# Hierarchical Regularizers for Reverse Unrestricted Mixed Data Sampling
Regressions

Alain Hecq Marie Ternes Ines Wilms<EMAIL_ADDRESS>author: Marie Ternes, Maastricht University, School of Business and Economics,
P.O. Box 616, 6200 MD Maastricht, The Netherlands, Email:
m.ternes<EMAIL_ADDRESS>Department of
Quantitative Economics, Maastricht University

###### Abstract

Reverse Unrestricted MIxed DAta Sampling (RU-MIDAS) regressions are used to
model high-frequency responses by means of low-frequency variables. However,
due to the periodic structure of RU-MIDAS regressions, the dimensionality
grows quickly if the frequency mismatch between the high- and low-frequency
variables is large. Additionally the number of high-frequency observations
available for estimation decreases. We propose to counteract this reduction in
sample size by pooling the high-frequency coefficients and further reduce the
dimensionality through a sparsity-inducing convex regularizer that accounts
for the temporal ordering among the different lags. To this end, the
regularizer prioritizes the inclusion of lagged coefficients according to the
recency of the information they contain. We demonstrate the proposed method on
an empirical application for daily realized volatility forecasting where we
explore whether modeling high-frequency volatility data in terms of low-
frequency macroeconomic data pays off.

Keywords: mixed-frequency models, MIDAS models, forecasting, group lasso

#### Acknowledgements.

The third author was financially supported by the Dutch Research Council (NWO)
under grant number VI.Vidi.211.032. In addition, we acknowledge support
through the HiTEc Cost Action CA21163. We gratefully acknowledge the comments
by participants at the 16th International Conference on Computational and
Financial Econometrics (CFE 2022) and thank Claudia Foroni and Stephan Smeekes
for helpful discussions.

##  Introduction

In recent years, a growing body of literature on mixed-frequency (MF) models
arose to exploit the information available in series recorded at different
frequencies. So far, the literature has mostly focused on modeling low-
frequency (LF) variables by means of more timely, high-frequency (HF)
variables to improve the former’s forecast or nowcast accuracy. One commonly
used univariate MF model is the MIxed DAta Sampling (MIDAS) regression
(Ghysels et al.,, 2004) that links low-frequency to high-frequency data
through tightly restricted distributed lag polynomials. Various extensions to
the MIDAS framework have been proposed in the literature. Foroni et al.,
(2015), for instance, consider univariate unrestricted MIDAS (U-MIDAS) to
enable ordinary least squares estimation instead of non-linear least squares
(NLS). Ghysels, (2016) propose a multivariate extension to model relations
between high- and low-frequency series in a stacked mixed-frequency VAR (MF-
VAR) system; state-space alternatives of the latter are provided by, amongst
others, Mariano and Murasawa, (2003), Schorfheide and Song, (2015) and Koelbl
and Deistler, (2020). Recently, high-dimensional MF models have been proposed
for the univariate (e.g., Han and Tsay,, 2020; Mogliani and Simoni,, 2021;
Babii et al., 2022b, ), multivariate (for factor models see e.g., Marcellino
and Schumacher,, 2010; Foroni and Marcellino,, 2014; Andreou et al.,, 2019;
for Bayesian estimation e.g., Schorfheide and Song,, 2015; Ghysels,, 2016;
Götz et al.,, 2016; McCracken et al.,, 2021; Cimadomo et al.,, 2022;
Paccagnini and Parla,, 2021 and for sparsity-inducing regularizers e.g., Hecq
et al.,, 2022) and panel settings (e.g., Babii et al., 2022a, ).

A less explored research area concerns MF models suited for modeling HF
variables by means of LF variables. This can be of interest when fundamental
economic variables, usually available at a lower frequency, play an important
role for forecasting higher frequency variables as they contain more valuable
information for longer-term horizons. While the stacked MF-VAR of Ghysels,
(2016) can be used to predict HF variables using LF ones (despite its main
usage for LF variables)111Another example is Dal Bianco et al., (2012) who
model the euro-dollar exchange rate at weekly frequency exploiting a set of
monthly financial and macroeconomic variables using a state-space framework as
in Mariano and Murasawa, (2003)., dedicated MF models for HF components have
been proposed in the literature. Foroni et al., (2018) extend the univariate
MIDAS and U-MIDAS models to the Reverse (R-MIDAS) and Reverse Unrestricted
MIDAS (RU-MIDAS) model where the HF dependent variable is regressed on the LF
explanatory variables. Indeed, in RU-MIDAS models each HF component has its
own model equation, thereby allowing the (lagged) dynamics between the high
and low frequency variables to change for every HF component. We consider this
RU-MIDAS model as it can be more flexible and parsimonious than the
multivariate stacked MF-VAR and does not require the specification of a model
for the LF variable. Moreover, the RU-MIDAS can automatically incorporate HF
information released within an ongoing LF period (i.e., a conditional model),
whereas the MF-VAR would require one to use a structural MF-VAR that describes
the contemporaneous correlations among variables to update the forecast for
every HF period. Thus, the RU-MIDAS model is more appropriate for direct
forecasts of the HF variable than the MF-VAR.

Previous studies have, however, mostly considered RU-MIDAS models where the
frequency mismatch $m$ between the HF and LF variable is small, for instance a
setting with monthly HF components and a quarterly LF component (see e.g.,
Foroni et al.,, 2018). Our interest lies in settings where this frequency
mismatch $m$ is large, think for instance of a setting with daily HF
components and a monthly LF component; such settings only recently attracted
attention (see e.g., Foroni et al.,, 2023). In this case RU-MIDAS regressions
are, however, severely affected by the “curse of dimensionality”. There are
two main sources of this curse. First, the larger the frequency mismatch the
more equations, namely one for each HF component, need to be estimated. Since
each equation comes with its own separate set of parameters, this in turn
increases the total number of parameters in the RU-MIDAS model. Second, the
number of observations available for the estimation of each HF equation is
limited to the number of LF observations. Hence, the larger $m$, the fewer
observations are available for estimation since the sample size of the high-
frequency variable is divided by $m$. Without further adjustments, one would
be limited to use RU-MIDAS with a small frequency mismatch. In this paper, we
address this curse of dimensionality by proposing a pooled RU-MIDAS model with
a hierarchical convex regularizer. First, to address the issue of sample size
reduction, we propose to pool the regression coefficients regulating the
lagged HF dynamics across the $m$ different HF equations. Pooling relies on
the assumption that the lagged effect of a HF variable on itself does not
change with the HF component that is modelled. For instance, by pooling we
constrain the lagged effect of the previous day to be the same for each day
within a month. Similarly for the lagged effect of order two and so on. We
advocate to pool in RU-MIDAS models where the frequency mismatch $m$ is large
as it allows one to use the full HF sample size instead of the limited LF
sample size to estimate the HF coefficients, which is a considerable
difference when the frequency mismatch is large. As a by product, by pooling
we decrease the total number of parameters that need to be estimated.

Second, to further reduce the curse of dimensionality in the pooled RU-MIDAS
model one could, for instance, turn to distributed lag polynomials that
restrict the coefficients similar to restricted R-MIDAS models. However,
R-MIDAS (similar to the regular MIDAS) requires computationally expensive NLS
estimation. Alternatively, Foroni et al., (2023) consider RU-MIDAS in a
Bayesian framework, whose good performance has been demonstrated for
forecasting electricity prices (Foroni et al.,, 2023) and shipping freight-
cost indices (Bouri et al.,, 2022). We instead consider sparsity-inducing
regularizers which form an appealing alternative (see Hastie et al.,, 2015 for
an introduction) and have successfully been applied to regular MIDAS
regressions (Babii et al., 2022b, ) and MF-VARs (Hecq et al.,, 2022). However,
to the best of our knowledge they have not been explored yet as tool for
dimension reduction in RU-MIDAS regressions. In this paper, we adapt the
mixed-frequency hierarchical regularizer of Hecq et al., (2022) to the RU-
MIDAS framework. The regularizer is build upon the group lasso with nested
groups (see e.g., Nicholson et al.,, 2020) and encourages a hierarchical
sparsity pattern that prioritizes the inclusion of coefficients according to
the recency of the information the corresponding series contains about the
state of the economy.

In our empirical application, we consider HF volatility forecasting through LF
macroeconomic variables. The relationship between financial volatility and
macroeconomic or financial variables has been analyzed in the literature,
including the mixed-frequency context (e.g., Engle et al.,, 2013; Asgharian et
al.,, 2013; Conrad and Loch,, 2015; Conrad and Kleen,, 2020; Fang et al.,,
2020). Most studies find that macroeconomic variables only pay off when it
comes to long-term forecasting but not so much for short-term horizons. We
contribute to this literature by assessing the performance of the proposed
regularized pooled RU-MIDAS model in forecasting the daily realized volatility
of the S&P 500 through a diverse collection of monthly macroeconomic variables
including industrial production, prices or housing starts. We evaluate whether
the addition of these LF macroeconomic variables improves the forecasts of the
daily realized variance across a wide range of forecast horizons ranging from
day-ahead towards six months ahead. We find that the pooled RU-MIDAS model
leads to (slightly) better forecast performance compared to the original RU-
MIDAS model. However, the addition of macroeconomic variables does not improve
forecast accuracy of the daily S&P 500 realized variance compared to the
popular HAR model of Corsi, (2009).

The remainder of this paper is structured as follows. Section 2 introduces the
pooled RU-MIDAS with hierarchically structured parameters and describes the
regularized estimation procedure. Section 3 presents the results on the
empirical application for volatility forecasting and Section 4 concludes.
Additional results are available in the Appendix.

##  Regularized Estimation of the Pooled RU-MIDAS Model

We start by revising the approximate RU-MIDAS regression of Foroni et al.,
(2018) based on linear lag polynomials in Section 2.1.222We slightly deviate
from Foroni et al.,’s (2018) notation to facilitate the introduction of
hierarchical structures in Section 2.3. First, we make the LF variable
observable at the end of LF period, together with the last HF observation
within the LF period. In contrast, in Foroni et al., (2018) the LF variable is
observable at the beginning of the LF period together with the first HF
observation within the LF period. Second, we define the dummy variables in the
dummy notation (see equation (2)) differently. Third, the first lag of the LF
variable always corresponds to the previous LF period. Then, we introduce the
pooled RU-MIDAS model in Section 2.2, followed by the hierarchical sparsity
structure on its parameters in Section 2.3. Finally, the regularized estimator
for the pooled RU-MIDAS model is presented in Section 2.4.

###  RU-MIDAS Model

Let $x_{t}$ denote the high-frequency (HF) variable which we observe at
$t=\frac{1}{m},\frac{2}{m},\dots,\frac{m-1}{m},1,1+\frac{1}{m},...$. and
$y_{t}$ the low-frequency (LF) variable which is only observed every $m$
periods at $t=1,2,\dots,T$, where $m$ denotes the frequency mismatch. For
simplicity, we consider one lag for the LF variable and $m$ lags for the HF
variable.333We can easily extend the methodology to include multiple high and
low frequency variables with higher-order lags at the cost of more complicated
notation. The reverse unrestricted MIDAS (RU-MIDAS) model is given by

$\displaystyle x_{t}$ $\displaystyle=\alpha_{i}\
y_{t-\frac{i}{m}}+\beta_{1,i}\ x_{t-\frac{1}{m}}+\beta_{2,i}\
x_{t-\frac{2}{m}}+\dots+\beta_{m,i}\ x_{t-1}+\varepsilon_{t},$ (1)
$\displaystyle t$
$\displaystyle=\frac{i}{m},1+\frac{i}{m},\dots,\tau+\frac{i}{m},\dots,T-\frac{m-i}{m}$
$\displaystyle i$ $\displaystyle=1,\dots,m,$

where $\alpha_{i}$, $\beta_{j,i}$ ($i,j=1,\ldots,m$) are the unknown
parameters, $\varepsilon_{t}$ the error term and the variables are assumed to
be mean-centered such that no intercept is included. For illustrative
purposes, consider equation (1) for the monthly/quarterly setting with $m=3$.
Then, $\alpha_{i}$ and $\beta_{j,i}$ ($j=1,2,3$) correspond to the effects of,
respectively, the lagged quarterly and monthly variables on the $i$th
($i=1,2,3$) monthly HF variable. Hence, the subscript $i$ of the coefficients
refers to the effect of the corresponding regressor on month $i$, whereas $j$
indicates the number of months the monthly regressor is lagged by. For $i=1$
(first month so for $t=\frac{1}{3},1+\frac{1}{3},...$), $\alpha_{1}$ contains
the effect of the previous quarter onto the first month and
$\beta_{1,1},\beta_{2,1}$ and $\beta_{3,1}$ the effect of respectively month
three, two and one of the previous quarter on the first month. Similarly for
$i=2$ (second month so for $t=\frac{2}{3},1+\frac{2}{3},...$), $\alpha_{2}$
contains the effect of the previous quarter, however $\beta_{1,2}$ now
contains the effect of month one of the current quarter, i.e., the latest
available monthly observation (first monthly lag), and
$\beta_{2,2},\beta_{3,2}$ refer to respectively month three and two of the
previous quarter. For $i=3$ (third month so for $t=1,2,...$), the latest
available monthly observations are month two ($\beta_{1,3}$) and one
($\beta_{2,3}$) of the current quarter, followed by month three of the
previous quarter ($\beta_{3,3}$). The RU-MIDAS model thus easily allows for
the inclusion of HF information released within the current LF period. Figure
1 provides a visual summary of these lagged dynamics for the RU-MIDAS model
with the monthly/quarterly set-up. Finally, note that for ease of exposition,
we include the same LF lag for each HF response in equation (1), yet the model
can easily accommodate different LF lags across HF responses to take, for
instance, data releases into account.444We investigated such different LF lag
structures in our empirical application, yet none resulted in improved
forecast performance over the simple LF lag structure used in equation (1).

Qt-1M1tM2tM3t$\alpha_{3}$$\alpha_{2}$$\alpha_{1}$

(a) Lagged quarterly effect

M1t-1M2t-1M3t-1M1tM2tM3t${\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}\beta_{1,1}}$${\color[rgb]{0.15,1,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.15,1,0.8}\pgfsys@color@cmyk@stroke{0.85}{0}{0.20}{0}\pgfsys@color@cmyk@fill{0.85}{0}{0.20}{0}\beta_{2,1}}$${\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\beta_{3,1}}$${\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}\beta_{1,2}}$${\color[rgb]{0.15,1,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.15,1,0.8}\pgfsys@color@cmyk@stroke{0.85}{0}{0.20}{0}\pgfsys@color@cmyk@fill{0.85}{0}{0.20}{0}\beta_{2,2}}$${\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\beta_{3,2}}$${\color[rgb]{0.15,1,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.15,1,0.8}\pgfsys@color@cmyk@stroke{0.85}{0}{0.20}{0}\pgfsys@color@cmyk@fill{0.85}{0}{0.20}{0}\beta_{2,3}}$${\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}\beta_{1,3}}$${\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}\beta_{3,3}}$

(b) Lagged monthly effect

Figure 1: Lagged dynamics of the RU-MIDAS model for the monthly/quarterly
setting.

Foroni et al., (2018) propose to group the system of equations in equation (1)
into a single equation

$\displaystyle x_{t}$
$\displaystyle=\alpha_{1}D_{1}y_{t-\frac{1}{m}}+\alpha_{2}D_{2}y_{t-\frac{2}{m}}+\dots+\alpha_{m}D_{m}y_{t-1}$
(2)
$\displaystyle+\beta_{1,1}D_{1}x_{t-\frac{1}{m}}+\beta_{1,2}D_{2}x_{t-\frac{1}{m}}+\dots+\beta_{1,m}D_{m}x_{t-\frac{1}{m}}$
$\displaystyle+\beta_{2,1}D_{1}x_{t-\frac{2}{m}}+\beta_{2,2}D_{2}x_{t-\frac{2}{m}}+\dots+\beta_{2,m}D_{m}x_{t-\frac{2}{m}}$
$\displaystyle\ \vdots$
$\displaystyle+\beta_{m,1}D_{1}x_{t-1}+\beta_{m,2}D_{2}x_{t-1}+\dots+\beta_{m,m}D_{m}x_{t-1}+v_{t}$
$\displaystyle t$ $\displaystyle=\frac{1}{m},\frac{2}{m},\dots,1,\dots,T,$

through the use of dummy variables $D_{1}$, $D_{2}$,…, taking respectively the
value of one in the first ($t=\frac{1}{m},1+\frac{1}{m},...$), second
($t=\frac{2}{m},1+\frac{2}{m},...$), … and $m$-th ($t=1,2,...$) HF observation
within the LF period. We return to this dummy notation in Section 2.2 due to
its convenience for introducing the pooled RU-MIDAS model.

In a low-dimensional setting, the parameters of the RU-MIDAS model can be
estimated by simple OLS. However, if the frequency mismatch $m$ is large, the
model suffers from the curse-of-dimensionality since for one LF and $m$ HF
lags $(1+m)m$ parameters need to be estimated with only $T$ low-frequency
observations available, thereby making OLS unreliable. Indeed, first, the
larger $m$ the more equations $i=1,\ldots,m$ need to be estimated which in
turn increases the total number of parameters since each equation has its own
set of parameters. Second, a large $m$ drastically decreases the sample size
of the HF variable as the number of observations available for the estimation
of each equation is limited to the number of LF observations $T$. The curse
exacerbates if the number of LF and/or HF lags grows. Next, we propose to
counter this sample size reduction by pooling corresponding lagged HF
coefficients across equations.

###  Pooled RU-MIDAS Model

To overcome the reduction in sample size, we propose to pool the coefficients
$\beta_{i,j}$ of the lagged HF variable. For illustrative purposes, consider
again equation (1) for the monthly/quarterly setting. Assume that the lagged
effect of the previous month onto the current month is the same regardless of
the HF period $i=1,2,3$. Then we can restrict the coefficients by pooling,
thereby imposing the restrictions $\beta_{1,1}=\beta_{1,2}=\beta_{1,3}$.
Similarly, we assume that the lagged two months effect onto the current month
is the same regardless of the HF period, i.e.,
$\beta_{2,1}=\beta_{2,2}=\beta_{2,3}$ and correspondingly for the the third
month $\beta_{3,1}=\beta_{3,2}=\beta_{3,3}$. In sum, re-consider Figure 1(b)
where the constraints imposed under pooling for the monthly/quarterly set-up
are visualized through colour coding of the parameter labels, i.e., parameters
with the same font color are pooled together.

Generally, let us introduce the “pooled RU-MIDAS” model in dummy notation as

$\displaystyle x_{t}$
$\displaystyle=\alpha_{1}D_{1}y_{t-\frac{1}{m}}+\alpha_{2}D_{2}y_{t-\frac{2}{m}}+\dots+\alpha_{m}D_{m}y_{t-1}$
(3) $\displaystyle+\beta_{1}\ x_{t-\frac{1}{m}}+\beta_{2}\
x_{t-\frac{2}{m}}+\dots+\beta_{m}\ x_{t-1}+v_{t},$ $\displaystyle t$
$\displaystyle=\frac{1}{m},\frac{2}{m},\dots,T.$

Equation (3) differs from (2) in several aspects. In the RU-MIDAS model (2),
we consider a different equation for each HF variable which implies that the
parameters capturing the lagged LF and HF effects vary for each HF component.
In the pooled version (3), in contrast, we only allow the effect of the lagged
LF variable to vary for each HF component, whereas the lagged HF effects are
constrained to be the same. Therefore, we now have $Tm$ observations available
for the estimation of the lagged HF coefficients $\beta_{1},\dots,\beta_{m}$
instead of just $T$, the number of LF observations. Furthermore, we greatly
reduce the total number of to be estimated parameters from $(1+m)m$ to $2m$.
Since even the later number can be considerable if the frequency mismatch $m$
is large and/or not all $m$ lagged HF or LF coefficients are necessarily of
importance in modeling the HF variables, we further impose, in the next
section, a hierarchical sparsity structure on the lagged parameters in
equation (3).

###  Hierarchical Sparsity in the Pooled RU-MIDAS

To further reduce the dimensionality in equation (3), we resort to sparse
estimation. While one could opt for “patternless” sparsity through the lasso
(Tibshirani,, 1996), we propose to use the dynamic structure of the pooled RU-
MIDAS in equation (3) as an additional source of information to guide the
estimation. We use a hierarchical sparse estimator to encourage structured
sparsity patterns appropriate to the context of RU-MIDAS regressions,
similarly to the one introduced in Hecq et al., (2022) for stacked MF-VARs.

Consider the hierarchical sparsity pattern that naturally arises in the
autoregressive (AR) coefficients of a pooled RU-MIDAS regression. Let
$\bm{\beta}$ be the vector collecting the HF coefficients in equation (3),
namely $\bm{\beta}=[\beta_{1},...,\beta_{m}]^{\prime}$, and define
$\bm{\alpha}=[\alpha_{1},...,\alpha_{m}]^{\prime}$ similarly for the LF
coefficients. For each of the $2$ sub-vectors, we impose a hierarchical
priority structure for parameter inclusion. We prioritize the inclusion of
lagged coefficients according to the recency of the information the lagged
(nonseasonal) time series contains relative to $x_{t}$, in line with the
recency-based priority structure introduced in Hecq et al., (2022). The more
recent the information contained in the lagged HF or LF predictor, the more
informative, and thus the higher its priority of inclusion. For instance, for
the HF lags in $\bm{\beta}$, the first lag $\beta_{1}$ contains the most
recent information, followed by the second HF lag $\beta_{2}$, etc. Thus, for
$j<j^{\prime}$ we prioritize parameter $\beta_{j}$ over $\beta_{j^{\prime}}$
in the model. Restrictions encouraging decay across lags are commonly used in
the mixed-frequency literature, see for instance Babii et al., 2022b  for
univariate or Ghysels, (2016) for multivariate models. A similar recency
structure arises in the coefficients of the LF variable. The first coefficient
$\alpha_{1}$ contains the most recent information as it captures the effect of
the previous LF period onto the first HF period which is followed by
$\alpha_{2}$, the effect of the previous LF period onto the second HF period
etc. Therefore, for $i<i^{\prime}$ we prioritize parameter $\alpha_{i}$ over
$\alpha_{i^{\prime}}$. In the next section, we introduce the regularized
estimator that encourages such hierarchical sparsity structures for the
estimated parameters in the pooled RU-MIDAS model (3).

###  Hierarchical Group Lasso for the Pooled RU-MIDAS

We use a nested group lasso (Zhao et al.,, 2009) to attain the hierarchical
structures detailed in Section 2.3. Define
$\bm{\beta}^{(j:m)}=[\beta_{j},\dots,\beta_{m}]^{\prime}$ for $1\leq j\leq m$.
A nested structure then arises with
$\bm{\beta}^{(1:m)}\supset\dots\supset\bm{\beta}^{(m:m)}$. Similarly, let
$\bm{\alpha}^{(i:m)}=[\alpha_{i},\dots,\alpha_{m}]^{\prime}$ for $1\leq i\leq
m$. We define the hierarchical group lasso estimator for the pooled RU-MIDAS
model as

$\displaystyle(\widehat{\bm{\alpha}},\widehat{\bm{\beta}})$
$\displaystyle=\underset{\bm{\alpha},\bm{\beta}}{\text{argmin}}\Bigg{\\{}\frac{1}{2}\sum_{t=\frac{1}{m},\frac{2}{m},...}^{T}\Big{(}x_{t}-\sum_{i=1}^{m}\alpha_{i}D_{i}y_{t-\frac{i}{m}}-\sum_{j=1}^{m}\beta_{j}x_{t-\frac{j}{m}}\Big{)}^{2}+\lambda\cdot\mathcal{P}^{\text{Hier}}(\bm{\alpha},\bm{\beta})\Bigg{\\}},$
(4)

where $\lambda\geq 0$ is a tuning parameter and
$\mathcal{P}^{\text{Hier}}(\bm{\alpha},\bm{\beta})$ denotes the hierarchical
group penalty given by

$\displaystyle\mathcal{P}^{\text{Hier}}(\bm{\alpha},\bm{\beta})=\sum_{i=1}^{m}\lVert\bm{\alpha}^{(i:m)}\rVert_{2}+\sum_{j=1}^{m}\lVert\bm{\beta}^{(j:m)}\rVert_{2}.$
(5)

The hierarchical structure for the high-frequency lags is built in through the
condition that if $\bm{\beta}^{(j:m)}=\bf{0}$, then
$\bm{\beta}^{(j^{\prime}:m)}=\bf{0}$ where $j<j^{\prime}$, equivalently for
$\bm{\alpha}$. We apply the proximal gradient algorithm to efficiently solve
the optimization problem in equation (4), as detailed in Appendix A of Hecq et
al., (2022). For the selection of the tuning parameter $\lambda$, we rely on
the Bayesian Information Criterion (BIC). Finally, to reduce the bias
introduced by lasso-type estimators, we follow Belloni and Chernozhukov,
(2013) and apply the least squares on the variables retained by the
hierarchical group lasso. We do not implement a Monte Carlo study as the
performance of the pooled RU-MIDAS estimated by hierarchical lasso largely
depends on the underlying data generating process chosen. Instead, we focus on
an empirical study as discussed in the next section.

##  Empirical Application

We consider the regularized estimator of the pooled RU-MIDAS to forecast the
realized variance of the S&P 500 using several LF macroeconomic variables. In
finance, movements in volatility are closely tracked since they have a
considerable impact on capital investment, consumption, and economic
activities (see e.g., Tsay,, 2005; Bauwens et al.,, 2012). Therefore,
forecasting volatility and the understanding of its drivers is of interest to
both academics and practitioners. In recent years, the relationship between
financial volatility and macroeconomic variables has received increased
attention in the literature, we contribute to this strand by investigating the
predictive information contained in LF macroeconomic variables for forecasting
HF financial volatility.

Two problems, however, that affect the construction of such volatility
forecasts is that volatility is unobservable and that macroeconomic variables
are usually only available at a lower frequency. To overcome these
difficulties, Engle et al., (2013) propose a GARCH-MIDAS model to bridge the
gap between daily stock returns and low-frequency (e.g., monthly, quarterly)
explanatory variables. This model has been extensively used in the literature
(e.g., Asgharian et al.,, 2013; Conrad and Loch,, 2015; Conrad and Kleen,,
2020; Fang et al.,, 2020), but the evidence on which macroeconomic variables
have predictive ability for stock market volatility is mixed. Nevertheless,
coincident/lagging indicators such as inflation and industrial production
(Engle et al.,, 2013) and the expectations about them (Conrad and Loch,,
2015), as well as leading indicators such as term spread and housing starts
(Conrad and Loch,, 2015, Fang et al.,, 2020) seem to often improve forecast
performance. Additionally, the findings highlight that including macroeconomic
variables pays off particularly when it comes to long-term forecasting and not
so much for short-term horizons.

While GARCH models treat volatility as latent, accurate nonparametric
estimates of volatility have become available since the emergence of high-
frequency data (see e.g., Andersen et al.,, 2001; Barndorff-Nielsen and
Shephard,, 2002). Realized variances, computed as the sum of the squared
intraday returns for a particular day, make volatility “observable” and
volatility forecast models relying on such data have become popular
alternatives (see e.g., Hua and Manzan,, 2013; Haugom et al.,, 2014).555Apart
from assessing forecast accuracy, one could explore Granger causality
relations, see for instance Götz et al., (2016) who test Granger-causality
from a HF variable, namely the SP500 bipower variation to a LF macro variable,
namely industrial production. In this context, the heterogeneous
autoregressive (HAR) model (Corsi,, 2009) has become particularly popular
because of its simplicity and good performance to capture long memory features
and for forecasting. The HAR model explains the log-realized variance as a
linear function of the log-realized variances of yesterday and the average
over the last five days (last week) and last 20 days (last month) which we
will refer to in the remainder as the day-week-month (dwm) lag structure.
Recently, it has become popular to compare the forecast performance of models
with a more general lag structure selected by lasso-type estimators with the
parsimonious dwm lag structure embedded in the HAR (e.g., Audrino and Knaus,,
2016; Audrino et al.,, 2017; Ding et al.,, 2021; Wilms et al.,, 2021). Though
the results are mixed, there is some evidence that the dwm lag structure of
the HAR model is hard to beat by more sophisticated lasso-based lag
structures. So far, the addition of macroeconomic variables to such models
has, however, not been explored yet. We build on this strand of the literature
and investigate whether HF daily realized variances can be forecast more
accurately by accounting for information in LF macroeconomic variables.

In particular, we forecast the daily realized variance of the S&P 500 using a
wide set of LF macroeconomic variables over the period January 3rd 2000 to
December 31st 2019 to avoid the influence of the pandemic. The data and
forecast-set up are described in Section 3.1. We have two main objectives in
our empirical application that are subsequently investigated. First, we
compare the forecast performance between the original RU-MIDAS and the
proposed pooled RU-MIDAS model in Section 3.2, to investigate how the degree
of parameter flexibility in the mixed-frequency model influences the forecast
accuracy. Second, we analyze in Section 3.3 whether including LF macroeconomic
variables for forecasting HF realized variance pays off by comparing the
forecast performance between our pooled RU-MIDAS model and the HAR.

###  Data Description and Forecast Set-up

In this section, we describe the dataset used to perform our empirical study
and we detail the forecast set-up. Regarding the HF variable, we consider the
daily log realized variance based on five-minute returns of the Standard &
Poor’s 500 market index “SP500” taken from the Oxford–Man Institute of
Quantitative Finance.666Alternatively, we could have taken a more robust
estimator of the integrated variance such as the bipower variation or the
median realized volatility but this is not the focus of our paper. The sample
spans from January 3rd 2000 to December 31st 2019. Our mixed-frequency models
require a fixed frequency mismatch $m$, which we set equal to $m=20$ as in
Götz et al., (2016). To ensure $m=20$ in each month, we therefore interpolate
additional values for non-existing days for months with less than 20 trading
days and disregard excessive days for months with more than 20 trading days at
the beginning of the corresponding month. Figure 2 plots the log-realized
variances of the S&P500.

Figure 2: Log realized variances based on five-minute returns of the S&P 500
from January 2000 to December 2019.

Regarding the LF macroeconomic variables, we consider monthly variables on
various aspects of the economy: amongst others output, income, prices and
employment, consumer sentiment and financial variables, see Table 1 for an
overview. The monthly macroeconomic series are directly taken from the FRED-MD
dataset which is available at the Federal Reserve Bank of St. Louis FRED
database (see McCracken et al.,, 2021 for details). We apply the
transformation codes in column “T-code” to remove stochastic trends.777The
transformation codes correspond to the codes in the FRED-MD dataset with the
exception of “CPIAUCSL” where we found the first difference to be sufficient
to remove the stochastic trend during the period we consider. Furthermore, we
also consider the monthly news-based Economic Policy Uncertainty Index
(“Epui”) by Baker et al., (2016) as additional “forward-looking” explanatory
variable.

In our analysis, we consider twelve different pooled RU-MIDAS models: eleven
models where each model includes a different LF macroeconomic predictor, and
one model that includes them all. We can hereby evaluate the predictive
information contained in each of the macroeconomic variables separately as
well as jointly.

Table 1: Data description macroeconomic variables. Column T-code denotes the
data transformation applied to a time series, which are: (1) not transformed,
(2) $\Delta x_{t}$, (3) $\Delta^{2}x_{t}$, (4) $\text{log}(x_{t})$, (5)
$\Delta\text{log}(x_{t})$, (6) $\Delta^{2}\text{log}(x_{t})$.

Variable | FRED Code | Description | T-code
---|---|---|---
Cpi | CPIAUCSL | CPI : All Items | 5
Csent | UMCSENTx | Consumer Sentiment Index (University of Michigan) | 2
Epui | - | News-based Economic Policy Uncertainty Index | 1
Fedfunds | FEDFUNDS | Effective Federal Funds Rate | 2
Houst | HOUST | Housing Starts: Total | 4
Indpro | INDPRO | IP Index | 5
Oil | OILPRICEx | Crude Oil | 6
M2 | M2REAL | Real M2 Money Stock | 6
Termsp | GS10 - TB3MS | Term spread (10-Year Treasury Rate (GS10) - 3-Month Treasury Bill (TB3MS)) | 2
Unrate | UNRATE | Civilian Unemployment Rate | 2
Vix | VIXCLSx | VIX | 1

To compare forecast performance of the different models and forecast methods,
we perform a rolling window forecast exercise with a window size of 5 years
($T=1200$) and various forecast horizons ranging from daily ($h=1$), over
weekly ($h=5$), and monthly ($h=20$), to several months ($h=40,60,120$). To
forecast more than one day-ahead in time (i.e., forecast horizon $h>1$), we
use a direct forecast approach (Marcellino et al.,, 2006). For a given
forecast horizon $h$, the direct approach re-estimates model (3) but with
$x_{t+h-1}$ as the dependent variable. The advantage of using such direct
approach is that it does not require the forecast of the lagged low-frequency
variable. However, the model specification changes for each forecast horizon.
Finally, forecast performance is measured based on the out-of-sample mean
absolute forecast error (MAFE) but results with root mean squared forecast
error (RMSFE) are available upon request.

###  Forecast Comparison RU-MIDAS and Pooled RU-MIDAS

In this section, we compare the forecast performance of the original RU-MIDAS
and the proposed pooled RU-MIDAS model, where the parameters of both models
are estimated using the hierarchical estimator for comparability. The
estimation procedure for the pooled RU-MIDAS model is detailed in Sections 2.3
and 2.4, for the original RU-MIDAS model, we set hierarchical sparsity
structures for each HF equation $i=1,\dots,m$ individually. For instance, for
the high-frequency AR lags ($\bm{\beta}_{i}$) the first high-frequency lag
$\beta_{1,i}$ contains the most recent information, followed by the second HF
lag $\beta_{2,i}$, etc. Thus, for $j<j^{\prime}$ we prioritize parameter
$\beta_{j,i}$ over $\beta_{j^{\prime},i}$ in the model. As we only consider
one LF lag and therefore $\alpha_{i}$ is a scalar, the group-lasso penalty for
the LF lag boils down to a simple $\ell_{1}$-penalty.

Figure 3 depicts the out-of-sample MAFEs for the pooled RU-MIDAS (x-axis)
versus the original RU-MIDAS (y-axis) across the six forecast horizons
(panels). Each panel includes twelve MAFEs (dots), one for each model with a
particular macroeconomic variable as indicated in the coloured legend. The
black line, being the 45 degree line, is added to facilitate comparison
between the pooled RU-MIDAS model and the original one since dots above this
line imply that the pooled model attains better average forecast accuracy
(i.e. lower MAFEs). Across all forecast horizons (apart from $h=120$) and all
macroeconomic variables, the pooled model delivers better forecast accuracy.
This indicates that pooling corresponding lagged HF variables pays off for the
considered data set. At short forecast horizons ($h=1,5$), the MAFEs are
nearly indistinguishable across macroeconomic variables. For horizons
$h=20,40,60$ the dots corresponding the “Houst” (red) and “All” (light blue)
stand out. The model including Housing obtains higher MAFEs for both the
pooled as well as the original RU-MIDAS model, hence the green dot is
positioned in the upper right corner relative to the other dots. The model
including all macroeconomic variables demonstrates that the larger the
dimensionality of the model the more pooling pays off, thus the red dot is in
the upper left corner relative to the other dots. At the longest horizon of
six months ($h=120$), the forecast accuracy of both models decreases
considerable. Both models deliver comparable performance across almost all
macroeconomic variables (i.e. all dots except for “Houst” are close to 45
degree line).

For each dot (i.e., each pairwise comparison between pooled RU-MIDAS and
original RU-MIDAS) in Figure 3, we perform a Diebold-Mariano (DM) test
(Diebold and Mariano,, 1995) to test for significant difference in forecast
accuracy. At horizon $h=1,5,20$, the pooled version is more accurate at 1%
significance level across all twelve models. At horizon $h=40,60$, the pooled
version performs better at (at least) 5% significance level across all twelve
models with the exception of “Houst” for which the improvement is
statistically insignificant and for “all” which is significant at 10%. For
$h=120$, there is no significant improvement. In sum, we observe that the
addition of the pooling constraints to the RU-MIDAS model never leads to
significantly worse forecast performance compared to the original RU-MIDAS
model, often even better. We therefore continue with the pooled RU-MIDAS model
and investigate whether the addition of LF macroeconomic variables can improve
forecast performance over the standard HAR model.

Figure 3: Mean absolute forecast errors, averaged across time, of the pooled
RU-MIDAS (x-axis) versus the original RU-MIDAS (y-axis) model, and this for
the different macroeconomic variables (coloured dots) and the six forecast
horizons (panels).

###  Forecast Comparison Pooled RU-MIDAS and HAR

In this section, we assess the predictive information in the LF macroeconomic
variables when forecasting the S&P 500 daily realized variance. To this end,
we compare the forecast performance between the pooled RU-MIDAS model, which
includes macroeconomic variables, and the HAR that does not include
macroeconomic variables. The parameters in the HAR model are estimated by OLS.

We first evaluate overall forecast performance, as computed via the MAFEs
averaged across time. Figure 4 depicts the MAFE for the pooled RU-MIDAS
estimated by the hierarchical estimator (dots, one for each macroeconomic
variable) and the HAR (black line) across the six horizons. We clearly notice
that, for both models, the forecast performance deteriorates with the forecast
horizon (as expected). Moreover, the HAR consistently outperforms the pooled
RU-MIDAS model by delivering lower MAFEs. Particularly, the addition of
“Houst” (red dot) deteriorates forecast accuracy in comparison to the other
macroeconomic variables for which the difference is almost indistinguishable.
The model including all macroeconomic variables (light blue) performs
comparable to the models including only a single macroeconomic variable for
all horizon with the exception of its high MAFE at $h=120$. Hence, overall, we
find no evidence that the addition of macroeconomic variables, neither
individually nor jointly, improves forecast performance over the standard HAR.
Similarly, Fang et al., (2020) observed that the inclusion of past volatility
information rather than the inclusion of macroeconomic variables improves
forecasting accuracy. There are some exceptions at horizon $h=120$, namely for
the models including the macroeconomic variables “Epui”, “Fedfunds”, “Indpro”,
“Termsp”,“Unrate” and “Vix”888The models with macroeconomic variables “Cpi”,
“Csent”, “M2”, “Oil” also obtain a lower MAFE than the HAR, however, no
macroeconomic lags are selected in these models (see Table 3) so they only
differ from the HAR in terms of their HF lag structure., which display a
somewhat lower MAFE than the one obtained with the HAR model. It should,
however, be noted that differences are largely negligible.

Figure 4: Mean absolute forecast errors, averaged across time, of the pooled
RU-MIDAS with different macroeconomic variables (dots) versus the HAR (solid
black line) and this for the the six forecast horizons.

To further analyze the differences in forecast performance, Table 2 present
the MAFE of the HAR, and the pooled RU-MIDAS estimated by the hierarchical
estimator (results as displayed in Figure 4), as well as by OLS to better
assess the influence of the regularization. For each forecast horizon, we
consider these three estimation procedures, the two RU-MIDAS in combination
with the twelve models, each including different macroeconomic predictor,
leading to 25 models in total. For each horizon, we obtain the models included
in the 75% Model Confidence Set (Hansen et al.,, 2011), whose MAFEs are
displayed in bold.

Table 2: Mean absolute forecast errors, averaged across time, of the HAR, the
pooled RU-MIDAS estimated by OLS (p-RU-MIDAS OLS) and the hierarchical
estimator (p-RU-MIDAS HIER), and this for the different macroeconomic
variables (columns) and the six forecast horizons. For each horizon, forecast
methods in the 75% Model Confidence Set (MCS) are in bold.

Model Low-frequency variables None Cpi Csent Epui Fedfunds Houst Indpro M2 Oil
Termsp Unrate Vix All Horizon $h=1$ HAR 0.4866 p-RU-MIDAS OLS 0.4904 0.4899
0.4909 0.4976 0.4907 0.4923 0.4930 0.4921 0.4927 0.4915 0.4927 0.5608 p-RU-
MIDAS HIER 0.4894 0.4894 0.4894 0.4894 0.4894 0.4894 0.4894 0.4894 0.4894
0.4894 0.4894 0.4894 Horizon $h=5$ HAR 0.6097 p-RU-MIDAS OLS 0.6170 0.6189
0.6170 0.6253 0.6166 0.6168 0.6208 0.6184 0.6183 0.6147 0.6191 0.6834 p-RU-
MIDAS HIER 0.6210 0.6210 0.6211 0.6210 0.6211 0.6210 0.6210 0.6210 0.6210
0.6210 0.6217 0.6215 Horizon $h=20$ HAR 0.7398 p-RU-MIDAS OLS 0.7494 0.7506
0.7400 0.7565 0.7601 0.7473 0.7499 0.7455 0.7491 0.7433 0.7518 0.8325 p-RU-
MIDAS HIER 0.7526 0.7526 0.7526 0.7526 0.7677 0.7526 0.7526 0.7526 0.7526
0.7575 0.7527 0.7526 Horizon $h=40$ HAR 0.8131 p-RU-MIDAS OLS 0.8171 0.8229
0.8095 0.8201 0.8463 0.8188 0.8282 0.8184 0.8230 0.8208 0.8312 0.9541 p-RU-
MIDAS HIER 0.8259 0.8271 0.8260 0.8271 0.8528 0.8270 0.8259 0.8271 0.8271
0.8259 0.8237 0.8238 Horizon $h=60$ HAR 0.8487 p-RU-MIDAS OLS 0.8535 0.8577
0.8448 0.8438 0.9007 0.8596 0.8630 0.8555 0.8512 0.8581 0.8850 1.0103 p-RU-
MIDAS HIER 0.8534 0.8534 0.8534 0.8534 0.9013 0.8533 0.8534 0.8534 0.8534
0.8533 0.8533 0.8532 Horizon $h=120$ HAR 0.9850 p-RU-MIDAS OLS 0.9847 0.9925
0.9728 0.9170 1.0880 1.0185 0.9997 0.9914 0.9547 1.0082 1.0829 1.0924 p-RU-
MIDAS HIER 0.9726 0.9726 0.9758 0.9669 1.0814 0.9727 0.9726 0.9726 0.9748
0.9831 0.9756 1.0934

Table 3: Percentage of rolling windows in which at least one low-frequency
macroeconomic coefficient was selected (i.e. estimated as nonzero) across the
six forecast horizons (rows).

Horizon Cpi Csent Epui Fedfunds Houst Indpro M2 Oil Termsp Unrate Vix All
$h=1$ 0 0 0 0 0 0 0 0 0 0 0 0 $h=5$ 0 0 0 0 0.17 0 0 0 0 0 0 0 $h=20$ 0 0 0 0
5.08 0 0 0 0 1.69 0 0 $h=40$ 0 0 0 0 4.57 0.31 0 0.43 0.57 0 0 0 $h=60$ 0 0 0
0 5.78 0 0 0 0 0 0 0.35 $h=120$ 0 0 3.44 10.74 22.74 0.57 0 0 0.87 1.20 0.60
6.55

We start by discussing the results for horizon $h=1$. Here, many combinations
are included in the MCS with the HAR being the best performing one (on
average). However, care is needed when evaluating the performance of the
mixed-frequency models with macroeconomic variables included in the MCS vis-a-
vis the HAR. Indeed, Table 3 reveals that the pooled RU-MIDAS models do not
include any macroeconomic variables, since all their corresponding
coefficients are set to zero by the regularized estimator. Hence, the HAR
models and the pooled RU-MIDAS models only differ from each other in terms of
the HF lag structure. The HAR model includes the HF lags via the parsimonious
dwm lag structure, which cannot be improved upon for short forecast horizon.
While the pooled RU-MIDAS model estimated by OLS includes all 20 HF lags, the
pooled RU-MIDAS model estimated by the hierarchical estimator includes on
average around 4 HF lags (min. 3 lags, max. 6 lags) in the model. Our results
confirm the previous findings by Audrino and Knaus, (2016), Audrino et al.,
(2017), and Wilms et al., (2021) on the limited evidence that the more general
lag structure allowed by lasso-based approaches can beat the parsimonious day-
week-month lag structure embedded in the HAR in terms of out-of-sample
forecast performance.

Across other forecast horizons, the HAR maintains its top performance. Hence,
we do not find evidence that the considered LF macroeconomic variables contain
predictive information for forecasting the daily S&P 500 daily realized
variance. The only exception is $h=120$, where there is a small gain in terms
of average MAFE over the HAR when using the pooled RU-MIDAS models various
macroeconomic variables. It should, however, be noted that only in a minority
of rolling windows (see Table 3) this improved forecast accuracy over the HAR
stems from the inclusion of macroeconomic variables.

Finally, let us discuss the performance of the pooled RU-MIDAS models
estimated by OLS and the hierarchical estimator. We find that their
performance is, overall, largely similar. The reason for that is that we
employ a rolling window of five years to have sufficient low-frequency
observations. However, with only one lag of a single low-frequency variable
and twenty high-frequency lags included in the model, we need to estimate 40
coefficients while having a sample size of $T=1200$ high-frequency
observations. Hence, the curse of dimensionality is rather limited in our set-
up. With the current rolling-window set-up, the benefit of using the
regularizer only becomes noticeable for the setting where we include “all”
macroeconomic variables. Indeed, in such settings, the number of parameters
that we need to estimate increases to 240 and we observe a noticeable
improvement of using a regularization method (see last column in Table 2). A
similar advantage of the regularized pooled RU-MIDAS model over the OLS-based
one can be observed if one were to use a smaller rolling window size, for
instance of 2 years. Detailed results of this analysis are included in Table 4
of the Appendix A.

Overall, we can draw two main conclusions from our empirical analysis. First,
macroeconomic variables do not help to improve the forecast performance of the
daily S&P 500 realized variance over the simple HAR. This finding is robust
across a wide range of macroeconomic variables, forecast horizons as well as
rolling window sizes (i.e. Table 2 reports results for 5 years, while results
for 2 and 10 years are available in Appendix A).999Note that for rolling
window of 10 years, the advantage of using a regularized method in contrast to
simple OLS largely disappears due to the sample size. Moreover, there is a
minor benefit of including macroeconomic variables, as evidenced by the pooled
RU-MIDAS estimated by OLS often being included in the MCS together with the
HAR. Second, as expected, the advantage of using a regularized pooled RU-MIDAS
model over an OLS-based one increases when the dimensionality of the model
increases, either due to the inclusion of more predictors or due to a
reduction in effective sample size.

##  Conclusion

We consider mixed-frequency RU-MIDAS regressions with a large frequency
mismatch $m$ and propose two extensions to the framework to tackle their curse
of dimensionality. We (i) pool the regression coefficients regulating the
lagged HF dynamics across the $m$ different HF equations to increase the
sample size and (ii) we use a hierarchical regularizer to discriminate between
recent and obsolete information. We use the proposed regularized pooled RU-
MIDAS model to forecast the daily realized volatility of the S&P 500 through
the information contained in a diverse set of monthly macroeconomic variables.
While we find that the pooled RU-MIDAS model improves forecast performance
upon the original (non-pooled) one, we do not find evidence that the monthly
macroeconomic variables contain important information to improve forecast
performance upon the popular benchmark HAR model that only exploits past
volatility information.

The RU-MIDAS model with hierarchical sparsity structures is quite flexible and
can easily incorporate other lag priority structures. For instance, if
practitioners prefer the parsimonious dwm-lag structure used in the HAR,
hierarchical sparsity penalties can be applied with minimal adaptation to
prioritize the daily lag over the weekly lag, which in turn could be
prioritized over the monthly lag. Table 6 in Appendix A presents the MAFEs for
a regularized pooled RU-MIDAS with such dwm-lag structure and compares is to
the same model estimated by OLS. Our results remain robust to the usage of lag
structure.

## References

  * Andersen et al., (2001) Andersen, T. G., Bollerslev, T., Diebold, F. X., and Ebens, H. (2001).  The distribution of realized stock return volatility.  Journal of Financial Economics, 61(1):43–76.
  * Andreou et al., (2019) Andreou, E., Gagliardini, P., Ghysels, E., and Rubin, M. (2019).  Inference in group factor models with an application to mixed-frequency data.  Econometrica, 87(4):1267–1305.
  * Asgharian et al., (2013) Asgharian, H., Hou, A. J., and Javed, F. (2013).  The importance of the macroeconomic variables in forecasting stock return variance: A GARCH-MIDAS approach.  Journal of Forecasting, 32(7):600–612.
  * Audrino et al., (2017) Audrino, F., Camponovo, L., and Roth, C. (2017).  Testing the lag structure of assets’ realized volatility dynamics.  Quantitative Finance and Economics, 1(4):363–387.
  * Audrino and Knaus, (2016) Audrino, F. and Knaus, S. D. (2016).  Lassoing the HAR model: A model selection perspective on realized volatility dynamics.  Econometric Reviews, 35(8-10):1485–1521.
  * (6) Babii, A., Ball, R. T., Ghysels, E., and Striaukas, J. (2022a).  Machine learning panel data regressions with heavy-tailed dependent data: Theory and application.  Journal of Econometrics, forthcoming.
  * (7) Babii, A., Ghysels, E., and Striaukas, J. (2022b).  Machine learning time series regressions with an application to nowcasting.  Journal of Business & Economic Statistics, 40(3):1094–1106.
  * Baker et al., (2016) Baker, S. R., Bloom, N., and Davis, S. J. (2016).  Measuring economic policy uncertainty.  The quarterly Journal of Economics, 131(4):1593–1636.
  * Barndorff-Nielsen and Shephard, (2002) Barndorff-Nielsen, O. E. and Shephard, N. (2002).  Econometric analysis of realized volatility and its use in estimating stochastic volatility models.  Journal of the Royal Statistical Society: Series B (Statistical Methodology), 64(2):253–280.
  * Bauwens et al., (2012) Bauwens, L., Hafner, C. M., and Laurent, S. (2012).  Handbook of volatility models and their applications, volume 3.  John Wiley & Sons.
  * Belloni and Chernozhukov, (2013) Belloni, A. and Chernozhukov, V. (2013).  Least squares after model selection in high-dimensional sparse models.  Bernoulli, 19(2):521–547.
  * Bouri et al., (2022) Bouri, E., Gupta, R., Rossini, L., et al. (2022).  The role of the monthly ENSO in forecasting the daily Baltic Dry Index.  Technical report.
  * Cimadomo et al., (2022) Cimadomo, J., Giannone, D., Lenza, M., Monti, F., and Sokol, A. (2022).  Nowcasting with large Bayesian vector autoregressions.  Journal of Econometrics, 231(2):500–519.
  * Conrad and Kleen, (2020) Conrad, C. and Kleen, O. (2020).  Two are better than one: Volatility forecasting using multiplicative component GARCH-MIDAS models.  Journal of Applied Econometrics, 35(1):19–45.
  * Conrad and Loch, (2015) Conrad, C. and Loch, K. (2015).  Anticipating long-term stock market volatility.  Journal of Applied Econometrics, 30(7):1090–1114.
  * Corsi, (2009) Corsi, F. (2009).  A simple approximate long-memory model of realized volatility.  Journal of Financial Econometrics, 7(2):174–196.
  * Dal Bianco et al., (2012) Dal Bianco, M., Camacho, M., and Quiros, G. P. (2012).  Short-run forecasting of the euro-dollar exchange rate with economic fundamentals.  Journal of International Money and Finance, 31(2):377–396.
  * Diebold and Mariano, (1995) Diebold, F. X. and Mariano, R. S. (1995).  Comparing predictive accuracy.  Journal of Business & Economic Statistics, 13(3):134–144.
  * Ding et al., (2021) Ding, Y., Kambouroudis, D., and McMillan, D. G. (2021).  Forecasting realised volatility: Does the LASSO approach outperform HAR?  Journal of International Financial Markets, Institutions and Money, 74:101386.
  * Engle et al., (2013) Engle, R. F., Ghysels, E., and Sohn, B. (2013).  Stock market volatility and macroeconomic fundamentals.  Review of Economics and Statistics, 95(3):776–797.
  * Fang et al., (2020) Fang, T., Lee, T.-H., and Su, Z. (2020).  Predicting the long-term stock market volatility: A GARCH-MIDAS model with variable selection.  Journal of Empirical Finance, 58:36–49.
  * Foroni et al., (2018) Foroni, C., Guérin, P., and Marcellino, M. (2018).  Using low frequency information for predicting high frequency variables.  International Journal of Forecasting, 34(4):774–787.
  * Foroni and Marcellino, (2014) Foroni, C. and Marcellino, M. (2014).  A comparison of mixed frequency approaches for nowcasting Euro area macroeconomic aggregates.  International Journal of Forecasting, 30(3):554–568.
  * Foroni et al., (2015) Foroni, C., Marcellino, M., and Schumacher, C. (2015).  Unrestricted mixed data sampling (MIDAS): MIDAS regressions with unrestricted lag polynomials.  Journal of the Royal Statistical Society: Series A (Statistics in Society), 178(1):57–82.
  * Foroni et al., (2023) Foroni, C., Ravazzolo, F., and Rossini, L. (2023).  Are low frequency macroeconomic variables important for high frequency electricity prices?  Economic Modelling, 120:106160.
  * Ghysels, (2016) Ghysels, E. (2016).  Macroeconomics and the reality of mixed frequency data.  Journal of Econometrics, 193(2):294–314.
  * Ghysels et al., (2004) Ghysels, E., Santa-Clara, P., and Valkanov, R. (2004).  The MIDAS touch: Mixed data sampling regression models.  Cirano working papers, CIRANO.
  * Götz et al., (2016) Götz, T. B., Hecq, A., and Smeekes, S. (2016).  Testing for Granger causality in large mixed-frequency VARs.  Journal of Econometrics, 193(2):418–432.
  * Han and Tsay, (2020) Han, Y. and Tsay, R. S. (2020).  High-dimensional linear regression for dependent data with applications to nowcasting.  Statistica Sinica, 30(4):1797–1827.
  * Hansen et al., (2011) Hansen, P. R., Lunde, A., and Nason, J. M. (2011).  The model confidence set.  Econometrica, 79(2):453–497.
  * Hastie et al., (2015) Hastie, T., Tibshirani, R., and Wainwright, M. (2015).  Statistical learning with sparsity: the lasso and generalizations.  Chapman and Hall/CRC Press.
  * Haugom et al., (2014) Haugom, E., Langeland, H., Molnár, P., and Westgaard, S. (2014).  Forecasting volatility of the US oil market.  Journal of Banking & Finance, 47:1–14.
  * Hecq et al., (2022) Hecq, A., Ternes, M., and Wilms, I. (2022).  Hierarchical regularizers for mixed-frequency vector autoregressions.  Journal of Computational and Graphical Statistics, 31(4):1076–1090.
  * Hua and Manzan, (2013) Hua, J. and Manzan, S. (2013).  Forecasting the return distribution using high-frequency volatility measures.  Journal of Banking & Finance, 37(11):4381–4403.
  * Koelbl and Deistler, (2020) Koelbl, L. and Deistler, M. (2020).  A new approach for estimating VAR systems in the mixed-frequency case.  Statistical Papers, 61(3):1203–1212.
  * Marcellino and Schumacher, (2010) Marcellino, M. and Schumacher, C. (2010).  Factor MIDAS for nowcasting and forecasting with ragged-edge data: A model comparison for German GDP.  Oxford Bulletin of Economics and Statistics, 72(4):518–550.
  * Marcellino et al., (2006) Marcellino, M., Stock, J. H., and Watson, M. W. (2006).  A comparison of direct and iterated multistep AR methods for forecasting macroeconomic time series.  Journal of Econometrics, 135(1-2):499–526.
  * Mariano and Murasawa, (2003) Mariano, R. S. and Murasawa, Y. (2003).  A new coincident index of business cycles based on monthly and quarterly series.  Journal of Applied Econometrics, 18(4):427–443.
  * McCracken et al., (2021) McCracken, M., Owyang, M., and Sekhposyan, T. (2021).  Real-time forecasting and scenario analysis using a large mixed-frequency Bayesian VAR.  International Journal of Central Banking, 18(5):327–367.
  * Mogliani and Simoni, (2021) Mogliani, M. and Simoni, A. (2021).  Bayesian MIDAS penalized regressions: estimation, selection, and prediction.  Journal of Econometrics, 222(1):833–860.
  * Nicholson et al., (2020) Nicholson, W. B., Wilms, I., Bien, J., and Matteson, D. S. (2020).  High dimensional forecasting via interpretable vector autoregression.  Journal of Machine Learning Research, 21:1–52.
  * Paccagnini and Parla, (2021) Paccagnini, A. and Parla, F. (2021).  Identifying high-frequency shocks with Bayesian mixed-frequency VARs.  Working paper No. 26/2021, CAMA.
  * Schorfheide and Song, (2015) Schorfheide, F. and Song, D. (2015).  Real-time forecasting with a mixed-frequency VAR.  Journal of Business & Economic Statistics, 33(3):366–380.
  * Tibshirani, (1996) Tibshirani, R. (1996).  Regression shrinkage and selection via the lasso.  Journal of the Royal Statistical Society: Series B (Methodological), 58(1):267–288.
  * Tsay, (2005) Tsay, R. S. (2005).  Analysis of financial time series.  John Wiley & Sons.
  * Wilms et al., (2021) Wilms, I., Rombouts, J., and Croux, C. (2021).  Multivariate volatility forecasts for stock market indices.  International Journal of Forecasting, 37(2):484–499.
  * Zhao et al., (2009) Zhao, P., Rocha, G., and Yu, B. (2009).  The composite absolute penalties family for grouped and hierarchical variable selection.  The Annals of Statistics, 37(6A):3468–3497.

##  Appendix A Additional Results

Table 4: Sensitivity analysis for a rolling window of 2 years. Mean absolute
forecast errors, averaged across time, of the HAR, the pooled RU-MIDAS
estimated by OLS and the hierarchical estimator, and this for the different
macroeconomic variables (columns) and the six forecast horizons. For each
horizon, forecast methods in the 75% Model Confidence Set (MCS) are in bold.

Model Low-frequency variables None Cpi Csent Epui Fedfunds Houst Indpro M2 Oil
Termsp Unrate Vix All Horizon $h=1$ HAR 0.4672 p-RU-MIDAS OLS 0.4849 0.4833
0.4863 0.4890 0.4850 0.4851 0.4854 0.4867 0.4840 0.4833 0.4861 0.6944 p-RU-
MIDAS HIER 0.4714 0.4714 0.4714 0.4714 0.4714 0.4714 0.4714 0.4714 0.4714
0.4714 0.4714 0.4715 Horizon $h=5$ HAR 0.5812 p-RU-MIDAS OLS 0.6071 0.6047
0.6054 0.6066 0.6019 0.6083 0.6036 0.6088 0.6041 0.6016 0.6042 0.8594 p-RU-
MIDAS HIER 0.5973 0.5974 0.5973 0.5974 0.5973 0.5973 0.5973 0.5973 0.5973
0.5973 0.5973 0.5973 Horizon $h=20$ HAR 0.7154 p-RU-MIDAS OLS 0.7395 0.7371
0.7333 0.7501 0.7532 0.7354 0.7308 0.7379 0.7405 0.7399 0.7440 1.0054 p-RU-
MIDAS HIER 0.7264 0.7264 0.7264 0.7264 0.7264 0.7264 0.7264 0.7264 0.7264
0.7264 0.7264 0.7264 Horizon $h=40$ HAR 0.7940 p-RU-MIDAS OLS 0.8111 0.8161
0.8236 0.8439 0.8305 0.8072 0.8186 0.8175 0.8170 0.8197 0.8204 1.1524 p-RU-
MIDAS HIER 0.7920 0.7920 0.7920 0.7920 0.7920 0.7920 0.7920 0.7920 0.7920
0.7920 0.7920 0.7920 Horizon $h=60$ HAR 0.8342 p-RU-MIDAS OLS 0.8807 0.8557
0.8737 0.8748 0.8722 0.8694 0.8791 0.8539 0.8639 0.8683 0.8878 1.2647 p-RU-
MIDAS HIER 0.8321 0.8321 0.8321 0.8321 0.8273 0.8321 0.8321 0.8321 0.8321
0.8321 0.8321 0.8321 Horizon $h=120$ HAR 1.0250 p-RU-MIDAS OLS 1.0351 1.0312
1.0337 0.9833 1.0328 1.0889 1.0512 1.0417 1.0106 1.0691 1.1109 1.3989 p-RU-
MIDAS HIER 0.9646 0.9646 0.9646 0.9646 1.0198 0.9646 0.9646 0.9646 0.9646
0.9646 0.9646 0.9628

Table 5: Sensitivity analysis for a rolling window of 10 years. Mean absolute
forecast errors, averaged across time, of the HAR, the pooled RU-MIDAS
estimated by OLS and the hierarchical estimator, and this for the different
macroeconomic variables (columns) and the six forecast horizons. For each
horizon, forecast methods in the 75% Model Confidence Set (MCS) are in bold.

Model Low-frequency variables None Cpi Csent Epui Fedfunds Houst Indpro M2 Oil
Termsp Unrate Vix All Horizon $h=1$ HAR 0.5160 p-RU-MIDAS OLS 0.5161 0.5161
0.5156 0.5169 0.5154 0.5162 0.5173 0.5174 0.5166 0.5166 0.5156 0.5376 p-RU-
MIDAS HIER 0.5190 0.5190 0.5190 0.5190 0.5190 0.5190 0.5190 0.5190 0.5190
0.5190 0.5191 0.5191 Horizon $h=5$ HAR 0.6537 p-RU-MIDAS OLS 0.6559 0.6600
0.6567 0.6583 0.6533 0.6565 0.6593 0.6591 0.6564 0.6581 0.6546 0.6668 p-RU-
MIDAS HIER 0.6617 0.6617 0.6617 0.6617 0.6617 0.6617 0.6617 0.6617 0.6617
0.6617 0.6617 0.6618 Horizon $h=20$ HAR 0.7733 p-RU-MIDAS OLS 0.7751 0.7812
0.7698 0.7739 0.7714 0.7774 0.7746 0.7736 0.7762 0.7712 0.7717 0.7675 p-RU-
MIDAS HIER 0.7877 0.7877 0.7878 0.7877 0.7878 0.7870 0.7877 0.7877 0.7877
0.7878 0.7884 0.7876 Horizon $h=40$ HAR 0.8403 p-RU-MIDAS OLS 0.8395 0.8482
0.8285 0.8408 0.8413 0.8441 0.8449 0.8457 0.8421 0.8389 0.8374 0.8357 p-RU-
MIDAS HIER 0.8456 0.8452 0.8458 0.8452 0.8468 0.8478 0.8452 0.8458 0.8452
0.8478 0.8478 0.8483 Horizon $h=60$ HAR 0.8867 p-RU-MIDAS OLS 0.8856 0.8944
0.8738 0.8831 0.8917 0.8917 0.8858 0.8900 0.8822 0.8846 0.8819 0.8829 p-RU-
MIDAS HIER 0.8926 0.8926 0.8925 0.8925 0.8925 0.8925 0.8925 0.8926 0.8926
0.8926 0.8924 0.8925 Horizon $h=120$ HAR 0.9355 p-RU-MIDAS OLS 0.9238 0.9372
0.8929 0.9318 0.9554 0.9437 0.9286 0.9377 0.9179 0.9356 0.9392 0.8779 p-RU-
MIDAS HIER 0.9514 0.9514 0.9576 0.9443 0.9448 0.9514 0.9514 0.9514 0.9514
0.9514 0.9515 0.9515

Table 6: Mean absolute forecast errors, averaged across time, of the HAR, the
pooled RU-MIDAS with a dwm-lag structure estimated by OLS (p-MF-dwm OLS) and
the hierarchical estimator (p-MF-dwm HIER), and this for the different
macroeconomic variables (columns), the six forecast horizons and a rolling
window of 5 years. For each horizon, forecast methods in the 75% Model
Confidence Set (MCS) are in bold.

Model Low-frequency variables None Cpi Csent Epui Fedfunds Houst Indpro M2 Oil
Termsp Unrate Vix All Horizon $h=1$ HAR 0.4866 p-MF-dwm OLS 0.4894 0.4884
0.4900 0.4961 0.4892 0.4913 0.4920 0.4911 0.4915 0.4901 0.4909 0.5572 p-MF-dwm
HIER 0.4870 0.4869 0.4871 0.4871 0.4870 0.4869 0.4869 0.4870 0.4869 0.4869
0.4869 0.4871 Horizon $h=5$ HAR 0.6097 p-MF-dwm OLS 0.6142 0.6147 0.6131
0.6218 0.6138 0.6136 0.6166 0.6148 0.6144 0.6114 0.6165 0.6808 p-MF-dwm HIER
0.6102 0.6102 0.6107 0.6112 0.6105 0.6100 0.6102 0.6102 0.6101 0.6102 0.6102
0.6108 Horizon $h=20$ HAR 0.7398 p-MF-dwm OLS 0.7472 0.7475 0.7377 0.7544
0.7581 0.7446 0.7477 0.7429 0.7471 0.7412 0.7489 0.8289 p-MF-dwm HIER 0.7407
0.7416 0.7407 0.7407 0.7507 0.7408 0.7407 0.7407 0.7404 0.7454 0.7407 0.7409
Horizon $h=40$ HAR 0.8131 p-MF-dwm OLS 0.8172 0.8222 0.8105 0.8183 0.8463
0.8182 0.8270 0.8178 0.8221 0.8192 0.8317 0.9511 p-MF-dwm HIER 0.8167 0.8166
0.8167 0.8167 0.8361 0.8175 0.8166 0.8167 0.8170 0.8167 0.8161 0.8177 Horizon
$h=60$ HAR 0.8487 p-MF-dwm OLS 0.8529 0.8561 0.8436 0.8425 0.9017 0.8583
0.8621 0.8547 0.8495 0.8571 0.8840 1.0050 p-MF-dwm HIER 0.8509 0.8521 0.8515
0.8509 0.8817 0.8509 0.8507 0.8524 0.8505 0.8537 0.8512 0.8516 Horizon $h=120$
HAR 0.9850 p-MF-dwm OLS 0.9785 0.9854 0.9694 0.9121 1.0851 1.0127 0.9943
0.9848 0.9516 1.0015 1.0778 1.0896 p-MF-dwm HIER 0.9884 0.9883 0.9769 0.9497
1.0902 0.9886 0.9885 0.9883 0.9817 1.0009 1.0126 1.0878