Econistics Research and Consulting | Use of Moderator Model
Cross products (moderators or quadratic function) can be used to add controlling variables in the model. But in most cases they are backed by the theory
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Control Variables – Moderator Approach

17 Nov Control Variables – Moderator Approach

In continuation of the discussion on selecting control variables. Moderation approach is used when two different variables co-influence the dependent variable, or each one of them determine the marginal effect of other one (Hayes, 2017). In economics it is called catalyst effect or cross product effect. There are two types of moderation which we can use.

Strategy 3a: Self Moderation

Self-moderation approach is used when theory has depicted a nonlinear (quadratic) effect of the independent variable.

y = f(x2)

dy/dx = f(x)

This can be confirmed when follow pre symptoms occur

  • Theory itself explains nonlinear effect like (Law of diminishing returns, Kuznets curve etc.)
  • When the effect of IV is different when incidence is low and when incidence is high.

Or when following post symptom occur.

  • Important variable has opposite sign or it is insignificant.

Advantages of using self-moderation

Even though we have calculated single coefficient for all countries, making homogeneous policy. But having a cross product will enable to generate country wise policy implications (see Arshed et al., 2017). Consider you have estimated following model.

y = a + bx + cx2

Now type of coefficients will determine the type of line formed.


Type of a

Type of b





Exponential increase

Linear increase

Inverted U shape


Delayed increase

No relation

Delayed decrease


U shaped

Linear decrease

Exponential decrease

After estimating the model we can estimate the optimal x which is denoted as x*. Which is calculated by equating first derivative equal to zero.

dy/dx = b + 2ax

x* = b + 2ax = 0

x* = -b / 2a

That x* can be compared with country wise averages of x to see which country below or above the x* which has different implications and make following table (see Arshed et al., 2017)



Average x



















Here the x* value will be constant for all countries but the average value of x (or current value of x based on researchers choice) can be below or above the x*.  Based on the nature of the line formed we can form two categories of countries one which are below the turning point and one which is above turning point. 

   Strategy 3b: Cross product

Cross product approach is used when two different variables co-influence dependent variable, or in where one of them is catalysis to other in defining the marginal effect (see Kalim et al., 2019).

y = f(xz)

dy/dx = f(z)

There are two to three hints of cross product

  • There will be multicollinearity between the variables i.e. Correlation (X, Y)
  • Mostly from the theory you will come to know that both are related to each other both are either substitute or compliment.


  • When we have to estimate the type of coordination between monetary and fiscal policy
  • Cross price elasticity measurement
  • Estimation of determinates of elasticity of a particular factor

Advantages of using cross products

First advantage is that it captures any lingering multicollinearity between the two independent variables which have any economic meaning. Second it provides new information. Consider you estimate the following model where z is catalyst in how x explains y.

y = a + bx + d(x*z)

Here we can see how the slope shifts for different values of z[1]. we can plot following table in panel data.


P0 of z

P25 of z

P50 of z

P75 of z

P100 of z

























[1] If z is a binomial variable we can use the two categories, while for a continuous variable z we calculate the above mentioned percentiles. Most of them we already have calculated in descriptive statistics.

Your participation and suggestions are welcomed


Arshed, N., Anwar, A., Kousar, N., & Bukhari, S. (2017). Education Enrollment Level and Income Inequality: A Case of SAARC Economies. Social Indicators Research, 1-14.

Hayes, A. F. (2017). Introduction to mediation, moderation, and conditional process analysis: A regression-based approach. Guilford Publications.

Kalim, R., Arshed, N., & Shaheen, S. (2019). Does Competitiveness Moderates Inclusive Growth: A Panel Study of Low-Income Countries. Competitiveness Review: An International Business Journal.

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