Empirical Asset Pricing

Transcription

1 Department of Mathematics and Statistics, University of Vaasa, Finland Texas A&M University, May June, 2013 As of May 24, 2013

2 Part III Stata Regression

3 1 Stata regression Regression Factor variables Postestimation: test More about factor variables Robust standard errors Postestimation commands

4 Regression Stata has a very rich family of regression models and estimation methods. Full description is found from the manuals. The basic syntax in all alternatives are the same as in the basic regression: regress depvar [indepvars] [if] [in] [weight] [, options]

5 Regression options Description Model noconstant suppress constant term hascons has user-supplied constant tsscons compute total sum of squares with constant; seldom used SE/Robust vce(vcetype) Reporting level(#) beta eform(string) depname(varname) display_options vcetype may be ols, robust, cluster clustvar, bootstrap, jackknife, hc2, or hc3 set confidence level; default is level(95) report standardized beta coefficients report exponentiated coefficients and label as string substitute dependent variable name; programmer s option control column formats, row spacing, line width, and display of omitted variables and base and empty cells noheader suppress output header notable suppress coefficient table plus make table extendable mse1 force mean squared error to 1 coeflegend display legend instead of statistics

6 Regression Example: IBM FF-factor model (regression.do) IBM stock Fama-French 3-factor model (daily data) reg rete mkt_rf smb hml, cformat(%6.3f) pformat(%5.3f) sformat(%8.2f) Source SS df MS Number of obs = F( 3, 5813) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = rete Coef. Std. Err. t P> t [95% Conf. Interval] mkt_rf smb hml _cons Formating printed output, see help cformat

7 Factor variables gen dow = dow(date) // day-of-week 0 = Sun, 1 = Mon,... i.dow is called a factor variable and defines virtual indicator variable (dummy variable) for each unique value of dow where the smallest forms the reference class. list i.dow in 1/5, clean 1b dow dow dow dow dow b. is the reference class with all values equal to zero. It is dropped from the regression.

8 Factor variables The base value (reference value/class) can be defined by ib. Base operator ib#. ib(##). ib(first). ib(last). ib(freq). ibn. Description use # as base, # = value of variable use the #th ordered value as base use smallest value as base (default) use largest value as base use most frequent value as base no base level The i may be omitted. E.g., you can type ib2.dow or b2.dow. For example, ib(#2). means to use the second value as the base.

9 Factor variables In fact, bn., rather than i., defines virtual indicator value for each class. list bn.dow in 1/5, clean dow dow dow dow dow

10 Factor variables Testing for day of the week effects // estimate ff-model with weekday dummies reg rete mkt_rf smb hml bn.dow, noconstant Source SS df MS Number of obs = F( 8, 5809) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = rete Coef. Std. Err. t P> t [95% Conf. Interval] mkt_rf smb hml dow Monday (dow = 1) is statistically significant (Monday effect?).

11 Postestimation: test Given the Monday effect, we can test whether the rest of the weekday effects are jointly zero. This can be carried out by the postestimation command test test (spec) [(spec)...] [, test options] See help test for details.

12 Postestimation: test Example: Testing for linear hypotheses test 2.dow 3.dow 4.dow 5.dow ( 1) 2.dow = 0 ( 2) 3.dow = 0 ( 3) 4.dow = 0 ( 4) 5.dow = 0 F( 4, 5809) = 1.87 Prob > F = // note, e.g., i2.dow is // the same as 2.dow No empirical evidence of joint difference from zero. Note: Usually 2.dow 3.dow 4.dow 5.dow is equivalent to i( ).dow or i(2/5).dow but with test it does not seems to work.

13 Postestimation: test Equality of Tue through Fri effects test 2.dow = 3.dow = 4.dow = 5.dow // equality of coefficients ( 1) 2.dow - 3.dow = 0 ( 2) 2.dow - 4.dow = 0 ( 3) 2.dow - 5.dow = 0 F( 3, 5809) = 2.49 Prob > F =

14 More about factor variables Factor variable operators Operator Description i. unary operator to specify indicators c. unary operator to treat as continuous # binary operator to indicate interaction ## binary operator to specify full-factorial interactions E.g. i1.dow.##c.smb includes all the terms 1.dow, smb, and 1.dow#c.smb to the model. The c. means continuous variable.

15 More about factor variables Monday effect in risk in factor betas? gen mon = 1.dow // for convenience, define Monday dummy and interaction terms gen mkt_x_mon = mkt_rf * mon gen smb_x_mon = smb * mon gen hml_x_mon = hml*mon reg rete mkt_rf smb hml mon mkt_x_mon smb_x_mon hml_x_mon // regression with // monday effects in coefficient Source SS df MS Number of obs = F( 7, 5809) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = rete Coef. Std. Err. t P> t [95% Conf. Interval] mkt_rf smb hml mon mkt_x_mon smb_x_mon hml_x_mon _cons

16 More about factor variables The enhanced model is of the form r e t = α + β mkt r e mt + s SMB t + h HML t + δ mon t (3) +δ m (r e m mon) t + δ s (HML mon) t + δ h (HML mon) t + u t which we can write r e t = α + δ mon t + (β mkt + δ m mon t )r e mt (4) +(s + δ s mon t ) SMB t +(h + δ h mon t ) HML t + u t. We see that on e.g. on Monday β mkt β mkt + δ m and thus δ m can be interpreted as the shift in market beta on Mondays. The significance of the δ m -estimate above indicates a change in IBM stock s market risk in Mondays (decreases).

17 Robust standard errors In financial data homoscedastisity and independence of regression errors are typically not satisfied. Stata has several options to adapt these problems by using the vce() option (see help vce option). If the errors are not correlated White (1980, Econometrica ) type corrected standard errors are typically utilized. Generally the OLS standard errors depend on the diagonal elements of the matrix X ΩX (5) where Ω is the covariance matrix of the error terms, which is diagonal if the error terms are not correlated.

18 Robust standard errors Under this assumption the diagonal elements of (5) are of the form n ωi 2 x ij x ik (6) i=1 j, k = 1,..., p = number of regressors, where ω 2 i = var[u i ] is the variance of the ith observation. In literature there are four different practices to estimate ω 2 i HC 0 : ûi 2 HC 1 : HC 2 : HC 3 : n n p 1û2 i û 2 i 1 h i û 2 i (1 h i ) 2 Stata : vcr(robust) Stata : vce(hc2) Stata : vce(hc3) (7) h i is the ith diagonal element of X(X X) 1 X.

19 Robust standard errors Generally the different scalings of ûi 2 in (7) result to different bias corrections to estimate the error variances ωi 2. HC 0 performs worst, HC 1 does better, HC 2 does better still, and HC 3 does best of all (usually), Davidson and MacKinnon (1993, p. 554) Estimation and Inference in Econometrics.

20 Robust standard errors OLS standard errors reg rete mkt_rf smb hml mon mkt_x_mon // OLS standard errors Source SS df MS Number of obs = F( 5, 5811) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = rete Coef. Std. Err. t P> t [95% Conf. Interval] mkt_rf smb hml mon mkt_x_mon _cons

21 Robust standard errors Huber-White heteroscedastic consistent standard errors (HC 1 ) // White-Huber heteroscedastic consistent standard errors reg rete mkt_rf smb hml mon mkt_x_mon, vce(robust) Linear regression Number of obs = 5817 F( 5, 5811) = Prob > F = R-squared = Root MSE = Robust rete Coef. Std. Err. t P> t [95% Conf. Interval] mkt_rf smb hml mon mkt_x_mon _cons Notice the difference in the F -statistic and standard errors, in particular that of mkt x mon interaction and its t and p values!

22 Robust standard errors White-MacKinnon correction (HC 2 ) reg rete mkt_rf smb hml mon mkt_x_mon, vce(hc2) Linear regression Number of obs = 5817 F( 5, 5811) = Prob > F = R-squared = Root MSE = Robust HC2 rete Coef. Std. Err. t P> t [95% Conf. Interval] mkt_rf smb hml mon mkt_x_mon _cons It is notable that mkt x mon is at best borderline significant!

23 Robust standard errors White-MacKinnon correction (HC 3 ) Linear regression Number of obs = 5817 F( 5, 5811) = Prob > F = R-squared = Root MSE = Robust HC3 rete Coef. Std. Err. t P> t [95% Conf. Interval] mkt_rf smb hml mon mkt_x_mon _cons

24 Robust standard errors Stata regress has also Jackknife and Bootstrap standard errors and also cluster type robust standard errors. Jackknife reg rete mkt_rf smb hml mon mkt_x_mon, vce(jacknife, nodots) // nodots suppress intermediate results Linear regression Number of obs = 5817 Replications = 5817 F( 5, 5816) = Prob > F = R-squared = Adj R-squared = Root MSE = Jackknife rete Coef. Std. Err. t P> t [95% Conf. Interval] mkt_rf smb hml mon mkt_x_mon _cons

25 Robust standard errors Bootstrap reg rete mkt_rf smb hml mon mkt_x_mon, vce(boot, nodots) // nodots suppress intermediate results Linear regression Number of obs = 5817 Replications = 50 Wald chi2(5) = Prob > chi2 = R-squared = Adj R-squared = Root MSE = Observed Bootstrap Normal-based rete Coef. Std. Err. z P> z [95% Conf. Interval] mkt_rf smb hml mon mkt_x_mon _cons

26 Robust standard errors Finally Stata allows also to robustify standard errors with respect to correlation within clusters (autocorrelation corrections can be dealt with by time series regressions). Using dow as a cluster variable gives: reg rete mkt_rf smb hml mon mkt_x_mon, vce(cluster dow) Linear regression Number of obs = 5817 F( 3, 4) =. Prob > F =. R-squared = Root MSE = (Std. Err. adjusted for 5 clusters in dow) Robust rete Coef. Std. Err. t P> t [95% Conf. Interval] mkt_rf smb hml mon mkt_x_mon _cons Estimation seems to have some problems because F -statistic does not show up.

27 Robust standard errors The overall conclusion from the above estimation results is that while the Monday effect on IBM s returns seems evident, the Monday effect on IBM s market beta may eventually not exist. Using dow as a cluster variable to capture the potential similarity of regression residuals on particular weekdays (intraclass correlation among weekdays, e.g. Mondays) does not seem to work. In this case a better alternative is to use time series tools of which we will return later. Before that we look at some post estimation commands.

28 Postestimation commands Postestimation commands provide tools for model checking, testing, etc. purposes. Earlier we have already used test for testing linear hypotheses. Some regress postestimation commands The following postestimation commands are of special interest after regress: ================================================================================ Command Description -- dfbeta DFBETA influence statistics estat hettest tests for heteroskedasticity estat imtest information matrix test estat ovtest Ramsey regression specification-error test for omitted variables estat szroeter Szroeters rank test for heteroskedasticity estat vif variance inflation factors for the independent variables acprplot augmented component-plus-residual plot avplot added-variable plot avplots all added-variables plots in one image cprplot component-plus-residual plot lvr2plot leverage-versus-squared-residual plot rvfplot residual-versus-fitted plot rvpplot residual-versus-predictor plot ===============================================================================

29 Postestimation commands Some more More useful postestimation commands: ========================================================================== Command Description contrast contrasts and ANOVA-style joint tests of estimates estat AIC, BIC, VCE, and estimation sample summary estat(svy) postestimation statistics for survey data estimates cataloging estimation results hausman Hausmans specification test lincom point estimates, standard errors, testing, and inference for linear combinations of coefficients linktest link test for model specification lrtest1 likelihood-ratio test margins marginal means, predictive margins, marginal effects, and average marginal effects marginsplot graph the results from margins (profile plots, interaction plots, etc.) nlcom point estimates, standard errors, testing, and inference for nonlinear combinations of coefficients predict predictions, residuals, influence statistics, and other diagnostic measures predictnl point estimates, standard errors, testing, and inference for generalized predictions pwcompare pairwise comparisons of estimates suest seemingly unrelated estimation test Wald tests of simple and composite linear hypotheses testnl Wald tests of nonlinear hypotheses ============================================================================= For more information, see help (e.g. help estat).

30 Postestimation commands Postestimation commands apply to the last estimated model. We estimate the previous model wit HC 3 corrected standard errors. Examples predict res, residuals // generates residuals into res predict yhat, xb // generates fitted values into yhat avplots, msymbol(point) // added variable plots avplots, called added variable plots or partial-regression leverage plots, create two-dimensional scatter plots of projections of the multidimensional data. These can be used to identify visually outliers in the data.

31 Postestimation commands e( rete X ) e( mkt_rf X ) coef = , se = , t = e( rete X ) e( smb X ) coef = , se = , t = 6.84 e( rete X ) e( hml X ) coef = , se = , t = e( rete X ) e( mon X ) coef = , se = , t = 3.71 e( rete X ) e( mkt_x_mon X ) coef = , se = , t = 2.63