5.1.State with reason whether the following statements are true, false, or uncertain.
a... The t test of significance discussed in this chapter requires that the
sampling distributions of estimators ˆ β1 and ˆ β2 follow the normal
b... Even though the disturbance term in the CLRM is not normally distributed,
the OLS estimators are still unbiased.
c... If there is no intercept in the regression model, the estimated ui (= ˆui )
will not sum to zero.
d... The p value and the size of a test statistic mean the same thing.
e... In a regression model that contains the intercept, the sum of the residuals
is always zero.
f... If a null hypothesis is not rejected, it is true.
g... The higher the value of σ 2, the larger is the variance of ˆ β2
h... The conditional and unconditional means of a random variable are the
i... In the two-variable PRF, if the slope coefficient β2 is zero, the intercept
β1 is estimated by the sample mean ¯Y.
j... The conditional variance, var (Yi | Xi ) = σ 2, and the unconditional variance
of Y, var (Y) = σ 2
Y , will be the same if X had no influence on Y
my answers are
e.false , as it alwys sums up to zero
f.false , depends on alpha , in how no. out of 100.
j. false . as var. is only affected by ui
help me out where i m wrong ?
R^2 = 0 means the no explanatory variable is able to explain the variation in the Explanatory variable, which is same as testing for the multiple linear restriction that is all the slope coefficient are 0, under the alternative hypothesis that at least 1 slope coefficient is non 0.
So we can use the F-test to check the increase in R^2 from the restricted model (in which R^2 = 0 because all slope coefficient are 0) to the unrestricted model is statistically significant or not.