Modeling conditional distributions with mixture models: Applications in finance and financial decision-making

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1 Modeling conditional distributions with mixture models: Applications in finance and financial decision-making John Geweke University of Iowa, USA Journal of Applied Econometrics Invited Lecture Università di Venezia Italia June 3, 2005

2 Compound Markov normal mixture models s t = Ã st1 s t2! ; s t1 {1,...,m 1 }, s t2 {1,...,m 2 } P ³ s t1 = j s t 1,1 = i, s t 1,2, s t 2, s t 3,... = p ij P (s t2 = j s t1 = i, s t 1, s t 2,...)=r ij y t (x t,s t1 = i, s t2 = j) N ³ β + φ i + ψ ij,σ 2 σ 2 i σ2 ij

3 P = β, φ = Summary of parameterization φ 1. φ m1, Ψ = p 11. p 1m1. p m1 1 p m1 m 1 σ 2, σ = σ 2 1. σ 2 m 1, Σ =, R = ψ 11. ψ 1m2. ψ m1 1 ψ m1 m 2, r 11. r 1m2. r m2 1 r m2 m 2 σ 2 11 σ 2 1m 2.. σ 2 m 1 1 σ2 m 1 m 2 ;, θ = n β, φ, Ψ, P, R, σ 2, σ, Σ o

4 Restrictions on parameters Let π : π 0 P = π 0 E (y t s t1 = i) =β + φ i + m 2 X j=1 r ij ψ ij = β + φ i E (y t )=β + m 1 X i=1 π i φ i + m 1 X i=1 X m 2 π i j=1 r ij ψ ij = β Number of identified parameters in the model is m 1 (m 1 +3m 2 3). If φ = 0, itism 1 (3m 2 + m 1 4) + 1.

5 Prior distribution β N (0, 1), φ i σ2 iid N ³ 0, 4σ 2 (i =1,...,m 1 ) For i =1,...m 1 : ψ ij ³ σ 2 σ 2 iid i N ³ 0, 4σ 2 σ 2 i (j =1,...,m2 ) 2/σ 2 ij 5/σ 2 χ 2 (5) 2/σ 2 iid i χ 2 (2) (i =1,...m 1 ) iid χ 2 (2) (i =1,...m 1 ; j =1,...,m 2 ) ³ pi1,...,p im1 iid Dirichlet(1,...,1) ³ ri1,...,p im1 iid Dirichlet(1,...,1) If no serial correlation is permitted then φ = 0.

6 All results for S&P 500 daily returns, Jan. 2, Dec. 31, ,000 MCMC iterations 2. Every 10 th iteraton recorded 3. Of the 1,200 iterations recorded (a) (b) First 200 discarded Remaining 1000 used for analysis

7 Comparison of models by means of average log-likelihood, Z log p (y θ) p (θ y) dθ Θ m 1 =2 m 1 =3 m 1 =4 m 1 =5 m 2 = m 2 = m 2 = m 2 = m 5 = Serial correlation is prohibited. There are 2m 1 (2m 2 1) parameters in the models.

8 With serial correlaton prohibited: m 1 =2 m 1 =3 m 1 =4 m 1 =5 m 2 = m 2 = m 2 = m 2 = m 5 = With serial correlation permitted: m 1 =2 m 1 =3 m 1 =4 m 1 =5 m 2 = m 2 = m 2 = m 2 = m 5 =

9 Characterizing the persistent states Standard deviation in state i: X m 2 j=1 ³ ψ 2 ij + σ 2 σ 2 i σ2 ij 1/2 Invariant probabilities: π : π 0 P = π 0 States are not identified with respect to re-labeling.

10 MCMC simulations of component standard deviations, Markov mixture model

11 MCMC simulations of component invariant probabilities, Markov mixture model

12 State dynamics in the model Posterior mean of transition matrix is P = where states are ordered as low, medium and high volatility. Posterior mean of invariant probability vector is π 0 = ³

13 Component p.d.f.s and unconditional p.d.f., compound Markov normal mixture model

14 One-day predictive distribution given current filtered probabilties

15 Daily filtered persistent state probabilities, compound Markov normal mixture model

16 Daily filtered persistent state probabilities, compound Markov normal mixture model

17 Smoothly mixing regression models Begin with the normal mixture model j =1,...,m. y t (x t, v t,s t = j) N β 0 x t + α 0 j v t,σ 2 j k 1 p 1, Determination of latent states s t : ew t m 1 = Γ z t q 1 + ζ t ; ζ t iid N (0, I m ) es t = j iff ew tj ew ti i =1,...,m

18 Variable of interest: y t : Daily S&P 500 returns, The covariate vectors x t, v t and z t are interactive polynomials in the variables a t = Return in period t 1, a t = y t 1 b t = g b t 1 +(1 g) a t 1 κ = X s=0 g s y t 2 s κ

19 Gaussian priors: β: µ =0, τ 2 =1 α: µ =0, τ 2 =9 Γ µ =0, τ 2 =16 Inverse gamma priors: 2/σ 2 χ 2 (2) 2/σ 2 j χ 2 (2)

20 Models considered y t N β 0 x t + α 0 j v t,σ 2 j k 1 p 1, ew t m 1 = Γ z t +ζ t, es t = iff ew tj ew ti i q 1 (A) q =1, k > 1, p =1 Linear regression, normal mixture (B) q =1, k =1, p>1 Mixture of linear regressions (C) q>1, k =1, p =1 w t -weighted mixture of normals (D) q>1, k>1, p =1 Regression, w t dependent normal mixture (E) q>1, k =1, p>1 w t -dependent mixture of linear regressions

21 Some comparisons across models by means of average log-likelihood, Z log p (y θ) p (θ y) dθ Polynomials of order 3 in both a t and b t, Mixture of m =3normals, Θ b t parameters g =0.9, κ =1 Model Description Average log-likelihood A Linear regression, normal mixture B Mixture of linear regressions C w t -weighted mixture of normals D Regression, w t dependent normal mixture E w t -dependent mixture of linear regressions

22 Some comparisons across models by means of average log-likelihood, Z log p (y θ) p (θ y) dθ Θ Model C, w t -weighted mixture of normals, b t parameters g =0.9, κ =1 Mixture of m =3normals Polynomial orders a t b t Average log-likelihood

23 Some comparisons across models by means of average log-likelihood, Z log p (y θ) p (θ y) dθ Θ Model C, w t -weighted mixture of normals, Polynomials of order 3 in both a t and b t, b t parameters g =0.9, κ =1 Number of mixture components m Average log-likelihood

24 Some comparisons across models by means of average log-likelihood, Z log p (y θ) p (θ y) dθ Model C, w t -weighted mixture of normals, Polynomials of order 3 in both a t and b t, Mixture of m =3normals Θ g κ Average log-likelihood

25 All results for S&P 500 daily returns, Jan. 2, Dec. 31, ,000 MCMC iterations 2. Every 100 th iteraton recorded 3. Of the 1200 iterations recorded (a) (b) First 20 discarded Remaining 100 used for analysis

26 Sample distribution of state variables, S&P 500 return

27 Posterior means of population moments, S&P 500 return,

28 Posterior standard deviations of population moments, S&P 500 return,

29 Distribution of state varaibles ( ) and conditional c.d.f. of returns

30 Conditional c.d.f. of returns

31 Conditional c.d.f. of returns

32 Posterior quantiles of S&P 500 predictive distribution,

33 Posterior quantiles of S&P 500 predictive distribution,

34 Posterior quantiles of S&P 500 predictive distribution

35 Populaton quantiles of S&P 500 predictive distribution in selected MCMC replications

36 Returns and 1-day 5% return at risk in two models

37 Returns and 1-day 5% return at risk in two models

38 Returns and 1-week 5% return at risk in two models

39 Returns and 1-week 5% return at risk in two models