DDNMs: Coupled Systems of univariate DLMs Dimension … Structure … Volatility …

Key First Example: VAR(1) Written other ways: = (row j of )’ Lag 1 responses for each univariate series Cross-talk at lag 1: “networks” Feed-forward effects: “impulse” responses Contemporaneous links/association: V

Dynamics in Inter- & Cross-connections: Time-Varying VAR(1) Evolve in time Written another way: Time-varying lagged connections Time-varying volatilities and “co-volatilities” in contemporaneous connections Easy to add more predictors series-by-series

Parameter dimension – BIG with moderate q,d: scalability? VAR and TV-VAR(d) Models Univariate elements: Parameter dimension – BIG with moderate q,d: scalability? Parsimony: sparse lagged and cross-sectional dependencies?

Dynamic Dependency Network Models A Structured Subset of TV-VAR(d) Models Dynamic Dependency Network Models

Coupled DLMs: DDNMs Interest in: e.g. 500 stocks S&P, daily Scale up: large “Decoupled” univariate series/models Series-specific independent predictors Series-specific states, evolutions, discount factors, volatilities Sequential analysis: Fast? Parallelisation? Cross-series / multivariate structure : Sensitive co-volatility modelling Critical: coherent joint forecast distributions Computationally fast

Multi-Regression and DDNMs Extensions & generalizations of “multiregression DLMs” C.M. Queen & J.Q. Smith (RSS B, 1993; and later) Z. Zhao, M. Xie & MW (2015) Lopes, McCulloch & Tsay (2010, unpublished) J. Nakajima & MW (2013,14,15) Cholesky-style multivariate stochastic volatility (MSV)

Decoupled, Parallel Sets of Univariate DLMs Stock prices: Citibank, AT&T, Ebay, ….. Bank of America, GE Cross-series: directed & sparse graphical structure – lagged Idiosyncratic: volatility C T EBAY BOA GE Customised predictors external/independent factors lagged y – sparse set e.g. Citibank “info”

? Contemporaneous Coupling of Univariate DLMs Cross-series: directed graphical structure e.g. ? C T EBAY BOA GE A few parental predictors

Coupled Sets of Univariate DLMs Multiple univariate models “decoupled” in parallel Known predictors “Parents” Parental sets: *Current* values of (some) related series as predictors *Independent* across j: residuals & univariate volatilities:

Implied Coherent Multivariate Model dynamic regressions & precision/volatility Sparse Non-zeros in . Multivariate volatility (precision) matrix : - Cholesky-style: upper triangular form Notation: extend to row j and pad with 0 entries General conclusions from this example

Dynamic Graphical Model Structure Induced precision/volatility Non-zeros in . zero precision: conditional independence Non-zeros in . General conclusions from this example Links to & from parents Links to & from parents-of-parents

Parental-Induced Cross-Talk in Dynamic Regressions Non-zeros in . Non-zeros in . General conclusions from this example parents grandparents ……. great-grandparents

“Triangular” Set of Coupled Univariate DLMs Multiple univariate models “decoupled” in parallel Parallel states - independent evolutions 2 discount factors, series specific

Normal pdf in conditional model j Dynamic Dependency Network Models Parents of j Joint pdf: Normal pdf in conditional model j Compositional form of joint density - directed graphical model - General conclusions from this example Known & variances: Multiregression DLM TV-VAR with lagged in , & unknown volatility processes: Dynamic dependency networks

Summary Concepts: DDNMs - Multiple univariate models: decoupled, in parallel - Conducive to on-line sequential learning: Analytic, fast, parallel - Simultaneous parental sets define *sparse* multivariate stochastic volatility matrix - New dynamic graphical models for MSV: evolutions of - * Dependent on choice of order of named series j=1:m *

DDNMs: Model Forms, Filtering and Forecasting Parallel, conditionally independent analyses: Example: – independent, steady state evolutions – discounting for state and volatility Forecasting: 1-step: individual T distributions - marginal likelihoods for parameter assessment - series j specific (discount factors, TVAR model order, etc … ) 1-step: multivariate/joint forecast mean and variance matrix - analytic: recursively computed More steps ahead? General conclusions from this example

DDNMs: Recoupling Decoupled Models Forecasting: TVAR models: Simulate for more steps ahead DDNM: Simulate! For steps ahead k=1:K … General conclusions from this example

Example: FX, Commodities & Stock Indices Daily closing prices in $US Model log(price) General conclusions from this example

Model Structure and Hyper-parameters TVAR order - all possible parental sets - discrete ranges of TVAR order and discounts Discrete model space: parental set state discount factor volatility discount factor Sequential learning over time: 1-step forecast pdf: Student t Score and normalize: - Bayesian model uncertainty and averaging for forecasting

Parallel Model Uncertainty Analysis Sequential learning over time: Parallel! - across all models for each series - independently/parallel across series - computationally accessible - model space dimension reduction:

But First: More Time-adaptive Structure Learning Model probabilities converge/degenerate Sometimes fast Always to “wrong” model Lack of adaptability over time “Reality” may be outside the “span” of model set? - all are “poor models”

Power Discounting of Model Uncertainties Discount (“forgetting”) factor - Exponentially discount past information across models - “Synthetic” dynamics on model space - Intervention: Change model probabilities each time step Extend model uncertainty analysis: grid of + product of model probabilities over j [ Raftery et al, 2010 Koop & Korobilis 2012 ] [ P.J. Harrison, 1960s; West & Harrison, 1987 1st edn ]

Finance Example: Models, Analyses, Decisions 50:50 training/test data Train (to mid 4/2006): evaluate all Test (from mid 4/2006): use reduced subsets of models Forecasting and decisions: 5-day look-ahead portfolio Commodity Trading Advisors’ (CTA) benchmark (“market index”) Model m+1 series with CTA included Optimization of portfolio to: - minimize predicted portfolio risk - subject to specified %target exceedance over expected/forecast CTA benchmark - and decorrelated- in expectation- with CTA benchmark

Finance Example: Scoring Power Discounting Forecast RMSE & MAD Improves: 1,5-step forecasts - adaptability - Portfolio returns?

Finance Example: One Series

Finance Example:TVAR Model Order by Series

Finance Example: Model Discount Factors by Series

Finance Example: Parental Inclusion Probabilities

Finance Example: Parental Inclusion Probabilities (cont)

Finance Example: Parental Inclusion Probabilities (cont)

Finance Example: 1-Day Ahead Portfolios

Finance Example: 5-Day Ahead Portfolios

Finance Example: 5-Day Ahead Portfolios

DDNMs - Flexible, customisable univariate models: decoupled, in parallel - Parental structure: sparse multivariate stochastic volatility matrix - On-line sequential learning: Analytic, fast, parallel - Recoupling for coherent posterior and predictive inferences dependent on choice of order of named series j=1:m Scale-up in dimension? Choice/specification of order? 400 S&P stock price series?

Portfolios & Dynamic Multivariate Graphical Models Time series + conditional independence structures/constraints Multivariate dynamic linear models & multivariate stochastic volatility + Graphical models of cross-series structure High-dimensional asset allocation - portfolio problems Structure, parameter reduction: stability, parsimony, accuracy

Financial Series: Structure of Dependencies Multiple equities: managed funds, index funds, exchange rates, etc - period t returns Volatility … Structure … Precision matrix: defines conditional (in)dependencies

Utilities & Decisions in Dynamic Portfolios Models/choices of response variable - % returns - actual FX or commodity, stock prices - opportunities for more informed and improved forecasts using prices? Whatever model choice, derived 1-step forecasts for %returns: Rebalance portfolio decision: - want high % return - want to understand, control, reduce risk Portfolio allocation vector Choose (re)allocation vector each time step : Update/optimize sequentially via updated forecasts : Observed returns – cumulate k-step ahead variants

Standard Portfolio Examples Decisions: choice of utilities Risk = s.d. of portfolio Good things: high target return low risk stable portfolio weights a. Minimize risk, minimum variance: portfolio risk2 b. Target return: analytic – terms in c. Target constrained: b with non-negative weights. Requires non-linear optimization You & Me investors (401K, no shorting) Role of precision matrix

Gaussian Graphical Models – precision matrix zeros/structure: constrain, induce parsimony Zeros represent conditional independencies

Drawing Graphs Graph exhibits conditional independencies Zeros in precision matrix ~ missing edges International exchange rates example, p=11 No edge: [ Carvalho, Massam & W, Biometrika 2007 ]

Mean-Target Portfolio: Precisions & Graphical Models Precision matrix Optimal mean-target portfolio: Predicted return after “explaining” by other assets Residual variance (“nondiversifiable risk”) Sparsity, parsimony: Implications for portfolio decisions & revision over time

People & Links www.stat.duke.edu/~mw Zoey Zhao PhD (Duke 2015) Citadel, Chicago Amy Xie (Duke BA 2013) PhD student, Duke Dynamic dependence networks: Financial time series forecasting & portfolio decisions, ASMBI, 2016 www.stat.duke.edu/~mw