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Understanding Wald Statistic in Hypothesis Testing

Learn how to test hypotheses using the Wald Statistic in statistical analysis. Explore the concept of Wald testing using distribution theory and Likelihood Ratio test. Practice problem sets with multiple linear restrictions. Watch out for redundant restrictions.

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Understanding Wald Statistic in Hypothesis Testing

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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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