# #StackBounty: #regression #hypothesis-testing #multiple-regression #estimation #linear-model Multiple Linear Regression Coefficient Est…

### Bounty: 100

A multiple linear regression model is considered. It is assumed that $$Y_i = beta_1x_{i1} + beta_2x_{i2} + beta_3x_{i3} + epsilon_i$$ where $$epsilon$$-s are independent and have the same normal distribution with zero expectation and unknown variance $$sigma^2$$. 100 measurements are made, i.e $$i = 1,2,…, 100.$$ The explanatory variables take the following values: $$x_{i1} = 2$$ for $$1 leq i leq 25$$ and $$0$$ otherwise, $$x_{i2} = sqrt{2}$$ for $$26 leq i leq 75$$ and $$0$$ otherwise, $$x_{i3} = 2$$ for $$76 leq i leq 100$$ and $$0$$ otherwise.

a) Let $$hat{beta_1},hat{beta_2}, hat{beta_3}$$ be least squares estimators of $$beta_1, beta_2, beta_3$$. Prove that in the considered case $$hat{beta_1},hat{beta_2}, hat{beta_3}$$ are independent, and $$Var(hat{beta_1}) = Var(hat{beta_3}) = Var(hat{beta_3})$$ Do these properties hold in the general case? If not, give counterexamples.

b) Perform a test for $$H_0: beta_1 + beta_3 = 2beta_2$$vs.$$H_1: beta_1 + beta_3 neq 2beta_2$$ The significance level is 0.05. The least squares estimates of $$beta_1, beta_2$$ and $$beta_3$$ are $$0.9812, 1.8851$$ and $$3.4406$$, respectively. The unbiased estimate of the variance $$sigma^2$$ is $$3.27$$.

For a) I know the OLS estimator for $$hat{beta} = (X^TX)^{-1}X^Ty$$, and $$Var(hat{beta}) = sigma^2 (X^TX)^{-1}$$. But I don’t know how to attain explicit expressions for each of the coefficients from this. Although it seems quite clear that the estimators are independent, for instance $$P(hat{beta_3} = beta_3, hat{beta_1} = 0, hat{beta_2} = 0) = P(hat{beta_3} = beta_3)$$ but I don’t how to write a proper proof. I believe the estimators are generally dependent and have unequal variance, but can’t come up with any particular examples.

For b) not sure what test-statistic to use (t or F) and how to set it up. Also don’t know the standard errors of the coefficients

Get this bounty!!!

This site uses Akismet to reduce spam. Learn how your comment data is processed.