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 aï¬€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 = Î±i + Î² i,M RMt + Îµit , i = 1, . . . , N ; t = 1, . . . , T

where Rit is the continuously compounded return on asset i (i = 1, . . . , N) between

time periods t âˆ’ 1 and t, and RMt is the continuously compounded return on a

market index portfolio between time periods t âˆ’ 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 coeï¬ƒcient

Î² i,M multiplying RMt in (1) measures the contribution of asset i to the variance

(risk), Ïƒ 2 , of the market index portfolio. If Î² i,M = 1 then adding the security does

M

not change the variability, Ïƒ 2 , of the market index; if Î² i,M > 1 then adding the

M

security will increase the variability of the market index and if Î² 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 aï¬€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 Îµit has a similar interpretation as the error term in the CER model. In the single

index model, Îµit represents random news that arrives between time t âˆ’ 1 and t that

captures micro or & rm-speci& risk factors that aï¬€ect an individual asset s return

c

that are not related to macro events. For example, Îµit may capture the news eï¬€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 kâˆ’variable linear regression model

Rit = Î±i + Î² i,1 F1t + Î² i,2 F2t + Â· Â· Â· + Î² i,k Fkt + Îµit

where Fjt denotes the j th systematic factorm, Î² i,j denotes asset i0 s loading on the j th

factor and Îµit denotes the random component independent of all of the systematic

factors. The single index model results when F1t = RMt and Î² i,2 = Â· Â· Â· = Î² 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 aï¬€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[Îµit ] = 0 for i = 1, . . . , N and t = 1, . . . , T (news is neutral on average).

3. var(Îµit ) = Ïƒ 2 for i = 1, . . . , N (homoskedasticity).

Îµ,i

4. cov(Îµit , RMt ) = 0 for i = 1, . . . , N and t = 1, . . . , T .

2

The purpose of an event study is to measure the eï¬€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, eï¬€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(Îµit , Îµjs ) = 0 for all t, s and i 6= j

6. Îµit 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 aï¬€ecting

asset i in time t is independent of news aï¬€ecting asset j in time s.

That Îµit is unrelated to RMs and Îµjs implies that any correlation between asset i

and asset j is solely due to their common exposure to RMt throught the values of Î² i

and Î² 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 ] = Âµi = Î±i + Î² i,M E[RMt ] = Î±i + Î² i,M ÂµM

2. var(Rit ) = Ïƒ 2 = Î² 2 var(RMt ) + var(Îµit ) = Î² 2 Ïƒ 2 + Ïƒ 2

i i,M i,M M Îµ,i

3. cov(Rit , Rjt ) = Ïƒ ij = Ïƒ 2 Î² i Î² j

M

4. Rit ˜ iid N(Âµi , Ïƒ 2 ), RMt ˜ iid N (ÂµM , Ïƒ 2 )

i M

cov(Rit ,RMt ) Ïƒ iM

5. Î² i,M = = Ïƒ2

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(Î±i + Î² i,M RMt + Îµit )

= Î² 2 var(RMt ) + var(Îµit ) + 2cov(RMt , Îµit )

i,M

= Î² 2 Ïƒ2 + Ïƒ2

i,M M Îµ,i

since, by assumption 4, cov(Îµit , RMt ) = 0. For 3, by the additivity property of

covariance and assumptions 4 and 5 we have

cov(Rit , Rjt ) = cov(Î±i + Î² i,M RMt + Îµit , Î±j + Î² j,M RMt + Îµjt )

= cov(Î² i,M RMt + Îµit , Î² j,M RMt + Îµjt )

= cov(Î² i,M RMt , Î² j,M RMt ) + cov(Î² i,M RMt , Îµjt ) + cov(Îµit , Î² j,M RMt ) + cov(Îµit , Îµjt )

Î² i,M Î² j,M cov(RMt , RMt ) = Î² i,M Î² j,M Ïƒ 2

= M

3

Last, for 5 note that

cov(Rit , RMt ) = cov(Î±i + Î² i,M RMt + Îµit , RMt )

= cov(Î² i,M RMt , RMt )

= Î² i,M cov(RMt , RMt )

= Î² i,M var(RMt ),

which uses assumption 4. It follows that

Î² var(RMt )

cov(Rit , RMt )

= i,M = Î² i,M .

var(RMt ) var(RMt )

Remarks:

1. Notice that unconditional expected return on asset i, Âµi , is constant and con-

sists of an intercept term Î±i , a term related to Î² i,M and the unconditional

mean of the market index, ÂµM . This relationship may be used to create pre-

dictions of expected returns over some future period. For example, suppose

Î±i = 0.01, Î² i,M = 0.5 and that a market analyst forecasts ÂµM = 0.05. Then the

forecast for the expected return on asset i is

b

Âµi = 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, Î² 2 Ïƒ 2 , and variability due to speci& risk,

c

i,M M

Ïƒ2 .

Îµ,i

3. Since Ïƒ ij = Ïƒ 2 Î² i Î² j the direction of the covariance between asset i and asset j

M

depends of the values of Î² i and Î² j . In particular

â€¢ Ïƒ ij = 0 if Î² i = 0 or Î² j = 0 or both

â€¢ Ïƒ ij > 0 if Î² i and Î² j are of the same sign

â€¢ Ïƒ ij < 0 if Î² i and Î² j are of opposite signs.

4. The expression for the expected return can be used to provide an unconditional

interpretation of Î±i . Subtracting Î² i,M ÂµM from both sides of the expression for

Âµi gives

Î±i = Âµi âˆ’ Î² i,M ÂµM .

4

1.1.2 Decomposing Total Risk

The independence assumption between RMt and Îµit allows the unconditional vari-

ability of Rit , var(Rit ) = Ïƒ 2 , to be decomposed into the variability due to the market

i

2

index, Î² i,M Ïƒ M , plus the variability of the & speci& component, Ïƒ 2 . This decom-

2

rm c Îµ,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 Ïƒ 2 = Î² 2 Ïƒ 2 + Ïƒ 2 to give

i i,M M Îµ,i

Î² 2 Ïƒ2 + Ïƒ2 Î² 2 Ïƒ2 Ïƒ2

Ïƒ2 i,M M Îµ,i i,M M

+ Îµ,i

i

1= 2 = =

Ïƒ2 Ïƒ2 Ïƒ2

Ïƒi i i i

Then we can de&ne

Î² 2 Ïƒ2 Ïƒ2

i,M M

= 1 âˆ’ Îµ,i

2

Ri =

Ïƒ2 Ïƒ2

i i

as the proportion of the total variability of Rit that is attributable to variability in

the market index. Similarly,

Ïƒ2

1 âˆ’ Ri = Îµ,i

2

Ïƒ2i

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 1âˆ’Ri as the proportion of risk that

2

can be diversi& away. It is important not to confuse Ri with Î² i,M . The coeï¬ƒcient

ed

2

Î² 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 â‰ˆ 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 ] = Âµi|RM = Î±i + Î² i,M rMt

2. var(Rit |RMt = rMt ) = var(Îµit ) = Ïƒ 2

Îµ,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 Ã— 1) vectors Rt = (R1t , R2t , . . . , RNt )0 , Î± =

nie

(Î±1 , Î±2 , . . . , Î±N ) , Î² = (Î² 1 , Î² 2 , . . . , Î² N )0 and Îµt = (Îµ1t , Îµ2t , . . . , ÎµNt )0 . Then the single

0

index model for all N assets may be represented as

ï£« ï£¶ ï£« ï£¶ ï£« ï£¶ ï£« ï£¶

R1t Î±1 Î²1 Îµ1t

ï£¬ . ï£·=ï£¬ . ï£·+ï£¬ . ï£·R + ï£¬ . ï£· , t = 1, . . . , T

ï£¬ .ï£·ï£¬ .ï£·ï£¬ . ï£· Mt ï£¬ .ï£·

.ï£¸ï£ .ï£¸ï£ .ï£¸ .ï£¸

ï£ ï£

RNt Î±N Î²N ÎµNt

or

Rt = Î± + Î² Â· RMt + Îµt , t = 1, . . . , T.

Since Ïƒ 2 = Î² 2 Ïƒ 2 + Ïƒ 2 and Ïƒ ij = Î² i Î² j Ïƒ 2 the covariance matrix for the N

i i,M M Îµ,i M

returns may be expressed as

ï£« ï£¶ ï£« ï£¶ï£« ï£¶

Î² 2 Ïƒ2 Î² i Î² j Ïƒ2 Â· Â· Â· Î² iÎ² j Ïƒ2 Ïƒ2

Ïƒ2 Â·Â·Â·

0 0

Ïƒ 12 Â· Â· Â· Ïƒ 1N i,M M M M Îµ,1

1

ï£¬ ï£· ï£¬ ï£·ï£¬ ï£·

2

0 Ïƒ2

Ïƒ 2 Â· Â· Â· Ïƒ 2N 2 2

Â· Â· Â· Î² iÎ² j Ïƒ2 Â·Â·Â·

Ïƒ 12 Î² i Î² j Ïƒ M Î² i,M Ïƒ M 0

ï£¬ ï£· ï£¬ ï£·ï£¬ ï£·

Îµ,2

2 M

Î£=ï£¬ ï£·=ï£¬ ï£·+ï£¬ ï£·

. . .. . . . . . . .

...

...

ï£¬ ï£· ï£¬ ï£·ï£¬ ï£·

. . . . . .

. . .

.

. . . . . .

. . .

ï£ ï£¸ ï£ ï£¸ï£ ï£¸

Â· Â· Â· Î² 2 Ïƒ2

Â· Â· Â· Â· Â· Â· Ïƒ2 Â· Â· Â· Ïƒ2

Î² iÎ² j Ïƒ2 Î² i Î² j Ïƒ2

Ïƒ 1N 0 0

N Îµ,N

M M i,M M

The covariance matrix may be conveniently computes as

Î£ = Ïƒ 2 Î²Î² 0 + âˆ†

M

where âˆ† is a diagonal matrix with Ïƒ 2 along the diagonal.

Îµ,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 = Î±1 + Î² 1,M RMt + Îµ1t ,

(3)

R2t = Î±2 + Î² 2,M RMt + Îµ2t .

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 (Î±1 + Î² 1,M RMt + Îµ1t ) + x2 (Î±2 + Î² 2,M RMt + Îµ2t )

= (x1 Î±1 + x2 Î±2 ) + (x1 Î² 1,M + x2 Î² 2,M )RMt + (x1 Îµ1t + x2 Îµ2t )

= Î±p + Î² p,M RMt + Îµpt

where Î±p = x1 Î±1 + x2 Î±2 , Î² p,M = x1 Î² 1,M + x2 Î² 2,M and Îµpt = x1 Îµ1t + x2 Îµ2t . 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 = Î±i + Î² i,M RMt + Îµit (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 (Î±i + Î² i,M RMt + Îµit )

i=1

ÃƒN !

N N

X X X

= xi Î±i + xi Î² i,M RMt + xi Îµit

i=1 i=1 i=1

= Î±p + Î² p RMt + Îµpt

Â³P Â´

PN PN

N

where Î±p = i=1 xi Î² i,M and Îµpt =

i=1 xi Î±i , Î² p = xi Îµit .

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 aï¬€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 Ïƒ 2 denote the variance. Now consider adding one

99

stock, say IBM, to the portfolio. Let RIBM and Ïƒ 2 IBM denote the return and variance

of IBM and let Ïƒ 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 eï¬€ect (not change portfolio variance)? To answer this question,

consider a new equally weighted portfolio of 100 stocks constructed as

R100 = (0.99) Â· R99 + (0.01) Â· RIBM .

The variance of this portfolio is

Ïƒ2 22 22

100 = var(R100 ) = (0.99) Ïƒ 99 + (0.01) Ïƒ IBM + 2(0.99)(0.01)Ïƒ 99,IBM

= (0.98)Ïƒ 2 + (0.0001)Ïƒ 2IBM + (0.02)Ïƒ 99,IBM

99

â‰ˆ (0.98)Ïƒ 2 + (0.02)Ïƒ 99,IBM .

99

Now if

â€¢ Ïƒ 2 = Ïƒ 2 then adding IBM does not change the variability of the portfolio;

100 99

â€¢ Ïƒ 2 > Ïƒ 2 then adding IBM increases the variability of the portfolio;

100 99

â€¢ Ïƒ 2 < Ïƒ 2 then adding IBM decreases the variability of the portfolio.

100 99

Consider the & case where Ïƒ 2 = Ïƒ 2 . This implies (approximately) that

rst 100 99

(0.98)Ïƒ 2 + (0.02)Ïƒ 99,IBM = Ïƒ 2

99 99

which upon rearranging gives the condition

Ïƒ 99,IBM cov(R99 , RIBM )

= =1

Ïƒ2 var(R99 )

99

De&ning

cov(R99 , RIBM )

Î² 99,IBM =

var(R99 )

then adding IBM does not change the variability of the portfolio as long as Î² 99,IBM =

1. Similarly, it is easy to see that Ïƒ 2 > Ïƒ 2 implies that Î² 99,IBM > 1 and Ïƒ 2 < Ïƒ 2

100 99 100 99

implies that Î² 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 )

Î² 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 = Î±i + Î² i,M rMt , t = 1, . . . , T

create the T observed errors

b

bit = rit âˆ’ rit = rit âˆ’ Î±i âˆ’ Î² i,M rMt , t = 1, . . . , T

b b

Îµ

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 âˆ’ Î±i âˆ’ Î² i,M rMt )2

b

SSR(Î±i , Î² i,M ) = Îµit =

t=1 t=1

b

b

The minimizing values of Î±i and Î² i,M are called the (ordinary) least squares (OLS) es-

bb bb

timates of Î±i and Î² i,M . Notice that SSR(Î±i , Î² i,M ) is a quadratic function in (Î±i , Î² 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

âˆ‚SSR b

b bit

= âˆ’2 (rit âˆ’ Î±i âˆ’ Î² i,M rMt ) = âˆ’2

0= Îµ

b

âˆ‚ Î±i t=1 t=1

T T

X X

âˆ‚SSR b

b bit rMt

= âˆ’2 (rit âˆ’ Î±i âˆ’ Î² i,M rMt )rMt = âˆ’2

0= Îµ

b

âˆ‚ Î² i,M t=1 t=1

which can be rearranged as

T T

X X

b

b

rit = T Î±i + Î² i,M rMt

t=1 t=1

T T T

X X X

b 2

b

rit rMt = Î±i rMt +Î² 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

Â¯bÂ¯

b

Î±i = ri âˆ’ Î² i,M rM

PT

t=1 (rit âˆ’ ri )(rMt âˆ’ rM )

Â¯ Â¯

b

Î² i,M = PT

Â¯2

t=1 (rMt âˆ’ rM )

where

T T

1X 1X

ri =

Â¯ rit , rM =

Â¯ rMt .

T t=1 T t=1

b b

The equation for Î² i,M can be rewritten slightly to show that Î² i,M is a simple

function of variances and covariances. Divide the numerator and denominator of the

b 1

expression for Î² i,M by T âˆ’1 to give

PT

1

t=1 (rit âˆ’ ri )(rMt âˆ’ rM )

Â¯ Â¯ d

cov(Rit , RMt )

b T âˆ’1

Î² i,M = =

1 PT d

Â¯2 var(RMt )

t=1 (rMt âˆ’ rM )

T âˆ’1

b

which shows that Î² i,M is the ratio of the estimated covariance between Rit and RMt

to the estimated variance of RMt .

The least squares estimate of Ïƒ 2 = var(Îµit ) is given by

Îµ,i

T T

1 X2 1X b

Ïƒ2 (rt âˆ’ Î±i âˆ’ Î² i,M rMt )2

b Îµ,i b b

= eit =

T âˆ’ 2 t=1 T âˆ’ 2 t=1

The divisor T âˆ’ 2 is used to make Ïƒ 2 an unbiased estimator of Ïƒ 2 .

b Îµ,i Îµ,Î¹

2

The least squares estimate of R is given by

2

b

Î² i,M Ïƒ 2 Ïƒ2

bM b Îµ,i

b 2

= 1âˆ’

Ri = ,

d d

var(Rit ) var(Rit )

where

T

1X

(rit âˆ’ ri )2 ,

d

var(Rit ) = Â¯

T âˆ’ 1 t=1

and gives a measure of the goodness of & of the regression equation. Notice that

t

b 2

2

Ri = 1 whenever Ïƒ Îµ,i = 0 which occurs when bit = 0 for all values of t. In other

b Îµ

b2 b2

words, Ri = 1 whenever the regression line has a perfect & Conversely, Ri = 0

t.

when Ïƒ 2 = var(Rit ); that is, when the market does not explain any of the variability

b Îµ,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 Î± = 0.

3.2

Using the single index model regression,

Rt = Î± + Î²RMt + Îµt , t = 1, ..., T

Îµt âˆ¼ iid N(0, Ïƒ 2 ), Îµt is independent of RMt (4)

Îµ

consider testing the null or maintained hypothesis Î± = 0 against the alternative that

Î± 6= 0

H0 : Î± = 0 vs. H1 : Î± 6= 0.

If H0 is true then the single index model regression becomes

Rt = Î²RMt + Îµt

and E[Rt |RMt = rMt ] = Î²rMt . We will reject the null hypothesis, H0 : Î± = 0, if

the estimated value of Î± is either much larger than zero or much smaller than zero.

Assuming H0 : Î± = 0 is true, Î± âˆ¼ N (0, SE(Ë† )2 ) and so is fairly unlikely that Î± will

Ë† Î± Ë†

be more than 2 values of SE(Ë† ) from zero. To determine how big the estimated value

Î±

of Î± needs to be in order to reject the null hypothesis we use the t-statistic

b

Î±âˆ’0

tÎ±=0 = d ,

b)

SE(Î±

db

b

where Î± is the least squares estimate of Î± and SE(Î±) is its estimated standard error.

The value of the t-statistic, tÎ±=0 , gives the number of estimated standard errors that

b

Î± is from zero. If the absolute value of tÎ±=0 is much larger than 2 then the data cast

considerable doubt on the null hypothesis Î± = 0 whereas if it is less than 2 the data

are in support of the null hypothesis3 . To determine how big | tÎ±=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

tÎ±=0 âˆ¼ Student âˆ’ t with T âˆ’ 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 : Î± = 0 at the 5% level if |tÎ±=0 | > |tT âˆ’2 (0.025)|

where tT âˆ’2 is the 2 1 % critical value (quantile) from a Student-t distribution with

2

T âˆ’ 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

dÎ±

so that Î± = 0, Î± 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 =âˆ’0.0002 + 0.3390 Â·RMt , R2 = 0.20, Ïƒ Îµ = 0.0524

b

(0.0888)

(0.0068)

b

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

dÎ±

very close to zero, and the estimated standard error, SE(Ë† ) = 0.0068, is much larger

b

than Î±. The t-statistic for testing H0 : Î± = 0 vs. H1 : Î± 6= 0 is

âˆ’0.0002 âˆ’ 0

= âˆ’0.0363

tÎ±=0 =

0.0068

b

so that Î± is only 0.0363 estimated standard errors from zero. Using a 5% signi&cance

level, |t58 (0.025)| â‰ˆ 2 and

|tÎ±=0 | = 0.0363 < 2

so we do not reject H0 : Î± = 0 at the 5% level.

Testing Hypotheses about Î²

3.3

In the single index model regression Î² 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 diï¬€erent from the market:

H0 : Î² = 1 vs. H1 : Î² 6= 1.

The data cast doubt on this hypothesis if the estimated value of Î² is much diï¬€erent

from one. This hypothesis can be tested using the t-statistic

b

Î²âˆ’1

tÎ²=1 =db

SE(Î²)

which measures how many estimated standard errors the least squares estimate of Î²

is from one. The null hypothesis is reject at the 5% level, say, if |tÎ²=1 | > |tT âˆ’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 : Î² = 1 vs. H1 : Î² â‰¥ 1.

Notice that this is a one-sided test. We will reject the null hypothesis only if the

estimated value of Î² much greater than one. The t-statistic for testing this null

12

hypothesis is the same as before but the decision rule is diï¬€erent. Now we reject the

null at the 5% level if

tÎ²=1 < âˆ’tT âˆ’2 (0.05)

where tT âˆ’2 (0.05) is the one-sided 5% critical value of the Student-t distribution with

T âˆ’ 2 degrees of freedom.

Example 4 Single Index Regression for IBM cont d

Continuing with the previous example, consider testing H0 : Î² = 1 vs. H1 : Î² 6= 1.

Notice that the estimated value of Î² is 0.3390, with an estimated standard error of

0.0888, and is fairly far from the hypothesized value Î² = 1. The t-statistic for testing

Î² = 1 is

0.3390 âˆ’ 1

= âˆ’7.444

tÎ²=1 =

0.0888

b

which tells us that Î² is more than 7 estimated standard errors below one. Since

t0.025,58 â‰ˆ 2 we easily reject the hypothesis that Î² = 1.

Now consider testing H0 : Î² = 1 vs. H1 : Î² â‰¥ 1. The t-statistic is still -7.444

but the critical value used for the test is now âˆ’t58 (0.05) â‰ˆ âˆ’1.671. Clearly, tÎ²=1 =

âˆ’7.444 < âˆ’1.671 = âˆ’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 = Î± + Î²RMt + Îµt , t = 1, . . . , T

where it is assumed that Îµt âˆ¼ iid N(0, Ïƒ 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 Î² 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 coeï¬ƒcients are reported in

the column labeled Coeï¬ƒcient and the estimated standard errors for the coeï¬ƒcients

are in then next column. A standard way of reporting the estimated equation is

b

rt =0.0053 + 0.3278 Â·rMt

(0.0069) (0.0890)

where the estimated standard errors are reported underneath the estimated coeï¬ƒ-

cients. The estimated intercept is close to zero at 0.0053, with a standard error of

db

0.0069 (= SE(Î±)), and the estimated value of Î² is 0.3278, with an standard error of

db b

0.0890 (= SE(Î²)). Notice that the estimated standard error of Î² is much smaller

than the estimated coeï¬ƒcient and indicates that Î² 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, Ïƒ Îµ .

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 âˆ’ 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 Ïƒ 2 = var(Îµt ). From the

Îµ

b b

above table, Ïƒ Îµ = 0.052. Recall, Îµt captures the & speci& risk of IBM and so Ïƒ Îµ is

rm c

an estimate of the typical magnitude of the & speci& risk. In order to interpret the

rm c

b

magnitude of Ïƒ Îµ 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

Ïƒ Îµ = 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 âˆ’ R2 ).

Con& dence intervals for the regression parameters are easily computed using the

reported regression output. Since Îµt is assumed to be normally distributed 95%

con& dence intervals for Î± and Î² take the form

db

b

Î± Â± 2 Â· SE(Î±)

b db

Î² Â± 2 Â· SE(Î²)

17

The 95% con&dence intervals are then

Î± : 0.0053 Â± 2 Â· 0.0069 = [âˆ’.0085, 0.0191]

Î² : 0.3278 Â± 2 Â· 0.0890 = [0.1498, 0.5058]

Our best guess of Î± 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 Î² 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 Î² 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 coeï¬ƒcient âˆ’ 0

t-statistic =

std. error

and it measures how many estimated standard errors the estimated coeï¬ƒcient 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 coeï¬ƒcient 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 coeï¬ƒcient is not equal to zero. From the data in the table, the

b

t-statistic for Î± is 0.767 so that Î± = 0.0053 is 0.767 standard errors from zero. Hence

it is quite likely that the true value of Î± equals zero. The t-statistic for Î² is 3.684,

b

Î² is more than 3 standard errors from zero, and so it is very unlikely that Î² = 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 coeï¬ƒcient is zero, the data

would generate the observed value of the t-statistic. The p-value for the t-statistic

testing Î± = 0 is 0.4465 so that it is quite likely that Î± = 0. Alternatively, the p-value

for the t-statistic testing Î² = 0 is 0.001 so it is very unlikely that Î² = 0.

4.2 Analysis of the Residuals

The single index model regression makes the assumption that Îµt âˆ¼ iid N (0, Ïƒ 2 ). That

Îµ

is the errors are independent and identically distributed with mean zero, constant

variance Ïƒ 2 and are normally distributed. It is always a good idea to check the

Îµ

behavior of the estimated residuals, bt , and see if they share the assumed properties

Îµ

b

b b

of the true residuals Îµt . The &gure below plots rt (the actual data), rt = Î± + Î²rMt

tted data) and bt = rt âˆ’ rt (the estimated residual data).

b

(the & Îµ

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 diï¬€erence being due to the fact that the formula for the standard error of the

regression uses T âˆ’ 2 as a divisor for the error sum of squares and the standard

deviation of the residuals uses the divisor T âˆ’ 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

Ãƒ !

c 2

b2 + (K âˆ’ 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 â‰ˆ 0 and K â‰ˆ 3 and JB â‰ˆ 0. Therefore,

b c

if S is moderately diï¬€erent from zero or K is much diï¬€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 âˆ¼ Ï‡2 .

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, Ï‡2 (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 (Ï‡2 â‰¥ 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