- If an estimator is "mean square consistent", then:
- Its mean squared error vanishes if ithe sample size is sufficiently large.
- Its mean squared error will be zero.
- Its probability limit will equal the true value of the parameter.
- Both A and C.

- Consider two possible estimators of the (constant) variance of the error term in a standard linear regression model, for which all of the usual assumptions are satisfied: vhat1 = (e'e) / (n-k), and vhat2 = (e'e) / (n-k+2).
- Both of these estimators are weakly consistent and mean square consistent.
- Both of these estimators are weakly consistent and unbiased.
- The first estimator is unbiased and mean square consistent, while the second estimator is biased and only weakly consistent.
- Both of these estimators are weakly consistent, but neither of them is mean square consistent.

- If plim(β*) = β and plim(α*) = α, where α and β are unknown parameters. Then:
- α* is mean square consistent for α, and β* is mean square consistent for β.
- α* and β* are unbiased estimators of α and β respectively.
- plim(log(β*)) = log(β), and plim(α* / β*) = (α / β).
- plim(log(β*)) = log(β), and plim(α* / β*) = (α / β), provided that β > 0.

- The Lindeberg-Lévy Central Limit Theorem tells us that, under certain conditions:
- The sum of independent drawings from a normal distribution will also be normally distributed.
- The sum of n independent drawings from any distribution will be normally distributed if n is greater than about 20.
- The Student-t distribution will approach the Standard normal distribution as the degrees of freedom tends to infinity.
- The arithmetic average of independent random variables from any distribution will be approximately normally distributed, provided the number of terms used in constructing this average is sufficiently large.

- Suppose that we have 2 statistics. One of them is 't', which follows a Student-t distribution with p degrees of freedom. The other is 'F', and it follows an F distribution with p1 and p2 degrees of freedom.
- As p → ∞ and p2 → ∞, the distribution for 't' approaches the standard normal distribution, and the distribution for 'F' approaches the chi-square distribution with p1 degrees of freedom.
- As p and p2 → ∞ , the distribution for 't' approaches the standard normal distribution, and the distribution for (p1F) approaches the chi-square distribution with p1 degrees of freedom.
- As p and p2 → ∞, the distribution for 't' approaches a chi-square distribution with p1 degrees of freedom, and the distribution for (p1F) also approaches the chi-square distribution with p1 degrees of freedom.
- As p and p2 → ∞, the distributions for 't' and 'F' approach the same limiting (asymptotic) distribution.

- Suppose that β1 and β2 are 2 weakly consistent estimators of a parameter, β; Also, suppose that the asymptotic variance of β1 is (v1 / n), and that of β2 is (v2 / n). Then:
- We can't say anything about the relative asymptotic efficiencies of these 2 estimators, as both of the asymptotic variances converge to zero as n becomes infinitely large
- β1 will be asymptotically more efficient than β2 if v1 < v2.
- Both estimators must have the same asymptotic efficiency as they are both consistent estimators.
- Both estimators are asymptotically unbiased, but we can't tell anything about their asymptotic efficiency.

- The "generalized" Instrumental Variables (I.V.) estimator for the coefficient vector in a linear regression model collapses to the "simple" I.V. estimator if:
- The number of instruments is greater than or equal to the number of regressors.
- The instruments are non-random.
- The number of instruments is less than or equal to the number of regressors.
- The number of instruments exactly equals the number of regressors.

- If we apply a Hausman test of the hypothesis that the errors in a regression model are asymptotically uncorrelated with the regressors:
- We would use I.V. estimation if the p-value for the test is large enough (say, greater than 10% or 20%).
- We would use OLS estimation if the null hypothesis is rejected.
- We would use I.V. estimation if the null hypothesis is rejected.
- The test statistic will follow a chi-square distribution if the null hypothesis is true.

- The Wu test is used:
- Because it is (asymptotically) more powerful than the Hausman test.
- To determine if a model's regressors are asymptotically uncorrelated with the error term, or not.
- To determine whether OLS or IV estimation is more appropriate.
- Both B and C.

- The usual OLS estimator of the coefficient vector in the linear regression model is also an I.V. estimator, with the instrument matrix chosen to be:
- Non-singular.
- The regressor matrix, X.
- Non-random.
- The (X'X) matrix.

- If we are constructing several I.V. estimators with "valid" choices of instruments (i.e., in each case the instruments are asymptotically uncorrelated with the error term in the model), then:
- All of these estimators will be equivalent asymptotically, although their finite-sample properties may differ.
- The choice of instruments will affect the relative asymptotic efficiencies of the estimators, but not their finite-sample bias.
- The choice of instruments will affect the consistency of the estimators, but not their relative asymptotic efficiencies.
- The choice of instruments will affect the relative asymptotic efficiencies of the estimators, but not their coinsistency.

- Check the EViews regression output located here. The following is true:
- The estimator used in Equation 1 will be inconsistent, while that used in Equation 2 will be consistent - this is why there are some large differences in the results.
- The larger standard errors in Equation 2, as compared with Equation 1, reflect the inefficiency of OLS relative to I.V. estimation (at least asymptotically).
- Equation 2 has been estimated with an allowance for the possibility that the income variable, Y, may be correlated with the error term, even in very large samples.
- The use of I.V. estimation (in this particular example) reduces the estimates of both the short-run and long-run marginal propensities to consume.