Question

1.a. Conduct a two-sample t-test to find out if there is a significant difference between U.S. stock returns and U.S. corporate bond returns using the monthly data covering the sample period 1980-2017. 1.b. Conduct a two-sample t-test to find out if there is a greater returns for U.S. stock as compared to U.K. stock returns using the monthly data covering the sample period 1980-2017. 2. Estimate a multiple linear regression relationship with the U.K. stock returns as the dependent variable, and U.K. Bonds Returns, U.S. Stock Returns, and Japan Stock Returns as the independent variables using the monthly data covering the sample period 1980-2017 (Finding the determinants of U.K. stock returns). a. Show the estimated regression relationship b. Conduct a t-test for statistical significance of the individual slope coefficients. Provide the interpretation of the significant slope estimates. c. Conduct a test for the overall significance of the regression equation. (Test for the significance of the regression relationship as a whole) d. Present the R-Square (Coefficient of Determination) and its interpretation.

SUMMARY OUTPUT Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations 0.735992889 0.541685532 0.538636877 3.523412257 455 ANOVA Significance F MS Regression Residual Total 3 6617.396126 2205.798709 177.6801681 4.89572E-76 451 5598.90970512.41443393 454 12216.30583 Coefficients Standard Error t Stat 0.115186435 0.17010687 0.730332455 0.041983928 17.39552475 3.10678E-52 0.647824048 0.812840861 0.621729509 0.8389354 0.209729348 0.029769306 7.045154189 6.96789E-12 0.15122558 0.268233116 0.132722873 0.286735823 0.113600031 0.030384391 3.738762846 0.000208723 0.053887475 0.173312588 0.030500247 0.192197593 P-value Lower 95% Upper 95% Lower 99.0% Upper 99.0% Intercept RSUS RSJA RUK 0.677141581 0.498663306-0.219114035 0.449486905 -0.324841646 0.555214515

Answer 1

Based on the output obtained,

The fitted regression equation

Estimated UK Stock Returns$= \widehat{\beta _{0}}+ \widehat{\beta _{1}}(RSUS)+ \widehat{\beta _{2}}(RSJA)+ \widehat{\beta _{3}}(RUK)$

      

0.1 1 51 86435+0.730332455( RSUS)-0.209729348 (RSJA)一0.11 3600031( RUK)

To test the significance of each of these estimated regression coefficients:

To test:
Ho: ,1,2,3
Vs   
На : 3, 0.1 = 1.2,3

Coefficients Standard Errort Stat 0.1151864350.17010687 0.677141581 0.498663306 -0.219114035 0.449486905 0.730332455 0.041983928 17.39552475 3.10678E-52 0.647824048 0.812840861 0.209729348 0.029769306 7.045154189 6.96789E-12 0.15122558 0.268233116 0.113600031 0.030384391 3.738762846 0.000208723 0.053887475 0.173312588 P-value Lower 95% |Upper 95% Intercept RSUS RSJA RUK

The columns 't stat' and 'p-value' gives the result for the t test for significance for each regression coefficient:

Since, the p-values for the t test of significance of each of the 3 predictors were significant, we do not have sufficient evidence to support the null hypothesis. We may reject H₀ at 5% level.We may conclude that:

The individual predictors of the model were examined and the result indicated that RSUS (t = 17.395, p = .000), RSJA (t = 7.045, p = .000) and RUK (t =3.738, p = .0002) were significant predictors in the model.

To test the overall significance of the model:

To test: H₀: The fitted model is similar to the intercept only model. H_a: The fitted model is more efficient than the intercept model.

ANOVA Significance F 3 6617.3961262205.798709 177.68016814.89572E-76 df 3 Regression Residual Total 515598.909705 12.41443393 5412216.30583

Since, the p-value of the F test for overall significance 0.000 < 0.05, we do not have sufficient evidence to support the null hypothesis. We may reject H₀ at 5% level.We may conclude that:

Results of the multiple linear regression indicated that there was a overall significant effect of RSUS, RSJA and RUK on UK Stock returns, (F(3,451) = 177.6801681, p = 4.89572E-76).

d. The goodness of fit measure R² explains the amount of variation in the dependent variable that is explained by the predictors in the model.

Regression Statistics Multiple R R Square Adjusted R Square 0.538636877 Standard Error Observations 0.735992889 0.541685532 3.523412257

We find that the coefficient of determination, R² = 0.5417; i.e. the predictors of the model together explains about 54.17% of the variation in UK Stock returns. The model may be concluded to be a moderate fit to the data.It can be improved by increasing the number of potential predictors.