Question

4. The table below has four models relating the graduate students grade point average (GGPA) of 427 students to their college grade point av- erage (COLGPA) and to their verbal and quantitative scores (VERB QUAN) in the Graduate Records Exam (GRE). A number of" dummy" variables are also included: DCAM-1 if the studen lived on ca pus, DPRI-1 if the student graduated from a private college, and several dummy variables for major categories science (DSCI), social science (DSOC), humanities (DHUM), and arts (DARTS). (The values in parentheses are standard errors.) Estimated Models for the Grade Point Average Data Variable Model A Model B Model C Model D CONSTANT 0.367 0.363 0.423 0.422 0.224) 0.224 (0.220 0.221) 0.414 COLGPA 0.406 0.398 0.389 (0.063) 0.02 (0.061 (0.062) VERB 0.00073 0.00068 0.00074 0.00079 (0.00029) (0.00029) (0.00028) (0.00029) 0.0011 (0.0003) 0.0003 0.0003) 0.0003) 0.027 QUAN 0.0011 0.0010 0.0010 DSCI -0.026 (0.057) 0.057) 0.054 0.056 DSOC (0.073) 0.073) 0.0041 0.0068 DHUM 0.142) (0.141) DARTS 0.243 (0.189) (0.188) 0.041 (0.052) 0.229 DCAM -0.040 (0.052) 0.033 DPRI 0.029 (0.063) (0.063) ESS 6.20 96.421 97.164 0.213 96.932 R2 0.211 0.215 0.213 (a) In the relevant models, test whether there is any difference due to the student's lving on campus or off campus. State the null and alternative hypotheses, the test statistic and its distribution, and the p-value (b) In the relevant models, test whether there is any difference due to the student's having graduated from a private or other type of school. State the and alternative hypotheses, the test statistic and its distribution, and the p-value (c) In Model A, test whether the dummy variables for majors are jointly significant at the 5 percent significance level. Be sure to state the nul and alternative hypotheses, the test statistic and its distribution. Perform the same test with Model B as the unrestricted model. Do you get the same result? (d) In Model A, test the hypothesis that all of the dummy variables have zero regression coefficients.

Answer 1

Answer:

a)

Since the dummy variable, DCAM is used in model D, hence it is the relevant model.

Now, the hypothesis is tested by calculating t-value and corresponding p-value as shown below,

Null Hypothesis: $H_0: \beta_{\text{DCAM}}=0$

Alternate Hypothesis: $H_1: \beta_{\text{DCAM}}\neq0$

The t value is obtained using the formula,

β-0 t=35 SE

From the model summary,

6-0 -0.040 - 076023 t-SE-=-0.052 一一0.76923

The P-value is obtained from t distribution function for t = -0.76923 and degree of freedom = n -1 = 427 - 1 = 426. In excel use function =T.DIST.2T(x,deg_freedom).

P-value0.4422

The P-value = 0.4422 is greater than 0.05 at 5% significance level. the null hypothesis is not rejected. Now we can state that there is no significant effect of DCAM in regression model.

b)

Similarly for dummy variable DPI, (relevant model is model D) hypothesis is tested by calculating t-value and corresponding p-value

From the model summary,

0.033-0.5238 0.0630.5238 S Eg

The P-value is obtained from t distribution function for t = 0.5238 and degree of freedom = n -1 = 427 - 1 = 426.

P-value0.6007

The P-value = 0.6007 is greater than 0.05 at 5% significance level. the null hypothesis is not rejected. Now we can state that there is no significant effect of DPRI in regression model.

c)

The joint significance of major dummy variable is tested by comparing model A with model D. Where the model A is unrestricted model (i.e. all the variables included) and the model C is restricted model (dummy variables excluded)

The F-statistic is used to test the hypothesis,

Hypothesis,

$H_0:\beta_5=\beta_6=\beta_7=\beta_8=0$

H1 : Not all the coefficient are zero at 5% significant level

The F-statistic is,

(1 - R-R)/(n - k - 1)

Where, q is the number of restriction, q = 4

k is the number of parameter in unrestricted model = 10

n = 427

(0.215 0.213)/4 2649 0.2649 (1-RUR) (n-k-1) = (1-0.215) (427-10-1)

The P-value for F-statistic is obtained using the F-distribution table for numerator df = q = 4 and denominator df = n - k - 1 = 424

P-value0.9004

The P-value = 0.9004 is greater than 0.05 at 5% significance level. the null hypothesis is not rejected.

Similarly, comparing model B and model D,

The F-statistic is,

(0.213 0.211)/4 (1-R-R)/(n-k-1) = (1-0.213) (427-8-1) 0.26556

P-value0.9000

The P-value = 0.9079 is greater than 0.05 at 5% significance level. the null hypothesis is not rejected.

results are same

d)

The joint significance of dummy variable is tested by comparing model A with model C. Where the model A is unrestricted model (i.e. all the variables included) and the model C is restricted model (dummy variables excluded)

The F-statistic is used to test the hypothesis,

Hypothesis,

$H_0:\beta_5=\beta_6=\beta_7=\beta_8=\beta_9=\beta_{10}=0$

H1 : Not all the coefficient are zero at 5% significant level

The F-statistic is,

(1 - R-R)/(n - k - 1)

Where, q is the number of restriction, q = 6

k is the number of parameter in unrestricted model = 10

n = 427

(0215-0211)/6 0.35329 (1-RUR) (n-k-1) = (1-0.215) (427-10-1)

The P-value for F-statistic is obtained using the F-distribution table for numerator df = q = 6 and denominator df = n - k - 1 = 424

$\text{P-value}=0.9079$

The P-value = 0.9079 is greater than 0.05 at 5% significance level. the null hypothesis is not rejected.