Intercept-Only regression with an interaction term - What does it mean?

[Holger Luedeke]

If I estimate an intercept only model in Stata

reg c.y

I get an estimate of a constant that shows the mean of the continuous variable y.

In Stata, it is also possible to have an intercept-only model with an interaction term, e.g.

reg c.y##c.x

Now, y gets estimated by a constant, a coefficient for the effect of c.x, and a coefficient for the effect of c.y#c.x.

I am not sure what to make of this result. Is the constant plus the effect of c.y#c.x the constant that I get if x equals "zero"?
Who could give me a hint? I cannot find any explanation in the manuals.

[Mike Lacy]

I'm surprised that what you showed is acceptable syntax to Stata, but apparently it is, so you've taught me something.

From trying the following:

sysuse auto
regress c.price##c.weight

I see that this estimates an equation of the form:
y = a + b₁*X + y*b₂*X which is more nicely written as y = (a+b₁*X)/(1-b₂*X)

The constant is as usual the predicted value of y when X = 0.

Your question "Is the constant plus the effect of c.y#c.x the constant that I get if x equals "zero" is not one I can answer, since "c.y#c.x" is not an explanatory variable, and consequently can't be said to have an "effect."

[Holger Luedeke]

Thank you that was helpful. If I understand you correctly, reg c.y##c.x estimates a relationship between x and y such that the association plays a larger role in the extreme regions than on the middle ground - a theoretical relationship that may apply for a number of social influences. As you expressed your surprise that Stata accepts the syntax: Do you think there is something intrinsically wrong with this estimation approach?

[Mike Lacy]

What's strange about the syntax is that a) Almost always, a -reg- command includes a dependent variable and one or more independent variables; and b) I've never seen factor variable notation used with the dependent variable.

I'd would not say there is something wrong with what you are doing, but it is very strange and non-standard. If you want to model a relationship in which the association is stronger (I think you actually mean the effect or slope is larger), then the standard thing to do is to pick some simple f(x) that fits that shape. For example, you might choose
f(x) = x^3, see e.g.:

 twoway (function y = x^3, range(-10 +10))