Chapter 4. Simulated Method of Moments and its siblings

Transcription

1 Chapter 4. Simulated Method of Moments and its siblings Contents 1 Two motivating examples Du e and Singleton (1993) Model for nancial returns with stochastic volatility Simulated Method of Moments (SMM) Du e and Singleton (1993) cont d The limiting distribution Summary: SMM is based on computing the model-dependent moment with simulations Indirect Inference (IE) Du e and Singleton (1993) cont d Summary: IE is SMM using estimates from an auxiliary model as target moments E cient method of moments (EMM) Summary: EMM uses the score of the density implied by the auxiliary model to de ne the GMM criterion

2 1. Two motivating examples GMM requires the sample moment restrictions to have a closed form (that is, X t in T 1 P T m(x t; 0 ) must be observable) in order to compute the GMM criterion function. However, this requirement is sometimes violated in some economic/ nancial models Du e and Singleton (1993) The rm maximize the value of dividends by choices of capital, i.e., max d t max ff(k t ; z t ) r t k t g k t k t with f(k t ; z t ) = z t k t where f(k t ; z t ) is the production function, k t is the capital z t is a technological disturbance r t is the rental rate of capital The agent s problem is max E fc t;k tg ( 1 X ) t (c t 1) 1 v t ; 1 subject to c t + k t+1 + p t s t+1 = (d t + p t )s t + (r t + )k t ; where c t is consumption, 1

3 v t is a taste disturbance, is a subjective discount factor, p t is the price of a share of the rm, 1 is the depreciation rate. The consumption Euler equation is pt+1 + d t+1 MC t = E t MC t+1 p t with MC t = t (c t 1) v t : This leads to the conditional moment restriction pt+1 + d t+1 E t MC t+1 MC t p t = 0 or the unconditional restriction pt+1 + d t+1 E MC t+1 MC t z t = 0 p t for any z t belonging to the information set at time t. The taste disturbance v t is unobservable to the econometrician. In addition, it a ects the decision of the agent, therefore a ects c t+1 in a complicated manner. Consequently, the above moment restriction cannot be computed analytically Model for nancial returns with stochastic volatility Consider a model for daily returns on the SP500 index: x t = + t " t ; where the volatility process t satis es log t =! + log t 1 + t : 2

4 For simplicity, assume " t ; t i:i:d:n(0; 1) and " t and s are mutually independent for all t and s. If t was observable, then t could be expressed as a function of s (s t) and the four parameters parameters in the model can be estimated using restrictions of the form xt E t xt = 0; E t 2 xt = 1; E t xs s = 0 for all t 6= s: (In fact, if t was observable, then standard MLE could be easily implemented.) In practice, t is unobservable and x t is not a Markovian process conditioning on its own past. GMM can not be directly applied. 2. Simulated Method of Moments (SMM) The basic idea behind SMM is to generate simulated series from the economic model, and then match their moments with those computed from the data Du e and Singleton (1993) cont d The estimator can be computed in three steps. Step 1. Solve the model. The model is subject to two exogenous processes: z t and v t. Let u t = (z t ; v t ) 0 and assume it follows a stationary Markov process, with transition function u t = h(u t 1 ; " t ; ); t = 1; :::; T; (1) where " t is a sequence of two-dimensional i:i:d: shocks and is an unknown parameter. A simple example of (1) is a vector AR(1) process! " #! z t = z 0 z t 1 v t 0 v v t 1 + " 1t " 2t Let be a vector that contains all the unknown parameters, i.e., = (; ; ; ; ):! : 3

5 Let Y t be a vector that contains the state variable k t and the exogenous process u t : Y t = (k t ; u t ) 0 : Then, in equilibrium, Y t+1 will be a function of Y t and " t+1 : Y t+1 = H(Y t ; " t+1 ; ): Given, the equilibrium relationship H(:) can be evaluated using the analytical solution or by numerical methods. This is true for Du e and Singleton (1993), also true for a large family of dynamic stochastic equilibrium models. Step 2. Simulation. For any admissible, we can generate a simulated process Y t by taking independent draws from the distribution of " t and computing recursively Y t+1 = H(Y t ; " t+1 ; ): The simulated sample size T (T ) can be di erent from the observed sample size T. Multiple mutually independent series can be obtained by using the same transition function H, but using a sequence of shocks f" t g that have the same distribution as but are independent of f" t g. Step 3. Construct the estimator. Now we have generated series using the structure of the model. Clearly, the outcome depends on the value of. The idea is then the following: if the model is correctly speci ed with equal to its true value, then the stochastic properties of the simulated series, including its moments, will be the same as the the actual series that we observe in practice. Working in the reverse order, we can estimate by minimizing the di erence between the simulated data moments and actual data moments. Speci cally, for some chosen observation function g, we form g t = g(y t ; :::; Y t l+1 ): Therefore, in each period t; an observation is made of a nite "l-history" of state information. Likewise, a corresponding observation g t simulated states. can be formed for each l-history of 4

6 Then, the SMM estimator is de ned as the value of that minimizes the distance between the sample mean of fg t g T and the sample mean of gt T (T ). More precisely, let m T () = 1 T (T ) TX 1 X g t gt T T (T ) and the SMM estimator is given by ^ = arg min m T () 0 W T m T (); where W T is a weighting matrix. As in the GMM case, the optimal weighting matrix, denoted by S 1 T, satis es S T! p S 0 = 1X j= The limiting distribution E([g t E(g t )] [g t j E(g t j )] 0 ): The following theorem gives the limiting distribution of ^ using S T matrix. as the weighting Theorem 1. Suppose T=T (T ) = as T! 1. Then, under su cient regularity conditions (c.f. Du e and Singleton (1993)), p T (^ 0 )! d N(0; (1 + )V 0 ) where V 0 = G 0 0S G 0 = E S 0 = 1X j= 1 0 G 0 g 0 t ; 1 ; E([g t E(g t )] [g t j E(g t j )] 0 ): 5

7 2.3. Summary: SMM is based on computing the model-dependent moment with simulations. In the above, the key issue is which moments to match. Indirect inference and E cient method of moments (EMM) can be viewed as two answers to this question. They both attack this issue using the concept of an "auxiliary model". 3. Indirect Inference (IE) The method is rst proposed by Smith (1993) and further developed by Gourieroux, Monfort and Renault (1993). The basic idea underlying indirect inference is to use an auxiliary model to form a criterion function. The insight is that the parameters of the auxiliary model can be estimated using either the observed data or data simulated from the economic model. Indirect inference chooses the parameters of the economic model so that these two sets of estimates are as close as possible. For it to work well, two requirements need to be satis ed: (1) it is possible to simulate data from the economic model given the values of its parameters and (2) the auxiliary model captures important aspect of the data and is easy to estimate. In short, IE can be viewed as a SMM estimator using estimates from an auxiliary model as target moments. To put these ideas in concrete forms, we still consider the example in Du e and Singleton (1993) Du e and Singleton (1993) cont d The estimator can again be computed in three steps. Step 1. Solve the model. As before, the state evolution can be expressed as Y t+1 = H(Y t ; " t+1 ; ); (2) which can be used to simulate data given. Next, for some chosen observation function g, we form g t = g(y t ; :::; Y t l+1 ): 6

8 Likewise, a corresponding observation g t can be formed for each l-history of simulated states: Step 2. Choose an auxiliary model. g t = g(y t ; :::; Y t l+1 ): The auxiliary model, in turn, is de ned by a conditional probability density function for the observed data, p(g t jg t 1 ; ); which depends on an unknown parameter vector. For example, it can be an vector autoregression: g t = A 0 + qx A j g t j + e t ; e t i:i:d:n(0; 2 ): j=1 The auxiliary model needs to be informative about the statistical properties of the observed data. However, it is not required to be the "true data generating process". It also needs to be tractable such that its parameters can be estimated from the observed data, say using maximum likelihood: ^ = arg max TX log p(g t jg t 1 ; ): (3) Step 3. Simulation and matching parameters. First, use a random number generator to draw a sequence of random errors {" t }. The sequence is drawn only once and then held xed throughout the estimation procedure. Second, pick a parameter vector and then iterate on equation (2), to generate a simulated sequence of variables Y t : Third, maximize the average of the log likelihood: T (T ) X ~() = arg max log p(gt jgt 1; ): (4) Finally, vary the value of to make ~ () and ^ as close as possible. Note that the dimension of needs to be at least as large as that of the structural parameter vector for the above problem to have a unique solution. 7

9 3.2. Summary: IE is SMM using estimates from an auxiliary model as target moments The limiting distribution of IE follows from the fact that it is a special SMM estimator. We omit the details. 4. E cient method of moments (EMM) EMM proposes to use the score of the density implied by the auxiliary model to de ne the GMM criterion. Speci cally, contrasting with Step 3 under IE, is chosen to make the following quantity as close to zero as possible: T (T ) log p(gt jgt 1 ; ) =^; where p(g t jg t 1 ; ) is the density implied by the auxiliary model, and ^ is given by (3). The rest is the same as in Steps 1 and 2 in IE Summary: EMM uses the score of the density implied by the auxiliary model to de ne the GMM criterion. 8

10 References [1] Du e, D., and Singleton, K. (1993), "Simulated Moments Estimation of Markov Models of Asset Prices," Econometrica, Vol. 61, 1993, [2] Gallant, A.R. and Tauchen, G. (1996), "Which moments to match", Econometric Theory, 12, [3] Gouriéroux, C. and A. Monfort (1996), Simulation-Based Econometric Methods. Oxford University Press, [4] Gouriéroux, C, Monfort, A and Renault, E (1993), "Indirect Inference," Journal of Applied Econometrics, 8, S [5] Michener (1984), "Permanent Income in General Equilibrium". Journal of Monetary Economics, 13, [6] Smith, A.A. (2007), "Indirect Inference", the New Palgrave Dictionary of Economics, second edition. 9