L12: SDF 1

Lecture 12: Stochastic Discount Factor and GMM Estimation

• The following topics will be covered:• SDF

– Basic expression

– Risk free rate

– Risk correction

– Mean-variance frontier

– Time-varying expected returns

• GMM– GMM overview

– Applying GMM

• Also, more on “Hypothesis Testing”

L12: SDF 2

Stochastic Discount Factor Presentation

])('

)('[ 1

1

tt

ttt x

cu

cuEp

xt+1 is the payoff in t+1. β captures impatience and is called the subjective discount factor. U is utility function, ct denotes consumption in date t. To see this:

)]([)(max 1 ttt cuEcu

, s.t.

ttt pec

111 ttt xec

The first order condition is ])('[)(' 11 ttttt xcuEcup , equivalent to (1).

Why? In CLM, we have )]1)(('[)(' 11 tttt rcuEcu , known as the Euler equation.

L12: SDF 3

Stochastic Discount FactorIt states: the loss in utility if the investor buys another unit of the asset equates the increase in utility he obtains from the extra payoff at t+1. Stochastic Discount Factor Presentation:

][ 11 tttt xmEp )('

)(' 11

t

tt cu

cum

or even more simply:

p=E(mx) The variable mt+1 (m) is known as the stochastic discount factor, or pricing kernel. It is also known as the intertemporal marginal rate of substitution. m is always positive.

L12: SDF 4

Examples

)(1 mRE

)(( 11 ttt dpmEp

bt

at

et

et RRRwheremRE 1111)(0

)(1 1f

tmRE

Asset price

Stock return

Excess stock return

Risk free rate

See page 9 – 10 of Cochrane.

L12: SDF 5

Relating to EGS• Recall the consumption and saving in Chapter 6, EGS:

– β is a discount factor for delaying consumption

• The first order condition of this problem is similar to the SDF presentation

• Also on page 176, EGS, we have

• Z is the aggregate wealth in state z, q(z) denotes the expected payoff of the firm conditional on z. In equilibrium, the marginal utility of the representative agent in a given state equals the equilibrium state price itself.

)]([)(max 1 ttt cuEcu

Subject to wealth constraint on page 91

)(')()()()( zvzEqzzEqqP

L12: SDF 6

Risk-free rate

)(1 1f

tmRE For power utility

1)(

1ccu , we have ccu )('

)(

1 1

t

tf

c

cR or

)(

11 1

t

tf

c

cr .

Real interest rates are high when people are impatient, when β is low; they are high when consumption

growth is high.

Real interest rates are more sensitive to consumption growth if the power parameter γ is high.

With lognormal consumption and power utility function, we have

1)ln()2/()ln( ][ 122

1 tttt ccEft eeR

L12: SDF 7

Risk Corrections

),cov()(

),cov()()(

][

xmR

xEp

xmxEmEp

mxEp

f

The first term is the present value of E(x) (expected payoff). The second is a risk adjustment. An asset whose payoff co-varies positively with the discount factor has its price raised and vice versa.

The key u’(c) is inversely related to c. If you buy an asset whose payoff covaries negatively with consumption (hence u’(c)), it helps to smooth consumption and so is more valuable than its expected payoff indicates.

L12: SDF 8

Risk Corrections – Return Expression

)]('[

)),('cov()(

),cov()(

),cov()()(1

][1

1

11

t

ittfi

iffi

ii

i

cuE

RcuRRE

RmRRRE

RmREmE

mRE

All assets have an expected return equal to the risk-free rate, plus a risk adjustment.

Assets whose returns covary positively with consumption make consumption more volatile, and so must promise higher expected returns to induce investor to hold them, and vice versa.

L12: SDF 9

Expected Return-Beta Representation

mmifi

ifi

RRE

mE

m

m

mRRRE

,)(

))(

)var()(

)var(

),cov(()(

Where βis the regression coefficient of the asset return on m.

It says each expcted return should be proportional to the regression coefficient in a regression of that return on the discount factor m.

λis interpreted as the price of risk and β is the quantity of risk in each asset.

L12: SDF 10

Mean-Variance Frontier

)()(

)(|)(| have we1exceedcannοann As

)()(

)()(

)()()()()(1

),cov()()()(1

,

,

ifi

i

Rm

fi

i

Rm

ii

iii

RmE

mRRE

RmE

mRRE

mRREmEmRE

mRREmEmRE

i

i

Implications:

(1) Means and variances of asset returns lie within efficient frontier.

(2) On the efficient frontier, returns are perfectly correlated with the discount factor.

(3) The priced return is perfectly correlated with the discount factor and hence perfectly correlated with any frontier return. The residual generates no expected return.

L12: SDF 11

Sharpe Ratio and Equity Premium Puzzle

fi

fi

RmmE

m

R

RRE)(

)(

)(

)(

|)(|

Let Rmv denote the return of a portfolio on the mean-variance efficient frontier and consider power utility. The slope of the frontier (Sharpe ratio) is

)ln(])/[(

])/[(

)(

)(

)(

|)(|

1

1 cccE

cc

mE

m

R

RRE

tt

ttmv

fmv

Sharpe ratio is higher if consumption is more volatile or if investors are more risk averse.

Over the last 50 years, average real stock return is 9% with a standard deviation of 16%. The real risk free rate is 1%. This suggests a real Sharpe ratio of _____

Aggregate nondurable and services consumption growth has a standard deviation of 1%. So

L12: SDF 12

Time-varying Expected Returns

),()()()(

),()()(

)(

)(

),(cov)(

1111

1111

111

ttttttttf

ti

t

ttttttt

tt

t

tttft

it

RmRcRRE

RmRmE

m

mE

RmRRE

The relation above is conditional. Conditional mean or other moment of a random variable could be different from its unconditional moment. E.g,, knowing tonight’s weather forecast, you can better predict rain tomorrow than just knowing the average rain for that date.

It suggests a link between conditional mean of stock returns and conditional variance of stock returns.

Little empirical support.

L12: SDF 13

Estimating SDF -- GMM

ttt

ttt

ttt

pxbmb

conditionmomentpxbmEeI

xbmEpE

111t

11

11

)()(u

vectors. typicallyare p andx .0])([.,.

])([)(

.parameters for the solve tois do try to weAll utility.power say form, specific a

it takes Assume factor.discount stochastic theestimate tomove weNow

Define gT(b) as the sample mean of the ut error, when the parameter vector is b in a sample of size T:

])([)(1

)( 111

tttT

T

ttT pxbmEbu

Tbg

The first-stage estimate of b minimizes a quadratic form of the sample mean of the error.

)()'(minarg }{

^

1 bWgbgb TTb

For some arbitrary matrix W (often, W=I).

L12: SDF 14

Estimating SDF – Second StageDefine an estimate S of

)(#~])[var('

)1,0(~

)var(

|)(

)'(1

)var(

)()'(minarg

]')()([

21^^

^

^

11^

2

1^

2

^

binincludedbbb

N

b

b

b

bgdwheredSd

Tb

bgSbgb

bubuES

jjjj

ii

i

bb

T

TTb

jjtt

JT Test: )#(#~)](')([ 2^

1^

parametersmomentsbgSbgTTJ TTT

L12: SDF 15

Implementing GMM

Moment condition 01)( 11 tt RbmE (1)

Involving instrumental variable z, we have 1)()( 11 ttt RbmEzE

tttt zRbmEzE 11 )()( , i.e., ttt zRbmE )1)((0 11 (2)

We have two returns R=[Ra, Rb] and one instrument z. Combining (1) and (2), we have:

0

0

0

0

1

1

)(

)(

)(

)(

11

11

11

11

t

t

tbtt

tatt

btt

att

z

z

zRbm

zRbm

Rbm

Rbm

E (we have 2 securities and z has (1, zt)’, thus having 2*2 equations)

Using the Kronecker product, we denote the above as: 01)( 11 ttt zRbmE

or 01 tt zuE

L12: SDF 16

GMM Example

Adapt from Moore (1998, USC lecture note)

Starting from 0111 ttt RmE

Where tj

tjtjtj P

DPR

,

1,1,1,

. Assume a power utility function, we have

t

tt c

cm 1

1

1)/( 1,11,

tjtttj Rccu

In other words, 0)( 1, tjt uE

L12: SDF 17

GMM Example (2)N securities, 3 instruments

NTNN

T

T

uuu

uuu

uuu

u

....

.

...

...

.

.

.

....

...

21

22221

11211

2,1,

12

1

1

...

...

1

1

1

TmTm

mm

m

RR

RR

R

Z

N=1

]/,/,/[

1

...

...

1

1

1

],...,)[/1(/

2,1,

2,1,

12

1

11

TRuTRuTu

RR

RR

R

uuuTTuG

tmttmtt

TmTm

mm

m

TZ

L12: SDF 18

GMM Example (3)

If N=2, we have

2,2

1,2

2

2,1

1,1

1

/1

tmt

tmt

t

tmt

tmt

t

Ru

Ru

u

Ru

Ru

u

Tg to solve this problem, we do Wgg 'min),(

Hensen(1982) proves that T times the minimized value of g’Wg is asymptotically

distributed as 2 random variable with degrees of freedom 4 for the case of N=2 with 3 instrumental variables.

L12: SDF 19

Program GMM using SAS /* N=5, 7 instruments */proc model data=gmm;

parms beta 1.0 gamma 1.0;endogenous cons0 cons1;exogenous r1 r2 r3 r4 r5;instruments lrm1 lrm2 lrm3 lrm4 lrm5 lrm6;eq.m1=1-(1+r0)*(beta*(cons0/cons1)**(-gamma);eq.m1=1-(1+r0)*(beta*(cons0/cons1)**(-gamma);eq.m1=1-(1+r0)*(beta*(cons0/cons1)**(-gamma);eq.m1=1-(1+r0)*(beta*(cons0/cons1)**(-gamma);eq.m1=1-(1+r0)*(beta*(cons0/cons1)**(-gamma);fit m1-m6/gmm kernel=(parzen, 1,0);ods output EstSummaryStats=parms;

run;

L12: SDF 20

More on Hypothesis Testing

• Testing J linear Restrictions

• We can base a test of H0 on the Wald criterion:

• The chi-squared statistic is not usable when σ2 is unknown. As an alternative, we have the following F statistic

R'X)R(X'R'bRqRbm

qRbm

qRβ

1

2

0

]}[{][)(

bestimator squaresleast Given the

:

VarVarVar

let

H

mmm' 12 ]}[{][ VarJW

J

sF

)(][)'( 12 qRbR'X)R(X'qRb 1

L12: SDF 21

Examples of J Restrictions

.:0 qRH

Each row of R is a single linear restriction on the coefficient vector.

One of the coefficients is zero, :0j

00100 R and q=0.

Two of the coefficients are zero, :jk

01100 R and q=0.

A set of the coefficients sum to one, :1432

01110R and q=1.

A subset of the coefficients are all zero, ,01 ,02 and :03

00100

00010

00001

R and

0

0

0

q

or, equivalently, 00: .

L12: SDF 22

More Examples

Several constraints hold simultaneously: ,132 ,064 and ,065

0

0

1

110000

101000

000110

6

5

4

3

2

1

L12: SDF 23

Example: Test of Structural Change

Denote the first 14 years of the data in y and X as 1y and 1X and the remaining years

as 2y and 2X . An unrestricted regression that allows the coefficients to be different in

two periods is

2

1

2

1

2

1

2

1

0

0

X

X

y

y. We have 2211 eeeeee .

Under a restricted model, the two coefficient vectors are the same, then (7-19) may be

written

2

1

2

1

2

1

X

X

y

y. The residual sum of squares from this restricted

regression, ee then forms the basis for the test. The test statistic is then given in (7-14),

where J, the number of restrictions, is the number of columns in 2X and the denominator

degrees of freedom is .221 knn

is qR , where :R and q=0. See page 289, Greene (2000)

L12: SDF 24

Test Based on Loss of Fit

• Least squares vector b is chosen to maximize R2.

• The overall fitness of a regression:

• To see if the coefficient of a particular variable is a given value, we can also apply the F-stat, where F[1,n-K]=t2[n-K]

• To see if the constraints on a set of variables hold, we use

• or

)/()1(

)1/(),1(

2

2

KnR

KRKnKF

)/()1(

/)(),1(

2

2*

2

KnR

JRRKnKF

)/()1(

/)(),1(

2 KnR

JKnKF

ee'ee *

'*

L12: SDF 25

Exercises

• CR 1.7

• Read through Chapter 11, CR