- When considering exact linear restrictions on the regression model's coefficient vector, the number of these restrictions must be:
- Greater than the number of regressors in the model.
- Less than the number of regressors in the model.
- Exactly equal to the number of regressors in the model.
- Less than 'n', the sample size.

- If we wish to test the validity of J independent linear restrictions on the coefficients of the usual linear model, and if all of the usual assumptions about that model hold, then:
- We can use an F-test. The distribution of the test statistic is F with J and (n-k) degrees of freedom if the restrictions are valid, but it is Chi-square with J degrees of freedom if they are false.
- We can use a Wald test.
- The Wald test will be uniformly most powerful.
- We can use an F-test, and this test will be valid even if the sample size is small.

- Suppose we have a regression model that is non-linear in the parameters, and we want to test the hypothesis that the sum of two of the parameters is unity. Then:
- Using a Wald test would be foolish, as the results will depend on how we write the restriction.
- A Wald test would be a sensible choice as it will be optimal for any sample size.
- A Wald test would be a sensible choice and the test statistic will have a known null distribution if the sample size is sufficiently large.
- Because the restriction is linear, we can use an F-test, which will be exact and UMP even if the sample size is relatively small.

- Check the EViews regression output located here. The following is true:
- The F-statistic that is reported there would increase if any other regressors were added to the model.
- The null distribution for the F-statistic that is reported there is "F", with 2 and 208 degrees of freedom.
- Both A and B are true.
- Because of the relatively large sample size, the F-statistic that is reported there has a distribution that is approximately Chi-Square with 3 degrees of freedom, if the null hypothesis is true.

- Suppose that we have a standard linear regression model, y = Xβ + ε, and we want to test the null hypothesis that β1 = (β2 / β3). Then:
- The F-test is a better choice than the Wald test, as the results for the former test will be invariant to how we write the null hypothesis.
- The Wald test is asymptotically valid, but the F-test is not appropriate.
- The Wald test is not a good choice as the results will depend on how we write the null hypothesis.
- Both B and C.

- Check the EViews regression output located here. The following is true:
- The hypothesis being tested is that β2 = -β3, and we would reject this hypothesis at the 10% significance level.
- The hypothesis being tested is that β2 = β3, and we would not reject this hypothesis at the 10% significance level.
- The hypothesis being tested is that β2 = β3, and we would reject this hypothesis at the 10% significance level.
- The hypothesis being tested is that β2 = -β3, and the alternative hypothesis is that β2 +β3 ≠ 0.

- Check the EViews regression output located here. The following is true:
- In the Wald test output box, the F-statistic should not be used for an exact tests because OLS estimation has not been used.
- In the Wald test output box, the F-statistic should not be used fr an exact test because a lagged value of the dependent variable is used as a regressor.
- In the Wald test output box, the Chi-Square and F-statistics are the same because v1*F(v1,v2) → Chisquare with v1 degrees of freedom.
- All of the above are correct.

- The Restricted Least Squares estimator collapses to the OLS estimator if:
- The sample data are such that the OLS estimator happens to exactly satisfy the restrictions.
- The restrictions are actually valid in the population.
- The regressors are non-random.
- The error term in the model has a zero mean, so that the OLS estimator is unbiased.

- Suppose we apply the Restricted Least Squares estimator, and all of the usual assumptions about our regression model hold. However, suppose the restrictions themselves are false. Then:
- The RLS estimator will be biased.
- The RLS estimator will be biased and inconsistent, but the OLS estimator will be unbiased and consistent.
- The RLS estimator will be inconsistent.
- The RLS estimator will be biased but weakly consistent.

- If we construct the Restricted Least Squares (RLS) estimator and the restrictions are actually false, then:
- The RLS estimator will be inefficient relative to the OLS estimator.
- The RLS estimator will be biased, and it will exhibit greater variability then the OLS estimator.
- The RLS estimator may be inefficient, or efficient, relative to the OLS estimator, depending on the bias/variance trade-off.
- The RLS estimator will be less biased than if the restrictions were true.

- The rank of the restrictions matrix, 'R', in the set-up for exact linear restrictions on the regression model's coefficients, is less than J (the number of restrictions) then:
- There will be some restrictions that are either redundant, or in conflict with other restrictions.
- The Wald test statistic will not be defined.
- The Restricted Least Squares estimator will not be defined.
- All of the above.

- If we wanted to estimate the model y = Xβ + ε by Generalized Instrumental Variables, and we wanted to impose the restrictions Rβ = q on the estimates, then we would:
- Be wasting our time, becuase we can only incorportate such restrictions if we are using OLS as the basis for estimation.
- Derive the estimator by finding the b* that solves the problem: Min. (y-Xb*)'Mz(y-Xb*) subject to Rb* = q; where Mz is the same idempotent matrix used to construct the usual Generalized I.V. estimator.
- End up with an estimator that would be consistent (but probably biased) if the the restrictions were valid, but inconsistent if the restrictions were false.
- Both B and C.