Using Model Selection and Prior Specification to Improve Regime-switching Asset Simulations

Transcription

1 Using Model Selection and Prior Specification to Improve Regime-switching Asset Simulations Brian M. Hartman, PhD ASA Assistant Professor of Actuarial Science University of Connecticut BYU Statistics Department Seminar 04 April 2013

2 Two Truths and a Lie 1. My childhood tetherball pole fell into a sinkhole in our backyard 2. I am the first person in Washington state to ever eat one dozen Krispy Kreme doughnuts in a store 3. I am a monthly contributor to Where we live on WNPR, the Connecticut public radio station

3 Outline This talk is presented in two parts, 1. Model Selection in Regime-switching Models of Various Types Joint work with Chris Groendyke, Robert Morris University Supported by the Actuarial Foundation 2. Prior specification in Multivariate Regime-switching Lognormal Models Joint work with David A. Engler, Brigham Young University

4 Candidate Models For both the simulation study and the application, we focus on nine candidate models 1. Gamma - Gamma 2. Gamma - Lognormal 3. Gamma - Weibull 4. Lognormal - Lognormal 5. Lognormal - Weibull 6. Weibull - Weibull 7. Gamma (iid) 8. Lognormal (iid) 9. Weibull (iid)

5 Regime-switching Models In a regime-switching model, the current state, x t, only depends upon the previous state, x t 1. Additionally, the observation, y t, depends only upon the current state, x t.

6 Motivation Currently, the work on regime-switching models assumes all regimes follow the same distribution but have different parameters. What if the regimes differ in kurtosis or skewness in addition to mean and variance? Ex: Stock Index Good economy: High mean, low variance, thin tails, symmetric Poor economy: Low mean, high variance, heavy tails, possibly left-skewed

7 Model Selection Methods - AIC/BIC We begin with two standard likelihood-based model selection criteria AIC (Akiake 1974) BIC (Schwartz 1978) AIC = 2k 2L BIC = k log n 2L Because e 2 < 8, for any reasonable sample size, BIC will be more penalizing.

8 Model Selection Methods - DIC (Spiegelhalter 2002) Often in hierarchical models, simply counting the number of parameters in the model will overstate the complexity. DIC uses the effective number of paramters, p D, to account for that problem p D = D(θ) D(θ) Where D(θ) is the Bayesian deviance, a measure of model fit D(θ) = 2 log(p(y θ)) + 2 log(p(y)) DIC is a measure of penalized model fit (like AIC/BIC) DIC = D(θ) + 2p D = D(θ) + p D DIC can be calculated from MCMC output

9 Model Selection Methods - Parallel Model Selection (Congdon 2006) AIC, BIC, and DIC will all define which model is best among the candidate set. From a Bayesian perspective, we can make the model a parameter and estimate it with the others in a chain. Examples: Saturation method (Carlin 1995) Reversible Jump MCMC (Green 1995)

10 Model Selection Methods - Parallel Model Selection (Congdon 2006) Parallel model selection is slightly different. It samples from all models simultaneously. Assuming p (θ j k M = k) 1 θ j θ k j k implies that p(m = k Y, θ) = p(y M = k, θ k) p(θ k M = k)p(m = k) j p(y M = j, θ j) p(θ j M = j)p(m = j) p(y M = k, θ) p(θ M = k)p(m = k), which can be approximated through MCMC samples.

11 Parameter Estimation - Prior Specification The model selection techniques (DIC/Parallel) are sensitive to the choice of prior. For a fair comparison, the possible models should have equally informative priors. That is most easily done through non-informative, proper priors.

12 Parameter Estimation - MCMC Sampler The MCMC algorithm includes the following steps: 1. Initialize all parameters. 2. Draw the state vector (x j ) one randomly selected observation at a time from the following equation: p(x i θ, y, x 1:i 1, x i+1:n ) which reduces through the Markov property to ν x1 π x1,x 2 p x1 (y 1 ) if i = 1 p(x i θ, y i, x i 1, x i+1 ) π xi 1,x i π xi,x i+1 p xi (y i ) if 1 < i < N π xn 1,x N p xn (y N ) if i = N

13 Parameter Estimation - MCMC Sampler 3. Draw each row of the transition probability matrix from π r Dir(1 + n r1, 1 + n r2 ) where n jk = N i=2 1{x i 1 = j, x i = k}. 4. Draw the regime-specific parameters using only the observations assigned to that regime. p(θ k x, y, θ j k ) = p(θ k y i[k] ) p(y i[k] θ k )p(θ k ) If no observations were assigned to the regime, draw the parameters using the entire sample. 5. Continue steps 2-4 until convergence. Discard those observations and then continue steps 2-4 until a strong picture of the posterior distributions emerges.

14 Simulation Study 1. Acquire total return data from the S&P 500 from January October Fit all nine models using an EM algorithm. 3. Generate many data sets of varying size (20, 50, 100, 200, 500, and 1000) from each of the fitted models. 4. Use the model selection techniques to choose the best model and measure the correct selections

15 Fitted Models Model 7 Model 8 Model 9 Model 1 Model 2 Model 3 Model 4 Model 5 Model Returns Returns

16 Fitted Parameter Estimates Model Regime 1 Regime 2 [ Transition ] GA(980.1,967.5) GA(267.8,268.6) [ ] GA(267.3,268.2) LN(0.013,0.001) [ ] GA(996.1,985.8) WB(19.1,1.03) [ ] LN(0.013,0.001) LN(-0.008,0.004) [ ] LN(0.001,0.001) WB(19.16,1.03) [ ] WB(32.66,1.02) WB(22.11,1.07) GA(544.5,539.9) 8 LN(0.008,0.002) 9 WB(19.12,1.03)

17 Correctly Identified Models/Posterior Model Probabilities

18 Correctly Identified Models/Posterior Model Probabilities

19 Tabulated Model Probabilities True Model Chosen Model

20 Model Selection on S&P 500 Data Parallel AIC BIC DIC 1: GA-GA : GA-LN : GA-WB : LN-LN : LN-WB : WB-WB : GA : LN : WB

21 Importance of Model Uncertainty 4 Discounted Value of Index Months

22 Multivariate Regime-switching Lognormal Models Multivariate RSLN model: y i N C (µ, Σ) (1) Covariance matrices are difficult to work with directly. Spectral decomposition of the matrix (e.g. Banfield 1993, Yang 1994, Bensmail 1997) Matrix logarithm (Leonard 1992, Chiu 1996) Cholesky decomposition of the inverse (Pourahmadi 1999, 2000) The separation strategy (Barnard 2000) where the covariance matrix is rewritten as Σ = SRS

23 Separation Strategy S is a C C matrix with standard deviations on the diagonal. σ S = 0 σ σ C R is a C C matrix of correlation coefficients. 1 r r 1C r r 2C R = r C1 r C2... 1

24 Correlation structures Similar to the suggestions in Liechty (2004), we examine three different prior specifications for the correlation matrix. 1. Common Correlations Model 2. Grouped Correlations Model 3. Grouped Variables Model

25 Priors Common to All Models The return means and variances are given normal and inverse gamma priors, respectively. µ N C (0, τ 2 I) σ 2 j IG(α σ, β σ ) The prior variance of µ, τ 2 I is chosen to be large and the hyperparameters of σj 2 are chosen to be relatively informative using results from previous studies.

26 Common Correlations Model In the common correlations model, all correlation elements come from the same prior distribution. r ab N(λ, γ 2 ) λ N(µ λ, σ 2 λ ) γ 2 IG(α γ, β γ )

27 Positive-definite Bounds Correlation matrices are constrained to be positive-definite. When updating the correlation matrix one element at a time, the bounds of ab value for which the entire matrix will remain positive-definite are the roots of the following quadratic equation. f ab (1) + f ab ( 1) 2f ab (0) 2 x 2 + f ab(1) f ab ( 1) x + f ab (0) 2 where f ab (s) is the determinant of the correlation matrix with the ab and ba elements replaced by s

28 Computational Issues Because of the positive-definite requirement, the chain may mix slowly. We work to reduce that impact two ways Shadow Priors Griddy Gibbs Sampling

29 Shadow Prior (Liechty 2009) Likelihood functions of both λ and γ 2 will include complicated normalization constants Insert a shadow prior for δ ab It has a small enough variance that the effect of the bound essentially cancels in the acceptance probability. Basic Formulation r ab N(λ, γ 2 ) λ N(µ λ, σλ 2 ) γ 2 IG(α γ, β γ ) With Shadow Prior r ab N(δ ab, 0.01) δ ab N(λ, γ 2 ) λ N(µ λ, σλ 2 ) γ 2 IG(α γ, β γ )

30 Griddy Gibbs When sampling from a bounded space, mixing can be improved through the use of a Griddy Gibbs sampler 1. Compute the bounds 2. Within the bounds find many (say 100) equally-spaced points 3. Calculate the posterior density of all the points 4. Using the densities as relative probabilities, draw a new value for the parameter

31 Grouped Correlations Model Under the grouped correlations model, the correlation elements come from a mixture prior. r ab N(δ ab, ν 2 ), ν 2 = 0.01 δ ab θ ab = k N(λ k, γ 2 k ) λ k N(µ λ, σ 2 λ ) γ 2 k IG(α γ, β γ ) θ ab MN(p 1,..., p k )

32 Grouped Variables Model Under the grouped variables model, the individual assets are clustered instead of the correlation elements. r ab N(δ ab, ν 2 ), ν 2 = 0.01 δ ab θ a = g, θ b = h N(λ gh, γ 2 gh ) λ gh N(µ λ, σ 2 λ ) γ 2 gh IG(α γ, β γ ) θ i MN(p 1,..., p L )

33 Common Correlations Model Results Regime 1 Regime 2 µ σ 2 µ σ 2 Belgium France Germany Netherlands Switzerland Australia Canada Mexico USA

34 Common Correlations Results, Regime 1 Belgium France Germany Netherlands Switzerland Australia Canada Mexico USA Belgium France Germany Netherlands Switzerland Australia Canada Mexico USA

35 Common Correlations Results, Regime 2 Belgium France Germany Netherlands Switzerland Australia Canada Mexico USA Belgium France Germany Netherlands Switzerland Australia Canada Mexico USA

36 Grouped Correlations Model Results Regime 1 Regime 2 µ σ 2 µ σ 2 Belgium France Germany Netherlands Switzerland Australia Canada Mexico USA

37 Grouped Correlations Results, Regime 1 Belgium France Germany Netherlands Switzerland Australia Canada Mexico USA Belgium France Germany Netherlands Switzerland Australia Canada Mexico USA

38 Grouped Correlations Results, Regime 2 Belgium France Germany Netherlands Switzerland Australia Canada Mexico USA Belgium France Germany Netherlands Switzerland Australia Canada Mexico USA

39 Using Model Selection and Prior Specification to Improve Regime-switching Asset Simulations Brian M. Hartman, PhD ASA Assistant Professor of Actuarial Science University of Connecticut BYU Statistics Department Seminar 04 April 2013