Finite distributed lag models, in general, suffer from the multicollinearity due to inclusion of the lags of the same variable in the model. To reduce the impact of this multicollinearity, a polynomial shape is imposed on the lag distribution (Judge and Griffiths, 2000). The resulting model is called Polynomial Distributed Lag model or Almond Distributed Lag Model.
Imposing a polynomial pattern on the lag distribution is equivalent to representing \(\beta\) parameters with another $k$th order polynomial model of time. So, the effect of change in \(X_{t-s}\) on the expected value of \(Y_{t}\) is represented as follows:
$$
\frac{\partial E(Y_{t})}{\partial X_{t-s}}=\beta_{s}=\gamma_{0}+\gamma_{1}s+\gamma_{2}s^{2}+\cdots+\gamma_{k}s^{k}
$$
where \(s=0,\dots,q\) (Judge and Griffiths, 2000). Then the model becomes:
$$
Y_{t} = \alpha +\gamma_{0}Z_{t0}+\gamma_{1}Z_{t1}+\gamma_{2}Z_{t2}+\cdots +\gamma_{k}Z_{tk} + \epsilon_{t}.
$$
The standard function summary()
prints model summary for the model of interest.