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
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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))
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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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