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I am trying to apply BMA to estimate the beta of a certain stock by combining different models. One, simple model to estimate beta is this

$$\beta_{i,T} = \frac{{\rm cov}(r_{i,t},r_{M,t})}{{\rm var}(r_{M,t})}$$

with

  • $r_{i,t}$: return of the stock $i$ at day $t$
  • $r_{i,M}$: return of an market index $M$ at day $t$

I obtained data for the last 13 years for both variables:

Stock return vs Market return

Now, to estimate the beta factor for the next year $T$(251 days), I am using the above formula and use the past 251 days for the market return $r_M,t$ and the stock return $r_i,t$. This is to obtain a forecast for the beta which will be observed in 251 days. If I calculate them in a rolling windows approach (1 day step forward) I get this:

Forecast vs Realized (+ error)

It is not very surprising that the forecast looks just as a 251-day forward-shift because the Model is a very naive model which assumend that the 1-year forward beta will be as the actual observed.

Now I want to compute the Posterior Model Probabilites (PMP):

$$p(M_j|y_{1:T}) = \frac{\big(\int{p(y_{1:T}|\theta,M_j)p(\theta|M_j)d\theta}\big)p(M_j)}{\sum_{l=1}^2\big(\int{p(y_{1:T}|\theta,M_l)p(\theta|M_l)d\theta} \big) p(M_l)}$$

Assuming equal model priors it more or less reduces to computing:

$$\int{p(y_{1:T}|\theta,M_j)p(\theta|M_j)d\theta}$$

And here I am confused about how to do it. My data $y_{1:T}$ seem to be the realized data? What exactly are my $\theta$? $r_{i,t}$ and $r_{i,M}$? If I want to use Bayesian Information Criterion (BIC) here, how to compute the maximimum likelihood?

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  • $\begingroup$ One possibility could be to sample $N$ vectors from the prior $p(\theta|M_j)$ and run simple monte-carlo integration like $\int{p(y_{1:T}|\theta,M_j)p(\theta|M_j)d\theta} \sim \frac{1}{N}\sum_{n=1}^N p(y_{1:T}|\theta,M)$. First question, how to sample from $p(\theta|M_j)$ in this example? An secondly, how to compute $p(y_{1:T}|\theta,M)$ for the sampled $\theta$ ? $\endgroup$ – user3165675 Nov 6 '16 at 19:07
  • $\begingroup$ Well. To sample from $p(\theta|M_j)$ I can use this de.mathworks.com/matlabcentral/fileexchange/41689-pdfsampler and sample by the custom probality function of ${\rm cov}(r_{i,t},r_{M,t})$ and ${\rm var}(r_{M,t})$, my $\theta$'s. But how to compute $p(y_{1:T}|\theta,M)$ for each sampled $\theta$ ? $\endgroup$ – user3165675 Nov 6 '16 at 20:56
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I actually think that the approach to integrate over $\theta$ might be wrong here, because at least this model assuming a fixed $\theta$. Therefore, from my perspective it seems that the integration problem doesnt play a role here... But then, still, how to compute $p(y_{1:T}|M_j)$ ?

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