1. 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

2. 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

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.
Spring 2005
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
Spring 2005
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
Spring 2005
K. Ensor, STAT 421

6. 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

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

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

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

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