Question

Examine the relationship between the equity market and the bond market. To this end, we have estimated the following regression ??? = ?0 + ?1???? + ?? where ??? and ???? denote the daily returns at time ? of an equity index and a bond index, respectively, while ?? is a random error term. The estimated coefficients are ?̂ 0 = −0.004 and ?̂ 1 = −0.256, with their standard errors being ??(?̂ 0) = 0.007 and ??(?̂ 1) = 0.014. The sample size is ? = 1,000.

1. Briefly discuss the intuition behind the estimated coefficients. word count: [200 words]
2. Test whether there is convincing evidence that equity returns are negatively linearly related to bond returns. Use a significance level of ? = 5%. : [200 words]
3. Briefly discuss the intuition behind the significance level. word count : [200 words]
4. The regression’s R-square was equal to 0.32. Briefly discuss the intuition behind this measure. word count: [200 words]
5. We are concerned about the regression model suffering from heteroscedastic residuals. To this end, we have run White’s test and obtained a test statistic with a p-value of 0.04. Briefly discuss whether heteroscedasticity is a problem for this model (at the 5% significance level) and what its implications would be. [word count: 300 words]

Answer 1

i. If the daily returns of a bond index is increased by one unit then the expected daily returns of  equity index is decreased by 0.256 units. If there is no daily returns of a bond index then the expected daily returns of  equity index is −0.004 which is meaningless. Hence intercept term has no practical meaning.

ii.

Value of test statistic=−0.256/0.014=-18.2857

p-value=P(t<-18.2857|t~t998)=0.0000<0.05

So we reject H0 at 5% level of significance and conclude that equity returns are significantly negatively linearly related to bond returns.

iii. Since level of significance is upper bound of probability of type I error and we generally take small value of level of significance. It is a pre assigned value given by experimenter. We generally take 0.01,0.05,0.10 as a level of significance.

iv. R2=0.32 i.e. 32% of total variation in the sample of equity returns is explained by this regression equation.