Large Bayesian Vector Autoregressions

Abstract

Bayesian vector autoregressions are widely used for macroeconomic forecasting and structural analysis. Until recently, however, most empirical work had considered only small systems with a few variables due to parameter proliferation concern and computational limitations. We first review a variety of shrinkage priors that are useful for tackling the parameter proliferation problem in large Bayesian VARs. This is followed by a detailed discussion of efficient sampling methods for overcoming the computational problem. We then give an overview of some recent models that incorporate various important model features into conventional large Bayesian VARs, including stochastic volatility, non-Gaussian, and serially correlated errors. Efficient estimation methods for fitting these more flexible models are then discussed. These models and methods are illustrated using a forecasting exercise that involves a real-time macroeconomic dataset. The corresponding Matlab code is also provided [Matlab code is available at http://joshuachan.org/].

Appendices

Appendix A: Data

The real-time dataset for our forecasting application includes 13 macroeconomic variables that are frequently revised and 7 financial or survey variables that are not revised. The list of variables is given in Table 4.6. They are sourced from the Federal Reserve Bank of Philadelphia and cover the quarters from 1964Q1 to 2015Q4. All monthly variables are converted to quarterly frequency by averaging the three monthly values within the quarter.

Table 4.6 Description of variables used in the recursive forecasting exercise

Appendix B: Sampling from the Matrix Normal Distribution

Suppose we wish to sample from ${\mathcal N}(\text{vec}(\widehat {\mathbf {A}}), \boldsymbol \Sigma \otimes {\mathbf {K}}_{\mathbf {A}}^{-1})$. Let ${\mathbf {C}}_{{\mathbf {K}}_{\mathbf {A}}}$ and C _Σ be the Cholesky decompositions of K _A and Σ respectively. We wish to show that if we construct

$$\displaystyle \begin{aligned} {\mathbf{W}}_1 = \widehat{\mathbf{A}} + ({\mathbf{C}}_{{\mathbf{K}}_{\mathbf{A}}}^{\prime}\backslash \mathbf{U}) {\mathbf{C}}_{\boldsymbol \Sigma}^{\prime}, \end{aligned}$$

where U is a k × n matrix of independent ${\mathcal N}(0,1)$ random variables, then vec(W ₁) has the desired distribution. To that end, we make use of some standard results on the matrix normal distribution (see, e.g., Bauwens et al., 1999, pp. 301–302).

A p × q random matrix W is said to have a matrix normal distribution $\mathcal {M}\mathcal {N}(\mathbf {M},\mathbf {Q}\otimes \mathbf {P})$ for covariance matrices P and Q of dimensions p × p and q × q, respectively, if $\text{vec}(\mathbf {W})\sim \mathcal {N}(\text{vec}(\mathbf {M}),\mathbf {Q}\otimes \mathbf {P})$. Now suppose $\mathbf {W}\sim \mathcal {M}\mathcal {N}(\mathbf {M},\mathbf {Q}\otimes \mathbf {P})$ and define V = CWD + E. Then, $\mathbf {V} \sim \mathcal {M}\mathcal {N}(\mathbf {C}\mathbf {M}\mathbf {D}+\mathbf {E},(\mathbf {D}'\mathbf {Q}\mathbf {D})\otimes (\mathbf {C}\mathbf {P}\mathbf {C}'))$.

Recall that U is a k × n matrix of independent ${\mathcal N}(0,1)$ random variables. Hence, $\mathbf {U}\sim \mathcal {M}\mathcal {N}(\mathbf {0},{\mathbf {I}}_n\otimes {\mathbf {I}}_k)$. Using the previous result with $\mathbf {C} = ({\mathbf {C}}_{{\mathbf {K}}_{\mathbf {A}}}^{\prime })^{-1}$, $\mathbf {D} = {\mathbf {C}}_{\boldsymbol \Sigma }^{\prime }$, and $\mathbf {E} = \widehat {\mathbf {A}}$, it is easy to see that ${\mathbf {W}}_1\sim \mathcal {M}\mathcal {N}(\widehat {\mathbf {A}}, \boldsymbol \Sigma \otimes {\mathbf {K}}_{\mathbf {A}}^{-1}).$ Finally, by definition we have $\text{vec}({\mathbf {W}}_1)\sim {\mathcal N}(\text{vec}(\widehat {\mathbf {A}}), \boldsymbol \Sigma \otimes {\mathbf {K}}_{\mathbf {A}}^{-1})$.