Multi‐step forecasting in the presence of breaks

• DOI: http://doi.org/10.1002/for.2480
• Published date: 01 January 2018
• Date: 01 January 2018
• Author: Jari Hännikäinen

Received: 21 December 2015 Revised: 12 February 2017 Accepted: 7 May 2017

DOI: 10.1002/for.2480

RESEARCH ARTICLE

Multi-step forecasting in the presence of breaks

Jari Hännikäinen

School of Management, University of

Tampere, Tampere, Finland

Correspondence

Jari Hännikäinen, School of Management,

University of Tampere, Kanslerinrinne 1,

FI-33014 Tampere, Finland.

Email: jari.hannikainen@uta.fi

Abstract

This paper analyzes the relative performance of multi-step AR forecasting methods

in the presence of breaks and data revisions. Our Monte Carlo simulations indicate

that the type and timing of the break affect the relative accuracy of the methods.

The iterated autoregressive method typically produces more accurate point and den-

sity forecasts than the alternative multi-step AR methods in unstable environments,

especially if the parameters are subject to small breaks. This result holds regardless

of whether data revisions add news or reduce noise. Empirical analysis of real-time

US output and inflation series shows that the alternative multi-step methods only

episodically improve upon the iterated method.

KEYWORDS

density forecasting, intercept correction, macroeconomic forecasting, multi-step forecasting, real-time

data, structural breaks

1INTRODUCTION

The medium- and long-term prospects of the economy are

important for consumers, investors, and policymakers. For

example, it is well known that monetary policy affects the

economy with a long lag. As a result, central banks conduct

forward-looking monetary policy; that is, central banks' inter-

est rate decisions are based on their forecasts of future output

growth, unemployment, and inflation. Given the importance

of the medium- and long-term economic outlook, economists

provide forecasts of key macroeconomic time series several

periods ahead in time.

When generating a multi-step forecast, a forecaster has

to decide whether to use the iterated or direct forecasting

strategy. In the iterated approach, forecasts are made using

a one-period-ahead model, iterated forward for the desired

number of periods. A central feature of the iterated approach

is that the model specification is the same regardless of the

forecast horizon. Direct forecasts, on the other hand, are made

using a horizon-specific model. Thus a forecaster estimates a

separate model for each forecast horizon. The theoretical lit-

erature analyzing the relative merits of the iterated versus the

direct forecast methods includes, for example, Bao (2007),

Brown and Mariano (1989), Chevillon and Hendry (2005),

Clements and Hendry (1996b, 1998), Findley (1985), Hoque,

Magnus, and Pesaran (1988), Ing (2003), Proietti (2011), and

Schorfheide (2005). This literature emphasizes that the choice

between iterated and direct multi-step forecasts is not clear

cut, but rather involves a trade-off between bias and estima-

tion variance. The iterated method uses the largest available

data sample in the estimation and thus produces more efficient

parameter estimates than the direct method. In contrast, direct

forecasts are more robust to model misspecification. Which

element—the bias or the estimation variance—dominates in

the composition of the mean squared forecast error (MSFE)

values in practice depends on the sample size, the forecast

horizon, and the (unknown) underlying DGP, and therefore

the question of which method to use cannot be decided ex

ante on theoretical grounds alone. Hence the question of

which multi-step forecasting method to use is an empirical

one. In their empirical analysis of 170 US monthly macroe-

conomic time series, Marcellino, Stock, and Watson (2006)

and Pesaran, Pick, and Timmermann (2011) find that the

iterated approach typically outperforms the direct approach,

especially if the sample size is small, if the forecast horizon

is long, and if long lags of the variables are included in the

forecasting model.

Although the parameters in many of the macroeconomic

time series are unstable over time (Stock & Watson, 1996),

work on multi-step forecasting in the presence of breaks has

been virtually absent from the literature. However, structural

breaks play a central role in economic forecasting (see, e.g.,

102 Copyright © 2017 John Wiley & Sons, Ltd. wileyonlinelibrary.com/journal/for Journal of Forecasting.2018;37:102–118.

HÄNNIKÄINEN 103

Clements & Hendry, 2006; Elliott & Timmermann, 2008;

Rossi, 2013). Forecasting models often systematically under-

or over-predict in the presence of structural instability.There-

fore, one wayto improve their forecast accuracy in an unstable

environment is to use intercept corrections, advocated by

Clements and Hendry (1996a, 1998). Intercept corrections

are based on the idea that if the forecast errors are system-

atically either positive or negative then adjusting the mech-

anistic, model-based forecast by the previous forecast error

should reduce the forecast bias and hence improve forecast

performance.

Another issue that has been overlooked in the multi-step

forecasting literature is the fact that key macroeconomic data

are subject to revisions. The real-time nature of macroeco-

nomic time series is potentially important for the relative

performance of multi-step forecasting methods for at least

three reasons. First, because data revisions are usually quite

large, the parameters estimated on the final revised data may

differ considerablyfrom those estimated on the real-time dat a.

Second, data revisions can also affect the dynamic lag struc-

ture of the forecasting model. Finally, real-time forecasts are

conditioned on the first-release or lightly revised data actually

available at each forecast origin, whereas forecasts based on

the final revised data are conditioned on the latest available

observations of each forecast origin.

In this paper, we focus on linear autoregressive (AR) mod-

els. There are two reasons for our choice. First, despite their

parsimonious form, linear AR models typically perform well

in macroeconomic forecasting applications (see, e.g., Rossi,

2013; Stock & Watson, 2003, 2007). Second, it is stan-

dard practice to use the linear AR model as a benchmark in

forecasting exercises.

The main contributions of this paper are as follows. First,

we analyze the relative performance of multi-step forecasting

methods in the presence of breaks through Monte Carlo sim-

ulations. Our comparison includes the iterated and direct AR

models and various forms of intercept correction. We con-

sider several break processes, including changes in the inter-

cept, autoregressive parameter, and error variance. Second,

we take into account in our simulations that most macroeco-

nomic time series are subject to data revisions.1A novelty of

our simulation framework is that data revisions can either add

news or reduce noise (see, e.g., Mankiw & Shapiro, 1986).

The distinction between news and noise revisions allows us

to study whether the properties of the revision process matter

1We use the conventional approach to deal with real-time data; that is, we

estimate the parameters of the forecasting models using data from the latest

available vintage. An alternative estimation strategy is the real-time vintage

approach (RTV). In the RTV approach,t he forecastingmodel is estimated on

first-release data. Clements and Galvão (2013) show that the RTV approach

produces more accurate real-time forecaststhan the standard approach. We do

not use the RTV approach in this paper because it requires more data vintages

than the standard approach and thus reduces the length of the out-of-sample

forecasting period.

for the multi-period forecasting problem. Third, we consider

both point and density forecasts. The existing multi-step fore-

casting literature focuses exclusively on point forecasting.

However, density forecasts contain more information than

point forecasts. Density forecasts summarize the information

regarding the uncertainty around point forecasts. Finally, the

real-time accuracy of the multi-step forecasting methods for

four key US macroeconomic time series, namely real gross

domestic product (GDP), industrial production, GDP defla-

tor, and personal consumption expenditures (PCE) inflation,

is compared. To the best of our knowledge, there are no

other papers analyzing the relative performance of multi-step

forecasting methods in a real-time environment.

The remainder of this paper is organized as follows.Section

2 introduces the notation and the statistical framework.

Section 3 provides a brief overview of the multi-step forecast-

ing methods. Section 4 presents the Monte Carlo simulation

results and Section 5 presents the empirical results. Section 6

concludes.

2STATISTICAL FRAMEWORK

Key macroeconomic time series are published with a lag and

are subject to revisions. For instance, a forecaster at period

T+1 has access to the vintage T+1 values of GDP up

to time period T. In addition, because of data revisions, the

first-released value and the final value for a period may dif-

fer substantially.We incorporate the publication lag and data

revisions into our statistical framework. The statistical frame-

work used in this paper follows that adopted in Clements

and Galvão (2013), Jacobs and van Norden (2011), and

Hännikäinen (2017). It relates a data vintage estimate to the

true value plus an error or errors. More specifically, the period

t+svintage estimate of the value of yin period t, denoted by

yt+s

t,2where s=1,…,l,3can be expressed as the sum of the

true value ̃yt, a news component vt+s

t, and a noise component

𝜀t+s

t;thatis,yt+s

t=̃yt+vt+s

t+𝜀t+s

t.

In this framework, revisions either add news or reduce

noise. Data revisions are news if they are uncorrelated with

the previously published vintages, cov(yt+k

t,vt+s

t)=0∀k⩽s.

On the other hand, data revisions reduce noise if each vintage

release is equal to the true value plus a noise. Noise revisions

are uncorrelated with the true values, cov(̃yt,𝜀

t+s

t)=0.

We sta ck t he ldifferent vintage estimates of yt,vtand 𝜀t

into vectors yt=(yt+1

t,…,yt+l

t)′,vt=(vt+1

t,…,vt+l

t)′and

𝜺t=(𝜀t+1

t,…,𝜀

t+l

t)′, respectively. Using these vectors we

2Throughout this section, superscripts refer to vintages and subscripts to time

periods.

3Following Clements and Galvão (2013), weassume t hatwe obser veldiffer-

ent estimates of ytbeforethetruevalue,̃yt, is observed. In practice, however,

data may continue to be revised forever, so the true value may never be

observed.