Questions 9

  1. Derive small sample distribution of the OLS estimator. What assumption apart from the ones made in CLRM we are making for this derivation?
  2. (*) Prove, that the distribution of the residual sum of squares is {$\chi^2_{N-K}$} and independent from the distribution of {$b$}.
  3. What is the formula for the statistics used for testing hypothesis that {$\beta_k=\beta_k^{*}$}
  4. Explain how to build the confidance interval for parameter {$\beta_k$}. The value of the estimate of this parameter is equal to {$b_k$}, standard deviation of this estimate is equal to {$\hat{se(b_k)}$}, {$N$} is the number of observations, {$K$} is the number of estimated parameters and significance level is set to {$1-\alpha$}.
  5. What is the differance between confiance intervals for expected values of {$y$} and forecasts of {$y$}?
  6. How the sum of squares of residuals in the unrestricted and restricted model can be used to test the joint hypothesis {$H\beta=h$}?
  7. Explain what are advantages and disadvantages of imposing retrictions on parameters of the model.
  8. For some statistical hypothesis p-value equal to {$\alpha^{*}$} was obtained. Given that significance level is equal to {$\alpha$} explain what case the null hypothesis should be rejected.