Asymmetry and Long Memory in Dynamics of Interest Rate Volatility

Transcription

1 Asymmetry and Long Memory in Dynamics of Interest Rate Volatility Pei-Shih Weng a, a Department of Finance, National Central University, Jhongli, TY 32001, Taiwan May 2008 Abstract Empirically, the conditionally heteroskedastic volatility effect in short rate volatility process has been presented for broad discussion, and the long memory phenomena is also tested. However, these two effects have been never involved together to model the dynamics of short rate volatility. This paper introduces a asymmetry and long memory conditional variance model, says FIEGARCH, to model the dynamics of short-term interest rate volatility on the three-month U.S. Treasury bills. By finding the significant estimates, we recognize the necessity of the asymmetric volatility function with the long memory property. Our results also show that the nonlinearity is not a necessary fact for the short rate drift, instead, the asymmetric specification plays an important role in the mean function. Finally, according to the comparison for the prediction of the volatility dynamics, the FIEGARCH shows better forecasting ability, especially in monthly frequency. JEL Classification: C51; C52; G12 Keywords: Short-Term Interest Rate, Volatility, Long Memory, Asymmetry 1 Introduction The short-term riskless rate of interest and its estimated volatility, driving the changes in the entire term structure, are the fundamental variables in continuous time models. This makes the choice of a volatility model for the risk-free rate crucial to pricing interest rate derivatives and hedging interest rate risk. The use of an incorrect volatility model could The author is currently Ph.D. candidates. Contact: Tel.: ext ; address: @cc.ncu.edu.tw. 1

2 lead to incorrect inferences, mishedged or unhedged risks, or pricing errors. In continuous time diffusion models, the short-term interest rate is usually based on Brownian motion. In a general framework, the dynamics of the short rate can be described by the stochastic differential equation: dr t = µ(r t, t)dt + σ(r t, t)dw t (1) were r t is the instantaneous spot rate, W t is a standard Brownian motion, and µ(r t, t) and σ(r t, t) are the drift and diffusion functions that depend on the short rate and possibly other state variables. Many single-factor term structure models imply dynamics for the short-term riskless rate that can be nested within the following model proposed by Chan et al. (1992) (henceforth CKLS), dr t = (α 0 + α 1 r t )dt + σr φ t dw t (2) where α 0, α 1, σ, and φ are fixed parameters. CKLS estimate the parameters of the continuous time model given in (2) using a discrete time econometric specification: r t r t 1 = α 0 + α 1 r t 1 + ε t (3) E(ε t Ω t 1 ) = 0, E(ε 2 t Ω t 1 ) = h t = σ 2 r 2φ t 1 (4) However, a major limitation in CKLS model is that they restrict volatility to be a function of the interest rate level only, and not of the news arrival process. 1 Therefore, it fails to capture adequately the serial correlation in conditional variances. These weakness motivate the work of Brenner, Harjes, and Kroner (1996) who nest the GARCH and CKLS models under more general discrete time specifications, in which the current volatility is a function of the last period s unexpected news and the last period s volatility. 1 We usually present the so-called level effect for that the magnitude of the volatility is dependent on the level of interest rate. 2

3 Another earlier work of Longstaff and Schwartz (1992) nests a CIR term structure in a model including a stochastic volatility factor that they approximate by a discrete time GARCH specification. These studies confirm the presence of rather extreme conditionally heteroskedastic volatility effects in the interest rate dynamics, but with weak level effect. A further investigation about conditionally heteroskedastic volatility effects is returned from the Andersen and Lund (1997). They propose a continuous time stochastic volatility model that displays mean reversion, and find that the standard GARCH model fail to approximate the discrete time short rate dynamics, while the diffusion models with a stochastic volatility process allowing for asymmetric volatility responses to interest rate shocks perform reasonably well. Bali (2000) compares several diffusion and symmetric and asymmetric GARCH models whose relative performance is unknown. His work offer a comprehensive prospect for the modeling ability of eight kinds of GARCH-like conditional volatility models. 2 Within the comparison, the models allow asymmetric mean and asymmetric volatility response to interest rate level with a GARCH process outperform the corresponding symmetric specification. The findings, therefore, recognize the necessity of the asymmetric GARCH for the conditional volatility dynamics. Following though the study in Bali (2000), Bali (2007) further models the dynamics of interest rate volatility with skewed fat-tail distributions. The results also indicate that the interest rate models with asymmetric GARCH volatility process still show better in-sample estimation and prediction power, especially for EGARCH. So far, no matter which GARCH-like volatility process has been adopted in the interest rate modeling, a common property within those models is known as short memory. However, in the equity market, empirical autocorrelations for absolute returns and 2 The testing for the empirical performance of exponential GARCH (EGARCH) process of Nelson (1991) is not incorporated in the comparison of Bali (2000). That is due to the EGARCH has been tested with the same data set in Anderson and Lund (1997) (1997). 3

4 squared return provide evidence that their theoretical counterparts decay more slowly so that they are not geometrically bounded, such as Ding et al. (1993); Bollerslev and Mikkelsen (1996). In this sense, a long memory model is appropriate in equity market. Comparatively, in the interest rate volatility modeling, the long memory property has not been well-explored. Duan and Jacobs (2002) propose either symmetric long memory specification - fractional integrated GARCH (FIGARCH) or symmetric short memory specification - GARCH to investigate the short-term interest rate dynamics, and find long memory evidence. Though their long memory parameter is statistically significant, the value, however, is quite small and somewhat loses the economic implication. 3 Moreover, their model is specified without considering the asymmetric volatility fact. In addition to volatility, numerous studies have investigated the presence and significance of nonlinearity in the short rate drift, but the results are not conclusive. Ait- Sahalia (1996a, b) finds strong nonlinearity in the drift of the 7-day Eurodollar deposit rate. Stanton (1997) and Jiang (1998) find similar nonlinearities in the drift function of the 3-month Treasury bills yields. Conley et al. (1997) also identifies nonlinearity in the drift of the federal funds rate. Furthermore, Bali (2000) then documents the asymmetric mean-reverting drift in 3-month Treasury bills yields. A key feature of this study is in line with the different findings of previous literature to incorporate the conditionally asymmetric volatility effects and long memory phenomena in interest rate volatility process. We make this compatible by using the so-called fractional integrated EGARCH (FIEGARCH) whose relative performance in the interest rate volatility dynamics is so far unknown. In addition to the volatility modeling, we also explore the impact of possible asymmetry and nonlinearity in the drift term. Another advantage of this empirical work is the use of recent data sets with a comparatively long sample. We use weekly observations for the annualized yields on U.S. Treasury bills with 3 The largest value is ca. 0.02, which is revealed in FIGARCH(2,d,2) by using one-week Eurodollar rates. 4

5 maturity of three months for 1950s through June Our main contributions can be outlined along the following lines. First, we confirm the importance of asymmetric specifications of the diffusion function in short rate volatility with a longer sample, meanwhile, we incorporate the long memory effect in the volatility process. The short-term interest rate is newly modeling by the FIEGARCH. Second, we investigate the relevance of possible asymmetry and nonlinearity in the short rate drift, and suggest a suitable specification. Third, we compare the ability of the short and long memory models to forecast the realized weekly or monthly volatility of interest rate changes. The remainder of the paper is organized as follows. Section II provides the alternative short-term interest rate specification we propose. Section III describes the data and estimation procedure. Section IV presents the empirical results, and Section V concludes this paper. 2 Short-Term Interest Rate Specification Similar to CKLS, we use the following discrete time specification for the interest rate. r t r t 1 = µ t + h 1/2 t z t, z t i.i.d. D(0, 1). (5) where µ t is the conditional means/drift term, h t is the conditional variance/difussion term, and the standard residual z t follows a specific distribution. 2.1 The Diffusion Function Our discussion on the interest rate volatility is focused on two short memory models, GARCH and EGARCH, and one long memory model, FIEGARCH, respectively. We show the comparison between GARCH and EGARCH to confirm the asymmetry of volatility process with previous empirical findings. Furthermore, the FIEGARCH is obviously the corresponding long memory version of the EGARCH, and can be directly compared to 5

6 identify the importance of long memory specification. For simplicity, we skip the comparison between GARCH and FIGARCH, however, we drop this part also for the theoretic concern. Although, both of Baillie (1996) and Bollerslev and Mikkelsen (1996) show how to use the filter (1 + L) d, where L is a lag operator and d is long memory parameter, to define a long memery process for the conditional variance h t, by making either the GARCH or EGARCH model more general; However, the GARCH generalization cannot be recommended because the returns process then has infinite variance for all positive value of d. On the contrary, the EGARCH generalization may not have this drawback, as then log(h t ) is covariance stationary for d < 1. 2 In this study, the process of h t for each model is defined as: GARCH(1, 1) model of Bollerslev (1986) h t = β 0 + β 1 z 2 t 1 + β 2 h t 1, (6) EGARCH(1, 1) model of Nelson (1991) log(h t ) = β 0 + β 1 [ z t 1 E( z t 1 )] + β 2 (log(h t 1 ) β 0 ) + γz t 1, (7) FIEGARCH(1, d, 1) model of Bollerslev and Mikkelsen (1996) log(h t ) = β 0 + (1 β 2 L) 1 (1 L) d (1 + ψl){β 1 [ z t 1 E( z t 1 )] + γz t 1 }. (8) where the lag operator L shift any process {y t } backwards by one time period, so L y t = y t 1, while the differencing parameter d is between zero and one for volatility applications. In (8), since the filter (1 L) d can be represented as an infinite series by the binomial expansion, then (1 L) d = 1 j=1 a j L j, a 1 = d, a j = j d 1 a j 1, j 2. j 6

7 Generally, for the essential application, the infinity series has to be truncated by a suitably large number T. 4 Given a choice for T, the conditional variances for the FIE- GARCH(1,d,1) model can be computed from T log(h t ) = β 0 + b j [log(h t j ) β 0 ] + g(z t 1 ) + ψg(z t 2 ) (9) j=1 with the coefficients b j defined by b 1 = d + β 2 and b j = a j β 2 a j 1, j The Drift Function As discussed earlier, numerous studies have investigated the presence and significance of nonlinearity and asymmetry in the short rate drift, but the conclusions from these studies are far from consensual. In this study, we combine the nonlinear drift specification of Ait- Sahalia (1996a, b) and the the asymmetric drift specification of Bali (2000), then µ t (r t 1 ) = α 0 + α + 1 r + t 1 + α 1 r t 1 + α 2 r 2 t 1 + α 3 /r t 1. (10) where r t 1 + = r t 1, only if r t > r t 1, and rt 1 = r t 1, only if r t r t 1. When α + = α and α 2, α 3 = 0, the short rate follows the usual linear and symmetric mean-reverting drift. On the contrary, if α + α, the drift has asymmetric meanreverting property; and if α 2 or α 3 0, then the drift is nonlinear. 3 Estimation of Parameters and Model Fitting 3.1 Estimation of Parameters We estimate the parameters embedded in the discrete time form of the interest rate process, viz the equation (5) with appropriate drift and diffusion function, by maximizing the likelihood function. To carry out the estimation, we need to make a distribution 4 The choice of truncation limit seems to be arbitrary, however, the limit T = 1000 has been often selected for daily data. By taking into account our lower data frequency, we set the truncation limit T = 200 for our weekly short rate observations. 7

8 assumption for the error term, ε t = h t z t. In light of the empirical evidence of fatfailed errors, several authors (e.g., Bollerslev (1987), Nelson (1991), and Sentana (1995)) have chosen leptokurtic distributions such as the Student-t distribution or the GED. For simplicity, we assume that the error term is drawn from a normal density. That is to say, the standard residual z t follows a standard normal distribution, z t i.i.d. N(0, 1). Therefor, the logarithmic likelihood function is [ log L sn (θ) = 1 n log(2π) + 2 where θ is a vector involved the parameters estimated. ] n (log h t (θ) + zt 2 (θ)) t=1 (11) Anderson and Lund (1997) fit their discrete time stochastic volatility model using both normal and Student-t distribution for z t. However, they find that the distribution governing z t dose not affect the conditionally heteroskedastic characteristics of the series. They conclude that when changing the distribution assumption from normal to Student-t for z t, the parameters estimates and associated standard errors are little changes. Bali s (2007) study also confirms this conclusion somewhat, while modeling the interest rate volatility process with the skewed fat-tailed distribution. 3.2 Model Fitting With regard to model fitting, two measures the likelihood ratio and the Akaike information criteria (AIC) are adopted to compare the performance of the alternative drift and diffusion specification. The likelihood ratio is given as: LR = 2(LLF i LLF j ) X 2 p i p j where LLF i and LLF j are log-likelihood functions, p i and p j are dimensions of parameters for each model. The likelihood ratio follows an X 2 distribution with a degree of freedom of p i p j. 8

9 The Akaike information criteria is then given as: AIC = 2(p/N) 2(LLF/N) where N is the number of observations. The best model fitting will be the specification with the smallest value of AIC. When applying the FIEGARCH model to volatility, we estimate parameters by maximizing the likelihood function for a subperiod that excludes the first 200 weeks. To ensure a consistent comparison among models, particular for the computation of AIC, we excludes the same number of observations for all volatility specifications. 4 Data The data we applied are obtained from the Federal Reserve H.15 database and consist of weekly observations for the annualized yields on the 3-month U.S. Treasury bills. 5 The time period of investigation extends from January 6, 1954 through June 13, 2007, yielding a total of 2,789 weekly observations. The annualized yields on U.S. Treasury bills are constructed from the daily series available from the Federal Reserve H.15 database. Using the rates quoted on Wednesdays, we obtain the weekly observations of the annualized T- bill rates. The rates are calculated as averages of the bid rates quoted by a sample of primary dealers in the secondary market. 6 Table 1 provides summary statistics of the level and first differences of the three-month Treasury bill yields. Obviously, the empirical autocorrelations ρ τ for absolute interest rate changes provide evidence that they decay 5 In previous studies, fed funds and 3-month Eurodollar rates are also often used to estimate the interest rate process. However, Bali (2007) pointed out that there are fundamental problems in trying to estimate the interest rate processes from high-frequency data especially on fed funds and 3-month Eurodollar rates. Therefore, we follow the suggestion to choose the lower frequency (weekly) Treasury bill data, which is presumably less affected by microstructure effects. 6 While generating the weekly data set from the daily observations, we use the reported rates for Wednesdays. Wednesdays have 46 missing observations for the 3-month T-bill rates. When a Wednesday rate is missing, we use the Tuesday rate. This ensures a valid observation for all weeks over the full January 1954 to June 2007 sample for the 3-month Treasury bill data, and results in a total of 2,789 data points 9

10 Figure 1: Autocorrelations of absolute short-term interest rate changes. slowly, so that they might not be geometrically bounded and hence imply long memory property. Figure 1 plots the autocorrelations of absolute interest rate changes. Though the show decline shown by Figure 1 is typical of long-memory effects, yet, the property may not be concluded from the figure directly. Table 1: Summary Statistics Variables N Mean S.D. Sk. Ku. ρ 1 ρ 2 ρ 3 ρ 4 ρ 26 ρ 52 ρ 104 r t % 2.8% r t r t % 0.19% Note: ρ τ presents the autocorrelation coefficient of lag τ periods. Sk. is Skewness, and Ku. is Kurtosis. 10

11 5 Empirical Results 5.1 Nonlinearity and Asymmetry in the Short Rate Drift First, we test the linearity/nonlinearity and symmetry/asymmetry specification in the short rate drift. Table 2 presents the maximum likelihood estimates of the GARCH models with different drift functions. 7 Table 3 shows the tests for model fitting in the short rate drift. According to the LR test in Table 3, obviously, the model involves asymmetry meanreverting effect have quite significant improvement in estimation, however, the involving of nonlinearity does not seem to benefit the estimation, though the estimate of quadratic term (α 2 ) in Table 2 is consistently significant. The parameter estimates and model fit tests provided strong evidence that property of a asymmetric mean-reverting specification, compare to the property of nonlinearity, weights more on the estimation, and should be accounted for the short rate drift, rather than the nonlinearity. Therefor, the specification of asymmetric and linear drift, with the AIC being the smallest, is applied in follow-up estimate for volatility. 5.2 Asymmetry and Long Memory in the Short Rate Volatility In the last section, we recognize that the asymmetric drift is more appropriate for the short rate modeling. With this asymmetric drift specification, we further compare the estimation performance of the GARCH, EGARCH and FIEGARCH, respectively. The comparison between GARCH and EGARCH is presented to confirm the previous finding that the standard GARCH model fail to approximate the discrete time short rate dynamics, while the diffusion models with a stochastic volatility process allowing 7 We also estimate the each drift function with EGARCH model, the results are quite similar with GARCH. Further, while involving the nonlinearity in either symmetry or asymmetry case, the logarithmic likelihood value is consistently lower. 11

12 Table 2: The nonlinearity and asymmetry in the short rate drift Models α0 α1/(α + 1, α 1 ) α2 α3 β0 β1 β2 LogL Symmetry & Linear (3.0130) ( ) ( ) ( ) ( ) Symmetry & Nonlinear ( ) (1.5199) ( ) (0.4171) ( ) ( ) ( ) Asymmetry & , Linear (1.7180) ( ), ( ) ( ) ( ) ( ) Asymmetry & , Nonlinear ( ) (5.4165), ( ) ( ) (0.5896) ( ) ( ) ( ) Note: This table presents the maximum likelihood estimates of the GARCH models with a nonlinear/linear and asymmetric/symmetric drift for the 3-month Treasury yields. The parameter estimates with asymptotic t-statistics are presented in parentheses for each model. The maximized log-likelihood values are shown to test the presence of nonlinearity/linearity and asymmetry/symmetry in the short rate drift. While substituting GARCH with EGARCH, the results are qualitatively similar and hence ignored here. 12

13 Table 3: Tests for model fitting in the short rate drift Panel A: Symmetry & Linear Symmetry & Nonlinear Asymmetry & Linear Asymmetry & Nonlinear AIC Panel B: Symmetry & Linear V.S. Asymmetry & Linear V.S. Symmetry & Linear V.S. Symmetry & Nonlinear V.S. Symmetry & Nonlinear Asymmetry & Nonlinear Asymmetry & Linear Asymmetry & Nonlinear LR test Note: The LR ratio follows an X 2 distribution, indicates significance at the 10% level, indicates significance at the 5% level, and indicates significance at the 1% level. 13

14 for asymmetric volatility responses to interest rate shocks will perform well. In addition, the comparison between the short memory and long memory models the EGARCH and FIEGARCH will further show us that if the long memory property is a stylized fact in the short rate volatility process and worth to model within. Table 4 shows the estimates and Table 5 offers the results of tests for each model performance. The results of Table 4 confirm the presence of rather extreme conditionally heteroscedastic volatility effects in the interest rate dynamics. For example, the symmetric GARCH parameters β 1 and β 2 are found to be highly significant, and the sum of β 1 and β 2 turns out to be close to one. This implies the existence of substantial volatility persistence in the short rate process. Another notable point in Table 4 is that the asymmetry parameter, γ. It is estimated to be positive, implying that positive interest rate shocks (or unexpected rise in the short rate) have a larger impact than negative interest rate shocks (or unexpected decrease in the short rate) of the same size on conditional volatility. This is consistent with the hypothesis that volatility tends to be higher when the short rate is high. The most important finding in the Table 4 and Table 5 is to recognize the long memory property in volatility process. The FIEGARCH model has higher logarithmic likelihood value than the GARCH and the EGARCH model, and the difference is quite significant according to LR test. Furthermore, the FIEGARCH has the smallest AIC among the models. In addition, the corresponding differencing parameter, d, significantly differs from zero and less than 0.5, which also implies the convariance stationary of log(h t ). Therefore, our estimate for d, compared with previous literature, may be more reasonably received. 14

15 Table 4: Maximum likelihood estimates of the GARCH, EGRACH, and FIEGARCH for short rate volatility Models α0 α + 1 α 1 β0 β1 β2 γ d ψ LogL GARCH (1.7180) ( ) ( ) ( ) ( ) ( ) EGARCH (2.4428) ( ) ( ) ( ) ( ) ( ) (7.8690) FIEGARCH (2.2951) ( ) ( ) ( ) ( ) ( ) (5.7574) ( ) ( ) Note: This table displays the maximum likelihood estimates of the GARCH, EGARCH, FIEGARCH models for the 3-month Treasury yields. The parameter estimates with asymptotic t statistics are presented in parentheses for each model. The maximized log-likelihood values are reported in the last column. The parameters of the FIEGARCH are estimated from the expansion of the equation (8). log(ht) = β0 + (1 β2l) 1 (1 L) d (1 + ψl)g(zt 1), where g(zt 1) = β1[ zt 1 E( zt 1 )] + γzt 1. By expending the filter (1 L) d to a truncated period, T, the equation (8) can be computed from log(ht) = β0 + T j=1 bj[log(ht j) β0] + g(zt 1) + ψg(zt 2) with the coefficients bj defined by b1 = d + β2 and bj = aj β2aj 1, j 2, where a1 = d, aj = j d 1 j aj 1, j 2. 15

16 Panel A: Table 5: Tests for model fitting in the short rate volatility GARCH EGARCH FIEGARCH AIC Panel B: GARCH V.S. EGARCH EGARCH V.S. FIEGARCH LR test Note: The LR ratio follows an X 2 distribution, indicates significance at the 10% level, indicates significance at the 5% level, and indicates significance at the 1% level. 5.3 Forecasting Realized Volatility of Interest Rate Changes To provide a further measure of the relative performance of the GARCH, EGARCH, and FIEGARCH models, we test their predictive power for the next period s absolute value of interest rate changes, which provides a simple ex-post measure of the future interest rate volatility. This is done by estimating the conditional variance or standard deviation for each model, and then computing the proportion of the total variation in one-week-ahead realized volatility of interest rate changes, r t+1 r t, that can be explained by the current conditional volatilities. The predictive power of the models is measured by the following regression: σ t+1 = a + bσ e t + ɛ t, (12) where σ t+1 = r t+1 r t is the traditional measure of realized volatility of interest rate changes at time t+1, and σt e is the estimated conditional volatility of interest rate changes at time t. We present adjusted R 2 and square root of means squared errors (MSE) for the comparison across the models. The adjusted R 2 can provide information about how well each model is able to forecast the one-week-ahead volatility of interest rate changes. The same realized volatility measure is also used by Chan et al. (1992), Longstaff and Schwartz (1992), Brenner, Harjes, and Kroner (1996), Ahn and Gao (1999), and Bali 16

17 (2000). In addition to the weekly realized volatility of interest rate changes, we also consider a relatively longer frequency. When measuring the intertemporal relation between risk and return for the aggregate stock market, French, Schwert, and Stambaugh (1987), Schwert (1989), Campbell et al. (2001), Goyal and Santa-Clara (2003), and Bali et al. (2005) calculate the monthly realized variance of stock market returns as the sum of squared daily index returns. The estimate of this quadratic variation is referred to as integrated volatility. We compute the monthly integrated volatility of interest rate changes as the square root of the sum of squared daily interest rate changes: σ t = N d ( r d ) 2 (13) d=1 where N d is the number of trading days in month t, and r d is the interest rate change on day d. Then, the corresponding estimated integrated volatility as the square root of the sum of daily conditional variances: σ t = N d σd 2 (14) d=1 where σ 2 d is the estimated daily conditional variance of interest rate changes on day d. The one-month-ahead forecasting performance of GARCH, EGARCH, and FIEGARCH models is measured by the aforementioned regression. Table 6 shows the regression results. In the line with the maximum likelihood estimation, the FIEGARCH reveals the highest forecasting power. Figure 2 plots the current monthly integrated volatility of interest rate changes and current conditional volatility estimates of the FIEGARCH. Figure 2 shows the in-sample (instead of the one-month-ahead) performance of FIEGARCH model by presenting its contemporaneous relation with integrated volatility. Whereas in most periods, the FIEGARCH could track the integrated volatility. 17

18 Table 6: Forecasting realized volatility of interest rate changes weekly monthly (integrated) Models Adjusted R 2 Root MSE Adjusted R 2 Root MSE GARCH 20.66% 0.172% 25.12% 0.265% EGARCH 25.55% 0.167% 35.64% 0.264% FIEGARCH 34.23% 0.157% 49.58% 0.218% Note: This table provides the adjusted R 2 and MSE values that provide information about how well each model is able to forecast the weekly/monthly (integrated) volatility of changes in 3-month Treasury yields. The predictive power of the models is measured by the regression: σ t+1 = a + b σ e t + ɛ t, where σ t+1 is the weekly/monthly (integrated) volatility of interest rate changes at week/month t + 1, and σ e t is the estimated (integrated) volatility of interest rate changes at week/month t. For weekly forecasting, the proportion of the total variance in realized volatility captured by the FIEGARCH is about 10% more than that by the EGARCH; and is around 15% higher for monthly integrated volatility forecasting. While Bali (2007) presents the adjusted R 2 of EGARCH for monthly forecasting is 66.01%, our corresponding R 2 value for EGARCH is lower. However, this does not imply our estimation bias, but infer that the forecasting ability seems quite sensitive to the sample period chosen. For example, both Bali (2007) and CKLS (1992) have estimated the predictive power of the CEV model for monthly realized volatility of interest rate changes. 8 While CKLS (1992) presents a 18.01% adjusted R 2, Bali (2007) presents 41.12%. In addition, while the adjusted R 2 of the GARCH we regressed for monthly forecasting is 25.12%, the corresponding value in Bali (2000) is relatively higher, says 48.93%. 8 The data period in Bali (2007) is only up to

19 Figure 2: Forecasting monthly integrated volatility of interest rate changes 6 Conclusion This paper introduces a asymmetry and long memory conditional variance model, named FIEGARCH, to model the dynamics of short-term interest rate volatility on the threemonth U.S. Treasury bills. Empirically, the conditionally heteroskedastic volatility effect in short rate volatility process has been presented for broad discussion, and the long memory property in volatility has been tested alone. However, these two effects have been never involved together to model the dynamics of short rate volatility. In our study, the short-term interest rate volatility is newly modeling by the FIE- GARCH, and the importance of the asymmetric volatility function with the long memory property is recognized by significant estimates. In the comparison between EGARCH and FIEGARCH, the later shows higher logarithmic likelihood value. This paper also 19

20 tests the presence and significance of asymmetry and nonlinearity in the short rate drift. The results show that the nonlinearity is not a necessary fact for the short rate drift. On the contrary, the asymmetric specification play an crucial role in the mean function, due to its relevant effect on the maximum likelihood estimation. We further compare the predictive power of each model in capturing the dynamic behavior of the short rate volatility. According to the coefficients of determination, the FIEGARCH shows better forecasting ability, especially for monthly frequency. As for the further inference, the commonly used interest rate cap-floor pricing models are based on the assumption that forward rates follow the geometric Brownian motion. A key assumption of the geometric Brownian motion is that the ratio of two consecutive forward rates does not depend on past forward rates. However, this paper provides evidence for the presence of long memory and conditionally heteroscedastic volatility effects in the short interest rate process, and could be further inferred to the forward rate process. Therefore, the conditional variance of the forward rate process should not be assumed as i.i.d. and constant over time. 20

21 References [1] Ahn, D. and B. Gao. (1999). A Parametric Nonlinear Model of Term Structure Dynamics. Reviewof Financial Studies, 12, [2] Ait-Sahalia, Y. (1996a). Testing Continuous-Time Models of the Spot Interest Rate. Review of Financial Studies, 9, [3] Ait-Sahalia, Y. (1996b). Nonparametric Pricing of Interest Rate Derivatives. Econometrica, 64, [4] Andersen, T.G. and J. Lund. (1997). Estimating Continuous-Time Stochastic Volatility Models of the Short- Term Interest Rate. Journal of Econometrics, 77, [5] Bali, T.G. (2000). Testing the Empirical Performance of Stochastic Volatility Models of the Short-term Interest Rate. Journal of Financial and Quantitative Analysis, 35, [6] Bali, T.G. (2007). Modeling the dynamics of interest rate volatility with skewed fat-tailed distributions. Annals of Operationse Research, 151, [7] Baillie, R.T., T. Bollerslev, and H.O. Mikkelsen (1996). Fractionally integrated generalized autoregressive conditional heteroskedasticity. Journal of Econometrics, 74, [8] Bollerslev, T. (1986). Generalized Autoregressive Conditional Heteroscedasticity. Journal of Econometrics, 31, [9] Bollerslev, T. (1987). A Conditionally Heteroscedastic Time Series Model for Security Prices and Rates of Return Data. Review of Economics and Statistics, 59,

22 [10] Bollerslev, T. and H.O. Mikkelsen (1996). Modeling and pricing long memory in stock market volatility. Journal of Econometrics, 73, [11] Brenner, R.J., R.H. Harjes, and K.F. Kroner. (1996). Another Look at Models of the Short-term Interest Rate. Journal of Financial and Quantitative Analysis, 31, [12] Chan, K.C., G.A. Karolyi, F.A. Longstaff, and A.B. Sanders. (1992). An Empirical Comparison of Alternative Models of the Short-Term Interest Rate. Journal of Finance, 47, [13] Conley, T.G., L.P. Hansen, E.G.Z. Luttmer, and J.A. Scheinkman. (1997). Short- Term Interest Rates As Subordinated Diffusions. Review of Financial Studies, 10, [14] Ding, Z., C.W.J. Granger, and R. F. Engle. (1993). A long memory property of stock market returns and a new model. Journal of Empirical Finance, 1, [15] Duan, J.C. and K. Jacobsh. (1996). A simple long-memory equilibrium interest rate model. Economics Letters, 53, [16] Duan, J.C. and K. Jacobsh. (2002). Is long memory necessary? An emirical investigation of nonnegative interest rate processes. Working paper. [17] Jones, C.S. (2003). Nonlinear Mean Reversion in the Short-Term Interest Rate. Review of Financial Studies, 16, [18] Longstaff, F.A. and E.S. Schwartz. (1992). Interest Rate Volatility and the Term Structure: A Two-Factor General Equilibrium Model. Journal of Finance, 47, [19] Nelson, D.B. (1991). Conditional Heteroscedasticity in Asset Returns: A New Approach. Econometrica, 59,

23 [20] Sentana, E. (1995) Quadratic ARCH Models. Review of Economic Studies, 62, [21] Taylor, S.J. (2005) Asset price dymanics volatility, and prediction. Princeton University Press. 23