Page 361 - Special Topic Session (STS) - Volume 3

STS550 Matteo Mogliani
Further, let h = 0, 1/, 2/, 3/, ... be an (arbitrary) forecast horizon, where
h = 0 denotes a nowcast with high-frequency information fully matching the
low-frequency sample. The MIDAS approach plugs-in the high-frequency
()
lagged structure of predictors ,−ℎ in a regression model for the low-
frequency response variable t as follows:

⁄
= + ∑ ( 1 ; () + , (1)

,−ℎ
=1
2
⁄
where is i.i.d. with mean zero and variance σ < ∞, and ( 1 ; ) =

⁄
∑ −1 (; ) is a weighting structure which depends on the weighting
=0

function B(c; ), a vector of p + 1 parameters = ( ,0 , , 1, … , , ), and

a maximum lag length C. In this study, we consider the simply
⁄
polynomial approximation of ( 1 ; ) provided by the Almon lag

polynomial B(c; ) = ∑ . Under the so-called “direct method”,

,
=0
Equation (1) with Almon lag polynomials can be reparameterized as:
t = + () + ⋯ + () + (2)

1,−ℎ
,−ℎ
()
where , = 1, … , , is a vector of linear combinations of the observed
,
high-frequency regressors, () = () , with () a (C × 1) vector of high-
,
,
,
frequency lags and a ( + 1 × ) polynomial weighting matrix. The main
advantage of the Almon lag polynomials is that (2) is linear and
parsimonious, as it depends only on ( + 1) parameters and can be
estimated consistently and efficiently via standard methods.
Although appealing, the MIDAS regression in (2) may be easily affected by
over-parameterization and multicollinearity in presence of a large number of
potentially correlated predictors. To achieve variable selection and parameter
estimation simultaneously, Tibshirani (1996) proposed the least absolute
shrinkage and selection operator (Lasso). In a nutshell, the Lasso is a penalized
least squares procedure, in which the loss function ℒ () is minimized after

setting a constraint on the ℓ norm of the vector of regression coefficients,
1
where the amount of penalization is controlled by a parameter λ. To achieve
the oracle property, which guarantees that the estimator performs as well as
if the true model had been revealed to the researcher in advance by an oracle,
Zou (2006) proposed the Adaptive Lasso (AL), where a different amount of
shrinkage (i.e. a different penalty term) is used for each individual regression
coefficient. However, the AL may not be suited in the present framework, as
lags of high-frequency predictors are by construction highly correlated and
hence the Lasso estimator would tend to select randomly only one lag and
shrink the remaining polynomial coefficients to zero. The theoretical rationale
for a failure in the selection ability of the AL in our mixed-frequency setting is

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