Lecture 12: Stochastic Discount Factor and GMM Estimation

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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

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Stochastic Discount Factor Presentation
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Stochastic Discount Factor
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Examples
Asset price
Stock return
Excess stock return
Risk free rate
See page 9 10 of Cochrane.
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Relating to EGS

• Recall the consumption and saving in Chapter 6,
  EGS
• ß 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.

Subject to wealth constraint on page 91
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Risk-free rate
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Risk Corrections
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.
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Risk Corrections Return Expression
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.
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Expected Return-Beta Representation
Where ßis 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. ?is interpreted as the price of risk
and ß is the quantity of risk in each asset.
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Mean-Variance Frontier

• Implications
• Means and variances of asset returns lie within
  efficient frontier.
• On the efficient frontier, returns are perfectly
  correlated with the discount factor.
• The priced return is perfectly correlated with
  the discount factor and hence perfectly
  correlated with any frontier return. The residual
  generates no expected return.

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Sharpe Ratio and Equity Premium Puzzle
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
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
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Time-varying Expected Returns
The relation above is conditional. Conditional
mean or other moment of a random variable could
be different from its unconditional moment. E.g,,
knowing tonights 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.
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Estimating SDF -- GMM
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Estimating SDF Second Stage
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Implementing GMM
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GMM Example
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GMM Example (2)
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GMM Example (3)
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Program GMM using SAS
/ N5, 7 instruments / proc model
datagmm parms beta 1.0 gamma 1.0 endogenous
cons0 cons1 exogenous r1 r2 r3 r4
r5 instruments lrm1 lrm2 lrm3 lrm4 lrm5
lrm6 eq.m11-(1r0)(beta(cons0/cons1)(-gamma
) eq.m11-(1r0)(beta(cons0/cons1)(-gamma)
eq.m11-(1r0)(beta(cons0/cons1)(-gamma) eq
.m11-(1r0)(beta(cons0/cons1)(-gamma) eq.m1
1-(1r0)(beta(cons0/cons1)(-gamma) fit
m1-m6/gmm kernel(parzen, 1,0) ods output
EstSummaryStatsparms run
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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 s2
  is unknown. As an alternative, we have the
  following F statistic

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Examples of J Restrictions
Each row of R is a single linear restriction on
the coefficient vector.
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More Examples
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Example Test of Structural Change
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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 F1,n-Kt2n-K
• To see if the constraints on a set of variables
  hold, we use
• or

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Exercises

• CR 1.7
• Read through Chapter 11, CR