Interaction Effects in Econometrics
Hatice Ozer-Balli
∗
Massey University
Bent E. Sørensen
†
University of Houston and CEPR
25 June 2010
Abstract
We provide practical advice for applied economists regarding specification and in-
terpretation of linear regression models with interaction terms.
JEL classification: C12, C13
Keywords: Non-Linear Regression, Interaction Terms.
∗
School of Economics and Finance, Massey University, New Zealand, e-mail: h.ozer-
balli@massey.ac.nz, tel:+64 63505799 ext. 2666.
†
Department of Economics, University of Houston, TX, e-mail: bent.sorensen@mail.uh.edu, tel:
7137433841, fax: 7137433798
1 Introduction
A country may consider a reform that would strengthen the financial sector. Would
this help economic growth and development? This simple question is frustratingly hard
to answer using empirical data because economic development itself spawns financial
development, so while economic and financial developments are positively correlated
this does not answer the question asked. In a highly influential paper, Rajan and
Zingales (1998) provide convincing evidence that financial development is important for
economic development by asking the simple question: do industrial sectors that are more
dependent on external finance grow faster in countries with a high level of development.
This question involves interactions between financial development and dependency on
external finance. Since the publication of Rajan and Zingales’ highly influential study,
the estimation of models with interaction effects have become very common in applied
economics.
In Section 2, we discuss some practical issues related to the specification of regres-
sions with interaction effects and make recommendations for practitioners. In Section 3,
we illustrate our recommendations with Monte Carlo simulations and, in Section 4, we
revisit some prominent applied papers where interaction effects figure prominently, in-
cluding Rajan and Zingales (1998), and examine if the published results are robust.
Section 5 concludes.
2 Linear Regression with Interaction Effects
Many econometric issues related to models with interaction effects are very simple and
we illustrate our discussion using simple Ordinary Least Squares (OLS) and instrumental
variable (IV) estimation. Often applied papers use more complicated methods involving,
say, Generalized Method of Moments, clustered standards errors, etc., but the points we
are making typically carry over to such settings with little modification.
Let Y be dependent variable, such as growth of an industrial sector, and X
1
and X
2
1
independent variables that may impact on growth, such as the dependency on external
finance and financial development. Applied econometricians have typically allowed for
interaction effects between two independent variables, X
1
and X
2
by estimating a simple
multiple regression model of the form:
Y = β
0
+ β
1
X
1
+ β
2
X
2
+ β
3
X
1
X
2
+ , (1)
where X
1
X
2
refers to a variable calculated as the simple observation-by-observation
product of X
1
and X
2
. In the example of Rajan and Zingales (1998), the interest centers
around the coefficient β
3
—a significant positive coefficient implies that sectors that are
more dependent on external finance grows faster following financial development.
We refer to the independent terms X
1
and X
2
as “main terms” and the product
of the main terms, X
1
X
2
, as the “interaction term.” This brings us to our first simple
observations:
1. In a regression with interaction terms, the main terms should always be included.
Otherwise, the interaction effect may be significant due to left-out variable bias.
(X
1
X
2
is by construction likely to be correlated with the main terms.)
1
2. The partial derivative of Y with respect to X
1
is β
1
+ β
3
X
2
. The interpretation
of β
1
is the partial derivative of Y with respect to X
1
when X
2
= 0. A t-test for
β
1
= 0 is, therefore a test of the null of no effect of X
1
when X
2
= 0. To test for
no effect of X
1
one needs to test if (β
1
, β
3
) = (0, 0) using, for example, an F-test.
1
Some authors have referred to this as a multicollinearity problem. Althauser (1971) show that the
main terms and the interaction term in the equation (1) are correlated. These correlations are affected
in part by the size and the difference in the sample means of X
1
and X
2
. Smith and Sasaki (1979) also
argue that the inclusion of the interaction term might cause a multicollinearity problem. In our view,
collinearity is not a problem for regressions with interaction effects of a different nature than elsewhere
in empirical economics—if one asks too much from a small sample, correlations between regressors make
for fragile inference.
2
In applied papers, the non-interacted regression
Y = λ
0
+ λ
1
X
1
+ λ
2
X
2
+ υ, (2)
is often estimated before the interacted regression. In this regression, λ
1
= ∂Y /∂X
1
is
the partial derivative of Y with respect to X
1
, implicitly evaluated at X
2
= X
2
(the mean
value of X
2
).
2
The estimated β
1
-coefficient in (1) is typically very close to
ˆ
λ
1
−
ˆ
β
3
X
2
.
3. Estimating the interacted regression in the form
Y = β
0
+ β
1
X
1
+ β
2
X
2
+ β
3
(X
1
− X
1
) (X
2
− X
2
) + , (3)
results in the exact same fit as equation (1) and the exact same coefficient
ˆ
β
3
.
ˆ
β
1
will typically be close to
ˆ
λ
1
estimated from equation (2) because β
1
= ∂Y/∂X
1
is the partial derivative of Y with respect to X
1
, evaluated at X
2
= X
2
. If a
researcher reports results from (2), and wants to keep the interpretation of the
coefficient to main terms similar, is usually preferable to report results of the
regression (3) with demeaned interaction terms.
3
4. In the case where, say, X
2
is endogenous, X
1
is exogenous, and Z is a valid in-
strument for X
2
, X
1
Z will be a valid instrument for X
1
X
2
. Alternatively, one can
regress X
2
on Z and obtain
ˆ
X
2
and use X
1
ˆ
X
2
for the interaction term and obtain
a consistent estimate of β
3
.
2
Some social scientists suggest that the interaction term undermines the interpretation of the re-
gression coefficients associated with X
1
and X
2
(e.g., Allison (1977), Althauser (1971), Smith and
Sasaki (1979), and Braumoeller (2004)). The point is simply that researchers sometimes do not notice
the change in the interpretation of the coefficient estimate for the main terms when the interaction term
is added.
3
Because β
0
+ β
1
X
1
+ β
2
X
2
+ β
3
(X
1
− X
1
)(X
2
− X
2
) = (β
0
+ β
3
X
1
X
2
) + (β
1
− β
3
X
2
) X
1
+ (β
2
−
β
3
X
1
)X
2
+ β
3
X
1
X
2
, we get the exact same fit, with the changes in the estimated parameters given
from the correspondence between the left- and right-hand side of this equality. E.g.,
ˆ
λ
0
will be equal
to
ˆ
β
0
+ β
3
X
1
X
2
.
3
2.1 Robustness to misspecification
Often a researcher wants to test whether Y = f (X
1
, X
2
) and chose a linear specification
such as (2) for convenience. A more adequate specification may be a second order
expansion
Y = β
0
+ β
1
X
1
+ β
2
X
2
+ β
3
(X
1
− X
1
) (X
2
− X
2
) + β
4
X
2
1
+ β
5
X
2
2
+ . (4)
(We will refer to X
2
i
; i = 1, 2 as “second-order terms”—in applications one may wish to
enter the second-order terms in a demeaned forms for the same reasons as discussed for
the interaction term, but for notational brevity we use the simpler non-demeaned form
here.) The relevance of this observation is as follows.
5. If Y = f(X
1
, X
2
) can be approximated by the second order expansion (4) with
a non-zero coefficient to either X
2
1
or X
2
2
and corr(X
1
, X
2
) 6= 0, the coefficient
β
3
in the interacted regression (1) may be spuriously significant. For example, if
corr(X
1
, X
2
) > 0 the estimated coefficient
ˆ
β
3
will usually be positive even if β
3
= 0.
If quadratic terms are not otherwise ruled out, we recommend also estimating
the specification (4) in order to verify that a purported interaction term is not
spuriously capturing left-out squared terms.
The potential bias from leaving out second order terms is easily understood. If X
1
and X
2
are (positively) correlated, we can write X
2
= αX
1
+ w (where α is positive) so
the interaction term (we suppress the mean for simplicity) becomes αX
2
1
+ X
1
w where
the latter term has mean zero and will be part of the error in the regression. If X
2
1
is
part of the correctly specified regression with coefficient δ, the estimated coefficient to
the interaction term when estimating equation (1) will be α δ.
4
