ECMT 6002/6702: Econometric Applications
1. Practice problems
1. (Some basic algebra related to the OLS estimator) Suppose that observations (y1, . . . , yT) and (x1, . . . , xT ) are given and define
Verify the following:
Consider the following simple linear regression model:
yt = α + βxt + ut
(v) Show that the OLS estimator is given by
2. (Simple linear regression)
(i) Consider the following simple linear regression model:
yt = α + βxt + ut
Let (
, 
) be the OLS estimates and 
. Verify that the sample average of 
is exactly equal to zero.
(ii) Now consider the following regression model:
yt = βxt + ut
and let 
be the OLS estimate and 
. Is the sample average of 
still equal to zero?
(iii) Now consider the following regression model:
where
Let 
be the OLS estimate and 
. Show that the OLS estimate 
is equivalent to 
b in (i) and thus the sample average of 
is equal to zero.
(iv) Based on the results obtained in (i)-(iii), explain the role of the intercept.
2 Empirical application
Suppose that we are interested in the relationship between # of crimes and # of employed police officers on campus. The following model is considered:
Instructions:
1. Use the attached dataset. The dataset is considered in Wooldridge’s text book “Introductory Econometrics” and was collected from the FBI Uniform Crime Report.
2. Check summary statistics for yt and xt . If you are using R, use summary(y) and summary(x) if y and x are the vectors contain the observations.
3. Compute the OLS estimates of α and β. In R, lm(y ∼ x) can be used. You can easily do this in any statistical package but, I would recommend you compute those by your own code (see (1.2)). Depending on which software is used, the estimates can be slightly different. But they must be close to the following obtained by lm(y ∼ x) in R:
4. This computing exercise is not mandatory; any computing code related to this application will not be asked in the exams. But you are strongly encouraged to work on this by yourself with your preferred statistical software (this will be helpful for your assignments).