The exact contents of citations.db main_text.text for this paper — one flattened LaTeX string, title through conclusion, appendix excluded, unmodified except for removing email addresses. This is what our citation measures are computed over.
93,609 characters
Supervised Mixed-Frequency Learning for Macro-Financial Forecasting When Factors are Weak
\if01\hypersetup{pdfauthor={}}\fi
\def\spacingset#1{\renewcommand{1}
{#1}\small\normalsize} \spacingset{1}
\if11
{
\title{\bf Supervised Mixed-Frequency Learning for Macro-Financial
Forecasting When Factors are Weak}
\author{Ulrich Hounyo\hspace{.2cm}\\
Department of Economics, University at Albany - SUNY\\
and \\
Zhendong Li \\
Department of Economics, University at Albany - SUNY}
\maketitle
} \fi
\if01
{
\bigskip
\bigskip
\bigskip
\begin{center}
{\LARGE\bf Supervised Mixed-Frequency Learning for Macro-Financial
Forecasting When Factors are Weak}
\end{center}
\medskip
} \fi
\bigskip
\begin{abstract}
Factor-MIDAS regressions forecast a low-frequency target by extracting
common factors from a large panel of high-frequency predictors via
principal component analysis (PCA). While PCA mitigates the curse of
dimensionality, it relies on factor pervasiveness, an assumption often
violated when factors are weak, as is common in macro-financial
forecasting. We propose SsPCA-MIDAS, which integrates supervised scaled
PCA (SsPCA) into the mixed-data sampling framework. We establish
consistency and asymptotic normality under weak factors, permitting
inference on the prediction target. Simulations show that SsPCA-MIDAS
outperforms competing PCA-based and supervised methods, especially when
weak factors are prevalent. Applying machine-learning techniques such as
boosting to the cleaner factors it extracts yields further gains. An
extensive application to U.S. macro-financial forecasting shows that
SsPCA-MIDAS selects economically meaningful predictors and improves
forecasts of GDP, inflation, unemployment, asset prices, and volatility.
\end{abstract}
\noindent
{\it Keywords:} Supervised scaled PCA, weak factors, factor-MIDAS, boosting, macro-financial forecasting
\noindent
{\it J.E.L. Codes:} C38, C53, C55, C58, E44.
\vfill
\newpage
\spacingset{1.8}
\section{Introduction}
Mixed-Data Sampling (MIDAS) regressions are widely used in economic forecasting. Introduced by \cite{ghysels2004midas,ghysels2007midas}, they exploit high-frequency predictors through a parametric aggregation function to forecast a lower-frequency target, and have become a standard tool for nowcasting and prediction in finance and macroeconomics \citep{Clements2008}.
A notable extension is the ``factor-MIDAS'' (or factor-augmented MIDAS) regression, which incorporates common factors extracted from large panels of higher-frequency series. Exploiting the dimension-reduction properties of factor models, these regressions forecast effectively and are widely applied \citep{marcellino2010factor,kim2018methods,ferrara2019nowcasting}. \cite{koh2023inference} provides theoretical support by deriving the asymptotic distribution of the factor-MIDAS coefficient estimators with PCA-estimated factors, and \cite{beyhum2024factor} extend the literature with a factor-augmented sparse MIDAS regression combining sparse and dense dimension reduction for nowcasting.
PCA constructs orthogonal principal components capturing maximum variance. It is consistent under strong factors \citep{bai2002determining,bai2003inferential} and remains effective under weaker assumptions \citep{bai2023approximate}. But it is unsupervised, ignoring the target, which limits its ability to identify the most useful low-dimensional predictors for mixed-frequency forecasting: when the signal-to-noise ratio is low, the factor space spanned by the principal components becomes inconsistent or even nearly orthogonal to the true factor space \citep{johnstone2009consistency}. We call such factors \textsl{weak}. Weak factors have been defined in various ways, including large idiosyncratic variances, small nonzero loadings, and sparsity with many zero loadings \citep{lettau2020estimating,uematsu2022estimation,freyaldenhoven2022factor}.
Weak factors are hard to capture and can bias predictions substantially, the \textsl{weak factor problem}. First identified by \cite{kan1999gmm,kan1999two} for useless factors and generalized by \cite{kleibergen2009tests}, who shows weak factors distort inference throughout the model, it was further shown by \cite{giglio2023prediction,giglio2021test} to bias the estimation of all factors, including strong ones, impairing asset pricing and forecasting.
Importantly, the weak factor problem is not merely a statistical artifact; it is tied to the structure of macro-financial data. A factor is weak when its loadings are pervasive only across a limited subset of predictors, so it explains a small share of the panel's cross-sectional variance. Because principal components are ordered by explained variance, such factors are relegated to lower-ranked components that are discarded or swamped by noise. Yet the variance a factor explains need not coincide with its predictive content: signals that move only a narrow set of predictors can still carry first-order information about future outcomes. Many economically important signals fit this profile. The term spread and corporate bond spreads load on a few interest-rate and credit series, yet are among the most reliable predictors of output growth and recessions \citep{Ang2006,ludvigson2009macro}; the global financial cycle operates through a limited set of cross-country channels, so its factor is weak in variance terms while remaining informative for domestic outcomes \citep{rey2015dilemma,miranda2020us}.
To recover such weak but economically meaningful factors, a natural remedy is supervision, using the target to guide factor extraction. Supervised PCA originated in cancer diagnosis \citep{bair2006prediction} and entered macroeconomic forecasting via \cite{bai2008forecasting} through hard and soft thresholding.
\cite{huang2022scaled} propose Scaled PCA (sPCA), which scales each predictor by its predictive slope relative to the target before applying PCA, upweighting stronger predictors and downweighting weaker ones and so partially addressing weak factors. Since sPCA still uses the entire predictor set, however, irrelevant predictors are downweighted but not eliminated, leaving residual noise.
In parallel, \cite{giglio2023prediction,giglio2021test} propose Supervised PCA (SPCA), which uses supervised selection: guided by the target, it iteratively selects a subset of the most predictive predictors, extracts a latent factor by PCA, and projects it out before repeating, raising the signal-to-noise ratio. For complex predictor sets with diverse irrelevant factors, though, the selected subset may not remain sufficiently informative.
In a same-frequency setting, \cite{hounyo2023forecasting} propose Supervised Scaled PCA (SsPCA) to jointly address weak factors and irrelevant-factor noise, but without theoretical justification. SsPCA first scales each predictor by its predictive slope and then applies SPCA; the scaling amplifies relevant signals and shrinks irrelevant ones while the selection removes uninformative predictors entirely, the two steps being complementary. Table \ref{Table A.1} in Appendix \ref{Appendix A} summarizes how these PCA-based methods differ in their use of target information, required factor strength, and recovery and prediction properties, with details in Section \ref{Section 3.4}.
Our primary contribution is to establish the asymptotic theory of SsPCA, including consistency and inference on the prediction target, in a framework where the sample size and cross-sectional dimension may grow at different rates, and to show analytically why SsPCA outperforms existing supervised PCA methods.
We further extend this framework to factor-MIDAS, terming the procedure SsPCA-MIDAS (hereafter SsPCA unless the distinction matters), which is practically important given the growing availability of high-frequency data for mixed-frequency forecasting and nowcasting. Unlike the supervised dynamic PCA of \cite{gao2024supervised}, which scales and combines predictors and their lags in a same-frequency setting, MIDAS aggregates lagged information parsimoniously through a parametric weighting function \citep{ghysels2007midas}. Building on \cite{koh2023inference}, who develops inference for strong factor-MIDAS, we establish the corresponding theory for weak factors, accounting for supervised selection and the jointly estimated MIDAS weights.
In our setting, the weak factor problem arises from the factor loading
matrix, whose eigenvalues grow but potentially more slowly than the
cross-sectional dimension; the factors we examine are thus weaker than
in \cite{bai2023approximate}. PCA fails to recover such factors, leading
to biased predictions \citep{giglio2023prediction}, and we show that
sPCA is also biased here. When factors are extremely weak, with
eigenvalues of the same order as the idiosyncratic ones
\citep{onatski2009testing,onatski2010determining}, the factor space
cannot be consistently recovered \citep{onatski2012asymptotics}, and
SsPCA is no exception; the self-normalized score statistic of
\cite{chao2022selecting} could replace our correlation screening but
does not resolve this case. Like \cite{giglio2023prediction}, our results
do not require perfect recovery of the predictors correlated with the
factors and account for errors accumulated over the iterative process.
A growing literature applies machine learning to problems where
traditional econometric methods struggle \citep{kelly2023financial},
including within MIDAS regressions, to exploit modern datasets and the
sluggish update frequency of some indicators \citep{babii2022machine}.
In particular, \cite{lahiri2022boosting} use MIDAS and boosting to
forecast New York State tax revenues and find that boosting with PCA
factors performs best in their context. This motivates us to ask whether
the cleaner factors extracted by SsPCA can further enhance boosting for
mixed-frequency forecasting.
Through Monte Carlo simulations, we assess the finite-sample performance
of SsPCA against other PCA-based methods and PLS within the factor-MIDAS
framework, across three scenarios that vary factor strength from strong
to weak, with irrelevant factors adding noise in all cases. SsPCA
outperforms competing methods, especially when weak factors dominate,
and applying boosting to the refined factors it extracts yields further
gains in prediction accuracy.
As an extension, we show that the asymptotic consistency of factor estimation in boosting with PCA, established by \cite{bai2009boosting}, carries over to our setting. Specifically, boosting combined with SsPCA ensures consistent factor estimation for both component-wise and block-wise boosting. This suggests that SsPCA may benefit techniques using latent factors.
Finally, we conduct an extensive empirical application to U.S.
macro-financial forecasting, covering eight quarterly targets and over
1,500 monthly predictors from U.S. and international sources. SsPCA
consistently improves forecast accuracy relative to PCA-based methods and
other benchmarks across horizons from one quarter to two years, with
further gains from boosting. The selected predictors are economically
interpretable, with rates, spreads, leverage, and profitability measures
central and cross-country factors contributing, and the COVID-19 crisis
marks a clear structural break in predictor relevance, highlighting the
method's ability to adapt to evolving information sets.
The rest of the paper is organized as follows. Section \ref{Section 2}
introduces the factor-MIDAS model and develops the SsPCA procedure. Section \ref{Section 3} establishes the asymptotic theory of SsPCA. Simulation results are shown in Section
\ref{Section 4}, and Section \ref{Section 5} presents the empirical
application to U.S. macro-financial mixed-frequency forecasting. Section
\ref{Section 6} concludes the paper. The appendices collect additional details and material omitted from the main text.
\section{Methodology\label{Section 2}}
\subsection{Notation}
Throughout this paper, we use the following notations. For any low-frequency
time series of vectors $\left\{ z_{t}\right\} _{t=1}^{T}$, we use
the capital letter $Z$ to denote the matrix $\left(z_{1},z_{2},...,z_{T}\right)$,
$\overline{Z}$ for $\left(z_{1+h},z_{2+h},...,z_{T}\right)$, and
$\underline{Z}$ for $\left(z_{1},z_{2},...,z_{T-h}\right)$, for
some horizons $h$. For any time series of high-frequency vectors
$\ensuremath{\left\{ z_{t}\right\} _{t=1}^{T_{H}}}$, we use the bold
letter $\ensuremath{\bold{z}_{H}}$ to denote the matrix $\ensuremath{\left(z_{1},z_{2},...,z_{T_{H}}\right)}$
and $\ensuremath{\underline{\bold{z}}_{H}}$ for $\ensuremath{\left(z_{1},z_{2},...,z_{T_{H}-mh}\right)}$,
where $m$ is the multiple by which high-frequency data increases
when an additional unit of low-frequency data is added. We use $[N]$
to denote the set of integers: $\{1,2,...,N\}.$ For an index set
$I\subset[N]$, we use $|I|$ to denote its cardinality. We use $Z_{[I]}$
to denote a sub-matrix of $Z$ whose rows are indexed in $I$.
We write $z\apprle x$ if $z\leq Cx$ for some $C>0$, $z\apprle_{\textrm{P}}x$ if $z=O_{\textrm{P}}(x)$, $z\asymp x$ if $z\apprle x$ and $x\apprle z$, and $z\asymp_{\textrm{P}}x$ analogously. We denote by $\lambda_{\min}(Z)$, $\lambda_{\max}(Z)$, and $\lambda_{i}(Z)$ the minimum, maximum, and $i$th largest eigenvalues of $Z$, and by $\sigma_{i}(Z)$ its $i$th singular value. The operator ($l_2$) norm, Frobenius norm, and max ($l_\infty$) norm of $Z=(z_{ij})$ are $\parallel Z\parallel=\sqrt{\lambda_{\max}(Z'Z)}$, $\parallel Z\parallel_{\textrm{F}}=\sqrt{\textrm{Tr}(Z'Z)}$, and $\Vert Z\Vert_{\max}=\max_{i,j}\mid z_{ij}\mid$. Finally, $\mathbb{P}_{Z}=Z(Z'Z)^{-1}Z'$ and $\mathbb{M}_{Z}=\mathbb{I}_{d}-\mathbb{P}_{Z}$ for any $Z$ with $d$ rows, where $\mathbb{I}_{d}$ is the $d\times d$ identity.
\subsection{Model Setup}
We predict a vector of low-frequency targets $y_{t+h}$, $h$ steps ahead, from $N$ high-frequency predictors $x_{t}$ with sample size $T$, where each predictor is observed at most $m$ times between $t-1$ and $t$. A MIDAS regression aggregates the high-frequency variables through a lag polynomial. We consider the factor autoregressive distributed lag (ADL)-MIDAS model:
\begin{equation}
y_{t+h}=\alpha B(L^{1/m};\theta_{f})f_{t}+\alpha_{w}w_{t}+\varepsilon_{t+h}=\alpha\stackrel[j=0]{J}{\sum}b^{f}_{j}({\theta}_f)f_{t-j/m}+\alpha_{w}w_{t}+\varepsilon_{t+h}\qquad t=1,...,T-h,\label{equation 1}
\end{equation}
\noindent where $\alpha=(\alpha_{1},\alpha_{2},...,\alpha_{K})$,
$\theta_{f}=(\theta'_{f,1},\theta'_{f,2},...,\theta'_{f,K})'$ with $\theta_{f,k}=(\theta^f_{k,1},\theta^f_{k,2},...,\theta^f_{k,p_f})'$
a $p_f\times1$ vector of weighting parameters for the $k$-th factor,
for $k=1,...,K$. $B(L^{1/m};\theta_{f})=\sum_{j=0}^{J}b_{j}^{f}(\theta_{f})L^{j/m}$
and $L^{j/m}f_{t}=f_{t-j/m}$, where $f_{t-j/m}$ is $K\times1$ vector
of latent high-frequency factors. Here, $b^{f}_{j}({\theta}_f)$ is a $K\times K$
diagonal matrix weighting function that temporally aggregates the
regressors and their lags, and $b_{j}^{f}(\theta_{f})\equiv\textrm{diag}(b_{j,1}^{f}(\theta_{f,1}),b_{j,2}^{f}(\theta_{f,2}),...,b_{j,K}^{f}(\theta_{f,K}))$,
such that $b_{j,k}^{f}(\theta_{f,k})$ is the weight for the $j$-th lag
of the $k$-th factor. Here, $j$ represents the lag order of the
high-frequency variable and $j=0,...,J$. The $w_{t}$ is a $\mathcal{W}\times1$
vector of predetermined variables such as a constant and lags of $y_{t+h}$.
$\varepsilon_{t+h}$ is a vector of prediction errors.
We assume that the high-frequency predictors $x_t$ follow a linear factor model:
\begin{equation}
x_{i,t-j/m}=\beta{}_{i}f_{t-j/m}+e_{i,t-j/m},\qquad i=1,...,N,t=1,...,T,j=m-1,...,0,\label{equation 2}
\end{equation}
\noindent where $\beta'_{i}$, $i=1,...,N,$ are $K\times1$ vectors
of factor loadings and $e_{i,t-j/m}$ is an idiosyncratic error term
satisfying $\mathbb{E}(e_{i,t-j/m})=0$, $\mathbb{E}(f_{t-j/m}e'_{i,t-j/m})=0,$
and $\mathbb{E}(w_{t}e'_{i,t-j/m})=0$.
We define
\[
F_{t}(\theta_{f})\equiv B(L^{1/m};\theta_{f})f_{t}=\sum_{j=0}^{J}b_{j}^{f}(\theta_{f})f_{t-j/m},
\]
as the aggregated factor, so that equation (\ref{equation 1}) can
be written as
\begin{equation}
y_{t+h}=\alpha F_{t}(\theta_{f})+\alpha_{w}w_{t}+\varepsilon_{t+h},\qquad t=1,...,T-h.\label{equation 3}
\end{equation}
To identify $\alpha$, we assume that $b_{j,k}^{f}(\theta_{f,k})\in(0,1)$
and $\sum_{j=0}^{J}b_{j,k}^{f}(\theta_{f,k})=1$. A common weighting scheme
in the MIDAS regression model is the exponential Almon lag with two
parameters ($p=2$) such that
\[
b_{j,k}^{f}(\theta_{f,k})=\frac{exp(\theta_{1,f,k}j+\theta_{2,f,k}j^{2})}{\sum_{j=0}^{J}exp(\theta_{1,f,k}j+\theta_{2,f,k}j^{2})}.
\]
Further weighting schemes are discussed in \cite{ghysels2007midas} and \cite{ghysels2009multi}. Parameters are estimated by nonlinear least squares (NLS), since the MIDAS regression is nonlinear in them, except for the unrestricted MIDAS (U-MIDAS) of \cite{foroni2015unrestricted}, estimable by OLS.
For convenience, we use the high-frequency time index $t_{h}=1,...,T_{H}$ with $T_{H}=mT$, where $t_{h}=m((t-1)+n/m)$ for $n=1,...,m$, so (\ref{equation 2}) reads $x_{i,t_{h}}=\beta_{i}f_{t_{h}}+e_{i,t_{h}}$. Equations (\ref{equation 2}) and (\ref{equation 3}) then take the matrix form
\[
\bold{x}_{H}=\beta\bold{f}_{H}+\bold{e}_{H},\qquad\overline{Y}=\alpha\underline{F}(\theta_{f})+\alpha_{w}\underline{W}+\overline{\epsilon}.
\]
\subsection{Assumptions}
We now state the assumptions on the data generating processes, comparable to those in \cite{huang2022scaled} and \cite{giglio2023prediction} to ease comparison with these same-frequency benchmarks. Hereafter $C$ is a generic constant, and in all asymptotics $N,T\rightarrow\infty$ while $h$, $K$, and $\mathcal{W}$ are fixed.
\begin{assumption}
\noindent\label{assu:1}The factors $\bold{f}_{H}$ satisfy $\sup_{t_{h}}\mathbb{E}\parallel f_{t_{h}}\parallel^{4}\leq C$
and $\sideset{\frac{1}{T_{H}}}{_{t_{h}=1}^{T_{H}}}\sum f_{t_{h}}f_{t_{h}}'\overset{\textrm{P}}{\rightarrow}\mathbf{\Sigma}_{f}$,
where $\mathbf{\Sigma}_{f}$ is a $K\times K$ positive definite matrix
with $\lambda_{K}(\mathbf{\Sigma}_{f})\apprge1$ and $\lambda_{1}(\mathbf{\Sigma}_{f})\apprle1$.
In addition, $\bold{f}_{H}$, the prediction error $\epsilon$ and the
predetermined regressor $W$ satisfy the following bounds:
\[
\begin{array}{c}
\parallel T_{H}^{-1}\underline{\bold{f}}_{H}\underline{\bold{f}}_{H}'-\mathbf{\Sigma}_{f}\parallel\apprle_{\textrm{P}}T_{H}^{-1/2},\parallel\bold{f}_{H}\parallel_{\max}\apprle_{\textrm{P}}\log(T_{H})^{1/2},\parallel T^{-1}\underline{WW}'-\mathbf{\Sigma}_{w}\parallel\apprle_{\textrm{P}}T^{-1/2},\\
\parallel W\underline{F}'(\theta_f)\parallel\apprle_{\textrm{P}}T{}^{1/2},\parallel\epsilon\parallel\apprle_{\textrm{P}}T{}^{1/2},\parallel\epsilon\parallel_{\max}\apprle_{\textrm{P}}\log(T)^{1/2},
\parallel\overline{\epsilon}\underline{F}'(\theta_f)\parallel\apprle_{\textrm{P}}T{}^{1/2},\parallel\overline{\epsilon}\underline{W}'\parallel\apprle_{\textrm{P}}T{}^{1/2},
\end{array}
\]
where $\mathbf{\Sigma}_{w}$ is a $\mathcal{W}\times\mathcal{W}$
positive definite matrix with $\lambda_{\mathcal{W}}(\mathbf{\Sigma}_{w})\apprge1$
and $\lambda_{1}(\mathbf{\Sigma}_{w})\apprle1$, $\underline{F}(\theta_f)$
is the aggregated factor.
\end{assumption}
Assumption \ref{assu:1} imposes mild conditions on the time-series properties of $f_{t_{h}}$, $\varepsilon_{t}$, and $w_{t}$, satisfied by stationary, strongly mixing processes with sufficient moments, and ensures that the $K$ left-singular values of $\bold{f}_{H}$ are bounded away from zero and infinity. In our context it is therefore the factor loadings that determine factor strength.
\begin{assumption}
\noindent\label{assu:2}The $N\times K$ factor loading matrix $\beta$
satisfies $\parallel\beta\parallel_{\max}\apprle1$ and
$\lambda_{K}(\beta'_{[I_{0}]}\beta{}_{[I_{0}]})\apprge N_{0}$
for some index set $I_{0}\subset[N]$, where $N_{0}=\mid I_{0}\mid\rightarrow\infty.$
\end{assumption}
Assumption \ref{assu:2} implies that one can construct a subset $I_{0}$
of predictors on which all latent factors are pervasive, while still
allowing a certain proportion of irrelevant factors. It is considerably
weaker than pervasiveness across all predictors, where
$\lambda_{1}(\beta'\beta)\asymp...\asymp\lambda_{K}(\beta'\beta)\asymp N$:
these eigenvalues may now grow at different rates slower than $N$, since
$N_{0}/N$ can diminish rapidly, and no restriction is imposed on
$\beta_{[I_{0}^{c}]}$ (see \citet{giglio2023prediction} for discussion). It excludes only
the extreme case where all entries of $\beta$ vanish uniformly, i.e.,
$\sup_{I,\mid I\mid\rightarrow\infty}\mid I\mid^{-1}\lambda_{K}(\beta'_{[I]}\beta{}_{[I]})=o_{\textrm{P}}(1)$,
in which case the desired $I_{0}$ would not exist.
\begin{assumption}
\noindent\label{assu:3}The idiosyncratic error term $e_{i,t_{h}}$
satisfies $e_{i,t_{h}}=\sigma_{i}e_{i,t_{h}}^{*}$, where $e_{i,t_{h}}^{*}$
is i.i.d. with $\sup_{i,t_{h}}\mathbb{E}\parallel e_{i,t_{h}}^{*}\parallel^{8}\leq C$
and $C^{-1}\leq\sigma_{i}\leq C$, where $\sigma_{i}$ is the standard
deviation. Additionally, the idiosyncratic error matrix $\bold{e}_{H}$ satisfies
$\parallel\bold{e}_{H}\parallel_{\max}\apprle_{\textrm{P}}(\log T_{H})^{1/2}+(\log N)^{1/2}$
and $\parallel\bold{e}_{H,[I]}\parallel\apprle_{\textrm{P}}\mid I\mid^{1/2}+T_{H}^{1/2}$
for any given non-random subset $I\subset[N]$.
\end{assumption}
Assumption \ref{assu:3} constrains the time-series dependence and heteroskedasticity of $e_{i,t_{h}}$. Under the i.i.d.\ structure on $e^{*}_{i,t_h}$, the first inequality follows from a standard large deviation theorem \citep[see, e.g.,][]{fan2011high} and the second from random matrix theory. This structure is imposed only for transparency, as the analysis uses only the two bounds, which should continue to hold under weak time-series and cross-sectional dependence. We do not assume a uniform bound over all index sets, since the rate $|I|^{1/2}+T_{H}^{1/2}$ may not hold universally.
\begin{assumption}
\noindent\label{assu:4}Suppose that $\{f_{t_{h}}\}$, $\{e_{i,t_{h}}\}$
and $\{\beta_{i}\}$ can be weakly dependent. For any non-random subset
$I\subset[N]$, the factor loading $\beta_{[I]}$ and the idiosyncratic
error $\bold{e}_{H,[I]}$ satisfy:
(i) $\parallel\underline{\bold{e}}_{H,[I]}Z'\parallel\apprle_{\textrm{P}}\lvert I\rvert^{1/2}T^{1/2},\,\parallel\underline{\bold{e}}_{H,[I]}Z'\parallel_{\max}\apprle_{\textrm{P}}(\log N)^{1/2}T^{1/2},$
(ii) \mbox{$\parallel\beta'_{[I]}\bold{e}_{H,[I]}\parallel\apprle_{\textrm{P}}\lvert I\rvert^{1/2}T^{1/2},\,\parallel\beta'_{[I]}\bold{e}_{H,[I]}\parallel_{\max}\apprle_{\textrm{P}}\lvert I\rvert^{1/2}(\log T)^{1/2},\,\parallel\beta'_{[I]}\underline{\bold{e}}_{H,[I]}Z'\parallel_{\max}\apprle_{\textrm{P}}\lvert I\rvert^{1/2}T^{1/2},$}
(iii) $\parallel(e_{T_{H}})'_{[I]}\underline{\bold{e}}_{H,[I]}Z'\parallel\apprle_{\textrm{P}}\lvert I\rvert+\lvert I\rvert^{1/2}T^{1/2},\,\parallel\beta'_{[I]}(e_{T_{H}})_{[I]}\parallel\apprle_{\textrm{P}}\lvert I\rvert^{1/2},$
\noindent\textsl{where $Z$ can be either $\underline{\bold{f}}_{H}$,
$\underline{W}$ or $\overline{\epsilon}$. If $Z=\underline{\bold{f}}_{H}$,
then in the bounds all occurrences of $T$ are replaced by $T_{H}$.
If the variables have different frequencies, high-frequency ones are
aggregated by an appropriate weighting scheme.}
\end{assumption}
In Assumption \ref{assu:4}, the $l_{2}$-norm bounds are implied by \cite{bai2003inferential} when $I=[N]$, and the max-norm results follow from a large deviation theorem as in \cite{fan2011high}.
\cite{bai2023approximate} and \cite{wang2017asymptotics} show that PCA remains consistent under non-pervasive loadings as long as $N/(T\lambda_{K}(\beta'\beta))\to0$, the latter identifying a bias term in the borderline case $N\asymp T\lambda_{K}(\beta'\beta)$. Assumptions \ref{assu:2} and \ref{assu:3} are the key identification conditions of our weak factor model; since we do not require $e_{t_{h}}$ to be stationary, $N/(T\lambda_{K}(\beta'\beta))$ may diverge, so traditional PCA typically fails unless the errors are homoscedastic. We therefore assume a subset $I_{0}\subset[N_{0}]$ with $\mid I_{0}\mid/(T\lambda_{K}(\beta_{[I_{0}]}'\beta_{[I_{0}]}))\rightarrow0$, identifying the factors within it. All these settings require $\lambda_{K}(\beta'\beta)\to\infty$, in contrast to the extremely weak factor model of \citet{onatski2012asymptotics}, which restricts $\lambda_{K}(\beta'\beta)\lesssim1$.
\begin{assumption}
\noindent\label{assu:5} The prediction error $\varepsilon_{t+h}$ and the associated score vector $g_{\kappa,t}$ satisfy the following moment and central limit theorem (CLT) conditions:
(i) $\mathbb{E}(\varepsilon_{t+h})=0,$ $\mathbb{E}||\varepsilon_{t+h}||^{2}<C,$
(ii) $\mathbb{E}\parallel g_{\kappa,t}\parallel^{4}\leq C,$ and $\frac{1}{T}\sum_{t=1}^{T}g_{\kappa,t}g'_{\kappa,t}\overset{\textrm{p}}{\longrightarrow}\mathbf{\Sigma}_{\kappa}>0,$
(iii) $\mathbb{E}\parallel\frac{1}{\sqrt{T}}\sum_{t=1}^{T}g_{\kappa,t}\varepsilon_{t+h}\parallel^{2}<C,$ and as $T\rightarrow\infty,$ $\frac{1}{\sqrt{T}}\sum_{t=1}^{T}g_{\kappa,t}\varepsilon_{t+h}\overset{\textrm{d}}{\longrightarrow}\mathcal{N}(0,\Omega_{\kappa}),$
\noindent\textsl{where $g_{\kappa,t}=\partial g(F_{t},\kappa)/\partial\kappa,$ $g(F_{t},\kappa)=\alpha F_{t}(\theta)+\alpha_{w}w_{t},$ $\kappa=(\alpha'_{l},\theta')'$ with $\alpha_{l}=(\alpha,\alpha_{w})', \theta=(\theta_{f},\theta_{x})'$. Here, $\theta_x$ denotes the weighting parameters for the high-frequency predictors, $F_{t}=(f'_{t},f'_{t-1/m},...,f'_{t-J/m})',$ and $\Omega_{\kappa}$ is the long-run variance of the empirical score process.}
\end{assumption}
Assumption \ref{assu:5} imposes the regularity conditions on the prediction error and score vector required for joint NLS estimation of $\alpha_{l}$ and $\theta$: the moment and covariance conditions ensure the sample score function behaves well uniformly in the parameter space, while the CLT makes the aggregated score process asymptotically Gaussian, as in \cite{koh2023inference} and \cite{gonccalves2014bootstrapping}. The parameter $\theta_x$ is defined in the next subsection.
\subsection{Algorithm and Implementation}
We now describe the estimation algorithm. The literature offers two representative PCA regression algorithms \citep{stock2002forecasting,giglio2023prediction}; since the former is more prone to overfitting and instability \citep{giglio2023prediction}, we adopt the latter, constructing $\hat{f}_{T_{H}}$ from the in-sample estimated weights for all subsequent PCA computations within the MIDAS model.\footnote{For PCA-based methods at the same frequency, see \cite{hounyo2023forecasting}.} For notational convenience, superscripts on predicted values denote the forecasting method (e.g., $\widehat{y}_{T+h}^{\textrm{SsPCA}}$) and subscripts on estimated factors the extraction method (e.g., $\underline{\bold{f}}_{H,\textrm{SsPCA}}$); we omit the `SsPCA' label unless needed, so an unlabeled estimator is understood to use SsPCA.
Existing supervised methods each have a limitation: SPCA \citep{giglio2023prediction,giglio2021test} selects an informative subset but cannot downweight irrelevant factors, while sPCA \citep{huang2022scaled} upweights informative predictors but cannot fully exclude irrelevant ones. \cite{hounyo2023forecasting} combine the two in SsPCA, whose weighting amplifies relevant signals and shrinks irrelevant ones and whose selection removes the remaining noise; Section \ref{Section 3.4} compares them in detail, and we extend SsPCA to the mixed-frequency setting.
The weighting step requires further notations. Let ${\theta}_{x}=(\theta'_{x,1},\theta'_{x,2},...,\theta'_{x,N})'$ with $\theta_{x,i}=(\theta^{x}_{i,1},\theta^{x}_{i,2},...,\theta^{x}_{i,p_x})'$
a $p_x\times1$ vector,
for $i=1,...,N$. Define $b^{x}_{j}({\theta}_x)$ as an $N\times N$
diagonal matrix weighting function with $b^x_{j}(\theta_x)\equiv\textrm{diag}(b_{j,1}^x(\theta_{x,1}),b^x_{j,2}(\theta_{x,2}),...,b^x_{j,N}(\theta_{x,N}))$. So we have the aggregated predictors $X_{t}({\theta}_x)=\stackrel[j=0]{J}{\sum}b^{x}_{j}({\theta}_{x})\bold{x}_{t-j/m}$, and we assume that $b_{j,i}^x(\theta_{x,i})\in(0,1)$
and $\sum_{j=0}^{J}b_{j,i}^x(\theta_{x,i})=1$.
Formally, we define $\Upsilon_{i}$ as the regression slope of the
target variable $y_{t+h}$ on each aggregated (standardized) predictor
$X_{i,t}(\theta_{x,i})$, and collect these slopes to construct the
diagonal matrix $\Upsilon=\textrm{diag}(\Upsilon_{1},\Upsilon_{2},...,\Upsilon_{N})$.
The scaling coefficient $\widehat{\Upsilon}_{i}$ is estimated from the regression:
\begin{equation}
y_{t+h}=c_{i}+\Upsilon_{i}\stackrel[j=0]{J}{\sum}b_{j,i}^x(\theta_{x,i})x_{i,t-j/m}+u_{i,t+h},\qquad i=1,...,N,t=1,...,T-h,\label{equation 4}
\end{equation}
\noindent where $c_{i}$ and $u_{i,t+h}$ are the constant and error term.
Our goal is to form a panel of scaled high-frequency predictors $(\Upsilon_{1}x_{1,t_{h}},\allowbreak...,\allowbreak\Upsilon_{N}x_{N,t_{h}})$, which shares the same latent factor structure:
\begin{equation}
\Upsilon_{i}x_{i,t_{h}}=\Upsilon_{i}\beta{}_{i}f_{t_{h}}+\Upsilon_{i}e_{i,t_{h}},\qquad i=1,...,N,t_{h}=1,...,T_{H},\label{equation 5}
\end{equation}
i.e. the scaled predictors share the same factors $f_{t_{h}}$ as the originals, in matrix form $\Upsilon\bold{x}_{H}=\Upsilon\beta\bold{f}_{H}+\Upsilon\bold{e}_{H}$.
For a given $\theta_x$, the aggregated predictor is $\underline{X}_{i}(\theta_{x,i})=\left(X_{i,1}(\theta_{x,i}),...,X_{i,T-h}(\theta_{x,i})\right)$ with $X_{i,t}(\theta_{x,i})=\sum_{j=0}^{J}b_{j,i}^{x}(\theta_{x,i})x_{i,t-j/m}$. Regressing $\overline{Y}$ on each standardized $\underline{X}_{i}({\theta}_{x,i})$ gives $\hat{\Upsilon}_{i}({\theta}_{x,i})=\overline{Y}\underline{X}_{i}({\theta}_{x,i})'(\underline{X}_{i}({\theta}_{x,i})\underline{X}_{i}({\theta}_{x,i})')^{-1}$, collected into the diagonal matrix $\hat{\Upsilon}({\theta}_{x})$.
We employ $\mathbb{M}_{\underline{W}'}$ to remove the effect of $\underline{W}$ from $\overline{Y}$, and describe the
SsPCA algorithm within the framework of MIDAS as follows:
\begin{lyxalgorithm}
\textbf{\textup{(Double Supervised Learning: SsPCA-MIDAS)\label{Algorithm 1}}}
\noindent Inputs: $\overline{Y},\bold{x}_{H}=(\underline{\bold{x}}_{H},\bold{x}_{T_{H}-mh+1}^{T_{H}}),\underline{W},\textrm{ and }w_{T}.$
S1. Supervised weighting. Obtain the scaling coefficients $\hat{\Upsilon}_{i}$ by regressing
$\overline{Y}$ on each aggregated predictor $\underline{X}_{i}(\theta_{x,i})$.
Stack them to form the diagonal scaling matrix $\hat{\Upsilon}(\theta_{x})$,
and construct the scaled high-frequency predictors $\bold{x}_{H,\textrm{scaled}}(\theta_{x})=\hat{\Upsilon}(\theta_{x})\bold{x}_{H}$.
S2. Supervised selection. Initialization: $Y_{(1)}:=\overline{Y}\mathbb{M}_{\underline{W}'}$,
$\bold{x}_{(1)}:=\underline{\bold{x}}_{\textrm{H,scaled}}(\theta_x)$, and, for given
values of the parameter ${\theta}_{x}$, $X_{(1)}({\theta}_{x}):=\underline{X}_{\textrm{scaled}}({\theta}_{x})$.
For $k=1,2,...,K$ iterate
the following steps using $X_{(k)}({\theta}_{x})$, $Y_{(k)}$:
$\qquad$a. Supervised selection: select a subset $\widehat{I}_{(k)}({\theta}_{x}):$
\begin{equation}
\widehat{I}_{(k)}(\theta_x):=\left\{ i\mid T^{-1}\Vert(X_{(k)}(\theta_x))_{[i]}Y'_{(k)}\Vert_{\max}\,\ge\,\widehat{c}_{qN}^{(k)}\right\} \subset[N],\label{equation 6}
\end{equation}
where $\widehat{c}_{qN}^{(k)}$ is the $(1-q)$th-quantile of $\left\{ T^{-1}\mid(X_{(k)}(\theta_x))_{[i]}Y'_{(k)}\mid\right\} _{i\in[N]}$, and $q$ is a scaling factor between 0 and 1. This hints selecting predictors with sufficiently
high absolute values of correlation or covariance.
$\qquad$b. Apply the standard PCA, singular value decomposition (SVD), to this subset $\bold{x}_{(k),[\widehat{I}_{(k)}(\theta_x)]}$
but only getting the first left singular vector $\widehat{\varsigma}_{(k)}(\theta_{x,[\widehat{I}_{(k)}(\theta_x)]})$.
Hence, we can estimate the $k$-th latent factor\footnote{Note that $\widehat{\underline{\bold{f}}}_{H,\text{SsPCA}}$ can be
written as $\widehat{{\underline{\bold{f}}}}_{H,\text{SsPCA}}\equiv\left(\underline{\hat{f}}_{1,\text{SsPCA}},\underline{\hat{f}}_{2,\text{SsPCA}},...,\underline{\hat{f}}_{T_{H}-mh,\text{SsPCA}}\right)$,
where $\hat{f}_{t_{h},\text{SsPCA}}$ is a $K\times1$ vector of estimated
high-frequency factors. For a given value $\theta_f$, the corresponding matrix of aggregated estimated factors
${{\hat{\underline{F}}}}_{\text{SsPCA}}\left(\theta_f\right)=\left(\hat{F}_{1,\text{SsPCA}}\left(\theta_f\right),\hat{F}_{2,\text{SsPCA}}\left(\theta_f\right),...,\hat{F}_{T-h,\text{SsPCA}}\left(\theta_f\right)\right)$,
where $\hat{F}_{t,\text{SsPCA}}\left(\theta_f\right)$ is given by $\hat{F}_{t,\text{SsPCA}}\left(\theta_f\right)=\sum_{j=0}^{J}b^f_{j}(\theta_f)\hat{f}_{t-j/m,\text{SsPCA}}$.} as $\widehat{\underline{\bold{f}}}_{H,(k)}(\theta_x)=\widehat{\varsigma}{}_{(k)}(\theta_{x,[\widehat{I}_{(k)}(\theta_x)]})'\bold{x}_{(k),[\widehat{I}_{(k)}(\theta_x)]}$.
$\widehat{\underline{\bold{f}}}_{H,(k)}(\theta_x)$ can also be rewritten
as $\widehat{\underline{\bold{f}}}_{H,(k)}(\theta_x) =\widehat{\zeta}{}_{(k)}'(\theta_x)\bold{x}_{(k)}$,
where $\widehat{\zeta}_{(k)}(\theta_x)=(\mathbb{I}_{N}\sideset{-}{_{i=1}^{k-1}}\sum\widehat{\beta}_{(i)}(\theta_x)$
$\widehat{\zeta}{'}_{(i)}(\theta_x))_{[\widehat{I}_{(k)}(\theta_x)]}^{'}$
$\widehat{\varsigma}_{(k)}(\theta_{x,[\widehat{I}_{(k)}(\theta_x)]})$ is constructed recursively using $\widehat{\beta}_{(k-1)}(\theta_x)$
(defined in c).
$\qquad$c. Projection step: Project $\bold{x}_{(k)}$ onto this factor $\widehat{\underline{\bold{f}}}_{H,(k)}(\theta_x)$ and project
$Y_{(k)}$ onto the corresponding aggregated factor $\widehat{\underline{F}}_{(k)}(\theta)$, where $\theta=(\theta_f,\theta_x)'$,
separately. Then estimate coefficients $\widehat{\beta}_{(k)}(\theta_x)=\bold{x}_{(k)}\widehat{\underline{\bold{f}}}_{H,(k)}'(\theta_x)(\widehat{\underline{\bold{f}}}_{H,(k)}(\theta_x)$ $\widehat{\underline{\bold{f}}}_{H,(k)}'(\theta_x))^{-1}$
and $\widehat{\alpha}_{(k)}(\theta)=Y_{(k)}\widehat{\underline{F}}_{(k)}'(\theta)(\widehat{\underline{F}}_{(k)}(\theta)\widehat{\underline{F}}_{(k)}'(\theta))^{-1}$.
Then compute residuals of $\bold{x}_{(k)}$ and $Y_{(k)}$,
denoted by $Y_{(k+1)}=Y_{(k)}-\widehat{\alpha}_{(k)}(\theta)\widehat{\underline{F}}_{(k)}(\theta)$
and $\bold{x}_{(k+1)}=\bold{x}_{(k)}-\widehat{\beta}_{(k)}(\theta_x)\widehat{\underline{\bold{f}}}_{H,(k)}(\theta_x)$.
$\qquad$d. Iterate K times in this stage: compute the (univariate)
correlation of the $Y_{(k+1)}$ and $\bold{x}_{(k+1)}$; select
predictors with sufficiently high correlation; extract another one
latent factor; projection, etc. Stop at $k=\widehat{K}$, where $\widehat{K}$
is chosen based on a proper stopping rule.
$\qquad$$\blacktriangleright$ the algorithm terminates as soon as
\begin{equation}
\widehat{c}_{qN}^{(k+1)}<c\:for\:some\:threshold\:c.\label{equation 7}
\end{equation}
S3. Once all factors are collected $\widehat{\underline{F}}_{\textrm{SsPCA}}(\theta)=(\widehat{\underline{F}}{}_{(1)}'(\theta),\widehat{\underline{F}}{}_{(2)}'(\theta),...,\widehat{\underline{F}}{}_{(\widehat{K})}'(\theta))'$,
estimate the coefficients $\alpha_{l}=(\alpha,\alpha_{w})'$
and weighting parameters $\theta$ jointly using NLS, which involve solving the optimization problem $\left(\hat{\alpha}_{l},\hat{\theta}\right)=\underset{\alpha_l,\theta}{\textrm{argmin}}\parallel\overline{Y}-\alpha\underline{\hat{F}}_{\textrm{SsPCA}}(\theta)-\alpha_{w}\underline{W}\parallel^{2}$.
The resulting prediction for $\ensuremath{y_{T+h}}$ is given by $\widehat{y}_{T+h}^{\textrm{SsPCA}}=\hat{\alpha}\hat{F}_{T,\textrm{SsPCA}}(\hat{\theta})+\widehat{\alpha}_{w}w_{T}\equiv\widehat{\alpha}\sum_{j=0}^{J}$
$b^f_{j}(\hat{\theta}_f)\widehat{\zeta}_{\textrm{SsPCA}}'(\hat\theta_x)\bold{x}_{T-j/m,\textrm{scaled}}+\widehat{\alpha}_{w}w_{T}$, where $\widehat{\zeta}_{\textrm{SsPCA}}(\hat\theta_x):=(\widehat{\zeta}_{(1)}(\hat\theta_x),\widehat{\zeta}_{(2)}(\hat\theta_x),...,\widehat{\zeta}_{(\widehat{K})}(\hat\theta_x))$.
outputs: $\widehat{y}_{T+h}^{\textrm{SsPCA}}$, $\widehat{\zeta}_{\text{SsPCA}}(\hat{\theta}_x)$,
$\widehat{\alpha}_{l}$, $\hat{\theta}$, factors $\widehat{\underline{F}}_{\textrm{SsPCA}}(\hat{\theta})$ and their loadings
$\widehat{\beta}(\hat{\theta}_x)$,.
\end{lyxalgorithm}
Intuitively, at each iteration, the algorithm extracts one factor from the most informative subset and then projects it out. The procedure stops once the most correlated remaining predictors fall below the threshold $c$, signaling that no further predictive information can be recovered, so that the number of extracted factors $\hat{K}$ is determined endogenously by $c$.
We thus tune two parameters: $q$, controlling the subset size for PCA, and $c$, the stopping rule. Following \cite{giglio2023prediction}, we tune $\lfloor qN\rfloor$ rather than $q$ (many $q$ map to the same integer) and $K$ rather than $c$ ($K$ is integer-valued, interpretable, and identifiable from a cutoff among the largest eigenvalues). The proofs use $q$ and $c$; the simulations and applications use $\lfloor qN\rfloor$ and $K$, with the threshold set where the MSE is minimized.
The algorithms for the comparison methods PCA, sPCA, and SPCA appear in Appendix \ref{Appendix E} (Algorithms \ref{Algorithm 2}--\ref{Algorithm 4}). We also consider two supervised alternatives: a block-wise boosting algorithm adapted from \cite{bai2009boosting}, motivated by the evidence in \cite{lahiri2022boosting} that boosting with factors performs best in the MIDAS context, and Partial Least Squares (PLS), common in finance \citep{kelly2013market} but, to our knowledge, not previously used with MIDAS; their algorithms are in Appendix \ref{Appendix E} (Algorithms \ref{Algorithm 5}--\ref{Algorithm 6}). Tuning-parameter choices for all methods, which govern the bias-variance trade-off, are described in Appendix \ref{Appendix F}.
\section{Asymptotic Theory\label{Section 3}}
We now investigate the asymptotic properties of SsPCA. Our analysis builds on the framework of \cite{giglio2023prediction} and must additionally accommodate the supervised scaling coefficients \citep{huang2022scaled} and the NLS estimation of the MIDAS aggregation weights \citep{koh2023inference}, neither of which arises in their setting.
\subsection{Consistency in Prediction\label{Section 3.1}}
To establish prediction consistency, we first examine factor-estimation consistency. In our weak-factor framework, prediction consistency follows from consistent recovery of the factors relevant to the target, unlike the strong-factor case of \cite{stock2002forecasting}, where all factors are recovered via PCA.
Because SsPCA inherits SPCA's selection mechanism, Algorithm \ref{Algorithm 1} yields the selected subsets $\widehat{I}_{(k)}$, $k=1,2,\dots$. We iteratively construct their population counterparts for any chosen $c$ and $q$, which define the factor space recovered by SsPCA. Without loss of generality we set\footnote{Since $\mathbf{\Sigma}_{f}$ is typically taken as $\mathbb{I}_{K}$ and we assume $\sum_{j=0}^{J}b^f_{j}(\theta_f)=\mathbb{I}_{K}$, it follows that $\mathbf{\Sigma}_{F(\theta_f)}=\mathbf{\Sigma}_{f}$.} $\mathbf{\Sigma}_{F(\theta_f)}=\mathbf{\Sigma}_{f}=\mathbb{I}_{K}$; the general case replaces $\beta$ and $\alpha$ with $\beta^{*}=\beta\mathbf{\Sigma}_{f}^{1/2}$ and $\alpha^{*}=\alpha\mathbf{\Sigma}_{F(\theta_f)}^{1/2}$. We also impose $||\Upsilon||_{\max}\lesssim1$, so scaling rescales predictors without changing their directions.
Specifically, we begin by defining $\alpha_{i}^{(1)}:=\parallel(\hat{\Upsilon}\beta)_{[i]}\alpha'\parallel_{\max}$
and set $I_{1}:=\{i\mid\alpha_{i}^{(1)}\geq c_{qN}^{(1)}\}$, where
$c_{qN}^{(1)}$ is the $\lfloor qN\rfloor$th largest value among
$\{\alpha_{i}^{(1)}\}_{i=1,...,N}$. We then denote the largest singular
value of $(\hat{\Upsilon}\beta)_{(1)}:=(\hat{\Upsilon}\beta)_{[I_{1}]}$
by $\lambda_{(1)}^{1/2}$ and the corresponding left and right singular
vectors by $\varsigma_{(1)}$ and $\iota_{(1)}$. For $k>1$, we define
$\alpha_{i}^{(k)}:=\parallel(\hat{\Upsilon}\beta)_{[i]}\prod_{j<k}\mathbb{M}_{\iota_{(j)}}\alpha'\parallel_{\max}$,
$I_{k}:=\{i\mid\alpha_{i}^{(k)}\geq c_{qN}^{(k)}\}$, and $\lambda_{(k)}^{1/2}$,
$\varsigma_{(k)}$, $\iota_{(k)}$ represent the leading singular
value, left, and right singular vectors of $(\hat{\Upsilon}\beta)_{(k)}:=(\hat{\Upsilon}\beta)_{[I_{k}]}\prod_{j<k}\mathbb{M}_{\iota_{(j)}}$.
The process continues until step $\widetilde{K}$ (not necessarily equal to $K$ or $\widehat{K}$) when $c_{qN}^{(\widetilde{K}+1)}<c$. The $I_{k}$ are what SsPCA would select if applied directly to $\hat{\Upsilon}\beta\in\mathbb{R}^{N\times K}$ and $\alpha\in\mathbb{R}^{1\times K}$, being determined by $\alpha,\beta,\Upsilon,\mathbf{\Sigma}_{f},c,q$, and $N$, whereas the $\widehat{I}_{k}$ are random, obtained by applying SsPCA to $\underline{X}_{H,\textrm{scaled}}$ and $\overline{Y}$.
To ensure the singular vectors $\iota_{(j)}$ are well-defined and identifiable, the top two singular values of $(\hat{\Upsilon}\beta)_{(k)}$ must remain distinct at each stage $k$, and distinct $c_{qN}^{(k)}$ are needed so the $I_{k}$ are identifiable. We call two sequences $a_{N}$ and $z_{N}$ asymptotically distinct if $a_{N}\leq(1+\delta)^{-1}z_{N}$ for some $\delta>0$, and impose:
\begin{assumption}
\noindent\label{assu:6}For any given $k$, the following three pairs
of sequences of variables, $\sigma_{1}(\beta_{k})$ and $\sigma_{2}(\beta_{k})$,
$c_{qN}^{(k)}$ and $c_{qN+1}^{(k)}$, and $c_{qN}^{(\widetilde{K}+1)}$
and $c$ are asymptotically distinct, as $N\rightarrow\infty$.
\end{assumption}
This assumption is mild, excluding only corner cases commonly disregarded in the high-dimensional PCA literature (e.g. Assumption 2.1 of \citet{wang2017asymptotics} and Assumption 5 of \citet{giglio2023prediction}).
To identify the NLS estimator of the weighting parameter $\theta$ in the MIDAS framework, we impose $\sqrt{T}/(qN)\rightarrow0$, paralleling the $\sqrt{T}/N\rightarrow0$ of \cite{koh2023inference}, who establishes the asymptotics of the NLS estimator with latent strong factors, building on \cite{andreou2010regression} for observed factors. We discuss the role of this condition for inference, and the regime in which it fails, in Remark \ref{rem:1} below Theorem \ref{thm:4}.
We can show the following result:
\begin{thm}
\label{thm:1}Suppose that $y_{t}$ satisfies (\ref{equation 3})
and scaled predictors $\Upsilon_{i}x_{i,t_{h}}$ follows (\ref{equation 5}),
and that Assumptions \ref{assu:1}-\ref{assu:6} hold. If $\log(NT)(N_{0}^{-1}+T^{-1})\rightarrow0$,
then for any tuning parameters $c$ and $q$ that satisfy
\begin{equation}
c\rightarrow0,\;c^{-1}(\log NT)^{1/2}(q^{-1/2}N^{-1/2}+T^{-1/2})\rightarrow0,\;qN/N_{0}\rightarrow0,\;and\;\sqrt{T}/(qN)\rightarrow0,\label{equation 8}
\end{equation}
\noindent\begin{flushleft}
we have $\widetilde{K}\leq K,$ $P(\widehat{I}_{k}=I_{k})\rightarrow1,$
for any $1\leq k\leq\widetilde{K}$, and $P(\hat{K}=\widetilde{K})\rightarrow1.$
Moreover, the aggregated factors recovered by SsPCA-MIDAS are consistent.
That is, for any $1\leq k\leq\widetilde{K}$,
\begin{equation}
\parallel\underline{\hat{F}}_{(k)}(\hat{\theta})\parallel^{-1}\parallel\underline{\hat{F}}_{(k)}(\hat{\theta})-\underline{\hat{F}}_{(k)}(\hat{\theta})\mathbb{P}_{\underline{F}(\theta_f)'}\parallel\apprle_{\textrm{P}}q^{-1/2}N^{-1/2}+T^{-1}.\label{equation 9}
\end{equation}
\par\end{flushleft}
\end{thm}
Theorem \ref{thm:1} establishes consistency of the factors estimated by SsPCA-MIDAS, generalizing Theorem 1 of \citet{giglio2023prediction} to our setting, where $\hat{\theta}=(\hat{\theta}_x,\hat{\theta}_f)$ are the jointly estimated weighting parameters: $\hat{\theta}_x$ enters the estimated factors through predictor-side weighting, while $\hat{\theta}_f$ aggregates the latent factors.
Note that the assumptions of Theorem \ref{thm:1} do not ensure a consistent estimate of the number of factors $K$. Since SsPCA excludes factors uninformative for $y$, it recovers only the relevant factor space, so, like SPCA, it drops the remaining $K-\tilde{K}$ factors while $\hat{K}$ consistently estimates $\widetilde{K}$. Inequality (\ref{equation 9}) has a clear geometric reading: its left-hand side is $\sin\hat{\Theta}_{(k)}$, the sine of the angle between the stage-$k$ estimated factor and the true factor space $\mathbb{P}_{\underline{F}(\theta_f)'}$, which vanishes asymptotically.
Regarding the tuning parameters, condition (\ref{equation 8}) implies $c\rightarrow0$, $c\sqrt{N}\rightarrow\infty$, and $c\sqrt{qN}\rightarrow\infty$. The threshold $c$ must be small enough to allow iterations until the selected predictors lose predictive power, yet large enough to control screening errors: the sample-covariance error between $\bold{x}_{(1),\textrm{scaled}}(\theta_x)$ and $Y_{(1)}$ is of order $T^{-1/2}$, projection residuals add error when $\widetilde{K}>1$, and the factor-estimation error of order $(qN)^{-1/2}+T^{-1}$ contaminates screening, so $c$ should dominate $T^{-1/2}+(qN)^{-1/2}$. The subset size $\lfloor qN\rfloor$ should scale like $N_{0}$; since $N_{0}$ is not precisely defined, we require $qN/N_{0}\rightarrow0$ so the selected predictors lie within the $N_{0}$-subset supporting a strong factor structure.
The factor-estimation error is unaffected by the scaling $\Upsilon$, and $qN$ plays the role that $N$ plays in the strong-factor case of \cite{bai2003inferential}, where the factor space is recovered at rate $N^{-1/2}+T^{-1}$. Since Assumption \ref{assu:2} does not require equal factor strength, some factors could converge faster if the subset size were tuned to each factor's strength. As our goal is prediction rather than factor recovery, we adopt the $\hat{I}_{k}$ rule in (\ref{equation 6}), which reduces tuning sensitivity and yields more stable out-of-sample predictions than rules selecting a different number of predictors at each stage.
We define $\hat{\gamma}=\hat{\alpha}\hat{\zeta}'$. With no relevant factors omitted, the prediction of $\hat{y}_{T+h}$
is consistent, as we illustrate next.
\begin{thm}
\label{thm:2}Under the same assumptions as Theorem \ref{thm:1},
we have $\hat{\alpha}_{w}-\alpha_{w}\overset{\textrm{P}}{\longrightarrow}0$,
$\parallel\hat{\gamma}\hat{\Upsilon}\beta-\alpha\parallel\overset{\textrm{P}}{\longrightarrow}0,$
and consequently, $\hat{y}_{T+h}\overset{\textrm{P}}{\rightarrow}\mathbb{E}_{T}(y_{T+h})=\alpha F_{T}(\theta_f)+\alpha_{w}w_{T}$.
\end{thm}
Theorem \ref{thm:2} establishes prediction consistency by examining the estimation errors $\hat{\alpha}_{w}-\alpha_{w}$ and $\hat{\gamma}\hat{\Upsilon}\beta-\alpha$. For the latter, define $H=\hat{\zeta}'\beta\in \mathbb{R}^{\hat{K}\times K}$, with $\hat{\zeta}$ from Algorithm \ref{Algorithm 1}, so $\hat{\gamma}\hat{\Upsilon}\beta=\hat{\alpha}H$. The adjustment matrix $H$ arises from the indeterminacy of latent factor models, and the theorem implies $\parallel\hat{\alpha}H-\alpha\parallel=o_{\textrm{P}}(1)$, i.e., $\alpha$ is consistently estimated up to $H$. The proof also shows that for $k\leq\tilde{K}$, $\left\Vert \underline{\hat{F}}_{(k)}(\hat{\theta})\right\Vert ^{-1}\left\Vert \underline{\hat{F}}_{(k)}(\hat{\theta})-h_{k}\underline{F}(\theta_f)\right\Vert \lesssim_{\textrm{P}}\,q^{-1/2}N^{-1/2}+T^{-1}$, where $h_{k}$ is the $k$-th row of $H$; hence $\hat{\alpha}\underline{\hat{F}}(\hat{\theta})\approx\hat{\alpha}H\underline{F}(\theta_f)\approx\alpha\underline{F}(\theta_f)$, which with $\hat{\alpha}_{w}-\alpha_{w}=o_{\textrm{P}}(1)$ yields prediction consistency.
Since factors omitted by SsPCA are uncorrelated with $y_{t+h}$, consistency of $\hat{y}_{T+h}$ does not require $\tilde{K}=K$, and the result relies on neither normal errors nor equal factor strength. The extra condition $\sqrt{T}/(qN)\rightarrow0$ reflects the cost of estimating the MIDAS weights, which the richer high-frequency information helps offset in practice.
\subsection{Recovery of All Factors }
In this subsection we develop the asymptotic distribution of $\hat{y}_{T+h}$ from Algorithm \ref{Algorithm 1}. The conditions of Theorem \ref{thm:2} suffice for prediction consistency but not for convergence at the optimal rate $T^{-1/2}$, the main difficulty being the full recovery of all factors.
In the screening step, predictors are selected by their sample covariance with $y_{t+h}$; even if $y_{t+h}$ is independent of $X_{t,\textrm{scaled}}(\hat{\theta}_x)$, this covariance can be large, so a threshold $c>T^{-1/2}(\log N)^{1/2}$ is needed to control Type I error. But when the signal-to-noise ratio is low, e.g. $\alpha\asymp T^{-1/2}$ ($y_{t+h}$ close to noise), screening may stop prematurely because the relevant covariances are at best of order $T^{-1/2}(\log N)^{1/2}<c$, giving no factor discovery. Then $\hat{y}_{T+h}=0$, which is consistent since the bias $\left|\mathbb{E}_{T}(y_{T+h})-0\right|\asymp T^{-1/2}$, but the CLT fails.
This arises when the procedure fails to recover all factors driving the target. If all factors are identified, the bias is negligible and the CLT holds regardless of $\alpha$. To rule out asymptotically missing factors we require $\lambda_{\min}(\alpha^{\prime}\alpha)\gtrsim1$, so each factor contributes non-negligibly to the target\footnote{As in \cite{giglio2023prediction}, this condition is formulated for a general multi-target setting with $\alpha\in\mathbb{R}^{D\times K}$ and $D\geq K$, and the number-of-factors recovery argument remains valid at this generality.}, while no more factors than necessary are selected asymptotically, since the iteration stops once all covariances vanish. A consistent estimator of the number of factors then lets us recover the factor space and perform inference on the prediction target.
Inference in strong-factor models typically requires consistent estimation of the number of factors \citep{bai2002determining}, which is not guaranteed in finite samples. We adopt the PCA regression of \citet{giglio2023prediction}, robust to overestimating the number of factors and supported by finite-sample simulations. The next theorem gives the asymptotic results under conditions ensuring complete recovery of all factors:
\begin{thm}
\label{thm:3}Under the same assumptions as Theorem \ref{thm:2},
if we additionally require that $\lambda_{\min}(\alpha'\alpha)\gtrsim1$,
then for any tuning parameters $q$ and $c$ in equations (\ref{equation 6})
and (\ref{equation 7}) satisfying condition (\ref{equation 8}), we have
(i) $\hat{K}$ defined in Algorithm \ref{Algorithm 1} satisfies
$P(\hat{K}=K)\rightarrow1$; (ii) the factor space is consistently recovered in such a way that $\left\Vert \mathbb{P}_{\underline{\hat{F}}'(\hat{\theta})}-\mathbb{P}_{\underline{F}'(\theta_f)}\right\Vert =O_{\textrm{P}}(q^{-1/2}N^{-1/2}+T^{-1})$;
and (iii) the estimator $\hat{\gamma}$ constructed via Algorithm \ref{Algorithm 1}
satisfies $\left\Vert \hat{\gamma}\hat{\Upsilon}\beta-\alpha-T^{-1}\overline{\epsilon}\underline{F}'(\theta_f)\mathbf{\Sigma}_{F(\theta_f)}^{-1}\right\Vert =O_{\textrm{P}}(q^{-1}N^{-1}+T^{-1}).$
\end{thm}
Part $(i)$ shows that our procedure recovers the true number of factors asymptotically under weak factors; with Theorem \ref{thm:1} it implies $\tilde{K}=K$ under the strengthened assumptions, removing the need to distinguish the two. Unlike \cite{onatski2010determining}, our framework also recovers the space spanned by weak factors, as in $(ii)$, at the factor-estimation rate $(qN)^{1/2}\land T$ rather than $N^{1/2}\land T$ in the strong-factor case. Part $(iii)$ extends Theorem \ref{thm:2} by replacing $\alpha$ with $\alpha+T^{-1}\overline{\epsilon}\underline{F}'(\theta_{f})\mathbf{\Sigma}_{F(\theta_{f})}^{-1}$, the regression estimator of $\alpha$ under observable factors, and shows the latent-factor estimation error is at most $O_{\textrm{P}}(q^{-1}N^{-1}+T^{-1})$.
\subsection{Inference on the Prediction Target }
Without observable regressors $w$, we treat $\theta_f$ and $\theta_x$ as known (justified in Remark \ref{rem:1}) and decompose the prediction error as $\hat{y}_{T+h}-\mathbb{E}_{T}(y_{T+h})=(\hat{\gamma}\hat{\Upsilon}\beta-\alpha)F_{T}(\theta_f)+\hat{\gamma}\hat{\Upsilon}E_{T}(\theta_x)$, where the second term is of order $(qN)^{-1/2}$. By Theorem \ref{thm:3}$(iii)$, if $q^{-1}N^{-1}T\rightarrow0$ this term is $o_{\textrm{P}}(T^{-1/2})$ relative to the first, and $(\hat{\gamma}\hat{\Upsilon}\beta-\alpha)F_{T}(\theta_f)=T^{-1}\overline{\epsilon}\underline{F}(\theta_f)'\mathbf{\Sigma}_{F(\theta_f)}^{-1}F_{T}(\theta_f)+O_{\textrm{P}}(T^{-1})$, so root-$T$ inference on $\mathbb{E}_{T}(y_{T+h})$ is available. For a more accurate finite-sample approximation, we retain both terms without constraining the relative magnitudes of $qN$ and $T$.
For this purpose, we need the following assumption:
\begin{assumption}
\label{assu:7}As $N,T\rightarrow\infty$, $T^{-1/2}\overline{\epsilon}\underline{F}'(\theta_f),$
$T^{-1/2}\overline{\epsilon}\underline{W}'$, and $(qN)^{-1/2}\Psi\hat{\Upsilon}E_{T}(\theta_x)$
are jointly asymptotically normally distributed, satisfying:
\[
\left(\begin{array}{c}
vec(T^{-1/2}\overline{\epsilon}\underline{F}'(\theta_f))\\
vec(T^{-1/2}\overline{\epsilon}\underline{W}')\\
(qN)^{-1/2}\Psi\hat{\Upsilon}E_{T}(\theta_x)
\end{array}\right)\overset{\textrm{d}}{\longrightarrow}\mathcal{N}\left(\left(\begin{array}{c}
0\\
0\\
0
\end{array}\right),\Pi=\left(\begin{array}{ccc}
\Pi_{11} & \Pi_{12} & 0\\
\Pi'_{12} & \Pi_{22} & 0\\
0 & 0 & \Pi_{33}
\end{array}\right)\right),
\]
\noindent where $\Psi$ is a $K\times N$ matrix whose $k$th row
is equal to $\iota'_{(k)}(\hat{\Upsilon}\beta)'_{[I_{k}]}(\mathbb{I}_{N})_{[I_{k}]}$
and $\iota_{(k)}$ is the first right singular vector of $(\hat{\Upsilon}\beta)_{(k)}=(\hat{\Upsilon}\beta)_{[I_{k}]}\prod_{j<k}\mathbb{M}_{\iota_{(j)}}$
as defined in Section \ref{Section 3.1}.
\end{assumption}
Assumption \ref{assu:7} describes the joint asymptotic distribution of $\overline{\epsilon}\underline{F}'(\theta_f),$ $\overline{\epsilon}\underline{W}'$, and $\Psi\hat{\Upsilon}E_{T}(\theta_x)$, a high-level condition like those in \cite{bai2003inferential} and \cite{giglio2023prediction}. As the first two components are finite-dimensional, the CLT for mixing processes applies as $T\to\infty$. The $k$th row of $\Psi\hat{\Upsilon}E_T(\theta_x)$, $\iota_{(k)}'(\hat{\Upsilon}\beta)_{[I_k]}'(\hat{\Upsilon}E_T(\theta_x))_{[I_k]}$, is a weighted average of $\hat{\Upsilon}E_{iT}(\theta_{x,i})$, $i\in I_k$, converging at rate $(qN)^{1/2}$ since $|I_k|=qN$.
Since $\Pi_{33}$ is not restricted to be diagonal, Assumption \ref{assu:7} accommodates cross-sectional correlation and heteroskedasticity in the aggregated idiosyncratic components, with the i.i.d. case as a special instance. As shown in Appendix \ref{Appendix B.5}, $\Pi_{33}=q^{-1}N^{-1}\Psi\Sigma_{E(\theta)}\Psi'$ with $\Sigma_{E(\theta)}=Cov[E_t(\theta)]$ general, and the feasible estimator of $\Phi_2$ there does not presume cross-sectional independence.
To establish the CLT for our analysis, we proceed as follows:
\begin{thm}
\label{thm:4}Suppose the same assumption as in Theorem \ref{thm:3}
hold. If, in addition, Assumption \ref{assu:7} holds, we have
\[
\varPhi^{-1/2}(\hat{y}_{T+h}-\mathbb{E}_{T}(y_{T+h}))\overset{\textrm{d}}{\longrightarrow}\mathcal{N}(0,1),
\]
\noindent where $\varPhi=T^{-1}\varPhi_{1}+q^{-1}N^{-1}\varPhi_{2},$
and $\varPhi_{1}$ and $\varPhi_{2}$ are given by:
\[
\begin{array}{c}
\varPhi_{1}=(F_{T}'(\theta_f),w'_{T})\mathbf{\Sigma}_{F(\theta_f),w}^{-1}\left(\begin{array}{cc}
\Pi_{11} & \Pi_{12}\\
\Pi'_{12} & \Pi_{22}
\end{array}\right)\mathbf{\Sigma}_{F(\theta_f),w}^{-1}(F_{T}'(\theta_f),w'_{T})',\\
\varPhi_{2}=\alpha L(\Lambda/qN)^{-1}\Omega'\Pi_{33}\Omega(\Lambda/qN)^{-1}L'\alpha,
\end{array}
\]
\noindent where $\Pi_{ij}$ is specified by Assumption \ref{assu:7},
$\mathbf{\Sigma}_{F(\theta_f),w}=\textrm{diag}\left(\mathbf{\Sigma}_{F(\theta_f)},\mathbf{\Sigma}_{w}\right)$,
$\Lambda=\textrm{diag}(\lambda_{(1)},...,\lambda_{(K)})$, $L$
is a $K\times K$ matrix whose $k$th column is given by $\iota_{(k)}$, and
$\Omega=(\omega_{1},...,\omega_{K})$ is a $K\times K$ matrix with $\omega_{1}=\varrho_{1}$
and $\omega_{k}=\varrho_{k}\sideset{-}{_{i=1}^{k-1}}\sum\lambda_{(i)}^{-1}\iota'_{(k)}(\hat{\Upsilon}\beta)'_{[I_{k}]}(\hat{\Upsilon}\beta){}_{[I_{k}]}\iota{}_{(i)}\omega_{i}$,
where $\varrho_{k}$ is a $K$-dimensional unit vector with 1 at the
$k$-th entry and $0$ elsewhere, $\lambda_{(k)}^{1/2}$ is the largest singular value of $(\hat{\Upsilon}\beta)_{(k)}$
and $\iota_{(k)}$ is the corresponding right singular vector as defined
in Section \ref{Section 3.1}.
\end{thm}
The asymptotic rate of $\hat{y}_{T+h}$ is driven jointly by $T$ and $qN$, so feasible inference requires consistent estimators of both $\varPhi_{1}$ and $\varPhi_{2}$. Estimating $\varPhi_{1}$ is straightforward from the outputs of Algorithm \ref{Algorithm 1}, while $\varPhi_{2}$ is harder because it depends on the large covariance matrix of $e_{T}(\theta_x)$; details are in Appendix \ref{Prof B.5}.
\begin{remark}[Estimated MIDAS weights]\label{rem:1}
Theorem \ref{thm:4} treats the weighting parameters $\theta=(\theta_f,\theta_x)'$ as known, which is asymptotically justified under the condition $\sqrt{T}/(qN)\to 0$ maintained in Theorem \ref{thm:1}. By Theorem 2.1 of \citet{koh2023inference}, the NLS estimator has an asymptotic bias proportional to $c_0$, the limit of $\sqrt{T}/N$ in her setting and of $\sqrt{T}/(qN)$ in ours; the maintained condition sets $c_0=0$, so $\hat{\theta}-\theta=O_P(T^{-1/2})$ with no first-order bias (Lemma \ref{lem:16}(iii)), and its contribution is absorbed into the $O_P(T^{-1}+q^{-1}N^{-1})$ remainder in the proof of Theorem \ref{thm:4}. Since $\lfloor qN\rfloor\ll N$, this is more demanding than its strong-factor counterpart. When instead $\sqrt{T}/(qN)\to c_0>0$, the plug-in error induces an asymptotic bias, and the bootstrap correction of \citet{koh2023inference} could in principle be adapted, though a formal treatment requires a different asymptotic framework and is left for future research. The regime $\sqrt{T}\ll qN$ is precisely the one for which our procedure is designed: short samples of the low-frequency target combined with large high-frequency panels. In our simulations ($T\in\{30,60\}$, $|I_0|\approx 100$) and empirical application ($T=60$, $\lfloor qN\rfloor$ from a grid starting at $100$), $\sqrt{T}/(qN)$ is at most $0.08$; consistent with this, the standardized prediction errors in Figure \ref{Figure D.9} are well approximated by the standard normal.
\end{remark}
Finally, since the prediction in Algorithm \ref{Algorithm 1} (Step S3) projects
$y_{T+h}$ directly onto $X_{T}(\theta_x)$ and $w_{T}$,
it is straightforward to compute and readily applicable out of sample,
avoiding the need for complex inference on latent factors.
\subsection{Comparative Advantages of SsPCA\label{Section 3.4}}
We now explain why combining supervised scaling with supervised selection is needed for reliable factor recovery and prediction under weak factors. Table \ref{Table A.1} in Appendix \ref{Appendix A} summarizes the four methods across key dimensions, and we discuss them below.
The four methods differ in how they use target information and which
factors they can recover. Standard PCA requires pervasiveness, that the
eigenvalues of $\beta'\beta$ grow at rate $N$; under weak factors they
grow more slowly, $N/(T\lambda_K(\beta'\beta))$ may diverge, and the
principal components become inconsistent, potentially nearly orthogonal
to the true factors \citep{giglio2023prediction}. Being unsupervised, PCA
also cannot prioritize factors relevant to $y_{t+h}$ over those that
merely explain predictor variance. sPCA scales each predictor by its
predictive slope $\hat{\Upsilon}_i$ before applying PCA, but assigns a
nonzero weight of order $T^{-1/2}$ to every predictor; when only an
$N_0\ll N$ subset loads on the relevant factors, the accumulated noise
from the remaining predictors renders both the factors and the forecast
inconsistent (Propositions \ref{prop:3} and \ref{prop:4}). SPCA instead
selects the $\lfloor qN\rfloor$ predictors most covariant with the target
residual and extracts a factor from that subset, achieving consistent
recovery under Assumption \ref{assu:2}, where pervasiveness is required
only within $I_0\subset[N]$; because screening uses raw covariances,
however, strong irrelevant factors can be selected spuriously and
contaminate the extracted directions. SsPCA combines both steps: scaling
by $\hat{\Upsilon}_i\approx\beta_i\alpha'$ inflates the weights on
relevant predictors and shrinks those on irrelevant ones, so the
subsequent SPCA screening more accurately targets the factors driving
$y_{t+h}$. Proposition \ref{prop:2} gives conditions under which SsPCA
attains strictly lower asymptotic MSFE than SPCA, and Propositions
\ref{prop:3} and \ref{prop:4} show it remains consistent where sPCA
fails. Since its advantage over PCA and PLS follows from established
results, the formal comparison in Appendix \ref{Appendix A} focuses on
SsPCA versus SPCA and sPCA.
\subsection{Asymptotic Properties of Boosting with SsPCA Factors\label{Section 3.5}}
\citet[Proposition 1]{bai2009boosting} establish the consistency of boosting applied to PCA-extracted factors, showing via an error-decomposition argument that the cumulative factor-estimation error vanishes so boosting stays consistent with estimated factors. Building on their same-frequency framework, we extend the analysis to aggregated factors estimated by SsPCA in a mixed-frequency setting. Because boosting repeatedly refits using the estimated factors, the estimation error accumulates over iterations; Proposition \ref{prop:5} (Appendix \ref{Appendix A.3}) controls this through the stopping rule $M(q^{-1/2}N^{-1/2}+T^{-1})\rightarrow0$, under which boosting consistently estimates the sparse model structure with observed or estimated aggregated factors, component-wise or block-wise.
\section{Monte Carlo Simulations\label{Section 4} }
We assess the finite-sample performance of SsPCA-MIDAS against other PCA-based methods and PLS in the factor-MIDAS framework, and evaluate boosting applied to the factors from each PCA-based method. Weak factors are generated as in \citet[Section 3]{hounyo2023forecasting}, i.e. with statistically significant loadings on about $5\%$ of the $N$ cross-sectional units, consistent with our definition.
Specifically, we consider a 3-factor ADL-MIDAS DGP:
\begin{equation}
y_{t+h}=\alpha_{0}+\alpha_{1}y_{t}+\alpha\stackrel[j=0]{J}{\sum}b^f_{j}(\theta_{f,g})g_{t-j/m}+\varepsilon_{t+h},
\end{equation}
\begin{equation}
x_{i,t-j/m}=\beta{}_{i}f_{t-j/m}+e_{i,t-j/m},j=m-1,...,0,
\end{equation}
\noindent where $f_{t-j/m}=(g_{1,t-j/m},g_{2,t-j/m},h_{t-j/m})'$: $g_{1,t-j/m}$ and $h_{t-j/m}$ are strong factors (the latter unrelated to the target) and $g_{2,t-j/m}$ is potentially weak. All three are drawn i.i.d. $\mathcal{N}(0,1)$. The strong loadings $\beta_{i,1}$ and $\beta_{i,h}$ are i.i.d. $\mathcal{N}(0,1)$; for $g_{2,t-j/m}$, the exposure $\beta_{i,2}$ is drawn from a mixture, with probability $\pi$ from $\mathcal{N}(0,1)$ and
$1-\pi$ from $\mathcal{N}(0,0.1^{2})$. The parameter $\pi$ governs the strength of $g_{2,t-j/m}$: we set $\pi=0.5$ when it is strong and $\pi=0.05$ when weak.
The idiosyncratic errors $e_{i,t-j/m}$ are heteroskedastic but independent over $i$ and $t-j/m$, with standard deviations $\sigma_{i}$ ($i=1,...,N$) drawn from $U[0,1]$; for time heterogeneity we multiply each $\sigma_{i}$ by $\sigma_{t_{H}}\sim U[0.5,1.5]$. The measurement errors $\varepsilon_{t+h}$ are i.i.d. $\mathcal{N}(0,1)$. The high-frequency variable is observed at most $m=3$ times per low-frequency period (quarterly target, monthly predictors), and we set $J=11$, so the target loads on $11$ lagged monthly factors. The weighting functions $b^f_{j}(\theta_f)$ and $b^x_{j}(\theta_x)$ use the two-parameter exponential Almon lag with $\theta_{1}=7\times10^{-4}$, $\theta_{2}=-5\times10^{-2}$ (fast-decaying weights). We set $\alpha_{0}=0$, $\alpha_{1}=0.2$ (i.e. $\alpha_{w}=(0,0.2)$), and let $\alpha$ vary across three scenarios:
\textbf{Scenario 1.} Predicting both $g_{1,t-j/m}$ and $g_{2,t-j/m}$, the latter strong, $\alpha=(1,1)$.
\textbf{Scenario 2.} Predicting both $g_{1,t-j/m}$ and $g_{2,t-j/m}$, but the latter weak, $\alpha=(1,1)$.
\textbf{Scenario 3.} Predicting only the weak factor $g_{2,t-j/m}$, $\alpha=(0,1)$, making $g_{1,t-j/m}$ irrelevant.
In all scenarios $T\in\{30,60\}$, so $T_{H}\in\{90,180\}$ since $m=3$. We set $N=200$, $\pi=0.5$ in Scenario 1 and $N=2000$, $\pi=0.05$ in Scenarios 2 and 3, so $\pi N=100$ throughout.
For tuning, PCA, sPCA, and PLS adjust only the number of factors $K$, set between $1$ and $5$ and chosen by 3-fold CV MSE. SPCA and SsPCA add the subset size $\lfloor qN\rfloor$, searched from $10$ to $200$ in steps of $10$ in Scenario 1 and from $100$ to $2000$ in steps of $100$ in Scenarios 2 and 3, with the optimal $(\lfloor qN\rfloor,K)$ again chosen by 3-fold CV MSE.
For the boosting extensions, applied after factor extraction, we fix the boosting-specific parameters to avoid extra tuning cost: the shrinkage $\nu=0.1$ (slow learning) and $M\in\{50,100\}$ iterations. These choices align with the forecasting literature, where useful stopping iterations are typically small at such learning rates \citep{wohlrabe2014assessing,lahiri2022boosting}.
We compare all methods and their boosting extensions in Tables \ref{Table 1}, \ref{Table D.1}, \ref{Table D.2}, and \ref{Table D.3}, based on OOS MSFE and bias at $h=1$ and $h=4$. We use the first $2/3$ of the sample for training and the remaining $1/3$ ($P=T/3$) for testing. For each origin $s=T-P-h+1,\ldots,T-h$, the model is re-estimated through period $s$ with $h$-step error $\tilde{u}_{s+1}=y_{s+1}-\hat{y}_{s+1|s}$, giving $\mathrm{OOS\ MSFE}_h=P^{-1}\sum_{s=T-P-h+1}^{T-h}\tilde{u}_{s+1}^{2}$ and $\mathrm{Bias}_h=P^{-1}\sum_{s=T-P-h+1}^{T-h}\tilde{u}_{s+1}$.
\begin{table}[!t]
\caption{Finite Sample Comparison of Predictions (MSFE)}
\small
\begin{centering}
\setlength{\tabcolsep}{2.5pt}
\begin{tabular}{c|c|cccccc|cccc}
\hline
\multicolumn{12}{c}{{\small Scenario 1: $N=200,\pi=0.5$}}\tabularnewline
\hline
\hline
\multicolumn{1}{c}{} & {\small$T_{H}$} & Oracle & {\small PCA} & {\small SPCA} & {\small sPCA} & {\small SsPCA} & {\small PLS} & {\small Bo-PCA} & {\small Bo-SPCA} & {\small Bo-sPCA} & {\small Bo-SsPCA}\tabularnewline
\hline
\multirow{2}{*}{{\small$h=1$}} & {\small$90$} & 0.681 & 0.972 & {\small 0.779} & {\small 0.920} & {\small\textbf{0.765}} & {\small 1.020} & {\small 1.002} & {\small 0.852} & {\small 0.951} & {\small\textbf{0.828}}\tabularnewline
\cline{2-12}
& {\small$180$} & 0.703 & {\small 0.920} & {\small 0.825} & {\small 0.872} & {\small\textbf{0.814}} & {\small 0.948} & {\small 0.942} & {\small 0.859} & {\small 0.892} & {\small\textbf{0.843}}\tabularnewline
\hline
\multirow{2}{*}{{\small$h=4$}} & {\small$90$} & 0.702 & {\small 1.046} & {\small 0.802} & {\small 0.972} & {\small\textbf{0.789}} & {\small 1.040} & {\small 1.069} & {\small 0.886} & {\small 1.003} & {\small\textbf{0.877}}\tabularnewline
\cline{2-12}
& {\small$180$} & 0.718 & {\small 0.973} & {\small 0.853} & {\small 0.913} & {\small\textbf{0.852}} & {\small 0.990} & {\small 1.000} & {\small 0.891} & {\small 0.942} & {\small\textbf{0.888}}\tabularnewline
\hline
\multicolumn{12}{c}{{\small Scenario 2: $N=2000,\pi=0.05$}}\tabularnewline
\hline
\multicolumn{1}{c}{} & {\small$T_{H}$} & Oracle & {\small PCA} & {\small SPCA} & {\small sPCA} & {\small SsPCA} & \multicolumn{1}{c}{{\small PLS}} & {\small Bo-PCA} & {\small Bo-SPCA} & {\small Bo-sPCA} & {\small Bo-SsPCA}\tabularnewline
\hline
\multirow{2}{*}{{\small$h=1$}} & {\small$90$} & 0.687 & {\small 0.955} & {\small 0.821} & {\small 0.872} & {\small\textbf{0.783}} & {\small 0.933} & {\small 0.988} & {\small 0.894} & {\small 0.919} & {\small\textbf{0.852}}\tabularnewline
\cline{2-12}
& {\small$180$} & 0.698 & {\small 0.914} & {\small 0.856} & {\small 0.854} & {\small\textbf{0.839}} & {\small 0.905} & {\small 0.940} & {\small 0.885} & {\small 0.890} & {\small\textbf{0.872}}\tabularnewline
\hline
\multirow{2}{*}{{\small$h=4$}} & {\small$90$} & 0.706 & {\small 1.018} & {\small 0.834} & {\small 0.912} & {\small\textbf{0.806}} & {\small 0.963} & {\small 1.054} & {\small 0.912} & {\small 0.966} & {\small\textbf{0.879}}\tabularnewline
\cline{2-12}
& {\small$180$} & 0.722 & {\small 0.974} & {\small 0.888} & {\small 0.896} & {\small\textbf{0.864}} & {\small 0.941} & {\small 1.002} & {\small 0.927} & {\small 0.939} & {\small\textbf{0.908}}\tabularnewline
\hline
\multicolumn{12}{c}{{\small Scenario 3: $N=2000,\pi=0.05$}}\tabularnewline
\hline
\multicolumn{1}{c}{} & {\small$T_{H}$} & Oracle & {\small PCA} & {\small SPCA} & {\small sPCA} & {\small SsPCA} & {\small PLS} & {\small Bo-PCA} & {\small Bo-SPCA} & {\small Bo-sPCA} & {\small Bo-SsPCA}\tabularnewline
\hline
\multirow{2}{*}{{\small$h=1$}} & {\small$90$} & 0.770 & {\small 1.005} & {\small 0.862} & {\small 0.905} & {\small\textbf{0.794}} & {\small 1.012} & {\small 1.035} & {\small 0.910} & {\small 0.942} & {\small\textbf{0.855}}\tabularnewline
\cline{2-12}
& {\small$180$} & 0.793 & {\small 0.970} & {\small 0.909} & {\small 0.910} & {\small\textbf{0.870}} & {\small 0.989} & {\small 0.990} & {\small 0.938} & {\small 0.933} & {\small\textbf{0.901}}\tabularnewline
\hline
\multirow{2}{*}{{\small$h=4$}} & {\small$90$} & 0.785 & {\small 1.064} & {\small 0.881} & {\small 0.949} & {\small\textbf{0.822}} & {\small 1.039} & {\small 1.079} & {\small 0.949} & {\small 0.991} & {\small\textbf{0.895}}\tabularnewline
\cline{2-12}
& {\small$180$} & 0.814 & {\small 1.018} & {\small 0.951} & {\small 0.951} & {\small\textbf{0.888}} & {\small 1.028} & {\small 1.041} & {\small 0.987} & {\small 0.978} & {\small\textbf{0.932}}\tabularnewline
\hline
\end{tabular}
\par\end{centering}
$\,$
\label{Table 1}{\footnotesize\textbf{Notes: }}{\footnotesize The table
reports the OOS MSFE of PCA, SPCA, sPCA, SsPCA and PLS, and of boosting
applied to the factors each of them extracts (Bo-PCA, Bo-SPCA, Bo-sPCA
and Bo-SsPCA), for $h=1,4$, with the number of boosting iterations fixed
at $M=50$. All values are averages over 1,000 Monte Carlo repetitions.
The Oracle uses the true factors in the MIDAS regression.}{\footnotesize\par}
\end{table}
To assess sensitivity to the subset-size tuning parameter, Table \ref{Table D.6} of Appendix \ref{Appendix D} perturbs $\lfloor qN \rfloor$ around its CV optimum in Scenario 2 with $h=4$ and $T_H=180$, which features coexisting strong and weak factors. The MSFE is lowest at the CV-selected value and changes only gradually nearby, so SsPCA's performance is not driven by a narrow or idiosyncratic choice of $\lfloor qN \rfloor$.
The results in Tables \ref{Table 1} and \ref{Table D.1}--\ref{Table D.3} align with our theory: SsPCA performs best in almost all scenarios, and boosting with SsPCA factors outperforms the other boosting variants. As $g_{2,t-j/m}$ weakens and $g_{1,t-j/m}$ becomes irrelevant from Scenario 1 to 3, SsPCA's advantage grows, especially at smaller $T$. The Oracle, which uses the true factors in the MIDAS regression, is infeasible and bounds the attainable MSFE from below; SsPCA tracks it most closely among feasible methods, so its factor-estimation loss is modest.
SPCA and sPCA both beat PCA but cannot be ranked directly, since SPCA's extra tuning parameter adds overfitting risk; SsPCA combines their strengths, and although it tunes two parameters the gains outweigh the cost, as for boosting with SsPCA. Boosting does not uniformly help---it accumulates error over iterations and can require up to three tuning parameters---yet boosting with PCA-based factors performs strongly in some settings \citep{lahiri2022boosting}, as our empirical results confirm. PLS tends to overfit here, since it uses $y$ directly, especially with many predictors \citep{giglio2023prediction,hounyo2023forecasting}; preprocessing such as thresholding before PLS \citep{hounyo2023forecasting} can mitigate this, so PCA-based approaches are more robust to noisy factors.
For prediction bias, SsPCA and boosting with SsPCA are less uniformly dominant but still attain the lowest bias in nearly half of the configurations, reflecting the bias-variance trade-off. On factor recovery, measured by $d(\hat{\mathbf{f}}_H,\mathbf{f}_H)=||\mathbb{P}_{\hat{\mathbf{f}}_H'}-\mathbb{P}_{\mathbf{f}_H'}||$ (Tables \ref{Table D.4} and \ref{Table D.5} of Appendix \ref{Appendix D}), SsPCA or boosting with SsPCA achieves the smallest distance in Scenario 3, where $g_{2,t-j/m}$ is hardest to detect; in Scenarios 1 and 2 it is marginally behind sPCA, whose all-predictor scaling aids recovery when strong factors are present, but the gap is negligible.
Finally, Figure \ref{Figure D.9} in Appendix \ref{Appendix D} presents histograms of the standardized prediction errors implied by the CLT of Theorem \ref{thm:4}, using the settings of Table \ref{Table 1} with $T_H=180$. For SsPCA the histograms match the standard normal density closely, confirming the CLT; these designs feature small $\sqrt{T}/(qN)$, consistent with Remark \ref{rem:1}. SsPCA and Bo-SsPCA also exhibit the smallest standard deviation across all scenarios.
\section{Empirical Application\label{Section 5}}
We evaluate SsPCA and boosting with SsPCA against the alternatives from our simulations, within the factor-MIDAS setting, targeting eight U.S. variables: four financial (S\&P 500 index, CBOE Volatility Index (VIX), crude oil price, and housing price) and four macroeconomic (real GDP growth, inflation, IP growth, and unemployment), across multiple horizons.
\subsection{Data}
Our analysis combines three datasets. We draw the quarterly target series from FRED-QD \citep{mccracken2016fred} and use the $126$ monthly series of FRED-MD \citep{mccracken2020fred} for part of the predictor set; both are maintained by the Federal Reserve Bank of St. Louis\footnote{See https://www.stlouisfed.org/research/economists/mccracken/fred-databases; we use the `current' version downloaded on June 27th, 2025.} and span the major categories of U.S. macro-financial data. We apply the databases' recommended transformations for stationarity. Third, we use the Global Factor Data of \cite{jensen2023there}, containing $153$ stock-level characteristics for $93$ countries or regions in $13$ thematic clusters, with monthly capped value-weighted returns per factor\footnote{See https://jkpfactors.com/; we use the `current' version downloaded on June 27th, 2025.}. We select the eleven countries or regions most closely linked to the United States, including the U.S. itself, listed with justifications in Table \ref{Table D.7} of Appendix \ref{Appendix D}, obtaining $153$ monthly factor series each; all series are transformed to stationarity or pass stationarity tests.
For a balanced panel, we use monthly data from October 1996 to December 2024, i.e. quarterly targets from 1996 Q3 to 2024 Q4; coverage and missing-value treatment are described following Table \ref{Table D.7}. The final predictor set contains $1{,}555$ series ($126$ from FRED-MD, the rest from the Global Factor Data), letting us examine global effects on the U.S. economy and construct scenarios with weak or complex factor structures.
\subsection{Out of Sample Forecast Evaluation}
We evaluate out-of-sample performance for the eight targets over 2011 Q4 to 2024 Q4. At each origin we re-estimate the model on a rolling window (using the most recently available revised data, once per month) for horizons $h=1,2,4,8$; $h=1$ is a nowcast, since the current quarter's target is not yet fully observed. Balancing the shorter windows suited to volatile financial variables against the longer ones suited to slower macroeconomic variables, and consistent with the simulation design, we use a rolling window of $180$ months ($T_{H}$) for predictors and $60$ quarters ($T$) for targets; Appendix \ref{Appendix F} gives a worked example.
The specification follows the simulation, except that aligning monthly predictors with the quarterly target occasionally involves leading months $l=1,2,3$, so we write
\begin{equation}
y_{t+h}=\alpha_{0}+\alpha_{1}y_{t}+\alpha\stackrel[j=0-l]{J-l}{\sum}b^f_{j}(\theta_{f,g})g_{t-j/m}+\varepsilon_{t+h},
\end{equation}
fixing one quarterly lag (consistent with BIC, which selects one lag on average), with $J=11$ and the two-parameter exponential Almon lag. The benchmark is an AR model with BIC-selected lags (maximum $12$); all methods extract up to the first $10$ latent factors, with $K$ and $\lfloor qN\rfloor$ chosen by rolling-window CV MSE and $\lfloor qN\rfloor$ searched from $100$ to $1500$ in steps of $100$, and for boosting we fix $\nu=0.1$ and $M=100$. Performance is the OOS relative MSFE, $\textrm{RMSFE(method)}=\textrm{MSFE(method)}/\textrm{MSFE(AR,BIC)}$, with values below one indicating improvement over the benchmark.
\subsection{Results}
\subsubsection{Mixed-frequency Forecasting Performance}
Table \ref{Table 2} reports the OOS RMSFE across all targets and horizons. Across most macroeconomic variables, SsPCA and boosting with SsPCA perform best: the scaling step emphasizes predictors more strongly associated with the target, while subset selection removes unrelated high-variance components that can distort medium- and long-horizon forecasts.
\begin{table}[!t]
\setlength{\tabcolsep}{2.4pt}
\caption{OOS Performance of Different Methods in Forecasting Macro and Financial
Targets}
\begin{centering}
{\footnotesize{}
\begin{tabular}{c|c|c|ccccc|cccc}
\hline
\multicolumn{1}{c}{} & \multicolumn{1}{c}{{\footnotesize Targets}} & \multicolumn{1}{c}{{\footnotesize$h$}} & {\footnotesize PCA} & {\footnotesize SPCA} & {\footnotesize sPCA} & {\footnotesize SsPCA} & {\footnotesize PLS} & {\footnotesize Bo-PCA} & {\footnotesize Bo-SPCA} & {\footnotesize Bo-sPCA} & {\footnotesize Bo-SsPCA}\tabularnewline
\hline
\hline
\multirow{16}{*}{{\footnotesize Macro}} & \multirow{4}{*}{{\footnotesize GDP Growth}} & {\footnotesize$1$} & {\footnotesize 0.987} & {\footnotesize 0.954} & {\footnotesize\textbf{\textit{0.792}}} & {\footnotesize 0.830} & {\footnotesize 1.062} & {\footnotesize 0.991} & {\footnotesize 0.948} & {\footnotesize 0.816} & {\footnotesize\textbf{0.801}}\tabularnewline
\cline{3-12}
& & {\footnotesize$2$} & {\footnotesize 0.993} & {\footnotesize 0.948} & {\footnotesize\textbf{0.808}} & {\footnotesize 0.824} & {\footnotesize 1.119} & {\footnotesize 0.993} & {\footnotesize 0.957} & {\footnotesize 0.862} & {\footnotesize\textbf{\textit{0.792}}}\tabularnewline
\cline{3-12}
& & {\footnotesize$4$} & {\footnotesize 0.965} & {\footnotesize 0.868} & {\footnotesize 0.763} & {\footnotesize\textbf{\textit{0.623}}} & {\footnotesize 1.062} & {\footnotesize 0.961} & {\footnotesize 0.867} & {\footnotesize 0.744} & {\footnotesize\textbf{0.651}}\tabularnewline
\cline{3-12}
& & {\footnotesize$8$} & {\footnotesize 1.016} & {\footnotesize 0.970} & {\footnotesize 0.815} & {\footnotesize\textbf{\textit{0.742}}} & {\footnotesize 1.093} & {\footnotesize 1.018} & {\footnotesize 0.989} & {\footnotesize 0.857} & {\footnotesize\textbf{0.756}}\tabularnewline
\cline{2-12}
& \multirow{4}{*}{{\footnotesize Inflation}} & {\footnotesize$1$} & {\footnotesize 0.943} & {\footnotesize 0.780} & {\footnotesize 0.762} & {\footnotesize\textbf{0.603}} & {\footnotesize 1.086} & {\footnotesize 0.916} & {\footnotesize 0.782} & {\footnotesize 0.731} & {\footnotesize\textbf{\textit{0.582}}}\tabularnewline
\cline{3-12}
& & {\footnotesize$2$} & {\footnotesize 0.957} & {\footnotesize 0.822} & {\footnotesize 0.733} & {\footnotesize\textbf{\textit{0.618}}} & {\footnotesize 1.343} & {\footnotesize 0.962} & {\footnotesize 0.818} & {\footnotesize 0.721} & {\footnotesize\textbf{0.627}}\tabularnewline
\cline{3-12}
& & {\footnotesize$4$} & {\footnotesize 0.957} & {\footnotesize 0.855} & {\footnotesize 0.890} & {\footnotesize\textbf{\textit{0.812}}} & {\footnotesize 1.163} & {\footnotesize 0.959} & {\footnotesize 0.903} & {\footnotesize 0.899} & {\footnotesize\textbf{0.842}}\tabularnewline
\cline{3-12}
& & {\footnotesize$8$} & {\footnotesize 0.963} & {\footnotesize 0.798} & {\footnotesize 0.936} & {\footnotesize\textbf{0.708}} & {\footnotesize 1.347} & {\footnotesize 0.959} & {\footnotesize 0.856} & {\footnotesize 0.886} & {\footnotesize\textbf{\textit{0.699}}}\tabularnewline
\cline{2-12}
& \multirow{4}{*}{{\footnotesize IP Growth}} & {\footnotesize$1$} & {\footnotesize 1.016} & {\footnotesize 0.994} & {\footnotesize\textbf{0.907}} & {\footnotesize 0.922} & {\footnotesize 1.057} & {\footnotesize 1.017} & {\footnotesize 0.997} & {\footnotesize 0.915} & {\footnotesize\textbf{\textit{0.887}}}\tabularnewline
\cline{3-12}
& & {\footnotesize$2$} & {\footnotesize 0.986} & {\footnotesize 0.956} & {\footnotesize 0.843} & {\footnotesize\textbf{0.823}} & {\footnotesize 1.099} & {\footnotesize 0.995} & {\footnotesize 0.936} & {\footnotesize 0.852} & {\footnotesize\textbf{\textit{0.781}}}\tabularnewline
\cline{3-12}
& & {\footnotesize$4$} & {\footnotesize 0.943} & {\footnotesize 0.907} & {\footnotesize 0.704} & {\footnotesize\textbf{0.680}} & {\footnotesize 1.037} & {\footnotesize 0.963} & {\footnotesize 0.874} & {\footnotesize 0.721} & {\footnotesize\textbf{\textit{0.647}}}\tabularnewline
\cline{3-12}
& & {\footnotesize$8$} & {\footnotesize 1.022} & {\footnotesize 0.962} & {\footnotesize 0.835} & {\footnotesize\textbf{\textit{0.790}}} & {\footnotesize 1.060} & {\footnotesize 1.024} & {\footnotesize 0.985} & {\footnotesize 0.876} & {\footnotesize\textbf{0.818}}\tabularnewline
\cline{2-12}
& \multirow{4}{*}{{\footnotesize Unemployment}} & {\footnotesize$1$} & {\footnotesize 0.966} & {\footnotesize 0.882} & {\footnotesize\textbf{0.838}} & {\footnotesize 0.842} & {\footnotesize 0.987} & {\footnotesize 0.967} & {\footnotesize 0.865} & {\footnotesize 0.849} & {\footnotesize\textbf{\textit{0.804}}}\tabularnewline
\cline{3-12}
& & {\footnotesize$2$} & {\footnotesize 0.933} & {\footnotesize 0.835} & {\footnotesize 0.769} & {\footnotesize\textbf{\textit{0.700}}} & {\footnotesize 0.824} & {\footnotesize 0.929} & {\footnotesize 0.849} & {\footnotesize 0.797} & {\footnotesize\textbf{0.740}}\tabularnewline
\cline{3-12}
& & {\footnotesize$4$} & {\footnotesize 0.861} & {\footnotesize 0.771} & {\footnotesize 0.654} & {\footnotesize\textbf{0.588}} & {\footnotesize 0.761} & {\footnotesize 0.874} & {\footnotesize 0.772} & {\footnotesize 0.648} & {\footnotesize\textbf{\textit{0.572}}}\tabularnewline
\cline{3-12}
& & {\footnotesize$8$} & {\footnotesize 0.895} & {\footnotesize 0.752} & {\footnotesize 0.680} & {\footnotesize\textbf{\textit{0.591}}} & {\footnotesize 0.718} & {\footnotesize 0.894} & {\footnotesize 0.778} & {\footnotesize 0.716} & {\footnotesize\textbf{0.594}}\tabularnewline
\hline
\multirow{16}{*}{{\footnotesize Finance}} & \multirow{4}{*}{{\footnotesize S\&P500}} & {\footnotesize$1$} & {\footnotesize 0.873} & {\footnotesize 0.760} & {\footnotesize 0.786} & {\footnotesize\textbf{0.756}} & {\footnotesize 1.005} & {\footnotesize 0.875} & {\footnotesize 0.787} & {\footnotesize 0.817} & {\footnotesize\textbf{\textit{0.661}}}\tabularnewline
\cline{3-12}
& & {\footnotesize$2$} & {\footnotesize 0.855} & {\footnotesize 0.707} & {\footnotesize 0.677} & {\footnotesize\textbf{0.612}} & {\footnotesize 0.891} & {\footnotesize 0.854} & {\footnotesize 0.679} & {\footnotesize 0.675} & {\footnotesize\textbf{\textit{0.522}}}\tabularnewline
\cline{3-12}
& & {\footnotesize$4$} & {\footnotesize 0.856} & {\footnotesize 0.654} & {\footnotesize 0.523} & {\footnotesize\textbf{0.516}} & {\footnotesize 0.823} & {\footnotesize 0.842} & {\footnotesize 0.608} & {\footnotesize 0.523} & {\footnotesize\textbf{\textit{0.481}}}\tabularnewline
\cline{3-12}
& & {\footnotesize$8$} & {\footnotesize 0.917} & {\footnotesize 0.720} & {\footnotesize 0.728} & {\footnotesize\textbf{0.583}} & {\footnotesize 0.877} & {\footnotesize 0.921} & {\footnotesize 0.666} & {\footnotesize 0.720} & {\footnotesize\textbf{\textit{0.559}}}\tabularnewline
\cline{2-12}
& \multirow{4}{*}{{\footnotesize VIX}} & {\footnotesize$1$} & {\footnotesize 0.378} & {\footnotesize 0.334} & {\footnotesize 0.378} & {\footnotesize\textbf{0.322}} & {\footnotesize 0.435} & {\footnotesize 0.375} & {\footnotesize 0.347} & {\footnotesize 0.371} & {\footnotesize\textbf{\textit{0.314}}}\tabularnewline
\cline{3-12}
& & {\footnotesize$2$} & {\footnotesize 0.619} & {\footnotesize\textbf{0.450}} & {\footnotesize 0.505} & {\footnotesize 0.455} & {\footnotesize 0.575} & {\footnotesize 0.622} & {\footnotesize 0.483} & {\footnotesize 0.493} & {\footnotesize\textbf{\textit{0.427}}}\tabularnewline
\cline{3-12}
& & {\footnotesize$4$} & {\footnotesize 0.680} & {\footnotesize 0.507} & {\footnotesize 0.596} & {\footnotesize\textbf{0.485}} & {\footnotesize 0.655} & {\footnotesize 0.709} & {\footnotesize 0.555} & {\footnotesize 0.578} & {\footnotesize\textbf{\textit{0.448}}}\tabularnewline
\cline{3-12}
& & {\footnotesize$8$} & {\footnotesize 0.826} & {\footnotesize 0.534} & {\footnotesize 0.614} & {\footnotesize\textbf{0.465}} & {\footnotesize 0.528} & {\footnotesize 0.820} & {\footnotesize 0.587} & {\footnotesize 0.603} & {\footnotesize\textbf{\textit{0.458}}}\tabularnewline
\cline{2-12}
& \multirow{4}{*}{{\footnotesize Oil Price}} & {\footnotesize$1$} & {\footnotesize 0.971} & {\footnotesize 0.826} & {\footnotesize 0.712} & {\footnotesize\textbf{\textit{0.640}}} & {\footnotesize 1.059} & {\footnotesize 0.947} & {\footnotesize 0.814} & {\footnotesize 0.748} & {\footnotesize\textbf{0.642}}\tabularnewline
\cline{3-12}
& & {\footnotesize$2$} & {\footnotesize 0.968} & {\footnotesize 0.877} & {\footnotesize 0.675} & {\footnotesize\textbf{0.632}} & {\footnotesize 1.069} & {\footnotesize 0.970} & {\footnotesize 0.869} & {\footnotesize 0.698} & {\footnotesize\textbf{\textit{0.593}}}\tabularnewline
\cline{3-12}
& & {\footnotesize$4$} & {\footnotesize 0.929} & {\footnotesize 0.808} & {\footnotesize 0.760} & {\footnotesize\textbf{\textit{0.627}}} & {\footnotesize 0.974} & {\footnotesize 0.932} & {\footnotesize 0.813} & {\footnotesize 0.789} & {\footnotesize\textbf{0.653}}\tabularnewline
\cline{3-12}
& & {\footnotesize$8$} & {\footnotesize 0.993} & {\footnotesize 0.808} & {\footnotesize 0.858} & {\footnotesize\textbf{0.707}} & {\footnotesize 1.131} & {\footnotesize 0.996} & {\footnotesize 0.825} & {\footnotesize 0.849} & {\footnotesize\textbf{\textit{0.682}}}\tabularnewline
\cline{2-12}
& \multirow{4}{*}{{\footnotesize Housing Price}} & {\footnotesize$1$} & {\footnotesize 0.390} & {\footnotesize 0.368} & {\footnotesize 0.361} & {\footnotesize\textbf{0.347}} & {\footnotesize 0.454} & {\footnotesize 0.404} & {\footnotesize 0.356} & {\footnotesize 0.372} & {\footnotesize\textbf{\textit{0.339}}}\tabularnewline
\cline{3-12}
& & {\footnotesize$2$} & {\footnotesize 0.705} & {\footnotesize 0.608} & {\footnotesize 0.607} & {\footnotesize\textbf{0.531}} & {\footnotesize 0.744} & {\footnotesize 0.719} & {\footnotesize 0.628} & {\footnotesize 0.592} & {\footnotesize\textbf{\textit{0.515}}}\tabularnewline
\cline{3-12}
& & {\footnotesize$4$} & {\footnotesize 0.373} & {\footnotesize 0.284} & {\footnotesize 0.336} & {\footnotesize\textbf{\textit{0.283}}} & {\footnotesize 0.357} & {\footnotesize 0.395} & {\footnotesize\textbf{0.299}} & {\footnotesize 0.333} & {\footnotesize 0.304}\tabularnewline
\cline{3-12}
& & {\footnotesize$8$} & {\footnotesize 0.578} & {\footnotesize 0.315} & {\footnotesize 0.422} & {\footnotesize\textbf{0.282}} & {\footnotesize 0.352} & {\footnotesize 0.558} & {\footnotesize 0.334} & {\footnotesize 0.391} & {\footnotesize\textbf{\textit{0.281}}}\tabularnewline
\hline
\end{tabular}}{\footnotesize\par}
\par\end{centering}
$\,$
{\footnotesize\textbf{\label{Table 2}Notes: }}{\footnotesize The table
reports the out-of-sample RMSFE (relative to the AR,BIC model) of
PCA, SPCA, sPCA, SsPCA and PLS, and of boosting applied to the factors
each of them extracts (Bo-PCA, Bo-SPCA, Bo-sPCA and Bo-SsPCA), in
forecasting four macroeconomic and four financial target variables at
horizons $h=1,2,4,8$ over the sample 2011 Q4 to 2024 Q4. For each target
variable and horizon, italics mark the best value across all methods and
bold marks the best value within the non-boosted and the boosting methods
separately, so that the overall best value is both bold and
italic.}{\footnotesize\par}
\end{table}
In a few cases, mostly at $h=1$, conventional sPCA yields lower RMSFE than SsPCA, likely reflecting a trade-off between shrinkage aggressiveness and robustness under the coarse $\lfloor qN\rfloor$ grid: although SsPCA can in principle select all predictors and match sPCA, short-horizon forecasts of variables such as GDP growth offer scarce high-frequency signal, making the model more tuning-sensitive. Even then, boosting with SsPCA often reverses the result, refining the clean factors and recovering weak short-term signals; consistent with the simulations, it attains the best accuracy at a given horizon among all nine methods in many instances, by refitting residuals on factor-lag blocks to correct factor--target misalignment while early stopping controls variance.
In financial markets, SsPCA again performs best across most variables, with boosting on SsPCA frequently achieving the lowest RMSFE; its advantage is even more pronounced than for macroeconomic variables, with only two instances where SPCA slightly outperforms, matching \cite{hounyo2023forecasting} for same-frequency forecasting. A plausible explanation is that supervised scaling and selection are especially effective where predictive signal arises from a mixture of persistent risk-premium and rapidly shifting sentiment components, dispersed across many predictors and easily masked by volatility or cross-asset spillovers---precisely where the two supervision steps help isolate the structural drivers of asset prices, aided by the many asset-pricing and global financial factors in the predictor set.
\subsubsection{Predictors Selected by SsPCA}
Finally, we examine the predictors SsPCA selects. Figures \ref{Figure D.1}--\ref{Figure D.8} in Appendix \ref{Appendix D} display, as heatmaps, the top $50$ predictors extracted through the first factor for all eight targets at $h=1$ and $h=4$, and Tables \ref{Table D.8} and \ref{Table D.9} list the top $10$ per target, split at the March 2020 COVID break; details are in the figure and table notes.
Several patterns emerge. Global factors are selected persistently, consistent with international spillovers and the global financial cycle \citep{rey2015dilemma,miranda2020us}; selection shifts markedly around the onset of COVID-19 in March 2020 and the recovery in January 2023, indicating a break in the information set; the selected predictors are horizon-dependent; and Rates-group variables are consistently important. Interest-rate and credit-spread measures dominate the top ranks, alongside balance-sheet and real-activity indicators, and the pre- to post-COVID comparison shows a clear rotation from the rates block toward real-activity and supply-side measures and toward trade- and technology-linked Asian economies. SsPCA thus reallocates attention as the dominant source of fluctuations changes rather than relying on a fixed predictor set. A detailed, economically motivated discussion is given in Appendix \ref{Appendix D.1}.
\section{Conclusion\label{Section 6}}
We develop SsPCA-MIDAS to address weak factors in mixed-frequency forecasting and establish its asymptotic theory. Unlike conventional PCA-based approaches, SsPCA combines supervised weighting to emphasize relevant predictors with supervised selection to filter out irrelevant ones, improving factor estimation and preserving predictive relevance under weak factors.
Our analysis shows that SsPCA yields consistent factor estimates, improves factor recovery, ensures prediction consistency, and delivers asymptotic normality, permitting inference on the prediction target; it also explains why SsPCA often outperforms existing supervised PCA methods. The simulations confirm consistent gains in mixed-frequency forecasting, especially when weak factors dominate, together with better factor recovery and more stable, lower-variance standardized prediction errors, and boosting with the cleaner SsPCA factors yields further, asymptotically consistent, improvements. The empirical application to U.S. macro-financial forecasting confirms the method's practical relevance: SsPCA delivers systematic accuracy gains over PCA-based and other benchmarks, with further improvements from boosting, while selecting economically meaningful predictors and adapting to structural shifts such as the COVID-19 crisis.
Looking ahead, mixed-frequency machine-learning methods relying on latent factors could benefit from SsPCA, and future work could develop theoretical explanations for this effect, extend beyond factor-MIDAS to broader macroeconomic forecasting, financial modeling, and asset pricing, and explore computational efficiency, factor-selection strategies, and more complex data environments.