Interpretation of interaction of 3 continuous variables

[Michael Lee]

Dear all,
I’m working on a model with N=6 and T=42, with heteroskedasticity and auto-serial correlation.
I have a hypothesis about inversed u-shape effect of X1 variable on Y. And also a hypothesis about effect of interaction of X2 and X1^2 (effect of (X1^2)*X2) on Y.
So the model is as follows:

xtgls Y c.X1##c.X1 X2 c.X1#c.X1#c.X2 ctrlvariables, corr(psar1) panel(h) force

My question about interpretation of coefficients of this model.
1. If I want to test u-shape relationship, shall I just use a model without the “c.X1#c.X1#c.X2” term or with the presence of this term also I can test the hypothesis?
2. How can I interpret “c.X1#c.X1#c.X2” term? It confusing because of their continuous form.


Thank you in advance

[George Ford]

You can take the derivates of Y on X1 and X2 to get expressions.

But margins may be your best bet.

So you theory does not have X1 alone?

[Michael Lee]

No. it hasn't X1 alone. it could have. but its not logical. the squared form of X1 is first hypothesis. and interaction of that with X2 is second one.

[Michael Lee]

Any idea? Help in this regard?

[Michael Lee]

Is this interpretation right?

[ Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \beta_3 X_1^2 + \beta_4 (X_2 \cdot X_1^2) + \epsilon ]

1. Intercept ((\beta_0)):
   • The intercept represents the expected value of Y when all predictors are zero.
2. Linear Term for (X_1) ((\beta_1)):
   • The coefficient for the linear term represents the change in Y associated with a one-unit increase in X1, assuming X2, the squared term, and the interaction term are constant.
   • For example, if we’re modeling crop yield (Y) based on fertilizer application (X1), a positive (\beta_1) implies that, on average, each additional unit of fertilizer increases yield by the value of (\beta_1).
3. Linear Term for (X_2) ((\beta_2)):
   • The coefficient for X2 represents the change in Y associated with a one-unit increase in X2, assuming X1, the squared term, and the interaction term are constant.
   • If we’re predicting crop yield, a positive (\beta_2) means that, on average, each additional unit of soil moisture increases yield by the value of (\beta_2).
4. Squared Term for (X_1) ((\beta_3)):
   • The coefficient for the squared term captures the curvature in the relationship between X1 and Y.
   • If (\beta_3) is positive, it indicates a U-shaped curve (concave up) for X1 and Y.
   • If (\beta_3) is negative, it suggests an inverted U-shape (concave down).
5. Interaction Term ((\beta_4)):
   • The interaction coefficient represents how the effect of X1 depends on the value of X2.
   • Specifically, it tells us how the impact of X1 on Y changes as X2 varies.
   • If (\beta_4) is positive, it means that the effect of X1 increases when X2 increases.
   • If (\beta_4) is negative, the effect of X1 decreases as X2 increases.

[George Ford]

To clean up, let X = X1 and Z = X2.

Y = a + b*X^2 + c*Z + d*X^2*Z

dY/dX = 2*b*X + 2*d*X*Z

dY/dZ = c + d*X^2

Say X is fertilizerQ(uantity) and Z is irrigationQ(uantity).

The effect of fertilizer on yield = 2*b*FertilizerQ + 2*d*FertilizerQ*IrrigationQ (if no irrigation, then 2*b measures effect of FertlizerQ).

The effect of Irrigation is c + d*FertilizerQ^2 (no Fertilizer, then just linear in Irrigation by c).

Messy. Margins/marginsplot will allow you to see the relationships.

I would expect FertilizerQ and IrrigationQ to have a quadratic relationship (positive for a bit, max out at some point, then harmful). And, there would be an interaction between the two, as they are be substitutes (need less X if you have some Z).

Y = a + b*X + c* X^2 + d*Z + e*Z^2 + f*X*Z

Now you have

dY/dX = b + 2*c*X + f*Z

dY/dZ = d + 2*eZ + f*X

By f (which would be positive, I'd think), you get a shift in the relationship (e.g., a larger effect of X given some positive Z).

You could get milder non-linearity (without a downturn) using natural logs of X and Z.

[George Ford]

https://edis.ifas.ufl.edu/publication/SS548

https://www.researchgate.net/publication/261102821_Impacts_of_climate_change_on_farm_income_security_in_Central_Asia_An_integrated_modeling_approach/figures?lo=1