GMM Estimation. 1 Introduction. 2 Consumption-CAPM

Transcription

1 GMM Estimation 1 Introduction Modern macroeconomic models are typically based on the intertemporal optimization and rational expectations. The Generalized Method of Moments (GMM) is an econometric framework which allows to estimate the parameters of such models. As an example, we consider consumption based asset pricing models. 2 Consumption-CAPM Consider a model where a representative agent lives for two periods, t and t + 1, and maximizes life-time utility: u(c t ) + βe t u(c t+1 ), where E t is the expectation conditional on period 1 information. 1 The representative agent can buy an asset at price p t which pays a dividend of d t+1 in the second period. So the expected return earned on this asset is E t (R t+1 ) = E[(p t+1 + d t+1 )/p t ]. In the first period, the following budget constraint has to hold: c t + a t = w t, (1) where a t denotes asset holdings which are carried into the second period and w t is initial wealth in the first period. A similar constraint applies in the second period: Combining these two constraints gives c t+1 = R t+1 a t. (2) c t + c t+1 R t+1 = w t, (3) 1 So strictly speaking, we have u(c t ) + βe(u(c t+1 ) Ω t ), where Ω t is the information set available in the first period. 1

2 which is the lifetime budget constraint. The representative consumer maximizes utility subject to this constraint. The necessary condition is the following Euler equation: u (c t ) = βe t [R t+1 u (c t+1 )], (4) or E t (R t+1 M t+1 ) = 1, (5) where M t+1 = βu (c t+1 )/u (c t ) is the marginal rate of substitution of current for future consumption. In the context of asset pricing models the marginal rate of substitution is usually called a stochastic discount factor. Note that (5) has to hold for any asset with a potentially stochastic return R t+1. Suppose we also have an asset with a risk free return, R f. Then we have R f = 1/E t (M t+1 ). Equation (5) can be rewritten as: 2 Using the risk-free return, R f, we obtain E t (R t+1 )E t (M t+1 ) + cov t (M t+1, R t+1 ) = 1, (6) E t (R t+1 ) = 1 cov t(m t+1, R t+1 ). (7) E t (M t+1 ) E t (R t+1 ) = (1 cov t (M t+1, R t+1 ))R f, (8) or E t (R t+1 R f ) = cov t (M t+1, R t+1 )R f, (9) which can be rewritten more explicitly in terms of consumption E t (R t+1 R f ) = cov t (M t+1, R t+1 )R f = cov t (βu (c t+1 )/u u (c t ) (c t ), R t+1 ) βe t u (c t+1 ), (10) Since β and u (c t+1 ) are known in period t the expression on the right-hand-side can be written as 3 cov t (βu (c t+1 )/u u (c t ) (c t ), R t+1 ) βu (c t+1 ) = cov t(u β u (c t ) (c t+1 )), R t+1 ) u (c t ) βu (c t+1 ). (11) 2 For any two random variables X and Y, E(XY ) = E(X)E(Y ) + Cov(X, Y ). 3 Recall that cov(ax, by ) = ab cov(x, Y ) for any constants, a and b. 2

3 And therefore we obtain E t (R t+1 R f ) = cov t(u (c t+1 )), R t+1 ). (12) u (c t+1 ) Equation (12) holds that equilibrium excess returns of any asset is determined by the covariance between the expected return of the asset and the marginal utility of consumption in the next period. How is the C-CAPM related to the CAPM? Recall that according to the CAPM the return of an asset depends on how it covaries with the market return. The C-CAPM basically replaces the return on the market with the marginal utility of consumption. However, the intuition is essentially the same. The return on an asset depends on how well the asset can be used to insure against changes in future consumption (instead of market risk in the CAPM). If cov t (u (c t+1 ), R t+1 )) > 0, then the asset return is high when consumption is low (under the standard assumptions that u > 0 and u < 0). In this case the asset can be used to insure or hedge fluctuations in consumption, and in this sense the asset is not risky! Therefore the equilibrium return is below the risk-free rate, since investors can reduce their consumption risks by holding this asset. 3 GMM Estimation The C-CAPM has been estimated and tested using a wide range of methodologies. One way to test the C-CAPM is to estimate the Euler equation underlying the asset pricing equation using the generalized method of moments (GMM) estimator. In general, GMM is a method for estimating models by exploiting moment conditions. Consider a model with m parameters collected in the vector θ and data X t, where specific combinations of θ and X t should be equal to zero on average. This condition be equal on average is the moment condition we exploit. More specifically: E[f i (θ, X t )] = 0, (13) for i = 1,..., n. So (13) represents n orthogonality conditions. To derive the GMM estimator, replace (13) by its empirical counterpart: m i = 1 T T [f i (θ, X t )] = 0. (14) t=1 3

4 So we have n sample moment conditions. If n > m, we have more moment conditions then parameters to estimate. Therefore, we will not obtain a unique solution for θ. The GMM estimator is defined as ˆθ = argmin{ m(θ) W 1 m(θ)}, (15) where m = ( m 1, m 2,..., m n ) and W is a weighting matrix. The minimization is usually done numerically. GMM is an extremely general framework. Many estimators you are familiar with can be viewed as special cases of GMM. So how can we use GMM to estimate the Euler equation. First, we have to assume a specific utility function: So the Euler equation becomes u(c) = c1 σ 1 σ. (16) c σ t = βe t (R t+1 c σ t+1). (17) or equivalently ( ( ) ) ct+1 σ E t βr t+1 1 = 0. (18) c t Since we are assuming that expectations are formed rationally, any information contained in the information set is already used optimally to form expectations. Therefore, the term in brackets must be orthogonal to any variable in the information set. Suppose that z t Ω t, then we must have 4 E [( βr t+1 ( ct+1 c t ) σ 1) ) z t ] = 0. (19) So the key insight here is that the assumption of rational expectations gives rise to orthogonality conditions. Therefore GMM is a natural method for the estimation of models with rational expectations. Note that if n > m, we have more moment conditions then parameters to estimate. Thus, we have overidentifying restrictions. The validity of these overidentifying restriction, that is the validity of the instruments, can be assessed using the J-test. A valid instrumental variable has to satisfy the orthogonality condition. EViews reports a J-value 4 More formally, the unconditional moment condition is implied by the Law of Iterated Expectations. 4

5 which is the minimized value of the objective function. 5 The appropriate test statistic can be constructed as the J-value multiplied by the number of observations. Under the null hypothesis that the overidentifying restrictions are satisfied, the J-statistic times the number of observations is asymptotically χ 2 with degrees of freedom equal to the number of overidentifying restrictions. 5 See also the Eviews manual or help for more details. 5