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Unformatted text preview: )i) Why run regression with PWE if AWE not available? Proxy Variable: In (1) we run into Omitted Variable Bias problem. We can’t run (3) as AWE not available. Potential solution: Use PWE as proxy to avoid OVB. Past Exam Practice Question Introduction Omitting Relevant Variable Including Irrelevant Variables Past Exam Practice Question 2005 Question 2 2)b)i) Why run regression with PWE if AWE not available? Suppose AWE = λ + µPWE → perfect proxy. Get alternative true specification: LGEARN = β1 + β2 S + β3 AWE + u (15) = β1 + β2 S + β3 (λ + µPWE ) + u (16) = (β1 + β3 λ) + β2 S + β3 µPWE + u (17) Fit alternative regression: Estimate of β2 same as if AWE used. s.e.’s and t-statistics of S same. t-statistic PWE same as for AWE. R 2 same. If know the value of µ, then retrieve estimate of β3 also with PWE. Introduction Omitting Relevant Variable Including Irrelevant Variables Past Exam Practice Question 2005 Question 2 2)b)ii) Did using proxy actually help? Logic for using proxy assumes perfect linear relation PWE to AWE. In practice this generally holds at most approximatively. However with imperfect proxy (chapter 8), we get into problem of measurement error and inconsistent estimator. Whether to use proxy or not is a judgement call sometimes. Remember: correlation PWE to AWE here only 0.47 → clearly imperfect proxy Introduction Omitting Relevant Variable Including Irrelevant Variables Past Exam Practice Question 2005 Question 2 2)b)ii) Did using proxy actually help? S coefficient estimate: increases using PWE as proxy, but not all the way to what is estimated in (3) Proxy seems to eliminate SOME of the bias S coefficient s.e.: increases using PWE as proxy both compared to (1) and (3) [N.B. in (1) s.e. are wrong! Why?] PWE coefficient: clearly not a good estimate of impact of AWE on earnings Constant estimate: falls using PWE as proxy, but not all the way to what is estimated in (3) R2 : negligible improvement (remember if perfect proxy should be same in (2) and (3)) Introduction Omitting Relevant Variable Including Irrelevant Variables Past Exam Practice Question 2005 Question 2 2)c) Why s.e. for S coefficient larger in (2) than (3) Correlation S to PWE much larger than S to AWE (absolute terms) So regression (2) has more trouble disentangling effect of S Use of better variable should lead to sharper estimate Introduction Omitting Relevant Variable Including Irrelevant Variables Past Exam Practice Question 2005 Question 2 2)d) Link t-statistics and size of RSS change Remember: F-test for significance of coefficient estimate βi Restriction is to set repective coefficient βi = 0 −RSS F-Statistic: F = (RSSR U /(n−U )/1 RSS k) (N.B. This is F-stat for 1 coefficient’s significance check!) Intuition: reject restriction if RSSR much larger than RSSU suggesting that the restriction in fact does not hold. Improvement in fit from imposing restriction and F-test intrinsically related: is the improvement in fit statistically significant? → RSS reduc...
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