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

2. 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

3. 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

4. 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?

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

6. 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

7. 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Lopes, McCulloch & Tsay (2010, unpublished)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)

8. 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”

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

10. 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:

11. 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

12. 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

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

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

15. 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

16. 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 *

17. 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

18. 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

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

20. 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

21. 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:

22. 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”

23. 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, st edn ]

24. 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

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

26. Finance Example: One Series

27. Finance Example:TVAR Model Order by Series

28. Finance Example: Model Discount Factors by Series

29. Finance Example: Parental Inclusion Probabilities

30. Finance Example: Parental Inclusion Probabilities (cont)

31. Finance Example: Parental Inclusion Probabilities (cont)

32. Finance Example: 1-Day Ahead Portfolios

33. Finance Example: 5-Day Ahead Portfolios

34. Finance Example: 5-Day Ahead Portfolios

35. 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?

37. 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

38. 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

39. 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

40. 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

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

42. 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 ]

43. 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

44. 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