Granger Causality Analog for Binary Time Series Is there a generalized form of granger causality that can be applied to two binary time series? By binary time series I mean an ordered series of values that take values 0 or 1.
 A: Granger causality can be applied to binary data by using the appropriate univariate distributions. For instance, in the two-series case you could assume the following structure. Let $\mathbf{y}_t = (x_t, y_t)$ be the vector of binary data at time $t \in \mathbb{N}$. Assume an autoregressive structure (like VAR) as follows: each component is independently Bernoulli, and their conditional probability at each time, given the past, is a linear combination of the past (passed through a logit link):
\begin{align*}
x_t \mid y_{t-1}, \ldots, y_{1}, x_{t-1}, \ldots, x_1 &\sim \mathrm{Bernoulli}(\pi^{(x)}_t) \\
\mathrm{logit}(\pi^{(x)}_t) &= \alpha^{(x)}_0 + \sum_{i=1}^p \alpha^{(yx)}_i y_{t-i} + \sum_{i=1}^p \alpha^{(xx)}_i x_{t-i}\\
\end{align*}
and
\begin{align*}
 y_t \mid y_{t-1}, \ldots, y_{1}, x_{t-1}, \ldots, x_1 &\sim \mathrm{Bernoulli}(\pi^{(y)}_t) \\
\mathrm{logit}(\pi^{(y)}_t) &= \alpha^{(y)}_0 +  \sum_{i=1}^p \alpha^{(xy)}_i x_{t-i} + \sum_{i=1}^p \alpha^{(yy)}_i y_{t-i}.
\\
\end{align*}
Then for example the parameter $\alpha^{(xy)}_i$ measures the causality from $x$ to $y$ at lag $i$. You can compute normal or profile likelihood confidence intervals for this parameter to determine if the relation is significant. You can also compute a Granger ``causality measure'', the reduction of error, using the likelihood ratio statistic:
$$
g^{x\to y} = \log \frac{\ell\left(\alpha\right) \,\rvert_{\alpha^{(xy)}=0}}{\ell(\alpha)}
$$
If removing the influence of $x$ (via setting $\alpha^{(xy)}_i=0$) increases the error significantly, then this statistic will be large.
You can find details about the model above in e.g. the following paper:
Agaskar, A., and Y. M. Lu. 2013. ‘ALARM: A Logistic Auto-Regressive Model for Binary Processes on Networks’. In 2013 IEEE Global Conference on Signal and Information Processing, 305–8. https://doi.org/10.1109/GlobalSIP.2013.6736876.
