Question: When using Generalized Lienar Models in SPSS (dist: binomial, link: logit, r/n) how to overdispersion correction? In otherwords how to apply binary logistic regression models with correction on over-dispersion using SPSS?

Answer: 
The persistence is equal to the largest eigenvalue of the companion matrix (in absolute value).

For example, an AR(3) process can be written as $$\left[ 
\begin{array}{c}
Y\_{t} \\ 
Y\_{t-1} \\ 
Y\_{t-2}%
\end{array}%
\right] =\left[ 
\begin{array}{c}
c \\ 
0 \\ 
0%
\end{array}%
\right] +\left[ 
\begin{array}{ccc}
\phi \_{1} & \phi \_{2} & \phi \_{3} \\ 
1 & 0 & 0 \\ 
0 & 1 & 0%
\end{array}%
\right] \left[ 
\begin{array}{c}
Y\_{t-1} \\ 
Y\_{t-2} \\ 
Y\_{t-3}%
\end{array}%
\right] +\left[ 
\begin{array}{c}
\varepsilon \_{t} \\ 
0 \\ 
0%
\end{array}%
\right] $$

The $3 \times 3$ matrix in this expression is called companion matrix. If it has eigenvalues larger than 1 in absolute value (outside the unit circle), then the process is explosive / divergent (it goes to plus or minus infinity even if the variance of the error term is zero).

When people speak about the persistence of an AR(p) process, they usually refer to the largest eigenvalue (in modulus) of the companion matrix. It is the eigenvalue that determines whether the process is stationary or not.
In a simple AR(p) process, the so-called cumulative impulse response is given by 1/(1-p), where p is the sum of the autoregressive coefficients. Therefore, your intuition can be justified.
See: Andrews Donald W. K., Chen, Hong-Yuan, 1994, “Approximately median-unbiased
estimation of autoregressive models”, Journal of Business and Economic
Statistics, Vol. 12, No. 2, 187- 204