How to create and interpret the interaction term in the following: lnY=*β0 +*β1 lnGDP +*β2 lnFX +*β3 (lnGDP x lnFX) +*ξ

[Lutfi Ozturker]

Hello,

I regress a bank's one balance sheet item(Y) on GDP, FX (foreign exchange rate), and the interaction term as follows: (all in percentage changes whereas the data includes 63 consecutive quarters for all)

lnY= β0 + β1 lnGDP + β2 lnFX + β3 (lnGDP x lnFX) + ξ

My questions are:

1-) Is it the correct code: reg lnY lnGDP lnFX c.lnFX#c.lnGDP,r

2-) What is the interpretation for β3? (

3-) FX is much more than GDP in percentage terms. So, how can I create a regression to calculate the joint effect of, for example, 1% change in GDP and 10% change in FX at the same time on Y?

4-) I want to show the effects of GDP and FX separately on Y as well as the effect of a simultaneous change on both GDP and FX on Y as the third scenario. So, which regression(s) should I run to illustrate these three scenarios? (Only the aforementioned one or three different regressions?)

5-) In this regression, both regressors and regressand may be subject to a nonstationarity problem, being vulnerable to a spurious estimation problem. However, since my objective is to measure the percentage change of the Y in response to one percent change in i. GDP, ii. FX, and iii. GDP&FX (simultaneously); is it the right model estimating β1,β2,β3 in this double log form? Also, do you suggest adding some control variables (e.g. other balance sheet items and/or the total size of the balance sheet)?

Kind regards,
Lütfi

[Clyde Schechter]

1) It depends. If you have longitudinal (panel) data, then using -regress- is usually not adequate and you need to apply one of the panel data estimators such as -xtreg-. Also, while the specification of the regressors you have is fine, you can simplify it by using the ## interaction operator:

regression_command lnY c.lnFX##c.lnGDP perhaps_covariates, perhaps_options

2) So, an interaction model like this one stipulates that instead of having a single marginal effect of lnFX on lnY, there are infinitely many such marginal effects, one for each value of lnGDP. In particular, the marginal effect of lnFX on lnY is a linear function of lnGDP: marginal effect of lnFX = a0 + a1*lnGDP. The coefficient β3 is this a1. In words, it is the rate at which the marginal effect of lnFX on lnY increases with each unit increase in lnGDP.

3) You can't. It isn't defined in your model. You can only do this conditional on particular starting values of lnFX and lnGDP.

4) In an interaction model there is no such thing as the separate effects of lnFX and lnGDP. If, however, the interaction model is a correct specification of the data generating process, then models that omit the interaction term are wrong and should not be used. I think your best bet is to do something like this:

regression_command lnY c.lnFX##c.lnGDP perhaps_covariates, perhaps_options
margins, at(lnFX = (list of interesting values of lnFX) lnGDP = (list of interesting values of lnGDP))
marginsplot, xdimension(lnGDP) name(x_lnGDP, replace)
marginsplot, xdimension(lnFX) name(x_lnF, replace)

That will give you a good picture of what your model says is going on.

Added: I don't have the expertise in time-series analysis to advise you regarding 5).

[Lutfi Ozturker]

Thank you for your response.

I'm interested in how much Y changes when I apply a shock to the;

(i) GDP only,
(ii) FX only,
(iii) GDP and FX at the same time.


Since all four variables (lnY, lnGDP, lnFX, c..D.lnGDP#c.D.lnFX) are I(1), then, should I rather regress the following three regressions for each scenario: (D is the first difference)

D.lnY= β0 + β1 D.lnGDP (i)
D.lnY= β0 + β2 D.lnFX (ii)
D.lnY= β0 + β3 c.D.lnGDP#c.D.lnFX (ii)


If yes, then, how should I interpret β's? I guess it's not β percentage change in Y as a response to a percentage change in the regressors, anymore.

Lütfi

[Clyde Schechter]

It does not make sense to adopt all three of these models simultaneously. If the model with interaction is correct, then the other two models are wrong, and vice versa. Also the third model you wrote, the one with the interaction, is incorrect because it includes the interaction without also including the constituent ("main") effects. You can fix that most simply by replacing # with ##. Once corrected, understand that these are models that make qualitatively different claims about what is going on and are not consistent with each other unless the interaction coefficient turns out to be zero (or extremely close to zero).

beta1 and beta2 are ordinary regression coefficients and, given the log transformations of the variables, are often interpreted as the percentage change in Y associated with a percentage change in the regressors. This is, in fact, only approximately true, and the approximation is really only good up to a beta value of about 0.10, maybe 0.15 if you don't care much about precision. After that it's a rather poor approximation, and if you really want elasiticities, then you have to do the work of computing them directly.

Beta 3 is a different animal because of the interaction. I explained what beta 3 is in my 2) in post #2 of this thread. The coefficients of lnFX and lnGDP in the interaction model do have an interpretation, but it differs from what they mean in a non-interaction model. In D.lnY= β0 + β3 c.D.lnGDP##c.D.lnFX, the coefficient of D.lnGDP is (approximately, as noted above) the percentage increase in Y associated with a 1 percent increase in lnGDP if lnFX = 0 (equivalently, FX = 1). A similar intrpretation can be given to the coefficient of lnFX in the interaction model.

[Lutfi Ozturker]

Thank you.
One last question:
Is the interpretation of β2 exactly the same in the following two regressions? (β2 percent change in Y associated with a percentage change in the FX)
(i) lnY= β0 + β2 lnFX
(ii) D.lnY= β0 + β2 D.lnFX

[Clyde Schechter]

I'm not sure in what since you mean it. The second equation pertains to the relationship between the change in lnY and the change in lnFX over your data's recurrence interval. The first pertains to the values of lnY and lnFX themselves. The results are usually going to be different, perhaps dramatically so.

But it is still true that in the first equation beta2 is (approximately) the percent difference in Y associated with a 1 percent difference in FX, say, on average among different countries or over time within a country.

The second equation is more complicated because of the presence of first differences of the logarithms. The difference of two logarithms is the logarithm of the quotient. So equation 2 is equivalent to:

ln (Y2/Y1) = beta0 + beta2* ln(FX2/FX1)
Exponentiating both sides gives:

Y2/Y1 = K*(FX2/FX1)beta2

where the subscripts 1 and 2 on Y and FX refer to consecutive times 1 and 2 within a single country.

So a 1% difference in the ratio (FX₂/FX₁) is associated with (approximately) a beta2 percent difference in the ratio Y₂/Y₁.