13 Week 4a Empirical methods notes1. Motivation and overview2. Time series regressions3. Cross sectional regressions4. Fama MacBeth5. Testing factor models.13.1 Motivation and Overview1. Expected return, beta model, as in CAPM, Fama-French APT.1TS regression, define : = + 0+ =1 2 .2 Model: ()=(+) 0(+) because the point of the model is that should = 0.2. How do we take this to the data? Objectives:(a) Estimate parameters. .(b) Standard errors of parameter estimates. How do vary if you draw new data andtry again?(c) Test the model. Does ()=0?Arethe =0?Arethe that we see due to badluc k rather than real failure of the model?(d) Test one model vs. another. Can we drop a factor, e.g. size?13.2 Time-Series Regression1. Setup: test assets. factors ( =1forCAPM, =3forFF). time periods.2. * It’s useful to think of data in vector, matrix format assets time⎡⎢⎢⎢⎢⎣1121 11222 2...12 ⎤⎥⎥⎥⎥⎦There are -vectors across assets, =h12 i0()=h12 i0180and -vectors across factors =h12 i0()=h(1) (2) ()i0(I’ll use = 1 wherever possible). Vectors are column vectors.3. Time Series Regression (Fama-French)(a) Method: Run and interpret= + 0+ =1 2 .foreach(b) Estimates: How do we estimate in ()=()+0?i. A: If the factor is an excess return, then we should see = (). Estimate it thatway!ii. Why is = ()? When is an excess return, the model also applies to . =1of course, soModel: ()=1× i()eiERSlopei1()Efiii. *Note this does not work if is not an excess return, i.e. = ∆. Then, youcannot conclude that ()=, so the intercept in the time-series regression neednot be zero13.13You can still run a time series regression in this case. Write the time series regression= + + ( not because this intercept is not necessarily a pricing error.) The model is()=+ and should be zero if the model is right. Contrasting the model with the expected valueof the time series regression,we have()=+ ()=+ = + [() − ]181iv. *Important point: the time-series regression intercept alpha is only a measure ofmispricing if the right hand variable is an (excess) return. Intuition: how do youprofit from alpha? Buy ,short, then your portfolio return is + ,withmean and volatility (). But you can’t short ∆, you can only short thingsthat are returns!(c) Estimates:i. ˆˆ : OLS time-series regression.= + 0+ =1 2 for each .ii.ˆ : Mean of the factor,ˆ =1X=1=¯iii. Stop and admire. We are here to estimate the cross-sectional relationship between() and . We don’t actually run any cross sectional regressions. We estimatethe slope of the cross-sectional relationship by finding the mean of the factor. Weestimate the cross-sectional error as the time-series intercept.(d) Standard errors:i. Reminder of the standard error question: There are true parameters whic h wedon’t know. In our sample, we produce estimates ˆˆˆ. Due to good or bad luck,these will be different from true values. If we could rewind and run history over andover again, we’d get different values of ˆˆˆ. How much would these vary across thedifferent samples? If we thought = 0, how likely is it that the ˆ we see is just dueto luck? To answer these questions we need to know what (ˆ) ³ˆ´³ˆ´are.Watch it — this is how much the estimates would vary across “alternate universes”.These quantities are called standard errors.ii. Suppose are independent over time.Foreachasset, (+) = 0. I do notassume () = 0 — that would be a big mistake — ret urns are correlated witheachotheratapointintime!iii. ⇒(statistics) Then you can use OLS standard errors ˆˆ.iv. ⇒(statistics) Forˆ, our old friend: If are uncorrelated over time,(¯)=()√Applying it toˆ,(ˆ)=()√(e) Testi. Reminder of t statistic. ˆ (ˆ)=. This shouldn’t be too big if the true alpha iszero. “due to luck” it will only be over 2 in 5% of the samples.The intercepts are not free, but they aren’t zero either. If the model is righ t, we should see high intercepts wherewe see high . You can see that in the case the factor is a return, then ()=, and the alpha and int ercept are thesame thing. It’s easy enough to measure alphas this way and test if they are zero, but for reasons I don’t understandmodels with non-traded factors tend to be estimated by the cross-sectional method described below instead.182ii. Wewanttoknowifall are jointly zero. We know how to do (ˆ)(ˆ), (ˆ)−−from the i and jth time series regression — but how do you test all the together?iii. Answer: look atˆ0(ˆ ˆ0)−1ˆThis is a number. If the true is zero, this should not be “too big.” Precise forms,ˆ0(ˆ)−1ˆ = h1+¯0Σ−1¯i−1ˆ0Σ−1ˆ˜2 − − h1+¯0Σ−1¯i−1ˆ0ˆΣ−1ˆ˜ − −GO AL: Understand and use this form ula. Not memorize or derive it!iv. “˜” means “is distributed as,” meaning the number of the left follows the distributiongiven on the right. Procedure: compute the number on the left. Compare thatnumber to the distribution on the right, which tells you how likely it is to see anumber this large, if the true are all zero. (Note = number of factors only; i.e.1 for the CAPM. Don’t add another K for the constant. Also the F test wants y outo use 1 in
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