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The article discusses regression output, overall significance of the model, relation between lunch program and percentage of tenth grade on a standardized math exam, relation between inflation and 3 months T-bill rate, and test of serial autocorrelation. It also includes a line plot of T-bill rate, inflation rate and federal budget balance, and Newy-West standard error estimation model.

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Running Head: ECONOMETRICS AND BUSINESS STATISTICS

ECONOMETRICS AND BUSINESS STATISTICS

Name of the Student

Name of the University

Author Note

ECONOMETRICS AND BUSINESS STATISTICS

Name of the Student

Name of the University

Author Note

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1ECONOMETRICS AND BUSINESS STATISTICS

Table of Contents

Answer 1..........................................................................................................................................3

Answer a......................................................................................................................................3

Answer b......................................................................................................................................3

Answer c......................................................................................................................................4

Answer d......................................................................................................................................4

Answer e......................................................................................................................................4

Answer 2..........................................................................................................................................5

Answer a......................................................................................................................................5

Answer b......................................................................................................................................6

Answer c......................................................................................................................................6

Answer d......................................................................................................................................7

Answer e......................................................................................................................................7

Part f.............................................................................................................................................9

Part g..........................................................................................................................................10

Answer 3........................................................................................................................................11

Answer a....................................................................................................................................11

Answer b....................................................................................................................................11

Answer c....................................................................................................................................12

Answer d....................................................................................................................................13

Answer e....................................................................................................................................14

Answer f.....................................................................................................................................15

Table of Contents

Answer 1..........................................................................................................................................3

Answer a......................................................................................................................................3

Answer b......................................................................................................................................3

Answer c......................................................................................................................................4

Answer d......................................................................................................................................4

Answer e......................................................................................................................................4

Answer 2..........................................................................................................................................5

Answer a......................................................................................................................................5

Answer b......................................................................................................................................6

Answer c......................................................................................................................................6

Answer d......................................................................................................................................7

Answer e......................................................................................................................................7

Part f.............................................................................................................................................9

Part g..........................................................................................................................................10

Answer 3........................................................................................................................................11

Answer a....................................................................................................................................11

Answer b....................................................................................................................................11

Answer c....................................................................................................................................12

Answer d....................................................................................................................................13

Answer e....................................................................................................................................14

Answer f.....................................................................................................................................15

2ECONOMETRICS AND BUSINESS STATISTICS

Answer 1

Answer 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 t-Statistic 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

R-squared 0.189291 Mean dependent var 24.10686

Adjusted R-squared 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 Hannan-Quinn criter. 7.362280

F-statistic 31.44310 Durbin-Watson stat 1.906937

Prob(F-statistic) 0.000000

From the regression result, the fitted equation is obtained as

math 10i=âˆ’23.14+7.75 log ( expendi ) âˆ’0.32lnchprgiâˆ’1.26 log (enrol)

Answer b

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.

Answer 1

Answer 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 t-Statistic 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

R-squared 0.189291 Mean dependent var 24.10686

Adjusted R-squared 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 Hannan-Quinn criter. 7.362280

F-statistic 31.44310 Durbin-Watson stat 1.906937

Prob(F-statistic) 0.000000

From the regression result, the fitted equation is obtained as

math 10i=âˆ’23.14+7.75 log ( expendi ) âˆ’0.32lnchprgiâˆ’1.26 log (enrol)

Answer b

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.

3ECONOMETRICS AND BUSINESS STATISTICS

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.

Answer c

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 lunch-program

and studentsâ€™ performance is rejected at 5% level of significance. Therefore, lunch-program fails

to increase performance of the student rather it influences student performance negatively.

Answer d

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.

Answer e

Dependent Variable: MATH10

Method: Least Squares

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

Sample: 1 408

Included observations: 408

Variable Coefficient Std. Error t-Statistic Prob.

C 32.14271 0.997582 32.22061 0.0000

LNCHPRG -0.318864 0.034839 -9.152422 0.0000

R-squared 0.171034 Mean dependent var 24.10686

Adjusted R-squared 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

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.

Answer c

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 lunch-program

and studentsâ€™ performance is rejected at 5% level of significance. Therefore, lunch-program fails

to increase performance of the student rather it influences student performance negatively.

Answer d

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.

Answer e

Dependent Variable: MATH10

Method: Least Squares

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

Sample: 1 408

Included observations: 408

Variable Coefficient Std. Error t-Statistic Prob.

C 32.14271 0.997582 32.22061 0.0000

LNCHPRG -0.318864 0.034839 -9.152422 0.0000

R-squared 0.171034 Mean dependent var 24.10686

Adjusted R-squared 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

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4ECONOMETRICS AND BUSINESS STATISTICS

Log likelihood -1499.274 Hannan-Quinn criter. 7.366965

F-statistic 83.76683 Durbin-Watson stat 1.907745

Prob(F-statistic) 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.

Answer 2

Answer a

Figure 1: Line plot of T-bill rate, inflation rate and federal budget balance

Log likelihood -1499.274 Hannan-Quinn criter. 7.366965

F-statistic 83.76683 Durbin-Watson stat 1.907745

Prob(F-statistic) 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.

Answer 2

Answer a

Figure 1: Line plot of T-bill rate, inflation rate and federal budget balance

5ECONOMETRICS AND BUSINESS STATISTICS

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 co-movement with Treasury bill interest rate.

Answer b

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 t-Statistic 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

R-squared 0.590980 Mean dependent var 4.687295

Adjusted R-squared 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 Hannan-Quinn criter. 4.342366

F-statistic 39.01139 Durbin-Watson stat 0.444886

Prob(F-statistic) 0.000000

Estimated regression equation

i3t=2.06 +0.82inf t +0.19 Â¿t

Answer c

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.

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 co-movement with Treasury bill interest rate.

Answer b

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 t-Statistic 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

R-squared 0.590980 Mean dependent var 4.687295

Adjusted R-squared 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 Hannan-Quinn criter. 4.342366

F-statistic 39.01139 Durbin-Watson stat 0.444886

Prob(F-statistic) 0.000000

Estimated regression equation

i3t=2.06 +0.82inf t +0.19 Â¿t

Answer c

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.

6ECONOMETRICS AND BUSINESS STATISTICS

Answer d

The value of inflation coefficient is 0.82. This implies there is a positive relation between

rate of inflation and 3 months T-bill rate. That is as inflation increases return to 3 months T-bill

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 T-bill rate. The coefficient of federal budget is 0.19. The positive co-efficient

indicates that federal budget positively influences interest on T-bill 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 Tâ€“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 T-bill 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.

Answer e

Test of serial autocorrelation

Hypothesis

Answer d

The value of inflation coefficient is 0.82. This implies there is a positive relation between

rate of inflation and 3 months T-bill rate. That is as inflation increases return to 3 months T-bill

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 T-bill rate. The coefficient of federal budget is 0.19. The positive co-efficient

indicates that federal budget positively influences interest on T-bill 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 Tâ€“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 T-bill 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.

Answer e

Test of serial autocorrelation

Hypothesis

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7ECONOMETRICS AND BUSINESS STATISTICS

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 â€“ p)*R2 , where T is the number of observation and p is the number of lagged residual

terms. The test statistics follows a chi-square distribution with p degrees of freedom.

Decision rule

The null hypothesis is rejected if P value of the Breusch-Godfrey 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.

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 â€“ p)*R2 , where T is the number of observation and p is the number of lagged residual

terms. The test statistics follows a chi-square distribution with p degrees of freedom.

Decision rule

The null hypothesis is rejected if P value of the Breusch-Godfrey 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.

8ECONOMETRICS AND BUSINESS STATISTICS

Breusch-Godfrey Serial Correlation LM Test:

F-statistic 39.78589 Prob. F(2,52) 0.0000

Obs*R-squared 34.47237 Prob. Chi-Square(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 t-Statistic 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

R-squared 0.604778 Mean dependent var -7.64E-16

Adjusted R-squared 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 Hannan-Quinn criter. 3.512092

F-statistic 19.89295 Durbin-Watson stat 1.995246

Prob(F-statistic) 0.000000

Part f

Newy-West 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, Newey-West fixed

bandwidth = 4.0000)

Variable Coefficient Std. Error t-Statistic 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

Breusch-Godfrey Serial Correlation LM Test:

F-statistic 39.78589 Prob. F(2,52) 0.0000

Obs*R-squared 34.47237 Prob. Chi-Square(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 t-Statistic 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

R-squared 0.604778 Mean dependent var -7.64E-16

Adjusted R-squared 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 Hannan-Quinn criter. 3.512092

F-statistic 19.89295 Durbin-Watson stat 1.995246

Prob(F-statistic) 0.000000

Part f

Newy-West 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, Newey-West fixed

bandwidth = 4.0000)

Variable Coefficient Std. Error t-Statistic 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

9ECONOMETRICS AND BUSINESS STATISTICS

R-squared 0.590980 Mean dependent var 4.687295

Adjusted R-squared 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 Hannan-Quinn criter. 4.342366

F-statistic 39.01139 Durbin-Watson stat 0.444886

Prob(F-statistic) 0.000000 Wald F-statistic 21.38973

Prob(Wald F-statistic) 0.000000

The result of Newy-West 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, Newey-West fixed

bandwidth = 4.0000)

Variable Coefficient Std. Error t-Statistic Prob.

C 1.522099 0.677918 2.245255 0.0288

INF 0.832006 0.131128 6.345005 0.0000

R-squared 0.570322 Mean dependent var 4.687295

Adjusted R-squared 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 Hannan-Quinn criter. 4.342620

F-statistic 73.00279 Durbin-Watson stat 0.421116

Prob(F-statistic) 0.000000 Wald F-statistic 40.25909

Prob(Wald F-statistic) 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 T-bill 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

R-squared 0.590980 Mean dependent var 4.687295

Adjusted R-squared 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 Hannan-Quinn criter. 4.342366

F-statistic 39.01139 Durbin-Watson stat 0.444886

Prob(F-statistic) 0.000000 Wald F-statistic 21.38973

Prob(Wald F-statistic) 0.000000

The result of Newy-West 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, Newey-West fixed

bandwidth = 4.0000)

Variable Coefficient Std. Error t-Statistic Prob.

C 1.522099 0.677918 2.245255 0.0288

INF 0.832006 0.131128 6.345005 0.0000

R-squared 0.570322 Mean dependent var 4.687295

Adjusted R-squared 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 Hannan-Quinn criter. 4.342620

F-statistic 73.00279 Durbin-Watson stat 0.421116

Prob(F-statistic) 0.000000 Wald F-statistic 40.25909

Prob(Wald F-statistic) 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 T-bill 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

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10ECONOMETRICS AND BUSINESS STATISTICS

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.

Answer 3

Answer 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.

Answer b

Figure 2: Scatter plot between x and y

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.

Answer 3

Answer 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.

Answer b

Figure 2: Scatter plot between x and y

11ECONOMETRICS AND BUSINESS STATISTICS

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 non-constant 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.

Answer c

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 t-Statistic Prob.

C -0.124573 0.048523 -2.567308 0.0151

X 0.073173 0.005179 14.12755 0.0000

R-squared 0.861823 Mean dependent var 0.476735

Adjusted R-squared 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 Hannan-Quinn criter. -1.066774

F-statistic 199.5875 Durbin-Watson stat 1.774258

Prob(F-statistic) 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 scatter plot between X and Y reveals that there exists a linear relationship between X and Y.

This indicates presence of heteroskedasticity that is non-constant 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.

Answer c

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 t-Statistic Prob.

C -0.124573 0.048523 -2.567308 0.0151

X 0.073173 0.005179 14.12755 0.0000

R-squared 0.861823 Mean dependent var 0.476735

Adjusted R-squared 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 Hannan-Quinn criter. -1.066774

F-statistic 199.5875 Durbin-Watson stat 1.774258

Prob(F-statistic) 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

12ECONOMETRICS AND BUSINESS STATISTICS

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.

Answer d

Test of Heteroskedasticity: White test

Hypothesis

Null hypothesis: Variances for errors are equal

Alternative hypothesis: Variance of errors are not equal

Auxiliary regression and test statistics

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.

LM =n R2

The obtained test statistics follows a chi-square distribution with (P-1) degrees of freedom. P is

the number of parameters in the auxiliary regression.

Decision rule

The null hypothesis is rejected if the p value of the chi square statistics is less than

significance value of 0.05.

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.

Answer d

Test of Heteroskedasticity: White test

Hypothesis

Null hypothesis: Variances for errors are equal

Alternative hypothesis: Variance of errors are not equal

Auxiliary regression and test statistics

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.

LM =n R2

The obtained test statistics follows a chi-square distribution with (P-1) degrees of freedom. P is

the number of parameters in the auxiliary regression.

Decision rule

The null hypothesis is rejected if the p value of the chi square statistics is less than

significance value of 0.05.

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13ECONOMETRICS AND BUSINESS STATISTICS

Conclusion

From the result of White test, p value of the chi-square 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

F-statistic 6.423121 Prob. F(2,31) 0.0046

Obs*R-squared 9.961452 Prob. Chi-Square(2) 0.0069

Scaled explained SS 11.90755 Prob. Chi-Square(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 t-Statistic 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

R-squared 0.292984 Mean dependent var 0.017372

Adjusted R-squared 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 Hannan-Quinn criter. -4.399414

F-statistic 6.423121 Durbin-Watson stat 2.210357

Prob(F-statistic) 0.004636

Answer e

White consistent standard error

Dependent Variable: Y

Method: Least Squares

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

Sample: 1 34

Included observations: 34

White heteroskedasticity-consistent standard errors & covariance

Conclusion

From the result of White test, p value of the chi-square 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

F-statistic 6.423121 Prob. F(2,31) 0.0046

Obs*R-squared 9.961452 Prob. Chi-Square(2) 0.0069

Scaled explained SS 11.90755 Prob. Chi-Square(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 t-Statistic 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

R-squared 0.292984 Mean dependent var 0.017372

Adjusted R-squared 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 Hannan-Quinn criter. -4.399414

F-statistic 6.423121 Durbin-Watson stat 2.210357

Prob(F-statistic) 0.004636

Answer e

White consistent standard error

Dependent Variable: Y

Method: Least Squares

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

Sample: 1 34

Included observations: 34

White heteroskedasticity-consistent standard errors & covariance

14ECONOMETRICS AND BUSINESS STATISTICS

Variable Coefficient Std. Error t-Statistic Prob.

C -0.124573 0.040414 -3.082420 0.0042

X 0.073173 0.006212 11.78005 0.0000

R-squared 0.861823 Mean dependent var 0.476735

Adjusted R-squared 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 Hannan-Quinn criter. -1.066774

F-statistic 199.5875 Durbin-Watson stat 1.774258

Prob(F-statistic) 0.000000 Wald F-statistic 138.7696

Prob(Wald F-statistic) 0.000000

The White consistent standard error model gives the same result as that obtained from

heteroskadasticity consistent output in part (c).

Answer f

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

Variable Coefficient Std. Error t-Statistic Prob.

C -0.124573 0.040414 -3.082420 0.0042

X 0.073173 0.006212 11.78005 0.0000

R-squared 0.861823 Mean dependent var 0.476735

Adjusted R-squared 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 Hannan-Quinn criter. -1.066774

F-statistic 199.5875 Durbin-Watson stat 1.774258

Prob(F-statistic) 0.000000 Wald F-statistic 138.7696

Prob(Wald F-statistic) 0.000000

The White consistent standard error model gives the same result as that obtained from

heteroskadasticity consistent output in part (c).

Answer f

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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