17 Nov Control Variables – Moderator Approach
In continuation of the discussion on selecting control variables. Moderation approach is used when two different variables coinfluence 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
Selfmoderation approach is used when theory has depicted a nonlinear (quadratic) effect of the independent variable.
y = f(x^{2})
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 selfmoderation
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 + cx^{2}
Now type of coefficients will determine the type of line formed.

Type of a 

Type of b 
Positive 
Zero 
Negative 
Positive 
Exponential increase 
Linear increase 
Inverted U shape 
Zero 
Delayed increase 
No relation 
Delayed decrease 
Negative 
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)
COUNTRY 
x* 
Average x 
C1 


C2 


C3 


C4 


C5 


C6 


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 coinfluence 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.
Applications:
 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.
dy/dx 
P_{0 }of z 
P_{25 }of z 
P_{50 }of z 
P_{75 }of z 
P_{100 }of z 
Country1 





Country2 





Country3 





Country4 





[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
References
Arshed, N., Anwar, A., Kousar, N., & Bukhari, S. (2017). Education Enrollment Level and Income Inequality: A Case of SAARC Economies. Social Indicators Research, 114.
Hayes, A. F. (2017). Introduction to mediation, moderation, and conditional process analysis: A regressionbased approach. Guilford Publications.
Kalim, R., Arshed, N., & Shaheen, S. (2019). Does Competitiveness Moderates Inclusive Growth: A Panel Study of LowIncome Countries. Competitiveness Review: An International Business Journal.
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