Forecast Many Predictors II Slides

# Forecast Many Predictors II Slides - NBER Summer Institute...

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Revised July 23, 2008 12-1 NBER Summer Institute What’s New in Econometrics – Time Series Lecture 12 July 16, 2008 Forecasting and Macro Modeling with Many Predictors, Part II

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Revised July 23, 2008 12-2 Outline Lecture 11 1) Why Might You Want To Use Hundreds of Series? 2) Dimensionality: From Curse to Blessing 3) Dynamic Factor Models: Specification and Estimation Lecture 12 4) Other High-Dimensional Forecasting Methods 5) Empirical Performance of High-Dimensional Methods 6) SVARs with Factors: FAVAR 7) Factors as Instruments 8) DSGEs and Factor Models
4) Other High-Dimensional Forecasting Methods Recall the introductory discussion of optimal forecasting with many orthogonal predictors, in which the frequentist problem was shown to be closely linked to the Bayes problem: Frequentist: min δ ± r ( n G d ± ) = 2 () ( ) n Ed d dG d κ ± cdf of d i Bayes: min ± r ( G d ± ) = ) 2 ( E dd d G d ± subjective prior Empirical Bayes: min ± r ˆ ( G d ± ) = ) 2 ˆ ( E d G d ± estimated “prior” So far we have focused on a setup – the DFM – in which the DFM imposed structure on the coefficients in Y t +1 = P t + ε t +1, t = 1,…, T , The DFM said that, if P t are the principal components, then only the first r of them matter – the rest of the coefficients are exactly zero. Revised July 23, 2008 12-3

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Revised July 23, 2008 12-4 High-dimensional methods, ctd. The DFM implication that only the first r elements of δ are nonzero is an intriguing conjecture, but it might be false, or (more usefully) might not provide a good approximation. The methods we will discuss now address the possibility that the remaining n r (= 135 – 4 = 131, say) principal components matter – or equivalently, all the X ’s enter separately with some small but useful weight. This problem of prediction with many predictors has received a lot of attention in the stats literature so we will draw on it heavily: Empirical Bayes (parametric and nonparametric) Bayesian model averaging (BMA) Bagging, Lasso, etc Hard threhsholding methods including false discovery rate (FDR) (which is closely linked to Empirical Bayes, see Efron (2003))
High-dimensional methods, ctd. We will focus on methods for orthogonal regressors (some generalize to non-orthogonal, some don’t) Y t +1 = δ P t + ε t +1, t = 1,…, T , where P P / T = I n (e.g. P = principal components) Some (of many) methods: 1. Optimal Bayes estimator under the assumption i = d i / T , d i i.i.d. G ; The d i i.i.d G model is the opposite extreme from a DFM (exchangeability: ordering i doesn’t matter) 2. Hard thresholding (i.e. using a fixed t -statistic cutoff). 3. Information criteria AIC, BIC: here these reduced to hard thresholding with a cutoff c T , where c T (but not too quickly) Revised July 23, 2008 12-5

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Revised July 23, 2008 12-6 High-dimensional methods, ctd.
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## This note was uploaded on 12/26/2011 for the course ECON 245a taught by Professor Staff during the Fall '08 term at UCSB.

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Forecast Many Predictors II Slides - NBER Summer Institute...

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