Empirical Asset Pricing - Uwasalipas.uwasa.fi/~sjp/Teaching/eaptx/lectures/p3.pdf · Empirical...

Empirical Asset Pricing

Seppo Pynnonen

Department of Mathematics and Statistics, University of Vaasa, Finland

Texas A&M University, May – June, 2013

As of May 24, 2013Seppo Pynnonen Empirical Asset Pricing

Stata regression

Part III

Stata Regression

Seppo Pynnonen Empirical Asset Pricing

Stata regression

1 Stata regression

Regression

Factor variables

Postestimation: test

More about factor variables

Robust standard errors

Postestimation commands


Stata regression

Regression

Stata has a very rich family of regression models and estimationmethods.

Full description is found from the manuals.

The basic syntax in all alternatives are the same as in the basicregression:

regress depvar [indepvars] [if] [in] [weight] [, options]


Stata regression

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) vcetype may be ols, robust, cluster clustvar, bootstrap,

jackknife, hc2, or hc3

Reporting

level(#) set confidence level; default is level(95)

beta report standardized beta coefficients

eform(string) report exponentiated coefficients and label as string

depname(varname) substitute dependent variable name; programmer’s option

display_options 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


Stata regression

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 = 5817

-------------+------------------------------ F( 3, 5813) = 1065.30

Model | 6941.76055 3 2313.92018 Prob > F = 0.0000

Residual | 12626.3454 5813 2.17208763 R-squared = 0.3547

-------------+------------------------------ Adj R-squared = 0.3544

Total | 19568.106 5816 3.36452991 Root MSE = 1.4738

------------------------------------------------------------------------------

rete | Coef. Std. Err. t P>|t| [95% Conf. Interval]

-------------+----------------------------------------------------------------

mkt_rf | 0.866 0.017 51.20 0.000 0.833 0.899

smb | -0.242 0.034 -7.12 0.000 -0.309 -0.176

hml | -0.446 0.033 -13.38 0.000 -0.511 -0.381

_cons | 0.030 0.019 1.57 0.117 -0.008 0.068

------------------------------------------------------------------------------

Formating printed output, see help cformat


Stata regression

Factor variables

gen dow = dow(date) // day-of-week 0 = Sun, 1 = Mon, ...

i.dow is called a factor variable and defines virtual indicatorvariable (dummy variable) for each unique value of dow where thesmallest forms the reference class.

list i.dow in 1/5, clean

1b. 2. 3. 4. 5.

dow dow dow dow dow

1. 0 1 0 0 0

2. 0 0 1 0 0

3. 0 0 0 1 0

4. 0 0 0 0 1

5. 0 0 0 0 0

1b. is the reference class with all values equal to zero. It isdropped from the regression.


Stata regression

Factor variables

The base value (reference value/class) can be defined by ib.

Base operator Descriptionib#. use # as base, # = value of variableib(##). use the #th ordered value as baseib(first). use smallest value as base (default)ib(last). use largest value as baseib(freq). use most frequent value as baseibn. 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.


Stata regression

Factor variables

In fact, bn., rather than i., defines virtual indicator value for eachclass.

list bn.dow in 1/5, clean

1. 2. 3. 4. 5.

dow dow dow dow dow

1. 0 1 0 0 0

2. 0 0 1 0 0

3. 0 0 0 1 0

4. 0 0 0 0 1

5. 1 0 0 0 0


Stata regression

Factor variables

Testing for day of the week effects// estimate ff-model with weekday dummies

reg rete mkt_rf smb hml bn.dow, noconstant


-------------+------------------------------ F( 8, 5809) = 404.01

Model | 7000.01619 8 875.002024 Prob > F = 0.0000


-------------+------------------------------ Adj R-squared = 0.3566

Total | 19580.9687 5817 3.36616274 Root MSE = 1.4717

------------------------------------------------------------------------------


-------------+----------------------------------------------------------------

mkt_rf | 0.866 0.017 51.29 0.000 0.833 0.899

smb | -0.230 0.034 -6.75 0.000 -0.297 -0.163

hml | -0.447 0.033 -13.41 0.000 -0.512 -0.381

|

dow |

1 | 0.178 0.044 4.00 0.000 0.091 0.265

2 | 0.088 0.043 2.07 0.039 0.005 0.172

3 | -0.067 0.043 -1.57 0.116 -0.151 0.017

4 | -0.001 0.043 -0.02 0.984 -0.085 0.083

5 | -0.037 0.043 -0.87 0.386 -0.122 0.047

------------------------------------------------------------------------------

Monday (dow = 1) is statistically significant (Monday effect?).


Stata regression


Given the Monday effect, we can test whether the rest of theweekday effects are jointly zero.

This can be carried out by the postestimation command test

test (spec) [(spec) . . .] [, test options]

See help test for details.


Stata regression


Example: Testing for linear hypotheses

test 2.dow 3.dow 4.dow 5.dow // note, e.g., i2.dow is

// the same as 2.dow

( 1) 2.dow = 0

( 2) 3.dow = 0

( 3) 4.dow = 0

( 4) 5.dow = 0

F( 4, 5809) = 1.87

Prob > F = 0.1122

No empirical evidence of joint difference from zero.

Note: Usually 2.dow 3.dow 4.dow 5.dow is equivalent to i(2 3

4 5).dow or i(2/5).dow but with test it does not seems towork.


Stata regression


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 = 0.0588


Stata regression


Factor variable operators

Operator Description

i. unary operator to specify indicatorsc. 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, and1.dow#c.smb to the model.

The c. means continuous variable.


Stata regression


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


-------------+------------------------------ F( 7, 5809) = 460.98

Model | 6988.12246 7 998.303209 Prob > F = 0.0000


-------------+------------------------------ Adj R-squared = 0.3563

Total | 19568.106 5816 3.36452991 Root MSE = 1.4716

------------------------------------------------------------------------------


-------------+----------------------------------------------------------------

mkt_rf | 0.890 0.019 46.07 0.000 0.852 0.928

smb | -0.217 0.038 -5.68 0.000 -0.291 -0.142

hml | -0.445 0.037 -11.88 0.000 -0.519 -0.372

mon | 0.178 0.050 3.58 0.000 0.081 0.276

mkt_x_mon | -0.105 0.040 -2.64 0.008 -0.184 -0.027

smb_x_mon | -0.079 0.085 -0.93 0.354 -0.245 0.088

hml_x_mon | 0.013 0.082 0.16 0.876 -0.149 0.175

_cons | -0.005 0.021 -0.25 0.806 -0.047 0.037

------------------------------------------------------------------------------


Stata regression


The enhanced model is of the form

r et = α + βmktremt + s SMBt + h HMLt + δ mont (3)

+δm(r em × mon)t + δs(HML× mon)t + δh(HML× mon)t + ut

which we can write

r et = α + δ mont + (βmkt + δmmont)remt (4)

+(s + δsmont) SMBt

+(h + δhmont) HMLt + ut .

We see that on e.g. on Monday βmkt → βmkt + δm and thus δm canbe interpreted as the shift in market beta on Mondays.

The significance of the δm-estimate above indicates a change inIBM stock’s market risk in Mondays (decreases).


Stata regression


In financial data homoscedastisity and independence of regressionerrors are typically not satisfied.

Stata has several options to adapt these problems by using thevce() option (see help vce option).

If the errors are not correlated White (1980, Econometrica817–838) type corrected standard errors are typically utilized.

Generally the OLS standard errors depend on the diagonalelements of the matrix

X′ΩX (5)

where Ω is the covariance matrix of the error terms, which isdiagonal if the error terms are not correlated.


Stata regression


Under this assumption the diagonal elements of (5) are of the form

n∑i=1

ω2i xijxik (6)

j , k = 1, . . . , p = number of regressors, where ω2i = var[ui ] is the

variance of the ith observation.

In literature there are four different practices to estimate ω2i

HC0 : u2iHC1 : n

n−p−1 u2i Stata : vcr(robust)

HC2 :u2i

1−hi Stata : vce(hc2)

HC3 :u2i

(1−hi )2Stata : vce(hc3)

(7)

hi is the ith diagonal element of X(X′X)−1X′.


Stata regression


Generally the different scalings of u2i in (7) result to different biascorrections to estimate the error variances ω2

i .

HC0 performs worst, HC1 does better, HC2 does better still, and HC3does best of all (usually), Davidson and MacKinnon (1993, p. 554)Estimation and Inference in Econometrics.


Stata regression


OLS standard errorsreg rete mkt_rf smb hml mon mkt_x_mon // OLS standard errors


-------------+------------------------------ F( 5, 5811) = 645.29

Model | 6986.00399 5 1397.2008 Prob > F = 0.0000


-------------+------------------------------ Adj R-squared = 0.3565

Total | 19568.106 5816 3.36452991 Root MSE = 1.4715

------------------------------------------------------------------------------


-------------+----------------------------------------------------------------

mkt_rf | 0.891 0.019 46.29 0.000 0.853 0.928

smb | -0.233 0.034 -6.84 0.000 -0.300 -0.166

hml | -0.441 0.033 -13.23 0.000 -0.506 -0.376

mon | 0.184 0.049 3.71 0.000 0.087 0.281

mkt_x_mon | -0.104 0.040 -2.63 0.009 -0.182 -0.027

_cons | -0.005 0.021 -0.23 0.821 -0.047 0.037

------------------------------------------------------------------------------


Stata regression


Huber-White heteroscedastic consistent standard errors (HC1)

// 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) = 412.25

Prob > F = 0.0000

R-squared = 0.3570

Root MSE = 1.4715

------------------------------------------------------------------------------

| Robust


-------------+----------------------------------------------------------------

mkt_rf | 0.891 0.022 40.94 0.000 0.848 0.933

smb | -0.233 0.039 -6.05 0.000 -0.309 -0.158

hml | -0.441 0.043 -10.28 0.000 -0.525 -0.357

mon | 0.184 0.047 3.90 0.000 0.091 0.276

mkt_x_mon | -0.104 0.052 -1.99 0.047 -0.207 -0.001

_cons | -0.005 0.022 -0.22 0.823 -0.048 0.038

------------------------------------------------------------------------------

Notice the difference in the F -statistic and standard errors, inparticular that of mkt x mon interaction and its t and p values!


Stata regression


White-MacKinnon correction (HC2)

reg rete mkt_rf smb hml mon mkt_x_mon, vce(hc2)


F( 5, 5811) = 408.31

Prob > F = 0.0000

R-squared = 0.3570

Root MSE = 1.4715

------------------------------------------------------------------------------

| Robust HC2


-------------+----------------------------------------------------------------

mkt_rf | 0.891 0.022 40.87 0.000 0.848 0.933

smb | -0.233 0.039 -6.04 0.000 -0.309 -0.157

hml | -0.441 0.043 -10.25 0.000 -0.525 -0.357

mon | 0.184 0.047 3.89 0.000 0.091 0.276

mkt_x_mon | -0.104 0.053 -1.95 0.051 -0.209 0.001

_cons | -0.005 0.022 -0.22 0.823 -0.048 0.038

------------------------------------------------------------------------------

It is notable that mkt x mon is at best borderline significant!


Stata regression


White-MacKinnon correction (HC3)


F( 5, 5811) = 404.03

Prob > F = 0.0000

R-squared = 0.3570

Root MSE = 1.4715

------------------------------------------------------------------------------

| Robust HC3


-------------+----------------------------------------------------------------

mkt_rf | 0.891 0.022 40.78 0.000 0.848 0.933

smb | -0.233 0.039 -6.01 0.000 -0.309 -0.157

hml | -0.441 0.043 -10.22 0.000 -0.526 -0.356

mon | 0.184 0.047 3.89 0.000 0.091 0.276

mkt_x_mon | -0.104 0.055 -1.91 0.056 -0.211 0.003

_cons | -0.005 0.022 -0.22 0.823 -0.048 0.038

------------------------------------------------------------------------------


Stata regression


Stata regress has also Jackknife and Bootstrap standard errorsand also cluster type robust standard errors.

Jackknifereg rete mkt_rf smb hml mon mkt_x_mon, vce(jacknife, nodots) // nodots suppress intermediate results


Replications = 5817

F( 5, 5816) = 404.10

Prob > F = 0.0000

R-squared = 0.3570

Adj R-squared = 0.3565

Root MSE = 1.4715

------------------------------------------------------------------------------

| Jackknife


-------------+----------------------------------------------------------------

mkt_rf | 0.891 0.022 40.78 0.000 0.848 0.933

smb | -0.233 0.039 -6.02 0.000 -0.309 -0.157

hml | -0.441 0.043 -10.22 0.000 -0.526 -0.356

mon | 0.184 0.047 3.89 0.000 0.091 0.276

mkt_x_mon | -0.104 0.055 -1.91 0.056 -0.211 0.003

_cons | -0.005 0.022 -0.22 0.823 -0.048 0.038

------------------------------------------------------------------------------


Stata regression


Bootstrapreg rete mkt_rf smb hml mon mkt_x_mon, vce(boot, nodots) // nodots suppress intermediate results


Replications = 50

Wald chi2(5) = 3190.50

Prob > chi2 = 0.0000

R-squared = 0.3570

Adj R-squared = 0.3565

Root MSE = 1.4715

------------------------------------------------------------------------------

| Observed Bootstrap Normal-based

rete | Coef. Std. Err. z P>|z| [95% Conf. Interval]

-------------+----------------------------------------------------------------

mkt_rf | 0.891 0.018 50.55 0.000 0.856 0.925

smb | -0.233 0.042 -5.59 0.000 -0.315 -0.151

hml | -0.441 0.047 -9.44 0.000 -0.533 -0.349

mon | 0.184 0.044 4.18 0.000 0.098 0.270

mkt_x_mon | -0.104 0.056 -1.87 0.061 -0.213 0.005

_cons | -0.005 0.022 -0.22 0.823 -0.047 0.038

------------------------------------------------------------------------------


Stata regression


Finally Stata allows also to robustify standard errors with respectto correlation within clusters (autocorrelation corrections can bedealt 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)


F( 3, 4) = .

Prob > F = .

R-squared = 0.3570

Root MSE = 1.4715

(Std. Err. adjusted for 5 clusters in dow)

------------------------------------------------------------------------------

| Robust


-------------+----------------------------------------------------------------

mkt_rf | 0.891 0.015 57.75 0.000 0.848 0.933

smb | -0.233 0.049 -4.78 0.009 -0.368 -0.098

hml | -0.441 0.010 -46.35 0.000 -0.467 -0.415

mon | 0.184 0.038 4.84 0.008 0.078 0.289

mkt_x_mon | -0.104 0.015 -7.15 0.002 -0.145 -0.064

_cons | -0.005 0.034 -0.14 0.894 -0.100 0.090

------------------------------------------------------------------------------

Estimation seems to have some problems because F -statistic does not

show up.Seppo Pynnonen Empirical Asset Pricing

Stata regression


The overall conclusion from the above estimation results is thatwhile the Monday effect on IBM’s returns seems evident, theMonday effect on IBM’s market beta may eventually not exist.

Using dow as a cluster variable to capture the potential similarityof regression residuals on particular weekdays (intraclass correlationamong weekdays, e.g. Mondays) does not seem to work.

In this case a better alternative is to use time series tools of whichwe will return later. Before that we look at some post estimationcommands.


Stata regression


Postestimation commands provide tools for model checking,testing, etc. purposes. Earlier we have already used test fortesting linear hypotheses.

Some regress postestimation commandsThe 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

===============================================================================


Stata regression


Some moreMore 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).


Stata regression


Postestimation commands apply to the last estimated model.

We estimate the previous model wit HC3 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 leverageplots”, create two-dimensional scatter plots of projections of themultidimensional data. These can be used to identify visuallyoutliers in the data.


Stata regression


−20

−10

010

e(

rete

| X

)

−10 −5 0 5 10e( mkt_rf | X )

coef = .89060303, se = .01923992, t = 46.29

−15

−10

−5

05

10

e(

rete

| X

)

−4 −2 0 2 4e( smb | X )

coef = −.23311098, se = .03406098, t = −6.84

−15

−10

−5

05

10

e(

rete

| X

)

−4 −2 0 2 4e( hml | X )

coef = −.44099632, se = .03333577, t = −13.23

−15

−10

−5

05

10

e(

rete

| X

)

−.5 0 .5 1e( mon | X )

coef = .1836463, se = .04945237, t = 3.71

−15

−10

−5

05

10

e(

rete

| X

)

−10 −5 0 5 10e( mkt_x_mon | X )

coef = −.10413018, se = .03956648, t = −2.63


Empirical Asset Pricing - Uwasalipas.uwasa.fi/~sjp/Teaching/eaptx/lectures/p3.pdf · Empirical...

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