Regression specifications in applied econometrics frequently employ regressors that are defined as the product of two other regressors to form an interaction. Unfortunately, the interpretation of the results of these models is not as straight forward as in the linear case. In this paper, we present a method for drawing inferences for interaction models by defining the partial influence function. We present an example that demonstrates how one may draw new inferences by constructing the confidence intervals for the partial influence functions based on the traditional published findings for regressions with interaction terms.Interaction effects; dummy variables; linear transformation; Fieller method

J. Hirschberg

J. Lye

English

Research Papers in Economics

                                                                                                        ISSN  0819-2642 
                                                                                                      ISBN  978 0 7340 3725 1 
 
 
 
 
 
 
 
 
 
THE UNIVERSITY OF MELBOURNE 
 
DEPARTMENT OF ECONOMICS 
 
 
RESEARCH PAPER NUMBER 1015 
 
 
November 2007 
 
 
A Reinterpretation of Interactions in Regressions 
 
by 
 
J. Hirschberg & J. Lye 
 
 
 
 
   
Department of Economics 
The University of Melbourne  
Melbourne    Victoria     3010 
Australia. 
  
A Reinterpretation of Interactions in Regressions  
 
 
J. Hirschberg and J. Lye  
 
Department of Economics  
University of Melbourne 
 
November 2007  
 
Abstract 
  Regression specifications in applied econometrics frequently employ regressors 
that are defined as the product of two other regressors to form an interaction.  
Unfortunately, the interpretation of the results of these models is not as straight forward 
as in the linear case.  In this paper, we present a method for drawing inferences for 
interaction models by defining the partial influence function.  We present an example 
that demonstrates how one may draw new inferences by constructing the confidence 
intervals for the partial influence functions based on the traditional published findings 
for regressions with interaction terms. 
 
Key Words: Interaction effects; dummy variables; linear transformation, Fieller method 
JEL classification: C12; C51 
 
Corresponding author, jnlye@unimelb.edu.au, Department of Economics, University of 
Melbourne, Melbourne, Victoria 3010, Australia. 
   1  
1 Introduction 
  Regression models with interaction effects are often proposed to allow for the 
marginal effect of one explanatory variable to depend on another.  The interpretation of 
the marginal effects depends on the choice of a value for the interacted regressors.  We 
demonstrate that the graph of the partial influence function and its confidence interval 
can be used to both establish the level of the effect of these regressors as well as provide 
the appropriate tests of hypotheses.  Section 2 defines the partial influence function.  
Section 3 describes how confidence intervals for the partial influence function are 
equivalent to the confidence bounds found using the Fieller technique and can be used 
to draw appropriate inference from a regression with interaction terms.  An application 
of this technique using the information from a previously published paper is presented 
in Section 4.  Our conclusions are presented in Section 5 
 
2  The Partial Influence of Interacted Regressors 
  Consider the model:  
   
12 3 4 5 1 - 4  =   +   +   +   +   +  +   
= 
kk xwx wz z ββ β β β β
+
y ε
Xβε
L
    (1) 
where y is the () 1 T × vector of observations on the dependent variable, X is the 
() Tk × matrix of explanatory variables consisting of a constant,  x, w, xw, and other 
regressors j-4
j5
k
z
=
∑  and ε  is the ( ) 1 T ×  vector of disturbance terms where 
1 ~( , ) TT T ×× σ
2 ε 0I .  The OLS estimators for the parameters are 
1 ˆ ()
− ′′ = β XX XY, 
ˆ  () Tk
2 ′ σ= − εε .  Under the usual assumptions, the parameter estimates are 
asymptotically normally distributed according to  () ( )
1 2 ˆ ~, N
− ′ σ ββ XX .   
 When  x and w are both continuous variables, the marginal effect of a change in x 
is defined as:
1   2  
    
( ) E,
24 =   +   
yxw
x w
∂
∂ ββ      ( 2 )  
If x is a dummy variable and w is a continuous variable, we define the difference in the 
regression equation when x =1 and when x = 0 as: 
    
( ) E,
24 =   +   
yxw
x w
Δ
Δ ββ      ( 3 )  
The equations defined in (2) and (3) are the partial influence (PI) functions. 
  Typically when estimating a model such as (1) a researcher would test the 
significance of the interaction term using the t-test of  04 :0 , H β =  and a joint F-test of 
02 4 :0 . H β= β=  Alternatively, one could construct the partial influence function (as (2) 
or (3)) using the estimated values of the coefficients.  However, using the partial 
influence requires the choice of an appropriate value of the other regressor (w) at which 
to evaluate it.  One approach is to set the value of this other regressor to a particular 
value, such as the mean.  An alternative approach is to determine the value of w at 
which 
( ) E, 0
yxw
x
∂
∂ =  or changes sign.   
  From (2), the value of w that results in 
( ) E, 0
yxw
x
∂
∂ =  is 
2
4
* w
β
β =−  and is estimated 
by 
2
4
ˆ *
ˆ ˆ , w
β
β =−  where  ˆ
i β are the OLS estimates of  i β .
2  In the next section we show how a 
confidence interval for 
* w  can be constructed using the Fieller method and how the 
bounds of this interval are related to the usual regression parameter hypothesis tests. 
 
3  Ratios of Parameters in Linear Regression 
 The Fieller method (Fieller 1932, 1954) provides a general procedure for 
constructing confidence limits for statistics defined as ratios.  Zerbe (1978) defines a 
version of Fieller’s method in the regression context.   
 For 
2
4
* , w
β
β =−  a 100(1 )% −α  confidence interval is determined by solving the 
quadratic equation 
2 20 aw bw c ++ = , where 
2
22 2
44 ˆ ˆ at α =β− σ,  ()
2
2
24 2 4 ˆˆ ˆ bt α = −σ − β β ,   3  
2
22 2
22 ˆ ˆ ct α =β − σ ; 
2 ˆ i σ are the estimated variances of 
2 ˆ
i β (i=2,4),  24 ˆ σ  is the estimated 
covariance between  2 ˆ β  and  4 ˆ , β  and 
2
2 tα  is the  % α  critical value of the t-distribution 
with T-k  degrees of freedom.  The two roots of the quadratic equation 
()
2
, LU
bba c
a ww −± − = , define the confidence bounds of 
*. w  The parameters a, b and c are 
equivalent to:  ()
2
22 2
44 ˆ , at t α =σ −   ( )
2
2
24 2 4 ˆˆ br t t t α =− σσ −  and  ()
2
22 2
22 ˆ , ct t α =σ −  where 
ˆ
ˆ
i
i i t =
β
σ ;  24 2 4 ˆˆ ˆ r = σ σσ and r is the correlation between  2 ˆ β  and  4 ˆ . β  
  Using the expressions for a, b and c 
   () () ( )( )
22
2 22 2 2 2 2 2
24 2 4 4 ˆˆ 1 ba ct tr t rtt αα ⎡ ⎤ −= σ σ − + − − ⎣ ⎦.   (4) 
From (4) if  0, a >  that is, 
2
22
4 tt α > then 
2 > 0 ba c − , which implies that one can reject 
04 H: 0 β=  at the  % α  level of significance.  In this case a finite confidence interval can 
be constructed (see also Buonaccorsi 1979).  Alternatively, the resulting confidence 
interval may be the complement of a finite interval when  ( ) ( )
2  >  0,  0 , ba c a −<  or the 
interval may be the whole real line when  ( ) ( )
2 < 0,  0 ba c a − < .   
  It is also interesting to note that when ( ) 0 a <  and  
      () 1,
1
2
Tk − > F F         ( 5 )  
where F is the value of the test statistic from the hypothesis:  02 4 :0 H β= β= , the 
resulting confidence interval will be a complement of a finite interval.  (See also 
Scheffé 1970 and Zerbe 1982).  This is the case when the coefficient on the interaction 
term is not significant (since  0 a <  implies that 
2
22
4 tt α < ) but there may still be regions of 
w over which the partial influence function is significantly different from 0.   4  
  Hirschberg and Lye (2007) show that the confidence bounds can also be 
obtained by plotting the 100(1 )% −α  confidence bounds for the partial influence 
function defined in (2) and (3) defined by:  
   () ( )
2
22 2
24 2 2 4 4 ˆˆ ˆˆˆ CI      + 2  +  wt w w α =β+ β ± σ σ σ     (6) 
An estimate of the value of w
* (
* ˆ w ) satisfies the relationship 
*
24 ˆˆ ˆ + 0 w ββ= .  Similarly, 
the bounds defining the 100(1 )% −α  confidence interval on 
* w are found by solving:  
  () ()
2
2
22 2 2
24 2 2 4 4 ˆˆ ˆˆˆ  -   + 2  +   = 0 wt w w α β+ β σ σ σ      (7) 
The solution for (7) is equivalent to the solution of the quadratic equation, 
(
2 +   +   = 0 aw bw c ), where a, b and c are defined as above.   
4 Class  Sizes  Example 
  Numerous studies have examined the effect of class size on student 
performance.  Using interaction terms, the recent studies by Akerhielm (1995), Angrist 
and Lavy (1999), Bonesrønning 2003 and Wöβmann and West  (2006) have examined 
how class size effects on student performance differ across subgroups, such as gender, 
race, and socioeconomic class.   
In this example, we follow Bonesrønning (2003) who estimates an education 
production function using individual student data for Norwegian children aged 13 to 16.  
To allow for possible gender differences an interaction between gender and class size is 
used.  In estimation, predicted class size is used as an instrument for class size.   
Bonesrønning finds that the interaction term is significant at the 10% level of 
significance (Table 2).  In this case the partial influence function for the predicted 
performance difference between males and females is: 
   
( ) , 2.53 0.11
yxw
x CS
Δ
Δ =− +
E      (8)   5  
where CS is class size.  We find that this relationship changes sign when 
* 2.53
0.11 23 CS == .   
The upper and lower bounds of the Fieller interval corresponding to 
( ) , 0
yxw
x
Δ
Δ =
E are 
found by solving the equivalent expression to (7), 
  () () () () () ( ) 2
22 2 22 2.53 0.11 1.45 2 1.45 0.06 0.11 0 CS t r CS CS −− − + + = α . (9) 
  In order to apply (9) we require a value for r the correlation between the 
parameter for class size as well as for the interaction of class size and gender.  From the 
reported difference in the 
2 R  between models with and without the interaction term we 
can estimate the value of the F-statistic for the joint test of the coefficients on gender 
and gender × class size =0.  Although the model reported also includes a gender term, 
the 
2 R  is appropriate to use as the gender t-stat equals 1.  Then r is calculated from the 
formula for the F-statistic for two restrictions given by the following relationship based 
on the t-statistics of the two parameters:  
     ( )
()
22
12 1 2
2
2
½
1
tt r t t
F
r
⎛⎞ +−
⎜⎟ =
⎜⎟ − ⎝⎠
     ( 1 0 )  
in this case we use the negative root r = -0.996.  In simulations using similar data we 
have found that correlation between the coefficient for the interaction and the 
coefficients for the variables with which they are interacted are found to be negative in 
almost all circumstances. 
Figures 1 and 2 plot the 
( ) , yxw
x
Δ
Δ
E  function and the corresponding 90% and 95% 
confidence intervals.  In Figure 1 we find that the 90% Fieller interval has finite bounds 
(13, 24).  However, from Figure 2 we note that the 95% interval is a complement of a 
finite interval, with upper bound equal to 24.  The 95% confidence interval indicates 
that female students are significantly affected by class sizes above 24, which contrasts 
with Bonesrønning (2003, pg 959) who concludes that “… females are unaffected by   6  
class size”..  Similar results are obtained using the 90% confidence interval, which also 
indicates that there is a significant benefit for males for class sizes smaller than 13. 
 
(Figures 1 and 2 here) 
 
5. Conclusion 
  In this paper we have demonstrated that the partial influence function, defined 
when an interaction is present in a regression, can be examined by plotting the function 
at appropriate values of the interacted variable.  The new material in this paper is the 
development of the relationship between the Fieller Method for the construction of 
confidence intervals for normally distributed random variables and the confidence 
intervals we can construct for the partial influence function. 
  We also demonstrate how it may be possible to construct a partial influence 
function and the corresponding confidence bounds for the influence of the interaction 
effect from previously published results.  These bounds can then be used to consider 
new inferences that are not available from the traditional approach.   
 
Acknowledgements 
The research carried out in this paper was partially funded by a grant from the Faculty 
of Economics and Commerce at The University of Melbourne.  We wish to thank David 
Moreton for helpful comments.   
   7  
References 
Akerhielm, K.  (1995) “Does Class Size Matter?”, Economics of Education Review, 14, 
229-241. 
Angrist, J.  and V.  Lavy (1999), “Using Maimonides’ Rule to Estimate the Effect of 
Class Size on Scholastic Achievement”, Quarterly Journal of Economics, 114, 
533-575. 
Bonesrønning, H.  (2003), “Class Size Effects on Student Achievement in Norway: 
Patterns and Explanations”, Southern Economic Journal, 69, 952-965. 
Buonaccorsi, J.  P.  (1979), “On Fieller’s Theorem and the General Linear Model”, The 
American Statistician, 33, 162. 
Fieller, E.  C.  (1932), “The Distribution of the Index in a Normal Bivariate 
Population”, Biometrika, 24, 428–440. 
Fieller, E.  C.  (1954), “Some Problems in Interval Estimation”, Journal of the Royal 
Statistical Society.  Series B, 16, 175–185.   
Hirschberg, J.  and J.  Lye (2007), “Providing Intuition to the Fieller Method with two 
Geometric Representations using Stata and Eviews”, Department of Economics 
Working Paper #952, University of Melbourne. 
Scheffé, H.  (1970), “Multiple Testing versus Multiple Estimation.  Improper 
Confidence Sets.  Estimation of Directions and Ratios”, The Annals of 
Mathematical Statistics, 41, 1–29. 
Zerbe, G., (1978), “On Fieller’s Theorem and the General Linear Model”, The 
American Statistician, 32, 103–105. 
Zerbe, G.  (1982), “On Multivariate Confidence Regions and Simultaneous Confidence 
Limits for Ratios”, Communications in Statistics Theory and Methods, 11, 
2401–2425.     8  
Wöβmann, L.  and M.  West (2006), “Class-Size Effects in School Systems around the 
World: Evidence from between-Grade Variation in TIMSS”, European 
Economic Review, 50, 695-736.   9  
Footnotes 
1.  Note that a similar analysis can be performed on 
( ) E, yxw
w
∂
∂  
2.  We use the case of the continuous regressor w it can be show that these results 
are the same for the discrete case.   10  
Figure 1: 
( ) , yxw
x
Δ
Δ
E  and the 90% Confidence Bounds 
-4
-3
-2
-1
0
1
2
3
4
5 1 01 52 02 53 03 5
class size
P
I
 
 
f
u
n
c
t
i
o
n
  Upper bound
  Lower bound
* CS
90%   Confidence bounds
   11  
Figure 2: 
( ) , yxw
x
Δ
Δ
E  and the 95% Confidence Bounds 
 
-4
-3
-2
-1
0
1
2
3
4
5 1 01 52 02 53 03 5
class size
P
I
 
f
u
n
c
t
i
o
n
  Upper bound
* CS
95%   Confidence bounds
 
 
 

Class Size Effects on Student Achievement in Norway: Patterns and Explanations”,

Does Class Size Matter?”,

Multiple Testing versus Multiple Estimation. Improper Confidence Sets.

On Fieller’s Theorem and the General Linear Model”,

On Multivariate Confidence Regions and Simultaneous Confidence Limits for Ratios”,

Providing Intuition to the Fieller Method with two Geometric Representations using Stata and Eviews”,

Some Problems in Interval Estimation”,

The Distribution of the Index in a Normal Bivariate Population”,

Using Maimonides’ Rule to Estimate the Effect of Class Size on Scholastic Achievement”,

A Reinterpretation of Interactions in Regressions

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A Reinterpretation of Interactions in Regressions

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