**4. Empirical methodology and results**

#### **4.1. Preliminary observations**

100 Risk Management – Current Issues and Challenges

observations. [10]

series is antipersistent.

**3. Data description** 

2001 to April 9, 2012, representing 2571 observations.

in view of its nonperiodic (flooding) cycles. [7] This feature was described as the "Joseph effect" alluding to the biblical reference in which seven years of plenty where to be followed by seven years of famine. [8] In this sense, long memory process concern observations in the remote past that are highly correlated with observations in the distant future. The implications of long memory in financial markets was related to the use of Hurst's 'rescaled range' statistic to detect long memory behaviour in asset return data. [9] It was observed that if security prices display long memory then the arrival of new market information cannot be arbitraged way, which in turn means that martingale models for security prices cannot be derived through arbitrage. As such, long memory processes can be characterised as having fractal dimensions, in the form of non-linear behaviour marked by distinct but nonperiodic cyclical patterns and long-term dependence between distant

A variety of measures have been used to detect long memory in time series. For example, in the time domain, long memory is associated with a hyperbolically decaying autocovariance function. Meanwhile, in the frequency domain, the presence of long memory is indicated by a spectral density function that approaches infinity near the zero frequency; in other words, such series display power at low frequencies. [11] Finally, a pattern of self-similarity in the aggregated sequences of a time series is an indicator of long memory (this refers to the property of a self-similar process, in which, different time aggregates display the same autocorrelation structure). These notions have led several authors to develop stochastic models that capture long memory behaviour, such as the fractionally-integrated I(*d*) time series models. [12-13] In particular, fractional integration theory asserts that the fractional difference parameter which indicates the order of integration, is not an integer value (0 or 1) but a fractional value. Fractionally integrated processes are distinct from both stationary and unit-root processes in that they are persistent (i.e., they reflect long memory) but are also mean reverting and as a consequence provide a flexible alternative to standard I(1) and I(0) processes. [14] Specifically, the long memory parameter is given by *d* 0, 0.5 while when *d* > 0.5 the series is nonstationary and when *d* 0.5, 0 the

Since, non-zero values of the fractional differencing parameter imply dependence between distant observations, considerable attention has been directed to the analysis of fractional dynamics in financial time series data. Indeed, long memory behaviour has been reported in the returns of various asset classes. [15] Against this background, a rapidly expanding set of

The data analysed in this study are obtained from the global bond index (GBI) series for emerging markets (EM) compiled by JP Morgan. In particular, the fixed income data used comprise of daily total returns for Hong Kong, Mexico, and South Africa from December 31,

models has been developed to capture long memory dynamics is asset return data.

Table 1 presents the time series properties of the data using some basic methods. The results of the Augmented Dickey-Fuller (ADF) unit root test offer evidence in favour of stationary fixed income returns. While this test may be deficient in terms of its ability to capture an order of integration that may not be an integer, the finding of stationary bond returns is consistent with those of many previous studies. [15] However, based on the standard normality and Lagrange Multiplier ARCH tests, fixed income return data exhibit non-normality and ARCH effects. [16-17] These non-white noise characteristics of the data motivate estimation of GARCH(1,1) model using the assumption of the Student *t* distribution.


Note:

1/ '\*\*' and '\*' indicate statistical significant at the 1% and 5% levels, respectively.

2/ Normality test follows a Chi-squared distribution

3/ ARCH (x) test follows an F-statistic with parameters (x, n-x)

**Table 1.** Description of the Data

### **4.2. GARCH(1,1) model**

The GARCH (1,1) specification comprises a return (or mean) and a variance equation. In particular, the returns generating process can be described by:

$$
\sigma\_t = \mu + \varepsilon\_t \quad \text{where} \quad \varepsilon\_t \left| \Phi\_{t-1} - N\left(0, h\_t\right) \right. \tag{2}
$$

Long Memory in the Volatility of Local Currency Bond Markets:

time and frequency, they provide a means to collect information on both the frequency and time characteristics of a time series. In fact, wavelets are already widely used as detectors of patterns in areas as diverse as digital signal processing and exploration geophysics. In the empirical finance literature they have recently been used to determine time-dependence in asset return data by comparing the scale decompositions of observations that exhibit significant autocorrelation between observations widely separated in time. [18] In this manner, the scaling properties of daily returns of emerging market government bonds are

Mexico South Africa Hong Kong

4.0842 5.4248 2.1147 1.4692 3.3072\*\* 8.3955

A wavelet is defined as a wave-like function with an amplitude that oscillates around zero and has a finite or quickly decreasing time support. These functions are well suited to locally approximating variables in time or space as they have the ability to be manipulated by being either 'stretched' or 'squeezed' so as to mimic the series under investigation. [19] The power of wavelet analysis is that it makes it possible to decompose a time series into its high- and lowfrequency components, which are localised in time. Wavelets, also allow the selection of an appropriate trade-off between resolution in the time and frequency domains, while traditional Fourier analysis stresses resolution in the frequency domain at the expense of the time domain. [20] Wavelets therefore provide a convenient and efficient method to analyse complex signals. [21] Wavelet theory is applicable to several subjects. They are especially useful where a signal (e.g. long memory) lasts for a finite time or shows markedly different behaviour in different time periods. These methods, have emerged as a useful tool in the empirical finance literature

A discrete wavelet transform (DWT) is any wavelet transform for which the wavelets are discreetly sampled, as is often the case in econometric analysis. A wavelet transform is a

0.0031 [0.0007]\*\* 0.0756 [0.0118]\*\* 0.9026 [0.0123]\*\*

analysed in order to capture temporal dependencies in the volatility process.

0.0029 [0.0006]\*\* 0.1245 [0.0166]\*\* 0.8530 [0.0170]\*\*

where long-run and short-run relationships can be distinguished. [22]

1.2197 1.8322 1.1775 1.3357 3.1661\*\* 7.7588\*\*

1/ The Ljung-Box Q test applied to standardized residuals. 2/ The Ljung-Box Q test applied to squared standardised residuals 3/ The numbers in () and [] refer to lag lengths and standard deviations

4/ '\*\*' and '\*' indicate significance at the 1% and 5% levels

**4.4. The discrete wavelet transform** 

ω α β Q(5) 1/ Q(5) 2/ Sign bias test

Negative size bias test Positive size bias test

**Table 2.** GARCH Estimates

Joint test

Notes:

Evidence from Hong Kong, Mexico and South Africa 103

0.0003 [0.0001]\* 0.0510 [0.0089]\*\* 0.9402 [0.0095]\*\*

1.0838 1.3785 2.6833 2.2911 5.6812\*\* 11.3702\*\*

where *<sup>t</sup> r* denotes the returns process, which may include autoregressive and moving average components, and *εt* is the error term, which is assumed to be normally distributed with zero mean and variance *<sup>t</sup> h* , given the information set *t*<sup>1</sup> . The conditional variance is modelled as:

$$
\hbar h\_t = \alpha + \alpha \varepsilon\_{t-1}^2 + \beta h\_{t-1} \tag{3}
$$

For *<sup>t</sup> h* to be well-defined, ,, and are constrained to be non-negative. In addition, the unconditional variance is given by <sup>2</sup> / 1 and for a finite unconditional variance to exist α+β < 1. Furthermore, in the GARCH model the effect of a shock on volatility decays exponentially over time and the speed of decay is measured by the extent of volatility persistence (which is reflected in the magnitude and significance of the summation of the α and β parameters).

The GARCH (1,1) model estimates are reported in Table 2. The results confirm the previous findings on the importance of GARCH effects by showing that the GARCH and ARCH terms are all statistically significant. The parameters of the conditional variance equations are all positive and statistically significant. Furthermore, they satisfy the positivity constraint for the GARCH(1,1) model. Furthermore evidence of persistence in variance as measured by the GARCH model is reflected in the magnitude and significance of the ARCH and GARCH terms (indeed, as this sum approaches unity the greater the degree of persistence). Therefore, in order to have an indication of long memory in fixed income return volatility the level of volatility persistence (i.e., α+β) is assessed.

The results indicate that volatility in these markets is very persistent, with the level of volatility persistence being 0.9775 for Mexico, 0.9782 for South Africa and 0.9912 for Hong Kong, which underscores the highly persistent nature of shocks to volatility, which also in turn is suggestive of a long memory component to volatility behaviour in these fixed income markets. The models' appropriateness has also been checked by applying the Box-Pierce Q statistic test to standardised and squared standardised residuals. Basic diagnostics indicate that the GARCH models are well-specified.

#### **4.3. Wavelet analysis**

To estimate the long memory parameter *d* of emerging market local currency debt of Hong Kong, Mexico and South Africa this study considers methods based on the discrete wavelet transform. Wavelet analysis, plays an important role in the characterisation of time series, by detecting scaling structures in data. More precisely, since wavelets are localised both in time and frequency, they provide a means to collect information on both the frequency and time characteristics of a time series. In fact, wavelets are already widely used as detectors of patterns in areas as diverse as digital signal processing and exploration geophysics. In the empirical finance literature they have recently been used to determine time-dependence in asset return data by comparing the scale decompositions of observations that exhibit significant autocorrelation between observations widely separated in time. [18] In this manner, the scaling properties of daily returns of emerging market government bonds are analysed in order to capture temporal dependencies in the volatility process.


Notes:

102 Risk Management – Current Issues and Challenges

particular, the returns generating process can be described by:

,, and 

The GARCH (1,1) specification comprises a return (or mean) and a variance equation. In

where <sup>1</sup> 0, *t t tt t r N*

average components, and *εt* is the error term, which is assumed to be normally distributed with zero mean and variance *<sup>t</sup> h* , given the information set *t*<sup>1</sup> . The conditional variance is

> 2 *t tt* 1 1 *h h*

> >

to exist α+β < 1. Furthermore, in the GARCH model the effect of a shock on volatility decays exponentially over time and the speed of decay is measured by the extent of volatility persistence (which is reflected in the magnitude and significance of the summation of the α

The GARCH (1,1) model estimates are reported in Table 2. The results confirm the previous findings on the importance of GARCH effects by showing that the GARCH and ARCH terms are all statistically significant. The parameters of the conditional variance equations are all positive and statistically significant. Furthermore, they satisfy the positivity constraint for the GARCH(1,1) model. Furthermore evidence of persistence in variance as measured by the GARCH model is reflected in the magnitude and significance of the ARCH and GARCH terms (indeed, as this sum approaches unity the greater the degree of persistence). Therefore, in order to have an indication of long memory in fixed income

The results indicate that volatility in these markets is very persistent, with the level of volatility persistence being 0.9775 for Mexico, 0.9782 for South Africa and 0.9912 for Hong Kong, which underscores the highly persistent nature of shocks to volatility, which also in turn is suggestive of a long memory component to volatility behaviour in these fixed income markets. The models' appropriateness has also been checked by applying the Box-Pierce Q statistic test to standardised and squared standardised residuals. Basic diagnostics indicate

To estimate the long memory parameter *d* of emerging market local currency debt of Hong Kong, Mexico and South Africa this study considers methods based on the discrete wavelet transform. Wavelet analysis, plays an important role in the characterisation of time series, by detecting scaling structures in data. More precisely, since wavelets are localised both in

 

 

return volatility the level of volatility persistence (i.e., α+β) is assessed.

 

*r* denotes the returns process, which may include autoregressive and moving

 

*<sup>h</sup>* (2)

(3)

are constrained to be non-negative. In addition, the

/ 1 and for a finite unconditional variance

**4.2. GARCH(1,1) model** 

For *<sup>t</sup> h* to be well-defined,

and β parameters).

unconditional variance is given by <sup>2</sup>

that the GARCH models are well-specified.

**4.3. Wavelet analysis** 

where *<sup>t</sup>*

modelled as:

1/ The Ljung-Box Q test applied to standardized residuals.

2/ The Ljung-Box Q test applied to squared standardised residuals

3/ The numbers in () and [] refer to lag lengths and standard deviations

4/ '\*\*' and '\*' indicate significance at the 1% and 5% levels

#### **Table 2.** GARCH Estimates

A wavelet is defined as a wave-like function with an amplitude that oscillates around zero and has a finite or quickly decreasing time support. These functions are well suited to locally approximating variables in time or space as they have the ability to be manipulated by being either 'stretched' or 'squeezed' so as to mimic the series under investigation. [19] The power of wavelet analysis is that it makes it possible to decompose a time series into its high- and lowfrequency components, which are localised in time. Wavelets, also allow the selection of an appropriate trade-off between resolution in the time and frequency domains, while traditional Fourier analysis stresses resolution in the frequency domain at the expense of the time domain. [20] Wavelets therefore provide a convenient and efficient method to analyse complex signals. [21] Wavelet theory is applicable to several subjects. They are especially useful where a signal (e.g. long memory) lasts for a finite time or shows markedly different behaviour in different time periods. These methods, have emerged as a useful tool in the empirical finance literature where long-run and short-run relationships can be distinguished. [22]

#### **4.4. The discrete wavelet transform**

A discrete wavelet transform (DWT) is any wavelet transform for which the wavelets are discreetly sampled, as is often the case in econometric analysis. A wavelet transform is a

scaling function used to transform a signal into a father and mother wavelet, where the former, are representations of a signal's trend component (i.e., the approximation coefficients) and the latter represent the deviations from the trend component (i.e., the detail coefficients). The discrete wavelet series approximation to a continuous signal *f(t)* is given by

$$f(t) \approx \sum\_{k} a\_{j,k} \Phi\_{l,k} \left( t \right) + \sum\_{k} d\_{j,k} \mu\_{l,k}(t) + \sum\_{k} d\_{j-1,k} \mu\_{l-1,k} \left( t \right) + \dots + \sum\_{k} d\_{1,k} \mu\_{1,k} \left( t \right) \tag{4}$$

Long Memory in the Volatility of Local Currency Bond Markets:

is a random error term. Since the variance of

(5)

0.1288

0.2958

0.1989

0.0310 0.9396

0.0281 0.9164

0.0664 0.9172

 <sup>2</sup> 2( ) , ln ln ln 2 *J j Var dj k d*

where *Var d j k*, is the variance of the detail coefficients associated with the value of the

the detail coefficients decomposes the variance of the initial time series over different scales,

To estimate *d* , this study uses multi-resolution analysis via the Haar wavelet to generate the respective detail coefficients of each volatility series at each dimension of scaling parameter. From here, the variance of the detail coefficients at each scale are computed and then the regression specified in equation (5) is performed, where the slope coefficient provides an estimate of *d* . To check for the robustness of the results, and, therefore, avoid spurious conclusions of long memory dynamics the Daubechies 4 (D4) wavelet is also examined.

The regression results of this analysis are presented in Table 3 and 4. The slope coefficient of the regression given in equation (3) provides an estimate, *d*. Table 3 shows that, when the Haar wavelet is used, this study is able to find evidence of long memory across the three volatility measures used. The long memory parameter, *d*, ranges from 0.2363 (Mexico) to 0.3423 (Hong Kong). Furthermore, the evidence obtained is significantly different from zero for all the fixed income markets. These results indicate that volatility realisations have a predictable component insofar as distant observations in the volatility series are associated with each other, albeit over long lags. The significant size of *d* obtained from this model illustrates the importance of modelling long memory in fixed income data. Furthermore, the result of *d* (0, 0.5) from these models is in contrast to the findings from the unit root tests

Volatility series Identifier Parameter Estimate Standard Error <sup>2</sup> *R*

1.4167\*\* 0.3423

1.0344\* 0.2363\*\*

1.0151\* 0.2679\*\*

**Table 3.** Estimates of the Long Memory Parameter using the Haar Wavelet

Notes: To estimate the long memory parameter *d*, the following regression is performed on the respective volatility

corresponding to the value of the scaling parameter *j* = 1,. . . , *J*, and ε is the error term. '\*\*'and '\*' indicate statistical

In econometric analysis, it is important to perform diagnostic checks in order to assess the validity of the initial estimates of *d* . Therefore, to avoid spurious evidence of long memory (due to the choice of wavelet employed) in the volatility process of the time series, equation

where *Var d i k*, is the variance of the detail coefficients

scaling parameter *j*, where, *j* =1, . . ., *J,* and

that led to a conclusion of *d* = 0.

Hong Kong Intercept

Mexico Intercept

South Africa Intercept

series: <sup>2</sup> , ln ln 2 , *J j Var d d i k* 

significance at the 1% and 5% levels, respectively.

Slope (*d*)

Slope (*d*)

Slope (*d*)

this permits an analysis of the dynamics of the series at each scale.

Evidence from Hong Kong, Mexico and South Africa 105

where *j* is the number of multi-resolution scales, and *k* ranges from 1 to the number of coefficients in the corresponding scale and the coefficients , , 1, , , , *Jk Jk k ad d* are the wavelet transform coefficients. Applications of wavelet analysis with respect to time series analysis make use of a DWT. The DWT maps the vector **f** = ' 1 2 ,,, *<sup>n</sup> f f f* to a vector of wavelet coefficients, which contains *j k*, *a* and , , *j k d j* = 1, 2, . . . , *J*, which are the approximation and detail coefficients, respectively.

In the empirical literature, Haar and Daubechies wavelets represent typical wavelets and have been used in the characterisation of the time series properties of asset return data. The Haar wavelet is the simplest wavelet and provides a basis for studying more complex wavelets.2 Since a wavelet is used to decompose a given function into different scale components, it follows that each scale component can then be studied with a resolution that matches its scale. [23]

The data used in this study are discreetly sampled, accordingly, the discrete wavelet transform is used, which permits the generation of the approximation coefficients, *j k*, *a* ,

which capture the trend of a time series and the detail coefficients, *dj k*, , reflecting the

deviations from the trend at each scale. Because the original function can be represented in terms of a wavelet expansion, data operations can be performed using the corresponding wavelet coefficients. This leads to a continuum of time-scale representations of the signal, all with different resolutions. Hence, multi-resolution analysis, which allows the computation of the coefficients corresponding to the wavelet transform of the observed time series.

The analysis of fractionally integrated processes through the use of wavelets is based on the result that the detail coefficients of a zero mean long memory process are asymptotically normally distributed with variance <sup>2</sup> 2( ) 2 *J jd* , where <sup>2</sup> is a finite constant, *j* is the scaling parameter and *d* measures long memory in the relevant volatility series. [24] To estimate *d* using wavelet theory, the logarithmic variance transformation regression estimator is widely used. This procedure is based on the exploitation of the variance of the detail coefficients at each scale, which generates a statistically consistent estimator of the long memory parameter. The estimator of the parameter of the fractional integration, *d*, is based on the following least squares regression:

<sup>2</sup> The Haar transform assumes a discrete signal and decomposes the signal into two sub-signals of half its length reflecting the trend process and fluctuations from the trend process.

Long Memory in the Volatility of Local Currency Bond Markets: Evidence from Hong Kong, Mexico and South Africa 105

$$d\ln Var\left(d\_{j,k}\right) = \ln \sigma^2 + d \ln 2^{-2(l-j)} + \varepsilon \tag{5}$$

where *Var d j k*, is the variance of the detail coefficients associated with the value of the scaling parameter *j*, where, *j* =1, . . ., *J,* and is a random error term. Since the variance of the detail coefficients decomposes the variance of the initial time series over different scales, this permits an analysis of the dynamics of the series at each scale.

104 Risk Management – Current Issues and Challenges

detail coefficients, respectively.

normally distributed with variance <sup>2</sup> 2( ) 2 *J jd*

on the following least squares regression:

reflecting the trend process and fluctuations from the trend process.

matches its scale. [23]

by

scaling function used to transform a signal into a father and mother

make use of a DWT. The DWT maps the vector **f** = '

where the former, are representations of a signal's trend component (i.e., the approximation coefficients) and the latter represent the deviations from the trend component (i.e., the detail coefficients). The discrete wavelet series approximation to a continuous signal *f(t)* is given

> , , , , 1, 1, 1, 1, ( ) *jk Jk jk Jk j kJ k k k k kk k f t a td td t d t*

where *j* is the number of multi-resolution scales, and *k* ranges from 1 to the number of coefficients in the corresponding scale and the coefficients , , 1, , , , *Jk Jk k ad d* are the wavelet transform coefficients. Applications of wavelet analysis with respect to time series analysis

coefficients, which contains *j k*, *a* and , , *j k d j* = 1, 2, . . . , *J*, which are the approximation and

In the empirical literature, Haar and Daubechies wavelets represent typical wavelets and have been used in the characterisation of the time series properties of asset return data. The Haar wavelet is the simplest wavelet and provides a basis for studying more complex wavelets.2 Since a wavelet is used to decompose a given function into different scale components, it follows that each scale component can then be studied with a resolution that

The data used in this study are discreetly sampled, accordingly, the discrete wavelet transform is used, which permits the generation of the approximation coefficients, *j k*, *a* ,

which capture the trend of a time series and the detail coefficients, *dj k*, , reflecting the deviations from the trend at each scale. Because the original function can be represented in terms of a wavelet expansion, data operations can be performed using the corresponding wavelet coefficients. This leads to a continuum of time-scale representations of the signal, all with different resolutions. Hence, multi-resolution analysis, which allows the computation of the coefficients corresponding to the wavelet transform of the observed time series.

The analysis of fractionally integrated processes through the use of wavelets is based on the result that the detail coefficients of a zero mean long memory process are asymptotically

, where <sup>2</sup>

parameter and *d* measures long memory in the relevant volatility series. [24] To estimate *d* using wavelet theory, the logarithmic variance transformation regression estimator is widely used. This procedure is based on the exploitation of the variance of the detail coefficients at each scale, which generates a statistically consistent estimator of the long memory parameter. The estimator of the parameter of the fractional integration, *d*, is based

2 The Haar transform assumes a discrete signal and decomposes the signal into two sub-signals of half its length

(4)

 

1 2 ,,, *<sup>n</sup> f f f* to a vector of wavelet

is a finite constant, *j* is the scaling

wavelet,

To estimate *d* , this study uses multi-resolution analysis via the Haar wavelet to generate the respective detail coefficients of each volatility series at each dimension of scaling parameter. From here, the variance of the detail coefficients at each scale are computed and then the regression specified in equation (5) is performed, where the slope coefficient provides an estimate of *d* . To check for the robustness of the results, and, therefore, avoid spurious conclusions of long memory dynamics the Daubechies 4 (D4) wavelet is also examined.

The regression results of this analysis are presented in Table 3 and 4. The slope coefficient of the regression given in equation (3) provides an estimate, *d*. Table 3 shows that, when the Haar wavelet is used, this study is able to find evidence of long memory across the three volatility measures used. The long memory parameter, *d*, ranges from 0.2363 (Mexico) to 0.3423 (Hong Kong). Furthermore, the evidence obtained is significantly different from zero for all the fixed income markets. These results indicate that volatility realisations have a predictable component insofar as distant observations in the volatility series are associated with each other, albeit over long lags. The significant size of *d* obtained from this model illustrates the importance of modelling long memory in fixed income data. Furthermore, the result of *d* (0, 0.5) from these models is in contrast to the findings from the unit root tests that led to a conclusion of *d* = 0.


Notes: To estimate the long memory parameter *d*, the following regression is performed on the respective volatility series: <sup>2</sup> , ln ln 2 , *J j Var d d i k* where *Var d i k*, is the variance of the detail coefficients corresponding to the value of the scaling parameter *j* = 1,. . . , *J*, and ε is the error term. '\*\*'and '\*' indicate statistical significance at the 1% and 5% levels, respectively.

**Table 3.** Estimates of the Long Memory Parameter using the Haar Wavelet

In econometric analysis, it is important to perform diagnostic checks in order to assess the validity of the initial estimates of *d* . Therefore, to avoid spurious evidence of long memory (due to the choice of wavelet employed) in the volatility process of the time series, equation

(4) is re-estimated using the Daubechies 4 (D4) wavelet. These results are presented in Table 4. The results are broadly similar in magnitude to those obtained using the Haar wavelet. The noticeable exception relates to the case of South Africa where the long memory parameter falls from 0.2679 (when the Haar wavelet is used) to 0.1784 (when the D4 wavelet is used). This notwithstanding, the results are all statistically significant. In sum, the results of this analysis suggest that bond return volatility in emerging markets is characterised by stochastic processes which have a long memory component.

Long Memory in the Volatility of Local Currency Bond Markets:

(6)

(7)

(8)

*r* is the proxy for volatility . Both the

= 0 the model reduces to a

is set to 0.94 following standard

=1 the model is equivalent to the prior period forecast of

is determined empirically by the value that minimizes the 'in-

2

<sup>2</sup> <sup>2</sup>

*t*

*f t t*

known property that volatility forecasts are additive, such that the sum of five daily volatility forecasts produces the weekly forecasts. And, the summation of weekly forecasts

In addition to the GARCH and long memory model the RiskMetrics model is also considered for comparative purposes. The RiskMetrics model was popularised by the investment bank JP Morgan and is widely used by financial institutions to model and forecast volatility, especially in the context of the Basle Committee adequacy criteria. This model is essentially an exponentially weighted moving average (EWMA). Under the EWMA, the fitted variance from the model, , *<sup>t</sup> h* which provides the multi-step ahead volatility forecast, is a weighted function of the immediately preceding volatility forecast

1 1 <sup>ˆ</sup> <sup>1</sup> *tt t hh h*

market practice, which is also consistent with previous research which indicates that this

Two standard symmetric measures are used to evaluate forecast accuracy, namely, the mean absolute error (MAE) and the root mean square error (RMSE). They are defined below:

1 *<sup>T</sup>*

*t T MAE h r* 

1 *<sup>T</sup>*

these evaluation criteria the model which minimises the loss function is preferred.

*t T RMSE h r* 

MAE and RMSE assume the underlying loss function to be symmetric. Furthermore, under

Table 5 reports out-of-sample performance of the estimated models based on the MAE and RMSE forecast error statistics. At the daily level, the results are not unexpected. That is, the GARCH model dominates forecast accuracy for South Africa on the basis of both the MAE and RMSE. For Mexico, the GARCH model dominates forecast performance on the basis of the MAE while the RiskMetrics models delivers the most accuracy when the RMSE is used as a criterion. For Hong Kong the GARCH process is preferred on the basis of the MAE, which surprisingly, the long memory model delivers the best performance when the RMSE

1

1

*f t t*

 

is the smoothing parameter, such that when

sample' sum of squared prediction errors. In this study

is the number of forecast data points and <sup>2</sup>

produces monthly forecasts.

and actual volatility is given below:

random walk process and when

value produces accurate forecasts. [25]

**5.1. Standard forecast evaluation** 

volatility. The value of

where 0 1 

where  Evidence from Hong Kong, Mexico and South Africa 107


Notes: To estimate the long memory parameter *d*, the following regression is performed on the respective volatility series: <sup>2</sup> , ln ln 2 , *J j Var d d i k* where *Var d i k*, is the variance of the detail coefficients corresponding to the value of the scaling parameter *j* = 1,. . . , *J*, and ε is the error term. '\*\*' and '\*' indicate statistical

significance at the 1% and 5% levels, respectively.

**Table 4.** Estimates of the Long Memory Parameter using the Daubechies 4 Wavelet

The analysis indicates robust evidence of long memory behaviour in the return volatility of emerging market debt. Further, wavelet methods provide a robust fit for the data as evidence by the <sup>2</sup> *R* readings presented in the final columns of both Table 3 and 4. If fixed income data exhibit long memory, then it displays significant autocorrelation between distant observations. This, in turn, implies that the series realisations may have a predictable component; and, perhaps, past trends in the data can be used to predict future volatility. Therefore, attention now turns to an exploration of the forecast performance of models with long memory relative to the standard volatility models.
