MinnesotaPrior {LaplacesDemon} R Documentation

## Minnesota Prior

### Description

The Minnesota prior, also called the Litterman prior, is a shrinkage prior for autoregressive parameters in vector autoregressive (VAR) models. There are many variations of the Minnesota prior. This Minnesota prior is calculated as presented in Lutkepohl (2005, p. 225), and returns one or more prior covariance matrices in an array.

### Usage

MinnesotaPrior(J, lags=c(1,2), lambda=1, theta=0.5, sigma)


### Arguments

 J This is the scalar number of time-series in the VAR. lags This accepts an integer vector of lags of the autoregressive parameters. The lags are not required to be successive. lambda This accepts a scalar, positive-only hyperparameter that controls how tightly the parameter of the first lag is concentrated around zero. A smaller value results in smaller diagonal variance. When equal to zero, the posterior equals the prior and data is not influential. When equal to infinity, no shrinkage occurs and posterior expectations are closest to estimates from ordinary least squares (OLS). It has been asserted that as the number, J, of time-series increases, this hyperparameter should decrease. theta This accepts a scalar hyperparameter in the interval [0,1]. When one, off-diagonal elements have variance similar or equal to diagonal elements. When zero, off-diagonal elements have zero variance. A smaller value is associated with less off-diagonal variance. sigma This accepts a vector of length J of residual standard deviations of the dependent variables given the expectations.

### Details

The Minnesota prior was introduced in Doan, Litterman, and Sims (1984) as a shrinkage prior for autoregressive parameters in vector autoregressive (VAR) models. The Minnesota prior was reviewed in Litterman (1986), and numerous variations have been presented since. This is the version of the Minnesota prior as described in Lutkepohl (2005, p. 225) for stationary time-series.

Given one or more J \times J matrices of autoregressive parameters in a VAR model, the user specifies two tuning hyperparameters for the Minnesota prior: lambda and theta. Each iteration of the numerical approximation algorithm, the latest vector of residual standard deviation parameters is supplied to the MinnesotaPrior function, which then returns an array that contains one or more prior covariance matrices for the autoregressive parameters. Multiple prior covariance matrices are returned when multiple lags are specified. The tuning hyperparameters, lambda and theta, can be estimated from the data via hierarchical Bayes.

It is important to note that the Minnesota prior does not technically return a covariance matrix, because the matrix is not symmetric, and therefore not positive-definite. For this reason, a Minnesota prior covariance matrix should not be supplied as a covariance matrix to a multivariate normal distribution, such as with the dmvn function, though it would be accepted and then (incorrectly) converted to a symmetric matrix. Instead, dnormv should be used for element-wise evaluation.

While the Minnesota prior is used to specify the prior covariance for VAR autoregressive parameters, prior means are often all set to zero, or sometimes the first lag is set to an identity matrix.

An example is provided in the Examples vignette.

### Value

This function returns a J \times J \times L array for J time-series and L lags.

### Author(s)

Statisticat, LLC software@bayesian-inference.com

### References

Doan, T., Litterman, R.B. and Sims, C.A. (1984). "Forecasting and Conditional Projection using Realistic Prior Distributions". Econometric Reviews, 3, p. 1–144.

Litterman, R.B. (1986). "Forecasting with Bayesian Vector Autoregressions - Five Years of Experience". Journal of Business & Economic Statistics, 4, p. 25–38.

Lutkepohl, H. (2005). "New Introduction to Multiple Time Series Analysis". Springer, Germany.

dmvn, dnormv, and LaplacesDemon.