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Hypothesis testing Ec402 PS 6 Notes on the Wald Statistic • Want to test hypotheses of the form 𝐻0 : 𝑅𝛽 = 𝑞 (with r restrictions = number of rows in matrix R), based on our estimate 𝛽. • Note: 𝐸 𝑅𝛽 = 𝑅𝛽 𝑉𝑎𝑟 𝑅 𝛽 = 𝐸 𝑅𝛽 − 𝑅𝛽 𝑅 𝛽 − 𝑅𝛽 𝑉𝑎𝑟 𝑅𝛽 = 𝐸 𝑅 𝛽 − 𝛽 𝛽 − 𝛽 = 𝑅𝑉𝑎𝑟 𝛽 𝑅′ 𝑉𝑎𝑟 𝑅𝛽 = 𝜎 2 𝑅 𝑋 ′ 𝑋 −1 ′ 𝑅 ′ ′ 𝑅 ′ • With A5N this implies: 𝑅𝛽~𝑁(𝑅𝛽, 𝜎 2 𝑅 𝑋 ′ 𝑋 −1 𝑅 ′ ) 𝑅𝛽 − 𝑅𝛽~𝑁(0, 𝜎 2 𝑅 𝑋 ′ 𝑋 −1 𝑅 ′ ) • And so under 𝐻0 : 𝑅𝛽 = 𝑞 we have: 𝑅𝛽 − 𝑞~𝑁(0, 𝜎 2 𝑅 𝑋 ′ 𝑋 −1 ′ 𝑅 ) (1) • If 𝜎 2 is known, use a 𝜒 2 distribution. • Why? • 𝜒 2 (𝑟) is the sum of r squared independent N(0,1) variables by definition. • From (1): [𝜎 2 𝑅 𝑋 ′ 𝑋 −1 𝑅 ′ ]−1/2 (𝑅 𝛽 − 𝑞)~𝑁(0, 𝐼) And so from the definition of 𝜒 2 (𝑟): 𝑅𝛽 − 𝑞 ′ 𝜎 2𝑅 𝑋′𝑋 −1 𝑅 ′ −1 𝑅𝛽 − 𝑞 ~𝜒 2 𝑟 (2) • If 𝜎 2 is unknown, use an F distribution. • Why? • F distribution is defined in terms of 2 independent 𝜒 2 distributions. Let 𝑦1 and 𝑦2 be independently distributed 𝜒 2 variables with 𝑛1 and 𝑛2 degrees of freedom. Then: 𝑦1 /𝑛1 𝐹= ~𝐹(𝑛1 , 𝑛2 ) 𝑦2 /𝑛2 • Can show that 𝜀′𝜀 2 ~𝜒 (𝑁 2 𝜎 − 𝐾) and is independent of 𝛽 (see Johnston & DiNardo p.495) • And combining this result with (2) we get: ′ 𝑅𝛽 − 𝑞 𝑅 𝑋 ′ 𝑋 −1 𝑅′ −1 𝑅𝛽 − 𝑞 / 𝑟 ~𝐹(𝑟, 𝑁 − 𝐾) 𝜀′𝜀/(𝑁 − 𝐾) (Note the 𝜎 2 s cancel) • Since 𝜀′𝜀 𝑁−𝐾 𝑅𝛽 − 𝑞 ′ = 𝑠 2 this gives us 𝑠2𝑅 𝑋′𝑋 −1 𝑅 ′ −1 𝑅 𝛽 − 𝑞 /𝑟 ~ 𝐹(𝑟, 𝑁 − 𝐾) • Note that this Wald test is equal to the Likelihood Ratio test given by: 𝐹= (𝑅𝑆𝑆𝑈 −𝑅𝑆𝑆𝑅 )/𝑟 𝑅𝑆𝑆𝑈 /(𝑁−𝐾) • See lecture notes for a proof