Forecasting Mixed‐Frequency Time Series with ECM‐MIDAS Models

• DOI: http://doi.org/10.1002/for.2286
• Date: 01 April 2014
• Published date: 01 April 2014
• Author: Alain Hecq,Thomas B. Götz,Jean‐Pierre Urbain

Journal of Forecasting,J. Forecast. 33, 198–213 (2014)

Published online 3 March 2014 in Wiley Online Library (wileyonlinelibrary.com)DOI: 10.1002/for.2286

Forecasting Mixed-Frequency Time Series with

ECM-MIDAS Models

THOMAS B. GÖTZ, ALAIN HECQ AND JEAN-PIERRE URBAIN

Department of Quantitative Economics, School of Business and Economics, Maastricht University,

The Netherlands

ABSTRACT

This paper proposes a mixed-frequency error correction model for possibly cointegrated non-stationary time series

sampled at different frequencies. We highlight the impact, in terms of model specification, of the choice of the

particular high-frequency explanatory variable to be included in the cointegrating relationship, which we call a

dynamic mixed-frequency cointegrating relationship. The forecasting performance of aggregated models and several

mixed-frequency regressions are compared in a set of Monte Carlo experiments. In particular, we look at both the

unrestricted mixed-frequency model and at a more parsimonious MIDAS regression. Whereas the existing literature

has only investigated the potential improvementsof the MIDAS framework for stationary time series, our study empha-

sizes the need to include the relevant cointegrating vectors in the non-stationary case. Furthermore, it is illustrated that

the choice of dynamic mixed-frequency cointegrating relationship does not matter as long as the short-run dynamics

are adapted accordingly. Finally,the unrestricted model is shown to suffer from parameter proliferation for samples of

relatively small size, whereas MIDAS forecasts are robust to over-parameterization. We illustrate our results for the

US inflation rate. Copyright © 2014 John Wiley & Sons, Ltd.

KEY WORDS ECM; MIDAS; forecasting

INTRODUCTION

In economics we are often concerned with the problem of dealing with variables that are available, and thus sam-

pled, at different frequencies. One example is the task of forecasting or nowcasting (see Giannone et al., 2008) the

quarterly gross domestic product using monthly variables such as the industrial production index or daily indicators

such as interest rates or stock prices. The classical way to deal with such a situation is the temporal aggregation

of the high-frequency variables, i.e. to sample at the common low frequency. However, this approach might lead to

a loss of information due to omitting the high-frequency observations (Andreou et al., 2010a). Hence, forecasting

performance might improve by making use of this extra information. In many cases, however, including all poten-

tial unrestricted high-frequency explanatory variables is unfeasible with regard to the number of observations for the

low-frequency dependent variable. This has motivatedthe introduction of mixed data sampling (henceforth MIDAS),

which aims at transforming a high-frequency variable so that all information in the high frequency can be preserved

(Ghysels et al., 2004). One of the virtues of the MIDAS approach is its parsimony, meaning that we can reduce the

number of parameters to estimate while still employing the information present in the high-frequency variables.

With the exception of Miller (2012), MIDAS is usually applied to stationary time series or to transformations

of non-stationary variables like first differences. For example, Clements and Galvão (2008, 2009) consider leading

and coincident indicators as well as the output growth. Galvão (2012) looks at financial variables such as the term

spread and the output growth. Andreou et al. (2010b) forecast US economic activity employing a large cross-section

of transformed daily series. Merely working with first differenced I.1/ variables might neglect a possible long-run

relationship between the variables. In this paper, we consider mixed-frequency I.1/ time series, a low-frequency

variable yand high-frequency variables x, which are possibly cointegrated. We compare the forecasting performance

of models for y when only the first difference terms of yand of xare included, omitting the presence of a long-run

relationship, with a mixed-frequency error correction model (ECM hereafter). We also compare our mixed-frequency

model with a model resulting from the temporal aggregation of high-frequency variables. In the usual case, i.e. the

common-frequency framework, Clements and Hendry (1998), among others, have compared models in terms of their

ability to predict the levels, first differences and long-run relationships. Concerning the first difference, the gain

of including the long-run relationship is apparent only for short horizons. In terms of forecasting and nowcasting

business cycle indicators such as inflation or the unemployment rate, however, this can be important.

When representing the relationship between a particular high-frequency regressor and a low-frequency regressand

in an error correction format we do not only consider the end-of-period observation to enter the long-run term (as done

Correspondence to: Thomas B. Götz, Department of Quantitative Economics, School of Business and Economics, Maastricht University, PO

Box 616, 6200 MD Maastricht, The Netherlands. E-mail: t.gotz@maastrichtuniversity.nl

Copyright © 2014 John Wiley & Sons, Ltd

Forecasting with ECM-MIDAS Models 199

in Miller, 2012), but allow for the ‘neighboring’ high-frequency observations to do so instead. Despite the fact that

it does not matter asymptotically whether a cointegrating relationship is modeled between yand the end-of-period

observation of xor, say, mid-period observation of x, since cointegration is a long-run property (Marcellino, 1999),

the choice of which particular high-frequency observation enters the disequilibrium error has an impact on the short-

run dynamics terms of the ECM at the model representation level. Hence, although irrelevant for cointegration, for

empirical analyses characterized by relatively small samples it is of interest to investigate whether an inappropriate

specification of the short-run dynamics has an effect on the forecast accuracy.

Seong et al. (2007) also model cointegrated multivariate time series of mixed frequencies. However, they regard

the high-frequency observations of yas missing data and cast the model in state-space form. Instead, this paper aims

at providing tools for working in a mixed-frequency framework without relying on a state-space form. It turns out that

most macroeconomic variables of interest (gross domestic product, inflation, etc.) are available at a lower frequency

than explanatory indicators, and therefore a regression approach makes sense. With the exception of Miller (2012),

the evaluation of a mixed-frequency error correction model for forecasting purposes without adopting a state-space

model has not been considered in the current literature and is one of the contributions of this paper.

The rest of the paper is organized as follows. In the next section we introduce a mixed-frequency ADL model and

derive the short-run dynamics implied by alternative choices of the variables to be included in a long-run compo-

nent. A MIDAS framework is proposed to capture the high-frequency structure of the short-run dynamics. The model

is compared with the unrestricted approach and classical methods of temporal aggregation using Monte Carlo sim-

ulations in the third section. The fourth section illustrates the theoretical and Monte Carlo results via an empirical

illustration. A final section concludes.

THE MF-ADL MODEL AND ECM REPRESENTATIONS

For expository simplicity we consider a stable (see Boswijk, 1994) mixed-frequency autoregressive distributed lag

model, denoted MF-ADL(pl,qh,ql), for two non-stationary I.1/time series yt, which is sampled at low frequency,

and x.m/

t, which is available at high frequency. Note that lag orders indexed with lcorrespond to the variables’

low-frequency lag orders, whereas the one indexed with hrefers to the high frequency. The index trepresents the

low frequency and runs from 1to T. The number of high-frequency observations per t-period equals m, such that

a high-frequency observation is represented by x.m/

ti=m,whereidenotes the fraction of high-frequency units per

low-frequency time period we look into the past from the end-of-period observations. In a year/month example we

have mD12 and, for iD6, for example, x.12/

t1=2 denotes the value of xfor June in year t. Note further that

x.12/

tC1=12 x.12/

tC111=12 represents January’s x-value of year tC1. This notation is standard in the literature on

mixed-frequency data (see, among others, Ghysels et al., 2004; Clements and Galvão, 2008, 2009; Miller, 2012).

The MF-ADL(pl;q

h;q

l) model can then be represented as follows:

ytDcC˛1yt1C:::C˛plytplC

ˇ00x.m/

tCˇ10x.m/

t1=m C:::Cˇqh0x.m/

tqh=mC

ˇ01x.m/

t1Cˇ11x.m/

t11=m C:::Cˇqh1x.m/

t1qh=mC

:

:

:

ˇ0qlx.m/

tqlCˇ1qlx.m/

tql1=m C:::Cˇqhqlx.m/

tqlqh=m C.m/

t

(1)

or in terms of lag polynomials by

A.L/y.m/

tDcCB0.L1=m/x .m/

tCB1.L1=m/x .m/

t1C:::CBql.L1=m /x.m/

tqlC.m/

t(2)

where tis assumed to be a martingale difference sequence with respect to the past information and, for simplicity,

x.m/

tis assumed to be strongly exogenous for the long-run parameters (see Urbain, 1992). This assumption is made to

simplify the presentation; extensions to the weakly exogenous case are direct. In the second representation, A.L/ D

1˛1L:::˛plLpland Bi.L1=m/Dˇ0i Cˇ1i L1=m C:::CˇqhiLqh=m;i D0;:::;q

l, denote polynomials in Land

L1=m, respectively. The former represents the usual lag operator, i.e. overthe low-frequency index t, whereas the latter

stands for the lag operator in the high frequency such that Lj=mx.m/

ti=m Dx.m/

ti=mj=m. Obviously, L1=mx.m/

t.m1/=m D

x.m/

t1. The same logic holds for difference operators: represents the usual difference operator in the low frequency,

Copyright © 2014 John Wiley & Sons, Ltd J. Forecast. 33, 198–213 (2014)