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It is a stylized feature that the correlation between SPX and VIX is negative (around -0.7) thus we have an interesting process here. Nothing as simple as driftless Brownian motion. The volatility process jumps up and then fades out which is often modelled as Hawkes process (see e.g. here). Thus conclusions from the simple Gaussian model (with Chi-quared ...

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This is a vector autoregression (VAR) model with restrictions. I would also use a shorter lag and add conventional factors to the VIX return. Simple bootstrapping destroys the serial correlation in your model, but you might consider block bootstrapping. I would also try the following test: use a subset to calibrate your model and then perform an ...

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