BAYESIAN TIME SERIES

(1)

A (hugely selective) introductory overview - contacting current research frontiers -

Mike West

Institute of Statistics & Decision Sciences Duke University

June 5th 2002, Valencia VII - Tenerife

(2)

Topics

Dynamic linear models (state space models)

• Sequential context, Bayesian framework

• Standard classes of models, model decompositions

Models and methods in physical science applications

• Time series decompositions, latent structure

• Neurophysiology - climatology - speech processing Multivariate time series:

• Financial applications - Latent structure, volatility models Simulation-Based Computation

• MCMC - Sequential simulation methodology

(3)

Standard Dynamic Models

Dynamic Linear Models Linear State Space Models

y_t = x_t + ν_t x_t = F⁰_t^θ_t ^θ_t = G_t^θ_t−1 + ^ω_t

• signal x_t, state vector ^θ_t = (θ_t1, . . . , θ_td)⁰

• regression vector F_t and state matrix G_t

• zero mean measurement errors νt and state innovations ^ωt

– often zero-mean and normally distributed

(4)

Examples

“Slowly varying” level observed with noise:

y_t = x_t + ν_t x_t = x_t−1 + ω_t

Dynamic linear regression:

y_t = x_t + ν_t x_t = F⁰_t^θ_t ^θ_t = ^θ_t−1 + ^ω_t

Error models for ν_t, ^ω_t

• normal distributions

• mixtures of normals: outliers and abrupt “structural” changes

(5)

Simple regression example

yt = xt + νt xt = at + btXt

Ft = (1, Xt)⁰ and ^θt = (at, bt)⁰ “wanders” through time

X t 6 5

4 3

2 1

0 2.2

2 1.8 1.6 1.4 1.2 1

0.8 Region 1 Region 2

(6)

Sales Data Example

45 50 55 60 65 70 75 80

1/75 1/77 1/79 1/81 1/83 1/85

SALES

100 110 120 130 140 150 160 170 180

190 MARKET

(7)

Sales Data Example

Market

0 25 50 75 100 125 150 175 200

Sales

0 10 20 30 40 50 60 70 80

(8)

Sales Data Example

0.42 0.43 0.44 0.45 0.46 0.47

QTR YEAR

1 75

1 77

1 79

1 81

1 83

1 85 RATIO

(9)

Simple Regression Example

Relative to “static” model, dynamic regression delivers:

• improved estimation via adaptation for “local” regression parameters

• and increased (honest) uncertainty about regression parameters

• adaptability to (small) changes → improved point forecasts

• partitions variation: parameter vs observation error

→ increased precision of stated forecasts

i.e., improved prediction: point forecasts AND precision

(10)

General Dynamic Linear Model

yt = xt + νt xt = F⁰_t^θt θ_t = Gtθ_t−1 + ^ωt

y_t−1 yt y_t+1

x



x



x



x_t−1 x_t x_t+1

x



x



x



−→ ^θ_t−1 −→ ^θt −→ ^θ_t+1 −→

• Sequential model definition : Markov evolution structure

• CI structure : ^θt sufficient for “future” at time t

(11)

Bayesian Forecasting

Key concepts:

• Bayesian: modelling & learning is probabilistic

• Time-varying parameter models: often non-stationary

• Sequential view, sequential model definitions – encourages interaction, intervention

Statistical framework:

• Forecasting: “What might happen?” and “What if?”

• Data processing and statistical learning from observations

• Updating of models and probabilistic summaries of belief

• Time series analysis ... Retrospection: “What happened?”

(12)

Bayesian Machinery

• Inferences based on information D_t = {(y₁, . . . , y_t), I_t} Find and summarise

– p(^θt|Dt) and p(^θ₁, . . . ,^θt|Dt) – and update as t → t + 1 → · · ·

• Forecasts:

– p(y_t+1, . . . , y_t+k|D_t)

– and update as t → t + 1 → · · ·

• Implementation & computations:

– Linear/normal models: neat theory, Kalman filtering

– Extend to infer variance components, non-normal errors ..

∗ need approximations, simulation methods, MCMC

(13)

Commercial Applications

• Short term forecasting of consumer sales and demand

• Monitoring: stocks and inventories of consumer products

• Many items or sectors: Aggregation and multi-level models Standard models for commercial applications:

Data = Trend + Seasonal + Regression + Error

↑ ↑ ↑ ↑ ↑

y_t = x_1t + x_2t + x_3t + ν_t

(14)

Models in Commercial Applications

Component signals x_jt follow individual dynamic models

• Trends: e.g., “locally linear” trend

x_1t = x_1,t−1 + βt + ∂x_1t, βt = β_t−1 + ∂βt

• Seasonals:

x_2t = ^X

j

(aj,t cos(2 ∗ πj/p) + bj,t sin(2 ∗ πj/p)) where (aj,t, bj,t) wander through time

Key concept: Model (De)Composition

• Modelling, prior specification, interventions: component-wise

• Posterior inference: detrending, deseasonalisation, etc

(15)

Special Classes of DLMs

Time Series DLMs – constant F, G

y_t = x_t + ν_t x_t = F⁰θ_t θ_t = Gθ_t−1 + ω_t

• Includes all “standard” point-forecasting methods (exponential smoothing, variants, ... )

• Polynomial trend and seasonal components in commercial models

• Includes all practically useful ARIMA models Multiple representations:

φ_t = E^θ_t ↔ G → EGE⁻¹ General class: Time-varying F_t, G_t

Includes non-stationary models, time-varying ARIMA models, etc., ...

(16)

Simulation-Based Computation

y_t = x_t + ν_t x_t = F⁰_t^θ_t ^θ_t = G_t^θ_t−1 + ^ω_t

• Normal error models p(νt), p(^ωt)

• Fixed time window t = 1, . . . , n

• State vector set: ^Θ = {^θ₁, . . . ,^θn}

• Require: full posterior sample ^Θ^∗ from p(^Θ|D_n)

Available via “Forward-filtering: Backward-sampling” algorithm

Carter and Kohn (1994) Biometrika

Fr¨uhwirth-Schnatter (1994) J Time Series Anal West and Harrison 1997

(17)

FFBS Algorithm

Forward-filtering:

• Standard normal/linear analysis: Kalman filter

• delivers normal p(^θt|Dt) at each t = 1, . . . , n Backward-sampling:

• at t = n : sample ^θ^∗_n from p(^θn|Dn)

• for t = n − 1, n − 2, . . . , 1 : sample ^θ^∗_t from normal distribution p(^θt|Dt,^θ^∗_t+1)

Builds up ^Θ^∗ as ^θ^∗_n, ^θ^∗_n−1, . . .

Exploits Markovian/CI model structure

(18)

MCMC in DLMs

DLM parameters: e.g., constant model

y_t = x_t + ν_t x_t = F⁰^θ_t ^θ_t = G^θ_t−1 + ^ω_t Parameters:

• Variances (variance matrices) of ν_t, ω_t

• Elements of F, G

• Indicators in normal mixture models for errors Add parameters to analysis: MCMC utilising FFBS

(19)

MCMC in DLMs

Parameters ^Φ (may depend on sample size)

Gibbs sampling: p(^Θ, ^Φ|Dn) iteratively resampled via

• Apply FFBS algorithm to draw ^Θ^∗ from p(^Θ|^Φ^∗, D_n)

• Draw new value of ^Φ^∗ from p(^Φ|^Θ^∗, Dn)

• Iterate

“Standard” Gibbs sampling: MCMC

May need “creativity” in sampling ^Φ^∗: Metropolis-Hastings, etc Often “easy”: as in Autoregressive DLM

(20)

MCMC: Autoregressive Model Example

Data: y_t = x_t + ν_t

State AR(d) xt = ^P^d_j₌₁ φjx_t−j + t

DLM for for xt : xt = (1, 0, . . . , 0)xt, xt = Gx_t−1 + ^ωt

G =







φ₁ φ₂ φ₃ · · · φ_d−1 φ_d

1 0 0 · · · 0 0

0 1 0 · · · 0 0

... . .. 0 ...

0 0 · · · 1 0







, ^ω_t =







_t 0 0... 0







• Parameters: {φ₁, . . . , φd; V (νt), V (t)}

• Conditional posteriors standard: linear regression parameters

(21)

Models in Scientific Applications

Historical interest in biomedical monitoring (change-points), engineering applications (control, tracking), environmental monitoring, ...

Goals:

• Exploratory “discovery” of interpretable latent processes

• Nonstationary time series: “hidden” quasi-periodicities

• Changes over time at different time scales

• Time:frequency structure (in time domain) State-space models:

• Stationary and/or nonstationary, time-varying parameters

• General decomposition theory for state space-space models

• DLM autoregressions and time-varying autoregressions

(22)

Autoregressive DLM

Latent AR(d) process: x_t = ^P^d_j=1 φ_jx_t−j + _t

Time series: y_t = x_0t + x_t + ν_t with trend, etc in x_0t DLM for for x_t : x_t = (1, 0, . . . , 0)x_t, x_t = Gx_t−1 + ^ω_t

G =







φ₁ φ₂ φ₃ · · · φ_d−1 φd

1 0 0 · · · 0 0

0 1 0 · · · 0 0

... . .. 0 ...

0 0 · · · 1 0







, ^ω_t =







t

0 0...

0







– x_t latent, unobserved

– G = G(^φ) with ^φ = (φ₁, . . . , φd)⁰ to be estimated

(23)

Time Series Decomposition

Eigenstructure: G = E⁻¹AE

Reparametrise to “diagonal” model: xt → Ext

Transforms to

x_t =

d^z

X

j=1

z_t,j +

d^a

X

j=1

a_t,j

• z terms: one for each pair of complex conjugate eigenvalues

• a terms: one for each real eigenvalue

• underlying latent processes zt,j and at,j follow “simple” models – at,j is AR(1) process - short-term correlations

– zt,j is quasi-periodic ARMA(2,1) - noisy sine wave with randomly time-varying amplitude & phase, fixed frequency

(24)

Time-varying Autoregression

TV-AR(d) model: x_t = ^P^d_j=1 φ_t,jx_t−j + _t

• AR parameter: ^φ_t = (φ_t,1, . . . , φt,d)⁰ “wanders” through time:

φ_t = ^φ_t−1 + ∂^φ_t

• stochastic “shocks” ∂^φ_t

• Innovations: _t ∼ N (0, σ_t²) – time-varying variance σ_t² Flexible representations:

• non-stationary process, time-varying spectral properties

• latent component structure

• other evolutions for ^φ_t (e.g., Godsill et al on speech processing)

(25)

DLM Form of TV-AR

(d)

:

x_t = (1, 0, . . . , 0)⁰x_t

xt = G(^φ_t)x_t−1 + (t, 0, . . . , 0)⁰

φ_t = ^φ_t−1 + ∂^φ_t with

G(^φ_t) =







φ_t,1 φ_t,2 · · · φ_t,d−1 φ_t,d

1 0 · · · 0 0

0 1 · · · 0 0

... . .. 0 ...

0 0 · · · 1 0







Analysis: Posterior distributions for {φ_t, σ_t : ∀t}

Component models: y_t = x_0t + x_1t + ν_t

→ infer latent TV-AR processes too: {x_0t, x_t : ∀t}

(26)

TVAR Decomposition

xt =

d^z

X

j=1

zt,j +

d^a

X

j=1

at,j

• z terms: one for each pair of complex conjugate eigenvalues

• a terms: one for each real eigenvalue

• underlying latent processes:

– a_t,j is TV-AR(1) process - short-term correlations + time-varying correlation – z_t,j is TV-ARMA(2,1) - noisy sine wave with

randomly time-varying amplitude & phase

+ time-varying frequency (2π/wavelength) Number of complex/real eigenvalues can vary over time too!

(27)

Time:Frequency Analysis

• Fit flexible (high order?) AR or TV-AR models

• Estimate latent components and their frequencies, amplitudes over time

• Time domain representation of spectral structure

• Often, some zt,j physically meaningful, some (high frequency) represent noise, model approximation

• at,j – noise, model approximation and and (possibly) low frequency

“trend”

(28)

Paleoclimatology Example

• deep ocean cores: relative abundance of δ¹⁸O

• δ¹⁸O ↓ as global temperatures ↑ (smaller ice mass)

• reverse sign: higher recent global temperatures

• “well known” periodicities: earth orbital dynamics → impact on solar insolation – Milankovitch; Shackleton et al since 1976

eccentricity: 95-120 kyear obliquity: 40-42 kyear

precession: 19-25 kyear (1 or 2?)

(29)

Oxygen Isotope Data

• Form of time variation in individual cycles ?

• Timing/nature of onset of “ice-age” cycle ↔ eccentricity component

∼ 1000 kyears ago ?

• Time scale: errors, interpolation, ... measurement, sampling error, etc

Models: High order TV-AR, p = 20, plus smooth trend (outliers?) Variance components estimated: changing AR parameters

Decomposition: Posterior mean of xt, φt,j at each t

4 dominant quasi-periodic components: order by estimated amplitude (innovation variance)

Others: residual structure &/or contaminations

(30)

reversed time in k years

oxygen level

0 500 1000 1500 2000 2500

3.0 3.5 4.0 4.5 5.0

oxygen isotope series

period

0 500 1000 1500 2000 2500

0 50 100 150

trajectories of time-varying periods of components

(31)

0 500 1000 1500 2000 2500

trend

comp 1

comp 2

comp 3

comp 4

(32)

Oxygen

• Wavelengths vary only modestly

• Estimated periods/wavelengths consistent with geological determinations

.... 108–120, peak 110 .... 40.8–41.6, peak 41.5 .... 22.2–23, peak 22.8

• “Switch” due to order of estimated amplitude

– Geological interpretation? Structural climate change ∼ 1.1m yrs?

(33)

Example: EEG Study

• Clinical uses of electroconvulsive therapy

• Measure seizure treatment outcomes via long, multiple EEG

(electroencephalogram) series – electrical potential fluctuations on scalp

• Many multiple series: one seizure – 19 channels, 256/sec Models to:

• Characterise seizure “waveforms” ... time varying amplitudes at ranges of frequencies (alpha waves, etc)

• Superimposed on “normal” waveform, noise, ...

• Identify/extract latent components: infer seizure effects

• Spatial connectivities: related multiple series

(34)

Why TVAR Models?

time

0 50 100 150 200

-300-200-1000100200

time

0 50 100 150 200

-300-200-1000100200

time

0 500 1000 1500 2000

-300-200-1000100200

(35)

EEG Decomposition Examples

time

0 100 200 300 400 500

data

delta theta alpha:theta

alpha beta beta

Central section of Ictal19-Cz

(36)

time

0 500 1000 1500 2000 2500 3000

beta

alpha

theta2

theta1

delta2

delta1

data

S26.Low Patient/Treatment

(37)

EEG Treatment Comparison

Two EEG series on one individual: S26.Low cf S26.Mod

• Repeat seizures with varying ECT treatment

• TVAR(20) with time-varying σ_t²

• Evident high frequency structure and “spiky” traces

→ higher order models

• Several frequency bands influenced by seizure – several components

• Identification issues

(38)

EEG Comparisons

time

0 1000 2000 3000 4000

0 2 4 6 8 10

theta range

delta range

S26.Low S26.Mod

-- -- --

•

-- - - - -

•

-- -- --

•

-- -- --

•

-- -- --

•

-- -- - -

•

- - - - - -

•

-- - - - -

•

- - - - - -

•

-- - - - -

•

-- - - - -

•

S26.Low vs. S26.Med: Low frequency trajectories

(39)

Sequential Simulation Analysis

Time t :

• States ^Θt = {^θ_t−k, . . . ,^θt} of interest

• Summarised via set of posterior samples: p(^Θ_t|D_t) Time t + 1 :

• Observe yt ∼ p(yt|^θt)

• Require updated summary, posterior samples: p(^Θ_t+1|D_t) Issues:

• Expanding/Changing state space and dimension

• Simulation-based summaries: discrete approximations

• Inference on parameters as well as state vectors

• New data may “conflict” with prior/predictions

(40)

Particle Filtering

Key Goal: sequentially update posteriors

· · · → p(θ_t|D_t) → p(θ_t+1|D_t+1) → · · · Numerical Approximations (points/weights):

{^θ^(j)_t , ω_t^(j) : j = 1, . . . , N_t} Theoretical update:

p(^θ_t+1|D_t+1) ∝ p(y_t+1|^θ_t+1)p(^θ_t+1|Dt) MC approximation to “prior”:

p(^θ_t+1|Dt) ≈

N^t

X

k=1

ω_t^(k)p(^θ_t+1|^θ^(k)_t )

• Mixture prior: sample and accept/reject ideas natural

(41)

Example: Auxilliary Particle Filter

APF state update from t → t + 1:

• for each k,

– “estimates” ^µ^(k)_t+1 = E(^θ_t+1|^θ^(k)_t )

– and weights g_t+1^(k) ∝ ω_t^(k)p(y_t+1|^µ^(k)_t+1)

• sample (aux) indicators j with probs g_t+1^(j)

• time t + 1 samples: ^θ^(j)_t+1 ∼ p(^θ_t+1|^θ^(j)_t )

• and weights:

ω_t+1^(j) = p(y_t+1|^θ^(j)_t+1) p(y_t+1|µ^(j)_t+1)

(42)

Multivariate Models in Finance

(Quintana et al invited talk, Valencia VII)

• Futures markets, exchange rates, portfolio selection

• Multiple time series: time-varying covariance patterns

• Econometric/dynamic regressions/hierarchical models

• Latent factors in hierarchical, dynamic models

• Common time-varying structure in multiple series

• Bayesian multivariate stochastic volatility

y_t = (y_1t, . . . , ypt)⁰

e.g., yit is p−vector of returns on investment i (exchange rate futures, etc.)

(43)

Dynamic Factor Models

Exchange rate modelling for dynamic asset allocation

• Monthly (daily) currency exchange rates

• Dynamic regression/econometric predictors

• Residual structure and residual stochastic volatility – Time-varying variances and covariances

– Dynamic factor models

• Dynamic asset allocation & risk management: portfolio studies

• Bayesian analysis: model fitting, sequential analysis, forecasting, decision analysis

(44)

Excess Returns on Exchange Rates (monthly)

Returns % -10010

7602 7710 7906 8102 8210 8406 8602 8710 8906 9102 9210 9406 9602 9710

DEM

Returns % -10010

7602 7710 7906 8102 8210 8406 8602 8710 8906 9102 9210 9406 9602 9710

JPY

Returns % -10010

7602 7710 7906 8102 8210 8406 8602 8710 8906 9102 9210 9406 9602 9710

GBP

Returns % -10010

7602 7710 7906 8102 8210 8406 8602 8710 8906 9102 9210 9406 9602 9710

AUD

Returns % -10010

7602 7710 7906 8102 8210 8406 8602 8710 8906 9102 9210 9406 9602 9710

CAD

Returns % -10010

7602 7710 7906 8102 8210 8406 8602 8710 8906 9102 9210 9406 9602 9710

CHF

Returns % -10010

7602 7710 7906 8102 8210 8406 8602 8710 8906 9102 9210 9406 9602 9710

FRF

Returns % -10010

7602 7710 7906 8102 8210 8406 8602 8710 8906 9102 9210 9406 9602 9710

NLG

(45)

Short Interest Rates (monthly)

IS -0.100.00.10

7602 7710 7906 8102 8210 8406 8602 8710 8906 9102 9210 9406 9602 9710

DEM

IS -0.100.00.10

7602 7710 7906 8102 8210 8406 8602 8710 8906 9102 9210 9406 9602 9710

JPY

IS -0.100.00.10

7602 7710 7906 8102 8210 8406 8602 8710 8906 9102 9210 9406 9602 9710

GBP

IS -0.100.00.10

7602 7710 7906 8102 8210 8406 8602 8710 8906 9102 9210 9406 9602 9710

AUD

IS -0.100.00.10

7602 7710 7906 8102 8210 8406 8602 8710 8906 9102 9210 9406 9602 9710

CAD

IS -0.100.00.10

7602 7710 7906 8102 8210 8406 8602 8710 8906 9102 9210 9406 9602 9710

CHF

IS -0.100.00.10

7602 7710 7906 8102 8210 8406 8602 8710 8906 9102 9210 9406 9602 9710

FRF

IS -0.100.00.10

7602 7710 7906 8102 8210 8406 8602 8710 8906 9102 9210 9406 9602 9710

NLG

Short Interest Rate

(46)

Stochastic Volatility Factor Models

y_t = dynamic regression_t + residual_t – residual_t ∼ N (0, Σ_t)

residual_t = X_tf_t + e_t ft ∼ N (ft|0, Ft) et ∼ N (et|0, Et)

f_t ... k−vector of latent factors e_t ... q−vector of “idiosyncracies”

F_t = diag(exp(λ_1,t), . . . , exp(λ_k,t)) E_t = diag(exp(λ_k+1,t), . . . , exp(λ_k+q,t)) Σ_t = X_tF_tX⁰_t + E_t

Factor and idiosyncratic latent log volatilities: λt = (λ_1,t, . . . , λ_k+q,t)⁰

(47)

Dynamic Factor Model: Factor Loadings Structure

• Constant factor loadings matrix X_t = X

– or “slowly varying” over time –

• Identification constraints on X :

X =







1 0 0 · · · 0

x_2,1 1 0 · · · 0

... ... ... · · · 0 x_k,1 x_k,2 x_k,3 · · · 1 x_k+1,1 x_k+1,2 x_k+1,3 · · · x_k+1,k

... ... ... · · · ...

x_q,1 x_q,2 x_q,3 · · · xq,k







• Series order defines interpretation of factors

(48)

Stochastic Volatility Models: Factors & Idiosyncracies

Multivariate SV models

• vector AR(1) model for latent log volatilities λ_t

• volatility persistence: AR(1) coefficients Φ (diagonal)

• marginal volatility levels: time-varying µ_t

• dependent innovations: shocks to volatilities related across series global/sector “effects” represented through correlations

λt = µt + Φ(λ_t−1 − µ_t−1) + ωt

ωt ∼ N (0, U) µ_t ∼ random walk

(49)

Dynamic Regressions and Shrinkage Models

• several predictors (e.g., interest rates)

• regression coefficients time-varying

• shrinkage models: coefficients related across series

e.g., sensitivity to short-term interest rates “similar” across currencies

Series j, time t :

β_jt = φ_jtβ_j,t−1 + (1 − φ_jt)γ_t + innovation_jt

• global or sector “average” γ_t

• time-varying degrees of “shrinkage” φ_jt

• multiple series, several predictors: state-space model formulations

(50)

Model Fitting & Analysis: MCMC

Inference based on a fixed sample over t = 1, . . . , T

• Bayesian analysis via posterior simulations

• Monte Carlo samples from joint posterior for

– model parameters, dynamic regression parameters, AND – latent processes: factors & volatilities

{ ft,^λt : t = 1, . . . , T }

• examples of highly structured, hierarchical models with many latent variables and parameters

• custom MCMC built of standard “modules”

• simple MC evaluation of forecast/predictive distributions

(51)

Model Fitting & Analysis: Sequential Analyses

Sequential particle filtering over t = T + 1, T + 2, , . . .

• “particulate” approximation to posterior distributions

• prior: cloud of particles, associated importance weights

• posterior: sampling/importance resampling to revise posterior samples

· · · → p(·|D_t−1) → p(·|D_t) → · · ·

• sample “regeneration” by local smoothing of particulate prior

• time-varying latent variables and model parameters

(52)

Volatilities (SD) of Currency Returns (monthly)

stdev 0.00.050.100.15

7602 7710 7906 8102 8210 8406 8602 8710 8906 9102 9210 9406 9602 9710

DEM

stdev 0.00.050.100.15

7602 7710 7906 8102 8210 8406 8602 8710 8906 9102 9210 9406 9602 9710

JPY

stdev 0.00.050.100.15

7602 7710 7906 8102 8210 8406 8602 8710 8906 9102 9210 9406 9602 9710

GBP

stdev 0.00.050.100.15

7602 7710 7906 8102 8210 8406 8602 8710 8906 9102 9210 9406 9602 9710

AUD

stdev 0.00.050.100.15

7602 7710 7906 8102 8210 8406 8602 8710 8906 9102 9210 9406 9602 9710

CAD

stdev 0.00.050.100.15

7602 7710 7906 8102 8210 8406 8602 8710 8906 9102 9210 9406 9602 9710

CHF

stdev 0.00.050.100.15

7602 7710 7906 8102 8210 8406 8602 8710 8906 9102 9210 9406 9602 9710

FRF

stdev 0.00.050.100.15

7602 7710 7906 8102 8210 8406 8602 8710 8906 9102 9210 9406 9602 9710

NLG

(53)

2 Latent Factors and their Volatilities (SD) (monthly)

Factor 1 -0.100.00.050.10

7602 8102 8602 9102 9602

DEM

Factor 1 0.00.040.080.12

7602 8102 8602 9102 9602

Factor 2 -0.100.00.050.10

7602 8102 8602 9102 9602

JPY

Factor 2 0.00.040.080.12

7602 8102 8602 9102 9602

(54)

3 Latent Factors and their Volatilities (SD) (monthly)

Factor 1 -0.10-0.050.00.050.10

7602 7710 7906 8102 8210 8406 8602 8710 8906 9102 9210 9406 9602 9710

DEM

Factor 1 0.00.020.040.060.08

7602 7710 7906 8102 8210 8406 8602 8710 8906 9102 9210 9406 9602 9710

Factor 2 -0.10-0.050.00.050.10

7602 7710 7906 8102 8210 8406 8602 8710 8906 9102 9210 9406 9602 9710

JPY

Factor 2 0.00.020.040.060.08

7602 7710 7906 8102 8210 8406 8602 8710 8906 9102 9210 9406 9602 9710

Factor 3 -0.10-0.050.00.050.10

7602 7710 7906 8102 8210 8406 8602 8710 8906 9102 9210 9406 9602 9710

GBP

Factor 3 0.00.020.040.060.08

7602 7710 7906 8102 8210 8406 8602 8710 8906 9102 9210 9406 9602 9710

(55)

Factor Loading Matrix

X

(monthly)

Factor 1 Factor 2 DEM 1.00 (0.00) 0.00 (0.00) JPY 0.55 (0.06) 1.00 (0.00) GBP 0.68 (0.05) 0.48 (0.14) AUD 0.23 (0.03) 0.35 (0.07) CAD 0.04 (0.02) 0.03 (0.08) CHF 1.04 (0.03) 0.23 (0.07) FRF 0.96 (0.01) -0.01 (0.02) ESP 0.99 (0.01) -0.01 (0.01)

(56)

Factor Loading Matrix

X

(monthly)

Factor 1 Factor 2

DEM 1 ·

FRF 1 ·

ESP 1 ·

CHF 1 0.2

GBP 0.7 0.5

JPY 0.5 1

AUD 0.2 0.3

CAD · ·

(57)

Idiosyncratic Volatilities (SD) (monthly)

Idiosyncratic 0.00.050.100.15

7602 7710 7906 8102 8210 8406 8602 8710 8906 9102 9210 9406 9602 9710

DEM

7602 7710 7906 8102 8210 8406 8602 8710 8906 9102 9210 9406 9602 9710

JPY

7602 7710 7906 8102 8210 8406 8602 8710 8906 9102 9210 9406 9602 9710

GBP

7602 7710 7906 8102 8210 8406 8602 8710 8906 9102 9210 9406 9602 9710

AUD

7602 7710 7906 8102 8210 8406 8602 8710 8906 9102 9210 9406 9602 9710

CAD

7602 7710 7906 8102 8210 8406 8602 8710 8906 9102 9210 9406 9602 9710

CHF

7602 7710 7906 8102 8210 8406 8602 8710 8906 9102 9210 9406 9602 9710

FRF

7602 7710 7906 8102 8210 8406 8602 8710 8906 9102 9210 9406 9602 9710

NLG

(58)

Financial Time Series

• Models/forecasts feed into portfolio decision management – Quintana et al invited talk, Valencia VII

• Live/Operational assessments of dynamic factor models

• Improved sequential particle filtering: parameters

• Time-varying loadings matrices X_t, parameters

• Uncertainty about number of factors

(59)

Bayesian Time Series, Currently

Applied aspects:

• Financial modelling and forecasting

• Natural/engineering sciences: signal processing

• Spatial time series: epidemiology, environment, ecology Models and methods:

• Highly structured multiple time series

• Spatial time series

• Computational methods: Sequential simulation methods

(60)

Links & Materials

• Books and papers: www.isds.duke.edu/˜mw – Copious references to broad literatures

• 1997 Tutorial – extensive and historical – at website

• Teaching materials, notes, from past time series courses

• Software: www.isds.duke.edu/˜mw

– TVAR (matlab, fortran), AR component models, BATS, links

• Encylopedia of Statistical Sciences (1998): Bayesian Forecasting (eds:

S. Kotz, C.B. Read, and D.L. Banks), Wiley.

• 1997 2nd Edition: West & Harrison/Springer book

Bayesian Forecasting & Dynamic Models

• Aguilar, Prado, Huerta & West (1999), Valencia 6 invited paper

(61)

Key Recent (since 1995) & Current Authors

– many/most are here! –

Omar Aguilar (Lehman Bros) Chris Carter (Hong Kong)

Sid Chib (Washington Univ/St Louis) Sylvia Fr¨uhwirth-Schnatter (Vienna) Simon Godsill & Co (Cambridge) Gabriel Huerta (Univ New Mexico) Genshiro Kitagawa (ISM Tokyo) Robert Kohn (Sydney)

Jane Liu (UBS NY) Hedibert Lopes (Rio)

Viridiana Lourdes (ITAM Mexico) Giovanni Petris (Univ Arkansas) Michael Pitt (Warwick) Nicolas Polson (Chicago)

Raquel Prado (Santa Cruz) Jos´e M Quintana (Nikko NY) Peter Rossi (Chicago) Neil Shephard (Oxford)

(62)

... and, of course, ...

Valencia VII Invited Papers

• Quintana, Lourdes, Aguilar & Liu

– Global gambling: multivariate financial time series

• Davy & Godsill

– Signal processing, latent structure, MCMC

... plus a number of contributed talks and posters