12 Week 5a Empirical methods overheads12.1 Motivation and overview1. Expected return, beta model, as in CAPM, Fama-French APT.1TS regression, define : = + 0+ =1 2 .2Model: ()=0 (+)(+) because the point of the model is that should = 0.2. How do we take this to the data? Objectives:(a) Estimate parameters. ˆˆˆ.(b) Standard errors of parameter estimates. How do ˆˆˆ vary if you draw new data andtry again?(c) Test the model. Does ()=0? Are the true =0?Arethe ˆ that we see due tobad luck rather than real failure of the model?(d) Test one model vs. another. Can we drop a factor, e.g. size?12.2 Time-series regression1. Time Series Regression (Fama-French)(a) Method:i. Run= + 0+ =1 2 .foreachii. Interpret the regression as a description of the cross section, without running moreregressions³´= 0 ()+ =1 2i()eiERSlopei1()Ef201(b) Estimates:i. ˆˆ : OLS time-series regression.= + 0+ =1 2 for each .ii.ˆ : Mean of the factor,ˆ =1X=1=¯(c) Standard e rrors: Assume independent over time (but not across portfolios).i. OLS standard errors ˆˆ.ii.ˆ :(ˆ)=()√(d) Test are jointly zero?i. Answer: look atˆ0(ˆ ˆ0)−1ˆPrecise forms,ˆ0(ˆ)−1ˆ = h1+¯0Σ−1¯i−1ˆ0Σ−1ˆ˜2 − − h1+¯0ˆΣ−1¯i−1ˆ0ˆΣ−1ˆ˜− −whereˆ = estimated alphas¯ = ()Σ= (0)Σ = ( 0) = sample length =numberoffactors = number of assets2= chi-squared distribution = F distributionDo not memorize! Understand, look up.ii. Basic idea: like the square of a t testˆ(ˆ)˜iii. If are also normal then a refinementisvalidinsmallsamples. − − h1+¯0ˆΣ−1¯i−1ˆ0ˆΣ−1ˆ˜− −“GRS Test.”202iv. Distribution of ˆ0(ˆ ˆ0)−1ˆ — if the true alphas are zero, how often should wesee this number be one, two, etc.?0.5 1 1.5 2 2.5 3 3.5 4 4.5500.10.20.30.40.50.60.70.80.91χ2 distributionsχ21χ22v. Fama and French “GRS” quote p. 57. The huge rejection because Σ is so small. − − h1+¯0ˆΣ−1¯i−1ˆ0ˆΣ−1ˆ˜− −12.3 Cross-sectional regression1. Another idea. The main point of the model is across assets,i()iERSlopei()=0 (+) =1 2Why not fitthisasacross sectional regression?2032. Two step procedure(a) TS (over time for each asset) to get ,= + + =1 2 for each .(b) Run CS (across assets) to get .()=()+ + =1 23. Estimates:(a)ˆ from TS.(b)ˆ slope coefficient in CS.(c) ˆ from error in CS: ˆ =1³P=1´−ˆˆ ˆ 6= isnottheinterceptfromthetimeseries regression any more.4. Standard errors.(a) (ˆ)fromTS,OLSformulas.(b) (ˆ). You can’t use OLS form ulas.(c) Answer: With no intercept in CS,2(ˆ)=1h¡0¢−10Σ(0)−1³1+0Σ−1´+ Σi(d) (ˆ)(ˆ)=1³ − (0)−10´Σ³ − (0)−10´³1+0Σ−1´(e) See notes for the form ula with an intercept in CS.5. Testˆ0(ˆ ˆ0)−1ˆ˜2−−16. General tool: how to correct OLS standard errors for correlated errors = + ; (0)=Ωˆ =(0)−10ˆ =(0)−10( + )(ˆ)=! (unbiased)(ˆ)=h(0)−100(0)−1i=(0)−10Ω(0)−1If Ω = 2then we’re back to our friend(ˆ)=2¡0¢−120412.4 Fama - MacBeth procedure.1. Run TS to get betas.= + 0+ =1 2 for each .2. Run a cross sectional regression at each time period,=()+0+ =1 2 for each 3. Estimates of are the averages across timeˆ =1X=1ˆ;ˆ=1X=1ˆ4. Standard errors use our friend 2(¯)=2() but applied to the monthly cs regressioncoefficients!2(ˆ)=1(ˆ)=12X=1³ˆ−ˆ´2(ˆ)=1(ˆ)=12X=1(ˆ− ˆ)(ˆ− ˆ)This one main point. These standard errors are easy to calculate.5. Testˆ0(ˆ ˆ0)−1ˆ˜2−16. Fact: if the are constant over time, the estimates are identical to those of cross-sectionalregressions! Standard errors are close.7. Other applications of Fama-MacBeth: any time you have a big cross-section, which may becorrelated with each other.(a)+1= + ln + ln + +1; () 6=0This is an obvious candidate, since the errors are correlated across stocks but not acrosstime. FF used FMB regressions in the second paper we read.(b)investment= + × Book/Market+ × profits+ (c) This is a big deal! Either FMB or “Cluster.”20512.5 Testing one model vs. another1. Example. FF3F.()=+ + + Drop size?()=+ + No, because the and change.2. Fallacy: see if =0.3. Example 2: CAPM w orks well for size portfolios, but shows up in FF model()=0+← works, higher with higher ()()=0++ ← works too, all =1and 04. Solution:(a) You can drop smb if the other factors price smb.= + + + We can drop smb from the three factor model if and only is zero. .(b)
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