Importance Sampling and the Method of Simulated Moments

Method of Simulated Moments (MSM) estimators introduced by McFadden (1989) and Pakes and Pollard (1989) are of great use to applied economists because of their ease of use even for estimating extremely complicated economic models. One simply needs to generate simulated data according to the model and choose parameters that make moments of this simulated data as close as possible to moments of the true data. This paper uses importance sampling techniques to address two caveats regarding these MSM estimators. First, if there are discrete parts of one's model, MSM objective functions are typically discontinuous in the parameter vector, making them hard to miminize or mimimize correctly. McFadden (1989) brie°y suggests the use of importance sampling to smooth simulated moments { we elucidate and expand on this technique. Second, often one's economic model is hard to solve. Examples include complicated equilibrium models and dynamic programming problems. We show that importance sampling can reduce the number of times a particular model needs to be solved in an estimation procedure, signi cantly decreasing computational burden. ¤Dept. of Economics, Boston University and NBER. Thanks to Steve Berry for helpful discussions. All errors are my own. Method of Simulated Moments (MSM) estimators (MacFadden (1989), Pakes and Pollard (1989)) have great value to applied economists estimating structural models due to their simple and intuitive nature. Regardless of the degree of complication of the econometric model, one only needs the ability to generate simulated data according to that model. Moments of these simulated data can then be matched to moments of the true data in an estimation procedure. The value of the parameters that sets the moments of the simulated data "closest" to the moments of the actual data is an MSM estimate. Such estimators typically have nice properties such as consistency and asymptotic normality, even for a nite amount of simulation draws. This paper addresses two computational problems that can arise with such estimators. The rst occurs when there is any discreteness in one's econometric model. In this case, the above simulation process typically results in an objective function that is not continuous in the parameter vector. This can be extremely problematic in optimization, particular when one is searching over many parameters. Not only can this make estimation take longer, but likely increases the probability of erroneously nding local extremum or non extremum. The second problem occurs when one's economic model is computationally time consuming to solve. Examples include dynamic programming problems with large state spaces and complicated equilibrium problems. In the above estimation procedure, one usually needs to solve such a model numerous times, typically once for every simulation draw, for every observation, for every parameter vector that is ever evaluated in an optimization procedure. If one has I observations, performs NS simulation draws, and optimization requires R function evaluations, estimation requires solving the model NS ¤ I ¤ R times. This can be unwieldly for complicated problems. This paper suggests using importance sampling to alleviate or remove these problems. Importance sampling is a technique most noted for its ability to reduce levels of simulation error. McFadden (1989) brie°y notes that importance sampling has an alternative use that of smoothing simulated moments, i.e. addressing our rst computational problem. The technique is quite simple for a simple multinomial choice model. This paper expands and develops this technique, noting that it can be applied to much more complex models. The key step in its application is nding the right change of variables to do the importance sampling over. We exhibit this smoothing technique with a number of examples. We next exhibit that importance sampling can be used to alleviate our second problem. What we show is that importance sampling can be used to dramatically reduce the number of times a complicated economic model needs to be solved within an estimation procedure. Instead of naively solving the model NS ¤ I ¤ R times, with importance sampling one only needs to solve the model NS ¤ I times or NS times. Since R can be