Question 1: Consider the following linear regression model:

y_i = β₁ + β₂x_i2 + β₃x_i3 + ε_i = x^'_iβ + ε_i

a. Explain how the ordinary least squares estimator for β is determined and derive an expression for b.

b. Which assumptions are needed to make b an unbiased estimator for β?

c. Explain how a confidence interval for β₂ can be constructed. Which additional assumptions are needed?

d. Explain how one can test the hypothesis that β₃ = 1.

e. Explain how one can test the hypothesis that β₂ + β₃ = 0.

f. Explain how one can test the hypothesis that β₂ = β₃ = 0.

g. Which assumptions are needed to make b a consistent estimator for β?

h. Suppose that x_i2 = 2 + 3x_i3. What will happen if you try to estimate the above model?

i. Suppose that the model is estimated with x_i2 = 2x_i2 - 2 included rather than x_i2. How are the coefficients in this model related to those in the original model? And the R²s?

j. Suppose that x_i2 = x_i3 u_i, where u_i and x_i3 are uncorrelated. Suppose that the model is estimated with u_i included rather than x_i2. How are the coefficients in this model related to those in the original model? And the R²s?

Question 2. Carefully read the following statements. Are they true or false? Explain.

a. Under the Gauss-Markov conditions, OLS can be shown to be BLUE. The phrase 'linear' in this acronym refers to the fact that we are estimating a linear model.

b. In order to apply a t-test, the Gauss-Markov conditions are strictly required.

c. A regression of the OLS residual upon the regressors included in the model by construction yields an R² of zero.

d. The hypothesis that the OLS estimator is equal to zero can be tested by means of a t-test.

e. From asymptotic theory, we learn that - under appropriate conditions - the error terms in a regression model will be approximately normally distributed if the sample size is sufficiently large.

f. If the absolute t-value of a coefficient is smaller than 1.96, we accept the null hypothesis that the coefficient is zero, with 95% confidence.

g. Because OLS provides the best linear approximation of a variable y from a set of regressors, OLS also gives best linear unbiased estimators for the coefficients of these regressors.

h. If a variable in a model is significant at the 10% level, it is also significant at the 5% level.