# Affine Term Structure Models: Forecasting the Yield Curve for Colombia/Modelos afines de la estructura a plazos de tasas de interes: Pronosticando la curva de rendimientos para Colombia/Modeles affines de la structure a termes des taux d'interet: Previsions de la courbe de rendements pour le cas de Colombie.

Abstract: Superior modeling of the yield curve is useful for asset pricing, financial planning, and risk management. In this article, we estimate five affine term structure models using daily data for Colombia. We find that a three-factor model outperforms the other models in one and five day ahead forecasts. The model factors closely mimic empirical proxies for the level, the slope, and the curvature of the yield curve in Colombia.Keywords: term structure, forecasting, interest rates, multifactor models

Classification JEL: G12, G17

Resumen: Modelar mejor la curva de rendimientos es util para la valoracion de activos, la planeacion financiera y la administracion de riesgos. En este articulo se estiman cinco modelos afines de la estructura a plazos de tasas de interes para Colombia usando datos diarios. Se encuentra que un modelo de tres factores tiene un desempeno superior a los demas modelos para pronosticos intramuestrales y para pronosticos (fuera de muestra) con horizontes de uno y cinco dias. Eos factores del modelo se asemejan a sus contrapartes empiricas del nivel, la pendiente y la curvatura de la curva de rendimientos de Colombia.

Palabras clave: estructura a plazos, pronosticos, tasas de interes, modelos multifactoriales

Clasificacion JEL: G12, G17

Resume: Modeliser la courbe de rendements est toujours une tache utile pour mieux evaluer les actifs, la planification financien et la gestion du risque. Cet article estime cinq modeles affines de la structure a termes des taux d'interet pour le cas de Colombie, lesquels sont estimes a partir des donnees quotidiennes. Nous montrons que l'un des modeles est beaucoup plus performant que les autres, en ce qui concerne les previsions etablies avec des echantillons, tandis que les previsions etablis en dehors des echantillons ont un horizon compris entre un et rinqjours. Ees modeles reproduisent bien les donnees empiriques de Colombie concernant le niveau, la pente et la courbure de la courbe de rendements.

Mots-cles: structure a termes, previsions, taux d'interet, modeles multifactonels

Classification JEL: G12, G17

Introduction

The term structure of interest rates (or yield curve, for short) plays a central role in an economy. Current yields have useful information for forecasting future short yields and, potentially, real economy activity, inflation, and other key economic variables (Piazzesi, 2010). Market participants use these forecasts for pricing financial assets, taking investment decisions, and managing financial risks. Central banks use them to inform monetary policy. Consumers use them to make saving and consumption decisions. Thus, superior modeling and forecasting of the yield curve serve policymakers in evaluating past, current, and future economic conditions and help market participants and consumers in taking better financial decisions.

In this paper, we model and forecast the daily yield curve for Colombia using non-arbitrage affine term structure models (ATSMs). The affine term structure modeling framework dominates the theoretical and empirical literature on term structure models (Piazzesi, 2010). To our knowledge, this is the first study to test the in-sample fit and out-of-sample forecasting capabilities of ATSMs using data for Colombia, which is a necessary step to determine the usefulness of ATSMs.

ATSMs specify the risk-neutral evolution of some unobservable factors responsible for the dynamics of the yield curve by making the yields of different maturities an affine (linear) function of those factors. These models provide a flexible structure for examining the dynamics of zero-coupon bond yields by ruling out arbitrage opportunities. They consistently link the cross-sectional and time-series properties of the yield curve (Piazzesi, 2010). The no-arbitrage conditions, in turn, improve the efficiency of the estimates.

Using daily data from 2002 to 2015, we estimate a battery of ATSMs following the estimation methods presented in Ait-Sahalia and Kimmel (2010). We find that a three-factor Gaussian model fits the data and forecasts the daily yield curve for Colombia remarkably well. As Dai and Singleton (2000) point out, Gaussian models are fully flexible regarding the signs and magnitudes of conditional and unconditional correlations of the underlying factors but at the cost of assuming constant conditional variances. (1) We find that allowing for conditional heteroscedasticity in the analyzed ATSMs has little effect on the accuracy of the forecasts for our sample data and severely complicates the maximization of the log-likelihood functions. For our daily data, the three-factor ATSM outperforms the one- and two-factor ATSM. The root mean squared errors (RMSE) of in-sample, one, and five day ahead forecasts of average yields are below twenty basis points. This makes the three-factor ATSM especially appealing for pricing financial instruments, taking investment decisions, and managing financial risk. It also offers the possibility of investigating the yield curve dynamics at higher frequencies than those traditionally used in the literature.

We show that the factors of the estimated three-factor ATSM closely mimic three widely used empirical proxies for the level, the slope, and the curvature of the yield curve. In particular, the third factor has a strong correlation with an average of the short- and long-run zero coupon yields. The second factor is highly correlated with the slope of the yield curve computed as the difference between ten-year and three-month interest rates. The first factor is highly correlated with an empirical proxy of the curvature computed as twice the four-year yield minus the slope of the yield curve. In addition, principal component analyses produce similar looking level, slope, and curvature factors as those of the three-factor ATSM. Our results are robust to the choice of estimation periods, the use of non-smoothed zero coupon bond yields to estimate the models, and the use of data at lower frequencies (e.g., weekly and monthly).

The article is organized as follows. Section I reviews the previous literature. Section II presents the ATSMs. Section III explains the methodology we use for estimation and forecasts. Section IV describes the data. Section V presents and discusses the main empirical results. The last section concludes.

I. Literature review

Before ATSMs, the dominant framework for explaining the term structure of interest rates was the expectation hypothesis, according to which expected returns are constant over time (Campbell, 1986). The liquidity preference and the preferred habitat theories of the term structure of interest rates can be seen as extensions of the expectation hypothesis, making additional predictions regarding term premiums as a function of the time to maturity of zero coupon bonds. Most tests of the expectation hypothesis reject the existence of constant risk premiums (Campbell & Shiller, 1991; Fama & Bliss, 1987). Thus, modeling time-varying risk premiums is at the heart of ATSMs.

ATSMs date back to the work of Vasicek (1977) and Cox, Ingersoll and Ross (1985). Duffie and Kan (1996) study ATSMs in detail and show how yields for every maturity can be represented as affine functions of some unobserved factors-latent variables. Dai and Singleton (2000) define a canonical representation of ATSMs according to which an [A.sub.m](N)-ATSM includes N factors, m of which affect the conditional volatility of the other factors. For instance, an [A.sub.0] (3)-ATSM implies that three homoscedastic latent factors explain the dynamics of the yield curve. Likewise, an [A.sub.1] (3)-ATSM implies that three latent factors explain the dynamics of the yield curve and that one of the factors determines the conditional volatility of all of them.

Ang and Piazzesi (2003) estimate a discrete Gaussian ATSM incorporating two observable macroeconomic factors: inflation and real activity. Their study finds that macroeconomic factors highly determine the movements of the short and middle ends of the curve, while latent factors are more influential in long yields. They also show, using out-of-sample forecast comparisons, that no-arbitrage cross-equation restrictions make ATSMs more accurate than unrestricted VARs, and that the inclusion of macroeconomic factors further improves their performance. We contribute to the literature in Colombia by estimating a benchmark model without macroeconomic factors as a natural step for future research incorporating them.

Estimating ATSMs is challenging. There exist different methods in the literature (e.g., quasi maximum likelihood, Kalman filtering, simulation, and method of moments, among others). Duan and Simonato (1999) propose a state-space representation of ATSMs and approximate the conditional mean and variance under the assumption that the diffusion process is Gaussian. They estimate the latent factors using the Kalman filter. This allows them to evaluate the likelihood function (quasi likelihood function for non Gaussian models) and estimate the parameters of various ATSMs.

Brandt and He (2006) argue that the quasi maximum likelihood method is skewed for multi-factor models. They present a correction for the quasilikelihood function, which is obtained by simulation and converges to the real likelihood function. This method reduces the skewness and variability of the estimated parameters, but it is computationally intensive.

More recently, Ait-Sahalia and Kimmel (2010) propose a new method to estimate ATSMs. They use closed-form approximations of the log-likelihood functions for the state variables following the methods presented in Ait-Sahalia (2008). They find that their proposed method generates superior parameter estimates.

ATSMs have been used for several purposes. For instance, Singleton and Umantsev (2002) price options on coupon bonds and swaptions using ATSMs. Ho, Huang and Yildirim (2014) propose a method for pricing inflation-indexed derivatives based on ATSMs. Duffee (2002) uses ATSMs to analyze the behavior of expected excess returns. Durham (2006) uses ATSMs to model observed and unobserved components of nominal U.S treasury curves and estimate inflation risk premiums.

Another branch of models currently being used for representing the yield curve are the dynamic models derived from the Nelson-Siegel equation, originally proposed by Nelson and Siegel (1987) as a curve-fitting tool. Diebold and Li (2006) re-parametrize the original model to depend on three dynamic factors associated with level, slope, and curvature. They produce forecasts by fitting autoregressive models to these factors and compare them to several other models. While the proposed model performs poorly for short forecast horizons (1 month), results improve as the horizon is enlarged.

These models are further analyzed by Diebold, Rudebusch, Glenn and Aruoba (2006), who include macroeconomic factors in their regressions and allow for correlated dynamic factors. They use variance decompositions to assess the effects of latent and macroeconomic factors on the different ends of the curve, finding a greater influence of the observable factors in short yields.

In a more recent study, Christensen, Diebold and Rudebusch (2011) adjust the dynamic Nelson-Siegel models (both in their independent and correlated factor versions) to be arbitrage free. They derive a model similar to traditional ATSMs, but in which the coefficients of the affine functions that describe yields match the Nelson-Siegel factor loadings. Their results show that the correlated specification has a superior in-sample performance, while the simpler model of uncorrelated factors produces better out-of-sample fore-casts. They also suggest that no-arbitrage restrictions improve the forecast accuracy of the models.

This kind of dynamic Nelson-Siegel models has been widely accepted amongst practitioners. It has paralleled the development of ATSMs as dynamic models of the yield curve. The main difference between these two types of models hinges on the no-arbitrage restrictions; but recent studies, such as Christensen et al. (2011), have brought them closer together. A complete review of dynamic Nelson-Siegel models can be found in Diebold and Rudebusch (2013).

For Colombia, most studies model the yield curve focusing on interpolation and curve-fitting of cross-sectional data. The Nelson-Siegel model and cubic splines are the main methodologies used. Few studies use dynamic models, most of them adopting the dynamic Nelson-Siegel framework. Melo-Velandia and Castro-Lancheros (2010) use the methodology in Diebold, Rudebusch, Glenn and Aruoba (2006) to relate monthly yield data for Colombia to macroeconomic factors. They find a strong relation between the model factors (level, slope, and curvature) and the interbank rate, inflation, the GDP gap, and the EMBI. They show, using Granger causality tests, that macroeconomic variables affect the yield curve factors. Maldonado-Castano, Zapata-Rueda and Pantoja-Robayo (2014) use the re-parametrization of the Nelson-Siegel model presented in Diebold and Li (2006) and apply the Kalman filter to estimate and forecast its factors.

Restrepo-Tobon and Botero-Ramirez (2008) calibrate one-factor arbitrage-free interest rate models to daily yield curves in Colombia. Their study concludes that these types of models can closely represent Colombia's term structure of interest rates. To our knowledge, no study has extended these results to multi-factor models and out-of-sample forecasts. We intend to examine these generalizations and forecast capabilities using arbitrage-free ATSMs with daily data for Colombia.

II. Affine term structure models

We denote the yield of a zero-coupon bond with maturity [tau] by [[gamma].sub.[tau]]. ATSMs assume that the short-term interest rate is an affine function (2) of a state vector X(t) of N underlying factors, which can be observable (macroeconomic variables) or latent (Piazzesi, 2010). Thus,

[mathematical expression not reproducible] (1)

with [[delta].sub.0] [member of] R and [[delta].sub.1] [member of] [R.sup.N]

The state vector is assumed to follow an affine diffusion process under the risk-neutral measure Q, that is,

[mathematical expression not reproducible] (2)

where [kappa],[SIGMA] [member of] [R.sup.N x N], [THETA] [member of] [R.sup.N], W is an N-dimentional independent brownian motion and S(t) is a N x N diagonal matrix with entries

[[S (t)].sub.i,i] = [[alpha].sub.i] + [[beta].sup.T.sub.i] X (t) (3)

with [[alpha].sub.i] [member of] R and [[beta].sub.i] [member of] [R.sup.N].

The market price of risk [LAMBDA](X) [member of] [R.sup.N] is also specified in order to obtain the physical dynamics. Following the literature, we assume that [LAMBDA](t) = [square root of S(t)[lambda]] where [lambda] is a vector of constants (see Dai & Singleton, 2000). Thus, the state process is also affine under the physical measure P (Duffie & Kan, 1996), that is,

[mathematical expression not reproducible] (4)

Under this structure, Duffie and Kan (1996) show that the yield for any maturity [tau] can be obtained as an affine function of the state vector, that is,

[[gamma].sub.[tau]] (t) = A([tau]) + B[([tau]).sup.T] X (t). (5)

The coefficients B ([tau]) and A ([tau]) are the solution to the following system of differential equations:

[mathematical expression not reproducible] (6)

[mathematical expression not reproducible] (7)

with a(0) = 0, 6(0) = 0, A([tau]) = -a([tau])/[tau], and B([tau]) = -b([tau])/[tau]. These equations come from imposing no-arbitrage restrictions (Duffie & Kan, 1996).

Dai and Singleton (2000) propose a canonical representation in which [SIGMA] is an identity matrix. We adopt their notation and representations. As an example, the following equations specify the physical dynamics of three-factor models:

The [A.sub.3] (3) model:

[mathematical expression not reproducible] (8)

The [A.sub.2] (3) model:

[mathematical expression not reproducible] (9)

The [A.sub.1] (3) model:

[mathematical expression not reproducible] (10)

The [A.sub.0] (3) model:

[mathematical expression not reproducible] (11)

See Ait-Sahalia and Kimmel (2010) for a full specification of all the models mentioned in this paper.

III. Methodology

To estimate the models, we follow the technique for maximum likelihood estimation of ATSMs proposed in Ait-Sahalia and Kimmel (2010) and Ait-Sahalia (2008). They approximate the log-likelihood function of ATSMs by a series of highly accurate expansions for the conditional distributions of the state processes. The resulting density expansion from this approach is in closed form. To estimate the parameter vector 8 of an [A.sub.m](N) model, N yields in a panel data of bond yields are assumed to be observed without error. All others yields are assumed to be observed with independent Gaussian errors.

A. The log-likelihood function

For a given parameter vector [theta], Equation (6) gives A([tau]) and B([tau]) for all [tau]. (3) We then form a vector [[GAMMA].sub.0]([THETA]) (N x 1), whose elements are A([tau]), and a matrix [GAMMA]([theta]) (N x N), whose columns are B([tau]), for all [tau] (maturities) observed without error. The same is done for maturities observed with errors, obtaining [[GAMMA].sub.e0] ([theta]) and [[GAMMA].sub.e]([theta]).

A vector containing yields observed without errors [[gamma].sub.ne](t) can then be expressed as

[[gamma].sub.ne](t) = [[GAMMA].sub.0]([theta]) + [GAMMA][([theta]).sup.T] X(t) (12)

using Equation (5). Equation (12) is a linear system with N equations and N unknown variables, which allows us to obtain time series of the values of each state variable in X (t).

Using these estimated state variables, we calculate the estimated yields observed with error using Equation (5).

As in Ait-Sahalia and Kimmel (2010), we denote by p X ([DELTA], x | [x.sub.0]; [theta]) the conditional density of X(t + [DELTA]) = x given X(t) = [x.sub.0] and [p.sub.[gamma]] ([delta],[gamma] | [[gamma].sub.0]; [theta]) the transition function of the vector of yields observed without errors. It follows from Equation (12) that

[mathematical expression not reproducible] (13)

where X(t) = ([GAMMA][([[theta]).sup.T]).sup.-1] ([[gamma].sub.ne] (t) - [[GAMMA].sub.0] ([theta])). As the conditional density of the state process is not known in closed form in most models, we use the approximations introduced in Ait-Sahalia (2008). The log-likelihood of yields observed without errors at times [t.sub.0],...,[t.sub.n] with constant time steps of [DELTA] is then:

[mathematical expression not reproducible] (14)

Estimates for yields observed with error are obtained using X (t) and Equation (5). Errors are obtained by comparing the estimates with their observed values and their log-likelihood is computed using the Gaussian density function. The final log-likelihood function, which we maximize to obtain parameter estimates, is computed by adding the log-likelihood of observation errors with the log-likelihood of yields observed without error. Figure 1 summarizes this process.

B. Optimization procedure

The log-likelihood functions of most ATSMs present multiple local maxima. This makes traditional optimization methods (e.g., gradient based methods) unreliable. In addition, the domain of feasible parameters is restricted for various models. This makes the estimation procedure a difficult task. We turn to heuristic algorithms to maximize the log-likelihood function. In particular, we use the 'differential evolution' heuristic approach (Storn & Price, 1997) given its ability to search for optimal parameter values in a continuous space. The algorithm works as follows:

Algorithm 1. Differential evolution Data: # Generations:ng, Population size:np, F [member of] [0, 2], CR [member of] [0, 1] P1 [left arrow] Random initial population for i = 1 to ng do P0 [left arrow] P1 for j = 1 to np do {a, b, c} [left arrow] random individuals from P0 V [left arrow] a + F * (b - c) for k = 1 to #Params do if rand [less than or equal to] CR then [U.sub.k] [left arrow] [V.sub.k] else [U.sub.k] [left arrow] P0[(j).sub.k] end end if f obj(U) [less than or equal to] P0(j) then P1(j) [less than or equal to] U end end end Source: authors' elaboration.

Solutions (sets of parameter values) are treated as vectors. We represent the j-th solution of a population P as P (j), and its i-th parameter value as P[(j).sub.i]. 'Evolution' is recreated by comparing individuals (solutions) from an initial population with new ones and preserving the best. New individuals are generated as linear combinations of individuals from the initial population. Before being compared with the initial individual, they can 'mutate' by changing some of their parameter values with a given probability. This 'evolutionary' process is repeated over numerous generations and the best individual from the last population (according to a given objective function) is taken as the final solution. (4)

C. Forecasts

ATSMs have mostly been used for monthly and yearly studies of the term structure of interest rates. We intend to evaluate their performance in forecasting the yield curve for Colombia at higher frequencies. Short-horizon forecasts of yield movements can be of help to investors and portfolio managers in building strategies and managing risk. For instance, accurate forecasts could give trading signals or be used to compute measures such as the value at risk of a portfolio.

With the estimated parameters, we find the 'true' state value for the last 'known' date (determined by the horizon) using Equation (5). We then simulate 10,000 state trajectories from this date up to the desired forecast date using Euler's numeric scheme on Equation (4). For each trajectory, we use Equation (5) to obtain yields for all maturities (observed with and without error). We take our forecast value as the mean of the 10,000 simulated points for every maturity.

We also report the simulated 95% confidence intervals for the forecast yields. The limits are obtained by computing the 2.5 and 97.5 percentile points of the 10,000 simulated values for every yield. We repeat this procedure for all days in the validation sample.

IV. Data

We use daily zero coupon yields estimated using the Nelson-Siegel model obtained from Infoval between January 8 2002 and February 3 2015. The sample includes 3.051 trading days. We use the first 2,000 observations for estimation, reserving the rest for out-of-sample forecasts and validations. Summary statistics for this dataset are presented in Table 1.

We consider maturities of 0.25, 0.5, 1, 2, 4, 5, 7, 8, 9 and 10 years. As pointed out above, for every [A.sub.m](N) model, we assume that the N yields observed without errors are:

* 3 months yield for one-factor models.

* 3 months and 4 years yields for two-factor models.

* 3 months, 4 years and 10 years yields for three-factor models.

The remaining yields are assumed to be observed with independent Gaussian errors.

In order to check the robustness of the models, we also work with a data set of bootstrapped zero rates obtained from Bloomberg[R]. These zero coupon yields are non-smoothed counterparts of the Nelson-Siegel yields. We use daily observations from April 29 2005 to May 22 2015. The first 1,700 observations are used for estimation. Table 2 presents summary statistics for the bootstrapped yield data set.

V. Empirical results

We estimate nine ATSMs ranging from one to three factors. Following the notation in Dai and Singleton (2000), we consider the following models: [A.sub.M](N) with N, M [member of] {1, 2, 3} and M [less than or equal to] N.

The estimation of the models [A.sub.2] (2), [A.sub.1] (3), [A.sub.2] (3) and [A.sub.3] (3) does not converge using our data. This can be due to the higher complexity that the feasible solution space acquires as conditional volatility is introduced. We tested numerous optimization procedures which either found no feasible solutions or stagnated on initial parameter values. Out of the tested methodologies, differential evolution performed best.

We report results for the models [A.sub.0] (1), [A.sub.1] (1), [A.sub.0] (2), [A.sub.1] (2) and [A.sub.0] (3).

A. In-sample Fit

The RMSE for the in-sample fit of each model and maturity is presented in Table 4. The parameter values used for the differential evolution heuristic are presented in Table 3. A plot of the in-sample fit for the model with the lowest RMSE is presented below (all others can be found in Appendix).

Figure 2, when compared with Figures A1-A4 (see the Appendix), depicts how the [A.sub.0] (3) model outperforms all other models based on its ability to fit the data. Table 4 shows that the in-sample RMSEs for this model are below 18.31 basis points for all maturities. Short and middle-maturity yields obtain higher in-sample errors because of their more platykurtic distributions (Tables 1 and 2). For long-term maturities, the RMSEs are below 7.31 basis points. Thus, based on the RMSE criterion, the [A.sub.0](3) model is the best model, which implies that, out of the models taken into account, a three-factor homoscedastic structure best describes the evolution of the yield curve in Colombia.

B. Out-of-sample forecasts

We conduct out-of-sample forecasts for one and five days. Tables 5 and 6 report the RMSEs between the mean forecast (of the 10,000 simulated trajectories) and the observed values. The [A.sub.0](3) model outperforms the other models considered in forecasting out-of-sample yields. RMSEs for in-sample, one- and five-day ahead forecasts are below 20 basis points. Again, shorter yields (0.5, 1 and 2 years) present higher errors, while longer maturities are more stable and easier to forecast.

The [A.sub.0] (2) model performs marginally better than the [A.sub.0] (3) in forecasting some of the shorter maturities. However, RMSEs for longer yields ([tau] > 5) are considerably bigger for the [A.sub.0] (2).

Apart from the [A.sub.0] (2) and [A.sub.0] (3), the models are not good at forecasting the specified maturities. One-factor models fall short when adjusting a high number of yields. In addition, the [A.sub.1] (2) model presents very large errors, which might be due to complications in the estimation procedure introduced by conditional heteroscedasticity.

Figures 3-6 show simulated confidence intervals for states and yields fore-casts. In line with previous results, short yields have wider confidence intervals and sometimes deviate from them. Longer maturities have narrower intervals and follow them more consistently.

C. Robustness tests

All the results discussed earlier are based on a data set of yields extracted from market data using the Nelson-Siegel method. These yield curves are smoother than what is normally observed in the market. In order to test the robustness of our results, we also use non-smoothed data obtained from zero coupon yields constructed using the bootstrap method, also known as the non-smoothed Fama-Bliss method (Fama & Bliss, 1987), which iteratively builds the discount rate function by computing the forward rates necessary to price successively longer maturity bonds." (5)

With the bootstrapped yields, our results change little (Tables 4-6). However, both the in- and out-of-sample forecasting errors tend to be slightly higher. The [A.sub.0] (3) model still outperforms the other models based on in-and out-of-sample forecasts for all maturities. However, the [A.sub.0] (2) model obtains lower errors than the [A.sub.0] (3) in forecasting shorter maturity yields. Similar to the Nelson-Siegel yields, the [A.sub.1] (1) model has lower RMSEs than the [A.sub.0] (1). The big errors in the [A.sub.1] (2) model also persist.

In order to evaluate the [A.sub.0] (3) model forecast capabilities, we compare its out-of-sample results with a random walk benchmark. We choose this simple model as a benchmark because it has remained hard to beat by models in the literature. For instance, Ang and Piazzesi (2003) find that it outperforms unrestricted vector auto-regressions (VARs), and only manage to slightly beat it (although not for all maturities) using an arbitrage-free model with macroeconomic factors. Duffee (2002) also documents random walks beating ATSMs' forecast capabilities. The one-day and five-day out-of-sample forecast RMSE comparisons are presented in Table 7.

Consistent with the literature, the [A.sub.0] (3) forecasts are outperformed by random walks. While the shorter end of the curve presents bigger differences in its forecast RMSEs, yields with maturities over four years obtain RMSEs less than two basis points over the random walk benchmark for both one-day and five-day ahead forecasts with the Nelson-Siegel data set.

The higher forecast errors for short yields can be attributed to the lack of macroeconomic factors in our setting. The literature shows macroeconomic factors may play a role in forecasting short and mid-maturity yields (Ang & Piazzesi, 2003).'

To further evaluate the performance of the [A.sub.0] (3) model against a random walk over time and maturity, we analyze the cumulative squared prediction error (CSPE), a metric introduced in Welch and Goyal (2008). The CSPE compares the forecasting performance of a model against a benchmark over time. Calling [[gamma].sub.i] the measured value of a yield observed at time i and [??], [mathematical expression not reproducible] (3) its forecast values by the random walk and [A.sub.0] (3) models, respectively, we compute the CSPE as follows:

[mathematical expression not reproducible] (15)

A positive slope in this metric means the [A.sub.0] (3) model outperforms the random walk over a period of time. We obtain the CSPE for all the maturities over the entire validation sample. Figure 7 presents the CSPE obtained for one-day and five-day forecast horizons using the [A.sub.0] (3) model estimated with daily observations.

In line with the results from Table 7, all the CSPEs from Figure 7 have a negative value at the end of the validation sample indicating that the random walk model has better accuracy overall. However, several interesting results can be drawn from the behavior of the CSPE metric over time and maturity. For one-day forecasts, CSPEs are very close to zero for the set of maturities which were assumed to be observed without error for the estimation of the model (0.25, 4 and 10 years). This is not the case for the rest of maturities, which have a steady decline in their CSPEs through most of the sample. This marked difference made by the assumption of an observation error in the estimation procedure repeats itself in the five-day forecasts: the [A.sub.0] (3) is more accurate when forecasting yields observed without error, even surpassing the random walk model for a long period of time around the end of 2012.

When analyzing the effect of maturity on the CSPE, the difference between short and long yields is highlighted again. Yields with maturities of less than four years have a much steeper descend in their CSPE than longer yields. This causes them to have significantly lower final values in both one-day and five-day forecasts. CSPEs for longer yields fall at a much slower rate and even stagnate after 2012, indicating the [A.sub.0] (3) model and the random walk are similarly accurate from 2013 to 2015.

In order to check the robustness of our results to the frequency of observation, we re-estimate the [A.sub.0] (3) model using weekly and monthly observations from the Nelson-Siegel dataset. We follow the same estimation and forecast methodologies that were used for daily observations. Table 8 reports the results for weekly data and Table 9 reports the results for monthly ones. Forecasts are obtained with a one-period ahead horizon (one week and one month, respectively).

Forecast RMSEs are below 26 basis points for the weekly estimations and below 38 basis points for the monthly estimations. As was the case with daily estimations, short term yields have higher errors in the weekly results, but this is not the case with the monthly forecasts.

We also compute CSPEs for both weekly and monthly forecasts. These results are presented in Figure 8. The distinction between yields assumed to be observed with and without error during estimation seems to lose relevance at these lower frequencies. For instance, the four-year yield CSPEs resemble those from the five-year yield very closely which is not the case for daily observations (see Figure 7). Maturity is the differentiating factor in Figure 8, with short yields again having poor performance but yields with maturities over four years showing better accuracy with the [A.sub.0] (3). One-month forecasts CSPEs also show better behavior than weekly ones, suggesting that lower frequency facilitates forecasting.

Although most studies focus their analyses and forecast tests on lower frequencies, we compare the out-of-sample RMSEs presented here with a few results from the literature as follows:

* Duffee (2002) reports RMSEs ranging from 28 to 52 basis points when forecasting U.S. yields with maturities up to 10 years with a three-month forecast horizon using various ATSMs and 'essentially affine' term structure models, which are a more flexible variation of ATSMs.

* Ang and Piazzesi (2003) also forecast U.S. yields with maturities up to 5 years and obtain RMSEs ranging from 18 to 30 basis points, making out-of-sample one-month forecasts and updating estimations at every observation. They manage to lower their errors by including macroeconomic factors in their model.

* Maldonado-Castano et al. (2014) use a dynamic Nelson-Siegel model, estimated by Kalman filtering, to produce one-day yield forecasts for the Colombian market. They achieve RMSEs ranging from 21 to 57 basis points for yields with maturities of 3 months, 3 years and 13 years.

Overall, our forecast RMSEs compare well against these results. Our monthly errors are in the range delimited by Duffee (2002), which should be taken as an upper bound considering the longer forecast horizon. One-month RMSEs from Ang and Piazzesi (2003) are lower, but close to our results. Finally, our one-day forecasts achieve lower errors than those reported by Maldonado-Castano et al. (2014) using Colombian data.

We also compare the modeled latent factors from the [A.sub.0] (3) model to three empirical proxies for the level, the slope, and the curvature of the yield curve. Following Diebold and Li (2006), we take the following proxies for the empirical factors:

* Level: ([[gamma].sub.0.25] + [[gamma].sub.4] + [[gamma].sub.10])/3

* Slope: [[gamma].sub.0.25] - [[gamma].sub.10]

* Curvature: 2[[gamma].sub.4] - [[gamma].sub.10] - [[gamma].sub.0.25]

We compare these proxies with the modeled latent factors from the [A.sub.0] (3) model estimated with the Nelson-Siegel dataset. Results for the non-smoothed yields data set are similar. Table 10 presents correlation coefficients between the estimated [A.sub.0] (3)-ATSMs latent factors and the empirical proxies for the level, the slope, and the curvature of the yield curve. The three estimated factors have high correlations with the level of the yield curve. The empirical proxies for the slope and the curvature show high correlations with the first and second estimated factors, respectively. Figure 9 depicts these relations.

We also compare the three [A.sub.0] (3) estimated factors with the first three principal components of the yield curve. Table 11 presents their corresponding correlation coefficients. Figure 10 depicts their relations. As with the empirical proxies, there is a strong correlation between the estimated factors and the three first principal components, which together explain close to 96% of the variance of the yield curve.

Conclusions

We estimate five ATSMs using daily data for Colombia. To our knowledge, this is the first paper applying ATSMs to the Colombian bond market. Our main empirical results indicate that a homoscedastic three-factor ATSM fits the data remarkably well. Our results hold under a series of robutsness tests, which include using an alternative data set and data observed at lower frequencies (e.g., weekly and monthly).

We find that the three estimated factors for the ATSM closely mimic the behavior of three empirical proxies for the level, the slope, and the curvature of the yield curve. According to principal components analyses, these estimated factors account for about 96% of the yield curve variance. One- and two-factor ATSMs are unable to describe the behavior of the yield curve in Colombia and should not be used in practice.

Our forecasts have similar errors to those reported in past studies and are close to the random walk benchmark. We encounter larger forecast errors in the short end of the curve, which are highly influenced by macroeconomic factors. Our results can serve as a benchmark for future research in which observable macroeconomic variables could be used. We think this is a natural next step in our research.

We find evidence supporting the assumption of normally distributed measurement errors on some of the yields disturbs forecasts for said maturities at high frequencies. We hypothesize that as our current estimation methodology is only concerned with the distribution of these errors, but not their minimization, other methodologies could yield better forecast results. These errors become less significant as the forecast horizon enlarges possibly due to the increment of variability between observations. This may explain why the existing literature, which focuses on lower frequency analyses, has not yet found this assumption to be problematic.

Therefore, the estimation of ATSMs applying different methodologies from the one used in this paper is also a necessary future development of our research. Because of its complexity, the literature on estimation of ATSMs is growing quickly. It is important to assess whether our results and choice of model hold using novel estimation procedures and to find less computationally intensive routines that can ease the use of ATSMs in the Colombian market.

Our work opens the possibility for future research on the relation between macroeconomic factors and the behavior of the yield curve using ATSMs. A promising area for further work is the identification of empirical macroeconomic factors not spanned by the yield curve and that can potentially be useful in forecasting it.

Appendix

Plots of the in-sample fit for the [A.sub.0] (1), [A.sub.1] (1), [A.sub.0] (2) and [A.sub.1] (2) models

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Mateo Velasquez-Giraldo and Diego Restrepo-Tobon (*)

(*) Mateo Velasquez-Giraldo: student in Mathematical Engineering at the Universidad EAFIT. E-mail: mvelas26@eafit.edu.co

Diego A. Restrepo-Tobon: Assistant Professor in the School of Economics and Finance at the Universidad EAFIT. He is currendy affiliated with the Research Group on Banking and Finance at the same institution. E-mail: drestr16@eafit.edu.co

doi: 10.17533/udea.le.n85a02

Original manuscript received on 29 June 2015; final version accepted on 29 October 2015

(1) Gaussian models allow nominal interest rates to take on negative values. However, this is a limitation of ATSMs in general. They cannot accommodate simultaneously unrestricted correlations among the underlying factors and positive interest rates (Dai & Singleton, 2000).

(2) A function F : [R.sup.N] [right arrow] [R.sup.M] is said to be affine if F(X) = A + B x X for some vector A and matrix B.

(3) We solve this ordinary differential equations system using the ode45(...) solver in Matlab[R]R2013A.

(4) The pseudo code for differential evolution is presented in Algorithm 1 and an implementation thereof in Matlab[R]R2013A is available upon request.

(5) There are other methods to compute zero coupon yields from market data. For instance, the smoothed Fama-Bliss method makes these discount rates smoothed by fitting a function to the 'non-smoothed' rates. The McCulloch method uses a cubic spline with an implicit smoothness penalty. The Fisher-Nychka-Zervos method employs a cubic spline to the forward rate function. The Nelson-Siegel method uses an exponential function for the discount rate function and applies it directly to bond prices. Bliss (1996) shows the non-smoothed Fama-Bliss method is the most appropriate.

Table 1. Summary statistics for the Nelson-Siegel data set Maturity (years) 0.25 0.50 1.00 2.00 4.00 5.00 7.00 Minimum 1.44 2.02 2.91 3.67 4.08 4.27 4.54 First quartilc 3.97 4.10 4.54 5.19 6.22 6.55 6.96 Median 5.23 5.39 5.76 6.60 8.03 8.48 8.85 Third quartile 7.54 8.12 8.99 9.66 10.68 10.98 11.55 Maximum 11.44 11.66 12.88 14.52 16.75 17.10 17.20 Mean 5.76 6.09 6.67 7.56 8.61 8.93 9.33 Variance 4.59 4.82 5.42 6.49 7.58 7.81 8.02 Standard deviation 2.14 2.20 2.33 2.55 2.75 2.79 2.83 Excess kurtosis 1.07 -1.31 -1.39 1.04 0.73 -0.72 0.74 Maturity (years) 8.00 9.00 10.00 Minimum 4.65 4.75 4.84 First quartilc 7.10 7.19 7.29 Median 8.93 8.98 9.02 Third quartile 11.69 11.84 11.90 Maximum 17.34 17.44 17.53 Mean 9.46 9.55 9.63 Variance 8.06 8.09 8.11 Standard deviation 2.84 2.84 2.85 Excess kurtosis 0.75 0.75 0.75 Note: the total number of observations is 3051. Source: authors' elaboration based on data from Infovalmer. Table 2. Summary statistics for the bootstrapped data set Maturity (years) 0.25 0.5 1 2 4 5 7 Minimum 2.81 3.00 3.37 3.77 4.12 4.25 4.67 First quartile 3.93 3.99 4.44 5.10 6.14 6.42 6.95 Median 4.77 4.92 5.23 6.01 7.23 7.69 8.33 Third quartile 7.42 7.57 8.06 8.66 9.33 9.72 9.80 Maximum 10.16 10.51 11.07 11.98 12.71 12.93 13.12 Mean 5.67 5.83 6.17 6.79 7.69 8.03 8.45 Variance 4.67 4.71 4.77 4.42 4.02 3.98 3.65 Standard deviation 2.16 2.17 2.18 2.10 2.00 1.99 1.91 Excess kurtosis -0.94 -0.89 -0.84 -0.75 -0.70 -0.80 -0.66 Maturity (years) 8 9 10 Minimum 4.78 4.86 4.97 First quartile 7.11 7.18 7.28 Median 8.37 8.40 8.41 Third quartile 9.84 9.82 9.82 Maximum 13.11 13.10 13.13 Mean 8.56 8.60 8.62 Variance 3.44 3.22 3.06 Standard deviation 1.85 1.79 1.75 Excess kurtosis -0.58 -0.48 -0.41 Note: the total number of observations is 2625. Source: authors' elaboration based on data from Bloomberg. Table 3. Differential evolution parameter values used for estimation Parameter Value ng 2000 np 200 F 0.4 CR 0.9 Note: the algorithm is stopped if the standard deviation of the objective functions of all solutions in a population were less than 0.1. Table 4. In-sample root-mean-square error (RMSE) between modeled and observed yields RMSE (basis points) Maturity [[LAMBDA].sub.0] (1) [[LAMBDA].sub.1] (1) (Years) NS BS NS BS 0.25 0.0000 0.0000 0.0000 0.0000 0.5 34.7293 14.3485 33.5475 13.5516 1 84.4390 34.1240 81.5798 30.6891 2 139.5312 54.1757 135.2037 50.1033 4 183.5011 78.8338 178.2805 76.2121 5 194.6745 86.6065 189.2136 84.6585 7 211.2276 92.6067 204.9971 93.3573 8 218.4159 90.3075 211.4492 90.2334 9 225.3108 90.6555 217.3645 86.9753 10 232.0046 98.9167 222.8885 89.4189 RMSE (basis points) Maturity [[LAMBDA].sub.0] (2) [[LAMBDA].sub.0] (2) (Years) NS BS NS BS 0.25 0.0000 0.0000 0.0000 0.0000 0.5 13.3769 18.1574 14.3687 16.4851 1 26.2842 33.4925 25.2418 32.6466 2 25.8577 32.7213 27.4021 31.8013 4 0.0000 0.0000 0.0000 0.0000 5 13.4699 30.9987 15.2345 29.7403 7 37.5874 56.8845 38.5727 56.1768 8 48.4840 57.7654 48.7305 57.1240 9 58.8277 57.9986 60.1145 56.3779 10 68.7309 61.6099 74.0213 57.7069 RMSE (basis points) Maturity [[LAMBDA].sub.0] (3) (Years) NS BS 0.25 0.0000 0.0000 0.5 9.8554 17.2051 1 18.3131 31.5518 2 16.0763 31.8009 4 0.0000 0.0000 5 4.8226 22.5503 7 7.3120 29.2402 8 5.8909 24.5676 9 3.3835 17.0939 10 0.0000 0.0000 Note: errors are reported in basis points. Columns marked with NS correspond to the Nelson-Siegel yields (2.000 observations); BS indicates bootstrapped rates (1.700 observations). Source: authors' elaboration using data from Infovalmer (for the NS set) and Bloomberg (for the BS set). Table 5. One-day out-of-sample forecast mean RMSE RMSE (basis points) Maturity [A.sub.0] (1) [A.sub.1] (1) [A.sub.0] (2) (Years) NS BS NS BS NS BS 0.25 8.1601 5.8908 7.9698 5.8018 8.2444 5.8321 0.5 37.9014 15.1482 36.4622 14.3558 11.4499 12.4465 1 100.5005 41.7525 95.2800 40.7645 17.0386 21.3431 2 191.4617 80.9970 182.8747 81.5551 13.4894 21.6089 4 299.7667 143.2359 279.1309 144.1055 6.1066 4.9791 5 333.5070 167.0939 304.3713 165.5840 8.2769 13.2989 7 378.8010 199.4466 333.1596 189.3652 16.1299 29.6337 8 394.2433 213.9765 341.5084 198.2714 21.1197 35.2649 9 406.5176 229.8556 347.5867 208.2402 26.7962 45.0049 10 416.4146 245.3361 352.1222 217.6140 33.0581 55.3252 RMSE (basis points) Maturity [A.sub.1] (2) [A.sub.0] (3) (Years) NS BS NS BS 0.25 91.5271 1531.2320 7.9759 5.8237 0.5 76.9352 1481.7873 12.0973 14.1969 1 74.3880 1410.7824 17.5101 25.1460 2 78.0349 1282.2888 14.2782 24.5432 4 50.3135 1100.0893 6.0965 5.0062 5 37.2752 1029.5393 6.8395 10.1583 7 30.2056 890.7205 7.5100 17.0006 8 34.7320 830.5686 7.2349 11.5639 9 43.2201 779.4247 6.7211 12.5920 10 55.2536 732.6228 6.4229 6.0447 Note: errors are reported in basis points. Columns marked with NS correspond to the Nelson-Siegel yields (1050 observations); BS indicates bootstrapped rates (924 observations). Source: authors' elaboration using data from Infovalmer (for the NS set) and Bloomberg (for the BS set). Table 6. Five-day out-of-sample forecast mean RMSE RMSE (basis points) Maturity [A.sub.0] (1) [A.sub.1] (1) [A.sub.0] (2) (Years) NS BS NS BS NS BS 0.25 15.1925 12.2025 12.4830 11.2027 16.1403 11.5939 0.5 33.1407 20.1558 39.0481 17.4795 13.5451 15.6078 1 95.5089 45.5055 97.3026 41.9420 15.8807 22.8493 2 187.4598 84.2295 184.7206 82.1965 14.8798 23.4962 4 297.0978 145.9109 280.6704 144.5052 14.9246 14.0779 5 331.2655 169.5456 305.7547 165.9627 16.4029 19.6076 7 377.1450 201.5111 334.2807 189.6990 21.1581 33.4648 8 392.7905 215.9600 342.5253 198.6720 24.9186 38.6090 9 405.2278 231.7549 348.5144 208.6865 29.6757 47.8283 10 415.2573 247.0460 352.9735 217.9718 35.2785 57.9955 RMSE (basis points) Maturity [A.sub.1] (2) [A.sub.0] (3) (Years) NS BS NS BS 0.25 428.0572 7642.6494 12.7942 11.5112 0.5 392.5900 7452.2251 14.9358 17.0529 1 354.2918 7112.1345 19.3979 26.6267 2 310.8762 6492.8519 17.9351 27.5118 4 235.1709 5490.7906 14.8562 14.2994 5 207.0488 5077.5200 15.4784 15.9843 7 175.7343 4360.6013 15.8685 21.7465 8 170.9744 4056.5068 15.7456 17.6619 9 172.1168 3786.4570 15.5817 18.2408 10 178.1510 3542.6739 15.6207 16.2657 Note: errors are reported in basis points. Columns marked with NS correspond to the Nelson-Siegel yields (1046 observations); BS indicates bootstrapped rates (920 observations). Source: authors' elaboration using data from Infovalmer (for the NS set) and Bloomberg (for the BS set). Table 7. Comparison of out-of-sample forecast RMSE between the [A.sub.0] (3) model and a random walk benchmark 1-day forecast RMSE (Basis points) Maturity Nelson-Siegel Bootstrapped (Years) [A.sub.0] (3) Random [A.sub.0] (3) Random walk walk 0.25 7.9759 7.9753 5.8237 5.8115 0.50 12.0973 7.2085 14.1969 5.1796 1.00 17.5101 6.3573 25.1460 4.3725 2.00 14.2782 5.3922 24.5432 4.8386 4.00 6.0965 6.0908 5.0062 4.9523 5.00 6.8395 6.1126 10.1583 5.0093 7.00 7.5100 6.0007 17.0006 5.2088 8.00 7.2349 6.0505 11.5639 5.3522 9.00 6.7211 6.1963 12.5920 5.5129 10.00 6.4229 6.4222 6.0447 5.9155 5-days forecast RMSE (Basis points) Maturity Nelson-Siegel Bootstrapped (Years) [A.sub.0] (3) Random [A.sub.0] (3) Random walk walk 0.25 12.7942 12.5073 11.5112 11.2754 0.50 14.9358 11.3416 17.0529 10.7431 1.00 19.3979 10.5281 26.6267 10.1987 2.00 17.9351 11.9459 27.5118 12.3527 4.00 14.8562 14.7843 14.2994 13.7999 5.00 15.4784 15.0885 15.9843 13.8300 7.00 15.8685 15.1914 21.7465 14.2843 8.00 15.7456 15.2614 17.6619 14.4649 9.00 15.5817 15.3958 18.2408 14.7716 10.00 15.6207 15.5942 16.2657 15.0288 Source: authors' elaboration using data from Infovalmer (for the NS set) and Bloomberg (for the BS set). Table 8. Modeled and forecast RSME for the [A.sub.0] (3) model estimated with weekly observations RMSE (basis points) Maturity (Years) In-sample Out-of-sample 1-week forecast [A.sub.0] (3) fit [A.sub.0] (3) Random walk 0.25 0.0000 23.7651 12.3279 0.50 10.3195 23.3404 11.3276 1.00 18.4792 25.4479 10.3468 2.00 15.5717 23.7895 11.9871 4.00 0.0000 16.9017 15.1659 5.00 4.6319 16.5392 15.6178 7.00 7.1523 16.4781 15.8800 8.00 5.8183 16.3442 15.9748 9.00 3.3527 16.3501 16.1189 10.00 0.0000 16.8046 16.3293 Note: 500 observations were used for estimation, and 124 for forecast validations. Source: authors' elaboration using data from Infovalmer (for the NS set). Table 9. Modeled and forecast RSME for the [A.sub.0] (3) model estimated with monthly observations Maturity (Years) In-sample [A.sub.0] (3) fit 0.25 0.0000 0.50 10.7852 1.00 19.9255 2.00 17.4827 4.00 0.0000 5.00 5.0450 7.00 7.6208 8.00 6.0756 9.00 3.5034 10.00 0.0000 RMSE (basis points) Maturity (Years) Out-of-sample 1-month forecast [A.sub.0] (3) Random walk 0.25 37.7977 20.3857 0.50 34.2667 19.7754 1.00 34.2831 19.7233 2.00 34.3616 23.9683 4.00 33.1740 31.3470 5.00 33.9651 32.9028 7.00 34.5065 33.9756 8.00 34.3494 34.0358 9.00 34.1842 33.9807 10.00 34.2838 33.8912 Note: 100 observations were used for estimation, and 49 for forecast validations. Source: authors' elaboration using data from Infovalmer (for the NS set). Table 10. Correlation coefficients between the [A.sub.0] (3) factors and the proxies for empirical factors Level Slope Curvature [X.sub.1] 0.7198 -0.7027 0.3667 [X.sub.2] 0.8928 -0.5102 0.7095 [X.sub.3] -0.9929 0.1166 -0.2949 Source: authors' elaboration using data from Infovalmer. Table 11. Correlation coefficients between the [A.sub.0] (3) factors and the principal components of the yield data PC1 PC1 PC3 [X.sub.1] 0.7554 -0.4945 -0.4201 [X.sub.2] 0.9332 -0.2361 0.2683 [X.sub.3] -0.9807 -0.1919 -0.0267 Source: authors' elaboration using data from Infovalmer.