Interpretation of Specific Interaction Term in OLS Fixed Effects Regression

[Marc Pelow]

Hi everyone,

I have a semi-STATA related question that I hope you don't mind me asking here.

I'm running an OLS Fixed Effects Regression in Stata to examine the influence of political authoritarianism on a continuous dependent variable. My model includes a continuous variable for authoritarianism (AUT), which cannot be smaller than 0. This variable is intended to capture the general level of political authoritarianism in a country.

In a second analysis, I want to investigate whether not only the general level of authoritarianism impacts my dependent variable but also the ideological orientation (Right-Left Ideology). The idea is that I expect the impact of authoritarianism on my y to be different for different levels of ideological orientation.
In this context, ARILE is a continuous variable reflecting right-wing authoritarianism with positive values and left-wing authoritarianism with negative values, with 0 representing the political center. I'm considering the interaction term to illustrate the combined effect of the level of authoritarianism and its ideological orientation.

The model is specified as follows:

y=β0+β1AUT+β2ARILE+β3(AUT×ARILE)+ϵ

In the regression results, authoritarianism (AUT) shows the expected positive coefficient, while the right-left ideology (ARILE) variable shows the expected negative coefficient. The interaction term shows a significant positive coefficient.

Here is my interpretation of the coefficients, but I'm uncertain due to the varying implications of the ARILE variable depending on its sign:

• AUT (+): An increase in the general level of authoritarianism leads to an increase in the dependent variable.
• ARILE (-): An increase in left-wing authoritarianism leads to an increase in the dependent variable.
• AUT x ARILE (+): The positive coefficient of this term indicates that higher levels of authoritarianism combined with either left-wing or right-wing ideological orientation further increase the dependent variable. -> This is the interpretation I am most uncertain about.

I would greatly appreciate any feedback or guidance on my interpretation.

Best regards,

[George Ford]

It shows that the effect of authoritarianism increases in right-wing authoritarian lean, since the value of ARILE is rising in right-wing lean.

Y = a1*AUT + a2*ARILE + a3*AUT*ARILE

dY/dAUT = a1 + a3*ARILE (if ARILE < 0, you pull down a1; with ARILE > 0, you add to a1).

dY/dARILE = a2 + a3*AUT

You could marginsplot to get a visual of the effect across a range of ARILE.

If ARILE = 0 , then the effect of AUT is a1
If ARILE = -1 , then the effect of AUT is a1 - a3 (lower Y)
If ARILE = 1, then the effect of AUT is a1 + a3 (higher Y)

IF AUT = 0.5, then the effect of ARILE = a2 + a3*0.5 (negative)
IF AUT = 0.8, then the effect of ARILE = a2 + a3*0.8 (negative, but closer to zero than at 0.5)
The effect of ARILE is 0 when AUT = 0.818, then turns positive (crosses the X axis) at higher values.

[Marc Pelow]

Dear George,

thank you for your reply. Just to make sure I am not misunderstanding the statements, could you please confirm that the correct interpretation would be:

AUT (+): An increase in the general level of authoritarianism leads to an increase in the dependent variable.
ARILE (-): An increase in right-wing authoritarianism leads to an increase in the dependent variable.
AUT x ARILE (+): The positive coefficient of this term indicates that higher levels of authoritarianism combined with right-wing ideological orientation further increase the dependent variable.

Thanks a bunch!

[George Ford]

I don't know the scale of your variables.

But here's a sketch.

AUT:
Say ARILE was -1. dY/dAUT = 0.109 + 0.143*(-1) < 0.
Say ARILE was 1. dY/dAUT = 0.109 + 0.143*(1) > 0.
cutover: -0.109/0.143 = -0.76

ARILE:
Say AUT was 0.5. dY/dARILE = -0.117 + 0.143*0.5 < 0.
Say AUT was 0.9. dY/dAUT = -0.117 + 0.143*0.9 > 0.
cutover: 0.117/0.143 = 0.818.

Holding AUT(ARILE) constant, and increase in ARILE(AUT) increases Y.

clear

set obs 1000

g AUT = runiform(3,10)/10
g ARILE = runiform(-1,1)
summ AUT ARILE

g Y = 1 + 0.109*AUT - 0.117*ARILE + 0.143*AUT*ARILE + rnormal(0,0.001)

reg Y AUT ARILE c.AUT#c.ARILE 
margins, at(AUT = (0.3(.1)1) ARILE = (-1(0.2)1)) nose  //this will take a few seconds
marginsplot , xdimension(ARILE) name(dYdAUT, replace)
marginsplot , xdimension(AUT) name(dYdARILE, replace)

[George Ford]

You could get a surface graph in R, or I think in Stata with Python integration with R.

[Cleo Rasmus]

Dear all,

I have a similar question. I have a model in which I use net FDI as my independent variable. I also interact net FDI with a dummy variable that takes the value of 1 if net FDI is negative and the value of 0 if net FDI is positive. I am not entirely sure how to interpret the interaction term.

My model is Y = beta0 + beta1FDI + beta2(FDI*Dummy variable) + control variables

Am I correct to say that beta1 is the effect of FDI on Y holding all other variables constant and that beta1 + beta2 is the effect of FDI on Y, if FDI is negative? And if both of my coefficients beta1 (e.g., -1,2) and beta2 (e.g., -1,5) are negative, is the effect on y positive or negative (e.g. -2,7 or +2,7), since in that case FDI and FDI*Dummy variable are negative.

I would be happy to receive advice on the interpretation of my coefficients.
I use an OLS fixed effects model.

Kind regards,
Cleo

[George Ford]

Cleo (and others),

It would be helpful if you'd provide an explanation for that specification. I think what you have is the Y,FDI relationship sharing the same Y intercept but have different slopes on either size of FDI==0. It might take a V-shape, for instance.

If you fill in the Dummy, you get what you want.

dY/dFDI if FDI > 0

= beta1 + beta2*0
= beta1

dY/dFDI if FDI < 0
= beta1 + beta2*1
= beta1 + beta2

If both beta are negative, then effect is negative, but could have a steeper/flatter relationship on either size of FDI=0

The sign of FDI does not matter except through beta2. If you standardize a variable with mean 0 and sd 1 or center a variable on its mean, then the sign of the coefficient is unchanged as is the general shape of the slope. The scale might matter in a linear model, but the interpretation is the same.

In Marc's case, the sign might matter, as the value of AUT and ARILE are in the derivative. Not in your case.

You probably need the DummyVariable as a standalone dummy variable so the intercept can vary on either side of FDI = 0.

I'd draw a picture of what you're up to so that you are sure you want to do what you are doing. Asymmetry is not unusual--you see a lot of these type tests in gas prices for example (are prices symmetrically related to oil prices depending on whether oil prices fall or rise?).