Rolling regression with multivariate stored in a pandas dataframe

Questions v5
1

I’m new to Pymc and I’m currently studying cases on pymc website and I found Rolling Regression — PyMC example gallery is very useful for me.

However I want to apply multivariate regression on this case, for example, I store all my Xs in a DataFrame:

image

and I hope if I could predict y.

image

Here’s the demo code when you only have one variable:

with pm.Model(coords={"time": prices.index.values}) as model_randomwalk:
    # std of random walk
    sigma_alpha = pm.Exponential("sigma_alpha", 50.0)
    sigma_beta = pm.Exponential("sigma_beta", 50.0)

    alpha = pm.GaussianRandomWalk("alpha", sigma=sigma_alpha, dims="time")
    beta = pm.GaussianRandomWalk("beta", sigma=sigma_beta, dims="time")
    # Define regression
    regression = alpha + beta * prices_zscored.GFI.values

    # Assume prices are Normally distributed, the mean comes from the regression.
    sd = pm.HalfNormal("sd", sigma=0.1)
    likelihood = pm.Normal("y", mu=regression, sigma=sd, observed=prices_zscored.GLD.to_numpy())

    trace_rw = pm.sample(tune=2000, target_accept=0.9)

My question is, how could I apply this code to multivariate regression? I know one approach is to build beta_1,beta_2,beta_3 and change the regression formula to

regression = alpha + beta_1 * x_1.values + beta_2 * x_2.values + beta_3 * x_3.values

Could I just define ONE beta with three dimensions to do the job? I have ~40 variables and it would be pathetic if I define 40 betas…