Assessing Regime Uncertainty Through Reversible Jump McMC August 14, 2008
1 Introduction Background Research Question 2 The RJMcMC Method McMC RJMcMC Algorithm Dependent Proposals Independent Proposals 3 Results Simulation Studies S&P 500 Data 4 Conclusion Future Work Works Cited
Background Thank you to the Committee on Knowledge Extension Research of the Society of Actuaries and the Department of Statistics at Texas A&M University for the funding to enable me to come and present here.
Background In regime-switching models, a Markov process switches between K states at random. The distribution of the current state (ρ t ) is only dependent upon the previous state. (ρ t 1 ) That is known as the Markov Property. The state (or regime) at time t determines the distribution of the random variable X t.
Background Often, the different regimes will be from the same family of distributions, but have different parameter values. In the RSLN model, each of the regimes follows a lognormal distribution. The RSLN model allows us to model various situations. (i.e. two regimes, one when the economy is good and the other when it is poor) [Hardy 2003]
Research Question How do we determine the optimal number of regimes?
Research Question Currently there are many options to determine which model is best (AIC, BIC, radj 2 ) or if one model is significantly better than another (Full vs. Reduced F Tests).
Research Question Currently there are many options to determine which model is best (AIC, BIC, radj 2 ) or if one model is significantly better than another (Full vs. Reduced F Tests). Wouldn t it be great if we could obtain the probability that a certain model is the best?
Research Question We can get those probabilities using a method first developed in [Green 1995] called reversible jump Markov chain Monte Carlo (RJMcMC). Advantages of having probabilities: Ease of explanation Ability to be incorporated into simulations
McMC When trying to find the distribution of the parameters in a model, we can use Bayes rule. P(θ y) = f(y θ)π(θ) P(y) It seems rather harmless, but let s expand it a bit.
McMC P(y) = f(y θ)π(θ) f (θ y) = f(y θ)π(θ) f(y θ)π(θ) Often, the integral in the denominator does not have a closed form.
McMC Luckily, Markov chain Monte Carlo (McMC) methods were developed to allow you to draw samples from the posterior distributions of the parameters When you get enough draws from the parameter s distribution, you will have a pretty good idea of what that distribution looks like. McMC is only works with likelihoods with a fixed number of parameters. Reversible Jump McMC expands the methodology to work with likelihoods of varying demension.
RJMcMC Algorithm Here are the basic steps of the RJMcMC algorithm. [Waagepetersen and Sorenson, 2001] 1 Select a starting value X 1 = (M 1, Z 1 ) where M i is the model index at iteration i and Z i is the parameter values with length n mi. 2 Generate a proposal value X p. 3 Satisfy certain conditions 4 Calculate the acceptance probability 5 If accepted X 2 = X p, otherwise X 2 = X 1 6 Repeat steps 2-5
RJMcMC Algorithm Select a starting value While this may seem like a difficult task, you can use other estimation methods (MLE, MOM) to find suitable starting values. You can also use information from other experts. Luckily, you don t even need to be that close because the algorithm will eventually bring you in to acceptable values.
RJMcMC Algorithm Proposal value You then generate X p = (m p, z p ). z p is generated by applying a deterministic mapping to the previous z and to a random component U. We can express it as z p = g mmp (z, U), where U is a random vector on R nmmp, n mmp 1, which has density q mmp (z, ) on R nmmp, and g mmp : R nm+nmmp R nmp is a deterministic mapping.
RJMcMC Algorithm Condition 1: Reversibility The condition of reversibility is: P(M n = m, Z n A m, M n+1 = m p, Z n+1 B mp ) = P(M n = m p, Z n B mp, M n+1 = m, Z n+1 A m ) for all m, m p 1,..., I, and all subsets A m and B mp respectively. C mp of C m and
RJMcMC Algorithm Condition 2: Dimension Matching The other crucial condition follows from the previous condition. n m + n mmp = n mp + n mpm This ensures that f m (z)q mmp (z, u) and f mp (z p )q mpm(z p, u p ) are joint densities on spaces on equal dimension.
RJMcMC Algorithm Acceptance Probability ( α mmp = min 1, p m p f mp (z p )p mpmq mpm(z p, u p ) p m f m (z)p mmp q mmp (z, u) g mmp (z, u) z u )
Dependent Proposals To propose new values in other models, we could let the new parameters depend on the current parameters of the current model. We could use moment matching or a similar method [Brooks et al 2003] to find an appropriate proposal. Unfortunately, the effectiveness of the sampler is highly dependent on the proposal function, especially in regime switching cases. Also, the inverse functions and Jacobian can get rather complicated.
Dependent Proposals Advantages of dependent proposals: Computationally quick Computationally sound Disadvantages: Results are highly dependent upon the choice of function (in RS situations, the function is not obvious) Jacobian and inverses can be very difficult to compute
Independent Proposals Instead, we can use the fact that MLE s are asymptotically multivariate normal [Gelman et al 2004] and generate draws independent of the current value of the parameters as follows: 1 Optimize the likelihood with respect to the parameters 2 Use the parameter estimates as the mean vector for the proposals 3 Use the inverse of the Hessian matrix as the covariance matrix for the proposals
Independent Proposals Advantages of independent proposals: Easy to set up (fewer errors) Jacobian equals 1 No inverses to compute Don t need extra information about the relationship between the parameters of different models Disadvantages: Numerical optimization can be unstable, especially with a large number of parameters Computationally intensive
Simulation Studies Does it actually work? To answer that question we ran two simulation studies and then applied it to real data.
Simulation Studies Simulations for µ with sample sizes of 75 and 750 Probability of One Regime 0.0 0.2 0.4 0.6 0.8 1.0 1 2 3 4 5 6 7 µ µ 1 varies σ 1 = 1 p 12 =.3 µ 2 = 4 σ 2 = 1 p 21 =.7
Simulation Studies Simulations for σ with sample sizes of 75 and 750 Probability of One Regime 0.0 0.2 0.4 0.6 0.8 1.0 1 2 3 4 5 6 7 µ 1 = 4 σ 1 varies p 12 =.3 µ 2 = 4 σ 2 = 1 p 21 =.7 σ
S&P 500 Data Now let s look at total return data from the S&P 500 index from January 1991 to March 2008. When we used the RJMcMC method, it returned a probability of 1 that the best model is the two regime RSLN model. (Remember that this is only compared to the one regime RSLN model)
Future Work To improve this project, I plan to: Explore other numerical optimization methods for the three regime case Include other models (ARCH, GARCH, SV, etc.) Try some other methods besides RJMcMC ([Chib et al 2001] or [Phillips and Smith 1996])
Works Cited Some interesting papers: Green, P.J. Reversible Jump Markov Chain Monte Carlo Computation and Bayesian Model Determination, Biometrika(1995), 82, 4, 711-732. Waagepetersen, Rasmus and Sorensen, Daniel, A Tutorial on Reversible Jump MCMC with a View toward Applications in QTL Mapping, International Statistical Review(2001), 69, 1, 49-61.