When you should include controls (avoid omitted v. bias): “determinants of x that might also affect y” “determinants of y that are in any way related to x” When you should NOT include controls: Including a control can remove significance:

Interaction Effects

Learning objectives Learn how to specify interaction models Learn how to interpret interaction effects

Why? Because sometimes, “it depends…” Theoretically, the impact of X on Y might be conditional on some value of Z For example, the impact of foreign aid on economic development might depend on conflict. Or, the impact of ethnic fragmentation on the number of parties might be contingent on how concentrated ethnic groups are geographically (ring any bells?) The key word here is theoretically Don’t just interact any and everything, just those things that you think should be contingent

How? So far we have specified additive regression models that look like this: What we’re suggesting now is that we include a multiplicative term, like this:

How? What difference does this make? Essentially this allows the slope of the regression line to vary across different values of Z Let’s say that Z is a dummy variable, equal to 0 or 1 Then for Z=0: And for Z=1:

Additive model y = b0 + b1*income + b2*married (married = 1; unmarried =0) Plugging either a 1 or a 0 in: Equation for married people = b0 + b1*x + b2 = (b0 + b2) + b1*x Equation for unmarried = b0 + b1*x Multiplicative model y = b0 + b1*income + b2*married + b3*income*married Equation for married people = b0 + b1*income + b2 + b3*income Rearrange for neatness: (b0 + b2) + (b1 + b3)*income Equation for unmarried = b0 + b1*income

For example: We might hypothesise that Development (Y) is increasing in Aid (X) if and only if a country is at Peace (Z): In which case, we should expect to find that is zero and is positive

Some things to remember: The constituent terms (X and Z) should both be included along with the interaction term (X*Z) = change in Y for one-unit change in X if Z=0 • = change in Y for one-unit change in X if Z=1 • Interaction effects should be informed by theory

Because: y = b0 + b1*x + b2*z + b3*x*z Multiplicative regression equation: y = b0 + b1*x + b2*z + b3*x*z + e Derivation by differencing of the marginal effect equation: dy/dx = b1 + b3*z Because: y = b0 + b1*x + b2*z + b3*x*z Can be thought of as: y = b0*x0 + b1*x1 + b2*z*x0 + b3*x1*z Which when differentiated with respect to x: dy/dx = b0*0*x-1 + b1*1*x0 + b2*z*0*x-1 + b3*1*x0*z And because anything multiplied by 0 is 0, and x0 = 1: dy/dx = b1 + b3*z Putting the numbers and variable names in for Exercise 2: ∂ legparties / ∂fragmentation = -0.34721 + 0.25583 * concentration Value of concentration Effect of fragmentation on legparties -0.34721 1 -0.34721 + 0.25583 = -0.09138 2 -0.34721 + 0.25583 * 2 = 0.16445 3 -0.34721 + 0.25583 * 3 = 0.42028

If you want more… Political Sociology: Rigby, Elizabeth, and Gerald C Wright. 2013. “Political Parties and Representation of the Poor in the American States.” American Journal of Political Science 57(3): 552–565. Gingrich, J, and B Ansell. 2012. “Preferences in Context: Micro Preferences, Macro Contexts, and the Demand for Social Policy.” Comparative Political Studies 45(12): 1624–1654. Comparative Government: Malesky, E. and Schuler, P. (2011) ‘Nodding or Needling: Analyzing Delegate Responsiveness in an Authoritarian Parliament’ American Political Science Review, 104(3), 1-21. -> I suggest replicating models 4 and 5 of Table 5 i.e. the fully specified OLS regression (Model 5) and the same model without the interaction (Model 4). Data: http://tinyurl.com/hjtm7l8