependents (pe). The graph also shows the residuals from a

regression of gfr on pe.

a. Comment on whether the two yield series look like I(1) series (i.e.

non-stationary series)? What about the residuals from the regression

of gfr on pe?

b. Explain how the residuals should behave in the case when the

variables are cointegrated.

c. Explain how you formally would test for stationarity and

cointegration.