The Dynamics of the Australian Short-Term Interest Rate.
This paper examines various models of the short-term interest rate in Australia. The analysis centres on three classes of models. First, the generalised diffusion model of Chan et al. (1992) is examined which allows the variance to be a function of interest rate levels. This model nests a number of the traditional term structure models. We find initial support for the generalised model. Second, we examine models which incorporate time-varying volatility dynamics. Third, a class of models that incorporates both time-varying volatility and the levels model is analysed. We extend these models by allowing an asymmetric reaction to news resulting in a threshold-type model. The paper examines each of the models and then proposes and performs prediction tests that allow different classes of models to be benchmarked. The second and third class of models appear to produce the most accurate estimates. The results indicate a number of important differences between the Australian market and overseas markets.
INTEREST RATES; STOCHASTIC PROCESSES; TERM STRUCTURE; VOLATILITY.
The behaviour of the short rate has received considerable attention in the literature.(1) An understanding of the dynamic process of the evolution of the short rate has important implications for the term structure which in turn affects the pricing of fixed interest securities and associated derivative products. Further, information on the term structure is often required in the determination of economic policy. Hence it is important to develop, analyse and
assess the accuracy of proposed interest rate models.
An understanding of interest rate dynamics also has practical implications. Models of interest rates are being increasingly used in risk management.(2) Interest rate volatility is becoming important for banks, portfolio managers and corporate treasurers as they seek to hedge interest rate risk, implement optimal hedging strategies, establish bond trading strategies and make portfolio allocation decisions.
Many term structure models are based on assumptions about the stochastic process of the instantaneous interest rate (e.g. Brennan & Schwartz 1980; Cox, Ingersoll & Ross 1985; Vasicek 1977). An array of models has been proposed in the literature to describe the interest rate process with varying levels of support (see Chan et al. (1992) for a summary). Empirical tests have tended to focus on specific models and not compared results across models. Some tests have sought to compare the empirical performance of models in a nested framework, but this research has only recently developed and is heavily biased to the US market (e.g. Chan et al. 1992). Tests have yielded inconsistent results that vary over time, sample and type of security. Thus we still do not have a deep understanding of the dynamic processes underlying the interest rate.
In this paper, we provide an analysis of the short-term interest rate in Australia using a variety of data series covering a 20-year period. The paper's contribution is to add to the literature in this area by examining a variety of recently proposed models using a rich data set and extending the more recent models to capture asymmetric news effects. The paper compares the performance of the various models using a variety of prediction tests and examines the robustness of the findings to various refinements in method. The paper adds to the international evidence on this complex issue and compliments the recent work of Brenner, Harjes and Kroner (1996) by performing prediction tests that allow different classes of model to be benchmarked.
The models commence with the generalised diffusion model of Chan et al. (1992) which incorporates, through parameter restrictions, many of the traditional term structure models such as Merton, Vasicek, Cox et al. and Brennan-Schwartz. The generalised model allows for the stylised features of mean reversion and the relationship between the instantaneous variance and the level of interest rates. The paper then examines the class of models which incorporate time-varying volatility dynamics. The analysis then extends these models to incorporate both levels and time-varying volatility dynamics. We also propose a threshold-type model that captures asymmetric news effects. Through application of these models, the paper describes the interest rate process.
The paper is organised as follows. The next section describes the general diffusion model commonly applied in the literature. Section 3 discusses prior empirical evidence. Section 4 introduces time-varying volatility models. The discussion of data is contained in section 5. Sections 6 and 7 contain the main results from the study while section 8 provides a sensitivity analysis. Finally, section 9 presents the summary.
2. The Diffusion Model
Chan et al. (1992) specify the well-known generalised diffusion model that allows for mean-reversion and specifies the instantaneous change in the interest rate as:
(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
In this model, [dW.sub.t] is the increment to a Weiner process, the parameters [Alpha], [Omega] and [Gamma] are assumed to be non-negative and [r.sub.0] is assumed to be a fixed positive constant. The model allows for mean reversion and specifies the instantaneous variance as a function of the level of interest rates. The parameter [Beta] captures the speed of adjustment while [Alpha] is the product of the speed of adjustment and the long-run mean. For estimation purposes equation 1 can be discretised as:(3)
(2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
From equation 2, the mean reversion feature can be easily seen if [Beta] is negative. Rewriting [Alpha] + [Beta]r as [Beta](r - [Alpha]*) shows that the more negative the value of [Beta], the faster r responds to deviations from [Alpha]*. Conditional heteroscedasticity is introduced through a positive value of [Gamma]. The generalised model allows for various volatility functions through [Gamma]. Most theoretical interest rate models can be captured by imposing restrictions on the model including Brennan and Schwartz (1980), Cox, Ingersoll and Ross (1985), Cox and Ross (1976), Merton (1973) and Vasicek (1977). Table 1 shows the parameter restrictions implied in equation 2 by various theoretical models.(4)
Table 1 Implied Parameter Restrictions of Alternative Interest Rate Models(a) Model [Alpha] [Beta] 1 Merton Free(b) 0.0 2 Vasicek Free Free 3 Cox, Ingersoll & Ross SR Free Free 4 Cox, Ingersoll & Ross VR 0.0 0.0 5 Black-Scholes 0.0 Free 6 Brennan-Schwartz Free Free 7 Cox-Ross CEV 0.0 Free 8 Unrestricted CKLS Free Free Model [[Sigma].sup.2] [Gamma] 1 Free 0.0 2 Free 0.0 3 Free 0.5 4 Free 1.5 5 Free 1.0 6 Free 1.0 7 Free Free 8 Free Free
Note: (a.) The generalised model is given by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
(b.) `Free' indicates no restriction on the parameter.
From table 1, the advantages of the specification in equation 2 can be seen. Alternative interest rate models can be nested within the generalised model. This approach has been used before in Chan et al. (1992) and Dahlquist (1996). Model 1 is a Brownian motion with drift (from Merton 1973). Model 2 is the Ornstein-Uhlenbeck process reflecting an elastic random walk (from Vasicek 1977). Both of these models assume a constant variance. The other models allow for changing variance by specifying the instantaneous variance as a function of the level of interest rates. Model 3 is the Cox-Ingersoll-Ross (1985) mean reverting square root process which restricts [Gamma] to 0.5 thereby imposing a linear relationship between the instantaneous variance and the level of the interest rate. Model 4 is from Cox, Ingersoll and Ross (1980) which allows a non-linear relationship between the instantaneous variance and the level of the interest rate but has no mean reversion features. Model 5 is from Black and Scholes (1973) and is the geometric Brownian motion model (GBM). Model 6 is from Brennan and Schwartz (1980) and extends model 5 to allow the full features of mean reversion. Model 7 is the constant elasticity of variance model proposed by Cox and Ross (1976) which does not impose restrictions on the variance but does restrict the mean reversion term. Finally, model 8 is the unrestricted model which allows for mean reversion and changing variance which can be non-linear with respect to the level of interest rates (as in Chan et al. 1992).
3. Prior Evidence
There is no clear consensus on the most appropriate model for short-term interest rates. Tests of equations 1 and 2 have yielded inconsistent results. Chan et al. (1992) examine US Treasury bill data sampled at monthly intervals between 1964 and 1989. They estimate the models using generalised method of moments (GMM) and find only weak evidence of mean reversion but a high value of [Gamma] of about 1.5. In relation to the models in table 1, Chan et al. find that the model rankings generally depend upon the level of [Gamma]. Specifically, those models with a value of [Gamma] less than one are rejected while those with a value of [Gamma] greater than one are preferred. Moreover in a forecasting test which examines the variation in the volatility of the ex-post yield changes explained by the conditional expected volatility of yield changes, model [Gamma] (in table 1) has the highest explanatory power of around 20%.
The work of Chan et al. (1992) has been re-examined by Duffee (1993) who shows that the results are sensitive to the interest rate series and the relationship between volatility and the level of rates is highly sensitive to the period 1979 to 1982, known as the Federal Reserve experiment. Ball and Torous (1994) provide similar evidence. Bliss and Smith (1994) demonstrate sensitivity of the result to the Crash of October 1987. This latter evidence shows that the value of [Gamma] is less than the Chan et al. estimate of 1.5.
The GMM approach to estimation is sometimes used as it does not require normality to hold in relation to interest rate changes and the estimators and standard errors are consistent even in the presence of conditional heteroscedastic residuals. The approach tests the expressions in equation 2 as a set of overidentifying restrictions on a system of moment equations (see Hansen 1982 & Chan et al. 1992). However, Broze, Scaillet and Zakoian (1993) show that the GMM estimator of the generalised model is not well behaved if [Gamma] [is greater than] 1. Indeed, Pagan, Hall and Martin (1995) show that the estimate of [Gamma] is sensitive to the estimation process with differences between GMM and maximum likelihood (MLE). We return to this issue when examining the empirics.
Dahlquist (1996) examines various European interest rates sampled at monthly intervals over various periods between 1976 and 1994. Strong evidence of mean reversion is found for Denmark and Sweden but weaker evidence exists for Germany and the UK. In the unrestricted model, the value of [Gamma] is less than one for all countries except for Sweden where the value of [Gamma] is 1.15. However, the hypothesis of constant variance cannot be rejected for Germany and the UK. Dahlquist generally argues for the Brennan-Schwartz model (model 6 in table 1) for Denmark and Sweden, and for the Vasicek and Cox-Ingersoll-Ross SR models (models 2 and 3 in table 1 respectively) for Germany and the UK. Nowman (1997) also finds support for the Cox-Ingersoll-Ross SR model in British interest rate data and finds an insignificant value for [Gamma] of only 0.29. But Nowman generally confirms the Chan et al. (1992) USA result with a value of [Gamma] of 1.36. In Asian markets albeit with a different model, Cheng (1996) finds strong evidence of mean reversion in Hong Kong interest rates.
Gray (1996a) has examined the 90-day bank-bill series in Australia using a weekly series over the period 1978 to 1995. Gray estimates the unrestricted discretised diffusion model as in equation 2 in addition to employing a regimeswitching model with a GARCH process. In the discretised diffusion model he finds only weak evidence of mean reversion with the [Beta] estimate not significantly different from zero and a value of [Gamma] of about 1.5.
4. Time-Varying Volatility
Existing models have been generally criticised for their restrictive assumptions about the behaviour of volatility (Engle & Ng 1993a). For instance, a high value of [Gamma] implies a high degree of sensitivity to the level of interest rates yet periods of high interest rates but relative stability have been observed in various markets. Similarly, periods of low interest rates but relatively high volatility have also been observed. Further criticism is made by Brenner, Harjes and Kroner (1996) who point out that volatility is restricted to be a function of only the level of interest rates, yet in practice news arrival is likely to have a significant impact on movements in rates. This point has much appeal as casual observation of bond traders suggests that their actions are, at least in part, driven by changes in expectations due to news arrival.
Elsewhere, the ARCH class of models has been extensively used to capture time-varying volatility features of financial time series (see Bollerslev, Chou & Kroner 1992). ARCH models have been applied to interest rate data using the following model:
(3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
The added flexibility of ARCH models allows the interest rate volatility to be dependent on its own past series and innovations in the conditional mean. The model can still capture mean reversion in the series by including [r.sub.t-1] in the set of regressors [X.sub.t-1]. The conditional heteroscedasticity captures leptokurtosis in the unconditional distribution.
The major problem with fitting ARCH models to short-term interest rate data is that the estimated parameters often imply an explosive volatility process such that the conditional variance process is not covariance-stationary (Bollerslev 1986). These conditions are met when the sum of the conditional variance parameters (a + b) exceed unity (with [Omega] [is greater than] 0). For example, Engle, Lilien and Robins (1987) fit an ARCH-in mean model to quarterly excess holding yields of six-month T-bills over three-month T-bills and find a + b [is greater than] 1. Similarly, Hong (1988) finds a + b = 1.073 in a sample of excess holding yields of three-month T-bills over one-month T-bills while Engle, Ng and Rothschild (1990) find a + b = 1.0096 for a portfolio of US Treasury instruments. Gray (1996a) finds a + b = 1.1192 in Australian bank bill data. Gray (1996b) notes that the consequent implied persistence in volatility may be an artefact of changes in the underlying economic fundamentals. Following Lameroux and Lastrapes (1990), structural shifts in the unconditional variance may lead to misestimation of the ARCH parameters resulting in too much implied persistence. Attempts have been made to employ regime switching models (Sanders & Unal 1988) but these models generally rely upon ex-post identification of break points and hence are of limited ex-ante use.
Recently, Gray (1996b) has argued that regime shifts driven by changes in the economic fundamentals may be mistaken for periods of volatility clustering. Gray proposes a regime-switching model using a Markov process with state-dependent transition probabilities governing the switching between regimes. He tests the model on weekly observations of one-month US T-bill data and in a forecasting test finds that it generally outperforms the constant variance and standard GARCH models. Moreover, the estimates of a + b are less than one in each of the regimes. Gray (1996a) also applies his model to Australian bank bill data with similar results. However, in an important difference between the US and Australian results, he finds only weak evidence of mean reversion, particularly in high volatility regimes, and the persistence of shocks is greater in the high volatility regime which is contrary to US findings where persistence is greater in low volatility regimes. Indeed, the insignificance of the GARCH parameters in the low volatility regime suggests a constant variance model may be appropriate in these periods.
Brenner, Harjes and Kroner (1996) have also extended the ARCH model to incorporate time-varying parameters and news arrival effects. They develop a model which incorporates the effects of both levels and shocks. Brenner, Harjes and Kroner introduce a time-varying parameter model which nests both the standard levels model (as in equation 2) and the ARCH model (as in equation 3). They refer to the model as TVP-LEVELS. However this model has been criticised by Koedijk et al. (1997) because of difficulty in establishing the stationarity conditions. Instead, Koedijk et al. (KNSW) propose a model in which the prediction error in the Brenner, Harjes and Kroner model is scaled.
The model specification is:
(4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],
where: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
The model incorporates volatility conditioned on both the level of interest rates and a GARCH process. Shocks induced by news arrival will enter the system through the scaled error which flows through to have an impact on volatility. The model also nests other models. The model collapses to the levels model in equation 2 if in the conditional variance equation of equation 4, [a.sub.1] = b = 0. However, if [Gamma] = 0 then the model collapses to the ARCH process as in equation 3.
Following the work of Nelson (1991), Engle and Ng (1993b) and Glosten, Jagannathan and Runkle (1993) in relation to Threshold ARCH models, Brenner, Harjes and Kroner allow for an asymmetry between negative and positive shocks to volatility through modifying the conditional variance to include an asymmetric term. We do the same except to the KNSW conditional variance, viz:
(5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],
where: [[Eta].sub.t-1] = 1 if [e.sub.t-1] [is less than] 0 else [[Eta].sub.t] = 0.
We refer to the model in equation 5 as the Threshold KNSW (T-KNSW) model. The parameter [a.sub.2] is a measure of the difference in the slope coefficient between negative and non-negative scaled squared errors. In equation 5, if [a.sub.2] [is greater than] 0 then negative shocks have a larger impact on volatility than positive shocks.
Initially we extend the work of Gray (1996a) in Australia by re-examining the generalised model in equation 2 using a richer data set. Second, the paper then extends this research by examining the various models which are nested within the generalised model. Third, the paper examines further models of Koedijk et al. (1997) which incorporate richer dynamics for the volatility process. The paper proposes and performs a series of tests designed to assess the prediction accuracy of the various models which allows them to be benchmarked. These tests also give some practical insight into the issue by examining the magnitude of the prediction errors.
The study uses data on 30-day bank accepted bill (BAB) rates.(5,6) The use of 30-day BAB rates as opposed to other proxies employed elsewhere such as 90-day BAB rates, provides a better approximation of the unobservable short-term rate.(7) The data are sampled on a daily basis from July 1976 to December 1995 providing 20 years of data. Initially we focus on the weekly sampling interval but examine both daily and monthly intervals later in the paper.(8)
Figure 1 plots the annualised 30-day BAB rate in Panel A and changes in the annualised rate in panel B. Panel A highlights the considerable range in the nominal rate from a high of over 21% in 1982 to a low of under 5% in 1993. There also appears to be considerable volatility in the early to mid-1980s which is reflected in panel B. This period was associated with a change in government and policy which resulted in substantial deregulation of financial markets including the floating of the Australian dollar in 1983. Arguably, there have been three general phases of government policy over the period comprising monetary targeting between 1976 and about 1984, a checklist approach between 1985 and 1989 and an anti-inflation policy between 1990 and 1995 (see Argy 1992).
[Figure 1 ILLUSTRATION OMITTED]
Table 2 presents the summary statistics of changes in interest rates. From table 2, it is apparent that the series is highly non-normal. The tests for normality indicate negative skewness, excess kurtosis and autocorrelation in rate changes. The high degree of excess kurtosis in the unconditional distribution indicates that stochastic volatility models may be well suited. We now turn to the modelling of the series.
Table 2 Summary Statistics of Weekly Changes in 30-day BAB Rates Data are drawn from July 1976 to December 1995 No. of observations 1,017 Mean (%) -0.28 Standard deviation (%) 60.99 Skewness -1.7903(*) Excess kurtosis 69.45(*) Jarque-Bera 204,707(*) ADF(10)(1) r -0.662 ADF(10)(1) [Delta] r -9.816(*) Ljung-Box (10)(2) [Delta] r 22.492(*) Ljung-Box (20)(2) [Delta] r 39.929(*) Ljung-Box (10)(2) [Delta] r(2) 14.218 Ljung-Box (20)(2) [Delta] r(2) 27.591
Notes: (1.) The ADF(10) test is the augmented Dickey-Fuller test with 10 lags where the test statistic is for all coefficients jointly equal to zero.
(2.) The Ljung-Box(n) test for first to nth order correlation is approximately distributed as 352 with n degrees of freedom.
(*) Significant at 5%.
6. Empirical Estimates
Initially the GMM approach of Chan et al. (1992) is used. The initial purpose is to compare the nested models within the generalised diffusion model. MLE cannot be used as the models are strictly non-nested and hence formal comparison tests are not possible to construct. Hence we use the more flexible GMM estimation for relativity purposes and avoid potential problems. However, MLE is subsequently employed in the later estimation. The parameter estimates are reported in table 3.
Table 3 GMM Estimates of the Discretised Diffusion Model
Model estimates are based on the discretised version of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] given by equation 2. Estimates are obtained under GMM with asymptotic Newey-West t-ratios reported in parentheses. Data are weekly changes in 30-day BAB rates: 7/1976 to 12/1995.
Model [Alpha] [Beta] [[Sigma].sup.2] [Gamma] Merton 0.0049 0.00 0.3731 0.0 (0.37) (6.35)(*) Vasicek 0.0482 -0.0045 0.3941 0.0 (1.15) (-1.11) (6.70)(*) CIR-SR 0.0667 -0.0061 0.1322 0.5 (1.50) (-1.42) (7.07)(*) CIR-VR 0.00 0.00 0.0968 1.5 (5.86)(*) Black-Scholes 0.00 0.0002 0.0352 1.0 (0.16) (6.22)(*) Brennan-Schwartz 0.0910 -0.0081 0.0419 1.0 (1.89) (-1.79) (7.34)(*) Cox-Ross 0.00 -0.0001 0.6795 -0.2855 CEV (-0.05) (0.29) (-0.18) Unrestricted 0.1331 -0.0117 0.0077 1.7039 CKLS (1.78) (-1.77) (0.50) (2.12)(*) Model D-statistic(a) Hansen(b) d.o.f. (p-value) (p-value) Merton 3.127 3.096 2 (0.105) (0.106) Vasicek 2.023 1.951 1 (0.102) (0.107) CIR-SR 1.293 1.264 1 (0.184) (0.189) CIR-VR 3.320 5.369 3 (0.138) (0.062) Black-Scholes 3.911 4.229 2 (0.071) (0.060) Brennan-Schwartz 0.562 0.558 1 (0.402) (0.404) Cox-Ross 3.156 3.208 1 CEV (0.046)(*) (0.045)(*) Unrestricted - - - CKLS
Notes: (a.) The D-statistic is from Newey and West (1987) and is approximately [Chi square].
(b.) The Hansen test statistic is from Hansen (1982) and is approximately [Chi square].
(c.) Significant at 5%.
From table 3, there is only weak evidence of mean reversion across all models. While the sign on [Beta] is generally negative it never reaches statistical significance. When restrictions are placed on [Gamma], all estimates of [[Delta].sup.2] are significant. However, when [Gamma] is unrestricted in the models of Cox-Ross CEV and unrestricted CKLS, the coefficient estimate on [[Delta].sup.2] is insignificant. Notwithstanding potential estimation problems with GMM, the value of [Gamma] in the unrestricted model is significant and about 1.7 which is generally consistent with other evidence (Chan et al. 1992). However, the coefficient estimate has a large standard error and while it is significantly different from zero, it is not significantly different from one (t-stat: 0.88).
A more formal comparison of the models can be made using the tests of Newey and West (1987) and Hansen (1982).(9) A significant test statistic suggests that the model is misspecified. Only one model being the Cox-Ingersoll-Ross CEV model can be rejected at the 5% level. However, other models are only marginally accepted such as Merton, Vasicek, Cox-Ingersoll-Ross VR and the Black-Scholes GBM. The two models which stand-out are Brennan-Schwartz and the Cox-Ingersoll-Ross SR process. These models are clearly accepted and cannot be distinguished from the unrestricted model.(10) This result is generally consistent with those documented in overseas markets.
Nevertheless, diagnostic tests of the residuals indicate that both skewness and excess kurtosis remain. Formal tests indicate non-normality including the presence of ARCH errors.(11)
Given the potential problems of GMM estimation in this area, the unrestricted model is also estimated under MLE. The results are reported in table 4. While there is little difference in the significance of the parameter estimates, the magnitudes differ when compared to table 3. While there is again little evidence of mean reversion, the magnitude of ([[Delta].sup.2] increases while [Gamma] falls to 1.14, but is not significantly different from one (t-stat: 0.43).
Table 4 MLE Estimates of the Discretised Diffusion Model
Model estimates are based on the discretised version of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] given by equation 2. Estimates are obtained under MLE with asymptotic White t-ratios reported in parentheses. Data are weekly changes in 30-day BAB rates: 7/1976 to 12/1995.
Model [Alpha] [Beta] [[Sigma].sup.2] [Gamma] Unrestricted 0.0450 -0.0036 0.0033 1.1448 (1.07) (-0.81) (1.21) (3.45)(*)
Note: (*) Significant at 5%.
Similar to the GMM diagnostics, tests of the residuals indicate that both skewness and excess kurtosis remain. Formal tests indicate non-normality including the presence of ARCH errors.(12)
Given the above findings in relation to the diagnostic tests, it is appropriate to examine further volatility models. First, a standard GARCH(1,1) model is estimated using equation 3 and imposing positivity and stationarity constraints. The MLE parameter estimates are reported in table 5 together with t-ratios computed from Bollerslev-Wooldridge robust standard errors.(13)
Table 5 Estimates of the GARCH Model
Model estimates are based on the GARCH (1, 1) model given by equation 3. Estimates are obtained under MLE with robust t-ratios reported in parentheses. Data are weekly changes in 30-day BAB rates: 7/1976 to 12/1995.
Parameter estimate (Robust t-ratio) [[Beta].sub.0] 0.0257 (0.56) [[Beta].sub.1] -0.0054 (-0.71) [Omega] 0.0002 (0.83) a 0.0822 (2.94)(*) b 0.9161 (32.85)(*) Log-likelihood -311.17 Mean [[Epsilon].sub.t]/[h.sub.t] 0.0118(*) Standard deviation [[Epsilon].sub.t]/[h.sub.t] 1.0834 Skewness [[Epsilon].sub.t]/[h.sub.t] -0.0190 Excess kurtosis [[Epsilon].sub.t]/[h.sub.t] 7.6951(*) Ljung-Box (20) [MATHEMATICAL EXPRESSION 16.00 NOT REPRODUCIBLE IN ASCII] Jarque-Bera [[Epsilon].sub.t]/[h.sub.t] 2,506.84(*)
Note: (*) Significant at 5%.
The GARCH model estimates reveal a negative but insignificant estimate on [[Beta].sub.1] implying only weak evidence of mean reversion. The mean reversion features are stronger in other markets but our results are consistent with other Australian research (Gray 1996a). Both the GARCH parameters are highly significant. The restrictions result in an integrated model (a + b ~ 1.0). A likelihood ratio test strongly indicates that the GARCH model is preferred over the constant variance model in which a = b = 0 (LR: 1,125.88 ~ [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]).(14)
The diagnostics from the GARCH model indicate better `behaved' residuals. The previously significant value of skewness is eliminated and the Ljung-Box test for autocorrelation in the squared residuals is insignificant as is the ARCH test (p-value: 0.66 - 20 lags). However, there is still non-normality in the standardized residuals and this appears primarily driven by excess kurtosis. The Jarque-Bera test is sensitive to departures from normality and it is the kurtosis estimate that drives this test statistic. The model was re-estimated under the assumption of a conditional t-distribution in an attempt to remove the excess kurtosis. However this distribution was also unable to fully capture the excess kurtosis.(15,16)
The GARCH model is somewhat restrictive because of its requirement that volatility is not dependent on the level of interest rates, which is contrary to the theoretical literature. The model is extended following the specification of KNSW given by equation 4. This model allows for time-varying volatility which is also a function of the level of interest rates. We also employ the augmented conditional variance in equation 5 which allows for asymmetry in the reaction to news arrival (T-KNSW model). The models are estimated under MLE with constraints on positivity and stationarity.(17) The results are reported in table 6.
Table 6 Estimates of the KNSW Models
Model estimates are based on the models given by equations 4 and 5. Estimates are obtained under constrained MLE with robust t-ratios reported in parentheses. Data are weekly changes in 30-day BAB rates: 7/1976 to 12/1995.
KNSW Parameter estimate (Robust t-ratio) [[Beta].sub.0] -0.0018 (-0.01) [[Beta].sub.1] -0.0004 (-0.17) [Gamma] 1.2075 (2.35)(*) [Omega] 0.0067 (0.29) [a.sub.1] 0.1617 (1.12) [a.sub.2] -- b 0.8373 (2.35)(*) Mean [[Epsilon].sub.t]/[h.sub.t] 0.0192 Standard deviation [[Epsilon].sub.t]/[h.sub.t] 1.0094 Skewness [[Epsilon].sub.t]/[h.sub.t] -0.2267 Excess kurtosis [[Epsilon].sub.t]/[h.sub.t] 7.0549(*) Ljung-Box (20) [MATHEMATICAL EXPRESSION 16.48 NOT REPRODUCIBLE IN ASCII] Jarque-Bera [[Epsilon].sub.t]/[h.sub.t] 2,113.63(*) T-KNSW Parameter estimate (Robust t-ratio) [[Beta].sub.0] -0.0004 (-0.02) [[Beta].sub.1] -0.0006 (-0.18) [Gamma] 1.1506 (0.74) [Omega] 0.0049 (0.11) [a.sub.1] 0.1886 (0.48) [a.sub.2] -0.0309 (-0.14) b 0.8351 (2.27)(*) Mean [[Epsilon].sub.t]/[h.sub.t] 0.0170 Standard deviation [[Epsilon].sub.t]/[h.sub.t] 1.0000 Skewness [[Epsilon].sub.t]/[h.sub.t] -0.2968 Excess kurtosis [[Epsilon].sub.t]/[h.sub.t] 7.2400(*) Ljung-Box (20) [MATHEMATICAL EXPRESSION 16.96 NOT REPRODUCIBLE IN ASCII] Jarque-Bera [[Epsilon].sub.t]/[h.sub.t] 2,230.00(*)
Note: (*) Significant at 5%.
The KNSW specification appears to offer a richer model than the GARCH model. Of note, the ARCH parameter (a[.sub.l]) loses its significance when compared with the standard GARCH (1,1) model although the GARCH parameter (b) remains significant. The drop in significance of the ARCH parameter is consistent with a reduced impact of news when the level parameter is added. Elsewhere Brenner, Harjes and Kroner (1996) in a similar specification in the US market also find a drop in the value of this parameter although it remains significant. The GARCH parameter (b) remains highly significant under both the standard and asymmetric specifications. Importantly the coefficient estimate on [Gamma] is significant. The value of this parameter is above one but not statistically different from one (t-stat: 0.40). This result is consistent with the results of Keodijk et al. (1997), however Brenner, Harjes and Kroner (1996) report a value of y around 0.5. The significance of [Gamma] implies that volatility is time-varying and an increasing function of interest rate levels.
In the threshold model, the asymmetric term ([a.sub.2)] is insignificant. In a comparison of the two models, the T-KNSW model is not preferred over the standard KNSW model in a likelihood ratio test (LR: 1.14). In the only other test of a similar augmented model, Brenner, Harjes and Kroner (1996) report a significant asymmetric parameter. We find no such evidence in the Australian market which is consistent with the lack of excess skewness in the residuals from the standard GARCH model or consistent with the standard KNSW model's asymmetric features which are already capable of capturing some of these effects.(18)
7. Comparison of the Models
The ARCH and ARCH-Levels class of models cannot be formally compared against the other non-constant volatility `levels' models nested in the generalised diffusion model of equation 1. However, in this section we compare the models by examining their performance in relation to implied predictions of volatility. These tests allow the models to be benchmarked against each other. For a visual comparison of the models, figure 2 plots the absolute interest rate change against the conditional standard deviation estimates obtained from three models--the unrestricted CKLS discretised diffusion model as in equation 2, the GARCH model as in equation 3 and the KNSW model as in equation 4.(19)
[Figure 2 ILLUSTRATION OMITTED]
The series which appears to most closely track the realised absolute rate change is from the KNSW model. The CKLS estimates closely follow the pattern of interest rate levels and tend to be too smooth. The GARCH model appears more accurate and its estimates closely follow those from the KNSW model. However, the KNSW model appears to do a slightly better job in accounting for the high volatility periods of the 1980s and early 1990s and is more responsive to sudden changes. Further, the GARCH model tends to exhibit greater volatility persistence. However, there are instances where even the KNSW model generally underestimates the level of realised volatility. As shown earlier in the paper in figure 1, the most volatile period was in late 1982 when rates fell from around 14% to 9% in one month. This volatility was essentially market driven although it occurredduring a period of on-going financial reform and a period of monetary targeting.(20)
To more accurately assess the relative performance of the various models we first examine some error metrics. Table 7 reports the mean absolute error, root mean squared error and mean percentage error in which the volatility prediction is used for scaling. Across all metrics, the worst performing model is the generalised CKLS while GARCH is marginally preferred over the KNSW predictions. However all models generate substantial errors. For instance, even GARCH has a mean absolute error of about 28 basis points and a mean percentage error of around 36%.
Table 7 Error Metrics of Deviations of Predicted Interest Rate Volatility from Actual Interest Rate Volatility
Interest rate volatility predictions are implied from parameter estimates obtained for the unrestricted CKLS discretised diffusion model as in equation 2, the GARCH model as in equation 3 and the KNSW model as in equation 4.
KLS Model GARCH Model KNSW Model Mean Absolute Error 0.7578 0.2827 0.3061 Root Mean Square Error 0.8469 0.4397 0.5063 Mean Percentage Error (%) 0.7317 0.3598 0.3923 scaled by prediction
A potential problem with the error metrics in table 7 is that they may favour models that have more parameters. An alternative method of comparison is to run a regression of the three series of conditional standard deviation estimates against the absolute interest rate change. This is a form of an encompassing test which forces the in-sample forecasts to compete against each other for explanatory power (Fair & Shiller 1990). The regression takes the form:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
If one series dominates all others then its parameter estimate will be one with a zero intercept. The results are reported in table 8. All coefficient estimates (except the intercept) are significantly different from zero but are also significantly different from unity implying that each has some explanatory power but no one coefficient dominates to the exclusion of the others. However, the largest magnitude is on the volatility estimates from the GARCH model. The parameter on the volatility estimates from the KNSW model has a negative sign implying that it is basically subsumed by the GARCH model estimates.
Table 8 Regression Estimates of Conditional Volatility Series on Realised Absolute Rate Changes
The regression model is: [[Sigma].sub.rt] = [Alpha] + [[Beta].sub.1] [[Sigma].sub.CKLS,rt] + [[Beta].sub.2] + [[Sigma].sub.GARCH,rt] + [[Beta].sub.3] [[Sigma].sub.KNSW,rt] + [[Epsilon].sub.rt]. Conditional volatility estimates are from the unrestricted CKLS discretised diffusion model as in equation 2, the GARCH model as in equation 3 and the KNSW model as in equation 4\). T-statistics reported in parentheses under the null [H.sub.o]: [Alpha] = 0 and [[Beta].sub.i] = 1.
[Alpha] [[Beta].sub.1] [[Beta].sub.2] (t-statistic) (t-statistic) (t-statistic) -0.054 0.122 0.707 (-1.73) (-26.91)(*) (-4.52)(*) [Alpha] [[Beta].sub.3] [R.sup.2] (t-statistic) (t-statistic) -0.054 -0.209 0.316 (-1.73) (-20.36)(*)
Note: (*) Significant at 5% in a test where [H.sub.o]: [Alpha] = 0 and [[Beta].sub.i] = 1.
8. Sensitivity Analysis
Prior studies have demonstrated that the results can be sensitive to the data. In this section, we briefly discuss the results from alternative tests which use different data and sample over different frequencies.
To date, we have focused on 30-day BAB rates sampled at weekly intervals. An often used security in Australian empirical studies is the 90-day BAB (e.g. Gray 1996a). However, BABs are typically traded as 30-day, 90-day and 180-day bills. Ideally we wish to examine the instantaneous risk-free interest rate, but data on such series are not available. While BABs are generally regarded in Australia as `government-guaranteed', Treasury notes (T-notes) are issued by the government and may be regarded as slightly less risky. That is, BABs may carry a small default risk premium. However, T-note data are available over a shorter sample period than BAB data, and are quoted as a weighted average of the actual yields and therefore the quotes may not represent actual traded yields.(21) T-notes are issued with two maturity dates--13 weeks and 26 weeks.(22) To examine the sensitivity of the results to the chosen proxy, the analysis is run on 90-day BAB rates, 180-day BAB rates, 13-week T-bill rates and 26-week T-bill rates. Table 9 presents the results from the unrestricted discretised generalised model as in equation 2.
Table 9 Estimates of the Discretised Diffusion Model Across Different Data Series
Model estimates are based on the discretised version of [dr.sub.t] = ([Alpha] + [Beta] [r.sub.t]) dt + [MULTIPLE EXPRESSION NOT REPRODUCIBLE IN ASCII] given by equation 2. Estimates are obtained under MLE with asymptotic White t-ratios reported in parentheses. Data are weekly changes in rates: 7/1976 to 12/1995 for BAB series and 1/1980 to 12/1995 for T-note series.
Data Series [Alpha] [Beta] [[Sigma].sup.2] [Gamma] 30-day BAB 0.0450 -0.0036 0.0033 1.1448 (1.07) (-0.81) (1.21) (3.45)(*) 90-day BAB 0.0118 -0.0007 0.0105 1.4451 (0.69) (-0.31) (2.86)(*) (10.13)(*) 180-day BAB 0.0117 -0.0100 0.0121 1.3277 (0.68) (-0.47) (3.22)(*) (10.23)(*) 13-week T-note 0.0139 -0.0007 0.0132 1.4342 (0.67) (-0.25) (2.43)(*) (8.31)(*) 26-week T-note 0.0118 -0.0009 0.0124 1.4204 (0.53) (-0.28) (2.34)(*) (7.99)(*)
Note: (*) Significant at 5%.
Across the three bank accepted bill series, the results are reasonably consistent in terms of both parameter magnitude and significance. The same conclusion applies to the two T-note series. Indeed, the T-note results are very similar to the 90-day BAB and 180-day BAB results with significant estimates of [Gamma] around 1.4. Hence any possibility of a default premium does not appear to affect the results.
The sampling interval can also have an affect on empirical results. This has been shown to be particularly the case for asset pricing tests in the equity market (e.g. Handa, Kothari & Wasley 1989, 1993). Further, in the case of stochastic volatility models, the sampling interval is known to affect the empirical estimates (e.g. Brailsford 1995). There are theoretical reasons for this as advanced by Nelson (1990) who shows that under certain conditions, as the sampling interval is reduced in the limit, the ARCH process approximates a diffusion process. Further, as unconditional normality becomes a more reasonable assumption over longer term frequencies such as monthly, ARCH effects tend to diminish (Bollerslev, Chou & Kroner 1992).
A priori, we may expect that as the sampling interval is reduced, we obtain closer discrete estimates of the diffusion process parameters. To analyse the impact of the return interval, the models are re-run using quarterly, monthly, weekly and daily series.
From table 10, the estimated value of the intercept ([Alpha]) increases monotonically with the increase in the sampling interval as expected. The mean reversion parameter is negative at all frequencies but is always insignificant. The volatility parameters are more difficult to interpret. The values of [Gamma] indicate the strongest influence of the level of interest rates on conditional volatility occurs at the monthly interval although while [Gamma] is significantly different from zero it is not significantly different from one for any of the daily, weekly or monthly frequencies. The value of [Gamma] falls quite sharply from the monthly to quarterly intervals and is no longer significant: This result implies a constant variance model for the quarterly frequency. This result is consistent with results from ARCH models run on low frequency data in equity markets where the ARCH effects disappear (see Bollerslev, Chou & Kroner 1992), although it may also be attributable to the small sample size. The strength of the mean reversion parameter is strongest at the quarterly frequency. These results imply that different models may be appropriate for different sampling frequencies as different dynamics are at work across different time intervals.
Table 10 Estimates of the Discretised Diffusion Model Across Different Sampling Intervals
Model estimates are based on the discretised version of [dr.sub.t] = ([Alpha] + [Beta] [r.sub.t]) dt + [MULTIPLE EXPRESSION NOT REPRODUCIBLE IN ASCII] given by equation 2. Estimates are obtained under MLE with asymptotic White t-ratios reported in parentheses. Data are weekly changes in 30-day BAB rates: 7/1976 to 12/1995
Data n [Alpha] [Beta] [[Sigma].sup.2] [Gamma] Series Daily 4,947 0.0115 -0.0010 0.0312 0.7894 (1.33) (-1.22) (1.68) (3.34)(*) Weekly 1,017 0.0450 -0.0036 0.0033 1.1448 (1.07) (-0.81) (1.21) (3.45)(*) Monthly 233 0.1055 -0.0069 0.0395 1.3343 (0.95) (0.50) (1.56) (5.41)(*) Quarterly 79 1.2386 -0.1090 0.7775 0.3990 (1.41) (-1.47) (0.81) (0.83)
Note: (*) Significant at 5%.
The behaviour of interest rates and their associated volatility is an increasingly important issue in financial markets. This paper adds to the international evidence on this issue by examining Australian interest rates over a 20-year period. In a comparison of traditional term structure models we find initial support for the generalised model of Chan et al. (1992). However this model is not preferred over other models which incorporate time-varying dynamics of volatility. The GARCH model and the Koedijk et al. (1997) model which incorporates both ARCH and levels effects are superior models in capturing the dynamics of the volatility process. An examination of volatility prediction errors marginally favours the GARCH model. However, while the models demonstrate some ability to capture the interest rate process, they all yield reasonably large errors which is indicative of the difficult nature of interest rate modelling.
(Date of receipt of final typescript: July 1998 Accepted by Garry Twite, Deputy General Editor.)
We acknowledge the constructive comments of Richard Heaney, Allan Hodgson, Tom Nguyen, Barry Oliver, Klm Sawyer, Mark Tippett and participants at the Quantitative Methods in Finance Conference (Cairns 1997), AAANZ Conference (Adelaide 1998), workshops at the ANU and Griffith University, and two anonymous reviewers.
(1.) For example, see Brenner, Harjes and Kroner (1996), Chan et al. (1992), Cox, Ingersoll and Ross (1985), Dahlquist (1996), Gray (1996a), Koedijk et al. (1997), Nowman (1997), Sanders and Unal (1988) and Vasicek (1977).
(2.) For instance, the Value-At-Risk methodology adopted by many financial institutions involves an understanding of the evolution of interest rates over time.
(3.) The discretised version in equation 2 is based on the Euler scheme and is popularly used as an approximation for the true process (e.g. Brenner, Harjes & Kroner 1996; Chan et al. 1992; Dahlquist 1996; Gray 1996b; Pagan, Hall & Martin 1995). However, the use of maximum likelihood upon the discretisation may not yield consistent estimators. We are grateful to Mark Tippett for alerting us to this issue (see Rhys & Tippett 1993). However, the use of high frequency data such as daily intervals, mitigates against the deficiencies of the approximation. Moreover, an advantage of this approximation is the ability to nest the various models that is not generally possible with other approximations.
(4.) There are many other permutations of parameter restrictions that can be imposed on the model. However, the imposition of other restrictions beyond those combinations in table 1 makes little theoretical sense and consequently can lack economic meaning. Hence, we do not examine other combinations of restrictions in the empirical tests.
(5.) Theories of the term structure generally revolve around the instantaneous risk-free rate. However, in Australia, sufficient time-series of yields on short-term government issued securities are not available. The default risk premium on BAB is typically very small as these bills are generally backed by one of the four major banks. Valentine (1991) notes that BABs carry a credit risk equivalent to that of Treasury notes as the market regards banks as being government guaranteed.
(6.) The data are initially obtained from the Reserve Bank of Australia and are verified and supplemented where missing observations exist from DataStream and various financial publications.
(7.) The robustness of the results to 90-day BAB and 180-day BAB series is examined in a later section of the paper.
(8.) Weekly observations are taken from Wednesday-to-Wednesday to minimise the impact of public holidays. The basic analysis was also repeated using observations Thursday-to-Thursday and Friday-to-Friday observations with little difference in the results.
(9.) We use the D-statistic from Newey and West (1987) which is approximately [chi square] and the Hansen test statistic from Hansen (1982) which is approximately [chi square].
(10.) There are other comparisons which can made across various models such as the Merton and Vasicek models and the Black-Scholes and Brennan-Schwartz models. However, the benchmark is the unrestricted model and we focus on this in the discussion.
(11.) For instance, tests on the residuals reveal the following statistics: skewness is -1.518, excess kurtosis is 67.97, Engle's ARCH test is significant 209.8 (10 lags) and the Ljung-Box test is significant in both levels and squares (10 lags).
(12.) Tests on the residuals reveal the following statistics: skewness is -1.653, excess kurtosis is 69.16, Engle's ARCH test is significant 204.2 (10 lags) and the Ljung-Box test is significant in both levels and squares (10 lags).
(13.) The robust standard errors follow from Bollerslev and Wooldridge (1992) which provide for asymptotic standard errors that are robust to departures from normality.
(14.) A threshold GARCH model was also estimated following Glosten, Jagannathan and Runkle (1993). However, the asymmetry parameter was insignificant.
(15.) For example, while the additional degree-of-freedom parameter estimate was significant, it is still more than two standard deviations away from the theoretical estimate given by 3([Upsilon]-2)/([Upsilon]-4) (see Bollerslev 1987). Further, the excess kurtosis in the standardised residuals remained high at 10.17.
(16.) Alternative distributions such as the GED or mixture of normals are left for future work.
(17.) The models were also estimated without constraints. However, for both the KNSW and T-KNSW specifications, the constrained model resulted in a lower value of the log-likelihood. Moreover, the unconstrained models breached the stationarity conditions (KNSW: [a.sub.1] + b = 1.09, T-KNSW: [a.sub.1] + b = 1.04). However, a strict interpretation of the constraints is difficult as volatility persistence is now a function of both the volatility parameter and the level of interest rates.
(18.) For instance, consider a negative shock. In the KNSW model, the volatility clustering features will increase volatility while the decrease in the level of interest rates will dampen volatility. Thus the forces offset each other whereas with a positive shock the forces work together.
(19.) Given the lack of superiority of the threshold model we concentrate only on the KNSW model.
(20.) For example, in 1983-1984 the Government allowed the entry of foreign banks into the previously restricted banking sector and in late 1983, the Australian dollar was floated (see Harper & Scheit (1992) for further discussion).
(21.) The Reserve Bank of Australia issues T-notes through a tendering process on a regular basis. The quoted T-note rates are computed as a weighted average of the yields on notes issued at tender.
(22.) T-notes with five weeks to maturity have also been issued since November 1991.
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T. J. Brailsford, Department of Commerce, Australian National University, Canberra ACT 0200; E-mail: Dean.email@example.com
K. Maheswaran, Department of Accounting and Finance, University of Melbourne, Parkville VIC 3052.3
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|Publication:||Australian Journal of Management|
|Date:||Dec 1, 1998|
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