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Multiple Regression Analysis

- y b0 b1x1 b2x2 . . . bkxk u
- 3. Asymptotic Properties

Consistency

- Under the Gauss-Markov assumptions OLS is BLUE,

but in other cases it wont always be possible to

find unbiased estimators - In those cases, we may settle for estimators

that are consistent, meaning as n ? 8, the

distribution of the estimator collapses to the

parameter value

Sampling Distributions as n ?

n3

n1 lt n2 lt n3

n2

n1

b1

Consistency of OLS

- Under the Gauss-Markov assumptions, the OLS

estimator is consistent (and unbiased) - Consistency can be proved for the simple

regression case in a manner similar to the proof

of unbiasedness - Will need to take probability limit (plim) to

establish consistency

Proving Consistency

A Weaker Assumption

- For unbiasedness, we assumed a zero conditional

mean E(ux1, x2,,xk) 0 - For consistency, we can have the weaker

assumption of zero mean and zero correlation

E(u) 0 and Cov(xj,u) 0, for j 1, 2, , k - Without this assumption, OLS will be biased and

inconsistent!

Deriving the Inconsistency

- Just as we could derive the omitted variable

bias earlier, now we want to think about the

inconsistency, or asymptotic bias, in this case

Asymptotic Bias (cont)

- So, thinking about the direction of the

asymptotic bias is just like thinking about the

direction of bias for an omitted variable - Main difference is that asymptotic bias uses the

population variance and covariance, while bias

uses the sample counterparts - Remember, inconsistency is a large sample

problem it doesnt go away as add data

Large Sample Inference

- Recall that under the CLM assumptions, the

sampling distributions are normal, so we could

derive t and F distributions for testing - This exact normality was due to assuming the

population error distribution was normal - This assumption of normal errors implied that

the distribution of y, given the xs, was normal

as well

Large Sample Inference (cont)

- Easy to come up with examples for which this

exact normality assumption will fail - Any clearly skewed variable, like wages,

arrests, savings, etc. cant be normal, since a

normal distribution is symmetric - Normality assumption not needed to conclude OLS

is BLUE, only for inference

Central Limit Theorem

- Based on the central limit theorem, we can show

that OLS estimators are asymptotically normal - Asymptotic Normality implies that P(Zltz)?F(z) as

n ??, or P(Zltz) ? F(z) - The central limit theorem states that the

standardized average of any population with mean

m and variance s2 is asymptotically N(0,1), or

Asymptotic Normality

Asymptotic Normality (cont)

- Because the t distribution approaches the normal

distribution for large df, we can also say that

- Note that while we no longer need to assume

normality with a large sample, we do still need

homoskedasticity

Asymptotic Standard Errors

- If u is not normally distributed, we sometimes

will refer to the standard error as an asymptotic

standard error, since

- So, we can expect standard errors to shrink at a

rate proportional to the inverse of vn

Lagrange Multiplier statistic

- With large samples, by relying on asymptotic

normality for inference, we can use more than t

and F stats - The Lagrange multiplier or LM statistic is an

alternative for testing multiple exclusion

restrictions - Because the LM statistic uses an auxiliary

regression its sometimes called an nR2 stat

LM Statistic (cont)

- Suppose we have a standard model, y b0 b1x1

b2x2 . . . bkxk u and our null hypothesis

is - H0 bk-q1 0, ... , bk 0
- First, we just run the restricted model

LM Statistic (cont)

- With a large sample, the result from an F test

and from an LM test should be similar - Unlike the F test and t test for one exclusion,

the LM test and F test will not be identical

Asymptotic Efficiency

- Estimators besides OLS will be consistent
- However, under the Gauss-Markov assumptions, the

OLS estimators will have the smallest asymptotic

variances - We say that OLS is asymptotically efficient
- Important to remember our assumptions though, if

not homoskedastic, not true

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