Bayesian VARs for absolute beginners- Part 2: Selecting priors
In the first part of this two-part article (which can be found here), we examined what Bayesian VARs are, how are they different from the OLS VARs, and what parameters we need to input to estimate a simple Bayesian VAR under the Minnesota-Litterman prior. We used this specific prior as it is the simplest (yet powerful) prior someone can use to estimate a Bayesian VAR.
In this second part, as the priors one can implement are numerous, we are going to present the most important families of priors and how they are different from the Minnesota prior which is the most fundamental of them all.
Normal-Wishart priors: This prior assumes that the variance-covariance matrix of the errors of the Bayesian VAR follows a Wishart distribution and given that it is assumed that the coefficients’ priors follow the Normal distribution (under the Bayesian setting coefficients are probabilistic), the resulting estimates follow a distribution that is a product of Normal and Wishart. Under this prior, the equations of this VAR are restricted to have a proportional relationship. This restriction limits the number of inputs the researcher needs to insert to 2 (compared to 4 in the Minessota prior).
At first, the researcher needs to insert μ1, which is a value corresponding to the persistence of the variables. So, when variables are stationary, this value is set at zero, whereas if the variables have a unit root it is set close to 1. It is assumed that all variables have the same level of persistence.
And second, the researcher defines λ1, which corresponds to the certainty of the priors we are imposing. The more certain we are that the prior describes the nature of the data. λ1 in the Normal-Wishart prior works in an opposite way compared to the Minnesota prior as 0 implies complete uncertainty and greater values imply greater certainty about the priors!!
Sum of coefficient priors: Many different priors have been extended to be able to include a hyper-parameter, commonly referred to as μ5. The higher the value is for μ5, the higher is the probability that the sum of the coefficients of the lags of variable i in the equation of variable i sum up to 1, for every variable in the model. In many econometric packages, the option to include μ6 is given irrespective of the prior you choose.
Initial observations dummy prior: This prior includes a hyperparameter called μ6. The higher the value of μ6 is, the more firmly it is imposed as a prior that the variables included are cointegrated (aka, they have a long-run relationship). This option has also been made available irrespective of the chosen prior in many econometric packages as well. Keep in mind however that in some packages high μ6 implies only co-trending, which is a weaker assumption than cointegration.
The GLP prior: This is a new type of prior that has taken its name from the economists who proposed it (Giannone, Lenza, and Primiceri). Under this prior, all hyperparameters instead of being decided by the researcher, are determined by optimization procedures, thus removing most of the subjectivity one can have.
There are also other families of priors where a training sample is used (for example the first ten years of the data used in the estimation) to determine priors, instead of being decided by the researcher.
The above list indicates the spectrum of possibilities Bayesian VARs offer compared to simple VARs. However, good knowledge of the data is needed to decide which one you should use but if you choose correctly then your model will have great forecasting gains.
If you do not know which prior to use or you cannot decide the hyperparameter values, you can simply estimate Bayesian VARs of different priors and hyper-parameters and compare their forecasting performance.
If you found this story useful and you want to keep reading such stories and support me in providing quality content, you can do so by following me!
You can also receive my stories straight to your email inbox by clicking here!
Furthermore, if you are an investment enthusiast, like me, and wish to get access to powerful investment platforms and help me keep delivering interesting stories in the process you can sign up on any of the platforms below:
(If you want to know more about those platforms and why they should be a part of your portfolio, you can read about them here.)
Freedom 24 -One of the top investment platforms, you can invest with low to zero fees in stocks, bonds, ETFs, etc. It also has a deposit account with 3% annual interest paid out daily, plus they give you free stock upon signing up through this link.
Nexo -One of the most regulated cryptocurrency investment platforms. Earn interest on crypto and fiat currencies, use your crypto to get collateralized loans, and get access to the first crypto credit card.
Mintos -Perhaps the only regulated P2P platform in the EU. They recently started turning loans into formal investment instruments (called “Notes”). They also plan to expand and offer instruments like ETFs in the future.
Revolut -The one that brings all other platforms together. Revolut is the best solution to move money between investment platforms and banking accounts. They recently became a fully- licensed bank themselves and offer a variety of products from insurance and crypto accounts to interest-bearing deposit accounts (in some countries).
Whether you believe it or not, economists run on coffee. So, if you do not feel like registering on any of the platforms above, you can support me by buying me my morning coffee!
(and help fight climate change in the process, as 1% of the donations go towards carbon removal!)
Subscribe to DDIntel Here.
Join our network here: https://datadriveninvestor.com/collaborate
