Technical Background

For notational simplicity we will assume a mixed frequency VAR model with no constant or exogenous variables. We also assume only two frequencies of data, low and high, and that there are m high frequency periods per low frequency period.

The VAR contains variables , observed at low frequency and variables observed at the high frequency. Let

• represent the i-th low frequency variable observed during low-frequency period

• represent the i-th high frequency variable observed in the t-th high-frequency period during low-frequency period

Stacking the and variables into the matrices and , respectively, and for brevity ignoring the intercepts and exogenous variables, we may write the VAR as:

(49.1) |

where is , is , and is , for all , , and is .

The U-MIDAS approach constructs these data and simply uses classical VAR least squares to estimate , the stacked matrix of coefficients.

Bayesian mixed-frequency VAR estimation (Ghysels 2016), analogously to Bayesian VAR estimation, requires you to specify prior distributions for and the residual covariance matrix, (
“Technical Background”).

(49.2) |

These priors coupled with the standard likelihood for a VAR model yields conditional posteriors for and of:

(49.3) |

(49.4) |

where

(49.5) |

The prior on is constructed like the Litterman prior in standard Bayesian VARs.

As with Litterman, the prior mean of is assumed to be all zero, other than the own first lag terms. However the Ghysels prior differs slightly by allowing a different hyper-parameter for the own lag term depending on whether a variable is high or low frequency. Another crucial difference is that the variables are assumed to follow an AR(1) process in the high-frequency, so that when estimating in low-frequency space, the parameter is exponential.

The prior variance of is similar to a Litterman prior, but with subtle differences to allow for the cross-frequency variances.

Thus, for the coefficient priors we have:

Element | Mean | Variance | Dimension |

, | |||

0 | , , | ||

0 | , | ||

0 | , , | ||

0 | , | ||

0 | , |

where , , and are hyper-parameters, and is a matrix with the (i,j)-th element equal to , , where the are taken from a univariate AR(1) estimation.

The covariance term between the i-th endogenous variable and an exogenous variable is set to .

is formed by taking a scaled identity matrix or by scaling an initial estimate of the variance taken from a classical least squares U-MIDAS estimation:

(49.6) |

where or .