12 12 1 Week 4a Empirical methods overheads Motivation and overview 1 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 and try again c Test the model Does 0 Are the 0 Are the that we see due to bad 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 regression 1 Time Series Regression Fama French a Method Run and interpret 0 1 2 for each E R ei Slope E f i 1 b Estimates i OLS time series regression 0 173 1 2 for each i ii Mean of the factor 1X 1 c Standard errors independent over time i OLS standard errors ii d Test are jointly zero i Answer look at 0 0 1 Precise forms h i 1 0 1 1 0 1 0 1 2 i 1 h 1 0 1 0 1 Do not memorize Understand look up ii If are also normal then a refinement is valid in small samples i 1 h 1 0 1 0 1 GRS Test iii Distribution of 0 0 1 2 distributions 1 2 1 2 2 0 9 0 8 0 7 0 6 0 5 0 4 0 3 0 2 0 1 0 0 5 1 174 1 5 2 2 5 3 3 5 4 4 5 5 12 3 Cross sectional regression 1 Another idea The main point of the model is across assets E Ri Slope i i 0 1 2 Why not fit this as a cross sectional regression 2 Two step procedure a TS over time for each asset to get 1 2 for each b Run CS across assets to get 1 2 3 TS vs CS E Ri Cross section Time Series Rf Factor market for CAPM i 175 4 Estimates a from TS b slope coe cient in CS c from error in CS 1 series regression any more P 1 6 is not the intercept from the time 5 Standard errors a from TS OLS formulas b You can t use OLS formulas c Answer With no intercept in CS 2 d i 1 h 0 1 0 0 1 1 0 1 1 0 1 0 0 1 0 1 0 1 e See notes for the formula with an intercept in CS 6 Test 0 0 1 2 1 7 Warning Cross section ei E R Time series i 12 4 Fama MacBeth procedure 1 Procedure 176 a Run TS to get betas 0 1 2 for each b Run a cross sectional regression at each time period 0 1 2 for each c Estimates of are the averages across time 1X 1X 1 1 d Standard errors use our friend 2 2 2 2 1 1 X 2 1 1 1 X 2 1 This one main point These standard errors are easy to calculate e Test 0 0 1 2 1 12 5 Testing one model vs another 1 Example FF3F Drop size 2 Fallacy test 0 Example 2 Suppose 1 1 2 2 We can drop as a factor But 12 12 0 3 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 Equivalently we are forming an orthogonalized factor and drop if 0 177 12 6 Summary of empirical procedures 1 The model is 0 the should be zero where the are defined from time series regressions 0 1 2 for each 2 Estimate parameters standard errors an overall test 0 1 3 Time series regression a from OLS time series regressions for each asset b c 2 GRS tests for 0 1 to test all together d Note must be a return or excess return for this to work True for CAPM FF3F 4 Cross sectional regression a from OLS time series regressions for each asset b from cross sectional regression 0 1 2 Ugly formulas for standard errors and c 0 1 to test all together d Can use if is not a return for example 5 Fama MacBeth a from OLS time series regressions for each asset b Cross sectional regressions at each time period 0 1 2 for each Then from averages of Standard errors from our old friend 1 similarly for c 0 1 to test all together 12 7 Comments 1 Summary of regressions a Forecasting regressions return on D P 1 1 1 2 178 b Time series regressions CAPM 1 1 1 1 2 c Cross sectional regressions 1 2 1 d Fama MacBeth cross sectional 1 2 e These are totally di erent regressions Don t confuse them In particular i ii iii iv TS is not about forecasting returns 2 in TS is not really relevant to the CAPM Cross sectional regressions aren t about forecasting returns either Only the average value of FMB cross section is really interesting 2 When is a factor important on tells you if hml is important for explaining variance of rmrf 0 tell you if adding hml helps to explain the mean of all the other returns 3 When is the TS intercept alpha only if the right hand variable is a return 0 0 179
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