Chapter 2 Asymptotic theory

Question 1

Continuous mapping theorem

Answer 1

If we apply a continuous function to a vector that converged to another vector, it now converges to the same vector but applying the continuous function to it.
Note that the inverse is a continuous function.

Question 2

Slutsky’s lemma

Answer 2

Question 3

How does Slutsky’s lemma apply to a matrix times rv?

Answer 3

Question 4

Kolmogorov (strong) Law of large numbers

Answer 4

Average converges in probability to the mean IF Z_I IS IID

Question 5

Stationarity

Answer 5

The distribution is invariate over time

Question 6

Ergodicity

Answer 6

The two groups become independent when n goes to infinity

Question 7

Ergodic theorem

Answer 7

iid is replaced by the ergodic theorem

Question 8

LLN to CLT 1- Classical Lindenberg-Levy CLT

Answer 8

z_i is iid.
In this case, the dependence that ergodicity restricts is not enough for the classical LL-CLT

Question 9

LLN to CLT 2- Martingale Difference CLT

Answer 9

Martingale difference sequence means that there is no serial correlation. If we have this + SE, we have MD-CLT
Note: the conditional variance remains unrestricted

Question 10

Linear regression assumptions (AT): 1

Answer 10

Linearity

Question 11

Linear regression assumptions (AT): 2

Answer 11

Stockastic assumption.
yi and xi are jointly stationary ergotic

Question 12

Linear regression assumptions (AT): 3

Answer 12

Predeterminedness assumption: E[x_iepsilon_i]=0

Question 13

Linear regression assumptions (AT): 4

Answer 13

Sigma_xx=E[xx’] is non singular (asymptotic full rank condition)

Question 14

Linear regression assumptions (AT): 5

Answer 14

x_i epsilon_i is a MDS with variance S=E[xx’epsilon^2], where S is a non-singular variance covariance matrix

Question 15

Consistency in OLS with Asymptotic Theory. Which assumptions were needed?

Answer 15

1 to 4.
Linearity for the formula, Stockastic assumption (StatErg) to apply ET, Predeterminedness to cancel out the second term, non-singularity for the ergodic theorem.
Thus, with infinite data we would find beta.

Question 16

Since we don’t have infinite data, we will have uncertainty. Convergence with finite data: CLT

Answer 16

With assumptions 1 to 5, OLS is asymptotically normal

Question 17

Since we do not know the asymptotic variance, we need an estimator for it. First, build the estimator for S

Answer 17

Question 18

What is the limiting distribution of b?

Answer 18

Question 19

What is the Robust SE for b?

Answer 19

Question 20

What is the limit of the robust t^* test for scalar hypothesis?

Answer 20

We need CMT for avarhat to converge to avar, and slutsky to combine them

Question 21

How do we test linear hypothesis? what is the limiting distribution of this test?

Answer 21

Question 22

What does specification testing tell us

Answer 22

If x_iepsilon_i is MDS and x_i includes a constant, then epsilon_i is also MDS, therefore it is serially uncorrelated

Question 23

What is autocovariance? And autocorrelation? How do we test MDS (H0)?

Answer 23

Assuming z_i is stationary.
H0: rho_1=—=rho_P=0
Where P is a fixed integer. If we don’t reject H0, then there is no serial correlation

Question 24

What are the estimators for autocovariance and autocorrelation? are they consistent? If MDS, what do they converge to?

Answer 24

Question 25

Propose a statistic to test serial correlation. What happens to it if there is serial correlation?

Answer 25

note: this test statistic doesn't always converge as we want it to

Question 26

Since we can't observe epsilon, how can we apply the BPS?

Answer 26

we can change rho tilde for rho hat if we can change gamma tilde for gamma hat. We substitute ei for epsilon i in the second equation and easily find that it converges to gamma tilde in probability. But we still beed

Question 27

Modified BPS

Answer 27

Where phi^{-1} is a standardizing matrix.