# The Forecasting performance of the Factor model with Martingale Difference
errors

Alessandro Giovannelli Luca Mattia Rolla Department of Engineering and
Information Science, Università degli Studi dell’Aquila. Email:
<EMAIL_ADDRESS>of Economics and Finance,
Università degli Studi di Roma ”Tor Vergata”. Email:
<EMAIL_ADDRESS>
###### Abstract

We compare the pseudo-real-time forecasting performance of two factor models
for a large set of macroeconomic and financial time series: (i) The standard
principal-component model used by Stock and Watson (2002) (ii) The factor
model with martingale difference errors introduced by Lee and Shao (2018). The
factor model with martingale difference error (FMMDE) makes it possible to
retrieve a transformation of the original series so that the resulting
variables can be partitioned into distinct groups according to whether they
are conditionally mean independent upon past information or not. In terms of
prediction, this feature of the model allows us to achieve optimal results (in
the mean squared error sense) as dimension reduction is performed. Conducting
a pseudo-real-time forecasting exercise based on a large dataset of
macroeconomic and financial monthly time series for the U.S. economy, the
results obtained from the empirical study suggest that FMMDE performs
comparably to SW for short-term forecasting horizons. In contrast, for long-
term forecasting horizons, FMMDE displays superior performance. These results
are particularly evident for Output & Income, Labor Market, and Consumption
sectors.

Keywords: Multivariate Time Series, Factor models, Nonlinear dependence,
Conditional mean.

##  1 Introduction

The present paper aims to evaluate the forecasting performance of the factor
model with martingale difference error (FMMDE, Lee and Shao, 2018), a recent
innovation in the factor modeling literature, to establish whether it can
represent a valid alternative to more traditional models in predicting
macroeconomic aggregates. To this end, we compare the forecasting performance
of FMMDE to that of the factor model defined in Stock and Watson (SW, 2002a,
2002b), which can be considered a well-established benchmark in the literature
and can therefore be used as a standard of reference. In macroeconomic
forecasting, dynamic factor models (hereafter DFM) proved to be extremely
successful in this contest, (see among others Stock and Watson, 2002a; Forni
et al., 2005; Boivin and Ng, 2005). The appeal of DFM is that the information
contained in a large dataset can be summarized using a small number of
factors. Those factors, estimated by means of linear principal components
analysis, are then used in a forecasting equation to predict the variable of
interest . This approach, known as diffusion index (DI), is now standard in
the forecasting literature due to its ability to condensate information from a
large number of predictors in a simple and parsimonious manner.

The use of FMMDE in the factor modeling literature is motivated by the search
for optimal prediction as dimension reduction for a multivariate stationary
time series is performed. Under the assumption that the error process model is
a martingale difference sequence - a more stringent condition than just
assuming that the error term follows a white noise process - it is possible to
show that there exists a contemporaneous linear transformation of the original
series such that the resulting factors can be separated into two distinct
segments, one of these being conditionally mean independent upon past
information. This has important implications in terms of optimal prediction,
as it reduces the modeling effort to the subset of factors that exhibit some
form of serial dependence in the conditional mean.

Another feature of FMMDE is represented by the fact that it relies upon
measures of statistical dependence that are suitable to detect nonlinear
relationships in the data. For this reason, the adoption of FMMDE appears to
be particularly fitting in the analysis of time series that present forms of
nonlinear dependence, as the model has a built-in advantage over traditional
factor models (including the SW model). In this sense, we provide some
simulated examples showing the advantage of FMMDE in such a nonlinear setting
in terms of forecast accuracy.

A further contribution of this paper is to propose a novel technique for
selecting the number of factors to use in the estimation of FMMDE. In doing
this, we depart from the methodology described in Lee & Shao’s original paper
by proposing a factor selection criterion that is, in our opinion, more
coherent with the specific characteristics of the model. Our approach will be
to include in the model the factors that have some form of conditional mean
dependence upon the past, that is, those factors that are not a martingale
difference sequence.

Our analysis considers a dataset of 123 US monthly macroeconomic time
series111We refer to the enclosed Appendix for more information about the
nature and composition of the series in the dataset. over the period from
January 1959 to December 2019 to evaluate the forecasting performance of the
two models. The study is carried out by employing a pseudo real time exercise
based on a large dataset proposed by (McCkracken & Ng, 2015). The dataset is
composed of 123 US monthly macroeconomic time series222We refer to the
enclosed Appendix for more information about the nature and composition of the
series in the dataset. observed over the period from January 1959 to December
2019. The forecast horizon considered are for 1,3, 6, 12 and 24 months ahead
and the loss function used top measure the forecast performance is the usual
mean squared error (MSE). Furthermore, the forecasting exercise was conducted
on all variables composing the dataset. In this way, it was possible to get a
more general picture of the different sectors that make up the dataset, beyond
the forecasting performance on the individual macroeconomic aggregates, such
as Personal Income, Industrial Production, Unemployment Rate, All Employees
and Consumer Price Index.

The main empirical findings can be summarized as follows. For short-term
forecasting horizons, the FMMDE approach has forecast performance comparable
to SW. In the longterm forecasting horizons, the improvement becomes more
evident especially for sectors related to Output & Income, Labor Market and
Consumption. This is true also for the main macroeconomic aggregates such as
Industrial production, Unemployment Rate, and Real Personal Income. If
instead, the prediction of nominal variables is concerned, SW model appears to
have an advantage over FMMDE.

The rest of the paper is structured as follows. We introduce in Section 2 the
main theoretical properties of the FMMDE, and the so called martingale
difference divergence matrix used in its implementation. In the same section,
we introduce the define factor selection techniques that are used to determine
the model dimensionality, including our proposed methodology. Section 3
briefly discusses the SW model and its main properties: inside it, we also
offer a first comparison of the characteristics of the two models through some
simulated examples. The composition of the dataset, estimation methodology and
forecasting procedure are extensively treated in Section 4. Finally, the
forecasting performance of the two competing models is discussed in Section 5:
we report results obtained for the recursive scheme of forecasting for both
individual series and the entire dataset as an aggregate.

##  2 Factor Model with Martingale Difference Error

Let $\mathbf{x}_{t}=(x_{1t},x_{2t},...,x_{nt})^{\prime}$ denote an $n\times 1$
vector of variables observed at times $t={1,...,T}$. Under the assumption that
the variables have a latent factor structure, the approximate factor model is
formulated as:

$\mathbf{x}_{t}=\mathbf{\Lambda
F}_{t}+\boldsymbol{\zeta}\mathbf{{}_{t}}\text{,}$ (1)

where $\mathbf{F}_{t}=(F_{1t},...,F_{rt})^{\prime}$ are $r<n$ unobserved
common factors, $\mathbf{\Lambda}$ is the $n\times r$ matrix of factor
loadings and $\boldsymbol{\zeta}_{t}=(\zeta_{1t},...,\zeta_{rt})^{\prime}$ is
the idiosyncratic disturbance not explained by the factors. If we further
assume that $\mathbf{x}_{t}$ is a mean zero $n$-variate strictly stationary
time series, the FMMDE can be defined as

$\mathbf{x}_{t}=E\left(\mathbf{x}_{t}|\mathcal{F}_{t-1}\right)+\boldsymbol{\zeta}_{t}\text{,}$
(2)

with $E(\mathbf{x}_{t}|\mathcal{F}_{t-1})=\mathbf{\Lambda}\mathbf{F}_{t}$, and
$\mathcal{F}_{t-1}=\sigma\left(\mathbf{x}_{t-1},...,\mathbf{x}_{1},...\right)$
denoting the information set at time $t-1$. By construction we shall now have
that the idiosyncratic component $\boldsymbol{\zeta}_{t}$ is a vector
martingale difference sequence.

Our goal is to identify a contemporaneous linear (invertible) transformation
for $\mathbf{x}_{t}$, say $\mathbf{M}$ $\in$ $R^{n\times n}$, such that
$\mathbf{Mx}_{t}$ $=$
$\left[\mathbf{F}_{a,t}^{\prime}\mathbf{,F}_{b,t}^{\prime}\right]^{\prime}$,
where $E\left(\mathbf{F}_{a,t}|\mathcal{F}_{t-1}\right)\neq
E\left(\mathbf{F}_{a,t}\right)$ ,
$E\left(\mathbf{F}_{b,t}|\mathcal{F}_{t-1}\right)$
$=E\left(\mathbf{F}_{b,t}\right)$ with $\mathbf{F}_{a,t}$ $\in$
$\mathbb{R}^{r}$ and $\mathbf{F}_{b,t}\in\mathbb{R}^{n-r}$. In other words, we
look for a transformation such that the resulting series can be separated into
two parts, with one part being conditionally mean dependent upon
$\mathcal{F}_{t-1}$, and another being conditionally mean independent upon the
past information set. In such a way, the modelling task for the series
$\mathbf{x}_{t}$ is reduced to that for the lower dimensional series
$\mathbf{F}_{a,t}$ while optimal results in terms of prediction accuracy (in
the mean squared error sense) are still attainable.

Let us introduce $\mathcal{B}$ as an $n\times\left(n-r\right)$ matrix for
which the relationship $\mathbf{\Lambda}^{\prime}\mathcal{B=}$ $0$ holds true.
Defining the matrix $\mathbf{M}$ as
$\mathbf{M}=\left[\mathbf{\Lambda},\mathcal{B}\right]^{\prime},$ we can write
accordingly

$\displaystyle
E\left(\mathbf{\Lambda}^{\prime}\mathbf{x}_{t}|\mathcal{F}_{t-1}\right)=E\left(\mathbf{F}_{a,t}|_{t-1}\right)=\mathbf{F}_{t}$
(3) $\displaystyle
E\left(\mathbf{F}_{b,t}|_{t-1}\right)=E\left(\mathcal{B}^{\prime}\mathbf{x}_{t}|_{t-1}\right)=E\left(\mathcal{B}^{\prime}\mathbf{x}_{t}\right)\text{.}$

As a result of obtaining the estimate $\widehat{\mathbf{M}}$ we can retrieve
the factor process $\mathbf{F}_{t}$ as
$\mathbf{\widehat{\mathbf{F}}}_{t}=\widehat{\Lambda}^{\prime}\mathbf{x}_{t}.$

###  2.1 Martingale Difference Divergence Matrix

The identification of these non-martingale difference factors is made feasible
by the introduction of the martingale difference divergence matrix,
mintroduced in Lee & Shao (2018), which allows us to retrieve the matrix
$\mathbf{M}$ and therefore to express $\mathbf{x}_{t}$ as a nonsingular linear
transformation of a $r$-dimensional dynamically dependent common factor
process, and a martingale difference error component of dimension $(n-r)$.

Such matrix can be regarded as a multivariate extension for the notion of
martingale difference divergence, introduced in Shao & Zhang (2014) as a
measure of conditional mean independence between random vectors: in our
specific case it can be used to measure the departure from the relationship of
conditional mean independence
$E\left(x_{h,t}|\mathbf{x}_{t-j}\right)=E\left(x_{h,t}\right)\text{ \ almost
surely,}$ for a fixed $j\in\mathbb{N}^{+}$, for each $h$-th element in the
vector $\mathbf{x_{t}}$.

Martingale difference divergence is defined as the non-negative square root of
the term

$MDD\left({x}_{h,t}|\mathbf{x}_{t-j}\right)^{2}=\frac{1}{c_{n}}\int_{\mathbb{R}^{n}}\frac{\left|G_{h}\left(\mathbf{s}\right)\right|^{2}}{\left|\mathbf{s}\right|_{n}^{1+n}}d\mathbf{s}$
(4)

where $G_{h}\left(\mathbf{s}\right)=$ cov$\left(x_{h,t},\exp
i\left\langle\mathbf{s},\mathbf{x}_{t-j}\right\rangle\right)$333Here we denote
$i=\sqrt{-1}$. For a complex-valued function $f(.)$ the complex conjugate of
$f$ is denoted by $f^{\ast}$ and $|f|^{2}=ff^{\ast}.$ and
$c_{n}=\pi^{\left(1+n\right)/2}/\Gamma\left(\left(1+n\right)/2\right)$. The
definition can be regarded as an extension of distance covariance (Székely,
Rizzo, & Bakirov 2007) using a similar weighting function. In particular,
martingale difference divergence is defined so that conditional mean
independence holds if and only if
$MDD\left({x}_{h,t}|\mathbf{x}_{t-j}\right)^{2}=0$. In this sense, the term
$G_{h}\left(\mathbf{s}\right)$ can reasonably be interpreted as a generalized
covariance component defined between $x_{h,t}$ and the lag of the entire
vector $\mathbf{x_{t}}$444The function $G_{h}\left(\mathbf{s}\right)$ will
denote che generalized covariance computed for the $h$-th element in
$\mathbf{x_{t}}$ and the complex exponential of its lag. This allows to
express serial dependence conditions in terms of characteristic functions..

Sample martingale difference divergence is defined from the
$\left(T-j\right)\times\left(T-j\right)$ pairwise Euclidean distance matrices
of elements $a_{rl}=\left|\mathbf{z}_{r}-\mathbf{z}_{l}\right|$ and
$b_{rl}=\frac{1}{2}\left|{x}_{h,r}-{x}_{h,l}\right|^{2}$, where the variable
$\mathbf{z}_{t}$ is simply defined as $\mathbf{z}_{t}=\mathbf{x}_{t-j}.$ Based
on the above quantities, sample martingale difference divergence $MDD_{T}$ can
be expressed as the nonnegative square root of

$MDD_{T}\left({x}_{i,t}|\mathbf{x}_{t-j}\right)^{2}=\left(T-\left|j\right|\right)^{-2}\sum_{r,l=1+\left|j\right|}^{T}A_{rl}B_{rl}$
(5)

defined for the two double centered matrices $A$ and $B$ with elements

$\displaystyle
A_{rl}=a_{rl}-\overline{a}_{r.}-\overline{a}_{.l}+\overline{a}_{..},\text{ \ \
}B_{rl}=b_{rl}-\overline{b}_{r.}-\overline{b}_{.l}+\overline{b}_{..}.$ (6)

where $\overline{a}_{r.}=\sum_{l=1}^{T}a_{rl}/T,\text{ \
}\overline{a}_{.l}=\sum_{l=r}^{T}a_{rl}/T,\text{ \
}\overline{a}_{..}=\sum_{r,l=1}^{T}a_{rl}/T^{2}$, and analogously for the
terms $\overline{b}_{r.}$, $\overline{b}_{.l}$, and $\overline{b}$.

In the case of multivariate time series objects, martingale difference
divergence matrix is defined as

$\mathbf{MDDM}\left(\mathbf{x}_{t}|\mathbf{x}_{t-j}\right)=\frac{1}{c_{n}}\int_{\mathbb{R}^{n}}\frac{\mathbf{G}\left(\mathbf{s}\right)\mathbf{G}\left(\mathbf{s}\right)^{\ast}}{\left|\mathbf{s}\right|_{n}^{1+n}}d\mathbf{s}\text{,}$
(7)

where $\mathbf{G}\left(\mathbf{s}\right)=$ cov$\left(\mathbf{x}_{t},\exp
i\left\langle\mathbf{s},\mathbf{x}_{t-j}\right\rangle\right)=\left(G_{1}\left(\mathbf{s}\right),...,G_{n}\left(\mathbf{s}\right)\right)^{\prime}$
for $\mathbf{s}\in\mathbb{R}^{n}$. For $h=1,...,n$, elementwise, we have
$G_{h}\left(\mathbf{s}\right)=$ cov$\left(x_{ht},\exp
i\left\langle\mathbf{s},\mathbf{x}_{t-j}\right\rangle\right)$. Accordingly,
the $(i,i)$th entry of the $n\times n$ matrix equals to
$MDD(x_{it}|\mathbf{x}_{t-j})^{2}$, whereas the $(i,k)$-th entry is

$\mathbf{MDDM}\left(\mathbf{x}_{t}|\mathbf{x}_{t-j}\right)_{ik}=\frac{1}{c_{n}}\int_{\mathbb{R}^{n}}\frac{G_{i}\left(\mathbf{s}\right)G_{k}\left(\mathbf{s}\right)^{\ast}}{\left|\mathbf{s}\right|_{n}^{1+n}}d\mathbf{s}=\mathbf{MDDM}\left(\mathbf{x}_{t}|\mathbf{x}_{t-j}\right)_{ki}^{\ast}\text{,}$
(8)

that is, the matrix is Hermitian and thus has real eigenvalues.

Assuming that
$E\left(|\mathbf{x}_{t}|_{n}^{2}+|\mathbf{x}_{t-j}|_{n}^{2}\right)$ $<\infty$,
Lemma 1 in Lee and Shao shows that the matrix is real, symmetric and positive
semi-definite. Therefore, for any real $n\times r$ matrix $\mathbf{\Lambda}$
we have the relationship

$\mathbf{MDDM}\left(\mathbf{\Lambda}^{\prime}\mathbf{x}_{t}|\mathbf{x}_{t-j}\right)=\mathbf{\Lambda}^{\prime}\mathbf{MDDM}(\mathbf{x}_{t}|\mathbf{x}_{t-j})\mathbf{\Lambda}\text{,}$
(9)

and subsequently, there exist $n-r$ linearly independent combinations that are
conditionally mean independent of $\mathbf{x}_{t-j}$, i.e.
$MDD\left(\mathcal{B}_{i}^{\prime}\mathbf{x}_{t}|\mathbf{x}_{t-j}\right)=0$
for $i=1,...,n-r$, if and only if
Rank$\left(\mathbf{MDDM}\left(\mathbf{x}_{t}|\mathbf{x}_{t-j}\right)\right)=r$.
As we want to determine the linear combinations of $\mathbf{x}_{t}$,
$\mathcal{B}^{\prime}\mathbf{x}_{t}$, that are conditionally mean independent
upon the information set$\ \mathcal{F}_{t-1}$, we need to approximate
$\mathcal{F}_{t-1}$ with its finite sample analogue
$\mathcal{F}_{t-1,t-k_{0}}=\sigma\left(\mathbf{x}_{t-1},...\mathbf{x}_{t-k_{0}}\right),$
where $k_{0}$ is a prespecified fixed integer. Lee and Shao (2018) introduced
the concept of cumulative martingale difference divergence matrix to quantify
the conditional mean independence of $\mathbf{x}_{t}$ on its recent past
$\mathcal{F}_{t-1,t-k_{0}}$, defined as

$\mathbf{\Gamma}_{k_{0}}=\sum_{j=1}^{k_{0}}\mathbf{MDDM}\left(\mathbf{x}_{t}|\mathbf{x}_{t-j}\right)\text{,}$
(10)

where $\mathbf{\Gamma}_{k_{0}}$ is also a positive semidefinite matrix, being
the summation of positive semidefinite matrices. The linearly independent
combinations of our interest live in the null space of
$\mathbf{\Gamma}_{k_{0}}$ and can therefore be estimated based on its eigen-
decomposition. Let $\left\\{\lambda_{i},\gamma_{i}\right\\}_{i=1}^{n}$ be the
$n$ pairs of eigenvalues and eigenvectors of $\mathbf{\Gamma}_{k_{0}}$, the
matrix $\mathbf{M}=\left[\mathbf{\Lambda},\mathcal{B}\right]^{\prime}$ can be
obtained as

$\mathbf{\Gamma}_{k_{0}}\mathbf{M=MD}\text{ ,}$ (11)

by the spectral analysis of $\mathbf{\Gamma}_{k_{0}}$, so that
$\mathbf{M=}\left[\gamma_{1},...,\gamma_{n}\right]$.

In particular, $\mathbf{\Lambda}$ will be the matrix collecting eigenvectors
corresponding to the $r$ factors that are not martingale difference sequences,
for which
$MDD\left(\Lambda_{i}^{\prime}\mathbf{x}_{t}|\mathbf{x}_{t-j}\right)^{2}\neq
0$, for every eigenvector $i=1,...,r,$ and for every value of $j=1,...,T\ $.
On the contrary, $\mathcal{B}$ will correspond to the remaining $n-r$
eigenvectors for which
$MDD\left(\mathcal{B}_{i}^{\prime}\mathbf{x}_{t}|\mathbf{x}_{t-j}\right)^{2}=0$,
for every eigenvector $i=r+1,...,n,$ and for every value of $j=1,...,T.$ We
can finally introduce the finite sample analogue of
$\mathbf{MDDM\left(.\right)}$. Let us define the lagged variable
$\mathbf{z}_{t}=\mathbf{x}_{t-j}$, Lee and Shao (2018) express the sample
martingale difference divergence matrix as

$\mathbf{MDDM}_{T}\left(\mathbf{x}_{t}|\mathbf{x}_{t-j}\right)=-\frac{1}{\left(T-j\right)^{2}}\sum_{h,l=1}^{T-j}\left(\mathbf{x}_{h}-\overline{\mathbf{x}}\right)\left(\mathbf{x}_{l}-\overline{\mathbf{x}}\right)^{{}^{\prime}}\left|\mathbf{z}_{h}-\mathbf{z}_{l}\right|_{n}\text{.}$
(12)

The sample estimate of the cumulative martingale difference divergence matrix
is simply characterized as
$\mathbf{\widehat{\Gamma}}_{k_{0}}=\sum_{j=1}^{k_{0}}\mathbf{MDDM}_{T}\left(\mathbf{x}_{t}|\mathbf{x}_{t-j}\right).$
This reduces our analysis to that of the matrix $\widehat{\mathbf{M}}$
$=\left[\widehat{\mathbf{\Lambda}},\mathcal{\widehat{\mathcal{B}}}\right]^{\prime}$
of the estimated eigenvectors of $\mathbf{\widehat{\Gamma}}_{k_{0}}$.

###  2.2 Determining the number of factors

An important issue in factor modelling is the determination of the true
dimension of the factor space. The methodology adopted in Lee and Shao’s
original paper is based upon the eigenvalue ratio estimator defined in Lam,
Yao, & Bathia (2011) defined as follows.

Let $\left\\{\widehat{\lambda}_{i},\widehat{\gamma}_{i}\right\\}_{i=1}^{n}$ be
the $n$ pairs of eigenvalues and eigenvectors obtained by the spectral
analysis of $\mathbf{\widehat{\Gamma}}_{k_{0}}$ rearranged in accordance with
the sequence of the eigenvalues put in decreasing order. We can estimate $r$
(i.e. the rank of $\mathbf{\Gamma}_{k_{0}}$) according to the ratio based
estimator defined as

$\widehat{r}=\left(\arg\min_{1\leq i\leq
R}\frac{\widehat{\lambda}_{i+1}}{\widehat{\lambda}_{i}}\right)\text{,}$ (13)

In the present paper we propose a novel approach for estimating the number of
factors in FMMDE. To do this we exploit the property the model that allows to
achieve a separation of the factors according to whether they are
conditionally mean dependent upon past information or not. In this sense, it
is possible to perform tests for the martingale difference hypothesis on the
individual factors in order to identify the subset of those that are not
martingale difference sequences. To this purpose, we employ the test statistic
for the martingale hypothesis introduced by Wang & Shao (2021), which is also
built around the definition of martingale difference divergence matrix and
takes the form

$\widehat{T}_{wn}^{F}\left(M\right)=T\sum_{j=1}^{M}\omega_{j}\left|\left|\mathbf{MDDM}_{T}\left(F_{it}|F_{it-j}\right)\right|\right|_{F}\text{,}$
(14)

where $\omega_{j}=\left(N-j+1\right)/\left(Nj^{2}\right)$ and
$\left|\left|\cdot\right|\right|_{F}$ indicates the Frobenius norm. Among all
the different test specifications proposed by Wang & Shao (2021) we choose
$\widehat{T}_{wn}^{F}$ as our statistic of reference, as it represents the
best compromise in terms of size and power properties. In particular, using
$\widehat{T}_{wn}^{F}\left(M\right)$, we test the dynamic properties of the
single factors in a sequential scheme starting from the factor corresponding
to the largest eigenvalue (i.e. we perform a sequence univariate dependence
tests on the single factors in the order dictated by the decreasing
eigenvalues). The testing sequence stops as soon as the first factor for which
the null hypothesis of conditional mean independence upon the past is not
rejected is identified: this factor will be associated to the
$\left(r+1\right)$-th eigenvector in the matrix $\widehat{\mathbf{M}}$, i.e,
the first eigenvector in the matrix $\widehat{\mathcal{B}}$.

In the rest of the analysis this will be our methodology of choice for
determining the number of factors555In our experiments we found that the
martingale testing procedure gives better results than the eigenvalue ratio
methodology when a recursive scheme of forecasting is considered. In our
estimates we shall assume the fixed truncation lag to be $M=6$ in
$\widehat{T}_{wn}^{F}\left(M\right)$. As the asymptotic null distribution of
$M_{wn}^{F}$ is non-pivotal, the critical values of the test statistics need
to be approximated by bootstrap methodology. To this purpose a wild bootstrap
procedure (Wu, 1986; Liu, 1988) is adopted. Specifically, the bootstrap sample
$\left\\{X_{t}^{\ast}\right\\}_{t=1}^{n}$ is generated according to the
formula $X_{t}^{\ast}=X_{t}w_{t}^{\ast}$ with $\left\\{w_{t}^{\ast}\right\\}$
being a sequence of iid random variables with mean zero, unit variance, and
bounded support. For instance, Wang et al. (2021) choose
$\left\\{w_{t}^{\ast}\right\\}$ as iid random variables with the distribution

$\displaystyle
P\left(w_{t}^{\ast}=\frac{1-\sqrt{5}}{2}\right)=\frac{\sqrt{5}+1}{2\sqrt{5}},\text{
\ \ \ \ \
}P\left(w_{t}^{\ast}=\frac{1+\sqrt{5}}{2}\right)=\frac{\sqrt{5}-1}{2\sqrt{5}}$
(15)

as defined in Mammen (1993). Throughout our testing procedure empirical
p-values are computed over 499 bootstrap replications for a significance level
of 0.05.

##  3 Stock and Watson’s Original Approach

The results obtained for FMMDE are compared with those of the factor model
defined in Stock & Watson (2002a, 2002b). This model is introduced by the
authors in the form of a dynamic factor model; specific assumptions, however,
allow to express its dynamic factors in the static form of Equation (1), so
that its estimation can in practice be performed through standard principal
component analysis. Their methodology follows the approach described in Bai
and Ng (2002) and Bai (2003). Their theory is developed within the framework
of large cross sections and large time dimension, also allowing for serial and
cross-section dependence for the idiosyncratic errors. The main difference
between this class of models and the FMMDE is that in the case of the latter
all factors are required to have some form of dependence. This is not
necessarily the case in Stock and Watson, where a factor can also be a white
noise process and, vice versa, the idiosyncratic components can display some
serial correlation.

In particular, letting
$\mathbf{S}=T^{-1}\sum_{t}\mathbf{x}_{t}\mathbf{x}_{t}^{\prime}$ and denoting
the spectral decomposition of the covariance matrix by

$\mathbf{S=VD}^{2}\mathbf{V}^{\prime}\text{,}$ (16)

where $\mathbf{V}=(v_{1},...,v_{n})$ is the $(n\times n)$ matrix of
orthonormal eigenvectors, $\mathbf{V^{\prime}V=I_{N}}$, we obtain an estimate
of the $r$ common factors as

$\widehat{\mathbf{F}}_{t}=\mathbf{D}_{r}^{-1}\mathbf{V}_{r}^{\prime}\mathbf{x}_{t}\text{,}$
(17)

with $\mathbf{D}$ being the matrix matrix containing the square root of the
ordered eigenvalues. Selection of the proper number of common factors is
performed according to one of the information criteria proposed by Bai and Ng
(2002).

###  3.1 Simulated examples

Our goal in this section is to assess the advantages given by the FMMDE in a
nonlinear setting in terms of forecast accuracy: for this reason we report
some results based on the simulated examples defined in Lee & Shao’s paper
(Examples 5 and 6 specifically). In this way we offer a further confirmation
of the qualities of the model with respect to the analysis contained the
original paper, where the adoption of simulated examples was limited to the
validation of factor selection procedure. Assuming that we know a priori the
exact number of factors in the latent structure generating the data (so that
we are not concerned with determining the true factor space dimension) we
study the performance of the two models, FMMDE and SW model, in terms of one
step ahead forecast error.

The first simulated model we consider, introduced in Example 5, is
characterized by a linear data generating process for the three components of
the latent factor structure. In particular,
$\mathbf{F}_{t}=\left(F_{1t},F_{2t},F_{3t}\right)^{\prime}$ is generated by
the following model,

$\displaystyle F_{1t}=w_{t},F_{2t}=w_{t-1},F_{3t}=w_{t-2},$ $\displaystyle
w_{t}=0.2z_{t-1}+z_{t},z_{t}\sim N\left(0,1\right).$ (18)

For the factor loading matrix $\mathbf{\Lambda}$, first $n/2$ elements of each
column are iid $U\left(-2,2\right)$ and are kept fixed once generated and the
other elements are set to be 0. The data $\mathbf{x}_{t}$ are generated as
$\mathbf{x}_{t}=\mathbf{\Lambda F}_{t}+\boldsymbol{\zeta}\mathbf{{}_{t}}$
where $\boldsymbol{\zeta}\mathbf{{}_{t}}$ is a random sample of
$N\left(0,\mathbf{\Sigma}\right)$ and is independent of $\mathbf{F}_{t}$,
where $\mathbf{\Sigma=}\left(\sigma_{i,j}\right)_{i,j=1}^{n}$ is defined as

$\sigma_{i,j}=\left\\{\begin{array}[c]{cc}\frac{1}{2}[\left(|i-j|+1\right)^{2H}-\left|i-j\right|^{2H}&\\\
+\left(\left|i-j\right|-1\right)^{2H}],\text{ }&\text{if }i\neq j\\\ 1&\text{\
if }i=j\end{array}\right.$ (19)

for Hurst exponent $H=0.9$. The above assumptions about the dependence
structure of error term makes possible to address the large $n$ case, where
$n$ can exceed the number of available observations $T$.

Forecast results are summarized in Table 1, which reports the ratio of the
(one step ahead) squared forecast error computed for the two models averaged
over 1000 simulated samples. These are obtained for each element (the $j$-th
element in this case) in vector $\mathbf{x_{t}}$ using a simple forecasting
equation of the form

$\widehat{x}^{m}_{j,T}=\sum_{i=1}^{3}\widehat{\beta}_{i}\widehat{F}^{m}_{i,T-1},$
(20)

where the correct number of underlying factors (assumed to be known a priori)
is used. In particular, the superscript $m$ stands for the for model for which
the factor component $\widehat{\mathbf{F}}_{t}$ is estimated, so that it is
ranging over SW, FMMDE. For each simulated sample we use the first $(T-1)$
observations to obtain an estimate of the underlying factor structure
$\mathbf{F}_{t}$, which is then used to obtain the one step ahead forecasts.
Forecasted values are then compared with the true value ${x}_{j,T}$, which is
a univariate component of the vector $\mathbf{x_{t}}=(x_{1T},\dots,x_{nT})$.
Specifically, for each model $m$ forecast errors obtained for the individual
series are then averaged as follows

$FE_{R}^{m}=\frac{\sum_{j}^{n}\left(\widehat{x}_{j,T}^{m}-x_{j,T}\right)^{2}}{n}$

where the subscript $R$ indicates that we are considering $R$-th simulated
sample. Finally, the values reported in the Table (1) are obtained as the
ratio

$AFE=\frac{\sum_{R=1}^{1000}FE_{R}^{MDDM}}{\sum_{R=1}^{1000}FE_{R}^{SW}}$

As expected, the two models offer an almost identical performance when the
underlying data generating process is assumed to be linear (left column).
Varying the process dimension $n$ or sample size $T$ does not have a relevant
influence on the results.

The second model under scrutiny, derived from Example 6 in Lee & Shao,
replaces the data-generating process of $\omega_{t}$ by a nonlinear one
defined as follows

$w_{t}=\left\\{\begin{array}[c]{cc}0.5+(0.05e^{-0.01w_{t-1}^{2}}&\\\
+0.9)w_{t-1}+z_{t},&\text{if }w_{t-1}<5\text{ , }z_{t}\sim
N\left(0,1\right)\\\ \left(0.9e^{-10w_{t-1}^{2}}\right)w_{t-1}+z_{t}&\text{\
if }w_{t-1}\geqslant 5\end{array}\right.$ (21)

The data $\mathbf{x}_{t}$ is again defined as $\mathbf{x}_{t}=\mathbf{\Lambda
F}_{t}+\boldsymbol{\zeta}$ where the error term
$\boldsymbol{\zeta}\mathbf{{}_{t}}$ is generated from
$N\left(0,0.25\mathbf{\Sigma}\right)$ and $\mathbf{\Sigma}$ is defined as in
the previous example.

Table 1: Ratio of the average one step ahead forecast errors (AFE) computed
for the two models over 1000 simulations, FMMDE (Numerator), SW model
(Denominator)

|  |  | Linear Factors |  |  | Nonlinear Factors
---|---|---|---|---|---|---
$T$ |  |  | 200 | 500 | 1000 |  |  | 200 | 500 | 1000
$n=100$ |  |  | 1.0048 | 1.0045 | 1.0030 |  |  | 1.0103 | 0.9708 | 0.9402
$n=300$ |  |  | 1.0046 | 1.0020 | 1.0027 |  |  | 0.9743 | 0.9598 | 0.9577
$n=500$ |  |  | 1.0025 | 1.0026 | 1.0019 |  |  | 0.9653 | 0.9172 | 0.9021

Results reported in Table 1 (right column) make apparent the advantages of
adopting FMMDE when the underlying factor structure contains significant
nonlinearities. Such advantage increases for increasing $n$ and $T$.

##  4 Data description and forecasting procedure

In our experiment we shall consider an $n$ dimensional covariance stationary
process $\mathbf{x_{t}}=(x_{1t},\dots,x_{nt})$ forming a large dataset of
economic indicators. Let $y_{t}$ be the variable to be predict, part of the
same set of variables: we are interested in forecasting $y_{t}$ at horizon $h$
by using all the information made available by $\mathbf{x_{t}}$. The dataset
consists of monthly observations on 123 U.S. macroeconomic time series
observed from January 1959 to December 2019. The series are grouped into 8
categories that are listed in the Appendix. As reported in McCracken & NG
(2015), the series are all transformed to be stationary by taking first or
second differences, logarithms or first or second differences of logarithms.
Let $y_{t+h}$ denote the variables to be predicted in a $h$-period ahead
forecast. When predicting real activity variables, $y_{t+h}$ is defined as the
$h$-period growth at an annual rate. As an example, let $IP_{t}$ the level of
Industrial Production index we obtain

$y_{t+h}=\frac{ln\left(IP_{t+h}/IP_{t}\right)}{h}.$

When predicting the nominal price and wage series, $y_{t+h}$ is defined as the
$h$-period growth of monthly changes. As an example, let $CPI_{t}$ be the
level of Consumer Price Index, we obtain

$y_{t+h}=\frac{ln\left(CPI_{t+h}/CPI_{t}\right)}{h}-\frac{ln\left(CPI_{t}/CPI_{t-1}\right)}{h}.$

The forecasts are obtained through a pseudo real-time forecasting procedure,
666As reported in D’agostino & Giannone (2012), the exercise is pseudo because
we use the last vintage of data and we do not take into account the real time
revision; it is in real-time because we use a moving estimation window. In
this exercise we use the datset. At the time we started the exercise, the
dataset used referred to the vintage dated March 2021. We decided to limit our
analysis to December 2019 because we intended to measure the predictive
capabilities of MDDM without interfering with COVID-SARS19 pandemic effects.
adopting a recursive scheme of estimation, so that the series used in the
estimation are characterized by a progressively increasing number of
observations, with both FMMDE and SW model being re-estimated for each
iteration of the exercise. Specifically, the size of the available sample is
expanded over time, starting from an original dimension of 120 monthly
observations, adding one observation at a time. In the end the exercise allows
us to obtain 589 subsequent estimates, for a total of 712 observations being
used in the last estimate. For each iteration of the exercise, a PCA is
conducted to extract the latent components of the SW model. Eigen-
decomposition of the matrix $\mathbf{\Gamma}_{k_{0}}$ is analogously performed
in order to obtain an estimate of the FMMDE factors. The selection of the
correct number of factors is performed in accordance with the two procedures
previously discussed, Bai & NG’s $IC_{p2}$ criterion in the case of SW model
and martingale testing procedure for FMMDE, as discussed in Section 2.

Based on the estimated factors, we implement the following forecasting
equation

$\widehat{y}_{t+h}^{m}=\widehat{\alpha}_{h}+\sum_{i=1}^{r}\widehat{\beta}_{hi}\widehat{F}_{it}^{m}+\sum_{j=1}^{p}\widehat{\gamma}_{hj}y_{t-j+1},$
(22)

where $m$ stands for the for model for which the factor component
$\widehat{\mathbf{F}}_{t}$ is estimated, so that it is ranging over SW, FMMDE.
In particular, the order p of the autoregressive component is estimated by
Bayesian information criterion (BIC), with $0<p\leqslant 6$. Following Stock &
Watson (2002a, 2002b), forecasting results obtained for the two factor models
are also compared to those obtained using a benchmark univariate
autoregression based on the equation

$\widehat{y}_{t+h}^{AR}=\widehat{\alpha}_{h}+\sum_{j=1}^{p}\widehat{\gamma}_{hj}y_{t-j+1},$

where the lag order $p$ is estimated by Bayesian information criterion (BIC),
with $0<p\leqslant 6$.

The estimated regression coefficients are first used at December 1969 to
obtain $\widehat{y}_{t+h}^{m}$ for $h=1,3,6,12$ and $24$ months ahead, where
all the forecasts are obtained for each horizon directly, not iterating one-
step ahead forecasts. In the subsequent iteration of the recursive exercise
the estimation sample is updated by one monthly observation, so that the
second set of observations ranges from the start of the sample to January
1970. The process continues until the end of the sample is reached: in our
case the last data considered are available at December 2019 when $h=24$. The
entire experiment delivers a total of 589 forecasts for each model.

The predictive performance of the models is evaluated in terms of the mean
square forecast error ($MSFE_{m}^{h}$), defined for model $m\in$ (AR, FMMDE,
SW) as

$MSFE_{h}^{m}=\frac{1}{T_{1}-T_{0}}\sum_{\tau=T_{0}}^{T_{1}}\left(\widehat{y}_{\tau+h}^{m}-y_{\tau+h}^{m}\right)^{2},$

where $T_{0}$ is the first point in time for out of sample evaluation and
$T_{1}$ is the last point in time for which we compute MSFE at horizon $h$.
Then, two different model specifications can be compared, at horizon
$h=1,3,6,12,24$, in terms of the ratio

$RMSFE_{h}^{m_{1}/m_{2}}=\frac{MSFE_{h}^{m_{1}}}{MSFE_{h}^{m_{2}}}.$ (23)

where RMSFE is root mean square forecast error.

###  4.1 Estimation of $k_{0}$ through cross validation

Since no explicit rule for the determination of the parameter is provided in
the original paper, we resort to a cross validation procedure, adapted for the
case of time series forecasting, for determining the optimal value of $k_{0}$.
This allows us to take into account the impact that the choice of $k_{0}$
appears to have on the FMMDE forecasting performance. The procedure is
extensively detailed in the enclosed appendix.

##  5 Empirical Results

In this section we investigate the ability of FMMDE and SW model in predicting
key macroeconomic aggregates, such as Real Personal Income (RPI), Industrial
Production index (IP), Unemployment Rate (UR), Nonfarm Employment (PAYEMS),
headline Consumer Price index inflation (CPI) and Core CPI inflation (CoCPI).
The same exercise will be extended to all series that compose the dataset,
reporting the results in an aggregate fashion to better appreciate the
proportion of the series that are most likely to be predicted by the two
methods.

Table 2: RMSE for selected series, estimated by Recursive Forecast over the
period 1976-2019

| MDDM vs AR |  | SW vs AR |  | MDDM vs SW
---|---|---|---|---|---
| $h=1$ | $h=3$ | $h=6$ | $h=12$ | $h=24$ |  | $h=1$ | $h=3$ | $h=6$ | $h=12$ | $h=24$ |  | $h=1$ | $h=3$ | $h=6$ | $h=12$ | $h=24$
RPI | 0.981 | 0.866 | 0.915 | 0.910 | 0.984 |  | 0.984 | 0.889 | 0.965 | 0.940 | 1.133 |  | 0.997 |  *0.975* |  **0.948** | 0.969 |  ***0.869***
IP | 0.904 | 0.893 | 0.968 | 0.936 | 0.848 |  | 0.912 | 0.891 | 1.001 | 1.001 | 1.068 |  | 0.992 | 1.002 | 0.967 |  *0.936* |  ***0.794***
UR | 0.912 | 0.853 | 0.869 | 0.909 | 0.836 |  | 0.920 | 0.876 | 0.895 | 0.920 | 0.860 |  | 0.991 | 0.974 | 0.971 | 0.988 | 0.972
PAYEMS | 0.937 | 0.833 | 0.881 | 0.827 | 0.745 |  | 0.902 | 0.824 | 0.877 | 0.887 | 0.939 |  | 1.038 | 1.010 | 1.005 |  *0.933* |  **0.794**
CPI | 1.250 | 1.304 | 1.324 | 1.123 | 1.074 |  | 1.208 | 1.253 | 1.267 | 1.066 | 1.003 |  | 1.035 | 1.041 | 1.046 | 1.054 | 1.071
cCPI | 1.139 | 1.007 | 0.952 | 0.864 | 0.890 |  | 1.085 | 0.974 | 0.925 | 0.837 | 0.826 |  | 1.049 | 1.033 | 1.030 | 1.032 | 1.077

NOTE: Entries are values of RMSEs computed for the single series, at the 1-,
3-,6- and 12-, 24-month ahead forecast horizon. All forecasts are direct.
RMSEs are calculated using recursive pseudo out-of-sample forecasts over the
period 1976-2019. In the third column, (*), (**), (***) respectively indicate
the significance level (0.1), (0.05), (0.01) of the DM test for which the
hypothesis $MSFE_{MDDM}<MSFE_{SW}$ can be accepted.

Table 2 presents results obtained for the individual in terms of root mean
square forecast error (RMSFE). We can immediately notice that, as far real
activity variables are concerned, both FMMDE and SW model display an higher
forecast accuracy than the simple autoregressive model. It is however
interesting to notice how FMMDE offers an improvement over the SW model for
three key macroeconomic aggregates such as IP, RPI and UR for all of the
considered horizons. This is not the case as we look at results obtained for
PAYEMS, as SW model has a slight advantage for shorter horizons ($h=1,3,6$).
These results are counterbalanced, however, by the accuracy shown by FMMDE
over longer horizons. In particular, An additional empirical evidence emerges
when comparing the results for long forecast horizons. In this case, as
confirmed by the Diebold Mariano test, for $h=24$ the gain is sizeable. This
is true for the real activity variables with the exception of UR.

Additional results are presented in Table 3 for all series combined. In Table
3 the distribution of the RMSE of the models is calculated. The table in
particular shows percentiles of the distributions of $h$-month ahead out-of-
sample RMSEs over the 123 series for the seven forecasting methods. We can
notice that both FMMDE and SW model improve at every horizon upon AR for more
than half of the series. Moreover, FMMDE model is able to outperform SW for
more than half of the series at almost all horizons (the only exception being
$h=1$).

Table 3: Distribution of RMSE estimated by Recursive Forecast over the period
1976-2019

| MDDM vs AR |  | SW vs AR |  | MDDM vs SW
---|---|---|---|---|---
| 10th | 25th | 50th | 75th | 90th |  | 10th | 25th | 50th | 75th | 90th |  | 10th | 25th | 50th | 75th | 90th
$h=1$ | 0.918 | 0.957 | 0.996 | 1.033 | 1.206 |  | 0.915 | 0.952 | 0.988 | 1.038 | 1.171 |  | 0.967 | 0.987 | 1.001 | 1.023 | 1.048
$h=3$ | 0.849 | 0.895 | 0.969 | 1.033 | 1.119 |  | 0.860 | 0.907 | 0.975 | 1.038 | 1.108 |  | 0.952 | 0.975 | 0.997 | 1.017 | 1.037
$h=6$ | 0.842 | 0.908 | 0.974 | 1.038 | 1.145 |  | 0.853 | 0.925 | 0.998 | 1.058 | 1.126 |  | 0.925 | 0.958 | 0.985 | 1.015 | 1.040
$h=12$ | 0.784 | 0.861 | 0.962 | 1.043 | 1.125 |  | 0.789 | 0.887 | 0.979 | 1.070 | 1.143 |  | 0.876 | 0.939 | 0.995 | 1.030 | 1.056
$h=24$ | 0.744 | 0.791 | 0.966 | 1.056 | 1.235 |  | 0.755 | 0.871 | 0.994 | 1.091 | 1.205 |  | 0.775 | 0.886 | 0.997 | 1.044 | 1.086

NOTE: Entries are percentiles of distributions RMSEs over the 123 variables
being forecasted, by series, at the 1-, 3-,6- and 12-, 24-month ahead forecast
horizon. All forecasts are direct. RMSEs are calculated using recursive pseudo
out-of-sample forecasts over the period 1976-2019. In the third column, grey
areas indicate in which percentiles MDDM is performing better than SW.

###  5.1 Forecast Results for groups of series

We split the variables in the dataset in eight groups: Output & Income (19
series in total), Labor Market (31 series), Housing Market (10 series),
Consumption, Business Inventories & Orders, Money & Credit (12 series),
Interest & Exchange Rates (21 series), Prices (16 series) and Stock Market (4
series). Table 4 reports the distribution of $RMSFE_{h}^{FMMDE/SW}$ computed
for different groups of variables. At group level we have confirmation of what
we have so far observed for individual series: FMMDE appears to have an
advantage over SW model in the prediction of real economic variables (roughly
represented by the first four groups in the dataset) but this is partly
balanced by the higher accuracy displayed by SW model in forecasting nominal
variables (roughly represented by the last four groups in the dataset).
Considering Output & Income, for instance, we can notice that FMMDE delivers
more accurate forecasts than SW model for more than half of the series in the
group, at all considered horizons. The relative performance of the model
improves as we take into account longer horizons ($h=12,24$), to the point
that FMMDE has an advantage over SW model for more than 75% of the series in
the group. Similar comments can be made about the model’s performance with
respect to the two categories of Labor market and Consumption, Business
Inventories & Orders.

Table 4: Distribution of RMSFEFMMDE/SW estimated by Recursive Forecast over
the period 1976-2019 for groups of variables

| $h=1$ |  | $h=12$ |  | $h=24$
---|---|---|---|---|---
| 10th | 25th | 50th | 75th | 90th |  | 10th | 25th | 50th | 75th | 90th |  | 10th | 25th | 50th | 75th | 90th
OUTPUT INCOME | 0,987 | 0,992 | 0,998 | 1,010 | 1,012 |  | 0,815 | 0,853 | 0,923 | 0,982 | 1,031 |  | 0,684 | 0,738 | 0,784 | 0,875 | 0,926
LABOR MARKET | 0,965 | 0,974 | 1,002 | 1,010 | 1,025 |  | 0,837 | 0,908 | 0,957 | 1,002 | 1,036 |  | 0,769 | 0,846 | 0,920 | 0,972 | 1,008
HOUSING | 0,957 | 0,977 | 0,987 | 0,993 | 1,010 |  | 0,931 | 0,941 | 0,985 | 1,013 | 1,026 |  | 0,942 | 0,956 | 1,000 | 1,020 | 1,100
CONS., INV. ORD | 0,951 | 0,967 | 0,992 | 1,000 | 1,001 |  | 0,886 | 0,909 | 0,954 | 0,981 | 1,032 |  | 0,779 | 0,818 | 0,907 | 0,981 | 1,039
MONEY CREDIT | 0,908 | 0,955 | 0,990 | 1,010 | 1,039 |  | 0,924 | 0,991 | 1,020 | 1,057 | 1,110 |  | 0,971 | 1,000 | 1,017 | 1,055 | 1,148
INT. EX. RATES | 0,976 | 1,002 | 1,026 | 1,052 | 1,068 |  | 0,967 | 0,999 | 1,025 | 1,039 | 1,055 |  | 0,944 | 1,017 | 1,045 | 1,073 | 1,098
PRICES | 0,982 | 1,000 | 1,012 | 1,042 | 1,053 |  | 0,975 | 0,995 | 1,028 | 1,054 | 1,068 |  | 0,988 | 1,003 | 1,020 | 1,063 | 1,080
STOCK MARKET | 1,000 | 1,014 | 1,033 | 1,037 | 1,038 |  | 1,000 | 1,021 | 1,047 | 1,060 | 1,067 |  | 1,047 | 1,089 | 1,131 | 1,137 | 1,144

NOTE: Entries are percentiles of distributions of RMSEs over the variables in
a single group, being forecasted, by series, at the 1-, 3-,6- and 12-,
24-month ahead forecast horizon. All forecasts are direct. RMSEs are
calculated using recursive pseudo out-of-sample forecasts over the period
1976-2019. Grey areas indicate in which percentiles MDDM is performing better
than SW.

Where FMMDE appears to lose ground with respect to SW model is in the
prediction of the last four groups: Money & Credit, Interest & Exchange Rates,
Prices and Stock Market. In particular, SW model is more accurate in its
forecasts for more than half of the series in these groups, for all the
reported horizons. Where FMMDE appears to lose ground with respect to SW model
is in the prediction of the last four groups: Money & Credit, Interest &
Exchange Rates, Prices and Stock Market. In particular, SW model is more
accurate in its forecasts for more than half of the series in these groups,
for all the reported horizons.

Table 5: Distribution of RMSFEFMMDE/AR estimated by Recursive Forecast over
the period 1976-2019 for groups of variables

| $h=1$ |  | $h=12$ |  | $h=24$
---|---|---|---|---|---
| 10th | 25th | 50th | 75th | 90th |  | 10th | 25th | 50th | 75th | 90th |  | 10th | 25th | 50th | 75th | 90th
OUTPUT INCOME | 0.886 | 0,906 | 0.930 | 0.978 | 0.997 |  | 0.869 | 0.906 | 0.911 | 0.950 | 0.999 |  | 0.784 | 0.809 | 0.855 | 0.978 | 1.138
LABOR MARKET | 0.894 | 0.929 | 0.965 | 0.991 | 1,022 |  | 0,703 | 0.807 | 0.847 | 0.912 | 1.038 |  | 0.644 | 0.737 | 0.768 | 0.851 | 0.980
HOUSING | 0.897 | 0,929 | 0.997 | 1.010 | 1.044 |  | 0.985 | 0.998 | 1.027 | 1.115 | 1.150 |  | 1.018 | 1.041 | 1.123 | 1.253 | 1.346
CONS., INV. ORD | 0.898 | 0.955 | 0.991 | 1.013 | 1.029 |  | 0,791 | 0,853 | 0,947 | 1,061 | 1,137 |  | 0,771 | 0.816 | 0.864 | 1.080 | 1.213
MONEY CREDIT | 0.989 | 1.000 | 1.026 | 1.058 | 1.125 |  | 0.944 | 0.984 | 1.011 | 1.036 | 1.121 |  | 0.953 | 0.983 | 1.012 | 1.058 | 1.183
INT. EX. RATES | 0.968 | 0.984 | 1.015 | 1.240 | 1.286 |  | 0.731 | 0.921 | 0.986 | 1.112 | 1.168 |  | 0.777 | 0.961 | 1.025 | 1.065 | 1.191
PRICES | 0.998 | 1.012 | 1,050 | 1,181 | 1,238 |  | 0.753 | 0.869 | 0.980 | 1.033 | 1.089 |  | 0.699 | 0.835 | 0.978 | 1.053 | 1.151
STOCK MARKET | 0.982 | 0.987 | 1.000 | 1.008 | 1.008 |  | 1,091 | 1.158 | 1.227 | 1.239 | 1.250 |  | 1.267 | 1.277 | 1.288 | 1.232 | 1.357

NOTE: Entries are percentiles of distributions of RMSEs over the variables in
a single group, being forecasted, by series, at the 1-, 3-,6- and 12-,
24-month ahead forecast horizon. All forecasts are direct. RMSEs are
calculated using recursive pseudo out-of-sample forecasts over the period
1976-2019. Grey areas indicate in which percentiles MDDM is performing better
than AR model.

## Conclusions

In this paper we have studied the qualities, in terms of predictive ability,
of the factor model with martingale difference error. Considering a large
dataset of macroeconomic and financial series we have compared the forecasting
performance of FMMDE to that more traditional model defined in Stock and
Watson (2002). The results show that in general the two competing methods have
comparable forecasting capabilities. However, there are some variables, such
as Real Personal Income, Industrial , and Unemployment where the FDDE
procedure seems to have an edge, especially for medium to long term horizons,
over the SW procedure. We attribute this result to the possible non-linearity
that may characterise the factor structure, as also highlighted by Monte Carlo
simulations. This is a research topic that certainly deserves to be further
investigated, for instance by attempting to identify the non-linear structure
of each series.

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