Method of using interactive dummy in PVAR model

[Binita Dey]

Dear Statalist members,

I am attempting to incorporate an interactive dummy variable into a Panel VAR (PVAR) model estimated using GMM, and I would appreciate some guidance on whether my approach and Stata commands are appropriate.

I consider the following equation:
Y_it = Y_(it-1) + D_it X_(it-1) + u_it
where (D_{it}) is a dummy variable constructed based on the change in (X). Specifically,

• (D_{it}=1) if (\Delta X_{it}>0) (i.e., (X) increases), and
• (D_{it}=0) if (\Delta X_{it}<0) (i.e., (X) decreases).

Thus, I intend (D_{it}X_{it-1}) to capture whether the effect of the lagged (X) on (Y) differs depending on whether (X) is increasing or decreasing.

I estimate the model as a Panel VAR using GMM, treating both (Y) and (X) as endogenous. My Stata commands are:
pvar y Dx1 Dx2 Dx3 Dx4, lags(1) instlags(1/6)
pvargranger
pvarirf, oirf mc(200) byoption(yrescale) porder(y Dx1 Dx2 Dx3 Dx4)
My questions are:

1. Is this specification an appropriate way to implement an interactive dummy variable in a PVAR framework?
2. Does the pvar command correctly estimate the model when the interaction term is constructed as (D_{it}X_{it-1}), while (Y) and (X) are treated as endogenous?
3. Are the pvarirf commands appropriate for obtaining impulse-response results for this specification?
4. If this approach is not appropriate, what would be the recommended way to incorporate and interpret an interactive dummy effect in a PVAR estimated using GMM?

I would particularly appreciate guidance on whether the commands above actually estimate the intended interactive effect, or whether a different specification or estimation procedure is required.

Thank you.

[Manh Hoang Ba]

Clearly, the key issue lies in how you construct Dx1, Dx2, Dx3, and Dx4. Along with y, these are the endogenous variables in the VAR, not x1, x2, x3, and x4.

If you define Dx1_it = D1_it * X1_it, then you will not obtain the terms of the form D1_it * X1_(i,t-1) in the Y_it equation as intended. Instead, the term that appears will be:
D1_(i,t-1) * X1_(i,t-1)

If you define Dx1_it = D1_it * X1_(i,t-1), then the term that appears in the Y_it equation will be:
D1_(i,t-1) * X1_(i,t-2)

If what you want is the term D1_it * X1_(i,t-1), then the endogenous variable Dx1 that you need to construct is:
Dx1_it = D1_(i,t+1) * X1_it

In Stata, you can generate it as follows:

 
 gen double Dx1 = F.D1 * x1

[Binita Dey]

I apologize for the confusion in my previous question. On reviewing my specification, I realized that there were some issues in how I had described the construction of the interaction term. I have revised the question below to clarify my methodology and the specific issue I am seeking guidance on.
The interaction term constructed by me is D_itX_it, and I intend to include it as an endogenous variable in my linear PVAR. In other words, the interaction term itself is treated as endogenous and is instrumented using appropriate lags within the GMM framework. My question is: Is this approach econometrically and technically valid in a linear PVAR estimated using GMM? In particular, can a constructed interaction term (Z_{it}=D_{it}X_{it}), where the dummy (D_{it}) is generated from the sign of (X_{it}), be included as an endogenous variable in a linear PVAR?

[Manh Hoang Ba]

Mathematically, I think this is fine, since Y and Z are just two ordinary variables. My only concern is that, since Z consists only of values indicating the same direction of movement (increase/decrease) as X, and these values do not necessarily occur consecutively (the opposite movement in X can occur in between them, for example, the value 4 in series: 3, 5, 4, 6, 7), the lags of Z may provide weak instruments.