Functional data analysis for Brazilian term structure of interest rate.
| Autor | Vaz, Lucelia Viviane |
-
Introduction
The term structure of interest rates (TSIR) is a crucial tool to guide the decision-making process of investors, regulators, risk managers, and others. The database analyzed in this paper can be viewed as observations of a single random function, since the term structure of interest rates defines a relation between the yield of a bond and its maturity. The term structure of interest rates and its sources of variability provide essential information about monetary policy, interest rate risk factors, and fixed-income trading decisions. It is also important to understand the dynamics of bond portfolio management, derivatives pricing, and risk management, among other objectives. Recently, modeling of TSIR evolution has been an active area of research. Many authors seek components, typically additive, answerable for well-defined characteristics of the interest rate curves (Cox et al., 1985). A seminal paper on this topic is that of Litterman and Scheinkman (1991). Using principal component analysis, the authors identify three components that explain around 98% of the variability of US bond prices. These components affect movements in the interest rate curves' level, slope, and curvature.
Term structure models adopted by major central banks can be classified as parametric and spline-based models, according to the Bank for International Settlements (BIS, 2005). Almeida and Faria (2014) find that parametric models fit the yield curve in a parsimonious way. Such methods are typically used in macroeconomic studies, in which smoothness and the ability to capture common movements are as important as model accuracy. This class of models includes the three-factor exponential model of Nelson and Siegel (1987), its four-factor extension proposed by Svensson (1994), and their corresponding dynamic extensions proposed by Diebold and Li (2006) and Pooter (2007).
Spline-based models are made up of several low-order polynomials, which are smoothly linked over the range of maturities. Therefore, splines might estimate a larger number of parameters, with the correspondent fitting curves being less smooth than standard models. Important benchmarks in this class include McCulloch (1975) and Vasicek and Fong (1982). More recent extensions include the penalized spline models of Waggoner (1997) and Chava and Jarrow (2004).
The analysis in this paper is similar to that of Litterman and Scheinkman (1991), and we also estimate a functional linear regression model to investigate the macroeconomic determinants of the yield curves for Brazil. One innovation on Litterman and Scheinkman (1991) is to consider the set of observations as points of a smooth function. The real-valued covariates for the functional regression model include (a) industrial capacity utilization, (b) expected inflation (Broad National Consumer Price Index - IPCA), (c) Selic overnight interest rate of reference, (d) variation of the logarithm of the nominal exchange rate(BRL/USD) (nominal exchange rate, hereafter), and (e) Brazil risk (EMBI+). Our approach is to use functional principal component analysis (PCA) to identify the yield curves' level, slope, and curvature. We then compute the scores of the yield curves on each principal component, and use these scores to assess the relation of level, slope, and curvature to macroeconomic variables.
The main difference between this paper and that of Litterman and Scheinkman (1991) is that noise present in the data is corrected by imposing smoothness restrictions in the estimation. From an economic perspective, Inoue and Rossi (2019) point out that applying functional data models to term structure and its associated shift provides a more general way to study the impact of monetary policy shocks. Actually, the scalar shocks considered (exogenous movement in the short-term interest rate, forward guidance, and others) can lead to an exogenous shift in the entire yield curve associated with unexpected monetary policy decisions.
Our proposed framework follows a typical analysis of functional data. It begins by using smoothing techniques to represent each observation as a functional object. This first procedure can correct potential problems induced by measurement errors and other types of local disturbances. This is not the case when observations of the yield curves are treated as a multivariate data set (models for repeated measures, longitudinal data of mixed effects, and structural equations). As observed by Levitin et al. (2007), we then set aside the original data, and use the estimated curves for the functional regression model. More specifically, we use cubic spline interpolation to obtain monthly yield curves. After that, the curves form the set of dependent variables in a functional linear regression model with the covariates mentioned above.
The set of dependent variables is composed of curves from January 4, 2010 to December 20, 2018, obtained from Bloomberg. The basic elements of our database are the interest rates of interbank deposits. Price quotations are expressed as a percentage rate per annum, compounded daily, based on a 252-day year. The contracts are those expiring for t = 1,2, ..., 39 months ahead.
We find that a large source of the variability in interest rate curves is due to level shocks. Moreover, it is not affected by the nominal exchange rate. All other variables: expected inflation, Selic reference rate, Brazil risk, and industrial capacity utilization, are positively related to the yield curves' level. Similar results for Brazilian yield curves are presented by Fernandes et al. (2020).
Slope changes are the yield curves' second-largest source of variability. The slope is not associated with expected inflation. The relation between scores on the second principal component and macroeconomic variables is negative for the nominal exchange rate and expected Selic. This means that higher values of the latter variables are related to lower scores, that is, lower sloped curves. On the other hand, the relationship is positive for Brazil's risk and industrial capacity utilization.
The Selic overnight reference rate, Brazil risk, (1) and the nominal exchange rate positively affect the yield curvature. Meanwhile, industrial capacity utilization and expected inflation negatively affect curvature. In other words, higher values of the last two variables are associated with lower scores of the third principal component (curves with lower second derivative).
This paper is organized as follows. The second section deals with the methodology of functional models, while also detailing underlying theory. In the third section, we provide descriptive statistics of the databases and the analysis of principal components for the yield curves. The fourth section is devoted to the results of the functional linear regression model of the yield curves against macroeconomic covariates and the traditional linear regression of the scores on the principal functional components against macroeconomic covariates. Our conclusions are presented in the fifth section.
-
Empirical model
To deal with functional data, we must create a suitable representation of each functional object. Here, we use a cubic spline. In the mathematical appendix, we present important considerations. Each vector in [R.sup.n] is a column vector.
2.1 Functional linear model
Let us consider y := [{[y.sub.t]}.sub.t[member of]Z] wherein (2) each [y.sub.t] is a real-valued random function (3) with common domain (0,N]. More specifically, [y.sub.t] is the yield curve for month t and domain (0,39]. Set [uci.sub.t] the industrial capacity utilization, [ipca.sub.t] the average expected inflation, [selic.sub.t] the Selic interest rate, [exr.sub.t] the nominal exchange rate, and [br.sub.t] the Brazil risk. Define (4) [F.sub.t] as the information generated by {[uci.sub.t] [ipca.sub.t], [selic.sub.t], [exr.sub.t],[br.sub.t]}. We assume that
[Please download the PDF to view the mathematical expression] (1)
where [mu] (n) is a function that plays the same role as the constant in traditional regression models and the functions [[beta].sub.1](n), [[beta].sub.2](n), [[beta].sub.3](n), [[beta].sub.4](n), and [[beta].sub.5](n) are coefficients related to each variable.
2.2 Functional principal component analysis
Functional principal component analysis is a key technique to explore features characterizing functions, mainly when the variance-covariance and correlation functions can be challenging to interpret. For [R.sup.K]- valued data, the principal component analysis is based on the spectral decomposition of the...
Para continuar a ler
PEÇA SUA AVALIAÇÃOCOPYRIGHT GALE, Cengage Learning. All rights reserved.
