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Hypothesis testing Ec402 PS 6

Hypothesis testing Ec402 PS 6. Notes on the Wald Statistic. Want to test hypotheses of the form (with J restrictions = number of rows in matrix R), based on our estimate . Note:. With A5N this implies: And so under we have : (1). If is known, use a distribution . Why?

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Hypothesis testing Ec402 PS 6

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  1. Hypothesis testingEc402PS 6

  2. Notes on the Wald Statistic • Want to test hypotheses of the form (with J restrictions = number of rows in matrix R), based on our estimate . • Note:

  3. With A5N this implies: • And so under we have: (1)

  4. If is known, use a distribution. • Why? • is the sum of J squared independent N(0,1) variables by definition. • From (1): And so from the definition of :

  5. If is unknown, use an F distribution. • Why? • F distribution is defined in terms of 2 independent distributions. Let and be independently distributed variables with and degrees of freedom. Then: • Can show that and is independent of (see Johnston & DiNardo p.495)

  6. And combining this result with (2) we get: (Note the s cancel) • Since this gives us

  7. Note that this Wald test is equal to the Likelihood Ratio test given by: • See lecture notes for a proof

  8. PS 6, question 2 • to (c) all involve multiple linear restrictions of the form (note change in notation between lecture and problem set) where R is an r x k matrix (where r = number of rows in R = number of restrictions), is k x 1 and hence is r x 1. General note: watch out for “redundant restrictions”

  9. PS 6, question 2 (d) (d): Carry out tests using Wald Statistic given by: (what happened to ?)

  10. PS 6, question 2 (d) (a) 2 restrictions (r=2): Restriction 1: Restriction 2: Hence:

  11. PS 6, question 2 (d) (b) 3 restrictions (r=3): Restriction 1: Restriction 2: Restriction 3: Hence:

  12. PS 6, question 2 (d) (c) 2 restrictions (r=2): Restriction 1: Restriction 2: Hence:

  13. PS 6, question 2 (e): See board Question 1: See board

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