Topic 6: Asymptotics

Question 1

What is consistency?

Answer 1

An estimator converges in probability to the correct population parameter as the sample size grows without bound.

Question 2

How should you start the proof for consistency of the OLS estimator in the simple linear regression model?

Answer 2

Start with the estimate, beta^j = betaj + [sum(xi-xbar)ui] \ [sum(xi-xbar)^2]

Question 3

As you divide both the numerator and denominator of the second term in the consistency proof by 1/n what happens?

Answer 3

As n grows without bound, the numerator goes to the Covariance of (x,u) and the denominator goes to the Variance of x

Question 4

What does Cov(x,u) converge on in probability?

Answer 4

Cov(x,u) = 0

Question 5

What is the constant elasticity in a regression?

Answer 5

Found in the parameter that is part of a log-log model, for a one percent change in x, y changes by beta1 percent. Take care if either parameter is already in percent units, then it is percentage points.

Question 6

If x and u are correlated, what is the inconsistency in the estimator equal to?

Answer 6

Cov(x,u)/Var(x)

Question 7

What happens to the distribution of an estimator, beta^j, as the sample size increases?

Answer 7

It becomes more tightly distributed around the parameter, betaj.

Question 8

Stata: How can you get a t value from a regression output in Stata?

Answer 8

Under the column t, it is the coefficient/std. err.

Question 9

What is asymptotic normality?

Answer 9

The idea that OLS estimators are approximately normally distributed in large enough sample sizes.

Question 10

What two properties are needed for statistical inference in multiple regression analysis?

Answer 10

Consistency and asymptotic normally together allow for hypothesis testing about parameters in the MLR model.

Question 11

Can we use the t stat, (beta^j - betaj)/se(beta^j), for a standard normal distribution?

Answer 11

Yes, as long as n is large enough, even without consistency

Question 12

What is root n convergence?

Answer 12

Standard errors can be expected to shrink at the same rate as the inverse root of the sample size, 1/sqrt(n)

Question 13

Is the root mean squared error of the regression (RMSE) the standard error of a parameter estimate?

Answer 13

No, it is the standard error of the regression in ML analysis. (for the population, not sample)

Question 14

A consistent parameter estimate will converge on what as the random sample size increases to infinity?

Answer 14

The true population parameter.

Question 15

If an unbiased population parameter estimate is averaged, the expected value will equal what as the number of random samples goes to infinity?

Answer 15

The true population parameter

Question 16

Stata: two ways to drop observations for a regression?

Answer 16

“drop in x/n” drops the observations in the x to n range. “reg y x1 x1…. if _n

Question 17

What is the convergence rule of thumb for shrinking standard errors?

Answer 17

sqrt(n1)/sqrt(n2) where n1 is the full sample and n2 is the partial sample.

Question 18

Consistency vs bias?

Answer 18

They are basically the same, except that inconsistency is expressed in terms of population variance of x1 and pop cov(x1, x2) while bias is in the sample.

Question 19

What is asymptotic efficiency?

Answer 19

For consistent estimators with asymptotically normal distributions, the estimator with the smallest asymptotic variance.