Recently you must have heard the word "Trackers", high frequency indicators that “track” in real time measures of economic activity. I have recently stumbled on an method to construct one that looks interesting. A short thread.



#EconTwitter

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Trackers are often constructed as the first principal components (PC) of high frequency indicators (e.g. electricity consumption, card payments, etc. etc.) correlated with economic activity. Say you have “k” such (standardized) predictors. Collect them in a Txk matrix X.

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The first PC of your X matrix is a linear combination Xw, where w is kx1 vector that maximizes w'X’Xw and w'w =1. The vector w is the eigenvector associated with the largest eigenvalue of (X’X).

For an example see the NY Fed WEI

https://www.newyorkfed.org/research/policy/weekly-economic-index#/

So far so good.

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But you might want your tracker to be very correlated with a particular set of business cycle indicators Y, of dimensions Txm. Then your set of weights w need to maximize w’(X’Y) (Y’X)w and w'w=1. This is called Partial Least Squares (PLS), proposed by Wold (1966).

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But you can do more than that. You can “sparsify” PLS, i.e. set to 0 those elements in w that are small. How do you sparsify PLS? This (beautiful) paper by Chon and Keles https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2810828/ shows you how to do that!

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The key is going back and forth between a singular value decomposition and a shrinkage step (i.e. the Elastic Net). Are you interested in the practical application? Get the paper and go directly to Section 3, where they present the estimation algorithm.

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This shrinkage step might buy you some gain in terms of bias/variance trade off, but do not take this for granted. This is an empirical question, "The illusion of sparsity", by @gprimice and co-authors, linked here https://faculty.wcas.northwestern.edu/~gep575/illusion4-2.pdf is a deep analysis of this issue

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You want a flavour of an actual economic application of sparse PLS? Check out this New York Fed nice short paper on estimating economic activity in China. That is where I learnt about this in the first place
https://www.newyorkfed.org/research/epr/2020/epr_2020_china-sparse-pls_groen

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