14 Week 5 Empirical methods notes1. Motivation and overview2. Time series regressions3. Cross sectional regressions4. Fama MacBeth5. Testing factor models.14.1 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. 1 is useful to understandvariationovertimeinagivenreturn, for example, to find goodhedges, reduce variance. 2 is for when we w ant to understand average returns in relation tobetas(crosssection)3. 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?Arethe =0?Arethe ˆ 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?4. Statistics reminder. There is a true value of , which we don’t know.(a) We see a sample drawn from this truth, generating estimates ˆ,ˆ,ˆ.Thesesamplevalues vary from the truth, but they’re our best guess.(b) “Standard errors:” If we ran history over and over again, with different luc k, how wouldthe estimates ˆ,ˆ,ˆ vary across these alternative universes. “(c) Test” If the true alpha (say) were zero, how often (in how many alternate universes)would we see an alpha as big as the ˆ we measure in this sample? If not many it’sunlikely that the true is zero.22514.2 Statistics reviewis a random variable (say, stock return) drawn from a distribution with mean and variance 2.You see a sample {}, meaning (12). You do not see the true . Your job is to learnwhat you can about them from the sample.EstimateYou might firm the sample meanˆ =1X=1as a good guess of the mean. This is the estimate. We don’t have to use this estimate — we coulduse the median if we wanted to. If so, we’d get different formulas for what follo ws.The sample mean ˆ is a number to you. But it is also a random variable — it could have comeout differently. It’s like flipping a coin just once. Quiz: do you really understand the differencebetween and ˆ?Standard errorThe standard error of the mean(ˆ)is your best guess of how differently ˆ might have come out if you run history all over again. Ifyou will, it’s your best guess of how much the sample mean varies across all alternative universeswhich are just like ours but different luck of the draw.How can we guess such a thing, given that we only see one ˆ? You can’t take the variance witha single observation! Answer: with assumptions. Start with2(ˆ)=2Ã1X=1!=12h2()+2( − 1)(−1+ i(What the...Yes, look at =3,andremember(12)=(23)=(32)=(−1),2(ˆ)=2µ13(1+ 2+ 3)¶=19h32()+2(−1)+1(−2)iNow, let’s add an assumption that the sample is also drawn independently.Thenallthecovterms are zero, and we get the familiar formula2(ˆ)=2(); (ˆ)=()√Make sure you understand this. The standard error of the (sample) mean, your best guess of howthe one sample mean you see w ould vary if you could run history all over again is related to thestandard deviation of x ,howmuchxvariesover time in the one sample you do see. It’s ratheramazing that we can connect these two quantities!(What if is correlated over time, you ask? Actually, that’s easy to solve. Just use the formulawith all the cov’s in it. This is the “generalized method of moments” formula, which Asset Pricingexplains in great detail. It’s important to recognize: the standard formula √ only works for that are uncorrelated over time. There is a better formula that corrects for this correlation. As an226extreme example, if all the are perfectly correlated, then you’re really seeing one observation not observations.)Hypothesis testThenextthingwemightwanttodoisahypothesis test to characterize our uncertainty aboutthe true . Whatwehavesofaristhatˆ is our best guess of the true and (ˆ)measuresouruncertain ty about that guess. The hypothesis test puts that observation another way: suppose yousaw ˆ = 10. What is the chance of seeing ˆ = 10 or larger, just by chance, if the the true =0?To answer that question, let’s add another assumption (not really needed, but easier for now),that the are normally distributed. Now , we can rely on a theorem from statistics (which is justalgebra) that the sum of normally distributed random variables is also normal. Thus, if the truevalues are and , then the sample mean has the distributionˆ˜³ (ˆ)=√´Equivalently, the sample mean divided by its standard error has a standard normal distributionˆ − (ˆ)=ˆ − √=˜ (0 1)Now you’re ready to do a hypothesis test: If the true value really was zero, form the numberˆ(ˆ)=ˆ³√´Compare the n umber you have to the normal (0,1) distribution (table or inmatlab) that tells you the c h ance of seeing something this big or bigger. (A number of about 2corresponds to about 2.5% chance of seeing a bigger number.)A fly in the ointment: What do you use for ? You don’t know that. You could try using yourbest guess, the sample standard deviationˆ2=1X(− ˆ)2(or 1( −1), the difference doesn’t matter here. ) Quiz: Do you understand the difference betweenˆ and ?. Using that guess, you’d form the numberˆ − ˆ√However, looking across samples, (alternate universes) ˆ now varies too, so this number will vary bymore than the last number we created. Fortunately, a smart statistician figured out the distributionof this number — the sample mean divided by the sample standard deviation. It’s called the distribution. So we writeˆ − ˆ√˜And you compare the number you create on the left hand side to this
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