Fitting Gaussian Random Walk to stock prices

Questions
2

Your model is unidentifiable as it is. Think of it this way: the variance in each observation in y comes from 2 places: from the GaussianRandomWalk controlled by σ (also the covariance with the previous time point) and the Normal distribution controlled by sd. With uniform prior, y could be generated completely from a GaussianRandomWalk with σ=var(diff(y)), or a Normal distribution with sd=var(y).

You can put more informative prior on the σ for the random walk (currently you use Exponential which put too much weight on the small value, thus pushing the variance of y all loaded onto the Normal), but I like to instead model the total variance and distribute them onto σ and sd:

with pm.Model() as model:
    smth_parm = pm.Uniform("alpha", lower=0, upper=1)
    mu2 = pm.Normal("mu", sd=100)
    tau2 = pm.Exponential("tau", 1.0/100)
    z2 = pm.GaussianRandomWalk("f",
                           mu=mu2,
                           tau=tau2 / (1.0 - smth_parm), 
                           shape=y.shape)
    obs = pm.Normal("obs", 
                    mu=z2, 
                    tau=tau2 / smth_parm, 
                    observed=y)
    trace = pm.sample(1000, njobs=4, tune=1000)

Which variance contributed more is controlled by alpha, you can interpreted it as a smoothing parameter.

You can find more explanations here:

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