The non-parametric regression model to be estimated looks like the following
x_t = b(x_t-1) + epsilon_t
Forfinding the optimal bandwith h in the kernel regression a cross-validation method (leave a point out and estimate it with the rest of the sample) is used and evaluated via the expected prediction error (EPE). This takes a while since the kernel has to be calculated each time for a range of potential h-values and every sample without x_i.
Therefore the use of FFT (Paper: link and similar post: link) can speed up the calculations. In my understanding the following is the way to go:
- Discretize the data (x_t-1 values) by using linear binning
- Take the Fourier-transform of the binned data
- Multiply the transformed binned data with the Fourier-Transform of the Gaussian-kernel
- Inverse-transform the product back to get the kernel estimation
But I fail to understand how implement this since once the data is binned (I assume here x_t-1) the connenction to the y-values of the kernel regression (here x_t) is broken?
