Introduction to Financial Econometrics
Chapter 6 The Single Index Model and Bivariate
Regression
Eric Zivot University of Washington
Department of Economics
March 1, 2001


1 The single index model
Sharpe s single index model, also know as the market model and the single factor
model, is a purely statistical model used to explain the behavior of asset returns.
It is a generalization of the constant expected return (CER) model to account for
systematic factors that may a¬ect an asset s return. It is not the same model as the
Capital Asset Pricing Model (CAPM), which is an economic model of equilibrium
returns, but is closely related to it as we shall see in the next chapter.
The single index model has the form of a simple bivariate linear regression model

(1)
Rit = ±i + β i,M RMt + µit , i = 1, . . . , N ; t = 1, . . . , T

where Rit is the continuously compounded return on asset i (i = 1, . . . , N) between
time periods t ’ 1 and t, and RMt is the continuously compounded return on a
market index portfolio between time periods t ’ 1 and t. The market index portfolio
is usually some well diversi& portfolio like the S&P 500 index, the Wilshire 5000
ed
index or the CRSP1 equally or value weighted index. As we shall see, the coe¬cient
β i,M multiplying RMt in (1) measures the contribution of asset i to the variance
(risk), σ 2 , of the market index portfolio. If β i,M = 1 then adding the security does
M
not change the variability, σ 2 , of the market index; if β i,M > 1 then adding the
M
security will increase the variability of the market index and if β i,M < 1 then adding
the security will decrease the variability of the market index.
The intuition behind the single index model is as follows. The market index RMt
captures macro or market-wide systematic risk factors that a¬ect all returns in one
way or another. This type of risk, also called covariance risk, systematic risk and
1
CRSP refers to the Center for Research in Security Prices at the University of Chicago.


1
market risk, cannot be eliminated in a well diversi& portfolio. The random error
ed
term µit has a similar interpretation as the error term in the CER model. In the single
index model, µit represents random news that arrives between time t ’ 1 and t that
captures micro or & rm-speci& risk factors that a¬ect an individual asset s return
c
that are not related to macro events. For example, µit may capture the news e¬ects
of new product discoveries or the death of a CEO. This type of risk is often called
& speci& risk, idiosyncratic risk, residual risk or non-market risk. This type of
rm c
risk can be eliminated in a well diversi& portfolio.
ed
The single index model can be expanded to capture multiple factors. The single
index model then takes the form a k’variable linear regression model

Rit = ±i + β i,1 F1t + β i,2 F2t + · · · + β i,k Fkt + µit

where Fjt denotes the j th systematic factorm, β i,j denotes asset i0 s loading on the j th
factor and µit denotes the random component independent of all of the systematic
factors. The single index model results when F1t = RMt and β i,2 = · · · = β i,k = 0. In
the literature on multiple factor models the factors are usually variables that capture
speci& characteristics of the economy that are thought to a¬ect returns - e.g. the
c
market index, GDP growth, unexpected in! ation etc., and & speci& or industry
rm c
speci& characteristics - & size, liquidity, industry concentration etc. Multiple
c rm
factor models will be discussed in chapter xxx.
The single index model is heavily used in empirical & nance. It is used to estimate
expected returns, variances and covariances that are needed to implement portfolio
theory. It is used as a model to explain the normal or usual rate of return on an
asset for use in so-called event studies2 . Finally, the single index model is often used
the evaluate the performance of mutual fund and pension fund managers.

1.1 Statistical Properties of Asset Returns in the single in-
dex model
The statistical assumptions underlying the single index model (1) are as follows:

1. (Rit , RMt ) are jointly normally distributed for i = 1, . . . , N and t = 1, . . . , T .

2. E[µit ] = 0 for i = 1, . . . , N and t = 1, . . . , T (news is neutral on average).

3. var(µit ) = σ 2 for i = 1, . . . , N (homoskedasticity).
µ,i

4. cov(µit , RMt ) = 0 for i = 1, . . . , N and t = 1, . . . , T .
2
The purpose of an event study is to measure the e¬ect of an economic event on the value of a &rm.
Examples of event studies include the analysis of mergers and acquisitions, earning announcements,
announcements of macroeconomic variables, e¬ects of regulatory change and damage assessments
in liability cases. An excellent overview of event studies is given in chapter 4 of Campbell, Lo and
MacKinlay (1997).

2
5. cov(µit , µjs ) = 0 for all t, s and i 6= j
6. µit is normally distributed
The normality assumption is justi& on the observation that returns are fairly
ed
well characterized by the normal distribution. The error term having mean zero
implies that & speci& news is, on average, neutral and the constant variance
rm c
assumptions implies that the magnitude of typical news events is constant over time.
Assumption 4 states that & speci& news is independent (since the random variables
rm c
are normally distributed) of macro news and assumption 5 states that news a¬ecting
asset i in time t is independent of news a¬ecting asset j in time s.
That µit is unrelated to RMs and µjs implies that any correlation between asset i
and asset j is solely due to their common exposure to RMt throught the values of β i
and β j .

1.1.1 Unconditional Properties of Returns in the single index model
The unconditional properties of returns in the single index model are based on the
marginal distribution of returns: that is, the distribution of Rit without regard to any
information about RMt . These properties are summarized in the following proposition.
Proposition 1 Under assumptions 1 - 6
1. E[Rit ] = µi = ±i + β i,M E[RMt ] = ±i + β i,M µM

2. var(Rit ) = σ 2 = β 2 var(RMt ) + var(µit ) = β 2 σ 2 + σ 2
i i,M i,M M µ,i

3. cov(Rit , Rjt ) = σ ij = σ 2 β i β j
M

4. Rit ˜ iid N(µi , σ 2 ), RMt ˜ iid N (µM , σ 2 )
i M
cov(Rit ,RMt ) σ iM
5. β i,M = = σ2
var(RM t ) M

The proofs of these results are straightforward and utilize the properties of linear
combinations of random variables. Results 1 and 4 are trivial. For 2, note that
var(Rit ) = var(±i + β i,M RMt + µit )
= β 2 var(RMt ) + var(µit ) + 2cov(RMt , µit )
i,M
= β 2 σ2 + σ2
i,M M µ,i

since, by assumption 4, cov(µit , RMt ) = 0. For 3, by the additivity property of
covariance and assumptions 4 and 5 we have
cov(Rit , Rjt ) = cov(±i + β i,M RMt + µit , ±j + β j,M RMt + µjt )
= cov(β i,M RMt + µit , β j,M RMt + µjt )
= cov(β i,M RMt , β j,M RMt ) + cov(β i,M RMt , µjt ) + cov(µit , β j,M RMt ) + cov(µit , µjt )
β i,M β j,M cov(RMt , RMt ) = β i,M β j,M σ 2
= M


3
Last, for 5 note that

cov(Rit , RMt ) = cov(±i + β i,M RMt + µit , RMt )
= cov(β i,M RMt , RMt )
= β i,M cov(RMt , RMt )
= β i,M var(RMt ),

which uses assumption 4. It follows that

β var(RMt )
cov(Rit , RMt )
= i,M = β i,M .
var(RMt ) var(RMt )


Remarks:

1. Notice that unconditional expected return on asset i, µi , is constant and con-
sists of an intercept term ±i , a term related to β i,M and the unconditional
mean of the market index, µM . This relationship may be used to create pre-
dictions of expected returns over some future period. For example, suppose
±i = 0.01, β i,M = 0.5 and that a market analyst forecasts µM = 0.05. Then the
forecast for the expected return on asset i is

b
µi = 0.01 + 0.5(0.05) = 0.026.

2. The unconditional variance of the return on asset i is constant and consists of
variability due to the market index, β 2 σ 2 , and variability due to speci& risk,
c
i,M M
σ2 .
µ,i

3. Since σ ij = σ 2 β i β j the direction of the covariance between asset i and asset j
M
depends of the values of β i and β j . In particular

• σ ij = 0 if β i = 0 or β j = 0 or both
• σ ij > 0 if β i and β j are of the same sign
• σ ij < 0 if β i and β j are of opposite signs.

4. The expression for the expected return can be used to provide an unconditional
interpretation of ±i . Subtracting β i,M µM from both sides of the expression for
µi gives
±i = µi ’ β i,M µM .




4
1.1.2 Decomposing Total Risk

The independence assumption between RMt and µit allows the unconditional vari-
ability of Rit , var(Rit ) = σ 2 , to be decomposed into the variability due to the market
i
2
index, β i,M σ M , plus the variability of the & speci& component, σ 2 . This decom-
2
rm c µ,i
position is often called analysis of variance (ANOVA). Given the ANOVA, it is useful
to de& the proportion of the variability of asset i that is due to the market index
ne
and the proportion that is unrelated to the index. To determine these proportions,
divide both sides of σ 2 = β 2 σ 2 + σ 2 to give
i i,M M µ,i


β 2 σ2 + σ2 β 2 σ2 σ2
σ2 i,M M µ,i i,M M
+ µ,i
i
1= 2 = =
σ2 σ2 σ2
σi i i i

Then we can de&ne
β 2 σ2 σ2
i,M M
= 1 ’ µ,i
2
Ri =
σ2 σ2
i i
as the proportion of the total variability of Rit that is attributable to variability in
the market index. Similarly,
σ2
1 ’ Ri = µ,i
2
σ2i
is then the proportion of the variability of Rit that is due to & speci& factors. We
rm c
2
can think of Ri as measuring the proportion of risk in asset i that cannot be diversi&ed
2
away when forming a portfolio and we can think of 1’Ri as the proportion of risk that
2
can be diversi& away. It is important not to confuse Ri with β i,M . The coe¬cient
ed
2
β i,M measures the overall magnitude of nondiversi& able risk whereas Ri measures the
proportion of this risk in the total risk of the asset.
2
William Sharpe computed Ri for thousands of assets and found that for a typical
stock R2 ≈ 0.30. That is, 30% of the variability of the return on a typical is due
i
to variability in the overall market and 70% of the variability is due to non-market
factors.

1.1.3 Conditional Properties of Returns in the single index model
Here we refer to the properties of returns conditional on observing the value of the
market index random variable RMt . That is, suppose it is known that RMt = rMt . The
following proposition summarizes the properties of the single index model conditional
on RMt = rMt :

1. E[Rit |RMt = rMt ] = µi|RM = ±i + β i,M rMt

2. var(Rit |RMt = rMt ) = var(µit ) = σ 2
µ,i

3. cov(Rit , Rjt |Rmt = rMt ) = 0

5
Property 1 states that the expected return on asset i conditional on RMt = rMt
is allowed to vary with the level of the market index. Property 2 says conditional
on the value of the market index, the variance of the return on asset is equal to the
variance of the random news component. Property 3 shows that once movements in
the market are controlled for, assets are uncorrelated.

1.2 Matrix Algebra Representation of the Single Index Model
The single index model for the entire set of N assets may be conveniently represented
using matrix algebra. De& the (N — 1) vectors Rt = (R1t , R2t , . . . , RNt )0 , ± =
nie
(±1 , ±2 , . . . , ±N ) , β = (β 1 , β 2 , . . . , β N )0 and µt = (µ1t , µ2t , . . . , µNt )0 . Then the single
0

index model for all N assets may be represented as
«  «  «  « 
R1t ±1 β1 µ1t
¬ . ·=¬ . ·+¬ . ·R + ¬ . · , t = 1, . . . , T
¬ .·¬ .·¬ . · Mt ¬ .·
. . . .
 
RNt ±N βN µNt
or
Rt = ± + β · RMt + µt , t = 1, . . . , T.
Since σ 2 = β 2 σ 2 + σ 2 and σ ij = β i β j σ 2 the covariance matrix for the N
i i,M M µ,i M
returns may be expressed as
«  « « 
β 2 σ2 β i β j σ2 · · · β iβ j σ2 σ2
σ2 ···
0 0
σ 12 · · · σ 1N i,M M M M µ,1
1
¬ · ¬ ·¬ ·
2
0 σ2
σ 2 · · · σ 2N 2 2
· · · β iβ j σ2 ···
σ 12 β i β j σ M β i,M σ M 0
¬ · ¬ ·¬ ·
µ,2
2 M
Σ=¬ ·=¬ ·+¬ ·
. . .. . . . . . . .
...
...
¬ · ¬ ·¬ ·
. . . . . .
. . .
.
. . . . . .
. . .
    
· · · β 2 σ2
· · · · · · σ2 · · · σ2
β iβ j σ2 β i β j σ2
σ 1N 0 0
N µ,N
M M i,M M

The covariance matrix may be conveniently computes as

Σ = σ 2 ββ 0 + ∆
M

where ∆ is a diagonal matrix with σ 2 along the diagonal.
µ,i


1.3 The Single Index Model and Portfolios
Suppose that the single index model (1) describes the returns on two assets. That is,

(2)
R1t = ±1 + β 1,M RMt + µ1t ,
(3)
R2t = ±2 + β 2,M RMt + µ2t .

Consider forming a portfolio of these two assets. Let x1 denote the share of wealth
in asset 1, x2 the share of wealth in asset 2 and suppose that x1 + x2 = 1. The return


6
on this portfolio using (2) and (3) is then

Rpt = x1 R1t + x2 R2t
= x1 (±1 + β 1,M RMt + µ1t ) + x2 (±2 + β 2,M RMt + µ2t )
= (x1 ±1 + x2 ±2 ) + (x1 β 1,M + x2 β 2,M )RMt + (x1 µ1t + x2 µ2t )
= ±p + β p,M RMt + µpt

where ±p = x1 ±1 + x2 ±2 , β p,M = x1 β 1,M + x2 β 2,M and µpt = x1 µ1t + x2 µ2t . Hence,
the single index model will hold for the return on the portfolio where the parameters
of the single index model are weighted averages of the parameters of the individual
assets in the portfolio. In particular, the beta of the portfolio is a weighted average
of the individual betas where the weights are the portfolio weights.

Example 2 To be completed

The additivity result of the single index model above holds for portfolios of any
size. To illustrate, suppose the single index model holds for a collection of N assets:

Rit = ±i + β i,M RMt + µit (i = 1, . . . , N)

Consider forming a portfolio of these N assets. Let xi denote the share of wealth
P
invested in asset i and assume that N = 1. Then the return on the portfolio is
i=1

N
X
Rpt = xi (±i + β i,M RMt + µit )
i=1
ÃN !
N N
X X X
= xi ±i + xi β i,M RMt + xi µit
i=1 i=1 i=1
= ±p + β p RMt + µpt
³P ´
PN PN
N
where ±p = i=1 xi β i,M and µpt =
i=1 xi ±i , β p = xi µit .
i=1


1.3.1 The Single Index Model and Large Portfolios
To be completed


2 Beta as a Measure of portfolio Risk
A key insight of portfolio theory is that, due to diversi&cation, the risk of an individual
asset should be based on how it a¬ects the risk of a well diversi& portfolio if it is
ed
added to the portfolio. The preceding section illustrated that individual speci& c
risk, as measured by the asset s own variance, can be diversi& away in large well
ed
diversi& portfolios whereas the covariances of the asset with the other assets in
ed

7
the portfolio cannot be completely diversi& away. The so-called beta of an asset
ed
captures this covariance contribution and so is a measure of the contribution of the
asset to overall portfolio variability.
To illustrate, consider an equally weighted portfolio of 99 stocks and let R99 denote
the return on this portfolio and σ 2 denote the variance. Now consider adding one
99
stock, say IBM, to the portfolio. Let RIBM and σ 2 IBM denote the return and variance
of IBM and let σ 99,IBM = cov(R99 , RIBM ). What is the contribution of IBM to the
risk, as measured by portfolio variance, of the portfolio? Will the addition of IBM
make the portfolio riskier (increase portfolio variance)? Less risky (decrease portfolio
variance)? Or have no e¬ect (not change portfolio variance)? To answer this question,
consider a new equally weighted portfolio of 100 stocks constructed as
R100 = (0.99) · R99 + (0.01) · RIBM .
The variance of this portfolio is
σ2 22 22
100 = var(R100 ) = (0.99) σ 99 + (0.01) σ IBM + 2(0.99)(0.01)σ 99,IBM
= (0.98)σ 2 + (0.0001)σ 2IBM + (0.02)σ 99,IBM
99
≈ (0.98)σ 2 + (0.02)σ 99,IBM .
99

Now if
• σ 2 = σ 2 then adding IBM does not change the variability of the portfolio;
100 99

• σ 2 > σ 2 then adding IBM increases the variability of the portfolio;
100 99

• σ 2 < σ 2 then adding IBM decreases the variability of the portfolio.
100 99

Consider the & case where σ 2 = σ 2 . This implies (approximately) that
rst 100 99

(0.98)σ 2 + (0.02)σ 99,IBM = σ 2
99 99

which upon rearranging gives the condition
σ 99,IBM cov(R99 , RIBM )
= =1
σ2 var(R99 )
99
De&ning
cov(R99 , RIBM )
β 99,IBM =
var(R99 )
then adding IBM does not change the variability of the portfolio as long as β 99,IBM =
1. Similarly, it is easy to see that σ 2 > σ 2 implies that β 99,IBM > 1 and σ 2 < σ 2
100 99 100 99
implies that β 99,IBM < 1.
In general, let Rp denote the return on a large diversi& portfolio and let Ri
ed
denote the return on some asset i. Then
cov(Rp , Ri )
β p,i =
var(Rp )
measures the contribution of asset i to the overall risk of the portfolio.

8
2.1 The single index model and Portfolio Theory
To be completed

2.2 Estimation of the single index model by Least Squares
Regression
Consider a sample of size T of observations on Rit and RMt . We use the lower case
variables rit and rMt to denote these observed values. The method of least squares
&nds the best & tting line to the scatter-plot of data as follows. For a given estimate
of the best & tting line
b
b b
rit = ±i + β i,M rMt , t = 1, . . . , T

create the T observed errors
b
bit = rit ’ rit = rit ’ ±i ’ β i,M rMt , t = 1, . . . , T
b b
µ

Now some lines will & better for some observations and some lines will & better for
t t
others. The least squares regression line is the one that minimizes the error sum of
squares (ESS)
T T
X X
bb b
b2 (rit ’ ±i ’ β i,M rMt )2
b
SSR(±i , β i,M ) = µit =
t=1 t=1

b
b
The minimizing values of ±i and β i,M are called the (ordinary) least squares (OLS) es-
bb bb
timates of ±i and β i,M . Notice that SSR(±i , β i,M ) is a quadratic function in (±i , β i,M )
given the data and so the minimum values can be easily obtained using calculus. The
& order conditions for a minimum are
rst
T T
X X
‚SSR b
b bit
= ’2 (rit ’ ±i ’ β i,M rMt ) = ’2
0= µ
b
‚ ±i t=1 t=1
T T
X X
‚SSR b
b bit rMt
= ’2 (rit ’ ±i ’ β i,M rMt )rMt = ’2
0= µ
b
‚ β i,M t=1 t=1

which can be rearranged as
T T
X X
b
b
rit = T ±i + β i,M rMt
t=1 t=1
T T T
X X X
b 2
b
rit rMt = ±i rMt +β rMt
i,M
t=1 t=1 t=1




9
These are two linear equations in two unknowns and by straightforward substitution
the solution is
¯b¯
b
±i = ri ’ β i,M rM
PT
t=1 (rit ’ ri )(rMt ’ rM )
¯ ¯
b
β i,M = PT
¯2
t=1 (rMt ’ rM )

where
T T
1X 1X
ri =
¯ rit , rM =
¯ rMt .
T t=1 T t=1
b b
The equation for β i,M can be rewritten slightly to show that β i,M is a simple
function of variances and covariances. Divide the numerator and denominator of the
b 1
expression for β i,M by T ’1 to give
PT
1
t=1 (rit ’ ri )(rMt ’ rM )
¯ ¯ d
cov(Rit , RMt )
b T ’1
β i,M = =
1 PT d
¯2 var(RMt )
t=1 (rMt ’ rM )
T ’1

b
which shows that β i,M is the ratio of the estimated covariance between Rit and RMt
to the estimated variance of RMt .
The least squares estimate of σ 2 = var(µit ) is given by
µ,i

T T
1 X2 1X b
σ2 (rt ’ ±i ’ β i,M rMt )2
b µ,i b b
= eit =
T ’ 2 t=1 T ’ 2 t=1
The divisor T ’ 2 is used to make σ 2 an unbiased estimator of σ 2 .
b µ,i µ,ι
2
The least squares estimate of R is given by
2
b
β i,M σ 2 σ2
bM b µ,i
b 2
= 1’
Ri = ,
d d
var(Rit ) var(Rit )
where
T
1X
(rit ’ ri )2 ,
d
var(Rit ) = ¯
T ’ 1 t=1
and gives a measure of the goodness of & of the regression equation. Notice that
t
b 2
2
Ri = 1 whenever σ µ,i = 0 which occurs when bit = 0 for all values of t. In other
b µ
b2 b2
words, Ri = 1 whenever the regression line has a perfect & Conversely, Ri = 0
t.
when σ 2 = var(Rit ); that is, when the market does not explain any of the variability
b µ,i d
of Rit . In this case, the regression has the worst possible &t.


3 Hypothesis Testing in the Single Index Model
3.1 A Review of Hypothesis Testing Concepts
To be completed.

10
Testing the Restriction ± = 0.
3.2
Using the single index model regression,

Rt = ± + βRMt + µt , t = 1, ..., T
µt ∼ iid N(0, σ 2 ), µt is independent of RMt (4)
µ

consider testing the null or maintained hypothesis ± = 0 against the alternative that
± 6= 0
H0 : ± = 0 vs. H1 : ± 6= 0.
If H0 is true then the single index model regression becomes

Rt = βRMt + µt

and E[Rt |RMt = rMt ] = βrMt . We will reject the null hypothesis, H0 : ± = 0, if
the estimated value of ± is either much larger than zero or much smaller than zero.
Assuming H0 : ± = 0 is true, ± ∼ N (0, SE(ˆ )2 ) and so is fairly unlikely that ± will
ˆ ± ˆ
be more than 2 values of SE(ˆ ) from zero. To determine how big the estimated value
±
of ± needs to be in order to reject the null hypothesis we use the t-statistic
b
±’0
t±=0 = d ,
b)
SE(±
db
b
where ± is the least squares estimate of ± and SE(±) is its estimated standard error.
The value of the t-statistic, t±=0 , gives the number of estimated standard errors that
b
± is from zero. If the absolute value of t±=0 is much larger than 2 then the data cast
considerable doubt on the null hypothesis ± = 0 whereas if it is less than 2 the data
are in support of the null hypothesis3 . To determine how big | t±=0 | needs to be to
reject the null, we use the fact that under the statistical assumptions of the single
index model and assuming the null hypothesis is true

t±=0 ∼ Student ’ t with T ’ 2 degrees of freedom

If we set the signi& cance level (the probability that we reject the null given that the
null is true) of our test at, say, 5% then our decision rule is

Reject H0 : ± = 0 at the 5% level if |t±=0 | > |tT ’2 (0.025)|

where tT ’2 is the 2 1 % critical value (quantile) from a Student-t distribution with
2
T ’ 2 degrees of freedom.

Example 3 single index model Regression for IBM
3
This interpretation of the t-statistic relies on the fact that, assuming the null hypothesis is true

so that ± = 0, ± is normally distributed with mean 0 and estimated variance SE(b )2 .
b

11
Consider the estimated MM regression equation for IBM using monthly data from
January 1978 through December 1982:
b
RIBM,t =’0.0002 + 0.3390 ·RMt , R2 = 0.20, σ µ = 0.0524
b
(0.0888)
(0.0068)

b
where the estimated standard errors are in parentheses. Here ± = ’0.0002, which is

very close to zero, and the estimated standard error, SE(ˆ ) = 0.0068, is much larger
b
than ±. The t-statistic for testing H0 : ± = 0 vs. H1 : ± 6= 0 is
’0.0002 ’ 0
= ’0.0363
t±=0 =
0.0068
b
so that ± is only 0.0363 estimated standard errors from zero. Using a 5% signi&cance
level, |t58 (0.025)| ≈ 2 and
|t±=0 | = 0.0363 < 2
so we do not reject H0 : ± = 0 at the 5% level.

Testing Hypotheses about β
3.3
In the single index model regression β measures the contribution of an asset to the
variability of the market index portfolio. One hypothesis of interest is to test if the
asset has the same level of risk as the market index against the alternative that the
risk is di¬erent from the market:

H0 : β = 1 vs. H1 : β 6= 1.

The data cast doubt on this hypothesis if the estimated value of β is much di¬erent
from one. This hypothesis can be tested using the t-statistic
b
β’1
tβ=1 =db
SE(β)

which measures how many estimated standard errors the least squares estimate of β
is from one. The null hypothesis is reject at the 5% level, say, if |tβ=1 | > |tT ’2 (0.025)|.
Notice that this is a two-sided test.
Alternatively, one might want to test the hypothesis that the risk of an asset is
strictly less than the risk of the market index against the alternative that the risk is
greater than or equal to that of the market:

H0 : β = 1 vs. H1 : β ≥ 1.

Notice that this is a one-sided test. We will reject the null hypothesis only if the
estimated value of β much greater than one. The t-statistic for testing this null


12
hypothesis is the same as before but the decision rule is di¬erent. Now we reject the
null at the 5% level if
tβ=1 < ’tT ’2 (0.05)
where tT ’2 (0.05) is the one-sided 5% critical value of the Student-t distribution with
T ’ 2 degrees of freedom.

Example 4 Single Index Regression for IBM cont d

Continuing with the previous example, consider testing H0 : β = 1 vs. H1 : β 6= 1.
Notice that the estimated value of β is 0.3390, with an estimated standard error of
0.0888, and is fairly far from the hypothesized value β = 1. The t-statistic for testing
β = 1 is
0.3390 ’ 1
= ’7.444
tβ=1 =
0.0888
b
which tells us that β is more than 7 estimated standard errors below one. Since
t0.025,58 ≈ 2 we easily reject the hypothesis that β = 1.
Now consider testing H0 : β = 1 vs. H1 : β ≥ 1. The t-statistic is still -7.444
but the critical value used for the test is now ’t58 (0.05) ≈ ’1.671. Clearly, tβ=1 =
’7.444 < ’1.671 = ’t58 (0.05) so we reject this hypothesis.


4 Estimation of the single index model: An Ex-
tended Example
Now we illustrate the estimation of the single index model using monthly data on
returns over the ten year period January 1978 - December 1987. As our dependent
variable we use the return on IBM and as our market index proxy we use the CRSP
value weighted composite monthly return index based on transactions from the New
York Stock Exchange and the American Stock Exchange. Let rt denote the monthly
return on IBM and rMt denote the monthly return on the CRSP value weighted index.
Time plots of these data are given in &gure 1 below.




13
Monthly Returns on IBM Monthly Returns on Market Index
0.2 0.2


0.1 0.1


0.0 0.0


-0.1 -0.1


-0.2 -0.2


-0.3 -0.3
78 79 80 81 82 83 84 85 86 87 78 79 80 81 82 83 84 85 86 87

IBM MARKET




Figure 1

Notice that the IBM and the market index have similar behavior over the sample
with the market index looking a little more volatile than IBM. Both returns dropped
sharply during the October 1987 crash but there were a few times that the market
dropped sharply whereas IBM did not. Sample descriptive statistics for the returns
are displayed in &gure 2.
The mean monthly returns on IBM and the market index are 0.9617% and 1.3992%
per month and the sample standard deviations are 5.9024% and 6.8353% per month,
respectively.. Hence the market index on average had a higher monthly return and
more volatility than IBM.




14
Monthly Returns on IBM Monthly Returns on Market Index

12 30


10 25


8 20


6 15

4 10


2 5


0 0
-0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 -0.2 -0.1 0.0 0.1

Series: IBM Series: MARKET
Sample 1978:01 1987:12 Sample 1978:01 1987:12
Observations 120 Observations 120

Mean 0.009617 Mean 0.013992
Median 0.002000 Median 0.012000
Maximum 0.150000 Maximum 0.148000
Minimum -0.187000 Minimum -0.260000
Std. Dev. 0.059024 Std. Dev. 0.068353
Skewness -0.036491 Skewness -1.104576
Kurtosis 3.126664 Kurtosis 5.952204

Jarque-Bera 0.106851 Jarque-Bera 67.97932
Probability 0.947976 Probability 0.000000



Figure 2

Notice that the histogram of returns on the market are heavily skewed left whereas
the histogram for IBM is much more sysingle index modeletric about the mean. Also,
the kurtosis for the market is much larger than 3 (the value for normally distributed
returns) and the kurtosis for IBM is just slightly larger than 3. The negative skewness
and large kurtosis of the market returns is caused by several large negative returns.
The Jarque-Bera statistic for the market returns is 67.97, with a p-value 0.0000, and
so we can easily reject the hypothesis that the market data are normally distributed.
However, the Jarque-Bera statistic for IBM is only 0.1068, with a p-value of 0.9479,
and we therefore cannot reject the hypothesis of normality.
The single index model regression is

Rt = ± + βRMt + µt , t = 1, . . . , T

where it is assumed that µt ∼ iid N(0, σ 2 ) and is independent of RMt . We estimate
this regression using the & & years of data from January 1978 - December 1982.
rst ve
In practice the single index model is seldom estimated using data covering more than
& years because it is felt that β may change through time. The computer printout
ve
from Eviews is given in & gure 3 below




15
Figure 3

4.1 Explanation of Computer Output
The the items under the column labeled Variable are the variables in the estimated
regression model. The variable C refers to the intercept in the regression and
MARKET refers to rMt . The least squares regression coe¬cients are reported in
the column labeled Coe¬cient and the estimated standard errors for the coe¬cients
are in then next column. A standard way of reporting the estimated equation is

b
rt =0.0053 + 0.3278 ·rMt
(0.0069) (0.0890)

where the estimated standard errors are reported underneath the estimated coe¬-
cients. The estimated intercept is close to zero at 0.0053, with a standard error of
db
0.0069 (= SE(±)), and the estimated value of β is 0.3278, with an standard error of
db b
0.0890 (= SE(β)). Notice that the estimated standard error of β is much smaller
than the estimated coe¬cient and indicates that β is estimated reasonably precisely.
The estimated regression equation is displayed graphically in &gure 4 below.




16
Market Model Regression
0.2



0.1
IBM




0.0



-0.1



-0.2
-0.3 -0.2 -0.1 0.0 0.1 0.2

MARKET

Figure 4

To evaluate the overall & of the single index model regression we look at the R2 of
t
the regression, which measures the percentage of variability of Rt that is attributable
b
to the variability in RMt , and the estimated standard deviation of the residuals, σ µ .
From the table, R2 = 0.190 so the market index explains only 19% of the variability
of IBM and 81% of the variability is not explained by the market. In the single index
model regression, we can also interpret R2 as the proportion of market risk in IBM
and 1 ’ R2 as the proportion of & speci& risk. The standard error (S.E.) of the
rm c
regression is the square root of the least squares estimate of σ 2 = var(µt ). From the
µ
b b
above table, σ µ = 0.052. Recall, µt captures the & speci& risk of IBM and so σ µ is
rm c
an estimate of the typical magnitude of the & speci& risk. In order to interpret the
rm c
b
magnitude of σ µ it is useful to compare it to the estimate of the standard deviation
of Rt , which measures the total risk of IBM. This is reported in the table by the
standard deviation (S.D.) of the dependent variable which equals 0.057. Notice that
b
σ µ = 0.052 is only slightly smaller than 0.057 so that the & speci& risk is a large
rm c
proportion of total risk (which is also reported by 1 ’ R2 ).
Con& dence intervals for the regression parameters are easily computed using the
reported regression output. Since µt is assumed to be normally distributed 95%
con& dence intervals for ± and β take the form
db
b
± ± 2 · SE(±)
b db
β ± 2 · SE(β)


17
The 95% con&dence intervals are then

± : 0.0053 ± 2 · 0.0069 = [’.0085, 0.0191]
β : 0.3278 ± 2 · 0.0890 = [0.1498, 0.5058]

Our best guess of ± is 0.0053 but we wouldn t be too surprised if it was as low as
-0.0085 or as high as 0.0191. Notice that there are both positive and negative values
in the con& dence interval. Similarly, our best guess of β is 0.3278 but it could be as
low as 0.1498 or as high as 0.5058. This is a fairly wide range given the interpretation
of β as a risk measure. The interpretation of these intervals are as follows. In
repeated samples, 95% of the time the estimated con& dence intervals will cover the
true parameter values.
The t-statistic given in the computer output is calculated as
estimated coe¬cient ’ 0
t-statistic =
std. error
and it measures how many estimated standard errors the estimated coe¬cient is away
from zero. This t-statistic is often referred to as a basic signi& cance test because it
tests the null hypothesis that the value of the true coe¬cient is zero. If an estimate is
several standard errors from zero, so that it s t-statistic is greater than 2, then it is a
good bet that the true coe¬cient is not equal to zero. From the data in the table, the
b
t-statistic for ± is 0.767 so that ± = 0.0053 is 0.767 standard errors from zero. Hence
it is quite likely that the true value of ± equals zero. The t-statistic for β is 3.684,
b
β is more than 3 standard errors from zero, and so it is very unlikely that β = 0.
The Prob Value (p-value of the t-statistic) in the table gives the likelihood (computed
from the Student-t curve) that, given the true value of the coe¬cient is zero, the data
would generate the observed value of the t-statistic. The p-value for the t-statistic
testing ± = 0 is 0.4465 so that it is quite likely that ± = 0. Alternatively, the p-value
for the t-statistic testing β = 0 is 0.001 so it is very unlikely that β = 0.

4.2 Analysis of the Residuals
The single index model regression makes the assumption that µt ∼ iid N (0, σ 2 ). That
µ
is the errors are independent and identically distributed with mean zero, constant
variance σ 2 and are normally distributed. It is always a good idea to check the
µ
behavior of the estimated residuals, bt , and see if they share the assumed properties
µ
b
b b
of the true residuals µt . The &gure below plots rt (the actual data), rt = ± + βrMt
tted data) and bt = rt ’ rt (the estimated residual data).
b
(the & µ




18
Market Model Regression for IBM
0.2

0.1

0.0
0.15
0.10 -0.1
0.05
-0.2
0.00
-0.05
-0.10
-0.15
1978 1979 1980 1981 1982

Residual Actual Fitted

Figure 5

Notice that the & tted values do not track the actual values very closely and that
the residuals are fairly large. This is due to low R2 of the regression. The residuals
appear to be fairly random by sight. We will develop explicit tests for randomness
later on. The histogram of the residuals, displayed below, can be used to investigate
the normality assumption. As a result of the least squares algorithm the residuals
have mean zero as long as a constant is included in the regression. The standard
deviation of the residuals is essentially equal to the standard error of the regression
- the di¬erence being due to the fact that the formula for the standard error of the
regression uses T ’ 2 as a divisor for the error sum of squares and the standard
deviation of the residuals uses the divisor T ’ 1.




19
Residuals from Market Model Regression for IBM
8
Series: Residuals
Sample 1978:01 1982:12
Observations 60
6
Mean -2.31E-19
Median -0.000553
Maximum 0.139584
4
Minimum -0.104026
Std. Dev. 0.051567
Skewness 0.493494
Kurtosis 2.821260
2

Jarque-Bera 2.515234
Probability 0.284331
0
-0.10 -0.05 0.00 0.05 0.10

Figure 6

The skewness of the residuals is slightly positive and the kurtosis is a little less
than 3. The hypothesis that the residuals are normally distributed can be tested
using the Jarque-Bera statistic. This statistic is a function of the estimated skewness
and kurtosis and is give by
à !
c 2
b2 + (K ’ 3)
T
JB = S
6 4
b c
where S denotes the estimated skewness and K denotes the estimated kurtosis. If
b c
the residuals are normally distribued then S ≈ 0 and K ≈ 3 and JB ≈ 0. Therefore,
b c
if S is moderately di¬erent from zero or K is much di¬erent from 3 then JB will get
large and suggest that the data are not normally distributed. To determine how large
JB needs to be to be able to reject the normality assumption we use the result that
under the maintained hypothesis that the residuals are normally distributed JB has
a chi-square distribution with 2 degrees of freedom:
JB ∼ χ2 .
2

For a test with signi&cance level 5%, the 5% right tail critical value of the chi-square
distribution with 2 degrees of freedom, χ2 (0.05), is 5.99 so we would reject the null
2
that the residuals are normally distributed if JB > 5.99. The Probability (p-value)
reported by Eviews is the probability that a chi-square random variable with 2 degrees
of freedom is greater than the observed value of JB :
P (χ2 ≥ JB) = 0.2843.
2

For the IBM residuals this p-value is reasonably large and so there is not much data
evidence against the normality assumption. If the p-value was very small, e.g., 0.05 or
smaller, then the data would suggest that the residuals are not normally distributed.

20