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http://www.archive.org/details/impactsofinvestm761leec
BR
FACULTY WORKING
PAPER NO.
761
Impacts of Investment Horizon on the Estimation of
Beta Coefficients, Jensen Measure and Efficient
Frontier:
Lognormal vs. Normal Distribution Approach
Cheng
F.
Lee
College of
Commerce and Business
Bureau
Economic and Business Research
of Illinois, Urbana-Champaign
of
University
Administration
^
I
I
1^5
I
BEBR
FACULTY WORKING PAPER NO. 761
College of Commerce and Business Administration
University of Illinois at Urbana-Champaign
March 1981
Impacts of Investment Horizon on the Estimation of
Beta Coefficient, Jensen Measure and Efficient
Frontier: Lognormal vs. Normal Distribution
Approach
Cheng F. Lee, Professor
Department of Finance
Acknowledgment
The author is grateful to 1979 summer research
support from the Investors in Business Education, College of
Commerce and Business Administration, University of Illinois at
Champaign-Urbana
:
UBRARY
I).
OF
I.
8RBAHA -CHAMPAIGN
Abstract
Based upon the lognormal distribution assumption the impacts of in-
vestment horizon on the estimation of beta coefficient, Jensen measure
and efficient frontier are analyzed in detail.
It is found that invest-
ment horizon is one of the most important factors in estimating beta coefficient, Jensen measure and efficient frontier.
the standard Jensen measure estimate is biased.
It is also shown that
An unbiased Jensen mea-
sure is derived in accordance with the property of intercept for a log-
linear regression.
Empirical results are used to support the analytical
results derived in this paper.
I.
Introduction
Portfolio theory developed by Markowitz (1957) and Tobin (1958,
1965) is essentially based upon the assumptions that portfolio's rates
of return are normally distributed and the investor's utility function
is quadratic.
Recently Elton and Gruber (1974) have developed a port-
folio theory in terms of lognormally distributed investment relatives.
Gressis, Philippatos and Hayya (1976) have shown that the investment
horizon will generally affect the portfolio analysis.
In addition, Levy
(1972), and Lee (1976A, 1976B) and Levhari and Levy (1977) have analyzed
the impact of investment horizon on the systematic risk estimates and
the investment performance measure estimates.
However, these analyses
have not explicitly taken the problem associated with alternative dis-
tribution assumptions into account.
The main purpose
of'
this paper is to analyze the possible impacts
of statistical distribution and investment horizon on the estimates of
beta coefficient, Jensen performance measure, and efficient frontier
parameter estimates of a portfolio.
Sharpe's (1963) diagonal model
is used to examine how the investment horizon can affect the composi-
tion of an efficient portfolio.
In the second section, a lognormal
capital asset pricing model (CAPM) with investment horizon parameter
is derived in accordance with the resiilts developed by Lee (1976) and
Bawa and Chakrin (1979); some portfolio theories on the efficient portfolio determination under a lognormal market are reviewed in accordance
with the results developed by Elton and Gruber (1974), Ohlson and Ziemba
(1976) and Levy (1973),
In the third section, the regression for
the bivariate lognormal distribution is discussed in accordance with
-2-
Heien (1968) and Goldberger (1968).
The implications of these results
on the Jensen investment performance measure in terms of investment ho-
rizon will be explored in some detail.
In the fourth section the im-
pacts of investment horizon on the beta estimates and the performance
measure estimates under both normal distribution and lognormal distribution capital asset pricing model will be explored and the implications
of these results to the efficient portfolio in terms of the Sharpe's
(1963) diagonal model will be examined.
Data of 45 firms randomly se-
lected from the New York Stock Exchange (NYSE) will be used to do some
To study the bias of Jensen measure, 464 firms will
empirical studies.
be used to do the empirical analysis.
Finally, the results of this pa-
per will be summarized and some concluding remarks will be indicated.
II.
Investment Horizon and the Lognormal CAPM
In a paper examining the relationship between investment horizon
and the systematic risk estimate, Lee (1976B) has derived a lognormal
CAPM in terms of a statistical concept.
Bawa and Chakrin (1979)
[BC]
have also derived a lognormal CAPM in terms of portfolio analysis identical to that derived by Lee (1976B) and drawn some possible implications
to the
risk-return relationship test.
However, BC do not consider the
problem associated with investment horizon.
Following Lee (1976B) and Jensen (1969), the risk-return relationship for the capital asset pricing model can be defined as
(1)
E( R.) =
H
H
R (l- 6j> + E
f
H
W HV
-3-
where H - the true investment horizon,
the rate of return on secu-
R.
rity j, „R » the market rate of return, JR,
HI
n m
the risk-free rate of in-
the system risk.
terest, and „6.
Equation (1) implies the risk return tradeoff is linear when the
true investment horizon is known.
generally is unknown.
However, the true investment horizon
If the observed horizon is defined as N, then the
relationship between JR. and
JR..
and JR
,
and JR
,
,
and JR, and JR
f
can
be defined as:
R* - (1 +
(2)
where
(3)
"
X
+ JR
R ) - (1
H ±
fi
- JR* A
X
i
)
(i - j, m,
f)
« H/N, after substituting (2) into (1), we have
[E( R*)]
A
- (JR*)* (1- g.) + [E( R*)]
H
N
N
X
Vj).
Equation (3) can be rewritten as:
(4)
e
[/f
r*) =
(n
x
u-
[e( r*)]
e.)
1/x
a
n
fi
h6j
.
]
If the observed rates of return, JR. and JR
are log normally dis-
tributed, following Aitchison and Brown (1957), we have
2
(5)
E(JR*) - exp(c /2 +
var(JR*) - exp(2u
where
p
» E(log JR
*
1
(a)
^)
+
a
2
[exp(aj)-l]
)
*
2
)
(b)
and a. - var(log JR ).
To apply the Cramer's rule to (5), we obtain
(6)
2
2
E(log JR*) = log[p R*/ /o R* +
w
var(log
m v
m
N
R
m)
m
2
2
y
R*]
= log[E(JR* )] - log[E
Mm
(a)
m
2
(Rj]
m
(b)
.
-4-
For deriving cov[(log vjR.)»
log
r»
(log
^
^.
and
be bivariate normally distributed with mean vector
jJl
u
(u n
1
,
W »
y~) and covariance matrix
Z
The bivariate distribution of R
i__
g(rlf r 2 )
(7)
= log
)], we let r
°11
a
°21
a
[
12
- v
]
-1
.
22
(r-, r_)' can be written as
,
exp {-i/2[(R- u ) V(R-u)]}.
2ir|wj
Using the definition of covariance and g(r., r ), it can be shown
2
that
2
(8)
cov[ R*,
N
From
(8)
,
- «p.(y
1
+
P
2
){exp[l/2(a
2
1
+o
2
+2cr
2
12
)] -
exp[l/2(o
2
1
+o
2
2
it can also be shown that
o
(9)
jjR*]
12
- logtCcovC^*,
R*)/exp(y
N
1
+ p^expd/ZCo^+Oj 2 ))) + 1].
After substituting equations (6a) and (6b) into (9), we have
o
(10)
12
=
logttK^Jx^)]}
From (5b) and (8)
,
- logClCjRj) e
r*)].
(n
the finite and the systematic risk can be de-
fined as
Derivation of the density function for a bivariate normal distribution can be found in Hogg and Craig (1969)
2
3
The derivation of this result is available from the author.
Following Jensen (1969) and Cheng and Deets (1973), the finite
systematic risk can be defined as:
N 6.
= cov( R.
N
,
,Jl
)/var(„R
).
) },
-5-
2
exp[l/2(a
(11)
= C,
8
J
where
C.
+a
1
{
2
2
+2a.,)] - «p[l/2(o- +c )]
£
i
2
±2-=
2
J
+
» exp(u,
}
[«p(oJ)-l]
2
+ a,).
u )/exp(2u,
2
Using the definition lognormal distribution, we have
E^*)
(12)
= exp(aJ/2 +
2
ECjjR*) =
exp(o /2 +
from equations (11) and (12)
^
(13)
u^
(a)
vj
(b)
we have
,
= (ECjjR^/ECjjR*))
Substituting equation (13) and
^
(14)
E
R*)
(h
(
eXp(a
12
)
- l)/(exp(aj) - 1)]
=1 into equation (4), we have
X
«p< g i2 )
R
: H f m
[(exp(a
2
)
1
-
j}
- 1
- HR f
5^0
]
[
E
y£>
Bawa and Chakrin (1979) have used equation (14) to explore the possible
Implications of the lognormal CAPM on the risk-return relationship tradeoff test.
If X is a nonnegative rational number, then both Jl
*X
and ^l
lognormal, hence
(15)
E( R
* X)
N
= exPt x2a 2 / 2
t
i
var^^)
E<jjR*
X
)
= exp(2X
yi
= exp[X 2 a 2 /2
2
varEjjR*^) = exp(2Xu
2
+
^1
X
+
+
2
^
(a)
2
)
[expCA
2
^
2
)
- 1]
(c)
Xu,,]
X
2
^
(b)
2
2
)
2
[exp(X a
2
)
- 1]
(d)
*X
are
-6-
Following the same procedure used in this section, we can obtain
3
=
^.{\)
(16)
[VJVeCj***)]
2
[exp(X a
12
- l/exp(A
)
2
2
)
- 1]
0;L
can be used to investigate the possible relationship between the
(A)
estimated systematic risk and the investment horizon.
This issue will
be explored in section IV of this paper.
III.
Jensen Performance Measure Under Bivariate Lognormal CAPM
Lee (1976A) has derived an approximate CAPM for Equation
(4)
in
terms of logarithm transformation as
log
(17)
N
R* - log
+
where „Y- " 1/2A
„(3.
HY j
^
(1° g
= a.
+
R
e!
(log
* 2
R
l0 S »*£ >
N m "
(1-8.) and
e.
N
R* - log
+
whether
A
y.
j
is the disturbance term.
H
y.
A
This spec-
is smaller than one.
of (17) can generally be used to test
is significantly different from zero.
the estimated
R*)
e
ification is subject to specification bias unless
Nevertheless, the estimated
N
If A is larger than one,
will be affected by the omitted higher order terms ex-
J
cept that all omitted terms are not correlated with the quadratic excess
market rate of return.
This is one source of specification error.
The
model defined in equation (17) has been used by Treynor and Mazuy (1966)
and Jensen (1972) in determining the unbiased measure of stock selection
ability.
Even the standard specification as defined in equation (14) is
correct, there still exists some problem in estimating so-called Jensen
performance measure in terms of a bivariate log normal regression as defined in Equation (18)
-7-
(18)
log(Z
where
Z
±
1
)
+
= o
^ l^
=
tt
t
log(Z
g.
and
with mean zero and variance a
\
2
=
+
)
e
N^tVft*
£
jt
±S normall y distributed
2
.
e
Based upon Heien (1968) and Goldberger (1968), the log-linear re-
gression as indicated in equation (18) can be written as
Z
(19)
where y
x
= y Z
J
exp(£
j
= anti log a.;
2
variance a..
)
2
e.
is normally distributed with mean zero and
Under this assumption, it is clear that
6
E(Z
(20)
X
)
2
= t Z J1 exp(l/2 o*
)
2
Both Goldberger (1968) and Heien (1968) have shown that the ordinary
least square (OLS) estimate of a
- 6
should be defined as
- 1/2 a~
(21)
a
= (log Z
where log
Z.
is average excess rates of return for jth security and
log
Z.
)
= a
- 111 a'
log Z„ is average excess market rates of return, therefore, the first
item of EHS in equation (21) is the so-called Jensen investment perfor-
mance measure plus the residual variance of jth security.
the unbiased estimated Jensen measure should be defined as
*
Unbiased Jensen Measure (UJM) = a. + 1/2 a
(22)
4
See Appendix A.
2
.
Therefore,
Equations (21) and (22) imply that the traditional Jensen measure esti-
mate is not independent of the non-systematic risk of an individual security or a portfolio.
In other words, a
biased against a security (or
portfolio) with larger residual variance (non-systematic risk).
Most
recently, Professor Roll (1978) has shown that the Jensen measure is
not an unambiguity measure for investment performance.
His conclusion
is essentially based upon the problem of index measurement.
Our results
here are based upon the assumption that the market index is free from
the measurement problem.
Merton (1970) and Jensen (1972, 385-386) have used the lognormal
and the discrete interval assumption to derive an equilibrium CAPM and
draw a similar conclusion to those derived in this section.
In sum,
Merton and Jensen concluded that the theoretical intercept of CAPM is
2
2
- 1/2 [c. - g.a ].
J
J
m
If Merton and Jensen's results are used,
then the
unbiased Jensen measure can be defined as
a
(22*)
2
Unbiased Jensen measure = a. + l/2[a. -
go 2
]
The difference between Merton and Jensen's results and the result de-
rived in this paper is to be studied in the future research.
IV.
Investment Horizon, Systematic Risk, Jensen Measure
and Efficient Portfolio
Cheng and Deets (1973), Levhari and Levy (1976) and Lee (1976A,
1976B) have shown that investment horizon does have some impact on
the estimates of systematic risk; Levy (1972) and Levhari and Levy
C1977) have demonstrated that both Sharpe investment performance measure
-9-
and Treynor performance measure can be affected by the investment horizon.
By using the Markowitz (1959) model, Gressis, Philippatos and
Hayya (1976) have shown that the efficient portfolio determination is
However, the lognormal dis-
not independent of the investment horizon.
tribution assumption has not been explicitly integrated with investment
horizon to investigate the impact of investment horizon on the estimated
systematic risk and the estimated investment performance measures.
The relationship between the estimated systematic risk and invest-
ment horizon under a lognormal assumption can be analyzed in accordance
There exist two components,
with the results indicated in equation (16).
A
i.e., [E( R* )/E( R*
N
N
the change of
(13).
A
2
X
and [exp(A a
)]
If A = 1,
g.(A).
12
l/exp^
-
2
^
2
- 1] to affect
)
then equation (16) reduces to equation
S.(A) with respect to the change of
then the change of
If X / 1,
)
can be analyzed as follows.
From equations (15a) and (15c)
2
E( R*
(23)
X
)
N
-fi-jL-
we have
,
2
°1 +X »1
/
£——
2
A
2
(a
2
-o
1
2
)
+ (u -p
1
)A
2
]
e
*A
E( R
N
Taking derivative of
E
(24)
"3A
[
—
fe"]
N m )
>
i
~-r—
<A
with respect to
then
>
= e
E( R
In equation (24),
A,
[X(0
1
~ a2
>
+
<V U 2 )]
the first item is always larger than zero, hence, the
sign is determined by the second term.
equation, it can be shown that
From the second tarn cf this
-10-
(i)
when
X
increases,
j^—] will increase if
[-
X >
a. 2
l-a 2 2
V?>
when
(ii)
X increases,
jr—] will reduce if
[
E
Vm
X
^2 -li l^
X < "
a
>
Now, the relationship between
and
_e
2
" 2
l
_a
2
is explored.
X
It can
-1-1
be shown that
,2
A
* -1)J
(e
(25)
,2
A O.
O. „
[2Xcx,,e
^""12
"][e
3X
i
.2
2
X
A
-
2
-l]-[2Xo/e
2* 2
-1)
(e
The numerator of equation (24) can be simplified as
(a-b) -
2X[c
(26)
a^(a-c)]
i2
X
2
(a
a = e
where
,2
12
+o
2
1
)
2
b = e
X
c a e
2
a
12
The sign of equation (26) is jointly determined by X
iu is a non-linear function of these variables.
equation (25) can hardly be determined.
2
and X a
2
2
,
o-
2
and
2
a.
and
Therefore, the sign of
2
If the magnitude of both X a
are relatively small, it can be shown that
,2
AC
0".
X
]
[e
'
-112
X
2
2
q
X
-1) is approximately equal to
-l)/(e
(e
—
a
a.
12 5
Under this cir-
j.
"l
cumstance, the impact of X on N 8(X) is essentially determined by the
term
*X
[E(-jR.
,
)/E(^l
*X
From equation (24), it has been shown that the
)].
relationship between
2
and (u_-u .) / (a. -a
X
2
)
is the key factor in deter-
mining the change of beta coefficient as the observation unit change.
2
If the magnitude of X o
tionship between
and
X
2
and X
N B(X)
2
a-
?
is not negligible,
becomes ambiguous.
then the rela-
6
To investigate the impact of investment horizon on the efficient
portfolio, 45 securities as listed in Appendix B are randomly selected
from the New York Stock Exchange (NYSE)
January, 1966-December, 1979.
in equation (26)
.
The sample period is during
Sharpe's (1963) diagonal model as defined
is used to formulate the efficient portfolio for 12
alternative investment horizons.
(26)
N Jt
w
j
jNmt
jt
„R.^ = rates of return for jth security in period
N jt
where
= market rates of return in period
„R
N mt
a. and 3.
t
are intercept and slope for jth security.
sents the observation horizon for R._ and R .
mt
Jt
the N represents either one month, two months,
t
Note that N repre-
In this empirical study,
..., or twelve months.
—
The information required for Sharpe's diagonal model is ^a^j N B.» N o.
and
a
2
.
2
The compositions of all possible efficient portfolios
See Apnendix B for the derivation.
*
Tnis is due to the fact that the higher order terms are not negligible.
-12-
under twelve different investment horizons are listed in Appendix C.
Appendix C indicates that number of possible efficient portfolio is not
independent of the investment horizons used to construct the efficient
These results also indicate that the estimated parameters
frontier.
for efficient portfolio are not independent of the different observation
horizons used (see Table
g
I)
.
The main reason for this difference is
due to the fact that the estimated beta coefficients are not independent
of observation horizons.
The results of estimated beta for 12 different
investment horizons are listed in Table II.
to those found by Cheng and Deets
others.
These findings are similar
(1973), Levhari and Levy (1977) and
These results have supported the results as indicated in equa-
tion (16).
To investigate how the bias associated with the estimated OLS
Jensen measure, data of 464 firms during the periods of 1966-72 and
1972-79 are used to estimate the bias Jensen measure.
Fisher index is
used to calculate the market rates of return and the monthly 90 day
treasury bill rate is used as the proxy of risk-free rate.
Both the stan-
dard Jensen measure and the unbiased Jensen measures as defined in equations
(22) and (22') are calculated.
The summary results of these three dif-
ferent kinds of Jensen measures are listed in Table IV.
ot
,
1
a
2
,
In table III,
and a^ are average biased and unbiased estimated Jensen measures
respectively.
By comparing
a.
with a_ and a,, it can be concluded that
45 firms used in investigating the impacts of investment horizon
on the composition of efficient frontier is randomly selected for these
464 firms.
g
The portfolios listed in Table I is one of possible set of efficient portfolios listed in Appendix C.
-13-
Table I
Estimated Parameters of Efficient Portfolios
Number
of
Periods, N
1
2
3
4
5
6
7
8
9
10
11
12
Coefficient of
variation
M(k)
a GO
.00030
.00312
.00357
.00849
.01040
.01270
.01549
.01998
.00629
.05316
.01170
.04140
.03241
.04544
.05871
.05404
.06504
.07375
.07095
.06620
.07643
.07625
.09079
.08693
<vv
108.03
14.56
16.45
6.37
6.25
4.76
4.58
3.31
12.15
1.43
7.75
2.10
Number of securities
included in the
portfolio
13
17
11
14
13
12
14
13
11
16
8
16
-14-
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-16-
Table III
1966-72
-.00223
.00738
.00042
.00808
.00151
.00839
1973-79
.00093
.00898
.00422
.01060
.00573
.01105
Jensen Performance Measure
a
l
= a±
a
3
+ ll2
\]
+ 1/2 [a 2 .
Bj <£]
-17-
the standard OLS Jensen measure estimate is an under -estimated Jensen
measure.
Therefore, the standard Jensen measure estimate does not
fairly evaluate the performance of mutual fund, portfolio and individual securities.
To study the possible ranking bias of using standard
biased Jensen estimates instead of unbiased Jensen measure estimates.
Both ranking correlation and product-moment correlation are used to
A
check the ranking differences caused by using
A
A
o,
instead of a„ and ou.
It is found that all correlation coefficients are higher than .945.
These results imply that the standard Jensen measure estimate can still
be used to rank the performance of mutual fund, portfolios and individual
securities.
V.
Summary
In this paper, the empirical relationships between investment hori-
zon and portfolio analysis are analyzed in accordance with both normal
and lognormal assumptions.
Especially, the lognormal distribution is
used to investigate possible impact of investment horizon on the esti-
mated beta in terms of an analytical functional relationship.
In addi-
tion, the possible bias of estimated Jensen measure in terms of a bi-
variate log linear specification is analyzed.
Finally, some empirical
efficient portfolio estimates in terms of different investment horizons
are used to show that investment horizon is an Important factor in portfolio analysis.
It is also shown that the standard Jensen measure does
exhibit estimation bias.
-18-
References
Aitchison, J., and J. A. C. Brown (1957). The Log Normal Distribution .
Cambridge: Cambridge University Press.
"Optimal Portfolio Choice and
Bawa, V. S., and L. M. Chakrin (1979).
Equilibrium in a Lognormal Securities market," Management Science ,
forthcoming
"Systematic Risk and the Horizon
K. Deets (1973).
, and M.
Problem," Journal of Financial and Quantitative Analysis , 8, pp.
299-316.
Cheng, P. L.
"Portfolio Theory When InvestElton, E. J., and M. J. Gruber (1974).
Journal of Finance ,
Distributed,"
are
Lognormally
ment Relatives
1265-1273.
29, pp.
Goldberger, A. S. (1968). "On the Interpretation and Estimation of
Cobb-Douglas Functions," Econometrica , 35, pp. 464-472.
Gressis, N. , G. C. Philippatos, and J. Hayya (1976). "Multiperiod
Portfolio Analysis and Inefficiency of the Market Portfolio,"
Journal of Finance , 31, pp. 1115-1126.
"A Note on Log-Linear Regression," Journal of
Helen, D. M. (1968).
American Statistical Association , 63, pp. 1034-1038.
Introduction to Mathematical Statistics
Hogg, R., and A. Craig (1969).
2nd ed. New York: John Wiley and Sons.
"Risk, the Pricing of Capital Assets, and the
Jensen, M. C. (1969).
Evaluation of Investment Portfolio," Journal of Business , April,
pp. 167-247.
Jensen, M. C. (1972), "Optimal Utilization of Market Forecasts and the
Evaluation of Investment Performance," Szegol/Shell (eds.),
Mathematical Methods In Investment and Investment (North Holland,
Amsterdam)
"Capital Markets: Theory and Evidence," The Bell Journal
,
of Economics and Management Science , 3, pp. 357-398.
"Investment Horizon and the Functional Form of
Lee, Cheng F. (1976A).
the Capital Asset Pricing Model," The Review of Economics and
Statistics , 58, pp. 356-363.
"On the Relationship Between the Systematic
Lee, Cheng F. (1976B).
Risk and the Investnent Horizon," Journal of Financial a t ,d Quantitative Analysis , 11, pp. 803-815.
.
-19-
Levhari, D., and H. Levy (1977).
"The Capital Asset Pricing Model and
the Investment Horizon," The Review of Economics and Statistics ,
49, pp. 92-102.
"Stochastic Dominance Among Lognormal Prospects,"
(1973).
International Economic Review , 14, pp. 601-614.
Levy, H.
"Portfolio Performance and Investment Horizon,"
Management Science , August, pp. 645-653.
Levy, H. (1972).
Markowitz, Harry M. (1957). Portfolio Selection .
Monograph 16, John Wiley and Sons, New York.
Cowles Foundation
Merton, R. C. (1970), "A Dynamic General Equilibrium Model of the Asset
Market and Its Application to the Pricing of the Capital Structure
of a Firm." Working paper No. 497-70, Sloan School of Management
School, MIT.
Ohlson, J. A., and W. T. Ziemba (1976).
"Portfolio Selection in a Lognormal Market When the Investor Has a Power Utility Function,"
Journal of Financial and Quantitative Analysis , 11, pp. 57-71.
"Ambiguity When Performance is Measured by the Securities
(1978).
Market Line," Journal of Finance , 33, pp. 1051-1069.
Roll, R.
"A Simplified Model for Portfolio Analysis,"
(1963).
Management Science , January, pp. 277-293.
Sharpe, W. F.
Tobin, J. (1965).
"The Theory of Portfolio Selection," in F. H. Hahn
and F. P. R. Brechlin (Eds.), The Theory of Interest Rates , London:
MacMillan, pp. 3-51.
Tobin, J. (1958).
"Liquidity Preference as Behavior Towards Risk,"
The Review of Economic Studies , 25, pp. 65-86.
Treynor, J. L. and K. K. Mazuy (1966).
"Can Mutual Funds Outguess the
Market?" Harvard Business Review, July-August, pp. 131-136.
M/E/179
-20-
APPENDIX A
If the regression of equation (18) is a normal linear regression
with the assumption that each
2
e.^.
jt
is N(0,o ).
e
theory it is known that if e. is normal then e
this circumstance, the expected
Z
From the distribution
J
is log-normal.
Under
as indicated in (20) is not equal
To make the expected disturbance for the multiplicative
to Y.Z.6..
specification as indicated in equation (19) equal to one, equation (19)
can be reparameterized as
(i)
z[ = Y Z exp(e
j 2
- l/2o*
:j
Equation (i) implies that
*
E(Z^) = Y Z E[Exp( £j - l/2c*)]
± 2
= Y Z
J 2
Taking the log transform of equation (i), we have
i
(ii)
log Z » a
1
where
u. = log e.
J
.
+
3.
- l/2a
u.
2
+
v
±
2
8
3
In equation (ii),
log Z
is N(-l/2a
2
2
,
a ).
If we applied the ordinary least
square (OLS) estimation method to equation (ii)
,
we actually estimate
the specification of equation (iii) instead of (ii)
(iii)
log Z
1
where a
= a
+
g
log Z
= a. - l/2a
.
2
+ log
e
= (log Z^ - 6. log Z
2
)
-21-
This is equation (21)
.
Further discussion on the issues discussed here
can be found in Heien (68) and Goldberger (1968).
-22-
APPENDIX B
2
\ a
e
A
e
2
12 l = \ a
2
u
2
+ yfCA
a
2
12 )
+
^\
+
1 ,.2„ 2.3
jj-(X a
)
1
2
2
)
+
.-•
(a)
2
°1
.
- 1 - X,2 a,
2
.
+
2
2
2.2
1 ,,2
yj-(X aj^ )
If A a., and X cn
2
,2
U
a. _
12 - ,
12
1 _
,2
e
l
*
2
a
,
- 1
.
+
...
,. v
(b)
approach zero, then the higher order terms are
negligible, and therefore
e
.
2
l
-23-
APPENDIX B
Sample List for 45 Firms
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
43.
44.
45.
NWT
TWA
PBI
WLA
SMC
JOY
BGG
TF
FIR
PVE
PFE
GR
CLL
ACF
UCL
HML
AD
RVS
TXT
NG
IGL
DG
REP
DE
JWL
CK
SUO
MLR
CCC
RHR
HFC
SOH
LOU
PAC
NFK
RCC
RGS
JCP
TU
CIT
ENS
NMK
NAS
AIC
SUN
Northwest Inds Inc
Trans World Airls Inc
Pitney Bowes Inc
Warner Lambert Co
Smith A
Corp
Joy Mfg Co
Briggs & Stratton Corp
Twentieth Centy Fox Film Corp
Firestone Tire & Rubr Co
Phillips Van Heusen Corp
Pfizer Inc
Goorich B F Co
Continental Oil Co
A C F Inds Inc
Union Oil Co Calif
Hammermill Paper Co
Amsted Inds Inc
Reeves Bros Inc
Textron Inc
National Gypsum Co
International Minerals & Chem
Associated Dry Goods Corp
Republic Corp
Deere & Co
Jewel Cos Inc
Collins & Aikman Corp
Sheel Oil Co
Midland Ross Corp
Continental Group Inc
Rohr Inds Inc
Household Fin Corp
Standard Oil Co Ohio
Louisville Gas & Elec Co
Pacific Tel & Teleg Co
Norfolk & Westn Ry Co
Royal Crown Cola Co
Rochester Gas & Elec Corp
Penney J C Inc
Trans UN Corp
C I T Final Corp
Enserch Corp
Niagara Mahawk Pwr Corp
National Svc Inds Inc
American lavt Co
Sun Inc
-24-
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