Real‐time inflation forecast combination for time‐varying coefficient models

• DOI: http://doi.org/10.1002/for.2563
• Author: Bo Zhang
• Published date: 01 April 2019
• Date: 01 April 2019

Received: 8 March 2018 Revised: 5 November 2018 Accepted: 10 November 2018

DOI: 10.1002/for.2563

RESEARCH ARTICLE

Real-time inflation forecast combination for time-varying

coefficient models

Bo Zhang

Research School of Economics, Australian

National University,Canberra, ACT,

Australia

Correspondence

Bo Zhang, Research School of Economics,

Australian National University,Canberra,

ACT 2601, Australia.

Email: bozhangyc@gmail.com

Abstract

Weuse real-time macroeconomic variables and combination forecasts with both

time-varying weights and equal weights to forecast inflation in the USA. The

combination forecasts compare three sets of commonly used time-varying coef-

ficient autoregressive models: Gaussian distributed errors, errorswith stochastic

volatility, and errors with moving average stochastic volatility. Both point fore-

casts and density forecasts suggest that models combined by equal weights do

not produce worse forecasts than those with time-varying weights. We also find

that variable selection, the allowance of time-varying lag length choice, and

the stochastic volatility specification significantly improve forecast performance

over standard benchmarks. Finally, when compared with the Survey of Profes-

sional Forecasters, the results of the best combination model are found to be

highly competitive during the 2007/08 financial crisis.

KEYWORDS

forecast combination, inflation forecasts, real-time data, time-varying coefficient model

1INTRODUCTION

Inflation is a core macroeconomic indicator that is closely

monitored by both central bankers and macroeconomic

researchers. Many researchers have consequently investi-

gated the time series properties of inflation. The consensus

from these studies is that the underlying trend and volatil-

ity of inflation have changed considerably over time; how-

ever, there is still no agreement on the best wayto forecast

inflation dynamics (e.g., Chan, 2013; Cogley & Sbordone,

2008; Koop & Korobilis, 2012; Stock & Watson,2007).

Since no single best model exists, forecast combina-

tion methods have proven to be a useful way for improv-

ing inflation forecast performance (e.g., Bunn, 1975;

Faria & Mubwandarikwa, 2008; Kascha & Ravazzolo,

2010; Raftery, Madigan, & Hoeting, 1997; Tibiletti, 1994).

Despite the existence of many sophisticated combination

methods—for example, the linear combination method,

the geometric combination method, the logarithmic com-

bination method, Bayesian model averaging, and com-

binations of experts' forecasts (Survey of Professional

Forecasters)—the empirical evidence suggests that fore-

cast accuracy is often best when simply averaging the

forecasts across the set of models (Timmerman, 2003). Two

commonly used types of averages are time-varying and

equal-weight methods. However, there is no consensus on

whether time-varying or equally weighted combinations

work better over a wide variety of models (e.g., Clark &

McCracken, 2009; Jore, Mitchell, & Vahey, 2010; Stock &

Watson, 2004).

With this in mind, our main objective in this paper is

to compare the forecast performance of both time-varying

and equally weighted combinations of a wide variety

of time series models with time-varying parameters and

various error structures. The main result is that models

combined by equal weights do not have worse forecast per-

formance than those with time-varying weights. We also

find that both combination strategies tend to provide better

Journal of Forecasting. 2019;38:175–191. wileyonlinelibrary.com/journal/for © 2018 John Wiley & Sons, Ltd. 175

176 ZHANG

forecast performance than univariate models in both point

forecasts and density forecasts. Finally,the forecast results

of our proposed models are also highly competitive when

compared with the quarterly reports from the SPF during

the financial crisis.

Another key contribution is that we investigate the tem-

poral relationship between inflation and other explana-

tory variables when conducting combination forecasts.

This is done by considering models with different pre-

dictors and lag structures. The present paper combines

forecasts based on one inflation predictor, which reduces

the number of models significantly. Furthermore, Koop

and Korobilis (2012) introduced dynamic model averag-

ing (DMA) and dynamic model selection (DMS), which

use a forgetting factor strategy to update time-varyingcoef-

ficients and averaging models with a set of explanatory

variables and different lag lengths. However, the compu-

tational burden must be considered when the number of

lags is greater than two with multiple explanatory vari-

ables. For instance, eight explanatory variables with three

lags could produce more than 400 million candidate mod-

els. Another interesting aspect of the forecasting results

from DMA and DMS is that high weights are given to

parsimonious models or parsimonious models that rarely

have more than two predictors selected. Moreover, quar-

terly data are generally widely used for inflation forecasts,

and most models use up to four lags (e.g., Clark & Ravaz-

zolo, 2015; Cogley & Sargent, 2005; Stock & Watson,2007).

In the present paper, the lag length can increase to four

without a heavy computational burden.

Finally, the present paper also compares the forecasting

results with the quarterly reports from the SPF.The results

of the proposed combination forecasts models are highly

competitive during the 2007/08 financial crisis. The SPF

collects forecasts from professional forecasters and their

forecasts are generally quite close to the actual values and

difficult to beat (Smith & Vahey, 2016; Tibiletti, 1994).

The remainder of the chapter proceeds as follows.

Section 2 describes the specifications of time-varying coef-

ficients models and the component models for inflation

forecast combination. Section 3 provides a brief introduc-

tion to the competing models for forecasts and the forecast

combination methodology. Section 4 discusses the fore-

casting results of the model combination for point and

density inflation forecasts. Section 5 concludes.

2COMPONENT MODELS

We consider three broad classes of time-varying coeffi-

cient models with different specifications of error terms:

(i) models with constant variance (TVC); (ii) models with

stochastic volatility (TVC-SV); and (iii) models with mov-

ing averagestochastic volatility (TVC-SVMA). Within each

class of model, parameters are estimated by different mea-

sures of economic activities, then they are employed in the

forecasting exercises. In addition, for each inflation predic-

tor, variouslag structures are considered, such as from one

to four single lags, and up to four more lags. By combin-

ing all lag structures with the eight predictors, this model's

averaging approach is conducted for TVC, TVC-SV and

TVC-SVMA. In the following subsections, the specifica-

tions of TVC, TVC-SV and TVC-SVMA are described first,

followed by the eight inflation predictor candidates.

2.1 Time-varying coefficient models

2.1.1 Constant variance

A generic TVC model can be described as a generalized

Phillips curve with time-varying coefficients:

t+k=1,t+Σ

n

=02+,txt−+

t,



t∼(0,2

),(1)

𝜷t=𝜷t−1+

t,



t∼(0,Q),(2)

𝜷t=1,t2,t··· n,t′,

Q0=

2

0𝜷1··· 0

⋮⋱⋮

0··· 2

0n

,Q=

2

1··· 0

⋮⋱⋮

0··· 2

n

,

where kis the forecasting horizon. Note that the sub-

scripts in the specification represent the order of the

coefficients and time points (e.g., 1,tto n,t), while the

superscripts indicate the underlying relationship between

variables (e.g., yand 𝜺y). xtis a vector of covariates that

may include lagged values of inflation or inflation predic-

tors. The model above incorporates both a time-varying

intercept and regression coefficients.

In Equation 2, the intercept and coefficients are assumed

to follow independent random walks (e.g., Clark, 2011;

Clark & Ravazzolo, 2015). By allowing the coefficients to

evolve gradually over time, this specification accommo-

dates a slowly changing relationship between inflation and

the explanatory variables. Atkeson and Ohanian (2001)

criticize the ability of the Phillips curve models to forecast

inflation compared with random walk forecasts. However,

the Phillips curve models they adopt all have constant

coefficients, and they do not perform as well as random

walk naive forecasts in some historical periods. It can be

expected that a time-varying Phillips curve performs better

than one with constant coefficients.

The covariance matrices Q0and Qof 𝜷0and 𝜷trespec-

tively are assumed to be diagonal matrices, where Q0and

𝜷0are initial values of Qand 𝜷t, respectively. It indicates

that 1,t…n,thave individual independent white noise