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### Fin econometricslecture

1. 1. Methods Lectures: Financial Econometrics Linear Factor Models and Event Studies Michael W. Brandt, Duke University and NBER NBER Summer Institute 2010 Michael W. Brandt Methods Lectures: Financial Econometrics
2. 2. Outline 1 Linear Factor Models Motivation Time-series approach Cross-sectional approach Comparing approaches Odds and ends 2 Event Studies Motivation Basic methodology Bells and whistles Michael W. Brandt Methods Lectures: Financial Econometrics
3. 3. Linear Factor Models Motivation Outline 1 Linear Factor Models Motivation Time-series approach Cross-sectional approach Comparing approaches Odds and ends 2 Event Studies Motivation Basic methodology Bells and whistles Michael W. Brandt Methods Lectures: Financial Econometrics
4. 4. Linear Factor Models Motivation Linear factor pricing models Many asset pricing models can be expressed in linear factor form E[rt+1 − r0 ] = β λ where β are regression coefﬁcients and λ are factor risk premia. The most prominent examples are the CAPM E[ri,t+1 − r0 ] = βi,m E[rm,t+1 − r0 ] and the consumption-based CCAPM E[ri,t+1 − r0 ] βi,∆ct+1 γVar[∆ct+1 ] λ Note that in the CAPM the factor is an excess return that itself must be priced by the model ⇒ an extra testable restriction. Michael W. Brandt Methods Lectures: Financial Econometrics
5. 5. Linear Factor Models Motivation Econometric approaches Given the popularity of linear factor models, there is a large literature on estimating and testing these models. These techniques can be roughly categorized as Time-series regression based intercept tests for models in which the factors are excess returns. Cross-sectional regression based residual tests for models in which the factors are not excess returns or for when the alternative being considered includes stock characteristics. The aim of this ﬁrst part of the lecture is to survey and put into perspective these econometric approaches. Michael W. Brandt Methods Lectures: Financial Econometrics
6. 6. Linear Factor Models Time-series approach Outline 1 Linear Factor Models Motivation Time-series approach Cross-sectional approach Comparing approaches Odds and ends 2 Event Studies Motivation Basic methodology Bells and whistles Michael W. Brandt Methods Lectures: Financial Econometrics
7. 7. Linear Factor Models Time-series approach Black, Jensen, and Scholes (1972) Consider a single factor (everything generalizes to k factors). When this factor is an excess return that itself must be priced by the model, the factor risk premium is identiﬁed through ˆ ˆ λ = ET [ft+1 ]. With the factor risk premium ﬁxed, the model implies that the intercepts of the following time-series regressions must be zero e ri,t+1 − r0 ≡ ri,t+1 = αi + βi ft+1 + i,t+1 e so that E[ri,t+1 ] = βi λ. This restriction can be tested asset-by-asset with standard t-tests. Michael W. Brandt Methods Lectures: Financial Econometrics
8. 8. Linear Factor Models Time-series approach Joint intercepts test The model implies, of course, that all intercepts are jointly zero, which we test with standard Wald test α (Var[ˆ ])−1 α ∼ χ2 . ˆ α ˆ N To evaluate Var[ˆ ] requires further distributions assumptions α With iid residuals t+1 , standard OLS results apply   −1 α ˆ 1 1 Eˆ T [ft+1 ] Var ˆ =  ˆ ˆ 2 ⊗Σ  β T ET [ft+1 ] ET [ft+1 ]2 which implies   2 1 ˆ ET [ft+1 ] Var[ˆ ] = α 1+ ˆ Σ . T σT [ft+1 ] ˆ Michael W. Brandt Methods Lectures: Financial Econometrics
9. 9. Linear Factor Models Time-series approach Gibbons, Ross, and Shanken (1989) Assuming further that the residuals are iid multivariate normal, the Wald test has a ﬁnite sample F distribution 2 −1   (T − N − 1)  ˆ T [ft+1 ] E 1+  α Σ−1 α ∼ FN,T −N−1 . ˆ ˆ ˆ N σT [ft+1 ] ˆ In the single-factor case, this test can be further rewritten as 2 2   ˆ e ET [rq,t+1 ] ˆ e ET [rm,t+1 ] e − ˆ e   (T − N − 1)  σT [rq,t+1 ]  ˆ σT [rm,t+1 ]  ,  N  2  ˆ e ET [rm,t+1 ]  1+   e σT [rm,t+1 ] ˆ where q is the ex-post mean-variance portfolio and m is the ex-ante mean variance portfolio (i.e., the market portfolio). Michael W. Brandt Methods Lectures: Financial Econometrics
10. 10. Linear Factor Models Time-series approach MacKinlay and Richardson (1991) When the residuals are heteroskedastic and autocorrelated, we can obtain Var[ˆ ] by reformulating the set of N regressions as a α GMM problem with moments ˆ rt+1 − α − βft+1 ˆ t+1 gT (θ) = ET = ET =0 (rt+1 − α − βft+1 )ft+1 t+1 ft+1 Standard GMM results apply α ˆ 1 −1 Var ˆ = DS −1 D β T with ∂gT (θ) 1 ˆ ET [ft+1 ] D= =− ˆ T [ft+1 ] ˆ 2 ⊗ IN ∂θ E ET [ft+1 ] ∞ t t−j S= E . t ft t−j ft−j j=−∞ Michael W. Brandt Methods Lectures: Financial Econometrics
11. 11. Linear Factor Models Cross-sectional approach Outline 1 Linear Factor Models Motivation Time-series approach Cross-sectional approach Comparing approaches Odds and ends 2 Event Studies Motivation Basic methodology Bells and whistles Michael W. Brandt Methods Lectures: Financial Econometrics
12. 12. Linear Factor Models Cross-sectional approach Cross-sectional regressions When the factor is not an excess return that itself must be priced by the model, the model does not imply zero time-series intercepts. Instead one can test cross-sectionally that expected returns are proportional to the time-series betas in two steps 1. Estimate asset-by-asset βi through time-series regressions e rt+1 = a + βi ft+1 + t+1 . 2. Estimate the factor risk premium λ with a cross-sectional regression of average excess returns on these betas ˆ ˆ ¯ie = ET [rie ] = λβi + αi . r Ignore for now the "generated regressor" problem. Michael W. Brandt Methods Lectures: Financial Econometrics
13. 13. Linear Factor Models Cross-sectional approach OLS estimates The cross-sectional OLS estimates are λ = (β β)−1 (β¯ie ) ˆ ˆˆ ˆr ˆˆ α = ¯e − λβ. ˆ r i A Wald test for the residuals is then αVar[α]−1 α ∼ χ2 ˆ ˆ ˆ N−1 with 1 Var[ˆ ] = I − β(β β)−1 β α ˆ ˆˆ Σ I − β(β β)−1 β ˆ ˆˆ T Two important notes Var[ˆ ] is singular, so Var[ˆ ]−1 is a generalized inverse and α α the χ 2 distribution has only N − 1 degrees of freedom. Testing the size of the residuals is bizarre, but here we have additional information from the time-series regression (in red). Michael W. Brandt Methods Lectures: Financial Econometrics
14. 14. Linear Factor Models Cross-sectional approach GLS estimates Since the residuals of the second-stage cross-sectional regression are likely correlated, it makes sense to instead use GLS. The relevant GLS expressions are λGLS = (β Σ−1 β)−1 (βΣ−1¯ie ) ˆ ˆ ˆ ˆ r and 1 Var[ˆ GLS ] = α Σ − β(β Σ−1 β)−1 β ˆ ˆ ˆ . T Michael W. Brandt Methods Lectures: Financial Econometrics
15. 15. Linear Factor Models Cross-sectional approach Shanken (1992) The generated regressor problem, the fact that β is estimated in the ﬁrst stage time-series regression, cannot be ignored. Corrected estimates are given by 1 λ2 Var[ˆ OLS ] = I − β(β β)−1 β α ˆ ˆˆ Σ I − β(β β)−1 β ˆ ˆˆ × 1+ T σf2 1 λ2 Var[ˆ GLS ] = α Σ − β(β Σ−1 β)−1 β ˆ ˆ ˆ × 1+ T σf2 In the case of the CAPM, for example, the Shanken correction for annual data is roughly 2 2 ˆ ET [rmkt,t+1 ] 0.06 1+ =1+ = 1.16 σT [rmkt,t+1 ] ˆ 0.15 Michael W. Brandt Methods Lectures: Financial Econometrics
16. 16. Linear Factor Models Cross-sectional approach Cross-sectional regressions in GMM Formulating cross-sectional regressions as GMM delivers an automatic "Shanken correction" and allows for non-iid residuals. The GMM moments are e   rt+1 − a − βft+1 e gT (θ) = ET [rt+1  (rt+1 − a − βft+1 )ft+1  = 0. rt+1 − βλ The ﬁrst two rows are the ﬁrst-stage time-series regression. The third row over-identiﬁes λ If those moments are weighted by β we get the ﬁrst order conditions of the cross-sectional OLS regression. If those moments are instead weighted by β Σ−1 we get the ﬁrst order conditions of the cross-sectional GLS regression. Everything else is standard GMM. Michael W. Brandt Methods Lectures: Financial Econometrics
17. 17. Linear Factor Models Cross-sectional approach Linear factor models in SDF form Linear factor models can be rexpressed in SDF form e E[mt+1 rt+1 ] = 0 with mt+1 = 1 − bft+1 . To see that a linear SDF implies a linear factor model, substitute in e e e 0 = E[(1 − bft+1 )rt+1 ] = E[rt+1 ] − b E[rt+1 ft+1 ] e e Cov[rt+1 , ft+1 ] + E[rt+1 ]E[ft+1 ] and solve for e b e E[rt+1 ] = Cov[rt+1 , ft+1 ] 1 − bE[ft+1 ] e Cov[rt+1 , ft+1 ] bVar[ft+1 ] = . Var[ft+1 ] 1 − bE[ft+1 ] β λ Michael W. Brandt Methods Lectures: Financial Econometrics
18. 18. Linear Factor Models Cross-sectional approach GMM on SDF pricing errors This SDF view suggests the following GMM approach ˆ e ˆ e ˆ e gT (b) = ET (1 + bft+1 )rt+1 = ET [rt+1 ] − bET [rt+1 ft+1 ] = 0. pricing errors = π Standard GMM techniques can be used to estimate b and test whether the pricing errors are jointly zero in population. GMM estimation of b is equivalent to Cross-sectional OLS regression of average returns on the second moments of returns with factors (instead of betas) when the GMM weighting matrix is an identity. Cross-sectional GLS regression of average returns on the second moments of returns with factors when the GMM weighting matrix is a residual covariance matrix. Michael W. Brandt Methods Lectures: Financial Econometrics
19. 19. Linear Factor Models Cross-sectional approach Fama and MacBeth (1973) The Fama-MacBeth procedure is one of the original variants of cross-sectional regressions consisting of three steps 1. Estimate βi from stock or portfolio level rolling or full sample time-series regressions. 2. Run a cross-sectional regression e ˆ rt+1 = λt+1 β + αi,t+1 for every time period t resulting in a time-series ˆ ˆ {λt , αi,t }T . t=1 3. Perform statistical tests on the time-series averages 1 T ˆ 1 T ˆ αi = ˆ t=1 αi,t ˆ and λi = T t=1 λi,t . T Michael W. Brandt Methods Lectures: Financial Econometrics
20. 20. Linear Factor Models Comparing approaches Outline 1 Linear Factor Models Motivation Time-series approach Cross-sectional approach Comparing approaches Odds and ends 2 Event Studies Motivation Basic methodology Bells and whistles Michael W. Brandt Methods Lectures: Financial Econometrics
21. 21. Linear Factor Models Comparing approaches Time-series regressions Source: Cochrane (2001) Avg excess returns versus betas on CRSP size portfolios, 1926-1998. Michael W. Brandt Methods Lectures: Financial Econometrics
22. 22. Linear Factor Models Comparing approaches Cross-sectional regressions Source: Cochrane (2001) Avg excess returns versus betas on CRSP size portfolios, 1926-1998. Michael W. Brandt Methods Lectures: Financial Econometrics
23. 23. Linear Factor Models Comparing approaches SDF regressions Source: Cochrane (2001) Avg excess returns versus predicted value on CRSP size portfolios, 1926-1998. Michael W. Brandt Methods Lectures: Financial Econometrics
24. 24. Linear Factor Models Comparing approaches Estimates and tests Source: Cochrane (2001) CRSP size portfolios, 1926-1998. Michael W. Brandt Methods Lectures: Financial Econometrics
25. 25. Linear Factor Models Odds and ends Outline 1 Linear Factor Models Motivation Time-series approach Cross-sectional approach Comparing approaches Odds and ends 2 Event Studies Motivation Basic methodology Bells and whistles Michael W. Brandt Methods Lectures: Financial Econometrics
26. 26. Linear Factor Models Odds and ends Firm characteristics So far H0 : αi = 0 i = 1, 2, ..., n HA : αi = 0 for some i We might, typically on the basis of preliminary portfolio sorts, a more speciﬁc alternative in mind H0 : αi = 0 i = 1, 2, ..., n HA : αi = γ xi for some ﬁrm level characteristics xi . We can test against this more speciﬁc alternative by including the characteristics in the cross-sectional regression ˆ ¯ie == λβi + γ xi + αi . r Michael W. Brandt Methods Lectures: Financial Econometrics
27. 27. Linear Factor Models Odds and ends Portfolios Stock level time-series regressions are problematic Way too noisy. Betas may be time-varying. Fama-MacBeth solve this problem by forming portfolios 1. Each portfolio formation period (say annually), obtain a noisy estimate of ﬁrm-level betas through stock-by-stock time series regressions over a backward looking estimating period. 2. Sort stocks into "beta portfolios" on the basis of their noisy but unbiased "pre-formation beta". The resulting portfolios should be fairly homogeneous in their beta, should have relatively constant portfolio betas through time, and should have a nice beta spread in the cross-section. Michael W. Brandt Methods Lectures: Financial Econometrics
28. 28. Linear Factor Models Odds and ends Characteristic sorts Somewhere along the way sorting on pre-formation factor exposures got replaced by sorting on ﬁrm characteristics xi . Sorting on characteristics is like sorting on ex-ante alphas (the residuals) insead of betas (the regressor). It may have unintended consequence 1. Characteristic sorted portfolios may not have constant betas. 2. Characteristic sorted portfolios could have no cross-sectional variation in betas and hence no power to identify λ. Michael W. Brandt Methods Lectures: Financial Econometrics
29. 29. Linear Factor Models Odds and ends Characteristic sorts (cont) Source: Ang and Liui (2004) Michael W. Brandt Methods Lectures: Financial Econometrics
30. 30. Linear Factor Models Odds and ends Conditional information In a conditional asset pricing model expectations, factor exposures, and factor risk premia all have a time-t subscripts Et [rt+1 − r0 ] = βt λt . This makes the model fundamentally untestable. Nevertheless, there are at least two ways to proceed 1. Time-series modeling of the moments, such as βt = β0 − κ(βt−1 − β0 ) + ηt . 2. Model the moments as (linear) functions of observables βt = γβ Zt or λt = γλ Zt . Approach #2 is by far the more popular. Michael W. Brandt Methods Lectures: Financial Econometrics
31. 31. Linear Factor Models Odds and ends Conditional SDF approach In the context of SDF regressions e Et [mt+1 rt+1 ] = 0 with mt+1 = 1 − bt ft+1 . Assume bt = θ0 + θ1 Zt and substitute in e Et (1 − (θ0 + θ1 Zt )ft+1 ) rt+1 = 0. Condition down e e E Et [(1 − (θ0 + θ1 Zt )ft+1 )rt+1 ] = E (1 − (θ0 + θ1 Zt )ft+1 )rt+1 = 0. Finally rearrange as unconditional multifactor model e E (1 − θ0 f1,t+1 − θ1 f2,t+1 )rt+1 = 0, where f1,t+1 = ft+1 and f2,t+1 = zt ft+1 . Michael W. Brandt Methods Lectures: Financial Econometrics
32. 32. Event Studies Motivation Outline 1 Linear Factor Models Motivation Time-series approach Cross-sectional approach Comparing approaches Odds and ends 2 Event Studies Motivation Basic methodology Bells and whistles Michael W. Brandt Methods Lectures: Financial Econometrics
33. 33. Event Studies Motivation What is an event study? An event study measures the immediate and short-horizon delayed impact of a speciﬁc event on the value of a security. Event studies traditionally answer the question Does this event matter? (e.g., index additions) but are increasingly used to also answer the question Is the relevant information impounded into prices immediately or with delay? (e.g., post earnings announcement drift) Event studies are popular not only in accounting and ﬁnance, but also in economics and law to examine the role of policy and regulation or determine damages in legal liability cases. Michael W. Brandt Methods Lectures: Financial Econometrics
34. 34. Event Studies Motivation Short- versus long-horizon event studies Short-horizon event studies examine a short time window, ranging from hours to weeks, surrounding the event in question. Since even weekly expected returns are small in magnitude, this allows us to focus on the information being released and (largely) abstract from modeling discount rates and changes therein. There is relatively little controversy about the methodology and statistical properties of short-horizon event studies. However event study methodology is increasingly being applied to longer-horizon questions. (e.g., do IPOs under-perform?) Long-horizon event studies are problematic because the results are sensitive to the modeling assumption for expected returns. The following discussion focuses on short-horizon event studies. Michael W. Brandt Methods Lectures: Financial Econometrics
35. 35. Event Studies Basic methodology Outline 1 Linear Factor Models Motivation Time-series approach Cross-sectional approach Comparing approaches Odds and ends 2 Event Studies Motivation Basic methodology Bells and whistles Michael W. Brandt Methods Lectures: Financial Econometrics
36. 36. Event Studies Basic methodology Setup Basic event study methodology is essentially unchanged since Ball and Brown (1968) and Fama et al. (1969). Anatomy of an event study 1. Event Deﬁnition and security selection. 2. Speciﬁcation and estimation of the reference model characterizing "normal" returns (e.g., market model). 3. Computation and aggregation of "abnormal" returns. 4. Hypothesis testing and interpretation. #1 and the interpretation of the results in #4 are the real economic meat of an event study – the rest is fairly mechanical. Michael W. Brandt Methods Lectures: Financial Econometrics
37. 37. Event Studies Basic methodology Reference models Common choices of models to characterize "normal" returns Constant expected returns ri,t+1 = µi + i,t+1 Market model ri,t+1 = αi + βi rm,t+1 + i,t+1 Linear factor models ri,t+1 = αi + βi ft+1 + i,t+1 (Notice the intercepts.) Short-horizon event study results tend to be relatively insensitive to the choice of reference models, so keeping it simple is ﬁne. Michael W. Brandt Methods Lectures: Financial Econometrics
38. 38. Event Studies Basic methodology Time-line and estimation A typical event study time-line is Source: MacKinlay (1997) The parameters of the reference model are usually estimated over the estimation window and held ﬁxed over the event window. Michael W. Brandt Methods Lectures: Financial Econometrics
39. 39. Event Studies Basic methodology Abnormal returns Using the market model as reference model, deﬁne ˆ ˆ AR i,τ = ri,τ − αi − βi rm,τ τ = T1 + 1, ..., T 2. ˆ Assuming iid returns 2 ˆ 1 (rm,τ − µm ) ˆ Var AR i,τ = σ 2i + 1+ . T1 − T0 σm ˆ due to estimation error Estimation error also introduces persistence and cross-correlation in the measured abnormal returns, even when true returns are iid, which is an issue particularly for short estimation windows. Assume for what follows that estimation error can be ignored. Michael W. Brandt Methods Lectures: Financial Econometrics
40. 40. Event Studies Basic methodology Aggregating abnormal returns Since the abnormal return on a single stock is extremely noisy, returns are typically aggregated in two dimensions. 1. Average abnormal returns across all ﬁrms undergoing the same event lined up in event time ¯ 1 N ˆ AR τ = AR i,τ . N i=1 2. Cumulate abnormal returns over the event window ˆ T2 ˆ CAR i (T1 , T2 ) = τ =T1 +1 AR i,τ or ¯ T2 ¯ CAR(T1 , T2 ) = τ =T1 +1 AR τ . Michael W. Brandt Methods Lectures: Financial Econometrics
41. 41. Event Studies Basic methodology Hypothesis testing Basic hypothesis testing requires expressions for the variance of ¯ ˆ ¯ AR τ , CAR i (T1 , T2 ), and CAR(T1 , T2 ). If stock level residuals i,τ are iid through time ˆ Var CAR i (T1 , T2 ) = (T2 − T1 )σ 2i . The other two expressions are more problematic because they depend on the cross-sectional correlation of event returns. If, in addition, event windows are non-overlapping across stocks ¯ 1 N 2 Var AR τ = 2 i=1 σ i N ¯ T2 ¯ Var CAR(T1 , T2 ) = τ =T1 +1 Var AR τ . Michael W. Brandt Methods Lectures: Financial Econometrics
42. 42. Event Studies Basic methodology Example Earnings announcements Source: MacKinlay (1997) Michael W. Brandt Methods Lectures: Financial Econometrics
43. 43. Event Studies Bells and whistles Outline 1 Linear Factor Models Motivation Time-series approach Cross-sectional approach Comparing approaches Odds and ends 2 Event Studies Motivation Basic methodology Bells and whistles Michael W. Brandt Methods Lectures: Financial Econometrics
44. 44. Event Studies Bells and whistles Dependence If event windows overlap across stocks, the abnormal returns are correlated contemporaneously or at lags. One solution is Brown and Warner’s (1980) "crude adjustment" Form a portfolio of ﬁrms experiencing an event in a given month or quarter (still lined up in event time, of course). Compute the average abnormal return of this portfolio. Normalize this abnormal return by the standard deviation of the portfolio’s abnormal returns over the estimation period. Michael W. Brandt Methods Lectures: Financial Econometrics
45. 45. Event Studies Bells and whistles Heteroskedasticity Two forms of heteroskedasticity to deal with in event studies 1. Cross-sectional differences in σ i . ¯ • Standardize ARi,τ by σ i before aggregating to AR τ . • Jaffe (1974). 2. Event driven changes in σ i ,t . • Cross-sectional estimate of Var [ARi,τ ] over the event window. • Boehmer et al. (1991). Michael W. Brandt Methods Lectures: Financial Econometrics
46. 46. Event Studies Bells and whistles Regression based approach An event study can alternatively be run as multivariate regression L r1,t+1 = α1 + β1 rm,t+1 + γ1,τ D1,t+1,τ + u1,t+1 τ =0 L r2,t+1 = α2 + β2 rm,t+1 + γ2,τ D2,t+1,τ + u2,t+1 τ =0 ... L rn,t+1 = αn + βn rm,t+1 + γn,τ Dn,t+1,τ + un,t+1 τ =0 where Di,t,τ = 1 if ﬁrm i had an event at date t − τ (= 0 otherwise). In this case, γi,τ takes the place of the abnormal return ARi,τ . Michael W. Brandt Methods Lectures: Financial Econometrics