  We propose a nonparametric method for detecting nonlinear causal relationship
within a set of multidimensional discrete time series, by using sparse additive
models (SpAMs). We show that, when the input to the SpAM is a $\beta$-mixing
time series, the model can be fitted by first approximating each unknown
function with a linear combination of a set of B-spline bases, and then solving
a group-lasso-type optimization problem with nonconvex regularization.
Theoretically, we characterize the oracle statistical properties of the
proposed sparse estimator in function estimation and model selection.
Numerically, we propose an efficient pathwise iterative shrinkage thresholding
algorithm (PISTA), which tames the nonconvexity and guarantees linear
convergence towards the desired sparse estimator with high probability.
