#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

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