Predictive distributions

1. 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. K. Ensor, STAT 421

2. Resampling • Block 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 K. Ensor, STAT 421

3. 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. K. Ensor, STAT 421

4. 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 K. Ensor, STAT 421

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

6. The prior • Conjugate prior • 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 K. Ensor, STAT 421

7. Normal  Normal = Normal K. Ensor, STAT 421

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

9. 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 K. Ensor, STAT 421

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

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

12. 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. K. Ensor, STAT 421

13. 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. K. Ensor, STAT 421