Koyck model with many regressors

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Permalink Reply by Eugene J Goldrick on November 11, 2009 at 8:38am

It sounds like you have more than one regressor and you want them to have different decay factors but still within the Koyck family. If that is the case, here's how to do it. Denote the dependent variable by "S". Let there be 2 regressors, "A" and "Z", with decay factors "a" and "b". Let the current impact of A and Z on S be denoted by "f" and "g", respectively. Written in terms of the lag operator, the equation

(1-aL)(1-bL)S(t) = f(1-bL)A(t) + g(1-aL)Z(t)

will impose a Koyck lag with a decay factor of "a" for variable A and a decay factor of "b" for variable Z.

We can rewrite this equation as

S(t) = (a+b)S(t-1) + (ab)S(t-2) + cA(t) -cbA(t-1) + dZ(t) - daZ(t-1)

Notice the non-linear constraints on the coefficients. They are the result of imposing the geometric decay pattern on the temporal impacts of A and Z.

If there are 3 regressors, A, Z, and W, the equation is

(1-aL)(1-bL)(1-cL)S(t) = (1-bL)(1-cL)fA(t) + (1-aL)(1-cL)gZ(t) + (1-aL)(1-bL)hW(t)

where "c" and "h" are the decay and current impact parameters for the regressor W.

With 3 separate decay rates, there are 3 lags of the dependent and independent variables in the transformed (estimated) equation with nonlinear constraints on their coefficients.

Of course, there is the issue of the disturbance term. If the disturbance term in the original equation is iid, the disturbance term in the transformed (estimated) equation will follow an MA process with as many lags as there are decay parameters and with constraints on its parameters that mimic those on the dependent variable.

Hope this helps.

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