Forecasting in long horizons using smoothed direct forecast

• DOI: http://doi.org/10.1002/for.2572
• Author: Yaein Baek
• Published date: 01 July 2019
• Date: 01 July 2019

Received: 6 July 2018 Revised: 9 November 2018 Accepted: 7 January 2019

DOI: 10.1002/for.2572

RESEARCH ARTICLE

Forecasting in long horizons using smoothed direct forecast

Yaein Baek

Department of Economics, University of

California, San Diego, CA

Correspondence

Yaein Baek, Department of Economics,

University of California, San Diego, 9500

Gilman Drive 0508, La Jolla, CA

92093-0508.

Email: yibaek@ucsd.edu

Abstract

This paper constructs a forecast method that obtains long-horizon forecasts with

improved performance through modification of the direct forecast approach.

Direct forecasts are more robust to model misspecification compared to iterated

forecasts, which makes them preferable in long horizons. However, direct fore-

cast estimates tend to have jagged shapes across horizons. Our forecast method

aims to “smooth out” erratic estimates across horizons while maintaining the

robust aspect of direct forecasts through ridge regression, which is a restricted

regression on the first differences of regression coefficients. The forecasts are

compared to the conventional iterated and direct forecasts in two empirical

applications: real oil prices and US macroeconomic series. In both applications,

our method shows improvement over direct forecasts.

KEYWORDS

forecasting, forecast comparisons, long horizon

1INTRODUCTION

In forecasting multiperiod time series we confront two

different methods. The “iterated” forecast specifies a

one-period-ahead model such as autoregression, then iter-

ates forward to obtain the multiperiod horizon forecast. In

contrast, the “direct" forecast has each horizon specified

in a model where the dependent variable on the left-hand

side of the regression is multiple periods ahead. The idea

of direct forecasting goes back to Cox (1961) and Weiss

(1991), where asymptotic properties of the direct forecasts

under general conditions are established. Direct forecast

methods are also used in estimating the impulse response

of a dynamic system, referenced as local projections by

Jordà (2005).

In theory, the iterative method would provide more

efficient estimates compared to the direct method if the

model is correctly specified. For instance, suppose we want

to forecast a time series by specifying an autoregression

model with four lags for estimation. If the data-generating

process (DGP) of this series is indeed a stationary autore-

gressive model with four lags or fewer, then the iter-

ated forecast will have smaller mean square forecast error

(MSFE) than the direct method. Under a Gaussian process,

the iterated method provides estimates that are asymp-

totically equivalent to the maximum likelihood estimator.

Analytic expressions are provided by Bhansali (1996,1997)

and Ing (2003) under a non-Gaussian assumption.

Instead of the true lags of four, suppose we use an

autoregressive model with two lags. Then the iterated

forecasts are biased and the compounding misspecifica-

tion error from recursive iteration would lead to larger

bias in long horizons. In contrast, direct forecast is more

robust to misspecification of an unknown DGP. The two

forecasting methods involve a bias and variance trade-off

(Pesaran, Pick, & Timmermann, 2011); thus a forecaster

who is particularly interested in predicting long horizons

would prefer the direct-forecast method over the iter-

ated approach. Furthermore, the direct method is flexible

because control variables can differ across horizons and

it is relatively easy to estimate for nonlinear dynamic sys-

tems.

Although obtaining long-horizon forecasts through the

direct forecast method seems attractive because of its

Journal of Forecasting. 2019;38:277–292. wileyonlinelibrary.com/journal/for © 2019 John Wiley & Sons, Ltd. 277

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robustness, direct method estimates tend to be erratic

across horizons. Because direct forecasting imposes less

structure than the iterated method, the obtained forecasts

are not “smooth” across horizons, contradicting what we

would expect in theory. For example, Figure 1 is a repli-

cation of Owyang, Ramey, and Zubairy's (2013) Figure 5,

which shows the response of government spending to a

news shock equal to 1% of gross domestic product, based

on quarterly data from 1920:Q1 to 2011:Q4 for Canada.

The direct method is used to estimate impulse responses

instead of standard vector autoregressions (VAR) due to

construction of government multipliers. The estimates

show jagged shapes across time whereas, in theory,

impulse responses are smooth. Hence a forecaster using

a direct method is likely to report unreliable multiperiod

forecasts.

This view motivates us to provide a smoothness mecha-

nism on direct forecasts to resolve the erratic behavior of

estimates. We expect improvement in long-horizon fore-

cast performance by imposing smoothness across multi-

period forecasts while maintaining flexibility of the direct

method.

The main goal of this paper is to develop a method

that can be implemented in direct forecasts to obtain

long-horizon forecasts with improved performance. A

smoothness prior is imposed across horizons of direct fore-

casts such that multiperiod forecasts show less jagged

shapes as the horizon length increases. The new method

is more robust to misspecification compared to the itera-

tive method, conducted through a restricted regression in

which we impose a smoothing parameter on the first dif-

FIGURE 1 Government spending response to a news shock: The

solid line is the impulse response estimated by the direct-forecast

method. The dotted line is the corresponding two-standard-error

band [Colour figure can be viewed at wileyonlinelibrary.com]

ferences of estimators, which is analogous to ridge regres-

sion (Hoerl & Kennard, 1970). The smoothing parame-

ter can be implemented as a prior distribution from a

Bayesian perspective. Shiller (1973) introduced the con-

cept of imposing a smoothness to the lag curve. We apply

our method to time series where long-horizon forecasts are

of interest: real oil prices and the US macroeconomic time

series from Marcellino, Stock, and Watson (2006). Both

results show that our method shows improvement over

the direct forecast approach in long horizons such as 3–5

years. For most series, forecasts based on our method have

uniformly less MSFE than direct forecasts across horizons.

The rest of this paper is organized as follows. Section 2

briefly introduces the direct and iterated forecast methods

and describes the estimation of our forecast model. Section

3 applies the method to forecasting real oil prices and the

macroeconomic series and evaluates performance. Section

4 includes an outline of further research and concluding

remarks.

2SMOOTHNESS MECHANISM ON

DIRECT FORECASTS

Section 2.1 introduces the difference in forecasts obtained

from the direct and iterated methods in addition to the

literature that compares the performance of the two fore-

casts. Section 2.2 describes the construction and intu-

ition of our forecast method, which is based on the

direct-forecast model. We explain how to choose the

smoothing parameter used for our estimation.

2.1 Direct versus iterated forecasting

Several researchers have evaluated the performance of

direct forecasts compared to iterated forecasts. Marcellino

et al. (2006) compared the performance of iterated and

direct forecasts using 170 US monthly macroeconomic

time series, spanning 1959–2002. A parametric bootstrap is

conducted by assuming an autoregression as the DGP.This

method allows examination of the spread of the distribu-

tion of MSFEs to see whether the direct method improves

on the iterated method, on average, over the population of

macroeconomic variables. Results show that iterated fore-

casts outperform the direct forecasts under correct speci-

fication. In contrast, Bhansali (1996) provided simulation

results in which the direct method had a smaller MSFE

than the iterated method if an underparametrized autore-

gressive model was fitted to a generated autoregressive

moving average process. Ing (2003) obtained asymptotic

expressions of MSFE of the two methods in an AR(p)pro-

cess with 1 ≤p≤∞and compared their performance.

If the fitted order kis such that k<p, the multistep