How do I create a predictor for a time series once I've confirmed Granger-causality? I have a set of time series data that I've found granger-causality (i.e. regressed Y vs. X, X-1, Y-1), and am wondering how I can create a predictor from this linear model?  Is it simply the coefficients I get back from my model? (I'm trying to code this programmatically, and am still a stats rookie, so if this is obvious, forgive me)
 A: Granger causality test was invented to test statistically that variable $x_t$ has a significantly sound information that helps to predict $y_t$. If however $y_t$ is potentially linked with some other variables than $x_t$ (both are a part of a vector, which could be modeled by vector autoregression) the testing scheme becomes a bit more complicated (involves some matrix manipulations). Note also that since testing for Granger causality involves $F$-test in the final stage it is sensitive to deviations from normality assumption (you may then consider some extensions from the wiki link above).

To make a predictive model you have to choose the horizon (denote it by $h$) up to which you would want to predict. Your complete predictive model is then:
$$y_t = \alpha_0 +\alpha_1 y_{t-1} + \dots + \alpha_p y_{t-p} + \beta_1 x_{t-1} + \dots + \beta_p x_{t-p} + \varepsilon_t, $$
for sequential predictions (note that in sequential case you have to make a predictive model for $x_t$), and in direct $h$ step ahead case:
$$y_t = \alpha_0 +\alpha_1 y_{t-h} + \dots + \alpha_p y_{t-h-p+1} + \beta_1 x_{t-h} + \dots + \beta_p x_{t-h-p+1} + \varepsilon_t, $$
then the $h$ step prediction after estimation and testing for Granger-(non)causality ($H_0: \beta_1 = ... = \beta_p = 0$) in direct case:
$$ y_{T+h}^{(h)} = \hat\alpha_0 +\hat\alpha_1 y_{T} + \dots + \hat\alpha_p y_{T-p+1} + \hat\beta_1 x_{T} + \dots + \hat\beta_p x_{T-p+1} $$
and in sequential:
$$ y_{T+1}^{(1)} = \hat\alpha_0 +\hat\alpha_1 y_{T} + \dots + \hat\alpha_p y_{T-p+1} + \hat\beta_1 x_{T} + \dots + \hat\beta_p x_{T-p+1}, $$
$$ y_{T+2}^{(2)} = \hat\alpha_0 +\hat\alpha_1 y^{(1)}_{T+1} + \dots + \hat\alpha_p y_{T-p} + \hat\beta_1 x^{(1)}_{T+1} + \dots + \hat\beta_p x_{T-p}, \dots $$
Thus estimation/testing step is separated from the prediction step which for linear models is straightforward. All this scheme is easily implementable in $R$.
