NBER WORKING PAPER SERIES

DIVIDEND YIELDS ANT) STOCK RETURNS:A TEST FOR TAX EFFECTS

Patrick J. Hess

Working Paper No. 619

NATIONAL BUREAU OF ECONOMIC RESEARCH1050 Massachusetts Avenue

Cambridge MA 02138

March 1981

This paper is based upon °w Ph.D. dissertation at the University ofChicago. I am thankful to the members of rr committee; G.Constantinides, E. Fama, J. Ingersoll, A. Madensky, U. Miller(chairman), and U. Scholes. I owe a particularly important debt •toMerton Miller for his help and encouragement. Finally, I wouldlike to thank the participants n the Finance workshop at OhioState University and the University of Chicago. Special thanks aredue to Steve Buser, Roger Gordon, Gallon Hite and P. fn Kim. Theresearch reported here is part of the NBERtS research program inTaxation. Any opinions expressed are those of the author and notthose of the National Bureau of Economic Research,

NBER Working Paper #649March 1981

Dividend Yields and Stock Returns:A Test For Tax Effects

ABSTRACT

This paper examines the empirical relation between stock returns and

dividend yields. Several equilibrium pricing models incorporating differ

ential taxation of dividends and capital gains are nested as systems of

time series regressions. Estimates of these models and tests of parameter

restrictions implied by the models are conducted within the context of

Zellner's seemingly unrelated regression. It is concluded that the data

fail to support these models as well as the hypothesis that dividends are

neutral. The inability to distinguish between these competing hypotheses

suggests the need for further research before definitive conclusions are

reached regarding the tax impacts of dividends.

Professor Patrick HessDepartment of EconomicsThe Ohio State University1775 S. College RoadColumbus, Ohio 43210

(614) 422—0280

DIVIDEND YIELDS AND STOCK RETURNS:

A TEST FOR TAX EFFECTS

By

Patrick J. Hess

Introduction

Th ffect of dividends on stock returns is a major issue in finance, with a

rich historical bacr'und. It has been argued (see Graham and Dodd [19511)

that low yield stocks require iigher returns because investors value dividends

more than retained earnings. Miller and Modigliani [19611 demonstrated that in

a world of no transaction costs, equal tax treatment of dividends and capital

gains, and investor rationality, the dividend policies of firms have no impact on

the welfare of security holders (given investment policies). Miller and Modigliani

pointed out, however, that the heavier taxation of dividends than capital gains

might lead to higher before-tax returns on shares with high dividend yields. But

they wained that such a relation would be weakened and possibly completely

offset by clientele effects. One important class of investors, pension funds, pay

no taxes and, therefore, have no reason to require higher returns on dividend

paying stocks. Other investors, notably corporations and casualty insurance

companies, face lower statutory tax rates on dividends than capital gains. On

the othe1 hand, some investors consuming wealth may find the tax penalties of

dividend paying stocks offset by lower transaction costs if part of their returns

are received as dividends. Recently, even the proposition that dividends are

taxed at higher rates than capital gains has been challenged. Miller and Scholes

[1978] have pointed to features of the tax code and currently available financial

instruments that investors could use to blunt any tax disadvantages of dividends.

Attempts to establish which of these hypothesized relations between

dividends 'and stock returns holds empirically have not resulted in a consensus.

—2--

Early tests of the general effects of dividends, such as Friend and Puckett

[1964], relied on cross sectional regressions of common stock prices on

dividends per share, and various alternative measures of earnings or retained

earnings per share. Friend and Puekett found little evidence of yield effects, but

their tests are subject to serious limitations. Black and Scholes (19741 also

tested for the general impact of dividends on common stock returns, using time

series methods that avoided many of the estimation difficulties of the earlier

cross sectional regressions of stock prices on dividends and retained earnings.

On the basis of their test, Black and Scholes [1974] found themselves unable to

reject the hypothesis that dividend yields have no significant effect on stock

returns.

Several authors have tested for the tax effects of dividends by comparing

returns on eum and ex days. Elton and Gruber [1970], in a well known study,

claim documentation of both a tax effect and a dependency between tax rates of

marginal investors and dividend yields (a clientele effect reflected in asset

prices). With a longer sample period and more appropriate statistical procedure,

Black and Scholes [1973] are unable to confirm either of these conclusions.

However, Black and Scholes do find evidence of unusual behavior in security

returns for several days surrounding ex dividend days.

Recent work, notably by Litzenberger and Ramaswamy [1979] and Blume

[19781, has sought to improve the efficiency of the Black—Scholes [1974] tests.

Both papers claims to have found evidence supporting the importance of

dividends. Litzenberger and Ramaswamy, at least, argue that the importance of

dividends is best explained by differential taxation of dividends.

In this paper I present a different estimation method that avoids some of

the shortcomings of the Litzenberger-Ramaswamy and Blume tests. I conclude

:ta fail to spor. lia itzenhe er.kTvJam; eoniec1: aential tha. . b attributed to he cz axaticn of ;ve: th

s. In fae. 'e eoe: seem to n a a seel seasc tne

a: a ae sap, any ra a:pte economic r] 51, .e S a a ods as:;

parJ a: a. ,r pas tictlar no cffe ts of divide ass act :form aeross a

:3curit1es',.

c vc a; of telated Works

This esew concentrats ca se papers of l3iuek a:•J ohc>ies 9?4j

iibenbers:er and tamaswamy [ ai: 0 These spars

represeat: ase state of the art viith respect Ic. iag rim hypothesis of a mrest,

AU are eassistent witi' modern theories of seearit;j a. im :d use estimation

tee 1iqu;a that avoid loony of the problems associated ath earlier crass-

sectional sudies

Black arid Scholes [1974]

Black and Scholes [1974] compare two motels of the relation between

civiciends yields and expected returns:

+ .[E(R) - + (o )/o (1)

E(R) = + .[E(R) (2)

where E(R;) is the expected return of security i, is the covariance of security

is return with the return of the market portfolio divided by the variance of the

return of the market portfolio, is the dividend yield of security i, mthe

dividend ieId of the rnarke portfolio, E(Rm) is the expected return of the

market portfolio, and y,y are parameters of the pricing equation.

—4-.

Black and Scholes estimate 1' as the average return of the minimum

variance portfolio satisfying the restrictions,

N N N= = 0, = 3)

1 i = 11

1 = 1

where N is the number of assets and w is the proportion of asset i in the

portfolio. To reduce the impact of errors in variables, Black and Scholes solve

(4) with twentyfive portfolios constructed from all New York Exchange stocks

that were listed for at least five years. These portfolios are constructed to

maintain dispersion in both beta and dividend yield. The ij elements of the" "2

covariance matrix oX portfolio returns ts approximated as a for i j and

fQ if i j where is the estimated betas of portfolio i, a is the

estimated variance of the equally weighted index of all New York Stock

Exchange common stocks and a is the estimated residual variance of portfolio i

in the regression of portfolio i's return on the returns of the equally weighted

index.

Black and Scholes estimate -y1 as the average monthly return of the

portfolio. For their overall sample period, January, 1939 to December, 1966, the

average return is .09 percent. During three ten year subperiods, the1

estimates range from .02 percent to .16 percent. None of the estimates are

statistically significant.

Litzenberger and Ramaswamy [19791

Litzenberger and Ramaswamy begin by deriving an after—tax capital asset

pricing model of the form,

E(R) - rf = a + b . + c(d1 rf) (4)

where rf is the risk-free rate of interest, d is the dividend yield of security i,

ana a, L : are parameters of the pricing &mation. iitzebe.rer arc Rameimy 'ume that ie tal gains tax s zero od dit ends are xe

'inar neome. Thc r&z'arnter eqmt ior

oadow pce et1eetng any icr;ri addi ionak do1ic of divionds. ;:

i) C WiiI OOV tiX ..;neters &i their dI Im'e va)ues.

- —

b E(R) C(

where E) is the expectea return on a zero beta portfolio with a divderiJ

yield CO&Oi to the risk—free rate of interest E(R) is the expected return of the

market ortfoio, and drn is the dividend yield of the market portfolio. The

model (5) is a more general form of Brennans [1970] after tax pricing equation,

E(R.)-rf = b . + i(d1 - rf) (5)

where

b' = E(R) rf- r(dm - rf)

and -r represents a tax differential between dividends and capital gains. While

(4) arid (5 differ from the Black and Scholes model, a portfolio satisfying the

constraints of Black and Scholes would have an expected return equal toCdff1 or

Tdm

—6—

In estimating equation (4), Litzenberger and Ramaswamy consider thestochastic version of their model,

Rt = E(Rt) +

it - rf = a + b it + c(dt - rft) + U. (6)

for securities 1 1,2 ...N and periods t 1, 2 ...T.4 If all thets, dt's, and ri'sin (6) could be estimated without error, the model could be estimated by pooling

cross sectional regressions over time. Litzenberger and Ramaswamy use three

different, although closely related techniques to estimate (6).

The first technique is a regression of individual securities' risk premiums

(returns less than one month T-Bill rate) on a constant, the securities' betas

estimates in the previous sixty months, and a measure of anticipated dividend

yield of the securities for the month less the one month T-Bill rate. Thesecurities' betas are estimated with the followingregression,

R. - r = a. + . (R - r ) + e. (7)ii rt it iT mi ft itfor securities i = 1,2 ...N and T = t—60 ... t-l, where R is the return in month rmT

of a value weighted index of all New York Stock Exchange common stocks. The

measure of anticipated dividends for any month equals the taxable dividends paidin the month if the dividends were announced prior to the beginning of themonth. If the taxable dividends were unannounced and nonrecurring, anticipated

dividends were assigned a value of zero and if dividends were unannounced but

recurring, anticipated dividends were set equal to the last recurring payment.5

The divnd yield is &nticipaied dividends divided by the c'osing pris of

;:evious

T1uecond technique usos the same regression bu with de ins enm,

instant :Limated hstas arM anticipated div s eks standordsd by ihn

.nnte.i sta;rard errots of the securities beta smatas Lit nba 'or anH

amasw. ny claim that this estimator is ga tiflied least squares if thn

oisturbannos in (? uncorrelated across seci :ios, osJ, indeed if alivn dbles snac obs€n'vable ha cmi vould be true.

The third estimation techni uc seo b Litzenherg€i nad Ramaswamy is

motivated by the errors in variables probien siooiaed with individual secur—

ities beta estimates. The estimation technique icndes an explicit adjustment

foi errors in the beta estimates. Lit.senberger and Ramasamv argue that their

method dealing with errors in cariables is superior to the portfolio rouping

procedure of Black—Scholes because it does not destroy cross sectional dis

persion. Moreover, the authors claim that the estimator is consistent and is a

maximum likelihood estimator if the joint distribution of security returns is

orrnal. These are remarkable claims since the errors in variables problem has

severely limited the ability to rnai<e reliable inferences in financial economics as

weil as onomics in generaL Litzenberger and Ramaswamys adjusted estimator

is analyied in Appendix A and it is shown there that this estimator is neither

consistent nor is it maximum likelihood.

Litzenberger and Ramaswamy derive estimates from each of these tech-

niques b averaging estimates over time. The standard errors of the estimates

itre estimated from the time series of monthly estimates. The resulting

estimates ot c are all quite close. turing the sample period January, 1936 to

Deeember 1977, they are .227, .234 and .236 for the OLS GLS, and adjusted

—8--

estimator. The t-values of these estimates relative to zero are 6.33, 8.24, arid

8.62 for the OLS, GLS, and adjusted estimator.6 The adjusted estimator is

computed for six subperiods. Litzenberger and Ramaswamy note that during the

overall sample period of Black and Scholes, January, 1936 to December, 1966,

the dividend yield of the market was .048 per year. Interpreting the Black—

Scholes as equivalent to cdm they infer an estimate of c equal to .225 from

the Black and Scholes estimate of y1 (.0009 x 12/.048). On the basis of this

calculation, they claim to have documented roughly the same effect as Black and

Scholes, but in a more precise way.7

B1urie [1978]

Blume [1978] proposes the following model,

at + b 8it + Ctd1t + (8)

for i = 1, 2 ... N and t = 1, 2 ...T. The coefficients of (8) are each assumed to be

normally, independent1y and identically distributed with means a, b, and c and

variances a , a , anda , respectively. Equation (8) closely resembles the

stochastic version of Litzenberger and Ramaswamy's model, equation (6). The

major differences are supression of the risk-free rate and the interpretation of

random coefficients period by period.

Blume estimates (8) by regressing quarterly returns of twenty-five port-

folios on a constant, the estimated portfolio betas, and a measure of the

portfolios' quarterly dividends yield. Portfolios are used to reduce the errors in

variables associated with estimated betas. Like Black and Scholes, Blume

attempts to maintain dispersion in both the portfolio betas and dividend yields.

The portfolio betas are estimated with sixty months of data prior to the

quarterly estimates. Although monthly data are available, Blume uses quarterly

'eturns o avoid smudging of tax effects across e and noc months. Blumso us tithe seres o quarteri esturaes b ca ain2 'erge tmi

.d the .andard error oi fle ver?.:e timate,.

•e estimates . mode far the fleriod J Lece.and for ?our non—overipping ten iear ubpeoi. The 'eiee

for tne overall Deriod is positwe cud siaifca . : .significance level. two remaining ubperiod he -itira ae s.b'e t.eignieant at the i5 ic ium ras o: teip": ee .s

being cansistent with differential videnos relative uo eapi i gains,aitnougik this is presumably the interore iaa f 2;e urer .nd Rarnaswamywould draw from a positive instead, ie argues dt rontant across

securities and. is positively related to dividend iieid so tk t a timu af

impiy proxying for the variable intercept. Blurne attemots to westg:t thalternative but the tests are not well motivated anc the connection betwren dn'

tests and the conjectured relationship is unclear.

Alternative Tests of the Tax

Effects of Dividends

In this section 1 will argue that the authors previously discussed have

overlooked important testable properties of models which assume differential

taxation of dividends.

Litzenbe'er and Ramaswamy

Recall that Litzenberger and Ramaswamy's model takes the form,

E(R.t) rf a + b + c(d1t - rf)

where r+ is the riskiess rate of interest in period t and d1t is the dividend yield

—10—

of security i in period t. The stochastic version of their model is,

R =E(Rt) + u

Rt - rf a + b + c(d. - rft) + (7)

The authors assume that the parameters a, b, and c are constant and they use

five years of data to estimate betas, Implicitly, then, they assume that betas

are constant for five year periods. Combining the assumptions of constant

parameters and betas with the stochastic version of their model results in the

equations,

R -r =1 +yd +yr +u (9)it ft oi lit 2ft it

- = a + b01 1

= C

where (9) holds for all securities and periods consistent with the assumption of

constant betas. The intercept of (9) is free across securities, but constant over

sixty month periods for each security.

The model (9) makes very precise predictions about securities' returns.

First, the model predicts that securities' risk premiums are constant over time,

except for changes in the riskiess rate and the securities' dividend yields. If the

dividend yield of a security, less the riskiess rate, is constant over time, the risk

emium of the secur!ty is also constant. Second, canges in the riskjess rathnor divi :' 'ici yields of securities oesuit in the came er i ng in riv-

c;oiurim. for all securtes. These very specifh mnh atoos my oc inched ertu ;. SUOh a tESt :!as::.i:e turther dvarto C AO cirho act matic

C I ccoiuc:es' betas because the beta terms are sccuuk;ed in i!mI lot

imnatico fj models uie tax effects of clividencic as svsi r Iir; 5551€!

regressions with re cted parameters. The system includes joint restrictiontciat Y souai zero for a securities and the y. sad me equ& ocross tilisecurities,

As noted, the modal (9) does not include tie ccnm on the market portfolio,

but can reaaily be expanded to include it. The detAfl ticos of a, ci, and c are,

a =E(R2)

- rfb =

E(R)-

E(R*) -C(dt -

where E(.RZ*) is the expected return of a zero beta portfolio with a dividend

yield equal to the riskless rate of interest, and dmt is the dividend yield of the

market portfolio in period t. Incorporating these terms into equation (4) and

simplifying,

E(R.t) - rf = E(R) - + [E(R) +E(R*) -

C(dt - rf)] +c(d1t - rf)

= E(R*)(1 -a1) + a E(R) - • C(d - rf) +

-

—12—

If,

Rit = E(R) + uRt E(Rt) + Ut

The stochastic version of the model is,

E(R7)(1- ÷ - - rf) +

c(d1t- rft) + it (10)

it—

Equation (10) holds across securities and over time periods consistent with

constant betas. Like (9), equation (10) involves restrictions on parameters for

each security, as well as restrictions on parameters across securities. The

restrictions implied by (10) are more difficult to incorporate than the restric-

tions of (9), because equation (10) includes restrictions on products of para-

meters. More important, estimation of (10) requires identification of the market

portfolio and this imposes an untestable restriction on the data. Nor can much

comfort be taken on the latter count from evidence that suggest particular

implications of asset pricing seem to be insensitive to alternative specifications

of the market portfolio.9 The concern of this study is with a particular problem

and generalization of results obtained with other problems is inappropriate.

Since determining the effects of alternative market portfolios is beyond the

scope of this paper, equation (10) is used only as a check on the sensitivity of

some of our tests.

13—

baennan

Ste. nan's model cinsely resemnws itze.oerr end ii;:: a t.sncm

model is

- r1 = b . + i(d1 r) (53

Using the same ntion, the stochastic verscn of Brenna aodei ;ay Ut

expressed as,

bt + T¼dr •f(7a)

+ uit

it is clear from (6) that if b', 'r , and beta can be taken as constant icx sixt

months, tim stochastic version. of Brennan's modec may be expressed as,

- = ÷ + 2 + (Ye)

= b'

= - T

The testable implications of this model are identical to the testable implications

of (9). The differences involve only the value of the intercept and can be ignored

if interest centers on tax effects.

—14--

As was the case for Litzenberger and Ramaswamy, a market return can be

explicitly incorporated into Brennan's model. The b parameter of his model is,

b =E(Rt) - rf - - rf)

Substituting this into (6) and rewriting, the stochastic version of the model

becorn es,

R. ft ' - + i R - . i(d rft) +

- rft) + Ljt (iDa)

it =3i U

with properties essentially the same as (10).

Blume

The model Blume estimates is,

R =a +b +cd (8)it t t I it it

a =a÷ut at

b =+tt bt

C =C+Ut CtThe parameters a, bt, and c are random over time, but the distribution of the

parameters is fixed over time. From our discussion of the previous models, it

would seem natural to express Blume's model as,

—15—

= 4 y d. bjit oi 1 1. it

Y•oi i= C

V. = u. + u + + d, uit it at i bt it ct

Notice, however, that the disturbance of (9h), is a composite disturbance,

and one of the terms included in the disturbances depends upon dividend yield,

d:t beeause dividend yields vary over times, Ue variance of the disturbances

and the covariances across securities are not constant. But because. most

quarterly dividend yields are small in absolute value (typically ranging from .0

.02), the departure from standard assumption are of little importance and will be

ignored Lor purposes of estimation. (A more detailed analysis is presented in

Appendix B.)

The coefficients , 6, and can be readily expressed in terms of the

market portfolio. If (8) holds for all assets, it holds for the market portfolio.

Since the beta of the market equals one, the market return equals,

R =a++&I +mt mt mtso that

MR --d -vn1t mt mt

Substitut ig this expression for into (8) and making use of the definition of

R. = (l - 8.) + 8.R - + d. +it i imt i mt itv. - 8.v (lOb)it lint

—1 6-

Both formulations of Blume's model, (9b) and (lob), closely resemble the

empirical formulations proposed for Litzenberger and Ramaswamy, (9) and (10),

and Brennan, (9a) and (lOa). In particular, the models share the implication that

the dividend yield coefficient is common across securities.

Synthesis of Formulations

Two fundamental equations have been proposed for testing the models of

Litzenberger and Ramaswamy, Brennan, and Blume. The equations are,

= y + y •d. + y .r + c (11)it 01 11 it 2i ft it

R. cx.+cx R ÷a d +it 01 lint 2imt

a.r +d ÷r.(1231 ft 4i it it

Table 1 catalogues the parameter values and restrictions on parameter values

implied by the alternative model.'°

Estimating and Testing the

Alternative Formulations

Equations (11) and (12) are systems of time series regressions with a

regression equation for each security. To estimate the systems properly and to

test the model's implications, the cross sectional dependence in security returns

must be taken into account. Zeilner [1971] has called systems like (ii) and (12)

"seemingly unrelated regressions" (SUR). The regressions have different in-

dependent variables, but are related by the contemporaneous correlation of the

disturbances. The disturbances are presumed to be contemporaneously cor-

related across equations, but the non-contemporaneous covariances and auto-

—17—

covariances are here all assumed to be zero. The general form of the SUR is.

ri (Xl \\.6.1

X)+

Lki

Eu 0

/°11'T IkT\( :Eu u —' .-\uk1T kk1TJ

18

TABLE 1

CATALOGUE OF PARAMETER VALUES AND RESTRICTIONS

Equation (11): Rt = oi + 1li dt + '2i rf +

Model : Litzenberger and Ramaswamy

Parameter Value: a + b'lj

1—c

Restrictions: + = = and = for all i

and j.

Model : Brennan

Parameter Val ues: = b , 1 = T, 12i = 1 - T

Restrictions: + 2I = and = for all i

and j

Model: Blume

Parameter Values: a +j, 'li =

c 0

Restrictions: = and 12i = 0 for all i and i

Equation 12: Rt = +a1 Rmt + a2 dmt + a3 rft + 4i d1t + it

Model: Litzenberger and Ramaswamy

Parameter Values: = E(R*t)(l - i' an i' i = - C

a31 3.C - Ca4 = C

19

TABLE 1 (Continued)

'esLricLons: +a21

la1 l—a1

a1 a4. = a21 for a ii i and j

Model

Paraneter Values: a01 O 'H Yj (X3i = (i_) +

T

Resftictions: 0, a1 a41 2i' a1 3i 2i +

for all i and J

Model: Blume

Parameter Values: a (l—) a , 0,

= E

Rest:'ictions:a01 = _____ a4 = —a2 a3 = 0l-a1 l_a1

a41=

U4J for all i and j

—20—

where k is the number of equations (securities), T is the number of observations,

and 1T is the TxT identity matrix. The Gauss—Markov estimator of the system is,

= [t (E' ® 1) X

The estimator of this system exploits the cross sectional information that

is ignored by single equation estimators. For example, if the independent

variables are orthogonal across equations and if the disturbance are multivariate

normal the SUR estimator say for equation (1) equals,11

s +(x' x - I u . . u )Jl i 1 1 1—i —l 2

Alternatively, if b is the multiple regression coefficient for U. when is the

dependent variable and the remaining k—i disturbances are the independent

variable, the estimator may be expressed as,

6* = 6 + (' x1) x(u - bu -b3u . . .

The estimator adjusts the disturbances for any linear dependence with the

remaining disturbances in the system.

Although the orthogonality condition does not hold for our problem, the

spirit of the adjustment is similar. Thus, if the market portfolio can be

approximated with the securities included in the system, the loss in efficiency

from not including the market portfolio in equation (11) is reduced. More

generally, if security returns are linear in other unobservable factors, the

seemingly unrelated estimator partially adjust for the excluded variables.

If the disturbances of the system are normally, independently, and iden-

tically distributed over time, the SUR estimator is the maximum likelihood

estimator. The feasible estimator involves an estimated covariance matrix

2 1—

rather .han the true one, In general, the estimator has the same asymptotic

distribuun as the maximum likelihood estimator.12

rnpiriea1Resuits

Two systems of regression equations have been sugges ed for testing t he

tax eff€c of dividends. The systems are,

- I d. + y •r. + a. (11)it ci .1 t 2i ft itR. = a - + R a. - a r +it oi ii mt 2 mt 3i' it

+ cit (12)

The specific form of (11) for Litzenberger and Ramaswarny and Brennan is,

'ft = yi1(dt — rf) + ci (ha)

y = y for all i and jii lj

for Blume, (11) becomes,

R +y d .c (hlb)it oi lilt it

= y for all i and jii lj

—22—

Equation (ha) and (hib) are estimated for a broad cross section of firms.

These firms are classified into systems of securities (equations) on the basis of

average dividend yields. These classifications are meant to control for any

dependency of effective tax brackets on dividends of the kind reported by Elton

and Gruber [1970], and by Litzenberger and Ramaswamy [1980]. If tax

induced clienteles effect asset prices, the tax parameters of (ha) and (lib)

would not be constant across securities, hence, our tests would be more likely to

reject a common effect across securities the larger the dispersion in dividend

yields.

Periods of Analysis and Samples of Firms

The empirical tests are conducted over nine sample periods between

January, 1926 and December, 1978. The sample periods include substantially

different statutory treatment of dividends and capital gains. During the first

sample period--January, 1926 to December, 1931--the statuory limit on dividend

income was 20 percent and the statutory limit on long term capital gains (two

year holding period) was 12.5 percent. Between 1935 and 1940, the statutory

limit on dividends increased from 54 to 75 percent, while the statutory limit on

long term capital gains (one to ten year holding periods) ranged from 30 to 60

percent. By 1945, the statutory limit on dividend income had risen to 90

percent, and 50 percent of the gain on assets held for more than six months were

excluded from taxable income subject to a maximum tax of 25 percent. Between

1945 and 1978, the statutory limit on dividend income decreased to 70 percent,

while the limits on long term capital gains have ranged between 25 and 30

percent.

—23—

For each of the nine sample periods, systems of thirty securities are

formed on the basis of average dividend yields over the sample periods. None of

the firms included in the systems have nontaxable cash distributions during the

period.

The securities are all sampled from the Center for Research in Security

Prices (CRSP) ionthly return file and include only those firms with complete

data during the period. Each system includes thirty securities.

Data and Defuitioris of Variables

Monthly returns from the CRSP return file are used in estimating the

models. After 1931 the riskless rate of interest is approximated with the one

month Treasury Bill rate. Prior to 1931, the shortest Government Bcnds with a

maturity of at least one month are used to proxy for the riskiess rate. The

source of this data is Ibbotson and Sinquefield [1977].

The dividend yield variable is actual cash dividends paid in the ex dividend

month divided by the closing price in the previous month. Note that this differs

from the definitions of Litzenberger and Ramaswamy and Blume (cf. pages 6 and

8). It is unclear on prior grounds which of the definitions is more meaningfuL

Blume's quarterly measures of dividend yields and returns are not used since our

primary interest is detection of tax effects. With this objective, it is unwise to

destroy time series variation in dividend yields, i.e., the variation due to dividend

yields being zero in months when no dividends are paid.

Estimates and Tests with Equations

(ha) and (hib)

The empirical results for equations (ha) and (lib) are reported in Tables 2

-24--

through 10. The first column of each table shows the range of average dividend

yield for each system. The second column lists the restricted estimates and

their asymptotic standard errors in parentheses. The third column shows an "F-

statistic" for the hypothesis that yii is equal across securities, and the final

column lists the approximate probability levels conditional on 11i being equal for

all securities in the system. The "F—statistic" reported in these tables are

distributed as F, with degrees of freedom equal to the number of restrictions and

thirty times the degrees of freedom per equation if the covariance matrix is

known.14 For an unknown covariance matrix, the asymptotic distribution is chi-

squared with the same degress of freedom; however, the F approximation is more

conservative with respect to the equality hypothesis.15' 17

The constants in equations (ha) and (hib) implicitly include a beta (times a

market risk premium). Since other authors have assumed stationary beta for

sixty month periods, a dummy variable is included in the estimated equations to

allow for shifts in beta. The dummy variable is assigned a value of zero for the

first half of the sample periods and one otherwise.18

The results of Table 2 thorugh 10 tell a very convincing story. The

restriction that the tax parameter 11i is equal across securities is badly violated

for virtually all the sample periods and yield groups. Since the restriction fails

to hold, it is not clear what, if any, interpretations can safely be drawn from the

restricted estimates. For what it may be worth, the behavior of the restricted

estimates across the dividend yield classification does appear roughly consistent

with the existence of clientele effects. The restricted estimates do appear to be

negatively related to dividend yield, though the relation is not strong or uniform.

TABLE 2

ESTIMATES AND TESTS WITH EQUATIONS (ha) and (lib)JANUARY, 1926 — DECEMBER, 1931

Equation (ha): it - rf + Yi.(d. - rft) + itAverage Annual Restricted DegreesDividend Yield Estimate Of

ytem (Percent) __________ F-Statistic Freedom

.96 - 4.00 .451

(.304)

-. 4.76

- 5.26

- 5.79

- 6.42

- 6.99

- 7.53

- 8.32

- 10.64

2 4.04

3 4.80

4 5.28

5 5.80

0 0.43

7 7.00

8 7.58

9 8.35

4.04

3 4.80

4 5.28

5 5.80

6 6.43

7 7.00

I

.529(.215)

.261

(.140)

.952

(.143)

.670

(.160)

.174

(.129)

.917

(.157)

.266

(.158)

.890(.126)

Equation (llb):

.894

(.310)

.629

(.208)

.376

(.136)

1.211

(.142)

.734

(.160)

.235

(.131)

1.035

(.158)

.331

(.160)

.839

(.125)

5.006

5.099

2.529

2.861

3.177

5.327

3.724

3.643

2.553

= +it 01

4.746

4.868

2.308

3.338

3.239

5.293

3.611

3.329

2,647

29, 20 70

29, 20 70

29, 20 70

29,2070

29, 20 70

29,2070

29, 20 70

29,2070

20,2070

.d. +E.ii it -it

29, 2070

29,20 70

29,2070

29,2070

29, 2070

29,2070

29,20 70

29,2972

29 , 20 70

ProbabilityLevel

.0001

.0001

.0001

.0001

.0001

.0001

.0001

.0001

.0001

.0001

.0001

.0001

.0001

.0001

.0001

.0001

.09 T1

.0001

.90 - 4.00

- 4.76

- 5.26

- 5.79

- 6.42

- 6.99

— 7.53

S 7.52

9 8,35

— 3.32

- 10.64

TABLL .

ESTIMATES AND TESTS WITH EQUATIONS (I lo) AND (110)

JANUARY 1935 - DECEMEIER)J1U

Equation (I Ia): jt -- rf1

Average Annua' Rest"ictedDividend Yield Estimates

JronL_.l.04•-4.778 .710

(.190)

4.788 - 5.313 850

(.183)

5.326 5.862 .584(.154)

4 5,875 - 6.440 .354(.123)

5 6.452 - 7.086 .356(.127)

6 7.134 - 8.096 .569

(.126)

7 8.120 — 9.612 .533(.116)

8 9.684 - 14.928 .566(.1 30)

o tm

3

-- + -01 1

F—Statistic

4.935

1.607

2.281

2525

3.994

4.606

4.362

3.879

Degrees OfFree doe

29 ,2070

29,2070

29,2070

2q, 2070

'9,2070

1)70

29,2070

29,2070

d fjt

29,2070

29,2070

29,20 70

29,20 70

29, 20 70

29,20 70

29,2070

29,20 70

irobahil I tyLevel

.000)

0215

.000 1

.0001

0001

0001

0001

000 I

.0001

.0 235

.000 1

.0001

.000 1

.000 1

.0001

.000

1 1.704

2 4.788

3 5.326

4 5.875

5 6.452

6 7.134

7 8.120

8 9.684

Equation

- 4.778

- 5.313

- 5.862

- 6.440

- 7.086

- 8.096

— 3.612

- 14.928

(lib)

.715(.190)

.851(.153)

.582

(.154)

.353

(.123)

.358(.127)

569(.126)

.535(.116)

.566(.130)

= +

4.932

1.593

2.760

2.626

3.977

4,613

4.360

3.874

TABLE 4

ESTIMATES AND TESTS WITH EQUATIONS (ha) AND (llb)

JANUARY 1941 - DECEMBER 1945

Equation (ha): Rt -' rf =V01 + Vii (d — rft) Ei

Sysle

Average AnnualDividend Yield

(Percnt)

RestrictedEstimatel F-Statistic

Degrees OfFreedom

ProbabilityLevel

1 2.336 - 4.400 1.430

(.133)

5.672 29,1710 .0001

2 4.466 - 5,068 .441

(.135)5.130 29,1710 .0001

3 5.084 - 5.504 .754

(.103)

4.545 29,1710 .0001

4 5.800 - 6.088 .663

(.103)

7.681 29,1710 .0001

5 6.440 - 6.824 .532

(.085)2.714 29,1710 .0001

6 6.864 - 7.288 .632

(.109)4.554 29,1710 .0001

7 7.290 - 7.760 .346

(.078)

5.633 29,1710 .0001

8 7.768 - 8.244 .611

(.091)

3.538 29,1710 .0001

9 8.788 - 10.164 .493

(.085)

6.975 29,1710 .0001

10 10.168-11.580 .260

(.083)

5.157 29,1710 .0001

Equation (lib): R1t + hi d1t +

1 2.336 - 4.400 1.460

(.133)

5.675 29,1710 .0001

2 4.468 - 5.068 , .459

(.135)

5.152 29,1710 .0001

3

4

5.084 - 5.504

5.800 - 6.088

.773

(.102)

.682

(.103)

,

4.565

7.686

29,1710

29,1710

.0001

.0001

5 6.440 - 6.824 .540

(.085)

2.715 29,1710 .0001

6 6,804 - 7.288 .642

(.109)

4.511 29,1710 .0001

7 7.290 - 7.760 .354

(.078)

5.629 29,1710 .0001

8 7.768 - 8.244 .616

(.091)

3.540 29,1710 .0001

9 8,783 - 10. 164 .501

(.085)

6.937 29,1710 .0001

10 10.168 - 11.580 .266

(.083)

5.178 29,1710 .0001

TABLE 5

ESIIMA1IS AND TESIS WITH EQUATiONS (ha) AND (lib)

OANIJARY 1146 DECEMBER 1950

Equatcn (la): .. = + . (d.

Averacie AnnuaJiviciendYi'.

(Percent;

60 — 5.1-140

3

4

5

b

8

9

0

4.808

5. 45

6.068

6.476

7.022

.. 388

7 064

8.896

9.920

5.436

6.468

- 7.020

— 7.376

- 1.856

- 8.892

— 9.904

- 13.072

rt• -it

:Stntl sti cOeqree-; 0-. Ereuom

nrohabclicyLevel

4.835 79,1710 0001

3.HI6 79,1710 .0001

3.792 29,1/10 .0001

5.015 29,1/10 .0001

1. 587 29,1710 .0001

5.04 10_i flD .0001

5.321 77;. 77:0 .0001

4.796 79 77 .flCH

7.938 29, 710 .6001

1-es 7777 uted[sO

• 845(.175)

• 996(.132)

.248

(.108)

.344(.097)

.411

(.019)

109(.086)

020

.420(.083)

.00/77076)

on Ilib):

8/8

(.176)

1.009

(.131)

.257

(.107)

- .064(.095)

.853

(.097)

.420

(.079)

.154

(.086)

.337

(.094)

.430

(.083)

Equati

1.760 - 4.880

2 4.888 5.436

3 5.452 - 6.060

4 6.068 - 6.468

5 6.476 - 7.020

6 7.022 - 7.376

7 1.388 - 7,856

8 7.964 - 8.892

9 8.896 - 9904

R. =..+yit Oi

4./71

3. 325

3. 801

4.979

3.090

5.105

5.750

4.801

7.89 1

d. +it --it

29,1710

29,1710

29,1710

29,1 710

29,1710

29,1710

29,1710

2Y, i710

29,1710

29, 1710

.0001

.0001

.000)

.0001

.000 1

.0001

0601

.0001

.0001

.000 110 9.920 - 13.072 .0178 2.919

(.076)

ESTIMATES AND TESTS WITH

TABLE 6

EQUATIONS (ha) AND (lib)

JANUARY 1951 - DECEMBER 1955

Equation (ha): it - rf =

.Y0 + -Yji (d1 rft) + itSys tern

Average Annual RestrictedDividend Yield Estimate

ftercegJ "h

.

F-StatisticDegrees OfFreedom

ProbabilityLevel

1 1.248 - 4.420 .424(.146)

4.856 29,1710 .0001

2 4.564 - 5.185 .355

(.140)

3.742 29,1710 .0001

3 5.200 — 5.648 .669

(.100)

2.332 29,1710 .0001

4 5.652 - 6.026 .325

(.075)

2.511 29,1710 .0001

5 6.026 - 2.348 .523(.086)

3.781 29,1710 .0001

6 6.360 - 6.680 .023

(.093)

4.535 29,1710 .0001

7 6.696 - 6.972 .0250

(.082)

3.661 29,1710 .0001

8 7.320 - 7.724 .037

(.070)

7.129 29,1710 .0001

9 7.756 - 8.104 .264

(.085)

2.327 29,1710 .0001

10 8.576 - 10.204 .260

(.095)

2.974 29,1710 .0001

Equation (lib): Rt = + r11 + it1 1.248 - 4.420 .360

(.144)

4.964 29,1710 .0001

2 4.564 - 5.188 .307

(.140)

3.698 29,1710 .0001

3 5.200 - 5.648 .652

(.100)

2.280 29,1710 .0001

4 5.652 - 6.026 .293

(.074)

2.508 29,1710 .0001

5 6.026 - 6.348 .489(.086)

3.750 29,1710 .0001

6 6.360 - 6.680 .006(.044)

4.590 29,1710 .0001

7 6.696 - 6.972 -.057(.081)

3.620 29,1710 .0001

8 7.320 - 7.724 .005(.069)

7.236 29,1710 .0001

9

10

7.756 - 8.104 .249(.085)

8.576 - 10.204 .245(.095)

2.395

2.885

29,1710

29,1710

.0001

.0001

TI-IDLE 7

ES1IMATES AND T[STv WIts-i [QU/I7IONS (lid) AND (I

JANU/1R9 956 - DECEMBER 1960

[quatien Ha): 2jt - rs- + (dAverage Aneea 9ev yr Lvi0vidend field

vysteri (Percent)

259 — 3. 764

7 3.788 - 4.472 3491.166)

.1 4.449 316 .309(.137)

4 4.331 — 5.160

5 5.112 - 5.404 47. 1 391

3 9.'08 - 5.108 --.13335151

7 5.712 — 6.348 .603 23581 106)

8 6.060 6.528 -140 5.147

9 6.572 - 8.632 454 3912(.133)

F-Stati stic

46

4.282

1 567

4.030

- rft)

0eqrcrFe edorn

79, 1710

29 1710

29 , 17 (7

29, 710

1710

9, 1710

i9, 710

Pyrhaii I StyLeve.i

2101)

.000

.0281

30111

0001

.0005

0001

.4800

.000 1

.0256

.0001

.0001

.0001

.0001

.0001

0001

1.256 -

2 3.788 -

3 4.449 —

4 4.237 —

5 5.172 -

6 5.408 -

7 5.712 -

8 6.068 -

9 6.572

Equ liti

3. 764

4.'172

4.836

5.160

5.404

5.708

6.048

6.528

0.632

on (Sib):

.022

(.217)

.169(.162)

- .076(.138)

.822

-.191(.137)

- .229(.097)- .022(.105)

-.226108)

4 57(.134)

1i dt f I.991

4.009

1 .582

4,633

2.465

4.301

2.772

5.922

3.922

it29.1710

29,1710

29, 1710

29,17 10

29,1710

29,1710

29,1710

29,1710

29,1710

•1•••1

TA8LE8

ESTIMATES AND TESTS WITh EQUATIONS (ha) AND (llb)JANUARY 1961 - DECEPER 1965

Equation (ha): Rit — rf • + V (die - rft) it

'x!a'!Average Annu',IDivides Yield

RestrictedEstimate

— ....!L.... t.tSJ1ci.cDegrees Of

trec4oi?i..Probability

Levol

1 1.040 - 2.480 1.401

(.464)

5.838 29,1710 .0001

2 2.560 - 3.132 1.709(.298)

3.078 29,1710 .0001

3 3.136 - .44R .139(.201)

5.022 29,1710 .0001

4 3.492 - 3.800 L1.(• IW

3.194 29.1710 .0001.

5 3.808 - 4.096 .486(.210)

7 810 29,1110. .0001

6 4.108 - 4.292 .460(.193)

.2J 29.1710 .0002

1 4.292 — 4.556.

-.311

.179)2.012 29.1110 .0012

8 4.568 — 4.768 .511(.1821

2.767 29,1710.

.GOOI

9 4.776 — 5.060 .175(.143)

5.585 29,1710 .0001

10 5.068 - 5.560 -.311(.140)

4.516 29,1710 .0001

.Equation (111,): + ii dit +

1 1.040 - 2.480 1.574

(.466)

4.492 29,1710 .0001

2 2.560 - 3.132 1.695

(.297)

3.059 29,1710.

.0001

3 3.136 — 3.448 .160(.202) .

4.827 29,1710 .0001

.

4 3.492 - 3.800. 1.136(.189)

3.560 29,1710 .0001

5 3.808 - 4.096 .483(.209)

2.893 29,1710 .0001

6 4.108 - 4.292 .434

(.193)

. 2.202 29,1710 .0003

7 4.292 - 4.556.

-:321(.179)

1.961 29,1710 .0017

8 4.568 - 4.768 .529

(.182)

2.736 29,1710 .0001

9 4.776 — 5.060. .177

(.143)5.593 29,1710 .0001

10 5.068 - 5.560 -.313(.140)

4.433 29,1710 .0001

TABLE 9

ESTIMATES AND TESTS WITH EQUATIONS (ha) ANT) (llb)FEBRUARY, 1966 - NOVEMBER, 1971

ProbabilityLevel

.0001

.0001

.0001

.0009

.0001

.0001

.00 20

.0001

Equation (ha): Rit - rf = + - rf) + itSystem

Average Annual RestrictedDividend Yield Estimate

(Percent) - F-Statistics

DegreesOf

Freedom

1 .28 - 1.12 7.064

(1.19)

4.725 29,2010

2 1.56 - 1.84 3.159

(.511)

2.903 29,2010

3 2.04 - 2.28 2.163

(.480)

3.575 29,2010

4 2.52 - 2.64 .070

(.389)

2.050 29,2010

5 2.96 - 3.12 .526

(.298)

3.159 29,2010

6 3.68 - 3.88 .637

(.243)

2.603 29,2010

7 4.24 — 4.40 .332

(.228)

1.939 29,2010

8 4.84 - 5.08 .087

(.171)

3.507 29,2010

9 5.28 - 5.56 —.092

(.099)

3.122 29,2010 .0001

10 5.96 - 9.32 .162

(.108)

5.334 29,2010 .0001

Equation (hib):R.t

= + i1d1 + it1 .28 - 1.12 5.688

(3.63)

5.468 29,2010 000l

2 1.56 - L84 2.626

(.509)

3.156 29,2010 .0001

3 2.04 - 2.28 1.949

(.482)

3.741 29,2010 .0001

4 2.52 — 2.64 -.218

(.380)

2.061 29,2010 .0008

5 2.96 - 3.12 .313

(.296)

3.123 29,2010 .0001

0 3.Od - .58 .549

(.2 94)

2.407 29,2010 .0001

7 4.24 — 4.)0 .253

(.228)

2.139 29,2010 .0001

8 4.84 - 5.08 .061

(.169)

3.659 29,2010 .0001

9 5.28 - 5.50 -.083(.099)

3.082 29,2010 .0001

10 5.90 - 9.32 .148

(.107)

5.222 29,2010 .0001

TABLE 10

ESTiMATES AND TESTS WITH EQUATIONS (ha) AND (hib)JANUARY, 1972 - DECEMBER, 1978

Equation (ila): Rjt - rft + Yi(d1 - rft) ÷cjAverage Annual Restricted

DegreesDividend Yield Estimate ofProbabilityem rc _ atistjcs dom _

1 .43 - 1.57 3.904 J.394 29,2430 .0800(.702)

2 1.61 2.20 1.963 3.293 29,2430 .0001(.486)

3 2.50 - 2.81 1.760 1.945 29,2430 .0019(.317)

4 5.42 - 3.66 1.176 2.976 29,2430 .0001(.329)

5 3.92 - 4.26 2.705 2.176 29,2430 .0003(.262)

6 4.72 - 4.95 .903 3.276 29,2430 .0001(.223)

7 5.21 - 5.64 .907 1.271 29,2430 .1515(.194)

8 5.88 - 6.24 .466 2.432 29,2430 .0001(.166)9 6.7o - 7.45 .253 3.087 29,2430 .0001

(.120)

10 8.29 - 11.82 .095 5.113 29,2430 .0001(.068)

Oquation (lib): = + Yi.d. +

1 .43 - 1.57 2.725 1.561 29,2430 .0300(.680)

2 1.61 - 2.20 1.1)94 2.503 29,2430 .0001(.462)

3 2.50 - 2.81 1.199 2.345 29,2430 .0001(.316)

4 3.42 - 3.66 .925 2.726 29,2430 .0001(.330)

5 3.92 - 4.26 2.615 2.049 29,2430 .0008(.266)

6 4.72 4.95 .768 3.185 29,2433 .0001(.223)7 5.21 - 5.64 .823 1.404 29,2430 .0745

(.197)

8 5.88 - 6.24 .378 2.283 29,2430 .0001(.165)

9 6.76 - 7.45 .196 3.190 29,2430 .0001(.121)

10 8.29 - 11,82 .061 5.205 29,2430 .0001(.068)

ModelMbeciflcation

EqLion (ha) imposes the rostriotion that the coefficicots on the rtskless

rntc of iiorest equal I—c, and (lIb) totally excludes the riskiers rate of interesL

The work of Fame and Setiwert 19771 suggests that 4ock returns we

egal ive. eiated to the riskks rate of interest. Since equation i Ia) and (.1 lL

fail to incctpora is behavior, it Ls possible that our tests have failed o hoid

ecause fect of th ;oss .e is not properly speciñO. tt is true Oi

course, thit the models of Li tzei :o Ramaswamv, irennan, ad Bhere

make spooific ;tatements about the ineo iscless rate on axpected

returns. However, if the shortcomings of rvdot due on nis

speciflcst on of the way the riskiess rate arid not divioc: : tO :'.•1 ion,

tax effeo may be present and masked by tne riskiess rate of

worthwhile, therefore, to serarately rxlore ttese two

Equation (11),-

R y +yd i-y r E [Ii)it o:i Ii it 2± ft itallows the risidess rate of interest i.o be f?ee. Referring to Table 1, on pages 20

and 21, ;ve see that (11) implies that = and = for all three

models, that 11i+ for Litzenberger and Ramaswamy and Brennan, and

that ye,. 0 for Blume. With equation (11) we thus can separate the hypothesis

that is equal across securities from the other hypotheses.

Equation (11) is estimated with a sample selected from the securities

included in the Dow Jones Industrials during the January, 1926 to December,

1978 period. The sample always consists of thirty securities but the composition

of the sample changes over time. For purposes of comparison, the same thirty

securities used for estimating (11) are also used to estimate (lie) for each of the

—35—

nine periods. These results are reported in Table 11. Table 12 reports test

statistics, restricted estimates, and unrestricted estimates for three of the

periods, along with their standard errors.

From the results of Tables 11 and 12, it appears that our earlier

conclusions are insensitive to the specification of the riskless rate. Generally,

the restricted estimates of equations (11) and (ha) are quite close and our

inferences regarding the equality of 11i across equations are unaffected. As for

the riskiess rate, the hypothesis of a common coefficient appears to be a

reasonable approximation for periods after 1960. Prior to 1960, we are generally

able to reject this hypothesis. The unrestricted estimates of Table 13 show a

wide variation in all sample periods but are predominately negative, consistent

with the Fama-Schwert findings. Given the observed variation, it is surprising

that the hypothesis of a common effect of the riskiess rate is not rejected in all

periods. But the smaller variation in the riskiess rate makes precision of these

estimates much lower than that of estimates for the dividend yield coefficient.

An additional check on the sensitivity of the test to model specification

can be obtained from equation (12) which differs from (11) by inclusion of the

market return and dividend yield as separate variables (cf. Table 1). These

estimates are reported in Table 13 for the same time periods and for the sample

of Dow Jones Industrials. Including the market return and yield as separate

variables seems to have little impact on the results. The probability levels,

restricted estimates, and unrestricted estimates are very similar to those of

equation (11).

Definition of Anticipated Dividend Yields

The results presented in Tables 2-13 provide little support for the simple

—36--

nx effec. models of dividends. We can reject y model predicting a common

c-.e cie on dividend yield, including the case of neutrality of dividends. it

2ossible that our results are due to the perfect foresight definition of dividend

yields, !iany dividend payments nre announced during the cx month, and

therefor announcement and tax effects may be convoluted in our estimates nf

the dividand i1,d coefficients. There is, of course no reason to suppose that

such announcement eieets would be the same across securities or across time

for the arne securities and because f this our tests may unfairly reject the

hypothesis of a common dividend yield ffec.. To control f or this possibility, we

consider a definition of anticipated dividend ioici tfat relies only upon

information announced prior to the ex month, namely the dnnition oC Litzen—

berger ane Ramaswamy. This choice allows us to make direct cornpaions

their results and may also be regarded as a naive definition from the set ct

possible definitions, i.e., many alternative definitions relying upon information

announced prior to the ex month will have forecasting errors that lie somewhere

between. those of the perfect foresight definition and the Litzenberger and

Ramaswamy definition.

January

January

Janu

ary

January

January

January

January

January

January Time Period

1926 -

Dec

embe

r193

1

1935 -

Dec

embe

r 1940

1941 -

Dec

embe

r 1945

1946 -

Dec

embe

r 1950

1951 -

Dec

embe

r 1955

1956 -

Dec

embe

r 1960

1961 -

Dec

embe

r 1965

1966 -

Dec

embe

r 1971

1972 -

Dec

embe

r 1978

F—Statistic

5.875

4. 145

7.675

5.992

2.735

6.025

4.229

1 .040

2.152

rft) + it

Degrees

Of

Freedom

29,2070

29,2070

29,1710

29,1710

29,1710

29,1710

29,1710

29,2070

29,2

430

Prob

abili

ty

Lev

el

.000

1

0001

.000 1

.0001

.000 1

.000.1

.000

1

.406

9

.000

4

TABLE 11

EST

IMA

TE

S AND TESTS WITH EQUATION (ha)

DOW JONES 30

Equation (ha):

R1t - rf

=

+ 11

i (d1t

Restricted

Estimate

327

(.150)

332

(.128)

.518

(.066)

.272

(.074)

- .0

28

(.077)

-.413

(.13

0)

.203

(.216)

.552

(.25

5)

.555

(.127)

TABLE 12

ESJU1ATES AND TETS WITH EQUATioN (11):

PERFECT FORESIGHT DEFINITIONS OF DIVIDEND IELD

Equation (11):

8jt =

+

d. +

2i

÷

DOW JONES 30

Restrictions

iU1i

ji arl9 -

Dem

ber_

1940

?L1

2i_=

4.712

29,2040

29,2040

2.728

4.278

L53

.0001

30,2040

29,2040

;o

.65b

.0001

6.702

.0001

.0001

O3i

.152

.358

62933

December 1950

F—Statistics

Degrees of Freedom

Probability

Level

Restricted Estimates

Standard Errors

F—Statistics

Degrees of Freedom

Probability Level

Restricted Estimates

Standard Errors

F. Statistics

Degrees of Freedom

Probability Level

Restricted Estimates

Standard Errors

Th_

Th.

g77uary 1941-

Dec

embe

r 1945

2i

2j 1i2i'

1.575

30,2040

.0247

li

'2i

1

7.393

29,1630

.000 1

490

.068

1.850

29, 1680

.0040

132. 799

22. 066

Jar,

uay

1951

-

Dec

embe

r 19

55

Th1 22

=

1

2.978

30,1680

.000 1

6.004

29,1680

29,1680

1.627

2.590

1.967

.0001

.0089

30,1680

29.1

680

29,1680

.225

-.918

.0176

.0001

.0017

.073

—5.990

Jaflu

aj9-

Dec

embe

r 19

65

ThT

h:

2i2j

.+

Y3.

r1

Dec

embe

r196

0

i.+.,=

1 JliJ i_

?

4.094

29,1680

.000 1

.171

.220

2.095

6.106

2.262

30,1680

.0005

29,1680

.0001

-.587

29,1680

.0002

—9.437

1.276

29, 1680

.1459

—6.627

7.524

Li21

1.

276

30,1

680

1459

2.847

29,1680

.000 1

1.158

29,2040

.2564

364

.249

1 . 19

0 29, 2040

.2228

—9.502

3.998

12i•

2it i

LiiJ

?I:J

1.510

30,2040

.03 77

2.058

29,2400

.0008

.510

I .166

29,2400

.2480

1 .591

2. 161

1. 151

30,2400

TABLE 12 (CONTD)

DOW JONES 30

Security

Number

:2

Unrestricted Estimates of

January 1961 -December1965 1 I

X2 j

ii and

21 With Equation (11)

January1966 -

Dec

embe

r 1971

2i

-1i

January 1972 -

Dec

embe

r 1978

12i

1

- 4.

579

20.625

--1

.Ljj

Ii 2

44.178

.439

1.237

-24.834

13.419

-1.465

1.226

3

-27.730

26.818

2.401

3.410

-19.433

18. 320

3.353

1.749

-O.512

7.303

1.449

1.08

6 4

- 6.

194

- .7

85

1.656

- 8.

706

12.329

1.661

1.427

- 4.

579

7.143

- .2

07

1.436

5

- 8.

207

16.013

.414

.630

-14.985

12.885

-1.432

1.614

- 5.

011

4.689

.24

.485

6

3.804

.167

.942

-11.207

10.711

.834

1.068

- 2.

923

4.307

.340

.447

7

33989

23.640

- .9

97

.670

-13.894

16.328

-2.134

1.914

- 4.

690

3.390

.560

.382

8

- 8.

527

36.414

5.323

3.815

- 2.

249

9.947

- .0

74

1.229

- 1.

612

7.971

- .3

41

.648

9

16.218

17.193

.694

.841

-10.942

7.931

- .8

35

1.112

-14.086

10.117

-1.507

1.818

10

- 2.

992

19.581

-1.047

1.356

- 5.

830

9.256

.152

- 3.

503

6.379

-1.605

.968

11

- 5.

965

24.784

1.171

.979

-14.037

13.276

1.545

2.175

-10.531

6.418

1.150

1.264

12

22.231

15.360

-1.441

.649

- .6

74

9.156

-1.229

- 1.

928

7.038

2.248

1.125

13

-31.835

20.867

1.279

1.323

- 1.

000

16.330

-1.338

- 8.

160

4.500

.576

.533

14

2.236

20.481

1.814

1.621

.322

15.530

2.021

1.492

-11.949

6.161

1.148

.842

15

-26.110

19.191

-2.141

.597

-30.154

17.676

- .5

82

2.945

6.714

.630

.903

16

-20.736

23.360

1.907

1.539

-18.726

12.994

.729

- 6.

124

3.193

- .4

42

.355

17

-11.022

24.947

- .6

57

1.411

13.856

12.041

- .5

90

1.339

1.464

- .8

20

6.362

- .6

84

.695

18

-20.424

20.418

- .2

58

.951

-10.159

13.400

- .2

15

- 4.

824

7.151

- .7

57

.988

19

- 2.

305

24.781

- .5

15

1.252

- 3.

519

16.643

-2.950

- 4.

676

7.615

- .7

63

.923

20

19.547

20.286

3.041

1.057

- 7.

663

12.694

2.004

- 1.

835

6.876

- .0

34

1.023

21

4.408

25.086

.248

3.214

- 7.

742

8.331

.585

- 3.

920

7.561

- .7

78

1.074

22

- 4.

834

19.602

4.841

1.373

-29.813

17.037

.525

1.804

- 7.

833

6.608

1.851

1.80

5 23

-13.620

21.152

- .2

19

1.344

-33.345

15.574

- .7

34

1.677

2.588

6.210

2.674

.964

24

- .3

05

21.884

2.954

2.085

6.653

17.525

2.160

1.719

- 5.

622

4.78

4 2.043

1.157

25

-18.010

16.350

-2.278

.927

- 6.

796

12.896

.630

1.302

-10.017

6.013

2.311

1.139

26

- 1.

085

19.013

-1.521

1.022

- 8.

112

11.406

.089

1.398

- 9.

642

6.296

- .3

58

.723

27

- 4.

097

20.875

- .0

83

1.232

-12.751

10.376

- .6

77

- 5.

290

5.584

- .1

56

.419

28

14.852

26.091

1.584

.774

-24.266

19.388

6.012

2.272

- 3.

606

6,535

2.293

.620

29

25.357

29.741

3.153

1.789

- .7

92

13.103

.794

1.399

1.497

6.897

- .0

28

.737

30

26.062

3.288

1.655

6.834

21.984

4.272

-13.154

7.479

-1.037

1.156

TA

BLE

13

ES

TIM

AT

ES

A

ND

TE

ST

S

WIT

H

EQ

UA

TIO

N (1

2):

PE

RF

EC

T

FO

RE

SIG

HT

D

EF

INflI

ON

OF

D

IVID

EN

D Y

IELD

Equ

atio

n (1

2):

=

+ a

i1t

+ a2

0m

t +

a3 ft

+

it

December 1965

DOW JONES 30

Restrictions

l935

-Oec

r.3r

194

0 Ja

nuar

y 1926 —

Dec

embe

r 1931

c.

a 3i

3j

a 41

2.63

12

4.58

74

29,2

010

29,2

010

.000

1 .0

001

-.99

7 .6

52

.543

.1

56

a3

=

4.

= a

..,1

.)3

F-Statistics

Degrees of Freedom

Probability Level

Restricted Estimates

Standard Errors

F-Statistics

Degrees of Freedom

Pro

babi

lity Level

Restricted Estimates

Standard Errors

F—Statistics

Degrees of Freedom

Probability Level

Restricted Estimates

Standard Errors

a31

= a3

.

ffL1t

29

,165

0 29

,201

0 29

,201

0 .0001

.0001

.4258

6.57

7 .656

12.608

.702

Januar11 -_December1955

a

'-a..

a _3

j 41

4

Janu

ary

1961

j9—

Dec

embe

r_19

60

2.534

3.593

'651

29,1650

29,1650

29,1650

:°.1650

29,1650

.0001

.0001

.0001

.0001

.0001

.0133

3.247

-.484

-1.995

.396

i.iqS

-.003

.152

Jay_

1966

_- D

ecem

ber

1971

2.109

29,1650

.000

5 -1

.891

.8

84

=

Janu

a197

2-D

ecen

iber

_197

8

a3j

=

3.29

3 29

237O

29,2370

29,1650

29,2010

29,2010

.4387

.0001

.0001

.Th10

1.67

0 .8

54

-, 0

4 -.

2.16

3 .5

78

.175

Q4j

—

4j

TABLE 13 (CONT'o)

DOW JONES 30

Unrestricted Estimates of a31 and a

141 With Equation (12)

Security

Number

03i

January 1961 — December 1965

January 1966 —

Dec

embe

r 1971

04j

"31

C31

"41

January 1972 -

Dec

embe

r 19

78

03

a41

041

1

.783

9.809

—

.003

3.615

— 4.

726

8.577

—1.462

3.995

.396

5.048

5.153

1.684

2

16.869

14.542

29.933

11.452

— 6.

236

13.714

3.607

1.805

2.376

6.601

.481

1.624

3

.754

12.163

—

.965

3.058

—

1.66

2 9.043

2.688

1.478

.742

3.616

.221

.900

4

— 4.

227

8.183

.269

.649

—15.616

11.456

—1.989

6.468

1.131

3.696

.367

.533

5

— 8.

274

6.446

3.936

1.425

—

1.02

9 7.913

—2.108

1.463

.032

2.808

1.349

1.021

6

12.211

10.916

—

.026

.712

—

8.66

7 12.161

—2.376

2.675

8.663

6.510

—

.448

.974

7

— 5.

551

19.450

- 2.

354

6.292

— 6.

783

8.638

—1.462

1.938

5.988

8.930

1.746

1.533

8

— 6.

501

6.557

2.157

2.320

— 3.

076

6.303

8.578

3.388

5.185

4.800

3,709

2,106

9

25.496

8.022

—

.814

1.907

—

.587

6.902

—1.597

2.143

- 1.

677

4.883

.045

2.140

10

11.844

12.648

.284

.983

— 8.

897

10.222

.550

2.162

4.850

6.160

2.346

1.121

11

—16.990

7.261

- 8.

267

1.857

3.374

7.580

—1.452

3.799

- .8

86

3.593

2.477

1.304

12

15.944

9.289

3.212

1.398

10.417

10.743

—2.644

1.569

•

- 1.

794

3.375

.776

.791

13'

—16.391

9.366

3.224

5.060

1.928

11.893

4.210

1.747

12.268

5.179

2.490

1.956

14

15

— 3.

736

— 2.

989

8.126

9.810

— 2.

942

7,865

1.332

4.239

—20.012

—16.585

13.279

9.938

4.695

—

.805

2.903

1.444

—

.695

6.

647

4.21

8 4.

856

—

.828

—1.513

.600

2.152

16

17

— 9.

519

.773

10.039

11.033

16.760

1.131

3.223

1.044

— 1.

004

1.148

8.496

9.881

—

.327

2.737

5.111

5.543

3.619

2.402

6.006

6.040

—1.279

—

.036

.992

.930

18

19

— 8.

430

'6.731

9.951

9.945

3.323

3.476

3.745

3.344

.729

3.425

10.865

9.395

—9.446

—

.300

3.665

1.343

8.893

2.897

5.220

6.516

2.888

-2.998

2.258

2,611

20

19.532

10.316

— 9.

774

6.888

.400

6.895 '

.127

.886

3.297

5.018

9.611

5.011

21

22

23

24

25

26

27

28

29

30

'

14.316

—

7.95

0 — 8.

168

—

5.19

3 —15.095

9.887

9.534

23.174

30.376

1.674

9.820

9.016

10.509

8.294

8.930

8.257

9.449

16.651

'

12.4

54

13.104

2.290

—

2.05

1 .319

— 3.

558

—

1.80

8 —

3.00

1 4.

128

— 1

.287

3.302

2.963

2.572

1.537

6.811

1.669

1.031

3.431

1.770

4.268

4.237

1.685

—10.870

—30.788

4.578

2.264

3.202

— 4.

279

—11.124

6.929

14.148

— 2.

167

10.536

12.012

11.606

9.260

7.927

7.414

15.546

8.793

17.070

8.906

—6.466

1.585

.802

2.152

2.454

.745

6.411

—3.027

1.652

1.191

6.199

2.165

1.849

1.442

5.938

1.058

2.409

1.450

4.917

1.297

11.017

1.637

— 2.

670

— 2

.208

1.749

6.469

10.953

— 5.

737

—26

.738

7.

737

4.372

3.432

4.692

5.386

4.201

4.516

5.767

6.735

14.0

84

6.085

' 5.

050

1.067

.013

.309

.295

2.602

.938

—

.566

4.

877

1.248

1.119

1.152

2.416

1.734

.434

.632

.806

2.250

2.39

0 .811

uar,y1926-mberw

1I2i'4.708 2.819 3.052

29.2040 29,2040 30,2040.0001 .0001 .0001

8.338SF 2.177

11"J 21 j 2i"13.364 1.548 i,78

29,1680 29,1680 29,2680.0001 .0166 .0244.040 1.981.045 7.798

January 1961- December 1965

Ii 1i +

1.546 1.373 1.35729,1680 29,1680 29.1680

.0321 - .0895 .0946

.531 . —2.977

.216 7.399

January 1926 - December 1931

li 1J 2i_j 'lI 2l 14.391 1.376 1.457

29,2040'- 29,2040 29,2040.0001 .0874 .0524

-.275 4.859.077 2,332

i,:i.,i 2i''2 1j2l 1

2.213 1.758 . 1.7956"9,1680 29,1680 30,1680

.0002 .0078 .0053-.046 -8.363

.052 3.463

965

t_u3.200 2.565 2.648

29,1680 29,1680 29,1680.0001 .0001 .0001.088 —18.354.132 9.276

DOW JONES 30

Restrict ions

j5-0ecernber 1943__'L1 ?±.Th1 l12i12.404 1.551 1.467

29,2040 29,2040 30,2040.0001 .0391 .0492.022 23.239.011 .681

January 1951 -December 1955

y.v. y.y. y.+y.1JJi iifl3040 1.866 2,029

29,1680 29,1680 30,1680.0001 0036 .0009.054 —6.088.077 3.446

1.808 1.241 1.02229.2040 29,2040 30,2040

.0053 .1764 .4400—.295 -.809

.174 4.215

UTILITIES

lestrictions'

January 1935- December 1940

'2i'2j 'i'1i 'v21 1

4.584 2.074 2.08929,2040 29,2040 30,2040.0001 .0007 .0005.085 —64.049.067 39.038

January 1951 - December1955

'n1i1fl j°2j 'rh 2i2.452 1.092 1.7138

29,1680 29,1580 30,1680.0001 .3369 .0096.436 —13.064.059 3.174

!9!ii4arL,i965-December 1971

'r1' 2i' 0ii213,387 1.063 1.026

29,2040 29,2040 30,2040.0001 .3747 .4275.196 4.609.067 3.974

Janua r,y 1941 . December 1945

S ' = . . +l1 ,,i,,,

3.678 2.505 4.40029,1680 29,1600 30,1663

.0001 .0001 .0002—.0411 148.063

.020 19.108

3anuj95E- December1960

5.124 2.563 3.09929,1680 29,1683 30,1680

.0001 .0001 .0001—.207 -7.730

.124 2.545

Janua2 1972-December 1978

j1ji' 2i'r2 Th 'r2 I

2.276 1.112 1.20029,2400 29,2400 30,2400

.0001 .3111 .2099.573 -3.006.132 2.244

j1_Cecer5'riijl' L1.i I1I'

4.092 2.931 7.67129,1680 29,1680 30.16-

.0001 .0001—.062 26,453

.062 17.188

!irLL' c.!i°9Lp.'-ij_Th t Th

3.871 114029,1680 29,1600 30.18::.0001 .2001.241 -10. 702.099 2.557

Janiiarj 1972

10.800 l.lSo29,2400 29,2-tOO

.0031 .25.29.050 -2.743043 1 6t2

TABLE 14

E51;CATES AND TESTS WJTH EQUATION (11): LI1ZCNBEPGER AND RArSASWAME DEFINITION OF DIVIDEND YIELD

Equation (11): oi 8it + ''2I rf

1.tt stcse Freecioe

tOabiit5 LevelRestrictec Esti.iiatet

Standard Errors

F-StatisticsDegrees of EreeomProbability LevelRestricted EstimatesStanda-e Errors

F-StatisticsDegrees of FreedomProbability LevelRestricted Estlrn.atesStandard Errort

F—Statj sticsDegrees of FreedomProbability LevelRestricted EstimatesStandard Errors

F- Sta t is ticsDecrees cf FreedomProbability Level

Restricted EstimatesStandard Errors

F-StatisticsDegrees of FreedomProbability LevelRestricted EstimatesStandard Errors

U' 0

0%

to 0%

—

0% —

'-a

Cr a

o 00

Co

— '.

4 to

C

r Cr t

O 0

000%

*00%

tO

°0

0(,a

cr0o

a Jo

to

00

Cr

C

a a

Cr 0

% 'D

a —

to

to V

. C

r ".

4 -a

0.t *

00 tO

-'00

0'. C

r -a

Ct a

0% 0

% -.

4 to

w a

'.4—

to

La

0'.a

(4-

to to

o to

Cr C

r Co

Cr t

o 0

Na

0% C

O tO

O '.

4 to

(4(0

Cr C

r a —

Cr a

Na C

r -o

0) - t

o C

o tO

(40C

C '-

a -.

a4 C

r 0)

Cr (

0 C

r (0(

0 - U

' Cr a

a 0 -a

- U

'OO

'..4C

r(40

0)tO

w0%

0

Jo

too

'-a CO

0

-o C

r O Cr 0

) Na 0 - C

r *0

toC

r Co

4030

% 0

% —

rO C

r C

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.1

tICS7*;cet of FrecoonFab1ty Level2e1trlcteC EsticatesSta'Car Errors

F-$t4tj sticsDegrees of Freedorn

Pro,abflity Level7estricted EstimatesStanoard Errors

F—StatisticsDegre of Freeaoei

Probability Levelstricte Estiieate

Standard Errors

F—St jt is ticsDegrees of Freedci,,Probabfflty LevelRe5trcted EstiratesStandard Errors

F—Stat sticsDegrees cf FreecomProbabiflty LevelRestricted Esti.mateStandard errors

Decrees of Freedom°'obability Levelestrcted EstimatesStôndard Errør

r1926ce931

2.91229,2010 29,2010

.0001 .0031-1.213

39 .157

.hniary 1946 - December 1950

2.20529,1650 29,IISD

.0003 .0291-1.622 —.0122.190 .041

Janua!y 1961 — Deteinber 1955

1.680 2.09329,1650 29,1650

.0134 .0006—1.494 .499

.974 .181

January 1926 — DecefYter 1q31

1.383 5.53629,2010 29,2010

.0818 .00013.630 -.2951.154 .080

j31.616 1.096

29,1650 29,1650.0205 .3310

-.0444.899 .052

ar1961De4143

1.256 3.52229,1550 29,1650

.0001-8.902 .0132.302 .147

3Ji1.184 2.706

29,2010 29.2010.229 .0001

12.167 .02525.682 .0106

Januar.y 1951 - Deceeter 1955

!Li541.698 2.470

29,1650 29,1650.0001 .0001

-.0271.598 .075

1971

a

1.026 1.66729,2010 29,2010

.4282 .01442.902 .4101.008 .230

January 1935 — December 1940

1.201 4.00629,2010 29,2010

.1303 .0001-5.850 .15516.356 .065

January 195! — December 1955

1.029 2.28629,1650 29,1650

.0001 .0001.325 -3.991.062 2.50

January 1966 - Decemoer 1971

d.3.

1.775 2.56529,2010 29.2010

.1488 .00019.589 -.1642.715 .070

r1941er1g4:

5.417 2.55329,1550 29,1650

.0001 .00011.969 -.0293.955 .016

a a a.313j _3.638 3.079

29,1650 29,1560.0001 .0001

3.903 .005.714 . .108

January1972 - bece er 1978

°41 °4J

1.040 2.32929,2370 29,2370

.4071 .00311.755 .863

.750 .165

January 1941 - ecemer 1945

2.472 3.261129,1650 29,1650

.0001 .000123.364 .05810.604 .061

4.2.354 4.419

29.1650 29,;653.0001 .0001.777 .343

2.462 .119

Janarv 1972 — :ecE'-er

1.28629,2370 29,237

.1409 .0031

.034 -.0111.579 .042

TABLE IS

ELIPATE ABO TESTS 1Tfl EQUATjO (12):LJTZEN9(pGCp AND RA)lA.SC.J.IY DEFINITION OF DIVIDEUD YIELD

Equation (12): •+ 07j R 4 074 dt * 03 rf6 + 044

dit

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cc'. 0

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r '.0

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-45—

UH .g Litzenberger and Ramaswamy's definition of anticipated dividend

yields, :atiorIs (11) and (12) are estimated with two groups of thirty securities:

the Dow Jones lndusttials sample and a sample of utilities and railroads. The

utility—railroad sample provides a check against peculiarities of the Dow Jones

IndustriaLs, ad this s;:mpie can be presumed to have been identified by investors

as high—yield stocks. The test with these securities are conducted for the same

periods as the previous tests are remrted in Tables 14 and 15. These results

indicate that the earlier conclusions rc;arding th dividend yield coefficient are

basically unaffected by the definition of atcipaed dividend yields. The F-

statistics and restricted estimates change somewhat, but the changes, on

balance, are not large enough to alter our conclusions. Further, the utility-

railroad sample exhibits the same pattern as our earlier samples of high—ield

stocks, and the variation of the unrestricted estimates for the Dow Jones sample

is similar to those earlier under the perfect foresight definition. In sum, the

failure to find a common tax effect cannot plausibly be attributed to an

obscuring of the tax effect by announcement effects impounded in the perfect

foresight definition.

Dividend Yields and Shifts in Expected Returns

Tables 2 through 15 reveal a complex relation between expected returns

and dividend yields. The unrestricted estimates highlight this complexity. In

many reseets, our results are unsettling since we have rejected the two simplest

models of the impact of dividends: neutrality and common tax effects. In fact,

the data 11 us both of these are bad approximations. But not why.

One pcssible explanation is that dividend yields are proxying for other

factors that affect returns. Imagine, for example, a world where dividends are

"steady" in the sense of not being adjusted immediately by firms to maintain

constant dividend yields. A decrease in the price of a firm's shares, on average,

will thus imply a higher dividend yield and an increase in price will, on average,

imply a lower dividend yield. For leveraged firms, changes in stock prices may

also result in changes in the riskiness of the firm's stock. Unless firms make

compensating adjustments in leverage, a decrease in the price of common stock

implies a higher leverage ratio and an increase a lower, leverage ratio. As a

result, both dividend yields and expected returns may be increasing or decreasing

simultaneously. Stated differently, equations have been omitted, namely, the

ones determining expected returns, and dividend yields are proxying for the

omitted equations.

To see the possible impact of this relation, consider equation (11). The

intercept of equation (11) may be expressed as,

oit — oi + "itwhereu it is the difference between the expected return of security i in period t

and its average during the sample period (ignoring taxes). Substituting this

expression into eqution (11),

R =y +y d +y r +E v (lic)it oi ii it 2i ft it itThe disturbance term in equation (11) consists of the true disturbance plus the

deviation of the expected return in period t from its average during the sample

period.

Our previous discussion suggests that, on average, it and d1t are both

negatively related to stock prices and, therefore, positively related to each

other. Further, the relation is not easily modeled. The best that we can do is

present videnee that establishes the plausibility of the connection. Never-

theless, this evidence had important implications for intercepting tests of the

relation between dividend yields and expected returns (and perhaps also other

anomalies of asset pricing).

To test for this relationsip, we generalize equation (ii) to,

R + v r + y .d y d. + y .d. + (lid)I 01 ii ft 21 it 3i it—i 4a it—2 itwhere dt is the diiiiend yield in month t. The Litzenberger and Ramaswamy

definition of anticipated dividend yields is used in estimating (lid). In effect,

(lid) includes the dividend yield terms of (ii) in both the cum and cx months,

Since the cum month dividend yield is taken from the previous ex month, there

can be no information effects in the coefficient and a fortiori in the two—

month lag coefficient y4. Clearly, there are no tax effects of the kind

hypothesized by Litzenberger and Ramaswamy in these coefficients since the

dividend has already been paid. If the ex month is included, the cum month

effects are purely statistical artifacts from their point of view. The tests

conducted with (lid) are tests of no effect of current or lagged dividend yields.

If both lagged and current dividend yields turn out to be non-zero, there is reason

to believe that dividend yields are proxying for shifts in expected returns. The

results of these tests for the Dow Jones Industrial sample are reported in Table

16.

F-Statistics

Degrees of Freedom

Probability Level

F—Statistics

Degrees of Freedom

Probability

Level

F—Statistics

Degrees of Freedom

Probability Level

TABLE 16

TEST OF CON AND EX MONTH EFFECTS

Equation Ilc: Rj = - +

li rf + 2i

+ y

djt_

l + 4i

d1t2 + it

Restrictions

January 1926 —

Dec

embe

r 1931

=0

4j0

January 1935 —

Dec

embe

r 19

40

=0

=0

January

2i

0

Yj =0

(3j =0

4i0

'2i =0

3.948

30,1980

.0001

2.920

3.316

30,1980

30,1980

.0001

.0001

3.313

30,1980

.0001

4.184

4.404

29,1980

29,1980

.0001

.0001

1.953

30,1620

.0016

1.655

2.328

29,1620

29,1620

.0034

.0001

January 1946 —

Dec

embe

r 1950

January 1951 -

Dec

embe

r 1955

-y3

0

14j

0

January 1956-December

960

2i =0

=0

4i =

0

2i =0

2i =0

=0

Y4j

0

2.522

30,1620

.0001

3.686

1.584

30,1620

30,1620

.0001

.0236

4.304

30,1620

.0001

2.762

3.558

30,1620

30,1620

.0001

.0001

2.367

30,1620

.0001

2.334

2.501

30,1620

30,1620

.0001

.0001

January 1961 —

Dec

embe

r 1965

13i0

4i°

January 1966 — December 1971

3i°

'(4j0

January 1972 - D

ecem

ber1

978

12i0

'r210

y0

131_

U

yO

1.496

30,1620

3.209

1.986

30,1620

30,1620

2.740

30,1980

2.828

3.149

30,1980

30,1980

5.392

30,2340

4.680

4.994

30,2340

30,2340

-49-

As can be seen both the lagged and current dividend yields appear to be non-

zero in all time periods. It would be difficult, on the basis of these results, to

conclude either that the current values dominate the lagged value or vice-versa.

The current and lagged values both contribute to the explanation of expected

returns, thus warning again of the perils of attaching too much economic

significance to the observed relations between expected returns and dividend19

yields.

—50--

Cone lusions

The purpose of this paper has been to test certain hypotheses on the

relation between dividend yields and expected stock returns, using a methodology

that is more powerful in a number of respects that those that have so far been

used in that context. The method employs systems of time series regressions

with the competing hypotheses taking the form of cross equation restrictions.

The major finding of this paper is that the relation between dividend yields

and stock returns is not constant across securities. In almost all cases examined,

the hypothesis that dividends have a common effect on expected returns can be

decisively rejected. This conclusion is inconsistent with the tax effect models of

Litzenberger and Ramaswamy and Brennan or the similar model of Blume. But

the results are also inconsistent with the hypothesis that dividends have no

effect on expected returns. Stated differently, the tests show a statistically

significant relation between yields and returns, but one that is not well described

either by dividend neutrality or by tax effect models. The unresolved question is

what does explain the results? The observed structure of dividend payments

suggests that dividend yields may be proxying for changes in the expected return

of securities over time. This possibility was tested by including the lagged

dividend yield in cum months along with ex month dividend yield. The cum

month dividend yields have no impact on' investors' tax liabilities and have no

informaitonal content since they are lagged values. Nevertheless, the cum

month dividend yields appear to be as important statistically as the ex month

dividend yields. The cum month results emphasize the need for greater caution

in interpreting the dividend tests here and more generally. They should

especially discourage attempts to appeal to existing empirical tests for justifying

—51—

either the dividend policies of particular firms or the security selection policies

of portfolio managers.

ERRORS IN VARIABLES

52

53

ERRORS IN VARIABLES:A GENERAL DISCUSSION

Errors in estimationof beta are critical for two of the

estimation techniques used byTo analyze

the effect of errors inbeta estimates on the estimate of c, Consider

the general model,

(Al)

with observational errors in the independentvariables,

X =X+v

The model in theobserved variables is,

Vand the least

squares estimator is,

= (x'xy1 x' (x + U - V)

= + (x'xy1 xi (u - V)

Assume that the disturbances in equation(Al) are independent

of the independentvariables in a

Probability limit sense, i.e.,

plim X'u =N- N

Further, assume that

54

plim XVN N —

plim XXN N

plim XX — MN N

where M and M are finite positive definite matrices and,

plim V'V =N N

is finite. With these assumptions the probability limit of the least

squares estimator is,

plim = plim (X'x1 plim X'u - plimN-±e N- N N-* N N-° N

plim (Xv + V'V)(,A2N±= N N

since the inverse of is a continuous function which does not

depend on N.

For sake of discussion, possible errors in d1t and rft are

ignored. Equation (A2) may be used to evaluate the effect of errors

in beta on the estimates of c. First, note that for Litzenberger and

Ramaswamy X, V, XX and V'V correspond to,

55

1

d1-rf 0 v 0

1 N d-r 0 vn F

N N N

(d_r)i=1 i=1XIX = N N N

•i=1 i=1 i=1 1 1

N N N

E (drf) (d-r) (d-r.i=1 1=1 i=1

00

vIv i1 0

0 0

where is the estimated beta of security i and v is the estimation

error of . Suppose that,

plim : v2/N =

then,

56

=

r0 (A3)

b1 eb

_J

From equation (A3), it follows that the degree of inconsis-

tency of Litzeiberger and Ramaswaniys estimators will depend only

upon the second column of M'. For C, it will only depend upon the

last element of the second column. To evaluate the effect of c, note

that M equals,

1 M1 (6) tI1 (d_r)

M =111 M2 (6) M1 (6d_rf)

M1 (d-rf) M1 (6d—jrf) M2 (d-Jrf)

where M1(13) is the ith moment of the estimated beta around zero,

M1(d-Irf)is the ith moment of the dividend yields less the risk free

rate around zero and t.1(6d_Jrf) is ith cross moment around zero. The

(3,2) element of the inverse of M is,

m32 - IMH [M1(d-irf) - M1() Ml(dird)]

= - ML' M1(5- 4-a)

57

where M1(-d-) is the first cross moment around the respective

first moments of betas and dividend yields. From equation (A2) and

(A3) it follows that,

plim C = c-1-Mj M1(-id-j)]eb (A4)

= c+ IMP' eb

By assumption, M is positive definite so that M > 0. The

parameter b is a market after tax risk premium. If b>0, it follows

that

> 0 if > 0

plim c-c < 0 if M1(-d-a) < 0

= 0 i f M1 (8- id- ) = 0

It is easy to imagine circumstances where the cross moment around the

mean is not equal to zero. If the cross moment is positive, the

Litzenberger and Ramaswamy estimate will be too large in a probability

limit sense. If dividend yieldsare negatively related to beta, the

probability limit of the estimate will be less than c. While the

magnitude of this effect is unknown for the Litzenberger and

Ramaswamy sample, a serious suspicion is cast on the results

obtained with the '0LS and 'GLS' estimationtechniques.

58

ERRORS IN VARIABLES: LITZENBERGER AND RAMASWA1y5 ADJUSTMENT

Litzenberger and Ramaswamy propose an adjusted estimator Toanalyze this estimator, consider again the model shown iti equation(A7) with errors in variables. The least

squares estimator of thismodel may be restated as,

= (X' + V'X + XV + VvY1 (x' + X'u + V'X + Vu)

Taking probbiiity limits and utflizjno the preyious assumptions,

plim (plim+ urn \/X

+ p1ir + plim V'VN>< N N N N N- N N N

(plim Xx plim X1u + plim VX+ plirn Vu)N — N N N - N N- TE

(M +(A5)

If were known, it would be possibleto adjust x'x by subtracting ,

Litzenberger and Ramaswamy propose an estimator in the form of

(AS). Their GLS estimator is calculated bystandardizing the observa-

tions by the estimated standard errors of the beta estimates. Because

of this, they state that,

_o a o

:= 0 1 0

L 0 oJ

and their adjusted estimator takes the form,

r°

°* (x '

59

Recall the Litzenberger andRamaswamy claim that this estimator is both

consistent and the maximum likelihood estimator if the joint distribu-

tion of security returns is normal.

An Examination of Litzenber9er andRamaswamy's Claim of Consistency

It is obvious that the properties of this adjusted estimator

depond upon taking on the value assumed by the authors. If s is

the estimated standard error of , it is necessary that

N v2plimi ___ = 1N-* N i=l s2

for the claims of Litzenberger andRamaswaniy to hold. Suppose that

s' = for all i and each v' is normal with a mean of zero. Each

term in the above sum then would bechi-squared with one degree of

freedom. Further, if each of the estimation errors are independent of

all others, 1/N times the sum will converge to 1 as N goes to infinity.20

Thus, Litzenberger and Ramaswamy's claim that

1-00 0

E 0 1 0

La a a

requires two assumptions: (1) the true standard errors of all the beta

estimates must be known, and (2) each of the estimation errors must be

independent of all other.

These are by no means trivialassumptions, although the authors

fail to mention either of them. is simply written down without anydiscussion of the required

assumptions. Neither of these assumptions

60

has merit. Conceptually, there is no more reason to condition on= Q for all i than there is to assume 6 = . for all i. TheH 1 1

assumption of independent estimation errors, moreover, im1jes the

disturbances of the market modelto independent across securities, which

is inconsistent with I indutry' effects that ha'ie been documented. 21

In summary, the assumptions requi red for Litzenberger and Ramaswamv sadjusted esti mator to be cn i stent are arbi tracy and empi ri callyunattracti ye.

fhe Maximum Ltei ihood Claim

Litsenbergey' and Ramaswamy also claim that their adjustedesimator, maximum likelihood and, indeed, there are conditions wherefliaximrm likelihood estimators exist with errors in variables. Themost irnportt of these conditions is prior know'edge of some of theparameters, If the distriHtjon of the variables are normal, it iswell known that not all the parameters are identifiable. If isknown, the other parameters can be identified and the estimator willbe maximum likelihood if other conditions hold.

In particular, each

row of X must be normally, independently, and identically distributed.

If X is assumed to benonstochastic, then each row of the observed

variable will have a different mean vector (the value of each row ofX). But the X matrix is unobservable and there will be T unknown mean

vectors instead of one. In short, if X is assumed to benonstochastic,

there will be 1-1 more parametersto identify per independent variable.

All this must be accomplished withonly T observation per independent

variable.

In addition, it is required that each row of V and u be

normally, independently, and identically distributed with a mean vector

61

of zero. Finally, each u. and each row of X and V must be mutually

independent.

The assumptions made by Litzenberger and Ramaswamy are not

jointly consistent with these conditions. For purposes of estimating

betas, Litzenberger and Ramaswamy assume that the market model holds,

Rjt - rft = +(Rmt

- rf) + it (A6)

In the stochastic version of their model, equation (7), the distri-

bution is defined as,

=R1t

-E(Rt) (A7)

substituting equation (A6) into (A7),

ut = i mt - E(Rt)] + itIf , Rt and it are all normally distributed, U will be the pro-

duct of two normally distributed random variables, plus a normally

distributed random variable. While sums of normals are normal,

products of normals are not normal. If is normal, then , Rt,and it can not all be normally distributed. If either or it are

not normal, the estimator is not maximum likelihood.However, Rt

simply a linear combination of the Rt and the normality of the

implies the normality of Rmt Clearly, all these conditions can not

be met simultaneously. it follows, then, that Litzenberger and

Ramaswamy's assumptions are not consistent with those required for their

adjustment estimator to be maximum likelihood.

62

Besides arbitrarily conditioning on s and the inconsistency of

Litzenberger and Ramaswamysassumptiois with those required for a

maximum likelihood estimator theauthors ignore the fact that betas

and standard errors of betas are functions of parameters in the jointdistribution of security returns. It s rnpiy does not make sense toCall e timsrs of parame cr maximum 1 kel i hoed when a i the para-meters arc being simultaneously estimated. Tks point may heii 'ustrated by noting that

Litznberger and Rarnasworny' s Andel impi ies

that unoondi tional means 01 security returns are nonstationary due to

the tax effe't of dividepH, c(d.t - 1k: beta of security i

eouals,

E[R. -

E(R)] i -

In estimating betas, it is necessary to aJjustE(2.t) for the tax

effects in period t, but this adjustment requires an estimate of

In summary. Litzenberger and Ramaswamy's claims regarding their

adjusted estimator are wrong. It has been shown that both the con-

sistency and maximum likelihood properties of this estimator depend

upon an arbitrary conditioning argument that the authors fail even to

mention. Further, it was shown that thassumptions made

by Litzenberger and Ramaswamv are not consistent with those required

for their adjusted estimator to be maximum likelihood. Finally,

Litzenberger and Ramaswamy ignore the fact that the betas and standard

errors of beta are parameters in the joint distribution of security

returns. In short, there is no reason to believe that their adjusted

estimator is any better than the OLS and GLS estimators they report.

APPENDIX B

BLUME'S RANDOM COEFFICIENTS

63

64

BLUMES RANDOM COEFFICIENTS

Blume's random coefficients may be modeled,

bt = b +Ubt

= +

Substituting these expressions into Blume's model,

+ 6 it + Ed +Uit + + %t it + Ut dt

If betas are presumed to be constant for t = Ti, T1 + 1 . . . r, then

+y1 d + it

(Bi)

=uit + + i ÷ d Uct

'yl =

for i=l, 2, ... N and t =T1, T1 +1 . . i-l, T2.

If the disturbances of (Bi) are independent and identifically

distributed over time, standard techniques may be used to estimate

(Bi). Independence of the disturbance vector over time is assumed.

From the definitions of the terms in the disturbances, it follows that,

65

E vj. =

E vj= +

2ia + i °ib + dt + + 2( Gab + d1t Gac)

+ ? a + 2d.t °bc + d?t

for i = 1, 2 ... N and t =T1, T1

+ 1 . . . T2. To simplify, assume

that Gib = Gic=

Gab=

°ac=

Gbc= 0. The variance of then

reduces to,

(B2)

From (B2) it is clear that the disturbances of (Bi) are not homosced-

astic. Define the part of the variance of the disturbances that is

constant as,

E ? + 2 + 2 a2

Equation (B2) may be written as,

E = + dt G

= ? (1 + x. dt) (B3)

where = The changes in the variance of u over time depends

upon d. Since is the ratio of one variance to three variances,

it is probably reasonable to approximate A1 with a value of about .33.

An average quarterly dividend yield is approximately .01; for a high

yielding stock the value would be .02. Thus, in an ex month the

variance of the disturbance would be about ? (1.000033) for an average

firm, and (1.000132) for a high yielding firm. In a non ex month,

66

the variance would, of course, be . Differences of this size can

probably be safely ignored for purposes of estimation.

Turning attention to the stationarity of the covariances across

equations, note that,

it jt EUtUIt + EUitU t+

Eujtubt+ d.t EUtUt +

Eutu. + EUt Eutubt + dt Eu tut +

EuU. + i Euu + r. . Eubt + 6i djt

EubtUCt + dt EUCtU.t + d.t Euu + d1t J

EUCtUbt + d.t dt Eut (B4)

+ 2 + + dt dt

where use is made of the assumption of the mutual independence of

Uat Ubt Uct and u for afl i and t. The covariance of security i

and j varies over time due to the dt d.t term in (B4). This will be

approximately the same magnitude as the nonstationary of the variance

and, therefore, ignored for purposes of estimation.

FOOTNOTES

1. It is possible that a relation more complicated than the ones tested herecould exist. See, for example, Constantinides, G. [1979].

2. Actually c is a weighted average of investors' marginal tax rates where theweights equal the global risk tolerance divided by the sum of the global risktolerances across all investors, see Litzenberger, R., and Ramaswamy, K.,[1979, 171i't2].

3. lbid.,p.2.

4. Ibid., p. 15.

5. Only dividends paid in the last twelve months are used to calculate theunannounced but recurring dividends. See Litzenberger, R., and Rama-swamy, K., [1979, 182].

6. Litzenberger, R., and Ramaswamy, K., [1979, Table 1].

7. The reliability of this calculation is questionable. First, Black and Scholesmeasure dividend yield for any year as dividends paid in the previous year,divided by the closing price of the previous year. Tax liabilities will dependupon dividends actually paid and not the previous years dividends. Apossible differential tax liability will not be proportional to the Black andScholes measure of dividend yield. Second, Black and Scholes estimateusing both ex and non-ex months. Since tax effects only occur in ex months,the estimates mixes tax effects with other dividend related effects.Finally, Black and Scholes use an equally weighted index and not a valueindex. In short, the relation between estimate and the presumed dividendtax bracket of the marginal investor is by no means clear. The most thatcan be said is that a positive estimate in a Black and Scholes test isconsistent with differential taxation of dividends.

8. Blume apparently expected a negative coefficient, a preference by investorsfor dividends, see Blume, M., [1978, 1].

9. See, for example, Stambaugh, R., [1979].

10. Gordon, R. and Bradford, D. (1979) estimate a model similar to (12).Unfortunately, the authors never test any of the restrictions implied bytheir model.

11. See Hess, p. [1980], pp. 23—25.

12. Schmidt, P., [1976, 85).

13. The historical statutory rates are taken from Statistics of Income, IndividualIncome Tax Returns, U.S. Department of Treasury, various issues between1945—1969.

67

14. Theil, H., [1971, 313]

15. Ibid., p. 402.

16. Recently, attention has focused on the finite sampling properties of this teststatistic. Meisner has reported simulations that would sugge't the teststatistic is heavily weighted toward rejecting the restriction when thedegrees of freedom are small per equation. As expected, as the degrees offreedom grows per equation (23 per equation were exsmined by Meisner) thebias is substantially reduced. Since the degrees of freedom are quite largehere, the large sample aproximation will be assumed to be appropriate. SeeMeisner, J., [1979] and Laitinen, K., [1978].

17. The F—distribution is more onservative with respect to rejecting therestriction and that approximation is adopted here. See Theil. H., [1971,402-O3}.

18. Since none of our hypotheses involve dummy coefficients, these estimatesare not recorded.

19. This sample was also used to test for common cum month effects andcommon differences between cum and ex month effects. In general, theserestrictions were rejected at high probability levels.

20. This sum is distributed as chi-squared with N degrees of freedom. 1/N timesthe sum has a mean of 1 anda variance of 2/N. See Theil, H., [1971, 402].

21. See, for example, King, B. [19661 and Meyers, S., [1973].

22. See Schmidt, P., [1976, 105—112].

23. I am indebted to E. Han Kim for bringing this particular example to myattention.

68

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Blume, M. "Stock Returns and Dividends Yields: Some More Evidence." Un-published manuscript, October, 1978.

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Litzenberger, R., and Ramaswamy, K. "Dividends, Short Selling Restrictions, Tax-Induced Investor Clienteles, and Market Equilibrium." Journal of Finance(July, 1980)

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Miller, M. and Modigliani, F. "Dividend Policy, Growth and the Valuation ofShares." Journal of Business (October, 1961),

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