How to I estimate almon's polynomial distriubted lag model by using panel data?

[Hansol Kim]

The Almon's Polynomial Distributed Lag (PDL) model is originally designed for time series analysis, and currently, both the "almon" and "almon1" commands in STATA are tailored for time series data.
However, many researchers have been adapting Almon's PDL to analyze panel data.
Currently, my dependent variable is "inv" (investment amount), and the independent variable is "pat" (patent applications). The polynomial degree is set to 2, with a lag of 6.
The existing commands are structured for time series analysis, which may not recognize the panel structure even if I specify it using the "xtset" command.
If there is a way to estimate Almon's PDL by using Panel data, I would greatly appreciate your guidance.

[Jeff Wooldridge]

You can impose the restrictions that are used in any Almon polynomial fairly easily. I take the lag of 6 to mean 7 original coefficients (including the contemporaneous effect) but that isn't especially important. The parameters you will initially estimate are c0, c1, c2. The coefficients satisfy bj = c0 + c1*j + c2*j^2 where j is the lag. I'll show a simpler case with three lags:

b0 = c0
b1 = c0 + c1 + c2
b2 = c0 + 2*c1 + 4*c2
b3 = c0 + 3*c1 + 9*c2

Plug these into

y(t) = a0 + b0*x(t) + b1*x(t-1) + b2*x(t-2) + b3*x(t-3) + u(t)
= a0 + c0*x(t) + (c0 + c1 + c2)*x(t-1) + (c0 + 2*c1 + 4*c2)*x(t-2) + (c0 + 3*c1 + 9*c2)*x(t-3) + u(t)

Now collect terms multiplying the three coefficients:

y(t) = a0 + c0*(x(t) + x(t-1) + x(t-2) + x(t-3)) + c1*(x(t-1) + 2*x(t-2) + 3*x(t-3)) + c2*(x(t-1) + 4*x(t-2) + 9*x(t-2)) + u(t)

So define the three regressors multiplying c0, c1, and c2. Then regress y(t) on these three regressors. It doesn't matter if you use random effects, fixed effects, pooled OLS. To obtain the bj estimates, use the initial formulas and the lincom command.

You should check my math!

[Hansol Kim]

First of all, thank you so much for your help.

The methods you explained seem applicable when there is 'no end point constraint.'
I am also curious about the analysis methods for 'near(left) end point constraint,' 'far(right) end pint constraint,' and 'near and far end point constraint.'

Could you provide assistance for each of these methods?

[Jeff Wooldridge]

I believe these just impose more linear restrictions. You should look up what these are.

[Hansol Kim]

I understand what you said.

I had thought constraints should be imposed from the calculation of the bj estimates, but after reading your feedback, I realized that constraints should be applied after deriving the bj estimates.

Thank you for your advice!

[Hansol Kim]

I have one more question.
First, my analysis data is structured as follows: the independent variable is R&D investment, and the dependent variable is the number of patent applications. It's a panel data with 8 years.
As the dependent variable is a count variable, I performed a Poisson regression (xtpoisson) and specified the random effects option based on the Hausman test.
I also took the natural logarithm of the independent variable.
For the lag length, I conducted a stepwise regression analysis by gradually adding lagged variables and chose the model with the lowest AIC and BIC values.
Therefore, I intended to conduct Almon's Pooled Dynamic Least Squares (PDL) analysis with a lag of 6 years and a polynomial order of 2.

Following the approach my professor suggested, I estimated the lag coefficients (bj).
However, the lag coefficients decrease up to b(t-2) and then increase from b(t-3) to b(t-6), with the coefficients at b(t-5) and b(t-6) being greater than 1.
Considering prior research results and the somewhat unique lag distribution, as well as the fact that some lag coefficients are above 1, I'm wondering if there might be an issue with my analysis process. Could you provide some advice on this matter?"