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Multiplicative Interaction Models Thomas Brambor , William Roberts Clark, Matt Golder

Multiplicative Interaction Models Thomas Brambor , William Roberts Clark, Matt Golder. use multiplicative interaction models whenever the hypothesis to test is conditional include all constitutive terms in interaction model specifications

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Multiplicative Interaction Models Thomas Brambor , William Roberts Clark, Matt Golder

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  1. Multiplicative Interaction ModelsThomas Brambor, William Roberts Clark, Matt Golder • use multiplicative interaction models whenever the hypothesis to test is conditional • include all constitutive terms in interaction model specifications • Do not interpret constitutive terms as if they are unconditional marginal effects •  calculate substantively meaningful marginal effects and standard errors

  2. Variance • the variance is the expected value of the squared difference between the variable's realization and the variable's mean • covariance is a measure of how much two random variables change together

  3. Variance of Interaction Models

  4. X2 X2 is also an interaction term. It means that the effect of X is conditioned on X

  5. Continous Dependent Variable with Single Modifying Variable  מספר מפלגות נבחרות (משתנה תלוי) האפקט של גודל המחוז על מספר מפלגות נבחרות מספר מפלגות מתחרות (משתנה מתנה)

  6. Continous Dependent Variable with Single Modifying Variable –STATA code * **************************************************************** *; * Estimate Model: Y = b0 + b1X + b2Z + b3XZ + b4Controls + epsilon *; * **************************************************************** *; regress YXZXZ Controls; This indicates that you will be calculating the marginal effect of X across the modifying range of MV (or Z) in increments of 0.1. You may want to calculate the marginal effect of X for different increments in MV (or Z). To do this simply divide ( n-1) by a number different than 10. For example, you can divide by 100 to calculate the marginal effect of X across MV as MV increases in increments of 0.01. generate MV=(( n-1)/10); replace MV=. if n >60; * **************************************************************** *; * Grab elements of the coefficient and variance-covariance matrix *; * that are required to calculate the marginal effect and standard errors. *; * **************************************************************** *; matrix b=e(b); matrix V=e(V); scalar b1=b[1,1]; scalar b2=b[1,2]; scalar b3=b[1,3]; scalar varb1=V[1,1]; scalar varb2=V[2,2]; scalar varb3=V[3,3]; scalar covb1b3=V[1,3]; scalar covb2b3=V[2,3];

  7. * **************************************************************** *; * Calculate the marginal effect of X on Y for all MV values of *; * the modifying variable Z. *; * **************************************************************** *; gen conb=b1+b3*MV if n < 60; This line calculates the marginal effect (or conditional beta) of X for all values of the modifying variable MV so long as MV is less than 6. So, now we have the marginal effect of X for when MV=0, when MV=0.1, when MV=0.2 . . . , when MV=5.9. * **************************************************************** *; * Calculate the standard errors for the marginal effect of X on Y *; * for all MV values of the modifying variable Z. *; * **************************************************************** *; gen conse=sqrt(varb1+varb3*(MV2)+2*covb1b3*MV) if n < 60; * **************************************************************** *; * Generate upper and lower bounds of the confidence interval. *; * Specify the significance of the confidence interval. *; * **************************************************************** *; gen a=1.96*conse; gen upper=conb+a; gen lower=conb-a;

  8. * **************************************************************** *; * Graph the marginal effect of X on Y across the desired range of *; * the modifying variable Z. Show the confidence interval. *; * **************************************************************** *; graph twoway line conb MV, clwidth(medium) clcolor(blue) clcolor(black) || line upper MV, clpattern(dash) clwidth(thin) clcolor(black) || line lower MV, clpattern(dash) clwidth(thin) clcolor(black), xlabel(0 1 2 3 4 5 6, labsize(2.5)) ylabel(-4 -2 0 2 4, labsize(2.5)) yscale(noline) xscale(noline) legend(col(1) order(1 2) label(1 “Marginal Effect of X”) label(2 “95% Confidence Interval”) label(3 “ ”)) yline(0, lcolor(black)) title(“Marginal Effect of X on Y As Z Changes”, size(4)) subtitle(“ ” “Dependent Variable: Y” “ ”, size(3)) xtitle( Z, size(3) ) xsca(titlegap(2)) ysca(titlegap(2)) ytitle(“Marginal Effect of X”, size(3)) scheme(s2mono) graphregion(fcolor(white)); graph export h: figure1.eps, replace;

  9. Continous dependent variable with Two Modifying Variables  Limited dependent variable with Single Modifying Variable

  10. Source: Multiplicative Interaction Models – Thomas Brambor, William Roberts Clark, Matt Golder https://files.nyu.edu/mrg217/public/interaction.html

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