Suppose we wish to estimate the effect of the federally funded school lunch program on student performance. The file MEAP93.xls contains data on 408 high schools in Australia for the year 1993.
 math10: percentage of tenth graders at a high school receiving a passing scoreon a standardized mathematics exam.
 expend: expenditures per student (in dollars).
 lnchprg: percentage of students who are eligible for the federal free lunch
 enrol: number of student enrolment which measures school size
 Consider a regression model
Estimate the model using Eviews and write down the fitted equation (including the sample size, tstatistic, and Rsquared).
(b) Test the overall validity of the regression model at the 5% significance level.
(c) Does the lunch program increase student performance? Test the hypothesis at the 5% level.
(d) Based on the regression output, if expend increase by 10% what is the estimated percentage point change in math10, holding lnchprg and enroll constant.
(e) Now run the simple regression of math10 on lnchprg such that
Compare the Eviews output with that from part (a). Which model will you choose? Why? [Hint: should we use R^{2} or adjusted R^{2} for model comparison?]
Suppose a researcher wants to analysis the shortterm interest rate, and she develops a model given by
where is the threemonth Tbill rate, is the annual inflation rate based on the consumer price index (CPI), and is the federal budget surplus or deficit as a percentage of GDP. He collects the yearly data for the period from 1960 to 2016 (57 observations) from the Federal Reserve Economic Data. They are in the Excel file Interest_rate.xlsx.
 Plot the three variables in a line graph using Eviews and comment on the dynamics (any comovement between interest rate and the other two variables).
 Estimate the regression model using Eviews and provide the output.
 What does the coefficient of determination tell you?
 Interpret the coefficients ^{?B}_{1}and ^{?B}_{2} . Do the signs of the estimated coefficients agree with standard economic intuition?
 Conduct a test of H0: there is no second order serial correlation in the errors of this model at the 5% significance level.
 Reestimate the equation with NewyWest standard errors and provide the Eviews output. Compare the serial correlation consistent estimation with output from Part (b).
 Test the null hypothesis that a one percent increase in the annual inflation rate leads to a one percent increase in the shortterm interest rate on average at the 5% level.
In the file pubexp.xls there are data on public expenditure on education (EE), gross domestic product (GDP), and population (P) for 34 countries in the year 1980. It is hypothesized that per capita expenditure on education is linearly related to per capita GDP. That is
Import the data into Eviews.
 It is suspected that . Why might the suspicion
may be heteroskedastic with a variance related to about heteroskedasticity be reasonable?
 Draw a scatter plot between x and y using Eviews and comment on the graph (what is the expected relationship and any evidence of heteroskedasticity). Hint: To generate variable x using Eviews, click Genr, type in x=GDP/P in the box.
 Run the OLS regression and provide the Eviews output. Does the sign of the slope coefficient make sense? Explain.
 Test for the existence of heteroskedasticity using a White test (assuming a 5% level of significance).
 Reestimate the equation with White consistent standard errors and provide the Eviews output. Compare the heteroskedasticity consistent estimation with output from part (c).
Model Estimation with Eviews
a.
The regression output is given as
Dependent Variable: MATH10 

Method: Least Squares 

Date: 06/08/18 Time: 13:07 

Sample: 1 408 

Included observations: 408 

Variable 
Coefficient 
Std. Error 
tStatistic 
Prob. 
C 
23.13766 
24.99323 
0.925757 
0.3551 
LOG(EXPEND) 
7.746062 
3.041386 
2.546885 
0.0112 
LNCHPRG 
0.323927 
0.036319 
8.918835 
0.0000 
LOG(ENROL) 
1.255435 
0.581173 
2.160175 
0.0313 
Rsquared 
0.189291 
Mean dependent var 
24.10686 

Adjusted Rsquared 
0.183271 
S.D. dependent var 
10.49361 

S.E. of regression 
9.483400 
Akaike info criterion 
7.346718 

Sum squared resid 
36333.69 
Schwarz criterion 
7.386044 

Log likelihood 
1494.731 
HannanQuinn criter. 
7.362280 

Fstatistic 
31.44310 
DurbinWatson stat 
1.906937 

Prob(Fstatistic) 
0.000000 
From the regression result, the fitted equation is obtained as
The overall significance of the model can be tested to examine overall significance of the model.
Null hypothesis: Coefficients of all the independent variables are zero
Alternative hypothesis: At least one of the coefficients is significantly different from zero.
The significant p value F statistics from the regression model is obtained as 0.00. The p value is less than the significant value of 0.05. This indicates rejection of null hypothesis of neither of coefficients are statistically significant. The result thus suggests that the model has overall significance.
As obtained from the regression result the coefficient of lunch program is 0.32. The negative coefficient indicates an inverse relation between lunch program and percentage of tenth grade on a standardized math exam. This implies with increase in lunch program obtained grades of the students decreases. P value of the coefficient is 0.0000. As the p value is smaller than the level of significance 0.05, the null hypothesis of no significant relation between lunchprogram and students’ performance is rejected at 5% level of significance. Therefore, lunchprogram fails to increase performance of the student rather it influences student performance negatively.
The coefficient of log(expend) is 7.74. This indicates a 10% increase in expend will increase math10 by (7.74* 10) = 77.4 percent.
Dependent Variable: MATH10 

Method: Least Squares 

Date: 06/08/18 Time: 14:22 

Sample: 1 408 

Included observations: 408 

Variable 
Coefficient 
Std. Error 
tStatistic 
Prob. 
C 
32.14271 
0.997582 
32.22061 
0.0000 
LNCHPRG 
0.318864 
0.034839 
9.152422 
0.0000 
Rsquared 
0.171034 
Mean dependent var 
24.10686 

Adjusted Rsquared 
0.168992 
S.D. dependent var 
10.49361 

S.E. of regression 
9.565938 
Akaike info criterion 
7.359184 

Sum squared resid 
37151.91 
Schwarz criterion 
7.378847 

Log likelihood 
1499.274 
HannanQuinn criter. 
7.366965 

Fstatistic 
83.76683 
DurbinWatson stat 
1.907745 

Prob(Fstatistic) 
0.000000 
In both the model, the variable lunch program is negative and statistically significant. The magnitude of the coefficient in both the model is equivalent to 0.32. For the model in part (a) the value of adjusted R square is 0.18. In the new model, the R square value is 0.16. The R square value indicates goodness of fit of the model. This explains how much variation in the dependent variable is explained by the independent variable. Higher the R square value better is fitted model. In terms of R square value, model 1 is more acceptable as compared to model 2.
a.
Figure 1: Line plot of Tbill rate, inflation rate and federal budget balance
As shown from the figure above, interest rate and inflation rate moves in the same direction. Movement of interest rate follows the movement of inflation indicating a positive association between them. In most of the times, interest rate is above inflation rate. The variable federal budget does not show any clear comovement with Treasury bill interest rate.
Testing Overall Validity of Regression Model
The regression output for the concerned regression model is given as
Dependent Variable: I3 

Method: Least Squares 

Date: 06/08/18 Time: 12:04 

Sample: 1960 2016 

Included observations: 57 

Variable 
Coefficient 
Std. Error 
tStatistic 
Prob. 
C 
2.056980 
0.556708 
3.694899 
0.0005 
INF 
0.817576 
0.096280 
8.491619 
0.0000 
DEF 
0.190321 
0.115244 
1.651455 
0.1045 
Rsquared 
0.590980 
Mean dependent var 
4.687295 

Adjusted Rsquared 
0.575831 
S.D. dependent var 
3.109814 

S.E. of regression 
2.025369 
Akaike info criterion 
4.300576 

Sum squared resid 
221.5143 
Schwarz criterion 
4.408105 

Log likelihood 
119.5664 
HannanQuinn criter. 
4.342366 

Fstatistic 
39.01139 
DurbinWatson stat 
0.444886 

Prob(Fstatistic) 
0.000000 
Estimated regression equation
Value of R square gives coefficient of determination. It indicates how much of the variation in dependent variable can be explained by the independent variables. In the present model, the R square value is given as 0.59. This implies inflation and balance in federal budget can together explain 59 percent variation in interest rate.
The value of inflation coefficient is 0.82. This implies there is a positive relation between rate of inflation and 3 months Tbill rate. That is as inflation increases return to 3 months Tbill rate increases. The p value of the coefficient is 0.00. P value less than the level of significance value 0.05 indicates rejection of null hypothesis of no significant relation between inflation and interest on Tbill rate. The coefficient of federal budget is 0.19. The positive coefficient indicates that federal budget positively influences interest on Tbill rate. P value of the coefficient 0.10. The value is greater than significance level of 0.05. As the p value is greater than significant value, null hypothesis of no significant relation between T–bill rate and budget surplus is rejected.
Inflation and expected inflation affect the interest rate on Treasury bill. The period of high inflation is generally associated with a high interest rate. From the regression result, a positive significant relation is obtained between inflation and 3 months Tbill rate. The sign of inflation coefficient is thus consistent with standard economic intuition. The federal budget deficit is associated with an inflationary pressure. In presence budget deficit, central bank purchases securities issued by the government. This raises growth of monetary base crating inflationary pressure. This has a positive impact on interest rate. The variable budget deficit though is not statistically significant but it has the expected sign.
Test of serial autocorrelation
Hypothesis
Null hypothesis: There is no second order serial autocorrelation in the model at 5% level of significance
Alternative hypothesis: There exists a second order serial autocorrelation in the model at 5% level of significance.
Auxiliary regression
The auxiliary regression regress current values of residuals on all the explanatory variables and is related with lagged residual terms. The Breusch Godfrey test statistics is given as (T – p)*R^{2 }, where T is the number of observation and p is the number of lagged residual terms. The test statistics follows a chisquare distribution with p degrees of freedom.
Effect of Lunch Program on Student Performance
Decision rule
The null hypothesis is rejected if P value of the BreuschGodfrey test statistics is less than 0.05.
Conclusion
From the test result, p value of the LM statistics is obtained as 0.0000. As the p value is less than 0.05, the null hypothesis of no second order serial autocorrelation exists in the model is rejected. This implies the model has the problem of second order autocorrelation.
BreuschGodfrey Serial Correlation LM Test: 

Fstatistic 
39.78589 
Prob. F(2,52) 
0.0000 

Obs*Rsquared 
34.47237 
Prob. ChiSquare(2) 
0.0000 

Test Equation: 

Dependent Variable: RESID 

Method: Least Squares 

Date: 06/08/18 Time: 12:50 

Sample: 1960 2016 

Included observations: 57 

Presample missing value lagged residuals set to zero. 

Variable 
Coefficient 
Std. Error 
tStatistic 
Prob. 
C 
0.148451 
0.357065 
0.415752 
0.6793 
INF 
0.003478 
0.061905 
0.056180 
0.9554 
DEF 
0.040788 
0.074426 
0.548033 
0.5860 
RESID(1) 
0.738955 
0.140036 
5.276884 
0.0000 
RESID(2) 
0.062394 
0.140898 
0.442831 
0.6597 
Rsquared 
0.604778 
Mean dependent var 
7.64E16 

Adjusted Rsquared 
0.574377 
S.D. dependent var 
1.988872 

S.E. of regression 
1.297536 
Akaike info criterion 
3.442443 

Sum squared resid 
87.54723 
Schwarz criterion 
3.621658 

Log likelihood 
93.10962 
HannanQuinn criter. 
3.512092 

Fstatistic 
19.89295 
DurbinWatson stat 
1.995246 

Prob(Fstatistic) 
0.000000 
NewyWest standard error estimation model
Dependent Variable: I3 

Method: Least Squares 

Date: 06/10/18 Time: 10:12 

Sample: 1960 2016 

Included observations: 57 

HAC standard errors & covariance (Bartlett kernel, NeweyWest fixed 

bandwidth = 4.0000) 

Variable 
Coefficient 
Std. Error 
tStatistic 
Prob. 
C 
2.056980 
0.525294 
3.915866 
0.0003 
INF 
0.817576 
0.129667 
6.305189 
0.0000 
DEF 
0.190321 
0.178272 
1.067589 
0.2905 
Rsquared 
0.590980 
Mean dependent var 
4.687295 

Adjusted Rsquared 
0.575831 
S.D. dependent var 
3.109814 

S.E. of regression 
2.025369 
Akaike info criterion 
4.300576 

Sum squared resid 
221.5143 
Schwarz criterion 
4.408105 

Log likelihood 
119.5664 
HannanQuinn criter. 
4.342366 

Fstatistic 
39.01139 
DurbinWatson stat 
0.444886 

Prob(Fstatistic) 
0.000000 
Wald Fstatistic 
21.38973 

Prob(Wald Fstatistic) 
0.000000 
The result of NewyWest standard error consistent model gives exactly same result as that obtained in part (b).Part g
Dependent Variable: I3 

Method: Least Squares 

Date: 06/08/18 Time: 13:02 

Sample: 1960 2016 

Included observations: 57 

HAC standard errors & covariance (Bartlett kernel, NeweyWest fixed 

bandwidth = 4.0000) 

Variable 
Coefficient 
Std. Error 
tStatistic 
Prob. 
C 
1.522099 
0.677918 
2.245255 
0.0288 
INF 
0.832006 
0.131128 
6.345005 
0.0000 
Rsquared 
0.570322 
Mean dependent var 
4.687295 

Adjusted Rsquared 
0.562510 
S.D. dependent var 
3.109814 

S.E. of regression 
2.056927 
Akaike info criterion 
4.314760 

Sum squared resid 
232.7021 
Schwarz criterion 
4.386446 

Log likelihood 
120.9707 
HannanQuinn criter. 
4.342620 

Fstatistic 
73.00279 
DurbinWatson stat 
0.421116 

Prob(Fstatistic) 
0.000000 
Wald Fstatistic 
40.25909 

Prob(Wald Fstatistic) 
0.000000 
From the estimated regression result, the coefficient of inflation is obtained as 0.83. This means 1 percent increases in inflation rate increases return on Tbill rate by 0.8%. For 1 percent increase in inflation rate to cause a 1 percent increase in interest rate, the two variables need to be perfectly correlated that is having R square value equals to 1. From the regression coefficient and value of R square, the null hypothesis that 1 percent increase in inflation leads to a 1 percent increase in interest rate is rejected.
a.
The suspicion about heteroskadascity is reasonable as countries with a higher per capita GDP have access to a large amount of money to distribute. The people with a higher average income enjoy a higher flexibility regarding their spending on education. Countries with smaller per capita GDP has limited option for budget and hence, spending on education tend to vary less.
Figure 2: Scatter plot between x and y
The scatter plot between X and Y reveals that there exists a linear relationship between X and Y. This indicates presence of heteroskedasticity that is nonconstant variance of error terms. In presence of heteroskadascity, variation in Y differs depending on the variation in X. From the scatter plot it is seen that small values of X leads to small scatter in Y while large values are associated with large scatter in Y.
The output OLS regression is given as follows
Dependent Variable: Y 

Method: Least Squares 

Date: 06/08/18 Time: 15:27 

Sample: 1 34 

Included observations: 34 

Variable 
Coefficient 
Std. Error 
tStatistic 
Prob. 
C 
0.124573 
0.048523 
2.567308 
0.0151 
X 
0.073173 
0.005179 
14.12755 
0.0000 
Rsquared 
0.861823 
Mean dependent var 
0.476735 

Adjusted Rsquared 
0.857505 
S.D. dependent var 
0.359903 

S.E. of regression 
0.135858 
Akaike info criterion 
1.097394 

Sum squared resid 
0.590635 
Schwarz criterion 
1.007608 

Log likelihood 
20.65569 
HannanQuinn criter. 
1.066774 

Fstatistic 
199.5875 
DurbinWatson stat 
1.774258 

Prob(Fstatistic) 
0.000000 
The coefficient of X is obtained as 0.07. The positive coefficient indicates a positive association between per capita education and per capita GDP. That is an increase in per capita GDP leads to an increase in per capita expenditure on education. With 10% increase in per capita GDP expenditure on education increases by 0.7 percent. P value of the coefficient is 0.00. As the p value is less than significance value of 0.05, it can be said that the coefficient is statistically valid. This means average income increase, people have more income to spend on education ad hence, expenditure on education increases.
Test of Heteroskedasticity: White test
Hypothesis
Null hypothesis: Variances for errors are equal
Alternative hypothesis: Variance of errors are not equal
In order to test constant variance auxiliary regression analysis is undertaken. The auxiliary regression regress squares of residuals from the original regression on a set of regressors containing regressors of the original model with their squares and cross product. One can then inspect R square value. The Lagrange multiplier test statistics is obtained as product of sample size and R square value.
The obtained test statistics follows a chisquare distribution with (P1) degrees of freedom. P is the number of parameters in the auxiliary regression.
The null hypothesis is rejected if the p value of the chi square statistics is less than significance value of 0.05.
Conclusion
From the result of White test, p value of the chisquare statistics is 0.0000. As the p value is less than significance value of 0.05, the null hypothesis of homoskadastiity of error variance is rejected implying presence of hetetroskadascity in the model.
Heteroskedasticity Test: White 

Fstatistic 
6.423121 
Prob. F(2,31) 
0.0046 

Obs*Rsquared 
9.961452 
Prob. ChiSquare(2) 
0.0069 

Scaled explained SS 
11.90755 
Prob. ChiSquare(2) 
0.0026 

Test Equation: 

Dependent Variable: RESID^2 

Method: Least Squares 

Date: 06/11/18 Time: 12:27 

Sample: 1 34 

Included observations: 34 

Variable 
Coefficient 
Std. Error 
tStatistic 
Prob. 
C 
0.017677 
0.016112 
1.097134 
0.2810 
X^2 
0.000484 
0.000264 
1.834593 
0.0762 
X 
0.005206 
0.004548 
1.144759 
0.2611 
Rsquared 
0.292984 
Mean dependent var 
0.017372 

Adjusted Rsquared 
0.247370 
S.D. dependent var 
0.028968 

S.E. of regression 
0.025131 
Akaike info criterion 
4.445344 

Sum squared resid 
0.019578 
Schwarz criterion 
4.310665 

Log likelihood 
78.57084 
HannanQuinn criter. 
4.399414 

Fstatistic 
6.423121 
DurbinWatson stat 
2.210357 

Prob(Fstatistic) 
0.004636 
White consistent standard error
Dependent Variable: Y 

Method: Least Squares 

Date: 06/08/18 Time: 15:48 

Sample: 1 34 

Included observations: 34 

White heteroskedasticityconsistent standard errors & covariance 

Variable 
Coefficient 
Std. Error 
tStatistic 
Prob. 
C 
0.124573 
0.040414 
3.082420 
0.0042 
X 
0.073173 
0.006212 
11.78005 
0.0000 
Rsquared 
0.861823 
Mean dependent var 
0.476735 

Adjusted Rsquared 
0.857505 
S.D. dependent var 
0.359903 

S.E. of regression 
0.135858 
Akaike info criterion 
1.097394 

Sum squared resid 
0.590635 
Schwarz criterion 
1.007608 

Log likelihood 
20.65569 
HannanQuinn criter. 
1.066774 

Fstatistic 
199.5875 
DurbinWatson stat 
1.774258 

Prob(Fstatistic) 
0.000000 
Wald Fstatistic 
138.7696 

Prob(Wald Fstatistic) 
0.000000 
The White consistent standard error model gives the same result as that obtained from heteroskadasticity consistent output in part (c).
Confidence interval
Coefficient Confidence Intervals 

Date: 06/11/18 Time: 15:27 

Sample: 1 34 

Included observations: 34 

90% CI 

Variable 
Coefficient 
Low 
High 

C 
0.124573 
0.193030 
0.056116 

X 
0.073173 
0.062651 
0.083695 
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