# Variable Selection in High Dimensional Linear Regressions with Parameter
Instability††thanks: We are grateful to Elie Tamer (Editor), two anonymous
reviewers and an associate editor, for their constructive comments and helpful
suggestions. We have also benefited from discussions and comments by George
Kapetanios, Oliver Linton, Ron Smith, and seminar participants at Cambridge
University. The views expressed in this paper are those of the authors and do
not necessarily reflect those of the Federal Reserve Bank of Dallas or the
Federal Reserve System. This research was supported in part through
computational resources provided by the Big-Tex High Performance Computing
Group at the Federal Reserve Bank of Dallas. This paper in part was written
when Sharifvaghefi was a doctoral student at the University of Southern
California (USC). Sharifvaghefi gratefully acknowledges financial support from
the Center for Applied Financial Economics at USC.

Alexander Chudik
Federal Reserve Bank of Dallas, Dallas, USA M. Hashem Pesaran
University of Southern California, Los Angeles, USA and Trinity College,
Cambridge, UK Mahrad Sharifvaghefi
University of Pittsburgh, Pittsburgh, USA Corresponding author. Postal
address: 230 S Bouquet St., Pittsburgh, PA, USA, 15260. Email:
<EMAIL_ADDRESS>
###### Abstract

This paper is concerned with the problem of variable selection in the presence
of parameter instability when both the marginal effects of signals on the
target variable and the correlations of the covariates in the active set could
vary over time. We pose the issue of whether one should use weighted or
unweighted observations at the variable selection stage in the presence of
parameter instability, particularly when the number of potential covariates is
large. We allow parameter instability to be continuous or discrete, subject to
certain regularity conditions. We discuss the pros and cons of Lasso and the
One Covariate at a time Multiple Testing (OCMT) method for variable selection
and argue that OCMT has important advantages under parameter instability. We
establish three main theorems on selection, estimation post selection, and in-
sample fit. These theorems provide justification for using unweighted
observations at the selection stage of OCMT and down-weighting of observations
only at the forecasting stage. It is shown that OCMT delivers better
forecasts, in mean squared error sense, as compared to Lasso, Adaptive Lasso
and boosting both in Monte Carlo experiments as well as in 3 sets of empirical
applications: forecasting monthly returns on 28 stocks from Dow Jones ,
forecasting quarterly output growths across 33 countries, and forecasting euro
area output growth using surveys of professional forecasters.

Keywords: Lasso, One Covariate at a time Multiple Testing, OCMT, Parameter
instability, Variable selection, Forecasting

JEL Classifications: C22, C52, C53, C55

##  1 Introduction

There is mounting evidence that models fitted to many statistical
relationships are subject to parameter instabilities. In an extensive early
study, Stock and Watson (1996) find that a large majority of time series
regressions in economics are subject to breaks. Clements and Hendry (1998)
consider parameter instability to be one of the main sources of forecast
failure. This problem has been addressed at the estimation/forecasting stage
for a given set of selected regressors. However, the theory of variable
selection in the presence of time-varying parameters is still largely
underdeveloped. In this study, we investigate whether unweighted or weighted
observations should be used at the variable selection stage when it is known
or suspected that the parameters are subject to change. To address this issue
we allow the marginal effects of signals on the target variable as well as the
correlation of the covariates in the active set to vary over time, without
committing to any particular model of parameter instabilities. We provide
theoretical arguments in favor of using unweighted observations at the
selection stage, and recommend that one should only consider weighting the
observations post selection, at the estimation and forecasting stages.

For a given model specification, many different approaches have been proposed
in the literature, basically trading off bias and efficiency of forecasts by
considering differing estimation windows or down-weighting schemes. Typical
solutions are either to use rolling windows or exponential down-weighting. For
instance, Pesaran and Timmermann (2007), Pesaran and Pick (2011) and Inoue et
al. (2017) consider the choice of an observation window, and Hyndman et al.
(2008) and Pesaran et al. (2013), respectively consider exponential and non-
exponential down-weighting of the observations. There are also Bayesian
approaches to prediction that allow for the possibility of breaks over the
forecast horizon, e.g. Chib (1998), Koop and Potter (2004), and Pesaran et al.
(2006). Rossi (2013) provides a review of the literature on forecasting under
instability. There are also related time varying parameter (TVP) and regime
switching models that are used for forecasting. See, for example, Hamilton
(1988) and Dangl and Halling (2012). All these studies take the model
specification as given and then consider different ways of modeling and
allowing for parameter instability. But, to the best of our knowledge, none of
these studies considers the problem of variable selection in the presence of
parameter instability.

In the case of models with stable parameters it is reasonable to weight the
observations equally at variable selection and post variable selection stages
such as for inference or forecasting. However, under parameter instability
there is little discussion in the literature as to whether weighted or
unweighted observations should be used at the variable selection stage,
particularly when the number of potential covariates is large. There are a
number of recent studies that use machine learning techniques to allow for
parameter instability, in particular penalized regression, especially the
Least Absolute Shrinkage and Selection Operator (Lasso) initially proposed by
Tibshirani (1996). For example, Caner and Knight (2013) and Koo et al. (2020)
suggest recursive application of Lasso using rolling windows. Qian and Su
(2016) consider a linear regression model with a finite number of covariates
but allow for an unknown number of breaks and use group fused Lasso due to
Alaíz et al. (2013) to consistently estimate the number of breaks and their
locations. Lee et al. (2016) have proposed a Lasso procedure that allows for
threshold effects. Kapetanios and Zikes (2018) have proposed a time-varying
Lasso procedure, where all the parameters of the model vary locally. Fan et
al. (2014) suggest an extension of the screening procedure initially proposed
by Fan and Lv (2008) to the case where the regression coefficients vary
smoothly with an observable exposure variable. Also recently, Yousuf and Ng
(2021) propose an interesting boosting procedure for the estimation of high-
dimensional models with locally time varying parameters. It is important to
note that, in the case of both penalized regression and boosting procedures,
variable selection and estimation are carried out simultaneously in a single
step.

Chudik et al. (2018) propose an alternative procedure called one covariate at
a time multiple testing (OCMT), based on the statistical significance of the
net effect of the covariates in the active set on the target variable, one-at-
a-time subject to the multiple testing nature of the inferential problem
involved. The idea of using one-at-a-time regressions is not unique to OCMT
and has been used in boosting as well as in screening approaches. See, for
example, Bühlmann (2006) and Fan and Lv (2018) as prominent examples of these
approaches. What is unique about the OCMT procedure is its inferentially
motivated stopping rule without resorting to the use of information criteria,
or penalized regression after the initial stage. In the case of models with
stable parameters, Chudik et al. (2018) establish that OCMT selects an
approximating model that includes all the signals and pseudo-signals that are
correlated with the target variable and none of the noise variables. They then
provide convergence rates for consistency of the regression error and
coefficient norms of the selected model. Finally, using Monte Carlo studies,
they show that OCMT tends to perform better than penalized regression or
boosting procedures under various designs.

Lasso and OCMT exploit different aspects of the low-dimensional structure
assumed for the underlying data generating process. Lasso restricts the
magnitude of the correlations within signals as well as the correlations
between signals and the remaining covariates under consideration. OCMT limits
the rate at which the number of pseudo-signals, $k_{T}^{\ast}$, rises with the
sample size, $T$. Under parameter stability, the variable selection
consistency of Lasso has been investigated by Zhao and Yu (2006), Meinshausen
and Bühlmann (2006) and more recently by Lahiri (2021). These conditions, and
how they compare with the conditions that underlie OCMT, are discussed in
Section 4 of the paper. Although Lasso does not directly impose any
restrictions on $k_{T}^{\ast}$, its Irrepresentable Condition (IRC), by
restricting the magnitude of correlations within and between the signals and
pseudo-signals, does have implications for the number of pseudo-signals that
Lasso selects. OCMT requires $k_{T}^{\ast}$ not to rise faster than
$\sqrt{T}$. When this condition is violated, then the true signals must end up
as common factors for the pseudo-signals, and what matters is the number of
residuals (from the regressions of pseudo-signals on the common factors) that
are correlated with the residuals of the true signals from the same set of
common factors. Sharifvaghefi (2023) shows that such common factors can be
estimated from the principal components of the covariates in the active set
and the OCMT condition on the number pseudo-signals, now defined in terms of
the correlation of the residuals, is no longer restrictive.111Another
extension of OCMT is provided by Su et al. (2023) who allow for unknown
potentially non-linear relationship between the signals and the target
variable. Once the approximating model is selected, Theorem 2 of the paper
establishes the rate at which the estimated regression coefficients of the
approximating model tends to the true coefficients, and hence a model
consistent outcome. This theorem shows that the convergence rate depends on
$d$, defined by $k_{T}^{\ast}=\ominus(T^{d})$, and the regular convergent rate
of $\sqrt{T}$ is achieved only if $d=0$. A similar issue also arises for
Lasso, as shown by Lahiri (2021) who establishes that the Lasso procedure
cannot achieve both variable selection consistency and $\sqrt{T}$-consistency
in coefficient estimation. Another feature of OCMT is that it allows for a
natural separation of the problem of variable selection from the problems of
estimation, inference and forecasting. As noted above, the focus of the
present paper is on the application of OCMT to variable selection in the
presence of parameter instability, broadly defined. To the best of our
knowledge, there are no studies that investigate the variable selection
properties of Lasso under parameter instability.

To take account of the time variations in the coefficients of the signals, we
consider their time averages and distinguish between strong signals whose
average marginal effects tend to a non-zero value, semi-strong signals whose
average marginal effects tend to zero, but sufficiently slow, and weak signals
whose average marginal effects tend to zero quite fast. In this way we allow
for variety of time variations that could arise in practice. Strong signals
tend to have non-zero effects at all times, semi-strong signals could have
zero effects during some periods, with weak signals enter the model relatively
rarely. Weak signals are often indistinguishable from noise variables. In our
theoretical analysis we will focus on selection of strong and semi-strong
signals.

We provide three main theorems in support of our proposed variable selection
method. Under certain fairly general regularity conditions we show that the
probability of OCMT selecting the true approximating model that contains all
the signals (strong and semi-strong) and none of the noise variables tends to
unity as the number of time series observations ($T$) tends to infinity. Our
results apply both when $N$ (the number of covariates in the active set) is
fixed as well as when $N$ tends to infinity jointly with $T$, covering the
case where $N\gg T$. We also establish conditions under which (a) least
squares estimates of the coefficients of selected covariates will tend to zero
unless they are signals, and (b) the average squared residuals of the selected
model achieves the oracle rate for regression models with time-varying
coefficients. These theoretical findings provide a formal justification for
application of statistical techniques from the time-varying parameters
literature to the post OCMT selected model. Our Monte Carlo experiments
indicate that the OCMT procedure with weighted observations only at the
estimation stage has appealing finite-sample performance relative to Lasso and
Adaptive Lasso (A-Lasso by Zou (2006)), as well as boosting by Bühlmann
(2006), under many different settings. We also found that Lasso and A-Lasso
consistently outperform boosting.

Finally, we provide three empirical applications, forecasting monthly rates of
price changes of 28 stocks in Dow Jones using large number of financial,
economic and technical indicators, forecasting output growths across 33
countries using a large number of macroeconomic indicators, and forecasting
euro area output growth using ECB surveys of 25 professional forecasters. To
save space the third application is included in the online supplement. We
generate a large number of forecasts using OCMT with and without down-
weighting of the observations at the selection stage and compare the results
with the forecasts obtained using Lasso, A-Lasso and boosting. The empirical
results are in line with our theoretical and MC findings and suggest that
using down-weighted observations at the selection stage of the OCMT procedure
worsens forecast performance in terms of mean square forecast error and mean
directional forecast accuracy. Overall, based on the empirical results we also
find that OCMT with no down-weighting at the selection stage outperforms
boosting, Lasso and A-Lasso.

The rest of the paper is organized as follows: Section 2 sets out the model
specification. Section 3 explains the basic idea behind the OCMT procedure for
variable selection without down-weighting in the presence of parameter
instability. Section 4 provides a discussion of key assumptions of Lasso and
OCMT under parameter stability. Section 5 discusses the technical assumptions
and the asymptotic properties of the OCMT procedure under parameter
instability. Section 6 provides the details of the Monte Carlo experiments and
a summary of the main results. Section 7 presents the empirical applications,
and Section 8 concludes. The paper is also accompanied with three online
supplements. A theory supplement contains the mathematical proofs of the
theorems and related lemmas. A Monte Carlo supplement provides additional
summary tables, the full set of Monte Carlo results, as well as the
description of the algorithms used for Lasso, A-Lasso and boosting. Further
details of the empirical applications are given in an empirical supplement.

Notations: Generic finite positive constants are denoted by $C_{i}$ for
$i=1,2,\cdots$. $\lVert\mathbf{A}\rVert_{2}$ and $\lVert\mathbf{A}\rVert_{F}$
denote the spectral and Frobenius norms of matrix $\mathbf{A}$, respectively.
$\text{tr}(\mathbf{A})$ and $\lambda_{i}(\mathbf{A})$ denote the trace and the
$i^{th}$ eigenvalue of a square matrix $\mathbf{A}$, respectively.
$\left\|\mathbf{x}\right\|$ denotes the $\ell_{2}$ norm of vector
$\mathbf{x}$. If $\\{f_{n}\\}_{n=1}^{\infty}$ and $\\{g_{n}\\}_{n=1}^{\infty}$
are both positive sequences of real numbers, then we say
$f_{n}=\ominus(g_{n})$ if there exist $n_{0}\geq 1$ and positive constants
$C_{0}$ and $C_{1}$, such that $\inf_{n\geq n_{0}}\left(f_{n}/g_{n}\right)\geq
C_{0}$ and $\sup_{n\geq n_{0}}\left(f_{n}/g_{n}\right)\leq C_{1}$. Similarly,
if $f_{iT}$ and $g_{iT}$ are positive double sequences of real numbers for
$i=1,2,3,\cdots$; and $T=1,2,3,\cdots$, then $f_{iT}=\ominus(g_{iT})$ if there
exist $T_{0}\geq 1$ and positive constants $C_{0}$ and $C_{1}$, such that
$\inf_{T\geq T_{0}}\left(f_{iT}/g_{iT}\right)\geq C_{0}$ and $\sup_{T\geq
T_{0}}\left(f_{iT}/g_{iT}\right)\leq C_{1}$.

##  2 Model specification under parameter instability

We consider the following data generating process (DGP) for the target
variable, $y_{t}$, in terms of the signal variables $(x_{it}$, for
$i=1,2,...,k$)

$\textstyle y_{t}=\sum_{i=1}^{k}\beta_{it}x_{it}+u_{t},\text{ for
}t=1,2,\cdots,T$ (1)

with time-varying parameters, $\left\\{\beta_{it}\text{,
}i=1,2,...,k\right\\}$, and an error term, $u_{t}$. Intercepts and other pre-
selected variables can also be included.222See the working paper version of
the paper available at https://doi.org/10.24149/gwp394r2. Since the parameters
are time-varying we refer to the covariate $i$ as “signal” if its average
marginal effect,
$\bar{\beta}_{i,T}=T^{-1}\sum_{t=1}^{T}\mathbb{E}(\beta_{it})$, is not equal
to zero. The strength of the signal can be captured by the exponent
coefficient $\vartheta_{i}$ in
$\bar{\beta}_{i,T}=\ominus(T^{-\vartheta_{i}})$. For $\vartheta_{i}=0$, the
signal is strong and the average marginal effect, $\bar{\beta}_{i,T}$, does
not converge to zero. For $0<\vartheta_{i}<1/2$, the signal is semi-strong and
the average marginal effect converges to zero, but not too fast. For
$\vartheta_{i}\geq 1/2$, the average marginal effect tends to zero very fast,
making it infeasible for the OCMT procedure to distinguish such weak signals
from noise, unless weak signals are sufficiently correlated with at least one
strong or semi-strong signal. In this paper, we do not impose any restrictions
on the correlations among signals, and we focus only on the covariates with
strong and semi-strong signals, where $0\leq\vartheta_{i}<1/2$. For simplicity
of exposition, unless specified otherwise, we will refer to both strong and
semi-strong signals simply as signals.

The coefficients are allowed to vary continuously following a stochastic
process as in the standard random coefficient model,

$\beta_{it}=\beta_{i}+\sigma_{it}\xi_{it},$

or could change at discrete time intervals, for example

$\beta_{it}=\beta_{i}^{\left(s\right)},\text{ if }t\in[T_{s-1},T_{s})\text{
for }s=1,2,\cdots,S,$

where $T_{0}=1$ and $T_{S}=T$. It is assumed that both the structure of the
parameter instabilities and the identity of the $k$ signals are unknown. The
task facing the investigator is to select the signals from a set of covariates
under consideration, $\mathcal{S}_{Nt}=\\{x_{1t},x_{2t},\cdots,x_{Nt}\\}$,
known as the active set, with $N$, the number of covariates in the active set,
possibly much larger than $T$, the number of data points available for
estimation prior to forecasting. We assume the coefficients, $\beta_{it}$, for
$i=1,2,...,k$, are independently distributed of all the covariates in the
active set $\mathcal{S}_{Nt}$.

The application of penalized regression techniques to variable selection is
often theoretically justified under two key parameter stability assumptions:
the stability of $\beta_{it}$ and the stability of the correlation matrix of
the covariates in the active set. Under these assumptions, the application of
the penalized regression to the active set can proceed using the full sample
without down-weighting or separating the variable selection from the
forecasting stage. However, in the of presence parameter instability Lasso
must be adapted to simultaneously deal with selection and parameter change. We
are not aware of any machine learning technique that simultaneously addresses
both issues. As noted in the introduction, the problem has been recognized in
the empirical literature focusing on slowly varying parameters and/or the use
of rolling windows without making a distinction between variable selection and
forecasting. It is also worth highlighting that in this paper, we relax the
assumption of fixed correlation among the covariates in the active set, which
is very common in the penalized regression studies, and allow for time-varying
correlations.

In this paper we follow Chudik et al. (2018) and consider the application of
the OCMT procedure for variable selection even when the parameters are time-
varying, and provide theoretical arguments in favour of using the full sample
of data available without down-weighting. We first recall that OCMT’s variable
selection is based on the net effect of $x_{it}$ on $y_{t}$. However, when the
regression coefficients and/or the correlations across the covariates in the
active set are time-varying, the net effects will also be time-varying and we
need to base our selection on average net effects. The average net effect of
the covariate $x_{it}$ on $y_{t}$ can be defined as

$\textstyle\bar{\theta}_{i,T}=T^{-1}\sum_{t=1}^{T}\mathbb{E}(x_{it}y_{t}).$

By substituting $y_{t}$ from (1) we can further write $\bar{\theta}_{i,T}$ as

$\bar{\theta}_{i,T}=\sum_{j=1}^{k}\left(T^{-1}\sum_{t=1}^{T}\mathbb{E}(\beta_{jt})\sigma_{ij,t}\right)+\bar{\sigma}_{iu,T},$

where $\sigma_{ij,t}=\mathbb{E}(x_{it}x_{jt})$, and
$\bar{\sigma}_{iu,T}=T^{-1}\sum_{t=1}^{T}\mathbb{E}(x_{it}u_{t})$. In what
follows we allow for a mild degree of correlation between $x_{it},$ and
$u_{t},$ by assuming that $\bar{\sigma}_{iu,T}=O(T^{-\epsilon_{i}}),$ for some
$\epsilon_{i}>1/2$. In this case the average net effect of the $i^{th}$
covariate simplifies to

$\bar{\theta}_{i,T}=\sum_{j=1}^{k}\left(T^{-1}\sum_{t=1}^{T}\mathbb{E}(\beta_{jt})\sigma_{ij,t}\right)+O(T^{-\epsilon_{i}}).$

In line with our assumption about the average marginal effects, namely that
$\bar{\beta}_{i,T}=\ominus(T^{-\vartheta_{i}}),$ for some
$0\leq\vartheta_{i}<1/2$, we distinguish between covariates with strong and
semi-strong net effects, and the noise variables whose net effects, averaged
over time, tend to zero sufficiently fast. Specifically, for covariates with
strong or semi-strong net effects we set
$\bar{\theta}_{i,T}=\ominus(T^{-\vartheta_{i}}),$ for some
$0\leq\vartheta_{i}<1/2$, and for the noise variables we shall assume that
$\bar{\theta}_{i,T}=\ominus(T^{-\epsilon_{i}}),$ for some $\epsilon_{i}>1/2$.

In what follows, we first describe the OCMT procedure and then discuss the
conditions under which the approximating model (that includes all the signals
and none of the noise variables) is selected with probability one by OCMT.

##  3 Parameter instability and OCMT

The OCMT procedure begins with $N$ separate regressions, for each of the $N$
covariates in the active set $\mathcal{S}_{Nt}$. Specifically, the focus is on
the statistical significance of $\phi_{i,T}$ in the following simple
regressions:

$y_{t}=\phi_{i,T}x_{it}+\eta_{it},\text{ for }t=1,2,\cdots,T;\text{
}i=1,2,...,N,$ (2)

where

$\phi_{i,T}\equiv\left(T^{-1}\sum_{t=1}^{T}\mathbb{E}(x_{it}^{2})\right)^{-1}\left(T^{-1}\sum_{t=1}^{T}\mathbb{E}(x_{it}y_{t})\right)=\left[\bar{\sigma}_{ii,T}\right]^{-1}\bar{\theta}_{i,T}\text{,}$
(3)

with $\bar{\sigma}_{ii,T}=$ $T^{-1}\sum_{t=1}^{T}\sigma_{ii,t}$. Due to non-
zero cross-covariate correlations, knowing whether $\phi_{i,T}$ $\ ($or
equivalently $\bar{\theta}_{i,T}$) is zero does not necessarily allow us to
establish whether $\bar{\beta}_{i,T}$ is sufficiently close to zero or not.
There are four possibilities:

(I) Signals |  $\bar{\beta}_{i,T}=\ominus(T^{-\vartheta_{i}})$ and $\bar{\theta}_{i,T}=\ominus(T^{-\vartheta_{i}})$
---|---
(II) Hidden Signals |  $\bar{\beta}_{i,T}=\ominus(T^{-\vartheta_{i}})$ and $\bar{\theta}_{i,T}=\ominus(T^{-\epsilon_{i}})$
(III) Pseudo-signals |  $\beta_{it}=0$ for all $t$ and $\bar{\theta}_{i,T}=\ominus(T^{-\vartheta_{i}})$
(IV) Noise variables |  $\beta_{it}=0$ for all $t$ and $\bar{\theta}_{i,T}=\ominus(T^{-\epsilon_{i}})$

for some $0\leq\vartheta_{i}<1/2$, and $\epsilon_{i}>1/2$. To simplify the
exposition, we consider the covariates $x_{it}$, for $i=1,2,\cdots,k$, as
signals, and for $i=k+1,k+2,\cdots,k+k_{T}^{\ast}$, as pseudo-signals. The
remaining covariates in the active set, $\left\\{{x_{it},}\text{ for
}{i=k+k_{T}^{\ast}+1,k+k_{T}^{\ast}+2,\cdots,N}\right\\}$, are classified as
(pure) noise variables. We assume that the number of signals, $k$, is a finite
fixed integer but we allow the number of pseudo-signals, denoted by
$k_{T}^{\ast}$, to grow with $N$ and $T$. Notice, if the covariate $x_{it}$ is
a noise variable, then $\bar{\theta}_{i,T}$ converges to zero very fast.
Therefore, down-weighting of observations at the variable selection stage is
likely to be inefficient for eliminating the noise variables. Moreover, for a
signal to remain hidden, we need the terms of higher order,
$\ominus(T^{-\vartheta_{j}})$ with $0\leq\vartheta_{i}<1/2$, to exactly cancel
out such that $\theta_{i,T}$ becomes a lower order, i.e.
$\ominus(T^{-\varepsilon_{i}})$, that tends to zero at a sufficiently fast
rate (with $\epsilon_{i}>1/2)$. This combination of events seem quite
unlikely, and to simplify the theoretical derivations in what follows we
abstract from such a possibility and assume that there are no hidden signals
and consider a single stage version of the OCMT procedure for variable
selection. To allow for hidden signals, Chudik et al. (2018) extend the OCMT
method to have multiple stages.

The OCMT procedure

  1. 1.

For $i=1,2,\cdots,N$, regress $y_{t}$ on $x_{it}$;
$y_{t}=\phi_{i,T}x_{it}+\eta_{it}$; and compute the $t$-ratio of $\phi_{i,T}$,
given by

$t_{i,T}=\frac{\hat{\phi}_{i,T}}{s.e.\left(\hat{\phi}_{i,T}\right)}=\frac{\sum_{t=1}^{T}x_{it}y_{t}}{\hat{\sigma}_{i}\sqrt{\sum_{t=1}^{T}x_{it}^{2}}},$
(4)

where
$\hat{\phi}_{i,T}=\left(\sum_{t=1}^{T}x_{it}^{2}\right)^{-1}\left(\sum_{t=1}^{T}x_{it}y_{t}\right)$
is the Ordinary Least Square (OLS) estimator of $\phi_{i,T}$,
$\hat{\sigma}_{i}^{2}=T^{-1}\sum_{t=1}^{T}\hat{\eta}_{it}^{2}$, and
$\hat{\eta}_{it}=y_{t}-\hat{\phi}_{i,T}x_{it}$, is the regression residual.

  2. 2.

Consider the critical value function, $c_{p}(N,\delta)$, defined by

$c_{p}(N,\delta)=\Phi^{-1}\left(1-\frac{p}{2N^{\delta}}\right),$ (5)

where $\Phi^{-1}(.)$ is the inverse of a standard normal distribution
function; $\delta$ is a finite positive constant; and $p$ is the nominal size
of the tests to be set by the investigator.

  3. 3.

Given $c_{p}(N,\delta)$, the selection indicator is given by

$\hat{\mathcal{J}}_{i}=\mathbf{\mathit{I}}\left[\lvert
t_{i,T}\rvert>c_{p}(N,\delta)\right],\ \text{for}\ i=1,2,\cdots,N.$ (6)

The covariate $x_{it}$ is selected if $\hat{\mathcal{J}}_{i}=1$.

OCMT uses the t-ratio of $\phi_{i,T}$, defined by (4), to select the signals
(strong as well as semi-strong), $\\{x_{it}:i=1,2,\cdots,k\\}$, and none of
the noise variables,
$\\{x_{it}:k+k_{T}^{\ast}+1,k+k_{T}^{\ast}+2,\cdots,N\\}$. The selected model
is referred to as an approximating model since it can include pseudo-signals,
$\\{x_{it}:k+1,k+2,\cdots,k+k_{T}^{\ast}\\}$, that proxy for the true signals.
To deal with the multiple testing nature of the problem, the critical value
$c_{p}(N,\delta)$ used for the separate-induced tests is chosen to be an
appropriately increasing function of $N$, by setting $\delta>0$. The choice of
$\delta$ is guided by our theoretical derivations, to be discussed below in
Section 5.

Before presenting our technical assumptions and theoretical results under
parameter instability, it is instructive to discuss and compare the key
conditions under which Lasso and OCMT lead to consistent model selection under
parameter stability.

##  4 Lasso and OCMT under parameter stability

As formally established by Zhao and Yu (2006) and Meinshausen and Bühlmann
(2006), three main conditions are required for the Lasso variable selection to
be consistent. Here we follow Lahiri (2021) who also considers the convergence
of Lasso estimated coefficients to their true values. The key condition is the
“Irrepresentable Condition” (IRC) that places restrictions on the magnitudes
of the sample correlations across the signals,
${\mathbf{x}_{1t}=(x_{1t},x_{2t},\cdots,x_{kt})^{\prime}}$, and the rest of
the covariates in the active set, namely
$\mathbf{x}_{2t}=(x_{k+1,t},x_{k+2,t},\cdots,x_{Nt})^{\prime}$. Let

$\mathbf{R}=\begin{pmatrix}\mathbf{R}_{11}&\mathbf{R}_{12}\\\
\mathbf{R}_{21}&\mathbf{R}_{22}\end{pmatrix}$

be the $N\times N$ matrix of sample correlations of the covariates in the
active set, partitioned conformably to
$\mathbf{x}_{t}=(\mathbf{x}_{1t}^{\prime},\mathbf{x}_{2t}^{\prime})^{\prime}$.
The IRC can be written as

$\left\|\mathbf{R}_{21}\mathbf{R}_{11}^{-1}\text{sign}\left(\boldsymbol{\beta}_{0}\right)\right\|_{\infty}\leq
1,$ (7)

where $\left\|.\right\|_{\infty}$ is the $\ell_{\infty}$ norm of a vector,
$\text{sign}(.)$ is the sign function, and
$\boldsymbol{\beta}_{0}=\left(\beta_{01},\beta_{02},\cdots,\beta_{0k}\right)^{\prime}$
is the $k\times 1$ vector of the coefficients of the signals. The following
example provides more intuition on how IRC imposes restrictions on the
magnitudes of the sample correlations between the covariates in the active
set.

###### Example 1

Suppose the DGP for $y_{t}$ contains only two signals, $x_{1t}$ and $x_{2t}$.
Denote the sample correlation coefficient between $x_{1t}$ and $x_{2t}$ by
$\hat{\rho}$, and the sample correlation coefficients of $x_{1t}$ and $x_{2t}$
with the rest of the covariates in the active set,
$x_{3,t},x_{4,t},\cdots,x_{Nt}$, by $\hat{\rho}_{i1}$ and $\hat{\rho}_{i2}$,
for $i=3,4,\cdots,N$, respectively. Then, after some algebra, the IRC given by
(7) simplifies to

$\max_{i\in\\{3,4,\cdots,N\\}}\left|(\hat{\rho}_{i1}-\hat{\rho}\hat{\rho}_{i2})\text{sign}(\beta_{01})+(\hat{\rho}_{i2}-\hat{\rho}\hat{\rho}_{i1})\text{sign}(\beta_{02})\right|\leq
1-\hat{\rho}^{2}.$

There are two cases: (A)
$\text{sign}\left(\beta_{01}\right)=\text{sign}\left(\beta_{02}\right)$ and
(B)
$\text{sign}\left(\beta_{01}\right)\neq\text{sign}\left(\beta_{02}\right)$.
Under case (A) it follows that the IRC condition is met if

$\max_{i\in\\{3,4,\cdots,N\\}}\left|\hat{\rho}_{i1}+\hat{\rho}_{i2}\right|\leq
1+\hat{\rho},$

Similarly under case (B) it is required that

$\max_{i\in\\{3,4,\cdots,N\\}}\left|\hat{\rho}_{i1}-\hat{\rho}_{i2}\right|\leq
1-\hat{\rho}.$

Since the sign of signals are unknown, to ensure that the IRC holds, we need

$\max_{i\in\\{3,4,\cdots,N\\}}\left|\hat{\rho}_{i1}\right|+\left|\hat{\rho}_{i2}\right|+\left|\hat{\rho}\right|\leq
1.$

From the above example, it is clear that IRC places restrictions on the
magnitude of sample correlation among signals ($\hat{\rho}$ in the above
example), as well as the magnitude of sample correlation between signals and
pseudo-signals ($\hat{\rho}_{i1}$ and $\hat{\rho}_{i2}$). Notably, the IRC is
met for noise variables but need not hold for pseudo-signals. OCMT also has no
difficulty in dealing with noise variables, and is very effective at
eliminating them. However, for estimation of the approximate model, post OCMT
selection, it is necessary to restrict the number of selected covariates
relative to the sample size, $T$. To this end, OCMT assumes that the number of
pseudo-signals, $k_{T}^{\ast}$, could grow at an order less than the square
root of the number of observations, namely

$k_{T}^{\ast}=\ominus(T^{d})\text{ for some }0\leq d<\frac{1}{2}.$

It is important to note that OCMT does not place any restrictions on the
magnitude of correlations of signals and pseudo-signals. Instead, it limits
the number of covariates that are correlated with the signals
($k_{T}^{\ast}$). Clearly, the IRC could be violated even when the number of
pseudo-signals grows at an order less than $\sqrt{T}$. Hence the OCMT’s
requirement on the number of pseudo-signals allows for cases where the IRC
does not hold, and vice versa.

The condition on the number of pseudo-signals ($k_{T}^{\ast}$) in the OCMT
framework has been recently relaxed by Sharifvaghefi (2023). To illustrate how
this is done, suppose there are no noise variables and hence the signals,
$\mathbf{x}_{1t}=\left(x_{1t},x_{2t},\cdots,x_{kt}\right)^{\prime}$, are
correlated with all the remaining covariates in the active set. In this case
$k_{T}^{\ast}=\ominus(T)$ and a straightforward application of OCMT will not
be valid. But, we can model the correlation between the signals,
$\mathbf{x}_{1t}$, and the remaining covariates, $\mathbf{x}_{2t}$, as

$x_{it}=\sum_{j=1}^{k}\psi_{ij}x_{jt}+\xi_{it}=\boldsymbol{\psi}_{i}^{\prime}\mathbf{x}_{1t}+\xi_{it},\text{
for }i=k+1,k+2,\cdots,N.$

The signals thus act as strong factors for the pseudo-signals. Given that the
identity of signals and pseudo-signals are unknown and the number of pseudo-
signals is large, it is reasonable to propose the existence of latent factors,
$\mathbf{f}_{t}$, that are common across the covariates in the active set.
This idea can be formally expressed as:

$x_{it}=\boldsymbol{\psi}_{i}^{\prime}\mathbf{f}_{t}+\varepsilon_{it}\quad\text{for
}i=1,2,\cdots,N,$

where $\boldsymbol{\psi}_{i}$ is vector of factor loadings, and
$\varepsilon_{it}$ refers to the idiosyncratic components that are weakly
cross-correlated such that

$\sup_{j}\sum_{i=1}^{N}\left|cov(\varepsilon_{it},\varepsilon_{jt})\right|<C<\infty.$
(8)

Substituting $x_{it}$ into the DGP for $y_{t}$, given by (1), we obtain:

$y_{t}=\boldsymbol{\delta}_{0}^{\prime}\mathbf{f}_{t}+\sum_{i=1}^{k}\beta_{i0}\varepsilon_{it}+u_{t},$

with $\boldsymbol{\delta}_{0}=\sum_{i=1}^{k}\beta_{i0}\boldsymbol{\psi}_{i}$.
When the common factors, $\mathbf{f}_{t}$, and idiosyncratic components,
$\varepsilon_{it}$ , are known, this model would correspond to that presented
in working paper version of our work, where common factors $\mathbf{f}_{t}$
can be used as preselected variables. Since $\mathbf{f}_{t}$ and
$\varepsilon_{it}$ are not known, Sharifvaghefi (2023) shows that when both
$N$ and $T$ are large the OCMT selection can be carried out using the
principal component estimators of $\mathbf{f}_{t}$ and $\varepsilon_{it}$,
denoted by $\hat{\mathbf{f}}_{t}$ and $\hat{\varepsilon}_{it}$, using all the
covariates in the active set. The large $N$ is required for consistent
estimation of the common factors. As a result, the OCMT condition on the
number of pseudo-signals now relates to the number of $\varepsilon_{it}$ for
$i=k+1,k+2,...,N$ that are correlated with $\varepsilon_{it}$ for
$i=1,2,...,k$, which is bounded under condition (8).

For variable selection consistency of Lasso under parameter stability, the
literature further requires the penalty term, $\lambda_{T}$, to grow at an
order greater than $\sqrt{T}$ such that:

$\lim_{T\rightarrow\infty}\Pr\left(\left\|\frac{1}{\sqrt{T}}\sum_{t=1}^{T}\tilde{\mathbf{x}}_{2t}u_{t}\right\|_{\infty}>\frac{\lambda_{T}}{\sqrt{T}}\right)=0,$

where $\tilde{\mathbf{x}}_{2t}=\mathbf{M}_{x_{1}}\mathbf{x}_{2t}$,
$\mathbf{M}_{\tilde{x}_{1}}$ is the orthogonal projection matrix and $u_{t}$
is the error term in the data generating process. The exact choice of
$\lambda_{T}$ in practice is often unclear, with practitioners typically
relying on cross-validation methods.

A third condition required by Lasso for variable selection consistency is the
beta-min condition:

$\min_{j=1,2,\cdots,k}\left|\beta_{j0}\right|>(2T)^{-1}\lambda_{T}\left|\mathbf{R}_{11}^{-1}\text{sign}\left(\boldsymbol{\beta}_{0}\right)\right|_{j}$

where $\left|.\right|_{j}$ denotes the absolute value of the $j^{th}$ element
of a vector. Given that $\lambda_{T}$ must grow at an order greater than
$\sqrt{T}$, we can conclude from the beta-min condition that
$\beta_{i0}\gg\frac{1}{\sqrt{T}}$ for $i=1,2,\cdots,k$. For example, Lahiri
(2021) assumes that $\beta_{i0}\gg\sqrt{\frac{k\log(T)}{T}}$. The OCMT’s
requirement on the strength of signals (under parameter stability) is given by
$\beta_{i0}=\ominus(T^{-\vartheta_{i}})$, for some $0\leq\vartheta_{i}<1/2$.
This condition is essentially very similar to the Lasso’s beta-min condition.

##  5  Asymptotic properties of OCMT under parameter instability

We now provide the theoretical justification for using the OCMT procedure for
variable selection in models with time-varying parameters. We start with some
technical assumptions in Section 5.1 and then provide the asymptotic
properties of the OCMT procedure under parameter instability in Section 5.2.

###  5.1 Technical assumptions

Let $\mathbf{x}_{t}=(x_{1t},x_{2t},\cdots,x_{Nt})^{\prime}$ be the $N\times 1$
vector of covariates under consideration. Define the filtrations:
$\mathcal{F}_{t}^{x}=\sigma(\mathbf{x}_{t},\mathbf{x}_{t-1},\cdots)$,
$\mathcal{F}_{jt}^{\beta}=\sigma(\beta_{jt},\beta_{j,t-1},\cdots)$, for
$j=1,2,\cdots,k,$ and $\mathcal{F}_{t}^{u}=\sigma(u_{t},u_{t-1},\cdots)$, and
set $\mathcal{F}_{t}^{\beta}=\cup_{j=1}^{k}\mathcal{F}_{jt}^{\beta}$ and
$\mathcal{F}_{t}=\mathcal{F}_{t}^{q}\cup\mathcal{F}_{t}^{\mathrm{a}}\cup\mathcal{F}_{t}^{\beta}\cup\mathcal{F}_{t}^{u}$.
Also consider the following assumptions:

######  Assumption 1 (Martingale difference processes)

enter

  1. (a)

$\mathbb{E}\left[\mathbf{x}_{t}\mathbf{x}_{t}^{\prime}-\mathbb{E}(\mathbf{x}_{t}\mathbf{x}_{t}^{\prime})|\mathcal{F}_{t-1}\right]=0$
for $t=1,2,\cdots,T$.

  2. (b)

$\mathbb{E}\left[u_{t}^{2}-\mathbb{E}\left(u_{t}^{2}\right)|\mathcal{F}_{t-1}\right]=0$
for $t=1,2,\cdots,T$.

  3. (c)

$\mathbb{E}\left[\mathbf{x}_{t}u_{t}-\mathbb{E}(\mathbf{x}_{t}u_{t})|\mathcal{F}_{t-1}\right]=0$
for $t=1,2,\cdots,T$.

  4. (d)

$\mathbb{E}\left[\beta_{it}-\mathbb{E}(\beta_{it})|\mathcal{F}_{t-1}\right]=0$
for $i=1,2,\cdots,k$ and $t=1,2,\cdots,T$.

######  Assumption 2 (Exponential decaying probability tails)

enter
There exist sufficiently large positive constants $C_{0}$ and $C_{1}$, and
$s>0$ such that

  1. (a)

$\sup_{j,t}\Pr(|x_{jt}|>\alpha)\leq C_{0}\exp(-C_{1}\alpha^{s}),\ for\ all\
\alpha>0.$

  2. (b)

$\sup_{i,t}\Pr(|\beta_{it}|>\alpha)\leq C_{0}\exp(-C_{1}\alpha^{s}),\ for\
all\ \alpha>0.$

  3. (c)

$\sup_{t}\Pr(|u_{t}|>\alpha)\leq C_{0}\exp(-C_{1}\alpha^{s}),\ for\ all\
\alpha>0.$

######  Assumption 3 (Coefficients of signals)

enter

  1. (a)

The number of signals, $k$, is a finite fixed integer.

  2. (b)

$\beta_{it}$, $i=1,2,\cdots,k$, are distributed independently of
$x_{jt^{\prime}}$, $j=1,2,\cdots,N$, and $u_{t^{\prime}}$ for all $t$ and
$t^{\prime}$.

  3. (c)

$\bar{\beta}_{i,T}\equiv
T^{-1}\sum_{t=1}^{T}\mathbb{E}(\beta_{it})=\ominus(T^{-\vartheta_{i}})$, for
some $0\leq\vartheta_{i}<1/2.$

Before presenting the theoretical results, we briefly discuss the rationale
behind our assumptions and compare them with the assumptions typically made in
the high-dimensional linear regressions and the parameter instability
literature.

To establish that the OCMT procedure with the critical value function
$c_{p}(N,\delta)=\Phi^{-1}\left(1-\frac{p}{2N^{\delta}}\right)$ does not
select any of the noise variables with a probability approaching one as $N$
and $T$ go to infinity, we need to show that the t-statistic given by (4)
follows a distribution with exponentially decaying tails. We utilize the
concentration inequality of an exponential decaying rate to accomplish this
goal. Assumptions 1 and 2 place constraints on the sequence of random
variables, $x_{it}$ for $i=1,2,\ldots,N$, $\beta_{jt}$ for $j=1,2,\ldots,k$,
and $u_{t}$ such that they adhere to a martingale difference process and
exhibit exponential decaying probability tails. These assumptions are
sufficient to establish the exponential decaying concentration inequality, as
provided in Lemma S-3.1 in the online theory supplement. Notably, these
assumptions could be relaxed provided that the exponential decaying
concentration inequality holds. For example, Theorem 1 of Merlevède et al.
(2011) and Lemma D1 of the online theory supplement for Chudik et al. (2018)
establishes that this inequality can be achieved while allowing for weak time-
series dependence. In penalized regression literature, a commonly held
assumption is that the covariates are non-random and fixed. Moreover, error
terms $\\{u_{t}\\}_{t=1}^{T}$ are typically presumed to be independent, for
example see Zhao and Yu (2006), Javanmard and Montanari (2013), Lee et al.
(2015), Belloni et al. (2014), Javanmard and Lee (2020), and Lahiri (2021).
Additionally, in the Lasso literature it is often assumed that $u_{t}$
possesses an exponentially decaying probability tail. See, for example,
Javanmard and Montanari (2018), Hansen and Liao (2019), Fan et al. (2020), and
Javanmard and Lee (2020).

Assumption 3(a) posits that the number of signals is a finite fixed integer.
This is crucial to ensure that the random variable $y_{t}$ has a distribution
with an exponentially decaying probability tail. Under the premise that the
covariates $x_{it}$ for all $i$ and $t$ are non-random and fixed, which is a
common assumption in the penalized regression setting, it becomes permissible
for the number of signals to grow with the sample size at an order slower that
$\sqrt{T}$. Assumption 3(b) is common in the literature under parameter
instabilities and restrict the distribution of time-varying parameters to be
independent of the covariates. This assumption could be relaxed if the
covariates are treated as non-random and fixed. Finally, Assumption 3(c)
ensures that the average marginal effect of the signal variables does not tend
to zero too rapidly. This is an identification assumption needed to
distinguish signals from noise variables and is similar to the beta-min
condition already discussed in Section 4.

###  5.2 Theoretical results

####  5.2.1 Variable selection consistency

As mentioned in Section 1, the purpose of this paper is to provide the
theoretical argument for applying the OCMT procedure with no down-weighting at
the variable selection stage in linear high-dimensional settings subject to
parameter instability. We now show that under certain conditions discussed in
Section 5.1, the OCMT procedure selects the approximating model that contains
all the signals; $\\{x_{it}:i=1,2,\cdots,k\\}$; and none of the noise
variables; $\\{x_{it}:k+k_{T}^{\ast}+1,k+k_{T}^{\ast}+2,\cdots,N\\}$. The
event of choosing the approximating model is defined by

$\textstyle\mathcal{A}_{0}=\left\\{\sum_{i=1}^{k}\hat{\mathcal{J}}_{i}=k\right\\}\cap\left\\{\sum_{i=k+k_{T}^{\ast}+1}^{N}\hat{\mathcal{J}}_{i}=0\right\\}.$
(9)

Note that the approximating model can contain pseudo-signals. In what follows,
we show that $\Pr(\mathcal{A}_{0})\rightarrow 1$, as $N,T\rightarrow\infty$.

###### Theorem 1

Let $y_{t}$ for $t=1,2,\cdots,T$ be generated by (1), and consider the set
$\mathcal{S}_{Nt}=\\{x_{1t},x_{2t},\cdots,x_{Nt}\\}$ which contains $k$
signals, $k_{T}^{\ast}$ pseudo-signals, and $N-k-k_{T}^{\ast}$ noise
variables. Suppose $N=\ominus(T^{\kappa})$ with $\kappa>0$, and consider the
OCMT procedure with the critical value function $c_{p}(N,\delta)$ given by
(5), for some $\delta>0$. Under Assumptions 1-3, there exist finite positive
constants $C_{0}$ and $C_{1}$ such that, for any $0<\pi<1$, the probability of
selecting the approximating model $\mathcal{A}_{0}$, as defined by (9), is
given by

$\Pr(\mathcal{A}_{0})=1-O\left[T^{\kappa\left(1-\mathcal{X}_{NT}\left(\frac{1-\pi}{1+d_{T}}\right)^{2}\delta\right)}\right]-O\left[T^{\kappa}\exp\left(-C_{0}T^{C_{1}}\right)\right],$
(10)

where $d_{T}$ is a positive sequence approaching zero as $T$ goes to infinity.
Here,

$\textstyle\mathcal{X}_{NT}=\inf_{i\in\\{k+k^{\ast}+1,\cdots,N\\}}\frac{\bar{\sigma}_{\eta_{i},T}^{2}\bar{\sigma}_{x_{i},T}^{2}}{\bar{\omega}_{iy,T}^{2}},$

where $\bar{\sigma}_{x_{i},T}^{2}=T^{-1}\sum_{t=1}^{T}\mathbb{E}(x_{it}^{2})$,
$\bar{\omega}_{iy,T}^{2}=T^{-1}\sum_{t=1}^{T}\mathbb{E}(x_{it}^{2}y_{t}^{2}|\mathcal{F}_{t-1})$,
$\bar{\sigma}_{\eta_{i},T}^{2}=T^{-1}\sum_{t=1}^{T}\mathbb{E}(\eta_{it}^{2})$,
$\eta_{it}=y_{t}-\phi_{i,T}x_{it}$, and $\phi_{i,T}$ is defined in (3).

As evident from the above theorem, the choice of $\delta$ is crucial for model
selection consistency. Setting $\delta$ very small, such that
$\delta<\mathcal{X}_{NT}^{-1}\left(\frac{1+d_{T}}{1-\pi}\right)^{2}$, makes us
prone to the multiple testing problem, initially highlighted by Bonferroni,
potentially leading to the inclusion of many noise variables in the selected
model. Conversely, a very large $\delta$ renders the OCMT procedure
conservative, characterized by an excessively high threshold for the t-test,
increasing the likelihood of missing some signals in finite sample analysis.
Corollary S.1 in the online theory supplement shows that if the covariates in
the active set are generated by a stationary process then
$\mathcal{X}_{NT}=1$. As a result by setting $\delta>1$, OCMT consistently
selects the approximating model, $\mathcal{A}_{0}$. However, $c_{p}(N,\delta)$
is reasonably stable with respect to small increases in $\delta$ in the
neighborhood of $\delta=1$. The extensive Monte Carlo studies in Chudik et al.
(2018) also suggest that setting $\delta=1$ performs well in practice. In view
of these findings we do not recommend the use of cross-validation techniques
for setting $\delta$.

It is interesting that the asymptotic results regarding the probability of
selecting the approximating model are unaffected by parameter instability, so
long as the average net effects of the signals are non-zero or tend to zero
sufficiently slowly in $T$, as defined formally by Assumption 3. In the next
step, we focus on estimation of the coefficients of the selected model, and
derive the limiting property of its sum of squared residuals.

####  5.2.2 Properties of the post OCMT selected model

To establish the results, the following additional assumption is also needed.

######  Assumption 4 (Eigenvalues)

Denote the total number of signals ($k$) and pseudo-signals ($k_{T}^{\ast}$)
by $\tilde{k}_{T}$ and let $\mathbf{x}_{\tilde{k}_{T},t}$ be the
$\tilde{k}_{T}\times 1$ vector of signals and pseudo-signals. then

$\lambda_{\min}\left[T^{-1}\sum\limits_{t=1}^{T}\mathbb{E}(\mathbf{x}_{\tilde{k}_{T},t}\mathbf{x}_{\tilde{k}_{T},t}^{\prime})\right]>c>0.$

This assumption ensures that the post OCMT selected model can be consistently
estimated subject to certain regularity conditions to be discussed below.

The post OCMT selected model can be written as

$\textstyle y_{t}=\sum_{i=1}^{N}\hat{\mathcal{J}}_{i}x_{it}b_{i}+\eta_{t}$

where $\hat{\mathcal{J}}_{i}=\mathbf{\mathit{I}}\left[\lvert
t_{i,T}\rvert>c_{p}(N,\delta)\right],\ $defined by (6). Also
$\sum_{i=1}^{N}\hat{\mathcal{J}}_{i}=\hat{k}_{T}$, where $\hat{k}_{T}$ is the
number of covariates selected by OCMT. By Theorem 1 the probability that the
selected model contains the signals tends to unity as $T\rightarrow\infty$. We
can further write

$\textstyle
y_{t}=\sum_{i=1}^{N}\hat{\mathcal{J}}_{i}x_{it}b_{i}+\eta_{t}=\sum_{\ell=1}^{\hat{k}_{T}}\gamma_{\ell}w_{\ell
t}+\eta_{t},$ (11)

where
$\mathbf{w}_{t}=\left(w_{1t},w_{2t},\cdots,w_{\hat{k}_{T}t}\right)^{\prime}$.
The least squares (LS) estimator of selected coefficients,
$\boldsymbol{\gamma}_{T}=\left(\gamma_{1},\gamma_{2},\cdots,\gamma_{\hat{k}_{T}}\right)^{\prime}$,
is given by

$\textstyle\hat{\boldsymbol{\gamma}}_{T}=\left(T^{-1}\sum_{t=1}^{T}\mathbf{w}_{t}\mathbf{w}_{t}^{\prime}\right)^{-1}\left(T^{-1}\sum_{t=1}^{T}\mathbf{w}_{t}y_{t}\right),$
(12)

In establishing the rate of convergence of $\hat{\mathbf{\gamma}}_{T}$ we
distinguish between two cases: when the vector of signals,
$\mathbf{x}_{k,t}=\left(x_{1t},x_{2t},\cdots,x_{kt}\right)^{\prime}$ is
included in $\mathbf{w}_{t}$ as a subset, and when this is not the case. But
we know from Theorem 1 that the probability of the latter tends to zero at a
sufficiently fast rate. The following theorem provides the conditions under
which the estimates of the coefficients of the selected signals and pseudo-
signals of the approximating model tend to their true mean values, defined
formally below.

###### Theorem 2

Let the DGP for $y_{t}$, $t=1,2,\cdots,T$ be given by (1) and write down the
regression model selected by the OCMT procedure as (11). Suppose that
Assumptions 1-4 hold and the number of pseudo-signals, $k_{T}^{\ast}$, grow
with $T$ such that $k_{T}^{\ast}=\ominus(T^{d})$ with $0\leq d<\frac{1}{2}$.
Consider the least squares (LS) estimator of
$\boldsymbol{\gamma}_{T}=\left(\gamma_{1},\gamma_{2},\cdots,\gamma_{\hat{k}_{T}}\right)^{\prime}$,
given by (12).

  1. (i)

If $\mathbb{E}(\beta_{it})=\beta_{i}$ for all $t$, then,

$\left\|\hat{\boldsymbol{\gamma}}_{T}-\boldsymbol{\gamma}_{T}^{*}\right\|=O_{p}\left(T^{\frac{d-1}{2}}\right),$

where
$\boldsymbol{\gamma}_{T}^{*}=(\gamma_{1}^{*},\gamma_{2}^{*},\cdots,\gamma_{\hat{k}_{T}}^{*})^{\prime}$,
and

$\left\\{\begin{matrix}\gamma_{\ell}^{*}\in\boldsymbol{\beta}=(\beta_{1},\beta_{2},\cdots,\beta_{k})^{\prime},&\text{if
}w_{\ell t}\in\mathbf{x}_{kt}\\\
\gamma_{\ell}^{*}=0,&\text{otherwise}.\vspace{-0.25cm}\end{matrix}\right.$

  2. (ii)

If
$\mathbb{E}\left(\mathbf{x}_{\tilde{k}_{T},t}\mathbf{x}_{\tilde{k}_{T},t}^{\prime}\right)$
is a fixed time-invariant matrix, where $\tilde{k}_{T}=k+k^{\ast}_{T}$, then,

$\left\|\hat{\boldsymbol{\gamma}}_{T}-\boldsymbol{\gamma}_{T}^{\diamond}\right\|=O_{p}\left(T^{\frac{d-1}{2}}\right),$

where
$\boldsymbol{\gamma}_{T}^{\diamond}=(\gamma_{1T}^{\diamond},\gamma_{2T}^{\diamond},\cdots,\gamma_{\hat{k}_{T},T}^{\diamond})^{\prime}$
, and

$\left\\{\begin{matrix}\gamma_{\ell,T}^{\diamond}\in\boldsymbol{\bar{\beta}}_{T}=(\bar{\beta}_{1T},\bar{\beta}_{2T},\cdots,\bar{\beta}_{kT})^{\prime},&\text{if
}w_{\ell t}\in\mathbf{x}_{kt}\\\
\gamma_{\ell,T}^{\diamond}=0,&\text{otherwise},\end{matrix}\right.$

and $\bar{\beta}_{iT}=T^{-1}\sum_{t=1}^{T}\mathbb{E}(\beta_{it})$,
$i=1,2,\cdots,k$.

###### Remark 1

The above theorem builds on Theorem 1 and establishes that in the post OCMT
selected model estimated by LS only signals will end up having non-zero
limiting values, as $N$ and $T\rightarrow\infty$. This theorem also shows that
the convergence rate of the LS estimators depends on $d$, defined by
$k_{T}^{\ast}=\ominus(T^{d})$, and the regular convergent rate of $\sqrt{T}$
is achieved only if $d=0$. Similarly, Lahiri (2021) establishes that the Lasso
procedure cannot achieve both variable selection consistency and
$\sqrt{T}$-consistency in coefficient estimation.

###### Remark 2

The conditions of Theorem 2 are met in the case of random coefficient models
where $\beta_{it}=\beta_{i}+\sigma_{it}\xi_{it}$, and $\xi_{it}$ are
distributed independently of the signals, and the LS estimator of
$\boldsymbol{\gamma}_{T}^{\ast}$ is consistent, so long as $0\leq d<1/2$.
Interestingly, if signal and pseudo-signal variables are generated by a
stationary process, and hence they satisfy condition (ii) of Theorem 2, then
we can extend the random coefficient model to have time-varying means, and
still estimate $\boldsymbol{\gamma}_{T}^{\ast}$ consistently by LS.

Lastly, we consider the fit of the post OCMT selected regression in terms of
its residuals given by

$\textstyle\hat{\eta}_{t}=y_{t}-\sum_{\ell=1}^{\hat{k}_{T}}\hat{\gamma}_{\ell}w_{\ell
t}\text{, for }t=1,2,...,T.$ (13)

It is worth noting that even when all the signal variables are correctly
selected, the forecasts based on the selected model will be biased due to
parameter instability. The implications of parameter instability for the in-
sample fit of the selected regression is derived in Proposition S.1 of the
online theory supplement, abstracting from variable selection uncertainty. In
what follows we derive the asymptotic properties of the sum of squared
residuals (SSR) of the selected model, namely
$\sum_{t=1}^{T}\hat{\eta}_{t}^{2}$, taking account of the costs associated
with variable selection uncertainty and parameter instability. To this end we
need the following assumption on the cross correlation of parameter
heterogeneity.

######  Assumption 5 (Weak time dependence)

$h_{ij,t}=x_{it}x_{jt}(\beta_{it}-\bar{\beta}_{iT})(\beta_{jt}-\bar{\beta}_{jT})$
is weakly correlated over time such that

$\sum_{t=1}^{T}\sum_{t^{\prime}=1}^{T}cov(h_{ij,t},h_{ij,t^{\prime}})=O(T),\text{
for }i,j=1,2,..,k,$

where $cov(.,.)$ is the covariance operator.

###### Remark 3

Assumption 5 is a high-level assumption. Here is an example of conditions
under which this assumption holds. Suppose, Assumptions 1 and 3 hold, and the
cross products of coefficients of the signals follow martingale difference
processes such that

$\mathbb{E}\left[\beta_{it}\beta_{jt}-\mathbb{E}(\beta_{it}\beta_{jt})|\mathcal{F}_{t-1}\right]=0,\text{
for }i=1,2,\cdots,k,\ j=1,2,\cdots,k,\text{ and }t=1,2,\cdots,T.$

Then,
$\sum_{t=1}^{T}\sum_{t^{\prime}=1}^{T}\text{cov}(h_{ij,t},h_{ij,t^{\prime}})=O(T)$.
See Lemma S-2.8 in the online theory supplement for a proof.

The following theorem establishes the limiting property of SSR of the post
OCMT selected model.

###### Theorem 3

Let the DGP for $y_{t}$, $t=1,2,\cdots,T$ be given by (1) and write down the
regression model selected by the OCMT procedure as (11). The residual of the
selected model, estimated by LS, is given by (13). Suppose that Assumptions
1-5 hold and the number of pseudo-signals, $k_{T}^{\ast}$, grow with $T$ such
that $k_{T}^{\ast}=\ominus(T^{d})$ with $0\leq d<\frac{1}{2}$.

  1. (i)

If $\mathbb{E}(\beta_{it})=\beta_{i}$ for all $t$, then

$\textstyle
T^{-1}\text{SSR}=\bar{\sigma}_{u,T}^{2}+\bar{\Delta}_{\beta,T}+O_{p}\left(T^{-\frac{1}{2}}\right)+O_{p}\left(T^{d-1}\right),$
(14)

where
$\bar{\sigma}_{u,T}^{2}=T^{-1}\sum_{t=1}^{T}\mathbb{E}\left(u_{t}^{2}\right)$,
and
$\bar{\Delta}_{\beta,T}=T^{-1}\sum_{t=1}^{T}\text{tr}\left(\boldsymbol{\Sigma}_{\mathbf{x}_{k},t}\boldsymbol{\Omega}_{\beta,t}\right)$
are non-negative, with
$\boldsymbol{\Sigma}_{\mathbf{x}_{k},t}\equiv\left(\sigma_{ijt,x}\right)$,
$\boldsymbol{\Omega}_{\beta,t}\equiv\left(\sigma_{ijt,\beta}\right)$ for
$i,j=1,2,\cdots,k$, and $\sigma_{ijt,x}=\mathbb{E}\left(x_{it}x_{jt}\right)$,
$\sigma_{ijt,\beta}=\mathbb{E}\left[(\beta_{it}-\beta_{i})(\beta_{jt}-\beta_{j})\right]$.

  2. (ii)

Let $\tilde{k}_{T}=k+k_{T}^{\ast}$ and suppose that
$\mathbb{E}\left(\mathbf{x}_{\tilde{k}_{T},t}\mathbf{x}_{\tilde{k}_{T},t}^{\prime}\right)$
is time-invariant (fixed). Then,

$\textstyle
T^{-1}\text{SSR}=\bar{\sigma}_{u,T}^{2}+\bar{\Delta}_{\beta,T}^{\ast}+O_{p}\left(T^{-\frac{1}{2}}\right)+O_{p}\left(T^{d-1}\right),$
(15)

where
$\bar{\Delta}_{\beta,T}^{\ast}=T^{-1}\sum_{t=1}^{T}\text{tr}\left(\boldsymbol{\Sigma}_{\mathbf{x}_{k},t}\boldsymbol{\Omega}_{\beta,t}^{\ast}\right)$
is non-negative, with
$\boldsymbol{\Omega}_{\beta,t}^{\ast}\equiv\left(\sigma_{ijt,\beta}^{\ast}\right)$
for $i,j=1,2,\cdots,k$, and
$\sigma_{ijt,\beta}^{\ast}=\mathbb{E}\left[(\beta_{it}-\bar{\beta}_{i,T})(\beta_{jt}-\bar{\beta}_{j,T})\right]$.

###### Remark 4

The condition $d<\frac{1}{2}$ in Theorem 3 ensures that the number of pseudo-
signals grows sufficiently slowly in $T$, which in turn ensures that
$T^{1-d}<T^{-\frac{1}{2}}$ and hence from equations (14) and (15), we can
conclude that the average of squared residuals ($T^{-1}$SSR) of the Post OCMT
selected model convergences at the same rate of $T^{-\frac{1}{2}}$ under both
scenarios (i) and (ii).

Results (14) and (15) in Theorem 3 show that the SSR of the selected model
depends on ($i$) the unavoidable uncertainty due to the unobserved error term,
$u_{t}$, given by the term $\bar{\sigma}_{u,T}^{2}$, ($ii$) the cost (in terms
of fit) of ignoring the time variation in the coefficients of the signals,
$\boldsymbol{\beta}_{it}$, $i=1,2,\cdots,k$, as given by the term
$\bar{\Delta}_{\beta,T}$ and $\bar{\Delta}_{\beta,T}^{\ast}$, respectively,
and ($iii$) the $O_{p}\left(T^{-1/2}\right)$ term due to sampling uncertainty
(which will be present even in the absence of variable selection uncertainty),
and $(iv)$ the $O_{p}\left(T^{d-1}\right)$ term which is due to variable
selection uncertainty, and will be dominated by $O_{p}\left(T^{-1/2}\right)$
when $d<1/2$. Therefore, the cost of variable selection can be controlled when
using OCMT if the number of pseudo-signals, $k_{T}^{\ast}$, do not rise faster
than $\sqrt{T}$. However, to reduce the cost associated with parameter
instability more information about the nature of time variations in
$\boldsymbol{\beta}_{it}\boldsymbol{\ }$and $\sigma_{ijt,x}$ are required. For
example, $\bar{\Delta}_{\beta,T}$ (or $\bar{\Delta}_{\beta,T}^{\ast}$) could
be lower if $\boldsymbol{\Omega}_{\beta,t}$ is close to zero in some periods,
or if there are cancelling effects from negative $\sigma_{ijt,x}$
($\sigma_{ijt,x}^{\ast}$) when $\sigma_{ijt,\beta}$ is positive, namely
$\sigma_{ijt,x}\sigma_{ijt,\beta}<0$
($\sigma_{ijt,x}^{\ast}\sigma_{ijt,\beta}<0$), for some $i\neq j$ and some
$t$. This finding for the in-sample fit is similar to the results for mean
square forecast errors (MSFE) in the presence of breaks in the literature,
such as Proposition 2 of Pesaran and Timmermann (2007) or equation (20) of
Pesaran et al. (2013), where the main focus is to minimize the MSFE by
mitigating the cost of parameter instability at the expense of increased
sampling uncertainty by appropriate weighting of the observations.

##  6 Monte Carlo evidence

We use Monte Carlo (MC) techniques to compare finite sample performance of
OCMT with and without down-weighting at the selection stage, as well as
comparing the OCMT results with those of Lasso, A-Lasso, and boosting. In
these comparisons we consider the number of selected covariates
($\hat{k}_{T}$), the true positive rate (TPR), the false positive rate (FPR),
and the one-step-ahead mean square forecast error (MSFE) of the selected
models. Sub-section 6.1 outlines the MC designs, sub-section 6.2 provides a
summary of how the OCMT, Lasso, A-Lasso, and boosting procedures are
implemented, and finally sub-section 6.3 presents the main MC findings.
Details of Lasso, A-Lasso, and boosting procures and how they are implemented
are provided in Section S-1 of the online Monte Carlo supplement.

###  6.1 Simulation design

We consider the following data generating process (DGP):

$y_{t}=c_{t}+\rho_{y,t}y_{t-1}+\sum_{j=1}^{k}\beta_{jt}\tilde{x}_{jt}+\tau_{u}u_{t}\text{,}$

where the four signals $\tilde{x}_{jt},$ $j=1,2,3,4$ have non-zero, time-
varying means $\mu_{jt}=\mathbb{E}\left(\tilde{x}_{jt}\right)$. To simplify
the exposition of the DGP we consider the demeaned covariates,
$x_{jt}=\tilde{x}_{jt}-\mu_{jt}$ (so that $\mathbb{E}\left(x_{jt}\right)=0$),
and write the DGP equivalently as

$y_{t}=d_{t}+\rho_{y,t}y_{t-1}+\sum_{j=1}^{k}\beta_{jt}x_{jt}+\tau_{u}u_{t}\text{,}$
(16)

where

$d_{t}=c_{t}+\sum_{j=1}^{k}\beta_{jt}\mu_{jt}.$ (17)

Since $c_{t}$ is a free parameter, without loss of generality we also treat
$\left\\{d_{t}\,,\text{ }t=1,2,...,T\right\\}$ as free parameters.

For each MC replication, $r=1,2,...,R$, the target variable, $y_{t}$, is
generated as random draws using (16). The signal variables $x_{jt}$,
$j=1,2,3,4$, are unknown and belong to a set
$\mathcal{S}_{Nt}=\left\\{x_{1t},x_{2t},\cdots,x_{Nt}\right\\}$. The vector of
covariates $\mathbf{x}_{t}=\left(x_{1t},x_{2t},\cdots,x_{Nt}\right)^{\prime}$
is generated as
$\mathbf{x}_{t}=\boldsymbol{R}_{t}^{1/2}\boldsymbol{\varepsilon}_{t}$, where
$\boldsymbol{\varepsilon}_{t}=\left(\varepsilon_{1t},\varepsilon_{2t},\cdots,\varepsilon_{Nt}\right)^{\prime}$.
$\left\\{\varepsilon_{it}\right\\}$ are generated as AR(1) processes with
GARCH(1,1) innovations

$\varepsilon_{it}=\rho_{i\varepsilon}\varepsilon_{i,t-1}+\left(1-\rho_{i\varepsilon}^{2}\right)^{1/2}e_{\varepsilon_{i}t}\text{,
for }t=1,2,\cdots,T,\text{and }i=1,2,...,N,$

using the starting values $\varepsilon_{i,0}\sim IIDN\left(0,1\right)$. The
parameters were generated heterogeneously as independent draws,
$\rho_{i\varepsilon}\sim IIDU\left(0,0.95\right)$. $e_{\varepsilon_{i}t}\sim
IIDN\left(0,\sigma_{\varepsilon_{i},t}^{2}\right)$, with
$\sigma_{\varepsilon_{i},t}^{2}$ given by

$\sigma_{\varepsilon_{i},t}^{2}=(1-\alpha_{1\varepsilon_{i}}-\alpha_{2\varepsilon_{i}})+\alpha_{1\varepsilon_{i}}e_{\varepsilon_{i}t-1}^{2}+\alpha_{2\varepsilon_{i}}\sigma_{\varepsilon_{i},t-1}^{2},$

where $\alpha_{1\varepsilon_{i}}\sim IIDU(0,0.2)$, and
$\alpha_{2\varepsilon_{i}}\sim IIDU(0.6,0.75)$. The error terms,
$\left\\{u_{t}\right\\}_{t=1}^{T}$, in (16) are generated as
$IIDN(0,\sigma_{ut}^{2})$ with $\sigma_{ut}^{2}$ following the GARCH(1,1)
specification

$\sigma_{ut}^{{2}}=(1-\alpha_{1u}-\alpha_{2u})+\alpha_{1u}u_{t-1}^{{2}}+\alpha_{2u}\sigma_{u,t-1}^{{2}},$

using $u_{0}\sim\mathcal{N}(0,1)$, $\alpha_{1u}=0.2$ and $\alpha_{2u}=0.75$.

As our baseline DGP we consider a model with stable parameters, and set
$\beta_{jt}=1$ for $j=1,2,3,4$. We also set $c_{t}=0$ and $\mu_{jt}=1$ in
(17), which yields $d_{t}=4$. In addition, we set $\rho_{y,t}=0$ when the
baseline model is static and $\rho_{y,t}=0.3$ when the baseline model is
dynamic. In the dynamic case we set $y_{0}=(1-\rho_{y,1})^{-1}d_{1}$. In the
case of models with parameter instability we consider a mixed deterministic-
stochastic model and generate $\beta_{jt}$ as

$\beta_{jt}=b_{jt}+\tau_{\eta_{j}}\eta_{jt},\text{ for }j=1,2,3,4,$

where $b_{jt}$ are deterministic and $\eta_{jt}$ are AR(1) processes with
GARCH(1,1) innovations,

$\eta_{jt}=\rho_{\eta j}\eta_{j,t-1}+\left(1-\rho_{\eta
j}^{2}\right)^{1/2}e_{\eta_{j}t},\text{ }$

using the starting values $\eta_{j,0}\sim IID\mathcal{N}\left(0,1\right)$, and
$\rho_{\eta j}=0.5$, for all $j$. $\left\\{e_{\eta_{j}t}\right\\}$ follows a
normal distribution with mean zero, and variance $\sigma_{\eta_{j}t}^{{2}}$
given by

$\sigma_{\eta_{j}t}^{{2}}=(1-\alpha_{1\eta_{j}}-\alpha_{2\eta_{j}})+\alpha_{1\eta_{j}}e_{\eta_{j},t-1}^{{2}}+\alpha_{2\eta_{j}}\sigma_{\eta_{j},t-1}^{{2}}\text{,
for }j=1,2,3,4,$

where $\alpha_{1\eta_{j}}=0.2$ and $\alpha_{2\eta_{j}}=0.75$. We set
$\tau_{\eta_{j}}$ such that deterministic variations in $\beta_{jt}$ are quite
large relative to the stochastic variations. To this end we set
$\tau_{\eta_{j}}$ (using simulations) so that

$\frac{T^{-1}\sum_{t=1}^{T}b_{jt}^{2}}{T^{-1}\sum_{t=1}^{T}\mathbb{E}\left[\left(\beta_{jt}^{\left(r\right)}\right)^{2}\right]}=0.95,\text{
for }j=1,2,3,4.$

For the deterministic components of the slope coefficients ($b_{jt}$, for
$j=1,2,3,4$), we consider the following specifications

$b_{1t}=b_{2t}=\begin{cases}2&\text{if }t\in\\{1,2,\cdots,[T/3]\\},\\\
0&\text{if }t\in\\{[T/3]+1,[T/3]+2,\cdots,[2T/3]\\},\\\ 1&\text{if
}t\in\\{[2T/3]+1,[2T/3]+2,\cdots,T\\},\end{cases}$ (18)

and

$b_{3t}=b_{4t}=\begin{cases}0.5&\text{if }t\in\\{1,2,\cdots,[T/2]\\},\\\
1.5&\text{if }t\in\\{[T/2]+1,[T/2]+2,\cdots,T\\},\end{cases}$ (19)

where $[.]$ is the nearest integer function.

We also set $c_{t}=0$ in (17) and generate the intercept as
$d_{t}=\sum_{j=1}^{k}\beta_{jt}\mu_{jt},$ where

$\mu_{1t}=\mu_{2t}=\begin{cases}0.6&\text{if }t\in\\{1,2,\cdots,[T/3]\\},\\\
1.5&\text{if }t\in\\{[T/3]+1,[T/3]+2,\cdots,[2T/3]\\},\\\ 0.9&\text{if
}t\in\\{[2T/3]+1,[2T/3]+2,\cdots,T\\},\end{cases}$ (20)

and

$\mu_{3t}=\mu_{4t}=\begin{cases}0.9&\text{if }t\in\\{1,2,\cdots,[T/2]\\},\\\
1.1&\text{if }t\in\\{[T/2]+1,[T/2]+2,\cdots,T\\}.\end{cases}$ (21)

In this design, the jumps in $b_{jt}$ and $\mu_{jt}$, for $j=1,2$, have
opposite signs and the jumps in $b_{jt}$ and $\mu_{jt}$, for $j=3,4$, have the
same sign.

The $N\times N$ correlation matrix of the covariates,
$\mathbf{R}_{t}\equiv(r_{ij,t})$, are set as $r_{ij,t}=r_{t}^{|i-j|}$, for all
$i,j=1,2,\cdots,N$. We allow for a break in the correlation matrix and set
$r_{t}$ equal to 0.8 in the first half of the sample and 0.4 in the second
half of the sample. Also, we consider two possibilities for $\rho_{y,t}$. In
the static scenario we set $\rho_{y,t}=0$ for all $t$. In the dynamic scenario
we allow for a switch in $r_{y,t}$ and set it as

$\rho_{y,t}=\begin{cases}0.2&\text{if }t\in\\{1,2,\cdots,[T/2]\\},\\\
0.4&\text{if }t\in\\{[T/2]+1,[T/2]+2,\cdots,T\\}.\end{cases}$ (22)

For the static and dynamic models with parameter instabilities, the parameter
$\tau_{u}$ is calibrated by simulations to ensure that the R-squared of the
linear regression of $y_{t}$ on a constant term, the signal variables
$\left\\{x_{1t},x_{2t},x_{3t},x_{4t}\right\\}$, and (in experiments with
$\rho_{y,t}\neq 0$) the lagged dependent variable is equal to $30\%$ (low fit)
and $50\%$ (high fit). The same value of $\tau_{u}$ is used for the
corresponding static and dynamic models without parameter instabilities.

We base the MC results on $R=2,000$ replications, and consider
$N\in\left\\{20,40,100\right\\}$ and $T\in\left\\{100,200,500\right\\}$,
combinations. These choices of $\left(N,T\right)$ cover our empirical
applications. For each pair of $(N,T)$, there are four experiments in case of
the models with no parameter instabilities, and four experiments in the case
of models with parameter instabilities, corresponding to the two choices of
$\tau_{u}$ (low and high fit), $\rho_{yt}$ (static to dynamic). In total, we
carry out eight different experiments.

###  6.2 Selection and estimation methods using weighted and unweighted
observations

Let $\mathbf{w}_{t}=(\mathbf{x}_{t}^{\prime},y_{t})^{\prime}$,
$t=1,2,\cdots,T$ be the (unweighted) set of available observations, and denote
the corresponding set of down-weighted observations by
$\hat{\mathbf{w}}_{t}(\lambda)=\lambda^{T-t}\mathbf{w}_{t}$ where
$0<\lambda\leq 1$ is the down-weighting coefficient. We are not arguing for
the use of exponential down-weighting – but use it as an example. There are
also non-exponential type down-weighting schemes that one can use, e.g.
Pesaran et al. (2013). We will consider the following selection/estimation
methods: (1) OCMT with down-weighted observations
$\\{\hat{\mathbf{w}}_{t}(\lambda)\\}_{t=1}^{T}$ used at both selection and
estimation stages; (2) OCMT with the unweighted observations,
$\\{\mathbf{w}_{t}\\}_{t=1}^{T}$, used at the selection stage and down-
weighted observations, $\\{\hat{\mathbf{w}}_{t}(\lambda)\\}_{t=1}^{T}$, used
at the estimation stage; (3) OCMT using unweighted observations,
$\\{\mathbf{w}_{t}\\}_{t=1}^{T}$, at both selection and estimation stages;
(4,5 & 6) Lasso, A-Lasso, and boosting also using unweighted observations,
$\\{\mathbf{w}_{t}\\}_{t=1}^{T}$; and (7,8 & 9) Lasso, A-Lasso, and boosting
with down-weighted observations,
$\\{\hat{\mathbf{w}}_{t}(\lambda)\\}_{t=1}^{T}$ used as inputs.

We also implement a two-step procedures based on Lasso, A-Lasso and boosting.
In the first step, we apply Lasso, A-Lasso and boosting to the original
(unweighted) observations and select the variables with non-zero coefficients.
In the second step, we estimate the corresponding post-selected model by LS
using the weighted observations. Overall, the MSFEs of these procedures were
higher than that of direct application of Lasso, A-Lasso and boosting to the
weighted observations. The results are available in Section S-2 of the online
MC supplement.

We consider two sets of values for the down-weighting coefficient, $\lambda$:
(1) Light down-weighting with
$\lambda=\left\\{0.975,0.98,0.985,0.99,0.995,1\right\\}$, and (2) Heavy down-
weighting with $\lambda=\left\\{0.95,0.96,0.97,0.98,0.99,1\right\\}$. For each
of the above two sets of exponential down-weighting schemes (light/heavy) we
focus on simple average forecasts computed over the individual forecasts
obtained for each value of $\lambda$ in the set under consideration.

###  6.3 Simulation results

A summary of the main results are provided in Tables 1 to 3, with additional
summary tables highlighting the effects of down-weighting at the selection
stage, and the differences between static versus dynamic models provided in
the online MC supplement. Table 1 give the number of selected covariates
($\hat{k}_{T}$), TPR and FPR of OCMT, Lasso, A-Lasso and boosting without
down-weighting. Panel A of this table reports the results for different $N$
and $T$ combinations, averaged across the four experiments without parameter
instabilities, and panel B of the table gives the corresponding results for
the four experiments with parameter instabilities. The results show that all
the methods under consideration have higher average TPR for models with stable
parameters compared to the ones with parameter instabilities. This is to be
expected, as the models with parameter instabilities are subject to an
additional source of uncertainty.

We further observe that the lower average TPR of OCMT in the models with
parameter instabilities is associated with a lower average number of selected
covariates, and hence a lower average FPR. On the other hand, the other
procedures tend, on average, to select more covariates in the models with
parameter instabilities and hence have a higher average FPR relative to the
models without parameter instabilities. Lastly, OCMT selects fewer covariates
relative to Lasso, A-Lasso, and boosting, while maintaining the TPR at a
similar level. As a result, OCMT has the lowest average FPR among the
selection methods under consideration across all $N$ and $T$ combinations,
with one exception, namely in the case of models with stable parameters for
$T=500$ and $N=20$, where the FPR for A-Lasso (0.11) is slightly lower than
that of OCMT (0.13). Summary Tables S.1 and S.2 in the online MC supplement
provide further results on the effects of down-weighting on TPR and FPR. The
results consistently show that down-weighting of observations provides no
gains for OCMT in terms of average TPR and FPR. This is also true for other
methods in majority but not all cases.

Table 2 focusses on the one-step-ahead MSFEs and provides comparative results
on the effects of down-weighting across the methods (OCMT, Lasso, A-Lasso and
boosting). As in Table 1, Panel A of Table 2 gives average MSFEs for the four
experiments without parameter instabilities, and Panel B gives the
corresponding results for the experiments with parameter instabilities. As
expected, in the absence of parameter instabilities, using unweighted
observations gives the lowest MSFE across all the methods. Moreover, for all
$N$ and $T$ combinations and different down-weighting scenarios, the average
MSFE of each method is lower in the case of models with stable parameters as
compared to those with parameter instabilities. This observation is consistent
with our finding in Theorem 3 about the cost of time-variation in the
coefficients on the in-sample fit of the estimated model. As can be seen, for
models with parameter instabilities, down-weighting does improve the
forecasting performance of OCMT (with and without down-weighting in the
selection stage), Lasso, A-Lasso, and boosting. However, by comparing the
MSFEs of OCMT with and without down-weighting at the selection stage, we see
that the down-weighting at the selection stage always results in deterioration
of the forecast accuracy of OCMT, which is in line with our main theoretical
result. Last but not least, the results in Table 2 show that OCMT with down-
weighting only at the estimation stage has the lowest average MSFE among all
the methods for all choices of $N$, $T$, and different down-weighting
scenarios. In fact, in the case of experiments with parameter instabilities
OCMT with down-weighting (light or heavy) at the estimation stage only, always
beats Lasso, A-Lasso and boosting with light or heavy down-weighting in terms
of the one-step-ahead MSFE.

Table 3 compares the performance of OCMT with the down-weighting option at the
estimation stage to that of the other procedures, using the same set of down-
weighting parameter ($\lambda$). Specifically, we report the MSFE of Lasso,
A-Lasso, and boosting relative to that of OCMT. Since the relative MSFE
ranking of OCMT, Lasso, A-Lasso, and boosting does not appear to be affected
by no/light/heavy down-weighting options, as a summary measure, we simply
average relative MSFE values across individual experiments and the three
(no/light/heavy) down-weighting options. However, we provide the relative MSFE
results for the models without and with parameter instabilities separately, on
left and right panels of Table 3. Two observations stand out from this table.
First, the reported average relative MSFEs are greater than one for all the
$N$ and $T$ choices, indicating that OCMT outperforms Lasso, A-Lasso, and
boosting in all cases. Second, the degree to which OCMT outperforms Lasso and
A-Lasso tends to increase (particularly for $T>100$) with the degree of
parameter instability. This is less so if we compare OCMT with boosting.

Tables S.4, S.5, and S.6 in the online MC supplement provide further details
about the performance of the methods under consideration in static and dynamic
experiments. In Table S.4, we compare the number of selected covariates, the
TPR, and the FPR of each method without down-weighting across static and
dynamic models. For various $N$ and $T$ combinations the reported results are
averaged across four experiments (with/without parameter instabilities and
with/without high-fit). The results show that all the methods tend to select
fewer covariates in the dynamic models relative to the static ones, and hence
have a lower TPR and FPR. This is expected, as in the dynamic models, part of
the variation in the target variable is explained by its own lag rather than
the signal variables. Consequently, in Tables S.5 and S.6, which are about the
MSFE in static and dynamic models, respectively, we see that all the methods
have a higher MSFE in dynamic models relative to the static ones.
Additionally, the results in Tables S.5 and S.6 show that the MSFE for models
with stable parameters is always lower than the ones with parameter
instabilities, regardless of whether the model is static or not.

Overall, the results of our MC studies suggest that the OCMT procedure without
down-weighting at the selection stage is a useful method to deal with variable
selection in linear regression settings with parameter instability.

##  7 Empirical applications

The rest of the paper considers empirical applications whereby the forecast
performance of the proposed OCMT approach with no down-weighting at the
selection stage is compared with those of Lasso and A-Lasso. In particular, we
consider the following two applications:333We also consider forecasting euro
area quarterly output growth using the European Central Bank (ECB) survey of
professional forecasters as our third application. The results of this
application can be found in Section S-3 of the online empirical supplement.

  * •

Forecasting monthly rate of price changes for 28 (out of 30) stocks in Dow
Jones using a relatively large number of financial, economic, as well as
technical indicators.

  * •

Forecasting quarterly output growth rates across 33 countries using macro and
financial variables.

In each application, we first compare the performance of OCMT with and without
down-weighted observations at the selection stage. We then consider the
comparative performance of OCMT (with variable selection carried out without
down-weighting) relative to Lasso and A-Lasso, with and without down-
weighting. For down-weighting we make use of exponentially down-weighted
observations, namely $\hat{x}_{it}(\lambda)=\lambda^{T-t}x_{it}$, and
$\hat{y}_{t}(\lambda)=\lambda^{T-t}y_{t}$, where $y_{t}$ is the target
variable to be forecasted, $x_{it}$, for $i=1,2,...,N$ are the covariates in
the active set, and $\lambda$ is the exponential decay coefficient. We
consider the same two sets of values for the degree of exponential decay,
$\lambda$, as in the MC section: (1) Light down-weighting with
$\lambda=\left\\{0.975,0.98,0.985,0.99,0.995,1\right\\}$, and (2) Heavy down-
weighting with $\lambda=\left\\{0.95,0.96,0.97,0.98,0.99,1\right\\}$. For each
of the above two sets of exponential down-weighting schemes we again focus on
simple average forecasts computed over the individual forecasts obtained for
each value of $\lambda$ in the set under consideration.

For forecast evaluation we consider Mean Squared Forecasting Error (MSFE) and
Mean Directional Forecast Accuracy (MDFA), together with related pooled
versions of Diebold-Mariano (DM), and Pesaran-Timmermann (PT) test statistics.
A panel version of Diebold and Mariano (2002) test is proposed by Pesaran et
al. (2009). Let $q_{lt}\equiv e_{ltA}^{2}-e_{ltB}^{2}$ be the difference in
the squared forecasting errors of procedures $A$ and $B$, for the target
variable $y_{lt}$ ($l=1,2,...,L)$ and $t=1,2,...,T_{l}^{f}$, where $T_{l}^{f}$
is the number of forecasts for target variable $l$ (could be one or multiple
step ahead) under consideration. Suppose $q_{lt}=\alpha_{l}+\varepsilon_{lt}$
with $\varepsilon_{lt}\sim\mathcal{N}(0,\sigma_{l}^{2})$. Then under the null
hypothesis of $H_{0}:\alpha_{l}=0$ for all $l$ we have

$\overline{DM}=\frac{\bar{q}}{\sqrt{V(\bar{q})}}\overset{a}{\thicksim}\mathcal{N}(0,1),\text{
for }T_{Lf}\rightarrow\infty,\text{ where
}T_{Lf}=\sum_{l=1}^{L}T_{l}^{f},\bar{q}=T_{Lf}^{-1}\sum_{l=1}^{L}\sum_{t=1}^{T_{l}^{f}}q_{lt},\text{
and }$
$V(\bar{q})=\frac{1}{T_{Lf}^{2}}\sum_{l=1}^{L}T_{l}^{f}\hat{\sigma}_{l}^{2},\text{
with
}\hat{\sigma}_{l}^{2}=\frac{1}{T_{l}^{f}}\sum_{t=1}^{T_{l}^{f}}(q_{lt}-\bar{q}_{l})^{2}\text{
and }\bar{q}_{l}=\frac{1}{T_{l}^{f}}\sum_{t=1}^{T_{l}^{f}}q_{lt}.$

Note that $V(\bar{q})$ needs to be modified in the case of multiple-step ahead
forecast errors, due to the serial correlation that results in the forecast
errors from the use of over-lapping observations. There is no adjustment
needed for one-step ahead forecasting, since it is reasonable to assume that
in this case the loss differentials are serially uncorrelated. However, to
handle possible serial correlation for $h$-step ahead forecasting with $h>1$,
we can modify the panel DM test by using the Newey-West type estimator of
$\sigma_{l}^{2}$.

The $MDFA$ statistic compares the accuracy of forecasts in predicting the
direction (sign) of the target variable, and is computed as

$MDFA=100\left\\{\frac{1}{T_{Lf}}\sum_{l=l}^{L}\sum_{t=1}^{T_{l}^{f}}\mathbf{1}[\text{sgn}(y_{lt}y_{lt}^{f})>0]\right\\},$

where $\mathbf{1}(w>0)$ is the indicator function takes the value of $1$ when
$w>0$ and zero otherwise, $\text{sgn}(w)$ is the sign function, $y_{lt}$ is
the actual value of dependent variable at time $t$ and $y_{lt}^{f}$ is its
corresponding predicted value. To evaluate statistical significance of the
directional forecasts for each method, we also report a pooled version of the
test suggested by Pesaran and Timmermann (1992):

$PT=\frac{\hat{P}-\hat{P}^{\ast}}{\sqrt{\hat{V}(\hat{P})-\hat{V}(\hat{P}^{\ast})}},\text{
}$

where $\hat{P}$ is the estimator of the probability of correctly predicting
the sign of $y_{lt}$, computed by

$\hat{P}=\frac{1}{T_{Lf}}\sum_{l=1}^{L}\sum_{t=1}^{T_{l}^{f}}\mathbf{1}[\text{
sgn}(y_{lt}y_{lt}^{f})>0],\text{ and \
}\hat{P}^{\ast}=\bar{d}_{y}\bar{d}_{y^{f}}+(1-\bar{d}_{y})(1-\bar{d}_{y^{f}}),\text{
with}$
$\bar{d}_{y}=\frac{1}{T_{Lf}}\sum_{l=1}^{L}\sum_{t=1}^{T_{l}^{f}}\mathbf{1}[\text{sgn}(y_{lt})>0],\text{
and
}\bar{d}_{y^{f}}=\frac{1}{T_{Lf}}\sum_{l=1}^{L}\sum_{t=1}^{T_{l}^{f}}\mathbf{1}[\text{sgn}(y_{lt}^{f})>0].$

Finally,$\ \hat{V}(\hat{P})=T_{Lf}^{-1}\hat{P}^{\ast}(1-\hat{P}^{\ast}),$ and

$\hat{V}(\hat{P^{\ast}})=\frac{1}{T_{Lf}}(2\bar{d}_{y}-1)^{2}\bar{d}_{y^{f}}(1-\bar{d}_{y^{f}})+\frac{1}{T_{Lf}}(2\bar{d}_{y}^{f}-1)^{2}\bar{d}_{y}(1-\bar{d}_{y})+\frac{4}{T_{Lf}^{2}}\bar{d}_{y}\bar{d}_{y^{f}}(1-\bar{d}_{y})(1-\bar{d}_{y^{f}}).$

The last term of $\hat{V}(\hat{P^{\ast}})$ is negligible and can be ignored.
Under the null hypothesis, that prediction and realization are independently
distributed, PT is asymptotically distributed as a standard normal
distribution.

###  7.1 Forecasting monthly returns of stocks in Dow Jones

In this application the focus is on forecasting one-month ahead stock returns,
defined as monthly change in natural logarithm of stock prices. We consider
stocks that were part of the Dow Jones index in 2017m12, and have non-zero
prices for at least 120 consecutive data points (10 years) over the period
1980m1 and 2017m12. We ended up forecasting 28 blue chip stocks. 444Visa and
DowDuPont are excluded since they have less than 10 years of historical price
data. Daily close prices for all the stocks are obtained from Data Stream. For
stock $i$, the price at the last trading day of each month is used to
construct the corresponding monthly stock prices, $P_{it}$. Finally, monthly
returns are computed by $r_{i,t+1}=100\ln(P_{i,t+1}/P_{it})$, for
$i=1,2,...,28$. For all 28 stocks we use an expanding window starting with the
observations for the first 10 years ($T=120$). The active set for predicting
$r_{i,t+1}$ consists of 40 financial, economic, and technical variables.555All
regressions include the intercept as the only conditioning (pre-selected)
variable. The full list and the description of the indicators considered can
be found in Section S-1 of online empirical supplement.

Overall we computed 8,659 monthly forecasts for the 28 target stocks. The
results are summarized as average forecast performances across the different
variable selection procedures. Table 4 reports the effects of down-weighting
at the selection stage of the OCMT procedure. It is clear that down-weighting
worsens the predictive accuracy of OCMT. From the Panel DM tests, we can also
see that down-weighting at the selection stage worsens the forecasts
significantly. Panel DM test statistics is -5.606 (-11.352) for light (heavy)
versus no down-weighing at the selection stage. Moreover, Table 6 shows that
the OCMT procedure with no down-weighting at the selection stage dominates
Lasso, A-Lasso and boosting in terms of MSFE and the differences are
statistically highly significant.

Further, OCMT outperforms Lasso, A-Lasso and boosting in terms of Mean
Directional Forecast Accuracy (MDFA), measured as the percent number of
correctly signed one-month ahead forecasts across all the 28 stocks over the
period 1990m2-2017m12. See Table 6. As can be seen from this table, OCMT with
no down-weighting performs the best; correctly predicting the direction of
56.057% of 8,659 forecasts, as compared to 55.33%, which we obtain for Lasso,
A-Lasso and boosting forecast, at best. This difference is highly significant
considering the very large number of forecasts involved. It is also of
interest that the better of performance of OCMT is achieved with a much fewer
number of selected covariates as compared to Lasso, A-Lasso and boosting. As
can be seen from the last column of Table 6, Lasso, A-Lasso and boosting on
average select many more covariates than OCMT (1-16 variables as compared to
0.072 for OCMT).

So far we have focused on average performance across all the 28 stocks. Table
8 provides the summary results for individual stocks, showing the relative
performance of OCMT in terms of the number of stocks, using MSFE and MDFA
criteria. The results show that OCMT performs better than Lasso, A-Lasso and
boosting in the majority of the stocks in terms of MSFE and MDFA. OCMT
outperforms Lasso, A-Lasso and boosting in at least 23 out of 28 stocks in
terms of MSFE, under no down-weighting, and almost universally when Lasso,
A-Lasso and boosting are implemented with down-weighting. Similar results are
obtained when we consider MDFA criteria, although the differences in
performance are somewhat less pronounced. Overall, we can conclude that the
better average performance of OCMT (documented in Tables 6 and 6) is not
driven by a few stocks and holds more generally.

###  7.2 Forecasting quarterly output growth rates across 33 countries

We consider one and two years ahead predictions of output growth for 33
countries (20 advanced and 13 emerging). We use quarterly data from $1979Q2$
to $2016Q4$ taken from the GVAR dataset.666The GVAR dataset is available at
https://sites.google.com/site/gvarmodelling/data. We predict
$\Delta_{4}y_{it}=y_{it}-y_{i,t-4}$, and $\Delta_{8}y_{it}=y_{it}-y_{i,t-8},$
where $y_{it}$, is the log of real output for country $i$. We adopt the
following direct forecasting equations:

$\Delta_{h}y_{i,t+h}=y_{i,t+h}-y_{it}=\alpha_{ih}+\lambda_{ih}\Delta_{1}y_{it}+\boldsymbol{\beta}_{ih}^{\prime}\mathbf{x}_{it}+u_{iht},$

where we consider $h=4$ (one-year-ahead forecasts) and $h=8$ (two-years-ahead
forecasts). Given the known persistence in output growth, in addition to the
intercept in the present application we also condition on the most recent
lagged output growth, denoted by $\Delta_{1}y_{it}=y_{it}-y_{i,t-1}$, and
confine the variable selection to list of variables set out in Table S.2 in
the online empirical supplement. Overall, we consider a maximum of 15
covariates in the active set covering quarterly changes in domestic variables
such as real output growth, real short term interest rate, and long-short
interest rate spread and quarterly change in the corresponding foreign
variables.

We use expanding samples, starting with the observations on the first 15 years
(60 data points), and evaluate the forecasting performance of the three
methods over the period 1997Q2 to 2016Q4.

Tables 8 and 10, respectively, report the MSFE of OCMT for one-year and two-
year ahead forecasts of output growth, with and without down-weighting at the
selection stage. Consistent with the previous application, down-weighting at
the selection stage worsens the forecasting accuracy. Moreover, in Tables 10
and 12, we can see that OCMT (without down-weighting at the selection stage)
outperforms Lasso, A-Lasso and boosting in two-year ahead forecasting. In the
case of one-year ahead forecasts, OCMT and Lasso are very close to each other
and both outperform A-Lasso and boosting. Table 12 summarizes country-specific
MSFE and DM findings for OCMT relative to Lasso, A-Lasso and boosting. The
results show OCMT under-performs Lasso in more than half of the countries for
one-year ahead horizon, but outperforms Lasso, A-Lasso and boosting in more
than 70 percent of the countries in the case of two-year ahead forecasts. It
is worth noting that while Lasso generally outperforms OCMT in the case of
one-year ahead forecasts, overall its performance is not statistically
significantly better. See Panel DM test of Table 10. On the other hand we can
see from Table 12 that overall OCMT significantly outperforms Lasso in the
case of the two-year ahead forecasts.

Finally in Tables 14 and 14 we reports MDFA and PT test statistics for OCMT,
Lasso, A-Lasso and boosting. Overall, OCMT has a slightly higher MDFA and
hence predicts the direction of real output growth better than Lasso, A-Lasso
and boosting in most cases. The PT test statistics suggest that while all the
methods perform well in forecasting the direction of one-year ahead real
output growth, none of the methods considered are successful at predicting the
direction of two-year ahead output growth.

It is also worth noting that as with the previous applications, OCMT selects
very few variables from the active set (0.1 on average for both horizons, with
the maximum number of selected variables being 2 for $h=4$ and $8$). On the
other hand, Lasso on average selects 2.7 variables from the active set for
$h=4$, and $1$ variable on average for $h=8$. Maximum number of variables
selected by Lasso is $9$ and $13$ for $h=4$, $8$, respectively (out of
possible $15$). Again as to be expected, A-Lasso selects a fewer number of
variables as compared to Lasso (2.3 and 0.8 on average for $h=4,8$,
respectively), but this does not lead to a better forecast performance in
comparison with Lasso. Boosting on average selects $6.3$ variables from the
active set for $h=4$, and $3.8$ variable on average for $h=8$.

In conclusion, down-weighting at both selection and forecasting stages
deteriorates OCMT’s MSFE for both one–year and two-years ahead forecast
horizons, as compared to down-weighting only at the forecasting stage.
Moreover, light down-weighting at the forecasting stage improves forecasting
performance for both horizons. Statistically significant evidence of
forecasting skill is found for OCMT relative to Lasso only in the case of two-
years ahead forecasts. However, it is interesting that none of the big data
methods can significantly beat the simple (light down-weighted) AR(1) baseline
model.

##  8 Concluding remarks

The penalized regression approach has become the de facto benchmark in the
literature on variable selection in the context of linear regression models.
But, barring a few exceptions (such as Kapetanios and Zikes, 2018), these
studies focus on models with stable parameters, and do not consider the
implications of parameter instabilities for variable selection. Recently,
Chudik et al. (2018) proposed OCMT as an alternative procedure to penalized
regression. One clear advantage of the OCMT procedure is the fact that the
problem of variable selection is separated from the forecasting stage, in
contrast to the penalized regression techniques where the variable selection
and estimation are carried out simultaneously. Using OCMT one can decide
whether to use the weighted observations at the variable selection stage or
not, without preempting whether to down-weight and how to down-weight the
observations at the forecasting stage.

We have provided theoretical arguments for using the unweighted observations
at the selection stage of OCMT, and down-weighted observations at the
forecasting stage of OCMT. Our MC results as well as empirical applications
uniformly suggest that using down-weighted observations at the selection stage
of OCMT deteriorates the forecasting accuracy in terms of mean square forecast
error. Moreover, OCMT without down-weighting at the selection stage is found
to outperform penalized regression methods such as Lasso and A-Lasso, as well
as boosting.

Table 1: The number of selected variables ($\hat{k}_{T}$), True Positive Rate
(TRP), and False Positive Rate (FPR) averaged across Monte Carlo experiments
with and without parameter instabilities.

| $\hat{k}_{T}$ |  | TPR |  | FPR
---|---|---|---|---|---
$N\backslash T$ | 100 | 200 | 500 |  | 100 | 200 | 500 |  | 100 | 200 | 500
A. Without parameter instabilities
| OCMT
20 | 3.91 | 5.15 | 6.68 |  | 0.80 | 0.95 | 1.00 |  | 0.03 | 0.07 | 0.13
40 | 3.69 | 5.01 | 6.52 |  | 0.77 | 0.94 | 1.00 |  | 0.02 | 0.03 | 0.06
100 | 3.47 | 4.74 | 6.26 |  | 0.73 | 0.91 | 1.00 |  | 0.01 | 0.01 | 0.02
| Lasso
20 | 6.93 | 7.33 | 7.54 |  | 0.86 | 0.94 | 0.99 |  | 0.18 | 0.18 | 0.18
40 | 8.48 | 8.89 | 8.99 |  | 0.83 | 0.93 | 0.99 |  | 0.13 | 0.13 | 0.13
100 | 11.17 | 10.95 | 10.98 |  | 0.80 | 0.91 | 0.98 |  | 0.08 | 0.07 | 0.07
| A-Lasso
20 | 5.39 | 5.81 | 6.06 |  | 0.77 | 0.88 | 0.97 |  | 0.12 | 0.11 | 0.11
40 | 6.75 | 7.28 | 7.45 |  | 0.76 | 0.89 | 0.97 |  | 0.09 | 0.09 | 0.09
100 | 9.27 | 9.45 | 9.70 |  | 0.75 | 0.88 | 0.97 |  | 0.06 | 0.06 | 0.06
| Boosting
20 | 9.19 | 9.58 | 9.74 |  | 0.90 | 0.95 | 0.99 |  | 0.28 | 0.29 | 0.29
40 | 16.04 | 16.52 | 16.78 |  | 0.89 | 0.96 | 0.99 |  | 0.31 | 0.32 | 0.32
100 | 35.32 | 36.64 | 37.72 |  | 0.88 | 0.95 | 0.99 |  | 0.32 | 0.33 | 0.34
B. With parameter instabilities
| OCMT
20 | 3.25 | 4.55 | 5.93 |  | 0.69 | 0.90 | 0.99 |  | 0.02 | 0.05 | 0.10
40 | 3.10 | 4.41 | 5.85 |  | 0.66 | 0.88 | 0.99 |  | 0.01 | 0.02 | 0.05
100 | 2.96 | 4.23 | 5.71 |  | 0.61 | 0.85 | 0.99 |  | 0.01 | 0.01 | 0.02
| Lasso
20 | 7.60 | 8.39 | 9.20 |  | 0.78 | 0.89 | 0.97 |  | 0.22 | 0.24 | 0.27
40 | 10.16 | 11.71 | 12.83 |  | 0.75 | 0.88 | 0.97 |  | 0.18 | 0.20 | 0.22
100 | 14.54 | 16.61 | 19.82 |  | 0.72 | 0.85 | 0.96 |  | 0.12 | 0.13 | 0.16
| A-Lasso
20 | 5.84 | 6.58 | 7.40 |  | 0.68 | 0.81 | 0.93 |  | 0.16 | 0.17 | 0.18
40 | 7.97 | 9.40 | 10.51 |  | 0.68 | 0.82 | 0.94 |  | 0.13 | 0.15 | 0.17
100 | 11.58 | 13.70 | 16.73 |  | 0.66 | 0.81 | 0.94 |  | 0.09 | 0.10 | 0.13
| Boosting
20 | 9.76 | 10.39 | 10.95 |  | 0.84 | 0.92 | 0.98 |  | 0.32 | 0.34 | 0.35
40 | 17.50 | 18.72 | 19.50 |  | 0.84 | 0.92 | 0.98 |  | 0.35 | 0.38 | 0.39
100 | 37.33 | 40.37 | 43.22 |  | 0.83 | 0.91 | 0.98 |  | 0.34 | 0.37 | 0.39

Notes: There are $k=4$ signal variables out of N observed covariates. The
reported results for OCMT, Lasso, A-Lasso, and boosting in the table are based
on the original (not down-weighted) observations. Each experiment is based on
2000 Monte Carlo replications. See Section 6 for the detailed description of
the Monte Carlo design.

Table 2: The effects of down-weighting on one-step-ahead MSFE of OCMT, Lasso,
A-Lasso and boosting averaged across all MC experiments with and without
parameter instabilities.

Down-weighting†: | No | Light | Heavy |  | No | Light | Heavy |  | No | Light | Heavy
---|---|---|---|---|---|---|---|---|---|---|---
$N\backslash T$ | 100 |  | 200 |  | 300
A. Without parameter instabilities
| OCMT(Down-weighting only at the estimation stage)
20 | 29.35 | 30.16 | 31.47 |  | 24.08 | 24.76 | 25.90 |  | 22.97 | 24.00 | 25.53
40 | 26.84 | 27.22 | 28.03 |  | 29.45 | 31.00 | 32.77 |  | 22.98 | 23.70 | 25.12
100 | 27.37 | 27.57 | 28.29 |  | 26.72 | 27.22 | 28.68 |  | 22.91 | 24.45 | 26.59
| OCMT(Down-weighting at the variable selection and estimation stages)
20 | 29.35 | 30.30 | 32.63 |  | 24.08 | 25.31 | 28.44 |  | 22.97 | 25.07 | 30.62
40 | 26.84 | 27.30 | 29.63 |  | 29.45 | 31.82 | 38.12 |  | 22.98 | 27.52 | 41.30
100 | 27.37 | 28.12 | 31.56 |  | 26.72 | 28.88 | 36.87 |  | 22.91 | 33.50 | 62.79
| Lasso
20 | 29.54 | 31.02 | 33.03 |  | 24.29 | 25.26 | 27.02 |  | 23.02 | 24.31 | 26.21
40 | 27.39 | 28.66 | 31.70 |  | 29.49 | 31.38 | 35.20 |  | 23.10 | 24.84 | 27.83
100 | 28.13 | 30.70 | 34.19 |  | 27.09 | 29.29 | 32.73 |  | 23.12 | 26.34 | 30.70
| A-Lasso
20 | 30.85 | 32.20 | 34.40 |  | 24.95 | 25.82 | 28.01 |  | 23.14 | 24.72 | 26.99
40 | 29.34 | 30.47 | 33.70 |  | 30.48 | 32.62 | 36.27 |  | 23.58 | 25.71 | 29.36
100 | 32.28 | 34.76 | 37.97 |  | 29.51 | 31.97 | 35.50 |  | 23.86 | 28.35 | 33.14
| Boosting
20 | 31.69 | 35.26 | 40.62 |  | 25.07 | 28.70 | 34.11 |  | 23.54 | 28.08 | 32.73
40 | 30.11 | 34.24 | 38.90 |  | 31.83 | 38.98 | 43.98 |  | 23.91 | 30.32 | 35.23
100 | 35.20 | 39.43 | 42.23 |  | 31.33 | 36.17 | 39.97 |  | 24.78 | 33.06 | 37.88
B. With parameter instabilities
| OCMT(Down-weighting only at the estimation stage)
20 | 33.68 | 32.77 | 33.24 |  | 27.18 | 25.75 | 26.51 |  | 26.26 | 24.58 | 26.04
40 | 30.55 | 29.54 | 29.77 |  | 32.65 | 31.93 | 33.22 |  | 25.68 | 24.23 | 25.47
100 | 31.60 | 30.75 | 30.90 |  | 30.68 | 28.96 | 30.00 |  | 26.63 | 25.00 | 26.84
| OCMT(Down-weighting at the variable selection and estimation stages)
20 | 33.68 | 33.38 | 34.79 |  | 27.18 | 26.48 | 29.45 |  | 26.26 | 26.00 | 31.46
40 | 30.55 | 30.08 | 32.15 |  | 32.65 | 33.29 | 40.09 |  | 25.68 | 28.63 | 42.35
100 | 31.60 | 31.46 | 35.07 |  | 30.68 | 31.10 | 40.90 |  | 26.63 | 34.56 | 64.47
| Lasso
20 | 34.50 | 34.48 | 35.64 |  | 27.84 | 26.76 | 28.28 |  | 26.58 | 25.25 | 27.31
40 | 31.31 | 31.26 | 33.65 |  | 33.27 | 33.01 | 36.52 |  | 26.30 | 26.09 | 29.17
100 | 32.24 | 33.68 | 36.73 |  | 31.76 | 31.52 | 34.56 |  | 27.18 | 27.63 | 31.98
| A-Lasso
20 | 35.79 | 35.35 | 36.76 |  | 28.28 | 26.99 | 29.06 |  | 26.60 | 25.49 | 28.02
40 | 33.29 | 33.05 | 35.76 |  | 34.34 | 34.04 | 37.73 |  | 26.46 | 26.69 | 30.62
100 | 36.14 | 37.72 | 40.65 |  | 34.54 | 34.29 | 37.60 |  | 27.95 | 29.59 | 34.51
| Boosting
20 | 35.23 | 37.90 | 42.76 |  | 27.68 | 29.95 | 35.67 |  | 26.42 | 29.29 | 34.28
40 | 33.68 | 37.73 | 42.27 |  | 34.26 | 40.31 | 45.49 |  | 26.21 | 31.80 | 37.08
100 | 37.49 | 42.43 | 45.35 |  | 34.32 | 38.41 | 42.21 |  | 27.68 | 34.52 | 39.77

Notes: The reported results are averaged across four experiments (with/without
dynamics and with/with high-fit) for models with and without parameter
instabilities. See Section 6 for the description of the Monte Carlo design.
Full set of results is presented in the online Monte Carlo supplement.

†For each of the two sets of exponential down-weighting (light/heavy)
forecasts of the target variable are computed as the simple average of the
forecasts obtained using the down-weighting coefficient, $\lambda$.

Table 3: One-step-ahead MSFE of Lasso, A-Lasso and boosting relative to OCMT
averaged across MC experiments with and without parameter instabilities.

$N\backslash T$ | 100 | 200 | 500 |  | 100 | 200 | 500
---|---|---|---|---|---|---|---
| A. Without parameter instabilities |  | B. With parameter instabilities
| Lasso
20 | 1.038 | 1.024 | 1.015 |  | 1.067 | 1.047 | 1.033
40 | 1.072 | 1.027 | 1.051 |  | 1.090 | 1.062 | 1.081
100 | 1.128 | 1.083 | 1.080 |  | 1.119 | 1.111 | 1.116
| A-Lasso
20 | 1.077 | 1.052 | 1.031 |  | 1.093 | 1.061 | 1.042
40 | 1.144 | 1.068 | 1.089 |  | 1.157 | 1.101 | 1.107
100 | 1.268 | 1.179 | 1.149 |  | 1.244 | 1.211 | 1.187
| Boosting
20 | 1.191 | 1.182 | 1.160 |  | 1.168 | 1.186 | 1.175
40 | 1.260 | 1.215 | 1.241 |  | 1.275 | 1.226 | 1.262
100 | 1.408 | 1.304 | 1.279 |  | 1.338 | 1.297 | 1.302

Notes: This table reports MSFE of Lasso, A-Lasso and boosting relative to MSFE
of OCMT. Relative MSFE values are averaged across experiments and across the
three options for down-weighting: no down-weighting (for all methods), light
down-weighting of observations prior to Lasso, A-Lasso and boosting procedures
relative to OCMT with light down-weighting only at the estimation stage, and
heavy down-weighting of observations prior to Lasso, A-Lasso and boosting
methods compared with OCMT with heavy down-weighting only at the estimation
stage. See Section 6 for the description of the Monte Carlo design. Full set
of results is presented in the online Monte Carlo supplement.

Table 4: Mean square forecast error (MSFE) and panel DM test of OCMT of one-
month ahead monthly return forecasts across the 28 stocks in Dow Jones index
between 1990m2 and 2017m12 (8659 forecasts)

|  Down-weighting at† |  |  |
---|---|---|---|---
| Selection stage | Forecasting stage |  | MSFE
(M1) | no | no |  | 61.231
Light Down-weighting, $\lambda=\left\\{0.975,0.98,0.985,0.99,0.995,1\right\\}$
(M2) | no | yes |  | 61.641
(M3) | yes | yes |  | 68.131
Heavy Down-weighting, $\lambda=\left\\{0.95,0.96,0.97,0.98,0.99,1\right\\}$
(M4) | no | yes |  | 62.163
(M5) | yes | yes |  | 86.073
Pair-wise panel DM tests
| Light down-weighting |  | Heavy down-weighting
| (M2) | (M3) |  | (M4) | (M5)
(M1) | -1.528 | -5.643 | (M1) | -2.459 | -11.381
(M2) | - | -5.606 | (M4) | - | -11.352

Notes: The active set consists of 40 covariates. The conditioning set only
contains an intercept.

†For each of the two sets of exponential down-weighting (light/heavy)
forecasts of the target variable are computed as the simple average of the
forecasts obtained using the down-weighting coefficient, $\lambda$, in the
“light” or the “heavy” down-weighting set under consideration. See footnote to
Table S.4.

Table 5: Mean square forecast error (MSFE) and panel DM test of OCMT versus
Lasso, A-Lasso and boosting of one-month ahead monthly return forecasts across
the 28 stocks in Dow Jones index between 1990m2 and 2017m12 (8659 forecasts)

| MSFE under different down-weighting scenarios
---|---
| No down-weighting |  Light down-weighting† |  Heavy down-weighting‡
OCMT | 61.231 | 61.641 | 62.163
Lasso | 61.849 | 63.201 | 69.145
A-Lasso | 63.069 | 65.017 | 72.038
Boosting | 76.509 | 112.781 | 171.455
| Selected pair-wise panel DM tests
| No down-weighting | Light down-weighting | Heavy down-weighting
| Lasso | A-Lasso | Boosting | Lasso | A-Lasso | Boosting | Lasso | A-Lasso | Boosting
OCMT | -1.533 | -4.934 | -16.406 | -2.956 | -6.025 | -21.188 | -7.676 | -10.261 | -27.414
Lasso | - | -4.661 | -14.973 | - | -6.885 | -20.477 | - | -9.569 | -26.419
A-Lasso | - | - | -14.450 | - | - | -20.415 | - | - | -26.134

Notes: The active set consists of 40 covariates. The conditioning set contains
only the intercept.

† Light down-weighted forecasts are computed as simple averages of forecasts
obtained using the down-weighting coefficient,
$\lambda=\\{0.975,0.98,0.985,0.99,0.995,1\\}$.

‡ Heavy down-weighted forecasts are computed as simple averages of forecasts
obtained using the down-weighting coefficient,
$\lambda=\\{0.95,0.96,0.97,0.98,0.99,1\\}$.

Table 6: Mean directional forecast accuracy (MDFA) and the average number of
selected variables ($\hat{k}$) of OCMT, Lasso, A-Lasso and boosting of one-
month ahead monthly return forecasts across the 28 stocks in Dow Jones index
between 1990m2 and 2017m12 (8659 forecasts).

| Down-weighting | MDFA | $\hat{k}$
---|---|---|---
OCMT | No | 56.057 | 0.072
|  Light† | 55.330 | 0.072
|  Heavy‡ | 54.302 | 0.072
Lasso | No | 55.364 | 1.659
| Light | 54.221 | 2.133
| Heavy | 53.205 | 3.794
A-Lasso | No | 54.648 | 1.312
| Light | 53.840 | 1.623
| Heavy | 52.951 | 2.855
Boosting | No | 52.142 | 9.495
| Light | 49.902 | 15.004
| Heavy | 48.851 | 16.270

Notes: The active set consists of 40 variables. The conditioning set contains
an intercept.

† Light down-weighted forecasts are computed as simple averages of forecasts
obtained using the down-weighting coefficient,
$\lambda=\\{0.975,0.98,0.985,0.99,0.995,1\\}$.

‡ Heavy down-weighted forecasts are computed as simple averages of forecasts
obtained using the down-weighting coefficient,
$\lambda=\\{0.95,0.96,0.97,0.98,0.99,1\\}$.

Table 7: The number of stocks out of the 28 stocks in Dow Jones index where
OCMT outperforms/underperforms Lasso, A-Lasso and boosting in terms of mean
square forecast error (MSFE), panel DM test and mean directional forecast
accuracy (MDFA) between 1990m2 and 2017m12 (8659 forecasts).

| MSFE
---|---
| Down- | OCMT | OCMT significantly | OCMT | OCMT significantly
| weighting | outperforms | outperforms | underperforms | underperforms
Lasso | No | 23 | 4 | 5 | 2
|  Light† | 25 | 5 | 3 | 0
|  Heavy‡ | 26 | 14 | 2 | 0
A-Lasso | No | 24 | 9 | 4 | 2
| Light | 27 | 10 | 1 | 0
| Heavy | 28 | 24 | 0 | 0
Boosting | No | 28 | 23 | 0 | 2
| Light | 28 | 28 | 0 | 0
| Heavy | 28 | 28 | 0 | 0
| MDFA |  |
| Down- | OCMT | OCMT |  |
| weighting | outperforms | underperforms |  |
Lasso | No | 14 | 6 |  |
| Light | 24 | 4 |  |
| Heavy | 17 | 10 |  |
A-Lasso | No | 18 | 4 |  |
| Light | 21 | 3 |  |
| Heavy | 19 | 7 |  |
Boosting | No | 23 | 5 |  |
| Light | 21 | 4 |  |
| Heavy | 19 | 3 |  |

Notes: The active set consists of 40 variables. The conditioning set only
contains an intercept.

† Light down-weighted forecasts are computed as simple averages of forecasts
obtained using the down-weighting coefficient,
$\lambda=\\{0.975,0.98,0.985,0.99,0.995,1\\}$.

‡ Heavy down-weighted forecasts are computed as simple averages of forecasts
obtained using the down-weighting coefficient,
$\lambda=\\{0.95,0.96,0.97,0.98,0.99,1\\}$.

Table 8: Mean square forecast error (MSFE) and panel DM test of OCMT of one-
year ahead output growth forecasts across 33 countries over the period
1997Q2-2016Q4 (2607 forecasts)

|  Down-weighting at† |  MSFE ($\times 10^{4}$)
---|---|---
| Selection stage | Forecasting stage | All | Advanced | Emerging
(M$1$) | no | no | 11.246 | 7.277 | 17.354
Light down-weighting, $\lambda=\left\\{0.975,0.98,0.985,0.99,0.995,1\right\\}$
(M$2$) | no | yes | 10.836 | 6.913 | 16.871
(M$3$) | yes | yes | 10.919 | 6.787 | 17.275
Heavy down-weighting, $\lambda=\left\\{0.95,0.96,0.97,0.98,0.99,1\right\\}$
(M$4$) | no | yes | 11.064 | 7.187 | 17.028
(M$5$) | yes | yes | 11.314 | 6.906 | 18.094
Pair-wise panel DM tests (all countries)
| Light down-weighting |  | Heavy down-weighting
|  (M$2$) |  (M$3$) |  |  (M$4$) |  (M$5$)
(M$1$) | 2.394 | 1.662 |  (M$1$) | 0.668 | -0.204
(M$2$) | - | -0.780 |  (M$4$) | - | -1.320

Notes: There are up to 15 macro and financial variables in the active set.

†For each of the two sets of exponential down-weighting (light/heavy)
forecasts of the target variable are computed as the simple average of the
forecasts obtained using the down-weighting coefficient, $\lambda$, in the
“light” or the “heavy” down-weighting set under consideration.

Table 9: Mean square forecast error (MSFE) and panel DM test of OCMT of two-
year ahead output growth forecasts across 33 countries over the period
1997Q2-2016Q4 (2343 forecasts)

|  Down-weighting at† |  MSFE ($\times 10^{4}$)
---|---|---
| Selection stage | Forecasting stage | All | Advanced | Emerging
(M$1$) | no | no | 9.921 | 7.355 | 13.867
Light down-weighting, $\lambda=\left\\{0.975,0.98,0.985,0.99,0.995,1\right\\}$
(M$2$) | no | yes | 9.487 | 6.874 | 13.505
(M$3$) | yes | yes | 9.549 | 6.848 | 13.704
Heavy down-weighting, $\lambda=\left\\{0.95,0.96,0.97,0.98,0.99,1\right\\}$
(M$4$) | no | yes | 9.734 | 7.027 | 13.898
(M$5$) | yes | yes | 10.389 | 7.277 | 15.177
Pair-wise panel DM test (all countries)
| Light down-weighting |  | Heavy down-weighting
|  (M$2$) |  (M$3$) |  |  (M$4$) |  (M$5$)
(M$1$) | 3.667 | 2.827 |  (M$1$) | 0.943 | -1.664
(M$2$) | - | -1.009 |  (M$4$) | - | -3.498

Notes: There are up to 15 macro and financial variables in the active set.

†For each of the two sets of exponential down-weighting (light/heavy)
forecasts of the target variable are computed as the simple average of the
forecasts obtained using the down-weighting coefficient, $\lambda$, in the
”light” or the ”heavy” down-weighting set under consideration..

Table 10: Mean square forecast error (MSFE) and panel DM test of OCMT versus
Lasso, A-Lasso and boosting for one-year ahead output growth forecasts across
33 countries over the period1997Q2-2016Q4 (2607 forecasts)

| MSFE under different down-weighting scenarios
---|---
| No down-weighting |  Light down-weighting† |  Heavy down-weighting‡
| All | Adv.* | Emer.** | All | Adv. | Emes | All | Adv. | Emes
OCMT | 11.246 | 7.277 | 17.354 | 10.836 |  6.913 | 16.871 | 11.064 | 7.187 | 17.028
Lasso | 11.205 | 6.975 | 17.714 | 10.729 |  6.427 | 17.347 | 11.749 | 7.186 | 18.769
A-Lasso | 11.579 | 7.128 | 18.426 | 11.153 |  6.548 | 18.236 | 12.254 | 7.482 | 19.595
Boosting | 11.663 | 7.007 | 18.826 | 11.132 |  6.515 | 18.234 | 12.375 | 7.331 | 20.135
| Pair-wise Panel DM tests (All countries)
| No down-weighting | Light down-weighting | Heavy down-weighting
| Lasso | A-Lasso | Boosting | Lasso | A-Lasso | Boosting | Lasso | A-Lasso | Boosting
OCMT | 0.220 | -1.079 | -1.228 | 0.486 |  -1.007 | -0.942 | -1.799 | -2.441 | -2.603
Lasso | - | -2.625 | -1.941 | - | -3.626 | -2.766 | - | -3.157 | -3.260
A-Lasso | - | - | -0.381 | - | - |  0.202 | - | - | -0.714

Notes: There are up to 15 macro and financial covariates in the active set.

† Light down-weighted forecasts are computed as simple averages of forecasts
obtained using the down-weighting coefficient,
$\lambda=\\{0.975,0.98,0.985,0.99,0.995,1\\}$.

‡ Heavy down-weighted forecasts are computed as simple averages of forecasts
obtained using the down-weighting coefficient,
$\lambda=\\{0.95,0.96,0.97,0.98,0.99,1\\}$.

∗ Adv. stands for advanced economies.

∗∗ Emer. stands for emerging economies.

Table 11: Mean square forecast error (MSFE) and panel DM test of OCMT versus
Lasso, A-Lasso and boosting of two-year ahead output growth forecasts across
33 countries over the period1997Q2-2016Q4 (2343 forecasts)

| MSFE under different down-weighting scenarios
---|---
| No down-weighting |  Light down-weighting† |  Heavy down-weighting‡
| All | Adv.* | Emer.** | All | Adv. | Emes | All | Adv. | Emes
OCMT | 9.921 | 7.355 | 13.867 | 9.487 |  6.874 | 13.505 | 9.734 | 7.027 | 13.898
Lasso | 10.151 | 7.583 | 14.103 | 9.662 |  7.099 | 13.605 | 10.202 | 7.428 | 14.469
A-Lasso | 10.580 | 7.899 | 14.705 | 10.090 |  7.493 | 14.087 | 11.008 | 8.195 | 15.336
Boosting | 10.356 | 7.758 | 14.352 | 10.006 |  7.402 | 14.012 | 11.461 | 8.606 | 15.855
| Pair-wise Panel DM tests (All countries)
| No down-weighting | Light down-weighting | Heavy down-weighting
| Lasso | A-Lasso | Boosting | Lasso | A-Lasso | Boosting | Lasso | A-Lasso | Boosting
OCMT | -2.684 | -4.200 | -3.442 | -2.137 |  -4.015 | -3.569 | -3.606 | -4.789 | -5.279
Lasso | - | -5.000 | -2.601 | - | -4.950 | -3.441 | - | -4.969 | -5.178
A-Lasso | - | - | 2.541 | - | - |  1.052 | - | - | -2.676

Notes: There are up to 15 macro and financial covariates in the active set.

† Light down-weighted forecasts are computed as simple averages of forecasts
obtained using the down-weighting coefficient,
$\lambda=\\{0.975,0.98,0.985,0.99,0.995,1\\}$.

‡ Heavy down-weighted forecasts are computed as simple averages of forecasts
obtained using the down-weighting coefficient,
$\lambda=\\{0.95,0.96,0.97,0.98,0.99,1\\}$.

∗ Adv. stands for advanced economies.

∗∗ Emer. stands for emerging economies.

Table 12: The number of countries out of the 33 countries where OCMT
outperforms/underperforms Lasso, A-Lasso and boosting in terms of mean square
forecast error (MSFE) and panel DM test over the period 1997Q2 -2016Q4

|  |  | OCMT |  | OCMT
---|---|---|---|---|---
| Down- | OCMT | significantly | OCMT | significantly
| weighting | outperforms | outperforms | underperforms | underperforms
|  One-years-ahead horizon ($h=4$ quarters)
Lasso | No | 13 | 0 | 20 | 3
|  Light† | 12 | 1 | 21 | 3
|  Heavy‡ | 17 | 1 | 16 | 3
A-Lasso | No | 16 | 1 | 17 | 2
| Light | 14 | 2 | 19 | 2
| Heavy | 19 | 1 | 14 | 0
Boosting | No | 11 | 1 | 22 | 1
| Light | 15 | 1 | 18 | 3
| Heavy | 17 | 2 | 16 | 0
|  Two-years-ahead horizon ($h=8$ quarters)
Lasso | No | 24 | 1 | 9 | 0
| Light | 25 | 1 | 8 | 1
| Heavy | 25 | 1 | 8 | 0
A-Lasso | No | 25 | 2 | 8 | 0
| Light | 28 | 3 | 5 | 1
| Heavy | 30 | 3 | 3 | 0
Boosting | No | 25 | 2 | 8 | 0
| Light | 26 | 2 | 7 | 0
| Heavy | 32 | 3 | 1 | 0

Notes: There are up to 15 macro and financial covariates in the active set.

†Light down-weighted forecasts are computed as simple averages of forecasts
obtained using the down-weighting coefficient,
$\lambda=\\{0.975,0.98,0.985,0.99,0.995,1\\}$.

‡ Heavy down-weighted forecasts are computed as simple averages of forecasts
obtained using the down-weighting coefficient,
$\lambda=\\{0.95,0.96,0.97,0.98,0.99,1\\}$.

Table 13: Mean directional forecast accuracy (MDFA) and PT test of OCMT,
Lasso, A-Lasso and boosting for one-year ahead output growth forecasts over
the period 1997Q2-2016Q4 (2607 forecasts)

| Down- | MDFA |  | PT tests
---|---|---|---|---
| weighting | All | Advanced | Emerging |  | All | Advanced | Emerging
OCMT | No | 87.6 | 87.4 | 88.0 |  | 8.12 | 7.40 | 3.48
|  Light† | 87.4 | 87.1 | 87.8 |  | 7.36 | 6.95 | 2.53
|  Heavy‡ | 86.8 | 86.3 | 87.5 |  | 6.25 | 5.93 | 1.95
Lasso | No | 86.2 | 86.7 | 85.3 |  | 9.64 | 9.15 | 3.80
| Light | 87.1 | 87.1 | 87.1 |  | 8.12 | 8.22 | 2.26
| Heavy | 86.0 | 85.8 | 86.4 |  | 6.24 | 6.43 | 1.40
A-Lasso | No | 87.3 | 87.3 | 87.2 |  | 10.80 | 9.91 | 4.75
| Light | 86.5 | 86.6 | 86.4 |  | 8.25 | 8.36 | 2.48
| Heavy | 85.5 | 85.3 | 85.7 |  | 6.84 | 6.92 | 1.88
Boosting | No | 86.5 | 87.1 | 85.7 |  | 9.01 | 9.20 | 3.37
| Light | 86.2 | 86.7 | 85.3 |  | 7.67 | 6.26 | 8.64
| Heavy | 84.9 | 85.2 | 84.5 |  | 5.45 | 6.06 | 1.06

Notes: There are up to 15 macro and financial variables in the active set.

† Light down-weighted forecasts are computed as simple averages of forecasts
obtained using the down-weighting coefficient,
$\lambda=\\{0.975,0.98,0.985,0.99,0.995,1\\}$.

‡ Heavy down-weighted forecasts are computed as simple averages of forecasts
obtained using the down-weighting coefficient,
$\lambda=\\{0.95,0.96,0.97,0.98,0.99,1\\}$.

Table 14: Mean directional forecast accuracy (MDFA) and PT test of OCMT,
Lasso, A-Lasso and boosting for two-year ahead output growth forecasts over
the period 1997Q2-2016Q4 (2343 forecasts)

| Down- | MDFA |  | PT tests
---|---|---|---|---
| weighting | All | Advanced | Emerging |  | All | Advanced | Emerging
OCMT | No | 88.0 | 86.7 | 89.9 |  | 0.52 | 0.00 | 0.47
|  Light† | 87.7 | 86.6 | 89.3 |  | 1.11 | 0.39 | 0.94
|  Heavy‡ | 87.0 | 85.8 | 88.8 |  | 0.50 | 0.89 | 0.34
Lasso | No | 87.2 | 86.2 | 88.7 |  | 0.77 | 0.60 | 0.66
| Light | 87.5 | 86.3 | 89.4 |  | 0.07 | 0.79 | 0.88
| Heavy | 86.8 | 85.5 | 88.8 |  | 1.54 | 1.87 | 0.34
A-Lasso | No | 87.0 | 85.6 | 89.2 |  | 0.33 | 0.13 | 1.00
| Light | 87.1 | 85.9 | 88.9 |  | 1.03 | 1.82 | 1.10
| Heavy | 86.2 | 84.8 | 88.4 |  | 1.53 | 1.92 | 0.62
Boosting | No | 87.2 | 85.6 | 89.6 |  | 0.31 | 0.44 | 1.25
| Light | 87.2 | 86.2 | 88.7 |  | 1.19 | 2.36 | 1.20
| Heavy | 85.9 | 84.8 | 87.6 |  | 2.15 | 2.96 | 1.00

Notes: There are up to 15 macro and financial variables in the active set.

†Light down-weighted forecasts are computed as simple averages of forecasts
obtained using the down-weighting coefficient,
$\lambda=\\{0.975,0.98,0.985,0.99,0.995,1\\}$.

‡ Heavy down-weighted forecasts are computed as simple averages of forecasts
obtained using the down-weighting coefficient,
$\lambda=\\{0.95,0.96,0.97,0.98,0.99,1\\}$.

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Online Theory Supplement to

“Variable Selection in High Dimensional Linear Regressions with Parameter
Instability”

Alexander Chudik

Federal Reserve Bank of Dallas

M. Hashem Pesaran

University of Southern California, USA and Trinity College, Cambridge, UK

Mahrad Sharifvaghefi

University of Pittsburgh

This online theory supplement has three sections. Section S-1 provides the
proofs of Theorems 1 to 3, and additional propositions and corollaries.
Section S-2 establishes the main lemmas needed for the proof of the theorems
in Section S-1. Section S-3 contains the complementary lemmas needed for the
proofs of the main lemmas in the previous section.

Notations: Generic finite positive constants are denoted by $C_{i}$ for
$i=1,2,\cdots$ and $c$. They can take different values in different instances.
$\lVert\mathbf{A}\rVert_{2}$, $\lVert\mathbf{A}\rVert_{F}$,
$\lVert\mathbf{A}\rVert_{\infty}$ and $\lVert\mathbf{A}\rVert_{1}$ denote the
spectral, Frobenius, row, and column norms of matrix $\mathbf{A}$,
respectively. $\lambda_{i}(\mathbf{A})$ denotes the $i^{th}$ eigenvalue of a
square matrix $\mathbf{A}$. $\text{tr}(\mathbf{A})$ and
$\text{det}(\mathbf{A})$ are the trace and determinant of a square matrix
$\mathbf{A}$, respectively. $\left\|\mathbf{x}\right\|$ denotes the $\ell_{2}$
norm of vector $\mathbf{x}$. If $\\{f_{n}\\}_{n=1}^{\infty}$ is any real
sequence and $\\{g_{n}\\}_{n=1}^{\infty}$ is a sequence of positive real
numbers, then $f_{n}=O(g_{n})$, if there exists a positive constant $C_{0}$
and $n_{0}$ such that $\lvert f_{n}\rvert/g_{n}\leq C_{0}$ for all $n>n_{0}$.
$f_{n}=o(g_{n})$ if $f_{n}/g_{n}\rightarrow 0$ as $n\rightarrow\infty$. If
$\\{f_{n}\\}_{n=1}^{\infty}$ and $\\{g_{n}\\}_{n=1}^{\infty}$ are both
positive sequences of real numbers, then $f_{n}=\ominus(g_{n})$ if there exist
$n_{0}\geq 1$ and positive constants $C_{0}$ and $C_{1}$, such that
$\inf_{n\geq n_{0}}\left(f_{n}/g_{n}\right)\geq C_{0}$ and $\sup_{n\geq
n_{0}}\left(f_{n}/g_{n}\right)\leq C_{1}$. If $\\{f_{n}\\}_{n=1}^{\infty}$ is
a sequence of random variables and $\\{g_{n}\\}_{n=1}^{\infty}$ is a sequence
of positive real numbers, then $f_{n}=O_{p}(g_{n})$, if for any
$\varepsilon>0$, there exists a positive constant $B_{\varepsilon}$ and
$n_{\varepsilon}$ such that $\Pr\left(\lvert
f_{n}\rvert>g_{n}B_{\varepsilon}\right)<\varepsilon$ for all
$n>n_{\varepsilon}$.

##  S-1 Proof of the Theorems

This section provides the proofs of Theorems 1 to 3. The proofs are based on
lemmas presented in Section S-2. Among these, Lemmas S-2.6 and S-2.7 are key.
For each covariate $i=1,2,\cdots,N$, Lemmas S-2.6 establishes exponential
probability inequalities for the t-ratio multiple tests conditional on the
average net effect, $\bar{\theta}_{i,T}$, being either of the order
$\ominus(T^{-\varepsilon_{i}})$ for some $\varepsilon_{i}>1/2$, or of the
order $\ominus(T^{-\vartheta_{i}})$, for some $0\leq\vartheta_{i}<1/2$. For
DGP given by (1), Lemma S-2.7 provides asymptotic properties of LS estimator
of coefficients and SSR of a regression model that includes all the signals
and pseudo-signals. This lemma establishes that the coefficients of pseudo-
signals estimated by LS converges to zero so long as
$k_{T}^{\ast}=\ominus(T^{d})$ grows at a slow rate relative to $T$, i.e.
$0\leq d<1/2$. This lemma also shows that the SSR of the regression model
converges to that of the oracle model, which includes only the signals.

Additional notations and definitions: Throughout this section we consider the
following events:

$\textstyle\mathcal{A}_{0}=\mathcal{H}\cap\mathcal{G},\ \text{where}\
\mathcal{H}=\left\\{\sum_{i=1}^{k}\hat{\mathcal{J}}_{i}=k\right\\}\ \text{and
}\
\mathcal{G}=\left\\{\sum_{i=k+k_{T}^{\ast}+1}^{N}\hat{\mathcal{J}}_{i}=0\right\\},$
(S.1)

where $\\{\hat{\mathcal{J}}_{i}\text{ for }i=1,2,\cdots,N\\}$ are the
selection indicators defined by (6). $\mathcal{A}_{0}$ is the event of
selecting the approximating model, defined by (9). $\mathcal{H}$ is the event
that all signals are selected, and $\mathcal{G}$ is the event that no noise
variable is selected. To simplify the exposition, with slight abuse of
notation, we denote the probability of an event $\mathcal{E}$ conditional on
$\bar{\theta}_{i,T}$ being of order $\ominus(T^{-a})$ by
$\Pr[\mathcal{E}|\bar{\theta}_{i,T}=\ominus(T^{-a})],$where $a$ is a
nonnegative constant.

###  S-1.1 Proof of Theorem 1

To establish result (10), first note that
$\mathcal{A}_{0}^{c}=\mathcal{H}^{c}\cup\mathcal{G}^{c}$ and hence
($\mathcal{H}^{c}$ denotes the complement of $\mathcal{H}$)

$\Pr(\mathcal{A}_{0}^{c})=\Pr(\mathcal{H}^{c})+\Pr(\mathcal{G}^{c})-\Pr(\mathcal{H}^{c}\cap\mathcal{G}^{c})\leq\Pr(\mathcal{H}^{c})+\Pr(\mathcal{\
G}^{c}),$ (S.2)

where $\mathcal{H}$ and $\mathcal{G}$ are given by (S.1). We also have
$\mathcal{H}^{c}=\\{\sum_{i=1}^{k}\hat{\mathcal{J}}_{i}<k\\}$ and
$\mathcal{G}^{c}=\\{\sum_{i=k+k_{T}^{\ast}+1}^{N}\hat{\mathcal{J}}_{i}>0\\}$.
Let’s consider $\Pr\left(\mathcal{H}^{c}\right)$ and $\Pr\left(\mathcal{\
G}^{c}\right)$ in turn. We have
$\textstyle\Pr(\mathcal{H}^{c})\leq\sum_{i=1}^{k}\Pr(\hat{\mathcal{J}}_{i}=0)$.
But for any signal

$\textstyle\Pr(\hat{\mathcal{J}}_{i}=0)=\Pr\left[\lvert
t_{i,T}\rvert<c_{p}(N,\delta)|\bar{\theta}_{i,T}=\ominus(T^{-\vartheta_{i}})\right]=1-\Pr\left[\lvert
t_{i,T}\rvert>c_{p}(N,\delta)|\bar{\theta}_{i,T}=\ominus(T^{-\vartheta_{i}})\right],$

where $0\leq\vartheta_{i}<1/2$ and hence by Lemma S-2.6, we can conclude that