ESTUDIO PILOTO. ANÁLISIS

The vast empirical literature on the estimation of interest rate continuous-time models portrays an inconclusive and rather complex picture in which various aspects are identified as affecting the empirical findings. Once the theoretical models have been considered to be conceptually suitable, there are multiple estimation routes to translate them into practice.

Early empirical studies tested the single-factor theoretical models, with a focus on certain aspects such as the selection of the best model in capturing the dynamics of interest rates, the evidence for reversion to the mean and the determination of the sensitivity level of the volatility to the interest rate level. Initially, lacking a common framework for comparison, empirical testing was conducted on individual models only.

42 For example, the CIR model was empirically tested by Brown and Dybvig (1986) using monthly quoted prices of US Treasury bills, bonds and notes between 1952 and 1983, while Edsparr (1992) estimated the same model based on Swedish data. In a seminal paper Chan et al. (1992) presented a general framework, facilitating a multidirectional comparison among different classical models. Eight¹⁶ short-term rate single-factor models could be nested in the unrestricted CKLS model, so their relative performance in terms of explanatory power could be consistently evaluated.

Following the proposal of the more general CKLS framework, numerous subsequent empirical studies provided early evidence of discrepancies. In an extensive international study Episcopos (2000) emphasized this sensitivity of the empirical results, concluding that the choice of the estimation techniques, sample period, data frequency, and country should be taken into account. Several comparative empirical studies (Treepongkaruna and Gray, 2003; Ioannides (2003); Lo 2005) have confirmed and demonstrated that different estimation routes lead to different results in terms of parameter estimates and implications for pricing interest rate contingent claims. Therefore, assessing the relative empirical performance of such an impressive number of theoretical models becomes extremely complex, and the following survey of key empirical evidence on testing interest rate models tries to illustrate just that.

For the estimation of the parameters Chan et al. (1992) employed the Generalized Method of Moments (GMM) of Hansen (1982). Based on one-month monthly US Treasury bills rates between June 1964 and November 1989, in the case of the unrestricted model the estimates of the drift parameters did not support the mean reversion feature. The level effect parameter was estimated at around 1.5 implying a high degree of dependence of the local volatility on the level of interest rates. The relative performance of the nine models was examined using two statistical tests and a metric that measured the ability of the models to capture the volatility of the changes in the risk-free rate. Based on the goodness-of-fit measure provided by the GMM objective function which is² distributed, the models with  1 (Brennan and Schwartz (1980) and CEV (1975)) performed best, whereas Merton (1973), Vasicek (1977), Cox et.al. (1985a) and Cox et.al. (1985b) models were rejected against the unrestricted model. Employing the Newey and West (1987) hypothesis-testing method, the restrictions imposed by the nested models on the unrestricted CKLS model were evaluated and pair-wise comparison was conducted with no rejection being observed between models with similar diffusion

16 Another general framework was proposed previously by Marsh and Rosenfeld (1983), but it nested only three restricted models (Chan et al. 1992).

43 coefficient. Additionally, the hypothesis of a structural break at the point October 1979 marking the Federal Reserve experiment (1979-1982) was rejected. Following an experiment to value a 2-year call bond option Chan et al. (1992) found that the option values varied substantially from one model to another, a result with great implications for valuing interest contingent claims and hedging interest rate risk.

Adopting the CKLS framework, Tse (1995) conducted an extensive international comparative empirical exercise based on data from eleven countries. The estimation results by the GMM method are rather mixed with countries grouped in three categories according to the magnitude (high, medium and low) of the estimated level-effect parameter. The most sensitive volatility of interest rate changes occurred in the U.S (

( ) 1.73

Tse US

  ), France(_Tse(France) 1.63) and Holland (_Tse(Holland) 1.60 ), while the lowest elasticity of volatility estimate was observed in the UK (_Tse(UK)0.11) and Canada (_Tse(Canada) 0.36). In contrast with another Chan et al. (1992b) study on the Japanese market where _CKLS(Japan)2.44, Tse (1995) found _Tse(Japan)0.62. In another important empirical study, Dalhquist (1996) looked at six alternative interest rate processes (CKLS, Vasicek, CIR SR, GBM, Brennan-Schwartz and CEV) for Denmark, Germany, Sweden and the UK over similar time periods. Employing GMM methodology and monthly one-month maturity data sets (Euro-currency and US Treasury bills rates), Dalhquist (1996) found evidence of a positive relationship between interest rate level and volatility as indicated by Chan et.al. (1992). Moreover, the estimates of the level-effect parameter vary with higher values for Sweden (1.154) and Denmark (0.970) and are less pronounced for Germany (0.387) and especially for the UK (0.156).

However, in contrast with the CKLS results for the U.S. significant mean reversion was found for Denmark and Sweden. In terms of relative explanatory power, the best models that could not be rejected at the 5% level of significance against the unrestricted CKLS model, were the CIR and Brennan and Schwartz models for Denmark and Sweden, and the Vasicek (1977) and the CIR SR (1985) for the U.K. and Germany, respectively.

Moreover, in the case of Denmark, Dalhquist (1996) discovered parameter instability during August 1985 when arguably the Danish central bank had adjusted its monetary policy.

Applying for the first time in finance the Gaussian estimation method developed by Bergstrom (1983, 1985, 1986, 1990), Nowman (1997) estimated the eight single-factor continuous-time short rate models within the CKLS setting. Based on the U.S. Treasury (1964-1989) and the U.K. interbank rates (1975-1995), the final quasi-maximum

44 likelihood estimates for the true parameters of the initial model were rather different between the two markets. Regarding the mean reversion parameters, the empirical results for the US contradicted those in CKLS indicating a weak presence of mean reversion, while the level effect parameter was found to be insignificant, with an estimate of

( ) 1.3610

Nowman US

  . For the U.K., the evidence for mean reversion was still weak, but the estimate for the level effect was inferred as highly significant at_Nowman(UK)0.2898.

A larger but similar study was conducted by Nowman (1998) for US, Japan, France and Italy, covering the period 1981-1995. The mean-reversion effect was in general weak with some significant evidence only in the case of US. For France and Italy, the level- effect parameter was in excess of two, whereas for Japan and the US was close to one.

In another comparative empirical study, Shoji and Ozaki (1998) developed a statistical method of model selection that was applied to data on Japan, US and Germany.

An advantage of their method was that it could involve models that are not nested in a best unrestricted model as in the CKLS framework. Several continuous-time models for the term structure were estimated, including models with a nonlinear drift. According to their method, for Germany, the model with nonlinear drift outperformed the best models with linear drift.

In Episcopos (2000) various classical one-factor short rate models across a sample of ten countries¹⁷ based on one month interbank rates. Some of the results are surprising with the CEV model outperforming the other competing models when other studies such as Tse (1995) and CKLS reject it, while the level-effect parameter varies across the ten countries from 0.20 to 1.56. For seven out of ten countries the level effect is under unity suggesting a less sensitive volatility than that in the CKLS findings. Also, the data sets used provided significant evidence for structural breaks in the case of six countries.

Yu and Phillips (2001, 2011) proposed a new estimation approach to a non-linear CKLS type diffusion model of interest rates, which is related to that of Nowman (1997).

Their time changing technique has great empirical appeal as it allows for non-equidistant observations and it converts the continuous-time model into a Gaussian one. In an extensive study, Treepongkaruna and Gray (2003) tested the robustness of various one-factor short-term interest rate models (Vasicek, CIR and CKLS) over different estimation techniques (GMM and QMLE) and data sets (eight countries) covering different sub-periods, with different frequencies (daily and weekly). The stability of the parameters

17 The group of ten countries is: Australia, Belgium, Germany, Japan, Netherlands, New Zeeland, Singapore, Switzerland, the UK and the US.

45 across sub-periods is uniformly rejected, suggesting that more complex models permitting for parameters to change over time could be more appropriate. The mean reversion parameters are insignificantly different from zero for all the countries, models, frequencies and estimation methods other than for the Italian Lira where there is some evidence of mean reversion in the Vasicek model estimated by QMLE. However, this evidence can be eliminated if four observations during the European currency crisis (September 1992) were excluded. The dependence of the volatility on the level of the interest rate differs from country to country as many previous studies have shown. The results were sensitive when the estimation technique and sampling frequency changed, requiring therefore some robustness checks.

To shed some light in this direction, Lo (2005) investigated the estimation of the Cox et al. (1985) and Chan et al. (1992) models in a comparative analysis of three Gaussian exact and approximate estimation methods implemented by Nowman (1997), Shoji and Ozaki (1998), and Yu and Phillips (2001). The conclusions from both a simulation study and empirical analysis of short-term interest rates for Canada and the UK, indicate that the Nowman (1997) and Shoji and Ozaki (1998) methods perform in a similar fashion, while the performance of the Yu and Phillips (2001) method was crucially impacted by the window width parameter used in the approximation. In terms of the best fit the Shoji and Ozaki method gave the best performance for both data sets, one-month Canadian Treasury bills (January 1980 to June 2002) and the one-month sterling interbank middle rate (March 1975 to March 1995), respectively. With regard to Yu and Phillips (2001) method it was observed that a large window width leads to a disappointing model fit and the estimation bias of the drift parameters could be significant as it depends on the choice of the window width. Lo (2005) concluded that all these aspects are important as they significantly affect the empirical results and emphasise the relative nature of any empirical work involving estimation of short-term interest rates models.

In a more recent comparative analysis of alternative single-factor continuous-time short rate models, Sanford and Martin (2006) employ a Bayesian inferential approach to estimate four models nested in the CKLS framework and to determine the magnitude of the level effect parameter that supports empirically the Australian interest rates over the 1990-2000 period. Their findings suggest that for this particular data set the CIR model is the most appropriate as indicated by the highest posterior probabilities provided by the Bayes factors, relative to the other models considered; therefore, the pricing equations implied by the CIR model are reasonable enough in the Australian context.

46 Modelling the Drift Component

The issue regarding the existence or non-existence of mean reversion in the dynamics of interest rates remains controversial. Looking at the overall evidence in terms of mean reversion most of the relevant empirical studies do not support statistically such a phenomenon. Following Ball and Torous (1996) who indicated a considerable estimation bias of the mean reversion estimates for the CIR model under popular estimation methods such as GMM and MLE, Faff and Gray (2006) investigated this problem further and demonstrate that the GMM estimates for the drift parameters in the single-factor models were severely overestimated, and therefore unreliable. At the same time, they asserted that the diffusion parameters are estimated with high precision under both GMM and MLE estimation methods, respectively.

Recently, Barros et al., (2012) investigated the mean reversion property of short-term rates for ten new EU countries. Using long memory fractionally integrated models and daily data covering the period January 2000 to December 2008, they concluded that interest rates are non-stationary (or stationary of order one, I(1)) and non-mean-reverting, except for Hungary. Testing for structural breaks, Lithuania is the only country for which in 2007 a structural break was statistically detected, while for all the other countries there is evidence of a structural break around 2001/2003. Once the structural breaks were considered, the mean reversion appeared more evident in some countries in the first sub-period, while after the break point the interest rates were clearly non-stationary with a higher degree of integration in all instances.

The parameterization of the short rate processes by restrictively assuming specific forms for the drift and diffusion functions could be another reason for such diversity of results. The natural alternative was to consider the most general SDE where the drift or/and the diffusion were not subject to parameterization.

In a famous article Ait-Sahalia (1996a) tested the validity of various classic parametric specifications in comparison with a non-parametric estimation technique.

Based on 7-day Eurodollar deposit rates from 1 June 1973 to 25 February 1995 the empirical results indicated a certain degree of non-linearity in the drift component with values of the drift close to zero in the region of 4% - 17% and substantially higher outside this region. As a result, Ait-Sahalia (1996b) proposed a richer parametric model that involves a non-linear drift and nests four well-known short-rate models (Vasicek, CIR, BS, CKLS). The specification test instrumented by Ait-Sahalia (1996b) failed to reject only the model with a non-linear drift, suggesting that the main source of rejection of the classical parameterizations is the linearity of the drift. These important findings were

47 explored further by various authors including Stanton (1997), Duffee (1999) and Chapman and Pearson (2000). However, their research led to mixed results about what is the appropriate drift specification.

Stanton (1997), for example, presented a general non-parametric procedure that allowed the estimation of both components (drift and the diffusion) by deriving a family of approximations to the true parameters. The procedure also permitted the market price of risk to be estimated by examining the daily excess returns between three month and six-month Treasury Bills rates from January 1965 to July 1995. The findings suggested that the drift exhibits a similar nonlinear pattern as in Ait-Sahalia (1996a) with a rapid increase in mean reversion when the interest rates level is high. Using Monte Carlo simulation Stanton (1997) examined the economic significance of the price of risk and finds that the assumption of a specific functional form for the price of risk has important implications for the evaluation of interest rate contingent claims especially as the maturity increases.

The linearity of the drift is also examined by Chapman and Pearson (2000) who applied the techniques developed by Ait-Sahalia (1996b) and Stanton (1997) to data generated through a linear drift by Monte Carlo simulation. The unexpected non-linear pattern measured in the drift was explained as the possibility of a source of bias stemming from the estimation approach. However, opposite results were obtained by Connolly et al.

(1997) and Durham (2003), who pointed out that the stationarity of the short rate may be induced by the dynamics of the volatility while the drift is fairly stable.

More recently, Goard and Hansen (2004) employed the GMM method to conduct an empirical comparison within a general non-linear drift framework that nested three important models: the CKLS (1992) model, the Ahn and Gao (1999) model and Goard and Hansen (2004) model. Empirically Goard and Hansen’s model seemed to outperform the other models even for smaller sampling periods, which indicates that the particular form of the drift as second order Fourier was able to capture very well the time dependence of the long-term equilibrium mean and also to explain the periodicity of the yield curve. Using an arbitrage-free framework Mahdavi (2008) estimates using the GMM approach the short-term interest rates of seven industrialised countries and the Euro zone. With no single model performing consistently across all countries, the empirical results strongly reflect once again the particularities of each market. While for the US, UK, Sweden and Canada there is evidence of mean reversion and non-linear volatility, the drift for Australia is non-linear, whereas for Japan it is constant indicating a log-normal process. For Denmark the volatility structure is close to that reported by Chan

48 et al. (1992) for the U.S., in the case of the Euro-zone the volatility is an increasing function of the level of the interest rate.

Modelling the Volatility Component

Using several jump models Das (2002) examined seven different empirical features in the Fed Funds data and finds that the models captured well the effects of new information with evidence that the volatility of interest rate changes is substantially higher following the arrival of news. The inclusion of the jump process as an intrinsic feature of financial markets seems to render a linear drift. Otherwise the drift is nonlinear due to information effects. In summary, the nature of the drift is not exactly known, as the two scenarios are arguably equally supported by empirical evidence.

As emphasised by Chan et al. (1992), volatility is a crucial component in the dynamics of interest rates and its modelling has important implications for the pricing of interest rate sensitive products and for the hedging of interest rate risk- the better the model captures the volatility, the more efficient the hedge implied by the model. Most of the theoretical models assume a simple parameterization of the volatility as a function of the interest rate level. Simultaneously, while the literature lacked consensus on the degree of this relationship, there was clear evidence of another feature of the volatility that emerged from serial correlation based (GARCH) modelling in discrete-time.

Volatility clustering and high level of volatility persistence should be also taken in consideration when modelling the volatility of the interest rates. The two features, the level effect and the conditional heteroskedastic (GARCH) effect, were combined in a new class of models by Brenner et al. (1996), who extended the CKLS allowing for the volatility to be affected by information shocks. They concluded that the sensitivity of the volatility on the levels has been overestimated in the literature implying that modelling the volatility solely on the levels it is an important source of model misspecification. A similar study by Koedijk et al. (1997) reconfirmed that the inclusion of the GARCH effect renders a weaker level effect. The new models developed by Koedijk et al. (1997) (KNSW hereafter) in a discrete-time setting, were estimated using the QML method and the consistent estimators based on weekly and monthly one-month Treasury Bills rates (January 1968-July 1996) provided a superior fit relative to both, pure GARCH and pure CKLS type models. Additionally, the more flexible KNSW specifications were found to have important implications for bond option prices that differ from the prices implied by CKLS models.

49 In another comparative study, Vetzal (1997) examined two classes of continuous-time interest rate models, the standard univariate short rate models and their variants of stochastic volatility models with an E-GARCH effect. The iterative GMM estimation method provided lower estimates of the volatility from stochastic volatility models relative to those implied by the classical one-factor models nested in the CKLS model.

Consequently, this led to lower prices for bond options under the stochastic volatility process for the short rate. Vetzal also emphasised the advantage of the tractability possessed by Longstaff and Schwartz (1992) model when it comes to pricing interest rate contingent claims, and that this advantage should be taken into account against the easier

Consequently, this led to lower prices for bond options under the stochastic volatility process for the short rate. Vetzal also emphasised the advantage of the tractability possessed by Longstaff and Schwartz (1992) model when it comes to pricing interest rate contingent claims, and that this advantage should be taken into account against the easier