Page 287 - Special Topic Session (STS) - Volume 3

STS544 Jonathan W. et al.
apply. Even today, it requires special computer programing that is often not
included in available statistical and econometric programs. Even with a
program in hand, MLE requires some experience in setting starting values of
iterative nonlinear computations so that they converge successfully. Moreover,
as a model gets larger with more variables and more parameters to estimate,
the top of the likelihood tends to get flatter in all parameter dimensions, so
that convergence, if it can be achieved at all, starts to take an impractical
amount of time.
Therefore, other estimation methods have been developed to avoid these
problems. Although Bayesian methods can be equally or more
computationally intensive, they are, at least mathematically, simpler than MLE
because using them one doesn’t require numerically scaling a peak, but only
requires simulating a peak and computing some of its sample moments.
Properly programmed, a Bayesian method should compute conclusively,
although that may take a long time.
Therefore, there has been a need for quicker, reliable, linear methods for
estimating VARMA models using MFD. Chen and Zadrozny (1998) proposed
and illustrated the extended Yule-Walker (XYW) method for estimating a VAR
model with MFD, which is a linear generalized least squares (GLS) method.
Instead of estimating VAR or VARMA models using MFD with feedbacks both
from high-frequency to low-frequency variables and vice versa, Ghysels et al.
(2007) introduced the more easily implemented mixed-data sampling (MIDAS)
which regresses one or more low-frequency variables onto one or more high-
frequency variables using an exponential distributed lag with few parameters
to estimate. The basic “stacking” idea in MIDAS of estimating a model of
mixed-frequency data in low-frequency form originated with Friedman (1962).
Given the desire to estimate a VARMA model using MFD in a simple yet
effective way, in this paper we describe and illustrate a stacking method for
estimating a monthly VAR model using monthly-quarterly data. The stacking
introduces a relatively large number of additional, possibly insignificant,
parameters to be estimated. Here the number of these additional parameters
to be estimated is reduced by equating feedbacks of the same variables at the
same lags in different months of quarters. The restrictions are implied by
stationarity and are implemented using Theil and Goldberg’s (1961) linear
Bayesian mixed-estimation strategy. Ghysels (2016) addresses the parameter
proliferation problem arising from stacking slightly differently, partly with a
MIDAS-like exponential distributed lag.

2. Econometric Method
We propose estimating a stacked monthly VAR model using monthly-
quarterly data, first expressed as
(1) + −1 + ,

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