Rolling regression (Gaussian random walk)

I am trying to implement in brms the rolling regression I found in the PyMC3 examples:

Briefly, the model looks like

GLD_T = \alpha_T + \beta_T *GFI_T
\alpha_t \sim \mathcal{N}(\alpha_{t-1}, \sigma_\alpha)
\beta_t \sim \mathcal{N}(\beta_{t-1}, \sigma_\beta)

so that the regression coefficients vary as a Gaussian random walk.

I thought I could use the non linear formulas in brms but I fear it is not possible. Below you find my attempt, which does not work (Error: Explicit covariance terms can only be specified on ‘mu’.). Would you be able to help?


data <- read_csv('')

bf <- bf(
  GLD ~ a + GFI*b,
  nl = TRUE
  ) +
  gaussian() +
  lf(a ~ ar(p = 1)) +
  lf(b ~ ar(p = 1))

EDIT: Just realized my question is similar to Possible to fit this time series model in brms?

1 Like

I think you are at the edge of what brms can offer. For a clean solution I think you would need to use Stan. However, you might be able to hack around this limitation. If we define:

\bar\alpha_t = \alpha_t - \alpha_{t-1}


\alpha_t = \alpha_0 + \sum_i^t \bar \alpha_i \\ \alpha_t \sim N(0, \sigma_\alpha)

Similarly for \beta.
Sou you can then create a normal linear formula (possibly machine generated) of the form:

GLD ~ 1 + GFI +
  (1 + GFI | mm(a1, a2, a3,  ..., a15, 
                weights = cbind(t1, t2, t3, ..., t15), scale = FALSE))

Now aN would always be "N" for all N and tN would be 1 if the time of the observation is <= N and 0 otherwise. This way only the terms up to the current time point are included. The mm term means that all the coefficients share the same standard deviation.

I didn’t test this, so please double check the generated Stan code, but I think it could work.

Best of luck with your model!

Smart idea! I need to try tomorrow. I’ll let you know. Thanks!