ECMT 6002/6702: Econometric Applications 3

1    Practice problems

1. Consider the following linear regression model:

yt = β 1 + β2x2t + β3x3t + ut,    t = 1, . . . , 100.

Suppose that we want to test if

H0 = β2 = β3 = 0.                                                     (1.1)

(i) Write (a) the unrestricted and restricted model for the F-test of the above hypotheses and (b) how to estimate each model.

(ii) Suppose that URSS and RRSS are given 805 and 1015, respectively. Write the F-statistic and report the test result when 95% significance level is employed.

(Note) 5% quantiles of F (a, b)

(iii)  (Optional) Note that in the above F-table, the changes in the 5% quantile depending on the “df in the denominator” are very small. Can you explain why this is reasonable based on the χ2  approximation of the F distribution discussed in class?

2. Suppose that we are interested in the following hypothesis in the previous question.

H0 = β2 + β3 = 0.                                                     (1.2)

(i) Write (a) the unrestricted and restricted model for the F-test of the above hypothesis and (b) how to estimate each model.

(ii) Suppose that URSS and RRSS are given 805 and 1015, respectively. Write the F-statistic and report the test result when 95% significance level is employed.

2    Empirical application

We will consider the dataset “ECONMATH” given in Wooldridge’s textbook“Introductory Econo- metrics”. The dataset contains grade point averages and standardized test scores, along with per- formance in an introductory economics course, for students at a large public university.  We con- sider the following regression model:

yt         = β 1 + β2 x2t  + β3   x3t     + β4   x4t    + β5    x5t     + ut ,

where

test scores : scores of a certain test, which we want to predict

GPA : GPA measured at the beginning of the semester

ACT-M : students ACT scores in math

ACT-E : students ACT scores in English

Calculus : variable taking 1 (if the student took calculus) or 0 (otherwise).

For more detailed explanation, check Wooldridge’s textbook.

Instructions:

1. The dataset contains missing values.  This quite common in empirical analysis.  First check these missing values (coded as "NA") and exclude individuals associated with such missing observations from the dataset if necessary.  In R, this can be done by na.omit(dataset),

where dataset is the matrix where each variable takes each column.

2. Compute the OLS estimates of β 1 , β2 , β3  and β4 . In R, lm(y ∼ X ) can be used if y is the vector of yt  and X is the (T × 5) data (independent variables) matrix. I would recommend you compute those directly.

3. Implement t-tests to examine

βj  = 0                                                           (2.1)

for each j .  Report the test results.  In R, the test results are contained in the results given by summary(lm(y ∼ X)). The results must be close to

11 ) = 5.78,            22 ) = 17.29,     33 ) = 0.47,            4 ) = 7.87       (2.2)

You can also try direct computation as in Week 2 tutorial.

4. Implement F-test to examine

H0  : β2 = β3 = β4 = 0.                                                 (2.3)