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I am trying to estimate a linear regression model (in the context of econometrics) using Bayesian approach (Gibbs sampler). The choice of the explanatory variables and model specification can be backed up by the literature and are intuitively reasonable. Simulated samples look like all converged from trace plots (MC iterations 20000+). However, I computed 95% highest posterior density and the results show that all of the estimated coefficients in the model are not significantly different from 0. It is micro level data and the sample size is around 300.

Could anyone help to explain how to understand such situation and how to deal with such "problem" please?

Specifically,

  1. Model estimated is demand system: expenditure share = intercept + log(price) + log(real expenditure) + log(real expenditure)^2 + sociodemographic variables + error

  2. Have tried 5 and 6 groups (5 or 6 equations to be estimated), and each equation contains at least 8 or 9 (where without any sociodemograghic variables added for testing) explanatory variables.

  3. I am kind of lost in experimenting with data (e.g trying out different number of groups and including different sociodem variables). Is there a systematic framework to approach this question please? e.g. From the perspective of the nature of data, the model specification and the sampling method?

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I presume that by "significant from 0", you mean "significantly different from 0". However, you haven't conducted any significance tests. You've created posterior intervals (and, I imagine, checked to see if they contain 0). Posterior intervals and significance tests are apples and oranges, so don't try to treat one as the other.

Similarly, you sound like you are under the impression that if the posterior intervals for all the coefficients contain 0, then there must be something wrong with the model. This is not so.

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It may well be the case that the available covariates/predictors simply do not explain the response variable. What you need to do, rather than just looking at credible intervals, is a formal Bayesian variable selection. There is an R package that can do this very quickly using some priors that were developed for variable selection:

Package ‘mombf’

If you prefer a simpler, rather informal, way of selecting variables, you can also use stepwise variable selection based on AIC or BIC (highly criticized) using the stepAIC() R command (BIC selection can be obtained after some small tweaks to this command).

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