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

Stochastic Discount Factor Presentation L12: SDF

Stochastic Discount Factor L12: SDF

Examples Asset price Stock return Excess stock return Risk free rate See page 9 – 10 of Cochrane. L12: SDF

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. Subject to wealth constraint on page 91 L12: SDF

Risk-free rate L12: SDF

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. L12: SDF

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. L12: SDF

Expected Return-Beta Representation 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

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. L12: SDF

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 L12: SDF

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 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

Estimating SDF -- GMM L12: SDF

Estimating SDF – Second Stage L12: SDF

Implementing GMM L12: SDF

GMM Example L12: SDF

GMM Example (2) L12: SDF

GMM Example (3) L12: SDF

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); fit m1-m6/gmm kernel=(parzen, 1,0); ods output EstSummaryStats=parms; run; L12: SDF

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 L12: SDF

Examples of J Restrictions Each row of R is a single linear restriction on the coefficient vector. L12: SDF

More Examples L12: SDF

Example: Test of Structural Change L12: SDF

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 L12: SDF

Exercises CR 1.7 Read through Chapter 11, CR L12: SDF