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Chicago Booth BUSF 35150 - Empirical methods notes

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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.1TS 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⎡⎢⎢⎢⎢⎣1121 11222 2...12 ⎤⎥⎥⎥⎥⎦There are -vectors across assets, =h12 i0()=h12 i0180and -vectors across factors =h12 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()eiERSlopei1()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. ˆˆ : OLS time-series regression.= + 0+  =1 2 for each .ii.ˆ : Mean of the factor,ˆ =1X=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 ˆˆˆ. 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 ˆˆˆ. 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 ˆˆ.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 atˆ0(ˆ ˆ0)−1ˆThis is a number. If the true  is zero, this should not be “too big.” Precise forms,ˆ0(ˆ)−1ˆ = h1+¯0Σ−1¯i−1ˆ0Σ−1ˆ˜2 −  − h1+¯0Σ−1¯i−1ˆ0ˆΣ−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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