Very simple estimators for a class of polynomial factor ...
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Very simple estimators for a class of polynomial factor ...
My website at Research Gate has a number of relevant working papers: â· Consistent Partial Least Squares for linear and polynomial factor models. (2011).
Very simple estimators for a class of polynomial factor models Theo K. Dijkstra University of Groningen Faculty of Economics and Business (Econometrics and Finance)
The VI European Congress of Methodology Utrecht, July 24, 2014
1/18
Recursive systems of polynomial equations in latent variables Example of an equation: η3 = γ1 η1 + γ2 η2 + γ11 η12 − 1 + γ22 η22 − 1 +
(1) (2)
γ12 (η1 η2 − E (η1 η2 )) + γ122 η1 η22 − E η1 η22 +
(3)
ζ3
(5)
(4)
plus possibly other powers. The latent variables η are standardized, and the residual ζ3 is independent of (η1 , η2 ). The next equation, if any, expresses η4 similarly in terms of (η1 , η2 , η3 ) and the residual ζ4 is independent of (η1 , η2 , η3 ). Et cetera. Each equation can be handled separately. 2/18
Measurement equations
Each latent variable is measured by a unique vector of indicators. For ηi we have: yi = λi ηi + εi (6) yi has least two components. The indicators are standardized. The elements of all εi are mutually independent and independent of η. So Σii := Eyi yi| = cov (yi ) = λi λ|i + Θi (7) where the error covariance matrix Θi is diagonal. For i 6= j Σij := Eyi yj| = ρij λi λ|j
(8)
3/18
A helpful implication Take any vector wj of the same dimension as yj and calculate Σij wj = Eyi (yj| wj ) = ρij λ|j wj λi (9) Let Sij be the sample covariance/correlation matrix between yi and yj . Define for all i X bi ∝ w sij Sij wj (10) j6=i
sij = sign (wi| Sij wj ). Normalizations: wi| Sii wi = 1 and bi| Sii w bi = 1. w Partial Least Squares (PLS): iterate until a fixed point occurs. For all approaches q bi = λi / λ|i Σii λi wi := the probability limit of w (11) 4/18
How to get λi Take a number b ci such that the difference between the off-diagonal elements of bi| and Sii bi · b (12) ci w b ci w is as small as possible (sum of squares). This gives X
