PRACTITIONERS' CORNER*: A Note on the Computation of the Correct Estimated Covariance Matrix for a Ridge Regression Shortcut
| DOI | http://doi.org/10.1111/j.1468-0084.1987.mp49003007.x |
| Author | William Bishopp,Simon Power |
| Published date | 01 August 1987 |
| Date | 01 August 1987 |
OXFORD BULLETIN OF ECONOMICS AND STATISTICS, 49, 3 (1987)
0305-9049 $3.00
PRACTITIONERS' CORNER*
A Note on the Computation of the Correct Estimated
Covariance Matrix for a Ridge Regression Shortcut
Simon Power and William Bishop p
The purpose of this note is to provide simple computational procedures
for the calculation of the correct estimated covariance matrix for a
commonly used ridge regression shortcut.
Consider the linear regression model
Y = X(3 + u
for which Hoerl and Kennard's (1970) ordinary ridge estimator may be
defined as
bR = (X'X + kI)'X'Y
where Y is a Tx 1 vector of observations on the dependent variable, X
is a T X p matrix of observations on the explanatory variables, u is a
Tx 1 vector of random disturbances such that E[u] = O and E[uu'] =
c2I I is an identity matrix of order p, (3 is a p X I vector of unknown
parameters, and k is the ridge constant. This estimator is well known to
have bias vector k(X'X + kI)'(3 and covariance matrix
V(bR) = u2(X'X + kIf'X'X(X'X + kI)1
with the latter being estimated in practice by
î(bR) &2(X'X+ kI)'X'X(X'X + kIi', &2 = û'û/(Tp).
Appropriate choices of k are discussed by, among others, Judge et al.
(1985) and Vinod and Ullah (1981).
A commonly used trick in the computation of bR (see, e.g., Vinod
and Ullah, 1981, pp. 189.-90) when a ridge regression routine is
unavailable,' is to run ordinary least squares (OLS) on the augmented
model
X*(3 + u*
where Y is a (T + p) X 1 vector (Y'O')', O is ap Xl vector of zeros, X*
is a (T+p)xp matrix (X'k1121)', and u* =(u'k"2(3')'.1 This produces
* The purpose of Practitioners' Corner is to publish brief methodological notes of interest to
applied economists. The Editors welcome submissions of this sort.
'TSP Version 4.0, for example, does not provide a ridge regression subroutine.
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