My lecturer introduced the concept autocovariance generating function (ACGF) of a stationary process in class.

$G(L) = \sum_{k = -\infty}^{+\infty} \gamma_kL^k$ [1]

$\gamma_k$ is autocovariance at k-th lag.

She also mentioned that we can retrieve information of $\gamma_k$ from this by differentiating this polynomial k times then evaluate the derivative at $L = 0$. Finally, divide the result by $k!$.

However, I think it is mathematically wrong as it is not possible to evaluate the derivative at $L = 0$ at any term $L^k$ where k is negative. It is possible that she mentioned another kind of ACGF which is $G(L) = \sum_{k = 0}^{+\infty} \gamma_kL^k$. Am I correct?

Furthermore, I want to know how I can retrieve information about autocovariance from the first ACGF. I can do it by deriving the spectral density function from ACGF. However, I want to know if there is another way to do it.


1 Answer 1


Turning comment to answer.

If $ \langle\gamma_j\rangle_{j =-\infty}^\infty$ is absolutely summable, then the autocovariances can be summarized using a scalar-valued function, the acv generating function $$g_Y(z) := \sum_{j =-\infty}^\infty \gamma_jz^j.$$

If the linear process is $Y_t = \psi(B) \varepsilon_t,$ then acv generating function can be shown to be

\begin{align} g_Y(z) &= \sum_{j =-\infty}^\infty \gamma_jB^j\\ &= \sigma^2\psi(B) \psi(B^{-1}) ;\end{align} $\gamma_j$ can be found out by observing the coefficient of $B^j$ (also $B^{-j}$) .

Since the process is absolutely summable, one can also resort to population spectrum $$ s_Y(\omega) = \frac{1}{2\pi} \sum_{j =-\infty}^\infty \gamma_j\exp(-i\omega j) $$ to retrieve the acv s.


Time Series Analysis, James D. Hamilton, Princeton University Press, 1994.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.