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Null Hypothesis, Cobb-Douglas, Regression model

1. If the expected value of Y for the ith observation in a regression model is 35, how could it be that the observed value is 33?

2. In a study of the determination of prices of final output at factor cost in the United Kingdom, the following results were obtained on the basis of annual data for the period 1951-1969:

PFt = 2.033 + 0.273Wt - 0.521Xt + 0.256Mt + 0.028Mt-1 + 0.121PFt-1
(0.992) (0.127) (0.099) (0.024) (0.039) (0.119)

R2 = 0.92

where PF = prices of final output at factor cost
W = wages and salaries per employee
X = gross domestic product per person employed
M = import prices
Mt-1 = import prices lagged 1 year
PFt-1 = prices of final output at factor cost in the previous year.
Standard errors of the estimated coefficients are in parentheses

(a) At the 1% level of significance, test the null hypothesis that the slope coefficient for the variable ??wages and salaries per employee?? (Wt) is zero versus the alternative hypothesis that it is not equal to zero.

(b) At the 10% level of significance, test the null hypothesis that the slope coefficient for the variable ??wages and salaries per employee?? (Wt) is zero versus the alternative hypothesis that it is not equal to zero.

(c) At the 1 % level of significance, test for the overall significance of the above estimated regression.

3. In an application of the Cobb-Douglas production function the following results were obtained:

1n Yi = 2.3542 + 0.9576(1nX2i) + 0.8242(1nX3i)
(0.3022) (0.3571)

R2 = 0.8432 degrees of freedom = 12

where Y = output, X2 = labor input, and X3 = capital input, and the figures in parentheses are the estimated standard errors.

(a) The coefficients of the labor and capital inputs in the preceding equation give the elasticities of output with respect to labor and capital. Test the hypothesis that these elasticities are individually equal to unity.

(b) Test the overall significance of the estimated regression equation given previously. Use 0.05 as your significance level.

4. Open the data file schoolspend.xls, which contains data on 50 US states for seven variables. Totalspend is the amount of money that school districts in the state spend on education from kindergarten through 12th grade, measured in millions of dollars. Totalstateaid is the amount of money provided by the state to school districts, measured in millions of dollars. Income is the total state income in millions of dollars. Population is the total population of the state in thousands, and Elderly, minority, and schoolage are the fraction of the state??s population that are 65 or over, members of ethnic minorities, and age 6 to 17 respectively.

Decisions are made by school district officials, based on revenue raised locally and state aid. If more state aid is available, local officials could either spend more money on schools, or cut back on local revenue, or some of both.

However, we also know that bigger and richer states will be able to afford to spend more money on school aid as well. Finally, consider the fact that school spending is driven by factors other than financial ones. In particular, we??ll look to see if age profiles and ethnic composition of populations affect the way school spending decisions are made.

Estimate the regression totalspendi = ?? + ??1*totalstateaidi + ??2*incomei + ??3*populationi + ??4*elderlyi + ??5*minorityi + ??6*schoolagei + ui.

Test the Null Hypothesis that totalstateaid and income can be jointly dropped from the equation. That is, test the Null Hypothesis:

HO: ??1 = ??2 = 0

Against the Alternative Hypothesis:

HA: ??1 ?j 0 and/or ??2 ?j 0.

Use 0.01 as your significance level. What do you conclude?

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1. If the expected value of Y for the ith observation in a regression model is 35, how could it be that the observed value is 33?

2. In a study of the determination of prices of final output at factor cost in the United Kingdom, the following results were obtained on the basis of annual data for the period 1951-1969:

PFt = 2.033 + 0.273Wt - 0.521Xt + 0.256Mt + 0.028Mt-1 + 0.121PFt-1
(0.992) (0.127) (0.099) (0.024) (0.039) (0.119)

R2 = 0.92

where PF = prices of final output at factor cost
W = wages and salaries per employee
X = gross domestic product per person employed
M = import prices
Mt-1 = import prices lagged 1 year
PFt-1 = prices of final output at factor cost in the previous year.
Standard errors of the estimated coefficients are in parentheses

(a) At the 1% level of significance, test the null hypothesis that the slope coefficient for the variable ??wages and salaries per employee?? (Wt) is zero versus the alternative hypothesis that it is not equal to zero.

(b) At the 10% level of significance, test the null hypothesis that the slope coefficient for the variable ??wages and salaries per employee?? (Wt) is zero versus the alternative hypothesis that it is not equal to zero.

(c) At the 1 % level of significance, test for the overall significance of the above estimated regression.

3. In an application of the Cobb-Douglas production function the following results were obtained:

1n Yi = 2.3542 + 0.9576(1nX2i) + 0.8242(1nX3i)
(0.3022) (0.3571)

R2 = 0.8432 degrees of freedom = 12

where Y = output, X2 = labor input, and X3 = capital input, and the figures in parentheses are the estimated standard errors.

(a) The coefficients of the labor and capital inputs in the preceding equation give the elasticities of output with respect to labor and capital. Test the hypothesis that these elasticities are individually equal to unity.

(b) Test the overall significance of the estimated regression equation given previously. Use 0.05 as your significance level.

4. Open the data file schoolspend.xls, which contains data on 50 US states for seven variables. Totalspend is the amount of money that school districts in the state spend on education from kindergarten through 12th grade, measured in millions of dollars. Totalstateaid is the amount of money provided by the state to school districts, measured in millions of dollars. Income is the total state income in millions of dollars. Population is the total population of the state in thousands, and Elderly, minority, and schoolage are the fraction of the state??s population that are 65 or over, members of ethnic minorities, and age 6 to 17 respectively.

Decisions are made by school district officials, based on revenue raised locally and state aid. If more state aid is available, local officials could either spend more money on schools, or cut back on local revenue, or some of both.

However, we also know that bigger and richer states will be able to afford to spend more money on school aid as well. Finally, consider the fact that school spending is driven by factors other than financial ones. In particular, we??ll look to see if age profiles and ethnic composition of populations affect the way school spending decisions are made.

Estimate the regression totalspendi = ?? + ??1*totalstateaidi + ??2*incomei + ??3*populationi + ??4*elderlyi + ??5*minorityi + ??6*schoolagei + ui.

Test the Null Hypothesis that totalstateaid and income can be jointly dropped from the equation. That is, test the Null Hypothesis:

HO: ??1 = ??2 = 0

Against the Alternative Hypothesis:

HA: ??1 ?j 0 and/or ??2 ?j 0.

Use 0.01 as your significance level. What do you conclude?

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