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Unformatted text preview: ot only BLUE, but also the minimum variance unbiased estimator
o This means that OLS has the smallest variance amongst all unbiased estimators (not just the linear ones) While we assume normality here, sometimes that is not the case in practice
o One way we get around that is to rely on the CLT when we are dealing with large samples
o We can then drop the normality assumption VER. 9/25/2012. © P. KOLM 40 Outline of the Statistical Inference Lecture 1. Tests of a single linear restriction (t-tests), e.g.
b j = b j0 2. Tests of multiple linear restrictions (F-tests), e.g.
bi = b j = 0 3. Tests of linear combinations (t-tests, also possible using 2. above) e.g.
bi + b j = b 0 VER. 9/25/2012. © P. KOLM 41 Normal Sampling Distributions Under the CLM assumptions, conditional on the sample values of the
independent variables4 ˆ
bj N (bj ,Var (bj )) where
Var (b j ) = s2 ( SSTj 1 - Rj2 ) This means that VER. 9/25/2012. © P. KOLM ˆ
(b j - bj ) ˆ
std (b j ) N (0,1) 42 The t-Test ˆ
We can replace std (bj ) in
(b j ˆ
with se(b j ) = - bj ) ˆ
std (b j ) N (0,1) s ( SSTj 1 - Rj2 ) Then, under the CLM assumptions we have that
tº j - bj ) ˆ
se (b j ) tn -k -1 We refer to t as the “t-statistic” VER. 9/25/2012. © P. KOLM 43 Remarks Note this is a t -distribution with n - k - 1 degrees of freedom (v.s.
normal) because we have to estimate s by s Facts: The t -distribution . . .
o Looks like the standard normal except it has fatter tails
o Is a family of distributions characterized by degrees of freedom
o Gets more like the standard normal as degrees of freedom increase
o Is pretty much indistinguishable from a standard normal when df > 120 VER. 9/25/2012. © P. KOLM 44 How to Conduct the Test 1. Start with a null hypothesis. For example,
H 0 : bj = 0 ˆ
We only reject if b j is “sufficiently far” from zero: If we want to have only a 5% probability of rejecting H 0 , if it is really true, that is
P (reject H 0 | H 0 true) = 0.05 then we say our significance level is a = 0.05 Significance levels usually chosen to be 1%, 5% or 10% The exact rule on how to perform the test depends on the alternative hypothesis If this null is true…
o Then x j has no effect on y , controlling for other x ’s
o Then x j should be excluded from the model (efficiency argument, extraneous regressor)
→ We look at (1) one-sided and (2) two-sided alternatives, one at a time VER. 9/25/2012. © P. KOLM 45 2. Besides the null, H 0 , we need an alternative hypothesis, H 1 , and a
significance level H 1 may be one-sided, or two-sided
o H 1 : b j > 0 and H 1 : b j < 0 are one-sided
o H 1 : b j ¹ 0 is a two-sided alternative VER. 9/25/2012. © P. KOLM 46 The One-Sided Alternative: b j > 0 (1/2) Consider the alternative H 1 : b j > 0 Having picked a significance level, a , we determine the (1 - a)th percentile in a t -distribution with n - k - 1 df and call this c , the critical value Rejection rule:
o t > c : Reject the null hypothesis in favor of the alternative hypothesis if the observed t -statistic is greater than the critical value
o t £ c : If the t -statistic is less than or equal to the critical value then we do not reject the null VER. 9/25/2012. © P. KOLM 47 The One-Sided Alternative: b j > 0 (2/2) Model: yi = b0 + b1x i 1 +¼+bk xik + ui Hypothesis:
H 0 : bj = 0 H 1 : bj > 0 1-a Reject Fail to reject a 0 VER. 9/25/2012. © P. KOLM c 4...
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This document was uploaded on 02/17/2014 for the course COURANT G63.2751.0 at NYU.
- Fall '14