Again on Sign Changes Upon Deletion of a Variable From a Linear Regression
| Published date | 01 May 1988 |
| DOI | http://doi.org/10.1111/j.1468-0084.1988.mp50002010.x |
| Date | 01 May 1988 |
| Author | I. Visco |
OXFORD BULLETIN OF ECONOMICS AND STATISTICS, 50,2(1988)
0305-9049 $3.00
Again on Sign Changes Upon Deletion of a Variable
From a Linear Regression
I. Visco
In a recent note in this BULLETIN, Oksanen (1987) re-examines a problem
tackled by Learner (1975) and Visco (1978), namely the conditions for a sign
change of a coefficient estimate in a linear regression when one of the regres-
sors is deleted. With reference to the unrestricted OLS equation)
y=x1b1+x2b2+X3b3+û (1)
Learner proved that a necessary condition for the sign of the coefficient of X2
to change upon deletion of x1 is t1 > t2 I where t1 and t2 are the t-ratios of the
first two regressors in (1). Visco extended this result to obtain the full neces-
sary and sufficient conditions involving not only t-ratios but also the correla-
tion coefficient between b1 and b2. Furthermore, he also considered two
special cases of (1), i.e. those with 2 and 3 regressors only (besides the inter-
cept), and derived simple necessary and sufficient conditions involving only
sample correlation coefficients of the variables present in the regression.
Ingeniously Oksanen examines a different solution to the problem con-
sidering the OLS equation: y*=Xb+xb+û (2)
where y, x13 and x23 are the OLS residuals from regressions of y, x1 and x2
on X3. As is well known, b1 and b2 in (2) are identical to b1 and b2 in (1), so
that we are back in a two regressors equation to which Visco's sample cor-
relation results should apply, this time, however, in terms of partial correla-
tion coefficients r013, r023 and r123 (i.e. the correlation coefficients between
y*, x13 and x23).
Oksanen then shows that Leamer's necessary condition for a sign change
can be re-written as r0131> Ir02.3 I; he also claims that in order to have suf-
ficiency, the further condition (r013r123r023)> O should be added. But this is
not enough. The correct necessary and sufficient condition is as follows.
Since:
b2 = A( r023 - r013r123) = Br0213 (3)
= Cr,23 (4)
* The author is grateful for helpful comments to C. A. Bollino, D. E Hendry and E. H.
Oksanen.
'The notation here follows that of Oksanen; Y' x1 and x2 are N-vectors, X3 is an Nx(p - 1)
matrix of other regressors (including a vector of 1 s, the scalars b1 and b2 and the vector b3 are
OLS coefficient estimates, and û is the vector of OLS residuals.
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