Predictive distributions Spring 2005 Predictive distributions Consider a well fit AR model of order p. When we make the statement A 95% prediction interval is … We assume, generally, that The model is perfectly specified The parameters are estimated without error How could we come up with forecast bounds that incorporate the estimation error? Incorporate estimation error in our theoretical development of the predictors and their distributional properties. Come up with this needed predictive distribution through resampling methods. Simulate multiple future forecasts that take draws from the sampling distribution of our estimators. Spring 2005 K. Ensor, STAT 421 K. Ensor, Stat 421

Resampling Block bootstrap Stationary bootstrap Model based bootstrap Resample blocks of the time series and reconfigure the TS Stationary bootstrap blocks are of random length Model based bootstrap Resample the residuals Rebuild the process and then fit model again Spring 2005 K. Ensor, STAT 421

Bayesian Analysis of Time Series View our parameters (and even our model) as random variables. Prior distribution of parameters Distribution of series given the parameters Posterior distribution of parameters Our objective is to come up with the posterior distribution of the parameters and the future values of the process. Spring 2005 K. Ensor, STAT 421

Homework for next week Simulate a realization from an autoregressive model – AR(1) with parameters (.8) and noise variance 1 Fit the model Obtain forecasts 1 time unit in the future Plot the predictive distribution of these forecasts Develop a predictive distribution that accounts for the model estimation error (for fixed order) Through simulation Through resampling Plot the three densities on the same graph Spring 2005 K. Ensor, STAT 421

Bayesian Vector Autogregressive Models Prior distribution for parameters: Likelihood of X given the parameter Want the posterior of  given X Spring 2005 K. Ensor, STAT 421

The prior Conjugate prior Example A prior such that the distributional form of the prior is the same as that of the posterior. Conceptually good idea Yields closed form solutions to the posterior Example Normal prior and Normal likelihood yields Normal posterior Spring 2005 K. Ensor, STAT 421

Normal  Normal = Normal Spring 2005 K. Ensor, STAT 421

Mean and variance of posterior distribution Spring 2005 K. Ensor, STAT 421

Posterior – always this easy? NO In general, it is difficult to find this posterior distribution in closed form. Use computational techniques to simulate the posterior distribution (MCMC, Metropolis-Hasting algorithm) Break down the multivariate integral problem into a series of conditional hierarchical integrations which are done via Monte Carlo methods. Forecasts? Obtained via simulation as well See page 439 Spring 2005 K. Ensor, STAT 421

Example – Policy data Bayesian VAR forecast Vector AR forecast Spring 2005 K. Ensor, STAT 421

Set prior standard deviation for AR parameter to a small value Spring 2005 K. Ensor, STAT 421

Homework for next week Work through Example 10.1 using the Finmetrics BVAR routines Prepare a demonstration of your results that includes A description of the prior distributions assumed Graphical display of the posterior distributions And appropriate forecasts for the series. Spring 2005 K. Ensor, STAT 421

Caution The posterior depends on the choice of the prior. Thus the results should be considered in light of the knowledge imposed by the prior distribution. There is a class of “noninformative” priors that place little to no information on the prior parameters. This is a commercial for STAT 423. Spring 2005 K. Ensor, STAT 421