1. Suppose that Leon uses the OLS estimator to estimate a simple linear regression model.
Leon notices that the sample correlation between the OLS regression residuals and the
independent variable is equal to zero. ‘This zero correlation suggests that there is no
omitted variable which is correlated with the independent variable,’ says Leon. Do you
agree with Leon? Explain your answer.
2. Let ??1 denote the slope coefficient on ??1 in the population regression function. Suppose that
the OLS estimate of ??1 is 0.3 and all Gauss-Markov assumptions are satisfied. ‘Since the
OLS estimator is an unbiased estimator of ??1, we can conclude that the true value of ??1 is
0.3,’ says Richter. Do you agree with Richter? Explain your answer.
3. Let ??1 denote the slope coefficient on ??1 in the population regression function. ‘If the two
bounds of the asymptotic 95% confidence interval for ??1 are estimated to be 0.1 and 0.5,
there is a 95% chance that the true value of ??1 lies between 0.1 and 0.5,’ says Maria. Do
you agree with Maria? Explain your answer.
4. Consider a sample of ?? observations and let ?? index individual observations. Let ?????? =
∑ (???? ? ??0 ? ??1????)
2 ??
??=1 denote the sum of squared residuals for a simple linear regression
model of ?? on ??, evaluated at some estimates ??0 and ??1 of the true population parameters.
Let ?????? = ∑ (???? ? ???)
2 ??
??=1 denote the total sum of squares in the dependent variable, where ???
is the sample mean of ??. Define the R-squared as ??
2 = 1 ? ??????/??????. ‘Even when the OLS
estimator is inconsistent and the TSLS estimator is consistent, the OLS estimates produce
a higher value of the R-squared than the TSLS estimates,’ says Alucard. Do you agree with
Alucard? Explain your answer.
5. Consider a multiple linear regression model of ?? on ??1 and ??2, ?? = ??0 + ??1??1 + ??2??2 + ??,
where ?? denotes the population error term. Suppose that all Gauss-Markov assumptions
except homoskedasticity are satisfied. ‘If the Breusch-Pagan test rejects the null hypothesis
of homoskedasticity in the error term ??, we can be certain that there is heteroskedasticity,’
says Carmilla. Describe steps involved in obtaining a Breusch-Pagan test statistic, and
explain to what extent you agree with Carmilla.
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6. Consider a simple linear regression model ?? = ???? + ??, where ?? is an ?? × 1 vector of
observations on the dependent variable; ?? is an ?? × 2 matrix of observations on the
constant regressor and the independent variable; ?? is a 2 × 1 vector of unknown
parameters; and ?? is an ?? × 1 vector of population regression errors. Suppose that in a
sample of ?? = 3 observations, the vector ??, the matrix ??, and the matrix (??
′??)
?1 are given
by
?? = [
2
0
5
], ?? = [
1 3
1 ?3
1 5
], and (??
′??)
?1 = [
0.413 ?0.048
?0.048 0.029 ].
Using this information, compute the OLS estimate of each parameter in ??. You must show
your working.
7. Consider the following time series OLS regression
????? = 3.41 + 2.72??1?? ? 5.33??2??
???? = 1.60 ?????? ?? = 60
where ?? indexes the period of observation; ???? denotes the Durbin-Watson statistic; and ??
is the sample size. Given the model specification and the sample size, the Durbin-Watson
test has 5% critical values of ???? = 1.514 and ???? = 1.652. ‘The presence of autocorrelation
implies that the OLS estimator is inefficient but it does not imply that the OLS estimator is
biased. In this regression, since DW lies between the two critical values, I conclude that
there is no autocorrelation,’ says Morris. To what extent do you agree with Morris? Explain
your answer.
Page 4 of 4 ECON47815-WE01
8. Suppose that the structural equation of interest is a simple linear regression model with the
dependent variable ?? and the endogenous independent variable ??. Suppose further that an
instrumental variable ?? is available to address the endogeneity of ??. Using this notation and
any additional notation that you require, present the first and second stage regression
models that you can estimate to obtain the TSLS estimate of the coefficient on ?? in the
structural equation. As part of your answer, you must explain what instrument exogeneity
and instrument relevance refer to.
