Monday, May 23, 2011

Vector Autoregressions and Bayesian Vector Autoregressions

In a previous post I highlighted the basic algebra associated with ARIMA models:
  
Ex ARMA(2,1,1) : ( Y*t = b0 + b1Y*t-1  + b2Y*t-2  + b3 et-1  + et

The terms in this model consist only of a single observed time series and its associated error terms. Vector Autoregressions (VAR's) are class of models that allow for a multivariate analysis of time series. For example 2 series Y and Y2 (perhaps GNP and treasury yields). We could specify the two series as follows:

Y1,t  =  c1 + A1,1Y1,t-1 + A1,2Y2,t-1 + e1,t
Y2,t  =  c2 + A2,1Y1,t-1 + A2,2Y2,t-1 + e2,t    

 




More generally, writing one equation for each series:

Yt  =  c + A1Yt-1 + …+ ApYt-p + e
  
Or in matrix form: Y = AZ + U

Y =  series of K different endogenous variables
A=  coefficient matrix
Z =  matrix of lagged variables
U = matrix of error terms

Exogenous variables ‘x’ can be incorporated as follows:

  Yt  =  c + A1Yt-1 + …+ ApYt-p + B1Xt-1 + …+ BpXt-p + u

Problems with VAR’s
Given the large number of parameters in VAR’s, overparamaterization and over fitting becomes an issue. Some coefficients may turn out to be different from zero by coincidence, leading to spurious relationships in the model specification.

Bayesian Vector Auto regression (BVAR)
Assume that the model parameters are random variable. A prior distribution is specified based on prior information and combined with objective information from observed data to obtain the posterior distribution using Bayes Theorem.  The prior distribution specification acts as a barrier, or provides ‘shrinkage’ preventing the estimated parameters from depicting what are only spurious correlations.

Estimation

Recall , Bayes Theorem, for parameter θ  and data X,  in its simplest form:

p(θ |X) = p(X|θ)p(θ ) / p(X)  or  p(X|θ)p(θ ) / ∑i p(X| θi) p(θi)

The posterior distribution can be defined based on the prior and the likelihood function as follows:
p(θ|X) = p(θ ) L(θ|X) / ∫θ p(θ) L(θ |X) dθ α p(θ) L(θ |X)

This should look familiar- see also the post on Bayesian Econometrics.

p(θ|X) is then estimated based on the above formulation using MCMC methods. Various prior distributions have been discussed in the literature, including the Minnessota Prior, Conjugate Priors, Flat Priors, and the Sims-Zha Prior.

References:
The Usage of Different Prior Distributions in Bayesian Vector Autoregressive Models. Volkan Sevinc and Gul Ergun. Hacettepe Journal of Mathematics and Statistics Volume 38(1) 2009.

Advances in Bayesian Time Series Modeling and the Study of Politics: Theory, Testing, Forecasting, and Policy Analysis. Patrick T. Brandt and John R. Freeman. Political Analysis. (2006) 14:1-36.

Vector Autoregression. (presentation) Jamie Monogan. Washington University. St. Louis. November 15 & 17, 2010. Link: http://artsci.wustl.edu/~jmonogan/teaching/ts/14var.pdf

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