Suppose a Ph D student has a sample (Yi ,Xi ,Zi i 1, ,n) and estimates by OLS the equation Y Z X0e where is the coefficient of interest She is interested in testing H0 0 against H1 6 0 She obtains b 2 0 with standard error s(b) 1 0 so the value of the t ratio for H0 is T b s(b) 2 0 To assess significance, the student decides to use the bootstrap She uses the following algorithm 1 Samples (Y i ,X i ,Z i ) randomly from the observations (Random sampling with replacement) Creates a randomsample with n observations 2 On this pseudo sample, estimates the equation Y i Z i X0 i e i by OLS and computes standard errors, including s(b ) The t ratio for H0, T b s(b ) is computed and stored 3 This is repeated B 10,000 times 4 The 0 95th empirical quantile q 95 3 5 of the bootstrap absolute t ratios jT j is computed 5 The student notes that while jT j 2 1 96 (and thus an asymptotic 5 size test rejects H0), jT j 2 q 95 3 5 and thus the bootstrap test does not reject H0 As the bootstrap is more reliable, the student concludes that H0 cannot be rejected in favor of H1 Question Do you agreewith the studentsmethod and reasoning Do you see an error in her method

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Question: Suppose a Ph.D. student has a sample (Yi ,Xi ,Zi : i 1, ...,n) and estimates by OLS the equation Y Z X0e

Suppose a Ph.D. student has a sample (Yi ,Xi ,Zi : i Æ 1, ...,n) and estimates by OLS the equation Y Æ Z®Å X0¯Åe where ® is the coefficient of interest. She is interested in testing H0 : ® Æ 0 against H1 : ® 6Æ 0. She obtains b®

Æ 2.0 with standard error s(b®) Æ 1.0 so the value of the t-ratio for H0 is T Æ b®/s(b®) Æ 2.0. To assess significance, the student decides to use the bootstrap. She uses the following algorithm 1. Samples (Y ¤

i ,X¤

i ,Z¤

i ) randomly from the observations. (Random sampling with replacement).

Creates a randomsample with n observations.

2. On this pseudo-sample, estimates the equation Y ¤

Æ Z¤

i ®ÅX¤0 i ¯Åe¤

i by OLS and computes standard errors, including s(b®

¤). The t-ratio for H0, T ¤ Æb®

¤/s(b®

¤) is computed and stored.

3. This is repeated B Æ 10,000 times.

4. The 0.95th empirical quantile q¤

.95

Æ 3.5 of the bootstrap absolute t-ratios jT ¤j is computed.

5. The student notes that while jT j Æ 2 È 1.96 (and thus an asymptotic 5% size test rejects H0), jT j Æ

2 Ç q¤

.95

Æ 3.5 and thus the bootstrap test does not reject H0. As the bootstrap is more reliable, the student concludes that H0 cannot be rejected in favor of H1.

Question: Do you agreewith the student’smethod and reasoning? Do you see an error in her method?

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