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Re: st: coefficient interpretation in OLS

From   David Hoaglin <>
Subject   Re: st: coefficient interpretation in OLS
Date   Fri, 17 Aug 2012 10:08:57 -0400

Dear Lynn,

In interpreting the coefficients in a multiple regression, two facts
are important.

1. The definition of each coefficient includes the set of other
predictors in the model.

2. The coefficient of a predictor, say X1, tells how Y changes in
response to change in X1 after adjusting for the contributions of the
other predictors in the model (in the data at hand).  A coefficient is
a slope, so it gives change in Y per unit increase in X1, not
necessarily the change in Y when X1 is increased by 1 unit (unless the
values of X1 are only 0 and 1).  Some textbooks, unfortunately,
interpret the coefficient of X1 as telling how Y changes with an
increase of 1 unit in X1 when the other predictors are held fixed, but
that is simply not how OLS works; that interpretation is
oversimplified and often incorrect.

Terry Speed's column in the current issue of the IMS Bulletin
discusses both of these points.

David Hoaglin

On Fri, Aug 17, 2012 at 6:25 PM, Lynn Lee <> wrote:
> Dear all,
> When I run simple OLS regression or pooled OLS regression, I find if I add
> more variables to the model, the coefficient on specific explanatory
> variable can vary in magnitude. For example,
> Y1=beta+beta1*X1+beta2*X2+beta3*X3+error term;
> Y2=alpha+alpha1*X1+ alpha2*X2+ alpha3*X3+ alpha4*X4+error term.
> The absolute value of estimates of beta1 or alpha1 can increase or sometimes
> decrease.  I am not confident to explain this theoretically. Is it related
> to potential endogeneity issue?
> Best Regards,
> Lynn Lee
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