Chris Bracegirdle, David Barber - Duke Electrical and ...

Bayesian Conditional Cointegration

Chris Bracegirdle, David Barber

Presenter: Jianbo Yang

Duke University

December 2, 2012

Outline

1 Preliminary

2 Proposed Cointegration Model

3 Extension to Intermittent Cointegration

4 Summary

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Definition of CointegrationGiven two time series with the first order of integration x1:T and y1:T , we considera linear combination of x1:T and y1:T

εt = yt − α− βxt , t = 1, . . . ,T (1)

εt = φεt−1 + ηt , ηt ∼ N (0, σ2), (2)

where α and β are regression coefficients.CointegrationIf exist a certain (α, β) in (1) such that ε1:T is stationary process, i.e.,|φ| < 1, wecall x1:T and y1:T are cointegrated;If ε1:T is a random walk, i.e.,|φ| = 1, x1:T and y1:T are non-conintegrated.Illustration of Cointegrationa drunk person is walking with a dog:

First order of integration

Given a time series x1:T , if z1:T−1, with elements zt = xt+1 − xt ∀t = 1, . . . ,T − 1, is a

stationary process, x1:T is considered as first order of integration.3 / 20

Classical Approach for Cointegration Testing andEstimation (Engle-Granger 1986)

1 Using ordinary least squares (OLS) method to estimate α, β in

εt = yt − α− βxt , t = 1, . . . ,T

2 Using Dickey-Fuller (DF) test to check for a unit root in the residual

εt = φεt−1 + ηt , ηt ∼ N (0, σ2).

Specifically, test the hypothesis that φ = 1 against the alternative hypothesis|φ| < 1.

Comments:

OLS is consistent, but has bias when T is small

OLS assumes φ = 0 in the autoregression. However, in the hypothesis φ = 1,ε1:T is a random walk so OLS estimation is spurious.

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Dickey-Fuller (DF) Test

For a simple autoregressive (AR) model

εt = φεt−1 + ηt , ηt ∼ N (0, σ2), (3)

DF-test tests whether a unit root is present (i.e., φ = 1) in the AR model.

For convenience, model (3) is often written as

∇εt = (φ− 1)εt−1 + ηt = δεt−1 + ηt

where ∇ is the first difference operator and δ = φ− 1.The hypothesis is:

H0 : δ = 0 (φ = 1), H1 : δ 6= 0 (φ 6= 1)

Decision rule:if t∗ > critical value ⇒ not reject null hypothesis, i.e., unit root exists;if t∗ < critical value ⇒ reject null hypothesis, i.e., unit root does not exists,where t∗ is the DF test-statistic.

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ExamplesRandom Walkx1:T : Danish private consumption (in log) in 1982-2001y1:T : Number (in log) of breeding cormorants in Denmark in 1982-2001

Cointegrationx1:T : Danish interest rate in 1972-2003y1:T : Danish bond yield in 1972-2003

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Outline

1 Preliminary



4 Summary

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Expectation-MaximizationLikelihood on Observation y1:T

p(y1:T |x1:T )

=

∫φ

∫ε1:T

T∏t=1

p(yt |xt , εt)p(εt |εt−1, φ)p(φ)

=

∫φ

[∫ε1:T−1

T−1∏t=1

p(yt |xt , εt)p(εt |εt−1, φ)

∫εT

p(yT |xT , εT )p(εT |εT−1, φ)

]p(φ)

As p(yt |xt , εt) = δ(yt − α− βxt − εt), the red term isN (yT − α− βxT |φεT−1, σ

2).Iterating the above process for all T and considering εt = yt − α− βxt , ityields

p(y1:T |x1:T ) =

∫φ

p(ε1)T∏t=2

N (εt |φεt−1, σ2)p(φ) =

∫φ

p(ε1:T |φ)p(φ)

Taking φ as latent variable and α, β, σ as parameters, EM algorithm isapplied.

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Bayesian Cointegration Test

Compare likelihoods of cointegration model with the random walk (RW)model

Random Walk

Cointegration:p(y1:T |x1:T ;φ = 1)

p(y1:T |x1:T ; |φ| < 1)=

p(ε1:T |φ = 1)

p(ε1:T ||φ| < 1)≡ lRW

lC

Note: The parameters α, β, σ2 in random walk model are same as those inthe cointegration model.

If lRWlC< threshold, the series y1:T , x1:T are cointegrated; otherwise not

cointegrated.

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Simulation Result5,000 generated series pairs for each T . 50% cointegrated and another 50%incointegrated.

Figure: FP and FN for DF method and Bayes method. OLS/DF test (5% significance),EM/Bayes factor (threshold = exp 2).

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10,000 generated series pairs for T = 100. 50% cointegrated and another 50%incointegrated.

Figure: ROC curve for different decision thresholds in EM/Bayes and DF testsignificance values in OLS/DF. Thresholds in EM/Bayes is normalized bylRWlC< C ⇔ lRW

lC< 1, lRW ≡ lRW

C

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Outline

1 Preliminary



4 Summary

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Intermittent CointegrationTask: Cointegration may only happens for certain segments along time. Inanother words, it can be ’switched off’ at a certain time point.Example:

Interconnect gas pipeline between UK and Belgium provides a direct link ingas price at each end.When the maintenance of pipeline occurs at every September, the linkbetween price is temporally broken.

Figure: Gas price of UK and Belgium .14 / 20

Bayesian Model

Key features:

Allow time-varying φt arranged to be piece-wise constant.

Switch between cointegration (|φt | < 1) and random walk (φt = 1).

Use binary itI if it = 1, εt corresponds to random walkI if it = 0, εt corresponds to cointegration

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Probabilistic Model

Figure: Graphical Model. Dashed edges to φt are absent when it = 1.

Initial state is modeled as

p(φ2|i2) =

{p1(φ2) = δ(φ2 − 1) i2 = 1p0(φ2) = U(φ2|(−1, 1)) i2 = 0

piecewise-constant transition is modeled as

p(φt |φt−1, it , it−1) =

p1(φt) it = 1δ(φt − φt−1) it = 0, it−1 = 0p0(φ2) it = 0, it−1 = 1.

Transitions p(it |it−1) and p(it) are user-specified.

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EM Algorithm

Inference on posterior p(φt , it |ε1:T ) are obtained by applying recursion forfiltering p(φt , it |ε1:t) and correction smoothing p(φt , it |ε1:T )

Reference:Bracegirdle, C. and Barber, D. Switch-Reset Models : Exact and Approximate

Inference. In AISTATS, volume 15. JMLR, 2011.

Let {i2:T , φ2:T} be latent variables and α, β, σ2 be parameters, then EMalgorithm can be applied.

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Results on Gas Price Data

Figure: Left: the residuals ε1:T with estimated α, β, σ2 by EM algorithm. Right: filter(blue) and smoothed (red) posterior for random walk p(it = 1|−).

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Outline

1 Preliminary



4 Summary

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Summary

Two novel techiques for cointegrationI Estimate α, β, σ2, φ in a Bayesian framework by considering φ as a latent

variable.I Check for stationarity by comparing likelihood with RW.

Extension to intermittent cointegration problem.

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