slides_Ch4_W[1]

We know that o y o o u melissa tartari yale

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: mal random variables. We can show that zj jx and wj jx are independent RVs. Putting all these results together we conclude: ˆ βj βj z h i=q j t(n K 1 ) wj ˆ se βj jx nK1 M elissa Tartari (Yale) Econometrics 15 / 41 Hypothesis Testing: Food for Thought I Observe that ∑n=1 ui2 is the sum of n normal random variables. ˆ i However, fewer than n are independent and this is the reason for the degrees of freedom of the χ2 distribution. Let us verify this lack of independence by considering the simple case in which Y = βo + U . We know that ˆ ˆ βo = y ) βo = βo + u . Melissa Tartari (Yale) Econometrics 16 / 41 Hypothesis Testing: Food for Thought I Observe that ∑n=1 ui2 is the sum of n normal random variables. ˆ i However, fewer than n are independent and this is the reason for the degrees of freedom of the χ2 distribution. Let us verify this lack of independence by considering the simple case in which Y = βo + U . We know that ˆ ˆ βo = y ) βo = βo + u . We also know (recall chapter 3) that ui b Melissa Tartari (Yale) = ui = ui ˆ βo βo u. Econometrics 16 / 41 Hypothesis Testing: Food for Thought I Observe that ∑n=1 ui2 is the sum of n normal random variables. ˆ i However, fewer than n are independent and this is the reason for the degrees of freedom of the χ2 distribution. Let us verify this lack of independence by considering the simple case in which Y = βo + U . We know that ˆ ˆ βo = y ) βo = βo + u . We also know (recall chapter 3) that ui b = ui = ui ˆ βo βo u. We are now in the position of deriving cov (ui , uj ) for j 6= i and if we bb …nd that it is non-zero we conclude that ui and uj are not b b independent. Melissa Tartari (Yale) Econometrics 16 / 41 Hypothesis Testing: Food for Thought II cov (ui , uj ) = cov (ui bb u , uj = cov (ui , uj ) =0 = = M elissa Tartari (Yale) 2cov u) cov (ui , u ) cov (u , uj ) + cov (u , u ) ! ∑N=1 uk + Var (u ) ui , k N σ2 σ2 + N N σ2 6= 0. N 2 Econometrics 17 / 41 Hypothesis Testing Testing Simple Hypothesis Melissa Tartari (Yale) Econometrics 18 / 41 Hypothesis Testing Testing Simple Hypothesis Testing Hypothesis About a Single Parameter Melissa Tartari (Yale) Econometrics 18 / 41 Hypothesis Testing Testing Simple Hypothesis Testing Hypothesis About a Single Parameter one-sided (e.g. Ho : βj > 0) Melissa Tartari (Yale) Econometrics 18 / 41 Hypothesis Testing Testing Simple Hypothesis Testing Hypothesis About a Single Parameter one-sided (e.g. Ho : βj > 0) two-sided (e.g. Ho : βj = 0) Melissa Tartari (Yale) Econometrics 18 / 41 Hypothesis Testing Testing Simple Hypothesis Testing Hypothesis About a Single Parameter one-sided (e.g. Ho : βj > 0) two-sided (e.g. Ho : βj = 0) other hypothesis involving only one parameter (e.g. Ho : 2 + 3 βj > 0) Melissa Tartari (Yale) Econometrics 18 / 41 Hypothesis Testing Testing Simple Hypothesis Testing Hypothesis About a Single Parameter one-sided (e.g. Ho : βj > 0) two-sided (e.g. Ho : βj = 0) other hypothesis involving only one parameter (e.g. Ho : 2 + 3 βj > 0) Testing Hypothesis About a Linear Combination of Parameters (e.g. Ho : β j = β i ) Melissa Tartari (Yale) Econometrics 18 / 41 Hypothesis Testing Testing Simple Hypothesis Testing Hypothesis About a Single Parameter one-sided (e.g. Ho : βj > 0) two-sided (e.g. Ho : βj = 0) other hypothesis involving only one parameter (e.g. Ho : 2 + 3 βj > 0) Testing Hypothesis About a Linear Combination of Parameters (e.g. Ho : β j = β i ) (Aside: we also review the notion of Con…dence Interval) Melissa Tartari (Yale) Econometrics 18 / 41 Hypothesis Testing Testing Simple Hypothesis Testing Hypothesis About a Single Parameter one-sided (e.g. Ho : βj > 0) two-sided (e.g. Ho : βj = 0) other hypothesis involving only one parameter (e.g. Ho : 2 + 3 βj > 0) Testing Hypothesis About a Linear Combination of Parameters (e.g. Ho : β j = β i ) (Aside: we also review the notion of Con…dence Interval) Testing Joint Hypothesis Melissa Tartari (Yale) Econometrics 18 / 41 Hypothesis Testing Testing Simple Hypothesis Testing Hypothesis About a Single Parameter one-sided (e.g. Ho : βj > 0) two-sided (e.g. Ho : βj = 0) other hypothesis involving only one parameter (e.g. Ho : 2 + 3 βj > 0) Testing Hypothesis About a Linear Combination of Parameters (e.g. Ho : β j = β i ) (Aside: we also review the notion of Con…dence Interval) Testing Joint Hypothesis that combine multiple one-parameter simple hypothesis (e.g. Ho : βj = 0& βi = 1) Melissa Tartari (Yale) Econometrics 18 / 41 Hypothesis Testing Testing Simple Hypothesis Testing Hypothesis About a Single Parameter one-sided (e.g. Ho : βj > 0) two-sided (e.g. Ho : βj = 0) other hypothesis involving only one parameter (e.g. Ho : 2 + 3 βj > 0) Testing Hypothesis About a Linear Combination of Parameters (e.g. Ho : β j = β i ) (Aside: we also review the notion of Con…dence Interval) Testing Joint Hypothesis that combine multiple one-parameter simple hypothesis (e.g. Ho : βj = 0& βi = 1) about multiple linear restrictions (e.g. Ho : 2 βj βk = 0& βi = 1) Melissa Tartari (Yale) Econometrics 18 / 41 A Simple Hypothesi...
View Full Document

This note was uploaded on 02/13/2014 for the course ECON 350 taught by Professor Donaldbrown during the Fall '10 term at Yale.

Ask a homework question - tutors are online