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Sequential Monte Carlo Filtering:
an Example∗
Jesús Fernández-Villaverde
University of Pennsylvania
Juan F. Rubio-Ramírez
Federal Reserve Bank of Atlanta
January 5, 2004
Abstract This short note presents an example of how to use a Sequential Monte Carlo to evaluate the likelihood of a nonlinear and nongaussian process.
∗Corresponding author: Jesús Fernández-Villaverde, Department of Economics, 160 McNeil Building, 3718 Locust Walk, University of Pennsylvania, Philadelphia PA 19104. E-mail: [email protected]. We must notice that any views expressed herein are those of the authors and not necessarily those of the Federal Reserve Bank of Atlanta or of the Federal Reserve System.
This short note presents an example of how to evaluate the likelihood function of a non- linear and nonnormal stochastic process using a Sequential Monte Carlo Filter. More details, including technical requirements can be found in Fernández-Villaverde and Rubio-Ramírez (2004). Suppose that we are interested in evaluating the likelihood function L
yT^
of the nonlin- ear, nonnormal process:
xt = 0 .5 + 0. (^3) 1 +xt −x^12 t− 1
where wt ∼ N (0, 1) and vt ∼ t (2) given some observables yT^ = {yt}Tt=1. The first equation, known as the transition equation, accounts for the evolution of the state xt while the second one, known as the observation equation, relates the state with the observable. For simplicity only the first equation is nonlinear and only the second one has nonnormal innovations. We can handle the more general case where nonlinearities and nonnormalities are present in both equations without further problem. Both the nonlinear structure of the process and the presence of nonnormal innovations stop us from using the Kalman filter or other similar procedure to evaluate the likelihood function. However a Sequential Monte Carlo filter delivers a consistent evaluation of that function using simulation methods. See Fernández-Villaverde and Rubio-Ramírez (2004) for details. For simplicity we assume that x 0 is known.^1 Then we can proceed as follows:
- Fix a large integer N.
- Draw N samples from N (0, 1). Call each of those w^1 |^0 ,i^ for i = 1, ..., N. Use those values to generate N values of x^1 |^0 ,i^ with the transition equation:
x^1 |^0 ,i^ = 0.5 + 0. (^3) 1 +x^0 x 2 0
- Use y 1 and each of the N x^1 |^0 ,i^ from the previous step to find N values of v^1 |^0 ,i:
v^1 |^0 ,i^ = y 1 − x^1 |^0 ,i
(^1) We could solve the problem when x 0 is unknown paying a cost in terms of further notation. See again Fernández-Villaverde and Rubio-Ramírez (2004) for the general treatment.
Once we have the likelihood function standard inference is simple. For example, if instead of the process described above, we have the case
xt = α + β xt−^1 1 + x^2 t− 1
where wt ∼ N (0, σ) and vt ∼ t (2) where α, β, δ and σ are unknown parameters, we can apply our filter to compute p
yT^
α, β, δ, σ
for each particular value of the parameters. Then we can either maximize this function with respect to the four parameters to undertake classical inference or, after postulating some prior distribution π (α, β, δ, σ) , find the posterior distribution: p
α, β, δ, σ| yT^
∝ p
yT^
α, β, δ, σ
π (α, β, δ, σ)
to perform Bayesian inference.
References
[1] Fernández-Villaverde, J. and J. Rubio-Ramírez (2004), “Estimating Nonlinear Dynamic Equilibrium Economies: A Likelihood Approach”. Mimeo, University of Pennsylvania. Available at www.econ.upenn.edu/~jesusfv
