Chapter 2. GMM: Estimating Rational Expectations Models Contents 1 Introduction 1 2 Step 1: Solve the model and obtain Euler equations 2 3 Step 2: Formulate moment restrictions 3 4 Step 3: Estimation and inference 4 4.1 Threats to identi cation........................... 5 4.2 Constructing the weighting matrix..................... 6 5 Step 4: Hypothesis testing and model diagnosis 7
1. Introduction In this chapter we estimate a rational expectations model of consumption and leisure choice under uncertainty. The reference is Eichenbaum, Hansen and Singleton (1988). The discussion is organized into four steps. Step 1: Solve the model and obtain Euler equations, Step 2: Formulate population moment restrictions, Step 3: Carry out estimation and inference, Step 4: Conduct hypothesis testing and model diagnostics. The economic model consists of a representative agent who maximizes utility by choosing optimally to allocate consumption and leisure over time: max E X 1 t U(c fc t ;l t g t ; lt ) (1) t=0 with where (1 ) t l U(c t ; lt ) = (c t ) 1 ; is the discount factor, is a preference parameter between 0 and 1, is a preference parameter whose value is less than 1. If is 0, then U(c t ; lt ) reduces to the log speci cation, U(c t ; lt ) = log c t + (1 ) log lt ; which is separable in consumption and leisure. If is di erent from 0 then the utility is not separable in consumption and leisure. c t is the hypothetical consumption level. It is de ned as a linear function of current and past values of consumption: c t = c t + ac t 1 : 1
Or equivalently, c t = A(L)c t with A(L) = 1 + al (L is the lag operator). The introduction of the lagged consumption is to capture the e ect of habits, i.e., changing today s consumption may a ect tomorrow s utility, even if tomorrow s consumption stays the same. lt is the hypothetical leisure level, de ned as lt = B(L)l t : EHS considered two speci cations for B(L) : B 1 (L) = 1 + L=(1 L) B 2 (L) = 1 + bl: We consider only the second. The goal is to estimate = (; ; ; a, b) using, and only using, restrictions from (1). 2. Step 1: Solve the model and obtain Euler equations Let I t denote the information set at time t. Let MC t denote the marginal utility of c t and ML t the marginal utility of l t : Suppose the consumer can trade a one-period asset. It costs one unit of c t and has a random payo of r t+1 units of c t+1 at date t + 1. The rst order conditions for optimization imply (let! t denote the real wage): E (! t MC t ML t ji t ) = 0 (2) and E(r t+1 MC t+1 MC t ji t ) = 0; (3) These two expressions imply that! t MC t ML t and r t+1 MC t+1 MC t are orthogonal to all variables in the information set I t. They provide the basis for our estimation. 2
We now write out the moment restrictions more explicitly. MC t = @ P 1 t=0 t U(c t ; l t ) @c t = (1 + al 1 )(c t + ac t 1 ) 1 (l t + bl t 1 ) (1 ) ; and ML t = @ P 1 t=0 t U(c t ; l t ) @l t = (1 + bl 1 )(1 )(c t + ac t 1 ) (l t + bl t 1 ) (1 ) 1 : To simplify the notation, de ne functions H c () and H l () as follows: H c (c t ; c t 1 ; l t ; l t 1 ; ) = (c t + ac t 1 ) 1 (l t + bl t 1 ) (1 ) ; H l (c t ; c t 1 ; l t ; l t 1 ; ) = (1 )(c t + ac t 1 ) (l t + bl t 1 ) (1 ) 1 : Then, MC t ; ML t and MC t+1 can be re-written as MC t = (1 + al 1 )H c (c t ; c t 1 ; l t ; l t 1 ; ); ML t = (1 + bl 1 )H l (c t ; c t 1 ; l t ; l t 1 ; ); MC t+1 = (1 + al 1 )H c (c t+1 ; c t ; l t+1 ; l t ; ): 3. Step 2: Formulate moment restrictions The equations (2) and (3) depend on consumption, leisure, real wages and real interest rate. Among these variables, consumption and real wage are trending. The trends need to be removed otherwise they poses substantial di culty for inference. To this end two methods are often used. The rst is to pass the data through a lter such that the low frequency component, which accounts for the trend, can be removed and the second is to transform the Euler equations such that the restrictions only depend on stationary variables. EHS used the second method. They de ned a vector x t ; x t = (c t =c t 1 ; l t ; w t l t =c t ; r t 1 ) ; (4) and assumed that x t forms a strictly stationary stochastic process. 3
Now we use (2) and (3) to derive a set of moment restrictions that depends on the observables only through x t. De ne H w (x t ; x t+1 ; x t 1 ; ) = w t(1 + al 1 )H c (c t ; c t 1 ; l t ; l t 1 ; ) (1 + bl 1 )H l (c t ; c t 1 ; l t ; l t 1 ; ) ; H l (c t ; c t 1 ; l t ; l t 1 ; ) where L 1 is the inverse operator of L: Equation (2) implies E(H w (x t ; x t+1 ; x t 1 ; 0 )ji t ) = 0; (5) where 0 denotes the true value of. A similar strategy can be used to transform (3) to obtain where H r (x t ; x t+1 ; x t+1 ; x t 1 ; 0 ) E(H r (x t ; x t+1 ; x t+2 ; x t 1 ; 0 )ji t ) = 0; (6) = r t+1(1 + al 1 )H c (c t+1 ; c t ; l t+1 ; l t ; 0 ) (1 + al 1 )H c (c t ; c t 1 ; l t ; l t 1 ; 0 ) ; H c (c t ; c t 1 ; l t ; l t 1 ; 0 ) which again only depends on x t ; x t estimation equations. 1 and x t+1. We will use (5) and (6) to derive the 4. Step 3: Estimation and inference De ne " # " d 1;t+2 () d t+2 () = = d 2;t+2 () # H w (x t ; x t+1 ; x t 1 ; ) : H r (x t ; x t+1 ; x t+2 ; x t 1 ; ) Let z 1;t and z 2;t be some r 1 and r 2 dimensional vectors in I t. Using z 1;t as instruments for d 1;t+2 () and z 2;t for d 2;t+2 (), we have " # d 1;t+2 ()z 1;t m(x t ; ) = : d 2;t+2 ()z 2;t Then, (5) and (6) imply E (m(x t ; 0 )) = 0: (7) 4
In EHS, E d 1;t+2 ( 0 ) 0 0 B B E @d 2;t+2 ( 0 ) @ 1 V t 1 V t!! = 0; 11 CC AA = 0; V t 1 where Let V t = h ct c t 1 c t l t l t 1 l t w t w t 1 w t r t 1 i 0 m T () = 1 T TX m(x t ; ) and W T a symmetric positive de nite weighting matrix. The GMM estimator is then given by 4.1. Threats to identi cation t=1 ^ = arg min m T () 0 W T m T (): We have implicitly assumed that (7) has a unique solution = 0. To ensure identi - cation the dimension of (7) has to be no less than that of. This is necessary but not su cient. The failure of identi cation are typically due to the following two reasons. 1. The conditional moment restrictions (5) and (6) have multiple solutions. If this is the case, then identi cation can only be achieved by adding more structure to the model, i.e., we need more "Euler equations". 2. The instruments are uncorrelated with d t+2 (); i.e., some or all of the instruments are irrelevant. better instruments. If this is the case, then identi cation can be achieved by using In practice, we often face two similar situations: 1. Even though (7) has a unique solution, there are multiple parameter values that make (5) and (6) very close to zero. 5
2. The instruments are only weakly correlated with variables in d t+2 (): In both cases, the estimates will be very imprecise and the conventional inferential methods can be very misleading. These two situations are often referred to as "weak identi cation". Problem 1. (1) how to detect weak identi cation? (2) how to conduct inference if the identi cation is possibly weak? 4.2. Constructing the weighting matrix The optimal weighting matrix is given by S 1 0 ; with S 0 = lim T!1 V ar(t 1=2 m T ( 0 )) = lim T!1 V ar 0 @ 1 p T T X j=1 1 m(x t ; 0 ) A : Two issues arise. First, the optimal weighting matrix depends on unknown parameters 0 : Second, m(x t ; 0 ) are in general autocorrelated. We rst attack the second issue, assuming 0 is known. In EHS E (m(x t ; 0 )m(x t+k ; 0 ) 0 = 0 for jkj 2: It follows that the optimal weighting matrix is the inverse of S 0 = 1X k= 1 E(m(X t ; 0 )m(x t+k ; 0 ) 0 ) In general, there may exist correlations beyond order 2, i.e., 1X S 0 = E(m(X t ; 0 )m(x t+k ; 0 ) 0 ): k= 1 To address the rst issue two approaches have been adopted in practice: 1. (Two-Step Procedure) Obtain some preliminary estimate of with W T being the identify matrix. Denote the estimate as ^ 1 : Then, compute ^S T (^ 1 ) and solve ^ = arg min m T () 0 ^ST (^ 1 ) 1 m T (): 6
2. (CUGMM) Obtain the estimate in one step, i.e., treat S T () as a function of and solve ^ = arg min m T () 0 ^ST () 1 m T (): 5. Step 4: Hypothesis testing and model diagnosis Consider the following two hypotheses of interest: Habit: does consumption good acquisitions today give rise to consumption service in the future? Are the Euler equations (2) and (3) consistent with the data? The rst hypothesis is about a parametric restriction, i.e., H 0 : a = 0 H 1 : a 6= 0: This can be tested using the Wald, Gradient or and Distance test discussed in the previous chapter. The second hypothesis is about the possible misspeci cation of the model. It can be formulated as testing for overidentifying restrictions. Consider the objective function evaluated at ^, when all moment restrictions are used: J = T m T (^) 0 1 ^S T m T (^): If both restrictions are correct, J should be close to zero. If at least one moment restriction is misspeci ed, the statistic will diverge with T. Formally, under the null hypothesis, J! d 2 (dim(m T ) dim()): 7
References [1] Andrews, D.W.K., (1991), "Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation", Econometrica, 59, 817-858. [2] Hansen, L.P., (1982), "Large Sample Properties of Generalized Methods of Moments Estimators," Econometrica, 50, 1029-1054. [3] Hansen, L.P. and Singleton, K.J., (1982), "Generalized Instrumental Variables Estimation of Nonlinear Rational Expectations Models," Econometrica, 50, 1269-1286. [4] Eichenbaum, M.S., Hansen, L.P. and Singleton,.K.J., (1988), "A Time Series Analysis of Representative Agent Models of Consumption and Leisure Choice under Uncertainty", The Quarterly Journal of Economics, 103, 51-78. 8