Forecasting the Term Structure of Interest Rates Using Integrated Nested Laplace Approximations

• Author: Luiz Koodi Hotta,MÁrcio Poletti Laurini
• DOI: http://doi.org/10.1002/for.2288
• Published date: 01 April 2014
• Date: 01 April 2014

Journal of Forecasting,J. Forecast. 33, 214–230 (2014)

Published online in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/for.2288

Forecasting the Term Structure of Interest Rates Using

Integrated Nested Laplace Approximations

MÁRCIO POLETTI LAURINI1AND LUIZ KOODI HOTTA2

1

FEA-RP USP, Ribeirão Preto, SP, Brazil

2

IMECC-Unicamp, Campinas, SP, Brazil

ABSTRACT

This article discusses the use of Bayesian methods for inference and forecasting in dynamic term structure models

through integrated nested Laplace approximations (INLA). This method of analytical approximation allows accurate

inferences for latent factors, parameters and forecasts in dynamic models with reduced computational cost. In the

estimation of dynamic term structure models it also avoids some simplifications in the inference procedures, such as

the inefficient two-step ordinary least squares (OLS) estimation. The results obtained in the estimation of the dynamic

Nelson–Siegel model indicate that this method performs more accurate out-of-sample forecasts compared to the meth-

ods of two-stage estimation by OLS and also Bayesian estimation methods using Markov chain Monte Carlo (MCMC).

These analytical approaches also allow efficient calculation of measures of model selection such as generalized cross-

validation and marginal likelihood, which may be computationally prohibitive in MCMC estimations. Copyright ©

2014 John Wiley & Sons, Ltd.

KEY WORDS term structure; latent factors; Bayesian forecasting; Laplace approximations

INTRODUCTION

Procedures for inference and forecasting in dynamic term structure models are usually computationally complex

because of the presence of latent factors and the high dimensionality of the parameter vector. The estimation method

used in the procedure can significantly affect the performance of the estimated model, especially in out-of-sample

forecasts.

As an example of this problem, Yu and Zivot (2011) evaluated the predictive performance of the dynamic for-

mulation of the Nelson–Siegel model proposed by Diebold and Li (2006) in forecasting yields of US Treasury and

corporate bonds. They concluded that different ways of estimating the same model can dramatically affect the pre-

dictive results for the various maturities and classes of assets studied. The estimation in a single step of this model

using a state-space formulation with the Kalman filter has better results for the prediction of high-yield bonds over

short horizons of time, while the simplest method using two-step ordinary least squares (OLS) has a superior perfor-

mance for investment-grade bonds and securities with shorter time horizons, confirming that the estimation method

used is not only important for the procedures of inference and hypothesis testing but can also significantly influence

the construction of forecasts.

The aim of this article is to compare the predictive performance of a Bayesian estimation method based on inte-

grated nested Laplace approximations (INLA) for the estimation of latent factors, parameters and forecasts in the

dynamic version of the Nelson–Siegel model proposed in Diebold and Li (2006), in relation to the two-step procedure

of Diebold and Li (2006) and to the estimation methods using Bayesian Markov chain Monte Carlo (MCMC) (e.g.

Migon and Abanto-Valle, 2007; Laurini and Hotta, 2010; Hautsch and Yang, 2012). The INLA method, proposed in

Rue et al. (2009), gives accurate analytical approximations for the posterior distribution of latent factors and param-

eters in generalized Gaussian models without the need for procedures such as numerical simulation methods based

on MCMC.

Forecasts constructed using Bayesian methods have several advantages over frequentist methods. As stated in

Geweke and Amisano (2010), Bayesian estimates incorporate the uncertainty of parameters in a consistent manner,

avoiding all approximations used in Diebold and Li (2006) for the estimation of parameters, latent factors and out-

of-sample forecasts.1

In particular, these computational advantages of the INLA method allow us to test the hypothesis of a fixed decay

parameter, which was assumed in Diebold and Li (2006) to linearize the model, by comparing with estimates by

models that allow this parameter to be time varying. Another advantage of this method is that it gives accurate

credibility intervals for the observed sample, in contrast to the method of Diebold and Li (2006), which does not

control for the estimation in two steps.

Correspondence to: Márcio Poletti Laurini, FEA-RP USP,Ribeirão Preto, SP, Brazil. E-mail: mplaurini@gmail.com

1See also Lahiri and Martin (2010), Bauwens et al. (1999) and Geweke (2010) for more properties of Bayesian forecasts.

Copyright © 2014 John Wiley & Sons, Ltd

Forecasting the TSIR using INLA 215

The INLA method also overcomes the limitations generated by the use of Monte Carlo simulation methods in the

Bayesian approach using MCMC, such as problems of convergence of Markovchains and limited rate of convergence

of Monte Carlo approximations, whose accuracy is limited by the pnrate of convergenceof Monte Carlo procedures.

Besides the higher accuracy obtained, this approach is computationally efficient and fast and allows measures of

comparison of models such as the marginal likelihood and cross-validated log score of the model to be calculated

efficiently, overcoming some difficulties in the existing computational methods of Bayesian estimation based on

Monte Carlo simulations.

The results indicate that the INLA method provides forecast gains over the methodologies based on the classical

two-step estimation by ordinary least squares and Bayesian estimation using MCMC, especially for longer maturities

of the yield curve, with a much lower computational time than the MCMC methods and almost equivalent to that

of two-step OLS. The results also confirm that, for the sample used in this study, the same database analyzed in

Diebold and Li (2006), assuming a constant decay parameter in the formulation of the Nelson–Siegel model does not

dramatically decrease the out-of-sample predictive performance.

This paper is organized as follows. The next section briefly summarizes the INLA method; the third section shows

the dynamic formulation of the model used in the Nelson–Siegel estimation; the fourth section shows the results of

the estimations and forecasts and the fifth section examines the effects of using a fixed decay parameter versus a time-

varying decay. In that section we propose two methods to select the decaying parameter value. Final comments are

made in the sixth section.

INTEGRATED NESTED LAPLAC E APPROXIMATIONS

The INLA method2was developed by Rue et al. (2009) for accurate Bayesian inference in models known as Gaussian

Markov random fields (GMRF), through analytical approximations for the hyperparameters and the latent factors ,

as an alternative to methods based on numerical simulations, such as MCMC. A latent GMRF model is a hierarchical

model with the first stage specifying a distribution for the observed variable y, usually assumed to be conditionally

independent given the latent factors and additional parameters , and the second stage specifying the evolution of

the latent factors :

.yj;/ DY

j

.yjjj;/;j 2J(1)

iDOffsetiC

f1

X

kD0

!kifk.cki /C´T

iˇCi;iD0;:::;

1(2)

where Jis a subset of the latent factors, such that yj;j2Jare the observed values; .yj; / is the likelihood

function of observed variables; is a vector of unstructured random effects; Offset is a possible fixed and known a

priori component to be included in the linear prediction; and !kare known weights for each observed point. The

function fk.cki/is the effect of generic covariates with value cki for observation i, and ´iis the effect of covariates

with linear effects, with corresponding parameter ˇ.

The formulation described by equation (1) can represent various statistical objects, such as generalized additive

models, generalized additive mixed models for longitudinal data, geoadditive models and especially time series mod-

els and models formulated in state space, such as dynamic linear and stochastic volatility models, as described in

Ruiz-Cárdenas et al. (2012). The INLA method allows us to approximate the posterior distributions of the latent

factors, denoted by

.ijY/DZ.ij;Y /.‚;Y /d(3)

and the marginal posterior distribution of hyperparameters, written as

.jjY/DZ.jY/dj(4)

using a sequence of analytical approximations based on Laplace methods for the full conditional distributions .jY/

and .ijY/and numerical integration for the hyperparameters ,wherejmeans the vector with the omitted

jth element. The INLA method of Rue et al. (2009) uses a sequence based on three steps to obtain these posterior

2This presentation follows Laurini (2013), Rue et al. (2009) and Ruiz-Cárdenas et al. (2012).

Copyright © 2014 John Wiley & Sons, Ltd J. Forecast. 33, 214–230 (2014)