# interpretation of this interaction

I am looking for help with the interpretation of a complicated interaction term.

I am using a fixed effects model.

The interaction term is: Code:

c.log(corruption)#c.d.log(military expenditure as a % of GDP)

where (corr) = corruption is a continuous variable with log transformation where (dlm) = military expenditure as a % of GDP is a continuous variable with log transformation followed by being differenced

any help would be greatly appreciated

My full regression is:

xtreg d.ly d.lpop d.lk d.lm l.ly l.lm ic ec corr mip gs c.mip#c.dlm c.gs#c.dlm c.corr#c.dlm i.year, fe vce(cluster country_id)


where:

y = GDP per Capita
ly = log GDP per Capita
pop = popultation
lpop = log(population)
k = Gross fixed capital formation as share of GDP
m = Military expenditure as share of GDP
gs = government stability
corr = corruption
mip = military in politics
ic = internal conflict
ec = external confflict

• Is the outcome also logged? Do you have both main effects in the model as well? – Dimitriy V. Masterov Mar 30 '18 at 22:21
• @DimitriyV.Masterov Sorry for the sparse details : the dependent variable is (d.l.GDPpercapita) = differenced log transformed variable of GDP per Capita I also have the main effects in the model. My regression is: xtreg d.ly d.lpop d.lk d.lm l.ly l.lm ic ec corr mip gs c.mip#c.dlm c.gs#c.dlm c.corr#c.dlm i.year, fe vce(cluster country_id) – Stat-metrics Mar 30 '18 at 23:44
• @DimitriyV.Masterov as well as this, what would the interpretation of the main effect of military expenditure be? in percentage points? – Stat-metrics Mar 31 '18 at 0:11
• This has me a bit stumped. – Dimitriy V. Masterov Apr 2 '18 at 18:58

My best guess is

1. difference in effect of corruption on GDP growth in countries with more rapid acceleration of military spending ratios; or
2. difference in effect of an acceleration of military spending ratio on GDP growth for more corrupt versus less corrupt countries.

Mathematically, the interpretation is a derivative of a derivative. Setting aside subscripts, your linear model would be something like:

$GDP Growth = \beta_1 lnCorrupt + \beta_2 \triangle ln(Military/GDP) + \beta_3 *lnCorrupt*\triangle ln(Military/GDP) + \delta Z+ error$

Differentiating with respect to the log of corruption gives

$\frac{\partial\triangle lnY}{\partial lnCorrupt} = \beta_1 + \beta_3 *\triangle ln(Military/GDP)$

This represents the impact of corruption on the rate of economic growth. This quantity includes a main effect and a term that depends on the rate at which military spending ratio is accelerating.

Differentiating with respect to $\triangle ln(Military/GDP)$ gives the parameter of interest:

$\beta_3 = \frac{\partial \big[ \frac{\partial\triangle lnY}{\partial lnCorrupt}\big]}{\partial[\triangle ln(Military/GDP)]}$

Which looks like the difference in the effect of corruption per unit of excess acceleration in the ratio of military spending to GDP.

Whether this has any useful economic interpretation is unclear to me.

I would also stress that you may face econometric critiques by virtue of having GDP on both the left and right hand sides of your equation.

• Ah I see.. thank you for that the maths there has really made that much clearer. Effectively what you are saying is that: β3 measures the effect that the growth of military expenditure as a share of GDP, would have on the growth of GDP per capita, as the level of corruption changes? – Stat-metrics Apr 3 '18 at 18:08
• Yes. Notice that you can switch the order of differentiation so that $\beta_3$ measures what you said or the effect that an increase in corruption has on the growth of GDP per capita, as the rate of growth of military expenditure as a share of GDP changes. – Paul Spin Apr 3 '18 at 20:16
• slightly different question, how would one interpret the beta coefficient if the dependent variable is a log difference =d.log(GDP) i.e. growth rate, and the independent variable is in log(x%) ? thank you – Stat-metrics Apr 5 '18 at 18:43