HIAS-E-118

Forecasting GDP growth using stock
returns in Japan:
A factor-augmented MIDAS approach

Hiroshi Morita (a),(a)

Hosei University

March 2022

Hitotsubashi Institute for Advanced Study, Hitotsubashi University
2-1, Naka, Kunitachi, Tokyo 186-8601, Japan
tel:+81 42 580 8668 http://hias.hit-u.ac.jp/
HIAS discussion papers can be downloaded without charge from:https://hdl.handle.net/10086/27202
https://ideas.repec.org/s/hit/hiasdp.html

Forecasting GDP growth using stock returns in Japan:
A factor-augmented MIDAS approach∗
Hiroshi Morita†
Hosei University
This version: March 2022

Abstract
Asset prices reﬂect expectations of future economic conditions. In this study, we use
the property of asset prices, especially stock prices, to forecast the GDP growthrate in Japan. For optimal use of the rich time-series and cross-sectional information of stock prices, we combine MIDAS (mixed-data sampling) regression andfactor analysis to examine which dimensions of information contribute to the accuracy of the GDP growth rate forecast. Our results show that the use of factors
signiﬁcantly improves forecast accuracy and that extracting factors from a broader
set of stock prices further improves accuracy. This highlights the important role ofcross-sectional stock market information in forecasting macroeconomic activity.

Keywords: Forecasting; MIDAS regression; factor model; stock returns.
JEL classiﬁcation: C22; C53; E37.

∗

This research is supported by ﬁnancial aid from the Japan Center for Economic Research and Grantin-Aid for Young Scientists (19K13652). Any remaining errors are entirely the author’s responsibility.
†
Address: Hosei University, Faculty of Economics, 4342 Aihara, Machida, Tokyo 194-0298, Japan.
E-mail address: morita-h@hosei.ac.jp1

1

Introduction

Asset prices are a mirror reﬂecting expectations of future economic conditions as they are
determined in a forward-looking manner based on currently available information. Forecasting literature has supported this forward-looking nature of asset prices by demonstrating that the forecast accuracy for macroeconomic ﬂuctuations is improved using theinformation in asset markets (e.g., Stock and Watson, 2003; Forni et al., 2003; Andreou
et al., 2013).1 In asset markets, many asset prices are recorded with high frequency, andthis rich data environment can be used for forecasting macroeconomic activities. This
study explores how such rich information in the asset market contributes to the improvement of macroeconomic forecasts, focusing particularly on the relationship between GDP
growth rate and Japanese stock prices.
More precisely, information in the stock market is spread across both time-series andcross-sectional dimensions. We handle these two-dimensional information by combining
mixed data sampling (MIDAS) regression and factor model approach, following recent
literature of Marcellino and Schumacher (2010) and Andreou et al. (2013). In forecasting
literature, those two methods were originally developed separately. MIDAS regression,
originally introduced by Ghysels et al. (2007), handles data sampled at diﬀerent frequencies in a single equation using a function that optimally transforms higher-frequency
data into lower-frequency data. Intuitively, giving optimal weights attached to eachhigh-frequency observation by “letting the data speak for itself” will help improve forecast ability compared to using the time-aggregated data. Conversely, the factor model
approach for forecasting macroeconomic activity is adopted in Stock and Watson (2002)
and Boivin and Ng (2005) for the single-frequency case. These studies demonstrate thata multivariate model outperforms a univariate model in forecast accuracy. Considering
this strand of literature, this study employs the MIDAS model augmented by factors(hereinafter referred to as the Factor MIDAS model) in forecasting GDP growth ratewith stock prices.
This study addresses the following three main questions: First, do stock prices re1

In another strand of literature, asset prices, particularly for stocks, are used as a proxy for capturing
news shock in the time series macroeconomic analysis. For example, Beaudry and Portier (2006) regard
innovations in stock prices as an anticipated TFP shock, while Fisher and Peters (2010) use stock returnsof military contractors to proxy for capturing anticipated government spending shock.

2

ally contain the relevant information for future GDP growth in Japan? Second, which
dimension of information, time-series or cross-sectional, contributes to a more accurate
forecast of GDP growth rate? Third, will the ongoing COVID-19 pandemic change therole of stock prices as a leading indicator of economic activities? Hence, eleven alternative forecast models, distinguished by their information sets, are compared with thenaı̈ve forecast model in predictive ability. Five of 11 models are MIDAS speciﬁcations
that regress quarterly GDP growth rate on monthly stock returns. Three of these ﬁvemodels are MIDAS models augmented by factors extracted from the set of individual
stock returns. Furthermore, the three Factor MIDAS models are classiﬁed according to
the set of stocks used for extracting factors (components of Nikkei225, components of
TOPIX, and all listed ﬁrms). In contrast, six of the 11 models are single-frequency models using quarterly aggregated stock returns or quarterly factors as regressors. As in the
MIDAS model, three of six models exploit the cross-sectional information augmented by
the factors, and factors are estimated from three sets of stocks. One of the quarterly models is a simple AR model without stock returns that only contains lagged GDP growthrate to assess the role of stock returns. Finally, the naı̈ve forecast of the benchmark
is obtained from the in-sample mean of GDP growth rate. The forecast ability of eachmodel is evaluated by the relative root mean square error (RMSE) to the benchmark
model.
We provide the following answers to the above questions. First, we ﬁnd that stock
returns are informative in predicting the future GDP growth rate before the COVID-19
pandemic. All the models with stock returns outperform those without stock returns andthe naı̈ve forecast model. Second, results highlight the importance of cross-sectional information in the stock market for the forecast of GDP growth rate in Japan. Speciﬁcally,
the forecast accuracy of the models with factors (both MIDAS and quarterly speciﬁcations) always exceeds that of the corresponding models without factors. In contrast, theenrichment of time-series information does not always contribute to more accurate forecasts. Only one MIDAS model shows the improvement of the forecast compared with
its quarterly counterpart. Moreover, the best predicting performance is obtained in thequarterly model with factors extracted from a set of stock returns of all listed ﬁrms.
Forecast accuracy then tends to improve as the set of stock prices increases, that is,3

in the order of Nikkei225, TOPIX, and all listed ﬁrms. 2 These ﬁndings emphasize thecontribution of cross-sectional information to the forecast of GDP growth rate. Third,
as mentioned, the predictive ability of stock returns for GDP growth rate seems to belost after the COVID-19 pandemic. Including the sample period after the ﬁrst quarterof 2020, all models failed to provide forecasts that are signiﬁcantly better than the naı̈veforecast. The possible reasons for this lack of predictive power of stock prices may be
attributed to massive stock market intervention by the Bank of Japan (BOJ)’s ETF (exchange traded funds) purchases and unprecedented macroeconomic ﬂuctuations. Since
the outbreak of the COVID-19 pandemic, particularly in 2020, the BOJ had conductedmassive money injection to the stock market through the large-scale purchases of ETF tocope with a sharp drop in stock prices. Moreover, such interventions from the BOJ may
have distorted the price formation mechanism of stock prices. In contrast, GDP growth
rates themselves have also ﬂuctuated dramatically after the COVID-19 outbreak, making
forecasting diﬃcult. The GDP growth rate has been taking positive and negative values
in turn after the ﬁrst quarter of 2020, reﬂecting the infection situation and intensity of
regulation. These two speciﬁc situations under the COVID-19 pandemic may cause aloss of predictive power in stock returns to GDP growth rate, which had been reliably
observed before the pandemic.
Methodologically, this study adopts the Almon lag polynomial among several functional forms of MIDAS speciﬁcations (Ghysels et al., 2007; Ghysels, 2016). This is because
the MIDAS model with Almon lag polynomial can be estimated by OLS. Additionally,
the factors are estimated by the principal components approach based on the Stock andWatson (2002)’s EM algorithm. This allows us to estimate the factors from the datasetwith missing observations. Forecast ability is almost independent on the MIDAS speciﬁcation and factor estimation methods (e.g., Forni et al., 2003; Boivin and Ng, 2005;
Marcellino and Schumacher, 2010).
Several forecasting studies have emphasized the importance of exploiting the asset
prices, mixed-frequency data, and a large amount of information summarized in factors.
All of which are considered in this study. Stock prices have the information to improve2

As detailed in Section 3, components of Nikkei225, TOPIX, and all listed ﬁrms in this study are 225,
2180, and 5294, respectively.

4the forecast ability on GDP growth rate in Aylward and Glen (2000), Ferrara and Marsilli(2013) and Andreou et al. (2013), while the beneﬁts of the MIDAS model compared withsingle-frequency forecast model has also been reported in Ghysels et al. (2007), Clements
and Galvão (2008), Clements and Galvão (2009), and Foroni et al. (2015). Moreover, thepioneering works of Stock and Watson (2002) and Boivin and Ng (2005) highlighted thatthe multivariate forecasts using the factors outperform the univariate ones and subsequent
works of Marcellino and Schumacher (2010) and Andreou et al. (2013) show that the
combination of MIDAS and factor analysis provides a more accurate forecast. We follow
this strand of literature in the presented study and conduct comparison analysis to clarify
important information in the forecast of GDP growth rate. We believe no studies have
explained the role of cross-sectional information in the entire stock market in forecasting
GDP growth. Stock prices are an informative large dataset that is relatively easy toaccess.3 Hence, our ﬁndings, highlighting the role of heterogeneous information in thestock market, have important practical implications for the forecasting analysis of real
economic activities.4
The remainder of the paper is organized as follows. Section 2 explains the structure of
MIDAS models employed in this study. Section 3 details the dataset used for the analysis,and Section 4 discusses the design of forecasting. Section 5 presents the forecasting results.
Finally, Section 6 concludes the paper.

3

The other possible ﬁnancial variables are the prices of private and government bonds. However,most bonds are traded over-the-counter and not on the market. Hence only reference statistical prices
are available. Furthermore, unlike stock prices, the same company or public institution issues multiple
bonds. Their prices constantly change based on the period to maturity, so it is diﬃcult to use raw dataof bond prices for the analysis.
4
The forecasting analysis for the Japanese economy has relatively less literature compared with theanalyses for the U.S. and the euro area, only conducted by Shintani (2005), Urasawa (2014), Bragoli(2017), and Chikamatsu et al. (2021).

5

2MIDAS regression model

2.1

Basic MIDAS model

The MIDAS model with augmented distributed lag (ADL-MIDAS) for forecasting h-steps
qahead of quarterly variable yt+husing monthly data can be formulated asq= µh +
yt+h

py
X

q
ρj yt−j−1+ βh

px k−1
X
X

j=1

h

Θ
h
ωi+(j−1)∗kxmt−j−1,k−i + ut+h ,(1)

j=1 i=0

where xmt,k−i denotes the monthly variable at the i-th month counting backward from theend of quarter t and k is ﬁxed at 3 in the monthly/quarterly regression because of threeobservations in one quarter, and the quarterly lags for ytq and xm
t are represented bypy and px , respectively. By directly incorporating monthly data into the MIDAS model
without time-aggregation, the information in the time-series dimension can be used more
eﬀectively for the forecast than in a single-frequency model. Particularly, weights of eachhΘ
are speciﬁed as a function of a low dimension vector ofmonthly observation ωi+(j−1)∗kparameters Θh = (θ1h , θ2h , · · · , θPh ). By assuming that P ≪ px × k, the MIDAS model
can avoid a parameter proliferation problem, even if it contains many lagged monthly
variables. Here, we adopt the Almon lag polynomial speciﬁcation for MIDAS weight as
in Pettenuzzo et al. (2016) because of its simplicity.5
Almon lag weight can be provided ashΘ
β h ωi+(j−1)∗k=PXθph (i + (j − 1) ∗ k)p−1 .

(2)

p=1

Then, inserting equation (2) into equation (1) leads to the following formulation for the5

There are other types of MIDAS weights: exponential Almon polynomial, beta polynomial, step
function, and unrestricted (e.g., Appendix of Ghysels, 2016). Except for unrestricted MIDAS (UMIDAS),a nonlinear estimation technique is required to estimate the model. Besides this practical diﬃculty, no
substantial diﬀerence in forecasting ability can be found among the MIDAS speciﬁcations (e.g., Marcellino
and Schumacher, 2010; Foroni et al., 2015). Hence, we adopt the Almon lag polynomial by which theestimation can be conducted by OLS.

6

parameters Θh = (θ1h , θ1h , · · · , θPh ):qyt+h= µh +pyXρj yt−j−1 +px k−1 PXXXhθph (i + (j − 1) ∗ k)p−1 xmt−j−1,k−i + ut+h .

(3)

j=1 i=0 p=1

j=1

We deﬁne (P × N ) matrix Q and the (N × 1) vector of the lagged monthly data Xtm ,respectively, as follows:


1 1

1

···

1

2

···

1
..
.

22
..
.

···
..
.



0


Q = 0

 ..
.

0

1 2P −1 · · ·


xm
t,k



 m

 xt,k−1 



..





N 
.






m
,
N 2  and Xt =  xm
t,1



 m

.. 



.
x
 t−1,k 



..


N P −1.
mxt−(px −1),1


1

(4)

where N (≡ px × k) is the total number of weights. Using equation (4), the model (3) canbe simply rewritten as follows:qyt+h= µh +pyXq+ Θh X̃tm + uht+h ,ρj yt−j−1

(5)

j=1

where X̃tm = QXtm is a (P × 1) vector of transformed monthly regressors. As emphasizedin Ghysels (2016) and Pettenuzzo et al. (2016), parameters in equation (5) can be estimated by OLS because the Almon lag polynomial transforms the MIDAS speciﬁcation
into a linear regression model.
We apply this MIDAS model to forecast quarterly GDP growth using monthly stock
returns in Japan. Speciﬁcally, we estimate equation (5) using the market returns of the
Nikkei225 average and TOPIX (Tokyo stock price index), two of Japan’s leading stock
indices, as monthly stock returns. Additionally, we estimate the model including only
the quarterly average of these indices, as speciﬁed by the following:q
yt+h= µh +pyXqρj yt−j−1
+

px
X
j=1

j=1

7

ψj xqt−j−1 + uht+h ,(6)where xqt is a quarterly stock returns aggregated in advance, thus not making full useof the time series information in equation (6). The comparison of the forecast accuracy
between equations (5) and (6) highlights the importance of the high-frequency data in
the forecasting of GDP growth rate.

2.2

Factor-augmented MIDAS model

As the market returns comprise the (weighted) average of stock prices of speciﬁc ﬁrms,
information used in the basic MIDAS model is likely to be only a small part of theoriginal information in the stock market. This is because thousands of individual stock
price movements can be found behind market indices. Thus, this study further augments
the MIDAS model with factors extracted from a set of individual stock returns (i.e., theFactor MIDAS model), aiming to fully exploit the large number of individual movementsof stock prices. By combing the MIDAS model with factors, we use the information
not only in the time-series dimension but also the cross-sectional dimension and explore
whether additional cross-sectional information can contribute to improving the forecast
accuracy.
Denoting the r-th monthly factor at the i-th month counting backward from the endm, the Factor MIDAS model can be formulated as follows:of quarter t by fr,t−j,k−iqyt+h= µh +pyXqρj yt−j−1
+

j=1

R
X
r=1

βrh

px k−1XX

Θhmr
ωi+(j−1)∗kfr,t−j−1,k−i+ uht+h .

(7)

j=1 i=0

where the number of the factors is denoted by R, and the parameters βrh and Θhr are
assumed to be diﬀerent across the factors, allowing the diﬀerent weights on the factors
measured at monthly basis. Parallel to the basic model, equation (7) can be transformed
using the Almon lag weight as follows:qyt+h= µh +pyXqρj yt−j−1+

RXm
Θhr F̃r,t+ uht+h ,

(8)

r=1

j=1

h
h
h
m
m
m
m
m
m
where Θhr = (θr,1, θr,1, · · · , θr,P
) and F̃r,t= [fr,t,k, fr,t,k−1, · · · , fr,t,1, fr,t−1,k, · · · , fr,t−(p]′ .
x −1),1

Importantly, equation (8) can be also estimated by OLS as in the basic MIDAS model,

8

if the factors are given.
This study uses a two-step procedure in estimating the Factor MIDAS model: First,
factors are extracted from a set of individual stock returns by principal component.
These are then used to estimate the Factor MIDAS model.6 While the second step ofthe estimation is conducted via simple OLS as noted, the ﬁrst step of extracting factors
is somewhat more complicated as the dataset of individual stock returns contains themissing values owing to lack of trade or entry and exit from the stock market. To
handle the missing value problem in factor estimation, this study employs EM algorithm
proposed by Stock and Watson (2002), where factors are estimated by iterating the
replacement of missing values by ﬁtted values and estimation of factors and loadings
using principal component eigenvalue calculation. Appendix A presents the detailed
estimation procedure of the factors.
For comparison purposes, this study further considers the model with the quarterly
factors extracted from a set of quarterly aggregated individual stock returns, such as thefollowing:q
yt+h= µh +pyXqρj yt−j−1
+

px
R X
X

q
ψr,j fr,t−j−1+ uht+h ,(9)

r=1 j=1

j=1qwhere fr,tis the r-th factor extracted from quarterly aggregated individual stock returns.

This speciﬁcation exploits cross-sectional information, but not the time-series information
suﬃciently in terms of using quarterly stock returns. Therefore, the comparison among
the forecasts derived from equation (5), (6), (8), and (9) reveals the extent to which
time-series and cross-sectional information in the stock market contribute to the forecast
GDP growth rate.

3

Data

As noted, this study regresses the quarterly GDP growth rate in Japan on either monthly
market stock returns or factors extracted from a set of individual stock returns and theirlagged variables and a constant. The quarterly sample period is from 1994Q2 to 2021Q3.
6

The two-step procedure of factor forecasting is commonly used in the literature, such as Stock andWatson (2002) and Boivin and Ng (2005). Moreover, Marcellino and Schumacher (2010) report that theforecast results are independent of the factor estimation technique.

9Moreover, its monthly counterpart is from 1994M04 to 2021M09. Quarterly GDP data isderived from the System of National Accounts, published by the Cabinet Oﬃce, Government of Japan. GDP data in accordance with 2008 SNA is released only from 1994Q1;
hence, the sample period starts from 1994Q2 owing to taking its log diﬀerentials. The
time series of GDP used in this paper is the published data of Quarterly Estimates ofGDP (The 2nd preliminary), which is available in January 2022. All stock returns arecollected from the Nikkei NEEDS-FinancialQuest database, provided by Nikkei Media
Marketing Incorporated. Market stock returns used in the basic MIDAS model are thelog diﬀerentials of monthly closing prices in Nikkei225 and TOPIX. Nikkei225, a representative index of Japanese stocks, is an average stock price index of 225 stocks selectedfrom the 1st section of the Tokyo Stock Exchange (TSE), and TOPIX, another representative index, is an average stock price index of all stocks listed in the 1st section of
TSE.
Individual stock returns for the components of Nikkei225 and TOPIX are used inestimating the Factor MIDAS model. In addition to the components of two representative
indices, individual stock returns for the ﬁrms listed or formerly listed on any stock markets
are also used to extract the factors. As of January 28, 2022, when data were downloaded,
the number of individual stock returns in each set was 225 for the Nikkei225, 2180 for
the TOPIX, and 5294 for all listed ﬁrms, respectively. However, information from all theﬁrms cited above is not always used in estimating the factors owing to the missing values.
For instance, the Nikkei225 components in this study are those listed in the index at thetime of data collection (January 2022). Since some of the components are changed once
yearly, stocks that had not been included in the index in the past have missing values.
Similarly, ﬁrms listed in TOPIX and all listed ﬁrms are likely to have missing values intheir stock returns as some ﬁrms may exit from the stock market owing to bankruptcy,
or others may be newly listed in the middle of the sample period. As mentioned above
and detailed in Appendix A, the Stock and Watson (2002)’s EM algorithm allows us toestimate the factors from the dataset with missing values; however, the ﬁrms which havetoo many missing values in their stock returns should be removed from the dataset toensure factor estimation stability.7 Speciﬁcally, the series of data that contains more than7

In addition to “missing” observations, we also treat outliers as missing observations. In the same10

a quarter of missing values for the estimation period is not used in estimating the factor.
Finally, quarterly stock returns in the comparison model are constructed as the log
diﬀerentials of the 3-month average of closing prices for each month. The quarterly
individual stock returns are also constructed in the same manner. We then extract
quarterly factors from them. Hence, models using quarterly stock returns correspondto a special case of the MIDAS model wherein the same weights are assigned to eachmonthly observation.

4

Forecast design

The performance of each model is assessed by a recursive out-of-sample forecasting, where
the estimation and forecast are repeated with updated estimation and evaluation periods.
Speciﬁcally, the initial estimation period covers 1994Q2 to 2011Q4, and the evaluation
period is the period after 2012Q1. After initial estimation and forecasting, 2012Q1 data
are added to the estimation, and the evaluation period is shifted to one-quarter ahead.
This process is iterated until the evaluation period reaches the end of the sample period.
In the actual forecasting exercises, two cases are considered for the end of the sample
period: 2019Q4 and 2021Q3 (i.e., before and after COVID-19, respectively). This isbecause, during the pandemic, stock prices have consistently risen owing to monetaryintervention with exception of a large initial drop. Hence, whether such a huge policy
intervention to the stock market changes the role of stock returns as a leading indicatorfor real activities is of interest.
The gap between the forecasts and the actual GDP growth rate is measured by the
root-mean-square error (RMSE). For the h-period ahead forecasts, we compute the RMSE
by the following:vuu
RM SE(h) = t

TX
2 −h
1
(yt+h − ŷt+h )2 ,(T2 − h) − T1 + 1 t=T

(10)

1

where yt+h is actual GDP growth rate and ŷt+h is h-period ahead out-of-sample forecasts
manner, as in Stock and Watson (2002), observations with deviations from the median exceeding 10
times the quartile range are replaced by missing observations as outliers.

11

obtained from the model. Moreover, T1 and T2 correspond to the end of the initial
estimation period and the end of the ﬁnal evaluation period, respectively. By changing
the speciﬁcation of the models and the available data, we compute the 12 forecasts fromthe following:(1) MIADS model using Nikkei225(2) MIDAS model using TOPIX(3) MIDAS model with factors from the components of Nikkei225(4) MIDAS model with factors from the components of TOPIX(5) MIDAS model with factors from all listed ﬁrms(6) The model without stock returns(7) The model using quarterly Nikkei225(8) The model with quarterly TOPIX(9) The model with quarterly factors from the components of Nikkei225(10) the model with factors from the components of TOPIX(11) The model with factors from all listed ﬁrms(12) In-sample mean
In what follows, the naı̈ve forecasts obtained from (12) in-sample mean is regarded asa benchmark, and the relative RMSE for each model is computed to evaluate forecast
accuracy. This model is considered to perform well compared with naı̈ve forecasts when
the relative RMSE takes a value less than 1. Additionally, the signiﬁcance of the diﬀerencein the relative RMSE is tested by Diebold and Mariano (1995)’s test as modiﬁed by Harvey
et al. (1997).

5

Forecasting results

Table 1 reports the relative RMSEs of each model to in-sample mean forecast with respect to one-period ahead GDP growth for two sample periods. The numbers of the lag
lengths (py , px ), factors (R), and the parameters to specify the weights (P ) are chosen
so the RMSE for each model is minimized for all possible combinations of them with the12maximum value of those being four. Hence, our results are the best-performing ones foreach speciﬁcation.
Table 1: Relative RMSE ratio (one-period ahead forecasts: h = 1)(a) Excl. COVID-19 period(b) Incl. COVID-19 period

relative RMSE

relative RMSE

(P, R, py , px )(P, R, py , px )Basic MIDAS model(1) Nikkei225

0.915*(3,–,3,2)

0.978(1,–,1,1)(2) TOPIX

0.919*(3,–,3,2)

0.988(1,–,1,1)(3) Nikkei225

0.901**(3,1,3,2)

0.970(1,1,1,1)(4) TOPIX

0.891**(3,2,3,2)

0.951(1,4,1,1)(5) All listed ﬁrms

0.901*(3,2,3,2)

0.941(1,2,1,1)(6) Without stock returns

0.954**(–,–,3,–)1.060(–,–,1,–)(7) Nikkei225

0.920*(–,–,3,1)

0.992(–,–,1,–)(8) TOPIX

0.917*(–,–,3,1)

0.998(–,–,1,–)(9) Nikkei225

0.895**(–,1,3,1)

0.990(–,1,1,3)(10) TOPIX

0.886**(–,3,3,2)

1.000(–,2,1,1)(11) All listed ﬁrms

0.878*(–,3,3,4)

0.997(–,2,1,1)

Factor MIDAS model

Quarterly model

Quarterly factor model

Notes: The table shows the relative RMSEs to in-sample mean forecast for two sample periods: (a)excluding COVID-19 pandemic and (b) including COVID-19 pandemic. The null hypothesis that theprediction error in each model is greater than the error in in-sample mean forecast is tested by Diebold
and Mariano (1995) modiﬁed by Harvey et al. (1997). The 5% and 10% signiﬁcance levels are denotedby ∗∗ and ∗, respectively.

This table reveals the following ﬁve results. First, we ﬁnd that all forecast models
considered here work well before the COVID-19 pandemic in the sense of reducing forecast
errors signiﬁcantly compared with naı̈ve forecasts but not at all after the pandemic. The
relative RMSEs in the sample excluding the COVID-19 period take the values less than
one with statistically signiﬁcant diﬀerences. Conversely, signiﬁcant improvements in theforecasts cannot be observed at all in the estimation using the sample including the
COVID-19 period. This implies that, after the COVID-19 pandemic, the role of stock13

prices as forecasters for economic activities has changed. This ﬁnding contrasts with theperiod of the global ﬁnancial crisis reported by Ferrara and Marsilli (2013), who documentthat stock prices contributed to the forecast accuracy of GDP growth rate in the euro area
during the 2008–2009 global ﬁnancial crisis; however, this is considered reasonable giventhat the origin of the current economic turmoil is not the ﬁnancial market. Moreover, thisﬁnding is consistent with a disconnection between real economy and stock prices whichhave been observed during the pandemic. While the COVID-19 pandemic caused serious
damage to the real economy, the Nikkei225 reached a high of over 30,000 yen for theﬁrst time in 30 years in February 2021 owing to the massive monetary easing. Drastic
ﬂuctuations in GDP growth rate during the pandemic period in response to infection
status may also have been another factor that made forecasting diﬃcult. 8 In the following
discussion, we focus mainly on results obtained in the pre-COVID-19 period and examine
whether the information in the time series or cross-sectional direction is more eﬀective in
improving forecasting performance.
Second, the models including stock returns in some form perform better than themodel without stock returns. This result suggests that stock prices have relevant information about future GDP growth rate, which is consistent with Aylward and Glen
(2000), Andreou et al. (2013) and Ferrara and Marsilli (2013).
Third, using high-frequency data does not necessarily contribute to improving forecasting ability, at least in the presented analysis. No signiﬁcant diﬀerence in forecast accuracy between the MIDAS model and the corresponding quarterly model can be found;rather, the quarterly models, except for the case of using Nikkei225 without factors, show
better performance in point estimates. This suggests that the advantage of having moretime-series information may not outweigh the disadvantage of making the model more
complex. Moreover, the number of the factors in the quarterly models (both for TOPIX
and all listed ﬁrms) is three. In contrast, that in the MIDAS model is two. For conﬁrmation of the role of the third factor, we check the forecasting ability of the quarterly
8

The third possible reason for stock prices losing predictive power to GDP after the pandemic is theheterogeneous impact of COVID-19 on each ﬁrm or each industry. While the aviation industry has been
severely hit by travel restrictions, the IT industry, which provided systems for working from home, hasseen an improved business performance. Such heterogeneous movement of stock prices across industries
attributed to the sector-speciﬁc COVID-19 shock may have caused the gap between stock prices and real
economic activities.

14

model with two factors and ﬁnd that the MIDAS model outperforms the quarterly model
when the factors extracted from ﬁrms listed on TOPIX; however, when information on all
listed ﬁrms is exploited, the quarterly model still provides better forecasts than MIDAS
model.9 Consider that the additional cross-sectional information contained in the third
factor is critical in the sense of compensating for the lack of time-series information in
the comparison between MIDAS and quarterly models, then Factor MIDAS model ofusing all listed ﬁrms provide better performance than the quarterly model having only
two factors. However, this is not the case. Hence, the noise in the high-frequency stock
returns for each ﬁrm, which may be irrelevant to macroeconomic activity, may aﬀect undesirable eﬀects on the forecasting when the factors are extracted considering too muchinformation for both time-series and cross-sectional dimensions.
Fourth, the use of factors improves the forecasting performance compared with thatof the market index. This suggests that relevant information about future GDP growth
rates may have been lost by cross-sectional aggregation. Relative RMSEs in the Factor
MIDAS model are all smaller than these counterparts in the basic MIDAS model. Thisis the same as in the case of quarterly models. Except for the case of Nikkei225, morethan the two-factor model is chosen as the best performance model for each speciﬁcation.
This highlights the fact that incorporating information about the heterogeneity of stock
prices into forecasts is beneﬁcial. The forecasting abilities of the Factor MIDAS modelusing TOPIX, and all listed companies have improved from the one- to the two-factor
models, and the same thing can be observed for the forecast accuracy of the quarterly
factor model from the one- to the three-factor models. Therefore, the heterogeneous
information aggregated into factors may have contributed to the improvement of theforecast accuracy.10 More interestingly, even in the one-factor model of Nikkei225, the9

To be concrete, the relative RMSE of quarter factor model using TOPIX components in the case of(P, R, py , px ) = (−, 2, 3, 2) is equal to 0.916 while that of quarterly factor model using all listed ﬁrms in
the case of (P, R, py , px ) = (−, 2, 3, 4) is 0.888.
10
Estimated factor loadings in the respective models are shown in Figures B.1 and B.2 of Appendix B.
Factor loadings are obtained in the ﬁnal estimation of out-of-sample forecasting and normalized so thatthe total sum of the absolute value of factor loadings is equal to 100. Factor loadings of the ﬁrst factor
in all speciﬁcations show the same sign (and roughly the same values). This suggests that the ﬁrst factor
represents the average movement of stock prices. Conversely, while the second and third-factor loadings
show diﬀerent signs and magnitudes for each ﬁrm, they often take on the same sign continuously when
seen from the x-axis direction. As the ﬁrms plotted in the x-axis are aligned by industry owing to thedata source, this sequence of factor loadings with the same sign indicates that stock price ﬂuctuationsin speciﬁc industries are aggregated in each factor.

15

model with factor improves the forecast accuracy over the model with the market indicatoraggregated in advance. Figure 1 presents the quarterly stock returns of Nikkei225 andthe ﬁrst factor derived from the quarterly stock returns of the ﬁrms listed in Nikkei225,
in the ﬁnal exercise of out-of-sample forecasting.11 Overall, both time series show thesame movement; however, a gap is sometimes observed between the two series.
330
Factor from the components of Nikkei225 (left axis)
Stock returns of Nikki225 (right axis)

2

20

1

10

0

0

-1

-10

-2

-20

-3

-30

-4

1995

2000

2005

2010

2015

-40

Figure 1: Nikkei225 and the ﬁrst factor obtained from its components
Notes: This ﬁgure plots both the ﬁrst factor obtained from the components of Nikkei225 andstock returns of Nikkei225 (market aggregated index). The series of the factor, shown here, isobtained in the ﬁnal estimation period for 1994Q2 to 2019Q4.

Additionally, Figure 2 depicts the estimated factor loadings in the quarterly factor
model and the weights assigned to each stock comprising the Nikkei225. Factor loadings
are normalized so that the total sum of the absolute value of factor loadings equals 100
for comparison purposes with the weights for Nikkei225. The total number of the ﬁrms
used for extracting the factor is 178 and not 225 owing to the missing observations asmentioned above. The compositional weights of the 225 stocks in Nikkei225 are quite
skewed as is well-known. From Figure 2, we conﬁrm that the estimated weights in thefactor model are substantially less biased compared with the ones in the Nikkei225 index,
suggesting the importance of utilizing the information as unbiased as possible considering
11

Although the estimated factor and its loadings are diﬀerent depending on the estimation period, we
demonstrated the result obtained in the ﬁnal estimation of the out-of-sample forecasting. This is because
ﬁnal estimation makes use of the most information.

16

forecast accuracy.
(a) Estimated factor loadings(b) Weights on stocks listed in Nikkei225
10

0.8

8

percents

percents

0.6

0.4

0.2

0

6
4
2

0

50

100

0

150

0

50

100

150

200

Figure 2: The Factor loadings and weights assigned to each stock of Nikkei225
Notes: The ﬁgures show estimated factor loadings for the ﬁrst factor obtained from quarterly stock returns of the Nikkei225 (Figure 2a) and the weights assigned to stocks listed in
Nikkei225(Figure 2b). Weights of the Nikkei225 are the values adopted at the end of January
2022. From Figure 2b, we ﬁnd that a large weighting is assigned to several ﬁrms (e.g., DAIKIN(3.1%), Softbank group (3.8%), Tokyo Electron (7.1%), and Fast Retailing (8.7%)). Comparedwith the weights of Nikkei225, no substantial diﬀerence in the estimated factor loadings can befound.

Fifth and most importantly, richer cross-sectional information tends to improve forecast accuracy. Speciﬁcally, models with the factors from the components of TOPIX andall listed ﬁrms provide better forecast performance than those using Nikkei225 components. Moreover, the quarterly model with the factors from all listed ﬁrms performs best
overall. Hence, much more rich variations of information for the cross-sectional direction
are key in forecasting the GDP growth rate.
To summarize all the results, forecasting analyses support the use of cross-sectional
information in the stock market to forecast GDP growth rate. Speciﬁcally, augmentationof forecast models with factors provides better forecasts on future GDP growth rates.
Moreover, forecast accuracy improves with a larger set of stock prices from which thefactors are extracted. Conversely, this study fails to detect the signiﬁcant contributionof the enrichment of time-series dimension in forecasting quarterly GDP growth rate.
Additionally, comparison analysis between two sample periods implies that stock prices
may have lost their role as a leading indicator of real economic activity during the COVID17

19 pandemic.

6

Conclusion

This study has explored whether stock prices contain relevant information on future GDP
growth rates. The Factor-MIDAS model is employed to exploit the time-series and crosssectional information in the stock market and clarify the contribution of stock market
information to the macroeconomic forecast. Although the analysis has conducted in aparsimonious way in the sense of using the information only in the stock market, thisstudy provides new insights on the role of stock prices as a leading indicator of real
economic activities. Our ﬁndings emphasize the role of cross-sectional heterogeneity incontributing to a more accurate GDP growth rate forecast. Additionally, stock prices
contain relevant information about future macroeconomic activity in the ﬁrst place. We
ﬁnd that forecasting ability improves using factors extracted from a larger set of individual stock prices. In contrast, enrichment of time-series information does not necessarily
contribute to the improvement of forecast accuracy at least in the current monthly andquarterly speciﬁcations. Furthermore, stock prices as a leading indicator of economic
activity may have been lost during the COVID-19 pandemic, owing to the drastic ﬂuctuation in GDP growth rate and massive monetary easing. Overall, these ﬁndings support
that asset prices are a mirror of expectations about the future economic condition andthat these expectations are accurate to some extent. The data-rich environment of thestock market can hence be highly useful in predicting future economic conditions, possibly
helping policymakers manage their policies in advance of actual economic ﬂuctuations.
Simultaneously, however, excessive intervention in the ﬁnancial market can lead to theloss of the role of stock prices as a forecaster of the real economy.
As an extension of this study, considering the case of a combination of daily stock
returns and quarterly GDP growth rates in the forecasting would be possible. Moreover, our method can be also applied to the relationship between stock prices and GDP
components. These tasks are then left to future work.

18

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21

A

Appendix: Factor estimation method

Factors are estimated from the dataset with missing observations using the EM algorithm
proposed by Stock and Watson (2002). Let vector Zt = (z1t , z2t , · · · , zM t )′ be deﬁned as
containing the normalized stock returns of ﬁrm i in period t as the i-th element. The
standard factor representation can then be speciﬁed as follows:
Zt = ΛFt + ξt ,(A.1)

where Λ = (λ1 , · · · , λM )′ is the (M × R) factor loadings, Ft = (f1,t , · · · , fR,t )′ is the(R × 1) vector of factors, and ξt is the (M × 1) vector of idiosyncratic disturbance. As
emphasized, elements of Zt are missing. First, the missing observations are replaced(0)with random variables drawn from N (0, 1) and the new matrix is deﬁned as Ẑt . Since
(0)

Ẑt

is the balanced dataset without missing observations, the factors F̂t(0)

and loadingsΛ̂(0) can be estimated using the usual principal component eigenvalue calculation. By(j+1)

setting j = 0, we update the elements of Ẑt(j+1)

and Ẑit(j+1)

dataset Ẑt(j+1)= Zit otherwise. Ft(j+1)

as Ẑit(j)(j)= λ̂′ i F̂t

if Zit is missingand Λ̂(j+1) are the reestimated by using the updated. This iteration of the algorithm is repeated until the estimated factors areconverged suﬃciently.

22

B

The estimated factor loadings(a) Nikkei225(b) TOPIX
0.14

First factor

0.4

0

50

100

150

0.08
0.06

0.06
0.04
0.02

0.02

0

0

-0.02

0

200

400

600

800

1000

0.3

1

0.3

0.2

0.5

0.2

0.1

0
-0.5
-1

0.1
0
-0.1

-1.5

-0.2

-2

-0.3

0

50

100

150

Second factor

0.4

Second factor

1.5

0

200

400

600

800

-0.4

1000

0.2
0.1

1

0.2
0.1
0
-0.1

-1

-0.2

-1.5

0

50

100

150

-0.3

0

500

1000

0

500

1000

-0.3

0.3

0

1000

-0.2

0.4

-0.5

500

0

2

0.5

0

-0.1

1.5

Third factor

Second factor

0.08

0.04

0.2

Third factor

0.1

0.1

Third factor

First factor

0.6

0.12

0.12

First factor

0.80(c) All listed ﬁrms

0
-0.1
-0.2
-0.3

0

200

400

600

800

1000

-0.4

Figure B.1: The factor loadings in each Factor MIDAS model
Notes: The ﬁgure presents factor loadings for each factor in the respective factor MIDAS model.
The x-axis indicates each ﬁrm listed in each group of stocks; the y-axis indicates the estimatedfactor loadings normalized so that the sum of absolute values for factor loadings is equal to 100,thus regarding the values taken on the y-axis as percentage.

23

(a) Nikkei225(b) TOPIX(c) All listed ﬁrms
0.12

0.14

0.8

0.12

0.1

0.4

First factor

0.1First factor

First factor

0.6

0.08
0.06
0.04

0.2

50

100

150

0

200

400

600

800

0

1000

0.4

0.3

0.3

0.2

1

0.2

0.5
0
-0.5

0.1
0
-0.1

-1

-0.2

-1.5

-0.3

0

50

100

150

Second factor

2

1

0

200

400

600

800

-0.5

0

500

1000

0

500

1000

0
-0.1

0.2

0.2
0

-1

0.1
0
-0.1
-0.2

-0.2

-1.5
-2

1000

0.3

Third factor

Third factor

0

500

0.1

-0.3

1000

0.4

0.5

0

-0.2

0.6

1.5

Third factor

0.04

1.5

Second factor

Second factor

0

0

0.06

0.02

0.02
0

0.08

0

50

100

150

-0.4

-0.3
0

200

400

600

800

1000

-0.4

Figure B.2: The factor loadings in each Quarterly factor model
Notes: This ﬁgure shows the factor loadings for each factor in the respective quarterly factor
model. The x-axis indicates each ﬁrm listed in each group of stocks; the y-axis indicates theestimated factor loadings normalized so that the sum of absolute values for factor loadings isequal to 100, thus regarding the values taken on the y-axis as percentage.

24