Essays on forecasting life expectancy and fiscal sustainability 2

MGARCH models estimate a conditional covariance matrix comprising time-varying conditional volatilities and correlations.. The MGARCH literature include several types of models including

The LC method is based on a linear fit to an additive model of age-specific death rates, with a fixed-age component and a time-varying component assuming homoskedastic Gaussian error structure It is computationally simple, yet reasonably accurate in capturing trends and variations in mortality rates Though significantly deviated from previous methods, the LC model and its extensions have been applied successfully to forecast life expectancy in developed countries, including the USA and the G7 countries See Lee and Miller (2002), Booth et al (2002) and Booth (2006)

The LC method and many other mortality models have commonly assumed the variance of error terms to be constant over time i.e a homoskedastic error

structure is adopted This assumption may not be satisfactory; since for time series data, it is often observed that volatility clustering are present, where large changes are followed by large changes and small changes followed by small changes, an evidence of heteroskedasticity In the presence of time-varying volatility in mortality data, the classical LC model will be inadequate

Although the importance of time-varying volatility has been accounted for

in various macroeconomic time series data, to the best of our knowledge, there is

no study of volatility clustering in the LC model This essay aims to fill this current gap, and seek evidence of such time-variation in volatility using the well-documented autoregressive conditional heteroskedasticity (ARCH)/generalized ARCH (GARCH) models introduced respectively by Engle (1982) and Bollerslev (1986) The countries included in this study are Japan, Australia, Hong Kong, Taiwan, U.S and U.K

In addition to time-varying volatility, the presence of time-varying correlations between the male and female mortality series of each country are also

of practical value and importance While extending the LC method with univariate GARCH model captures volatility clustering for individual series, volatility co-movements between related series will require the use of Multivariate GARCH (MGARCH) models

To the best of our knowledge, there has also been no study of time-varying correlation dynamics of mortality series using MGARCH-type models We adopt

the use of such models with the LC approach to simultaneously examine the presence of time-varying volatility and correlations in mortality forecasting We will adopt the use of both the Constant Conditional Correlation (CCC-) GARCH model of Bollerslev (1990) and time-varying Conditional Correlation (VC-) GARCH model of Tse and Tsui (2002) We then seek to compare the within-sample forecast performance of the classical LC model with the proposed model extensions in this paper in forecasting life expectancy at birth and at age 65

This essay is structured as follows: Section 1.2 offers a literature review Section 1.3 describes the LC-GARCH model while section 1.4 summarizes the data used and presents the estimation results In section 1.5, we offer an overview

of the MGARCH models, and Section 1.6 describes the LC method extended with CCC-GARCH and VC-GARCH models Estimation results are then reported in section 1.7 In section 1.8, we show that using the LC model with VC-GARCH structure offers the largest improvement in within samples forecasts of life expectancy at birth and at age 65 Concluding remarks are offered in the final section

1.2 LITERATURE REVIEW

While there are many approaches to forecast mortality rates, the LC method

is still one of the leading statistical models in the demographic literature for mortality trend fitting and life expectancy projections since its development in

1992

Proposed by Lee and Carter, it was initially used for projecting age-specific mortality rates in the United States but has since been adopted by many other countries as the basic mortality model for population projections See applications

in, for example, Canada (Lee and Nault, 1993), Japan (Wilmoth, 1996) and the seven most economically developed nations (G7) (Tuljapurkar et al., 2000)

The LC model was designed to examine the long-term patterns in the natural logarithm of central death rates using a single index of mortality It essentially describes the logarithmically transformed age-specific central rate of death as a sum of an age-specific component that is independent of time, and the product of a time-varying parameter (also known as the mortality index that summarizes the general level of mortality) and an additional age-specific component that represents how rapidly or slowly mortality at each age varies when the mortality index changes

The advantages of the parsimonious LC model lie in its simplicity This is because once data is fitted to the model and the model parameter values are estimated, they are held fixed as constants and only the mortality index needs to be

forecasted The LC model also adopts an extrapolative method where forecasts are carried out using time series methods and historical information With the use of logarithms, it also allows mortality rates to decrease exponentially without the need for restrictions

Like many other models, the LC model has its set of limitations as well As

it is based on extrapolation methods, its forecast accuracy fares unfavorably when historical data fails to hold in the future and/or structural changes occur It also does not account for any changes in social economic factors, such as medical advancement, lifestyle changes etc Furthermore, the LC model assumes a constant variance for the residual term; which can potentially be another limitation since it constrains the model's ability to capture the volatility of series if changes across periods are far from being constant

While many extensions to the LC model have likewise assumed a constant variance for the residual term, the presence of time-variation in volatility has been detected in mortality series and some studies have questioned the assumption of using constant variances Among them, Renshaw & Haberman (2003a, b) used heteroskedastic Poisson error structures in their mortality forecast, where ordinary least-squares regression is replaced with Poisson regression for the death counts Cossette et al (2007) suggested the use of a Binomial regression model where the annual number of deaths is assumed to follow a Binomial distribution and the death probability is expressed as a function of the force of mortality Delwarde, Denuit

and Partrat (2007) also proposed the use of Negative Binomial Distribution to account for the presence of heterogeneity in mortality

The use of ARCH and GARCH models (Engle, 1982; Bollerslev, 1986) has been proven to be capable in capturing the existence of non-constant variances in many applications Nonetheless, among these extensions, the use of ARCH/GARCH models in demography forecasting is limited While Keilman and Pham (2004) incorporated ARCH-type structure with the use of ARIMA models in their forecasts of fertility, mortality and net migration for 18 countries in the European Economic Area, their study did not attempt to incorporate the use of ARCH/GARCH models with the LC model This is our first objective in this essay

Univariate ARCH/GARCH models face two restrictions; firstly it does not accommodate the asymmetric effects of positive and negative shocks and secondly,

it assumes independence between conditional volatilities across different groups

In financial series, it has been established that volatility in the returns of financial variables exhibit an asymmetric character where negative shocks contribute more

to volatility than positive shocks of the same magnitude (see for instance, Nelson,

1991 and Glosten, Jagannathan and Runkle, 1993) It is not known if such asymmetric effects are also present in mortality data and this is the second objective of this essay which seeks to address the limitation of using univariate ARCH/GARCH models Hence, further to extending the LC model with

ARCH/GARCH type models, we will also examine the presence of asymmetric effects in mortality series using the LC model extended with EGARCH

Other variations of the LC method included the recent works of Girosi and King (2008), who adopted the Baynesian approach and Markov Chain Monte Carlo estimation to improve on mortality forecasting They developed a general Baynesian hierarchical framework for forecasting different demographic variables and incorporated exogenously measured covariates as proxies for systematic causes

of death Their methodology have not only generalized the LC model to an analysis involving several principal components; it also uses additional information about regularities along the dimensions of age, sex, country and death causes to improve on forecasting results The framework, however, only models a single population in isolation, and also requires a lot of extensive information which may not be easily available in many countries

The use of multivariate GARCH models allows one to account for the presence of co-movements across related series, which makes it possible to model separate series jointly and incorporate their interactions In the financial sector, the dependence in co-movements across asset returns is important since the co-movements or covariance of assets in a portfolio provides critical information for asset pricing Using MGARCH models allow us to extract such covariance information and improve the analysis for asset pricing models, hedging, portfolio selection and Value-at-Risk forecasts Studies have found the variances of financial time series to be interacting, see for instance, Cifarelli and Paladino

(2005) who studied the linkages in equity markets MGARCH models have been used to examine the volatility and correlation transmissions and spillover effects in contagion studies as well, see Tse and Tsui (2002)

MGARCH models estimate a conditional covariance matrix comprising time-varying conditional volatilities and correlations It has been proven in many financial applications that modeling the dynamics of the covariance matrix using a multivariate approach yields better results than working with separate univariate models for each individual series The importance of accounting for co-movements using MGARCH models is similarly valid for other markets In the area of mortality, the presence of co-movements between the male and female population is potentially large

The MGARCH literature include several types of models including VEC model by Bollerslev, Engle and Wooldridge (1988), Baba-Engle-Kraft-Kroner (BEKK) model by Engle and Kroner (1995), CCC-GARCH model by Bollerslev (1990) and time-varying conditional correlation GARCH models (VC-GARCH and DCC-GARCH models by Tse and Tsui (2002) and Engle (2002) respectively In the area of demography, the extension of the LC model with MGARCH models has not been attempted in the literature and this is the third gap which we seek to address in this essay Such an extension will allow us to examine the presence of co-movements between the mortality series of male and female populations

1.3 EMPIRICAL MODEL

1.3.1 LC Model

The LC Model is widely used in mortality forecasting and life expectancy forecasting It was introduced in 1992 and has been used by the United States Social Security Administration, the US Census Bureau, and the United Nations

The structure proposed by Lee and Carter (1992) is as follows:

where is the central death rate for an individual aged at time t; is the additive age-specific constant displaying the average shape across age of the mortality schedule; describes the relative sensitivity of the mortality at age to changes in the general level; is the error term which reflects any age-specific historical influences not captured in the model and is a time-specific index of the general level of mortality

is modeled to follow the random walk model with drift1 as described in (1.2) with a constant drift μ and is assumed to be independent, identically distributed Normal For different values of , the fitted model defines a set

of central death rates, which can then be used to derive a life table conveniently

1

Lee and Carter (1992) claimed that other ARIMA models might be preferable for different set of data, but the random walk model with drift is mostly used in practice

When changes linearly with time, mortality at each age changes at a constant exponential rate accompanied by the age-specific constants varying by age

As equation (1.1) is over-parameterized, is taken to be the arithmetic mean of over time while the sums of and are normalized to one and zero respectively, so as to ensure a unique solution Furthermore, given that all the parameters on the right-hand side of equation (1.1) are unobservable, the LC model requires a two-stage estimation procedure In stage one, the Singular Value Decomposition (SVD) estimation procedure is applied to the matrix of

to obtain estimates of and Following stage one, the time series of in equation (1.2) is re-estimated in stage two by solving the following

where is the total number of deaths in time t, and is the exposure to risk of

an individual aged x in time t The re-estimation of in stage two ensures that the mortality schedules fitted will reconcile with the actual total number of deaths and the population age distributions

1.3.2 Introduction to ARCH/GARCH models

The importance of accounting for ARCH/GARCH effects has been established in many empirical studies Although volatility is not directly observable, it has some commonly observed characteristics and facts The various stylized facts confirmed by numerous studies include volatility clustering and persistence, fat-tailed behavior, mean reversion and leverage effect The basic ARCH/GARCH model (Engle, 1982; Bollerslev, 1986) and its subsequent extensions have proven to be capable of capturing the existence of time-varying volatility and account for these characteristics

In the basic ARCH/GARCH model, the conditional variance is expressed as

a weighted average of its long run mean value, as well as shocks and the level of conditional variance in previous periods A GARCH process is an infinite-order ARCH process with a lag structure imposed on the coefficients

A low order GARCH model is preferred over a higher order ARCH model since the former is more parsimonious and allow for a slower, exponential decay rate A lower order process is generally preferred to higher order models since the latter may have many local maxima and minima The most popular and widely used model within the GARCH family thus far is still the model, where three parameters in the conditional variance equation are often adequate and good enough for model fit and short term forecasts Many empirical studies have supported the superiority of the

model According to Hansen and Lunde (2001), it is difficult to find

a volatility model that outperforms the simple

The potential of ARCH/GARCH models has been proven in financial applications, but it can be applied in any markets which require the forecasting of volatility Since the introduction of ARCH/GARCH models, the GARCH family

of models have been extensively used in many different areas to model volatility and forecasting, ranging from the earlier applications in asset pricing, risk management, GDP growth rates, interest rates, stock index returns, option pricing, foreign exchange rates, to the more recent applications in agriculture, internet traffic, etc See such applications in Christoffersen and Diehold (2000), Yang, Haigh and Leatham (2001) and Zhou, He and Sun (2006)

1.3.3 LC-GARCH Model Extension

We propose two model modifications and enhancements in this section First of all, the LC model assumes a random walk with drift specification in equation (1.2), which may overly restrictive Hence, instead of modeling using a random walk with drift process, we allow to follow a more general formulation

We adopt a stationary autoregressive process of order p, AR(p), where we allow the data to determine the order of p and values of in the conditional mean equation of (1.4)

Next, rather than a homoskedastic error structure, we assume that the residual term follows the well-documented structure pioneered by Bollerslev (1986) Conditional on , i.e the information set available at time t-

1, the conditional variance equation of is specified below

Conditional mean equation

where and is a sequence of identically and independently distributed normal random variables with zero mean and unit variance

The GARCH specification in (1.5) implies that the best predictor of conditional variance in the next period is a weighted average of its long-run mean value, the variance predicted for this period, as well as all shocks arriving in this period that is captured by the most recent squared residual Since last period's shocks enter the conditional-variance equation (1.5) in squared terms, shocks of the same magnitude will produce the same level of volatility irrespective of their sign, which means that negative and positive return shocks from the previous period will contribute equally to current period's volatility

Constraints and sum of all are imposed to ensure a positive conditional variance and a weakly stationary unconditional variance In particular, measures the extent to which a shock in this period feeds into next period's volatility, and the sum of parameters measures the persistence of any volatility shocks The greater it is, the greater is the persistence of shocks to volatility and the longer it takes to converge to the unconditional variance

Over the long run, forecasts of conditional variance will exhibit mean reversion towards its unconditional variance Forecast made m periods ahead will converge to the unconditional variance so long as

The value determines the speed of convergence to w; the larger the value, the longer the convergence period

As a check against the adequacy of the LC-GARCH model, we apply the Ljung–Box test on the standardized residuals and squared standardized residuals to check if the autocorrelations of the series are statistically different from zero The test statistic is

1.4 DATA AND ESTIMATION RESULTS OF LC-GARCH MODELS 1.4.1 Data

The data set used for this study consists of annual mortality tables by sex for Australia (1945-2007), Japan (1955-2007), Hong Kong (1971-2008), Taiwan (1970-2008), U.K (1945-2006) and U.S (1941-2006) There are a total of 111 (0,

1, 2 110+) ages for each of the countries, with the exception of Hong Kong where only 101 ages (0, 1, 2 100+) are available And except for Hong Kong which is obtained from the Census and Statistics Department of Hong Kong2, the rest are culled from the Human Mortality Database3 available at the University of California, Berkeley

Based on the published life tables, we follow the first stage of the LC method to estimate and by country and by sex, using the method of Singular Value Decomposition (SVD) During the estimation procedure, additional conditions have to be imposed to secure a unique solution since SVD often yields multiple solutions In particular, constraints and are imposed, with vector computed as the average of over time

In the second stage, we re-estimate the values of by matching the fitted life expectancy at birth with actual life expectancy Details of such estimation procedures are available in Lee and Carter (1992) and Lee and Miller (2002) Finally, extrapolation of using the fitted models enables us to obtain forecasts of

2

See http://www.censtatd.gov.hk

3 See http://www.mortality.org

future mortality rates, which can then be used to compute forecasts of life expectancy at various ages

1.4.2 Descriptive Statistics

Table 1.1 first exhibits some descriptive statistics and unit root tests for the time-varying mortality index, by country and by sex Under standard normal distributions, the skewness and kurtosis of any series should be 0 and 3 respectively The descriptive statistics revealed that the sample skewness is positive for Japan, Hong Kong males and females of U.S and U.K but negative for the rest

The plots of in Figure 1.1 suggest its non-stationary behavior We further apply the Augmented Dickey-Fuller (ADF) (Dickey & Fuller, 1981) and Phillips-Perron (PP) (Phillips & Peron, 1988) unit root tests on and the results in Table 1.1 confirm that are non-stationary at the 5% level of significance for all the countries As are I(1) series, we obtained its first difference, denoted as ,

to fulfill stationary conditions

Table 1.1 Descriptive Statistics and Unit Root Tests for Time-Varying Mortality

-22.7 -30.7 69.2 -88.1 44.6 0.5 2.2 3.4

25.9 39.7 79.6 -53.5 38.0 -0.6 2.2 5.6

40.1 55.1 77.9 -31.1 32.4 -0.6 2.0 6.8

-16.1 -11.4 48.9 -79.3 39.0 -0.09 1.8 2.2

8.1 8.2 60.0 -37.3 29.9 0.08 1.8 2.4

Unit root test

Augmented DF

PP

0.3 -4.7

-2.9 -1.8

1.0 -0.9

2.2 -3.0

-0.5 -3.3

-0.4 -4.6

1.9 1.1 32.7 -28.3 16.0 -0.03 2.4 0.7

-15.8 -14.3 55.7 -67.2 33.2 0.3 2.0 3.7

-6.7 -0.4 36.8 -52.6 25.6 -0.2 1.9 3.7

-6.0 -1.5 64.8 -74.5 36.0 0.02 2.3 1.4

16.1 25.8 60.4 -50.2 29.7 -0.6 2.3 5.1

Unit root test

Augmented

DF

PP

0.6 -1.5

-0.6 -2.1

-2.3 -3.1

-0.8 -2.0

-1.1 -2.1

4.0 -1.0

Figure 1.1 Plots of Time-Varying Mortality Index

1970 1973 1976 1979 1982 1985 1988 1991 1994 1997 2000 2003 2006

Taiwan Males

The descriptive statistics for by country and sex is given in Table 1.2 The sample skewness is negative for Japan males and females in Hong Kong, USA and UK Except for Japan males, the sample kurtosis for all the remaining series exceed 3 and the Jarque-Bera (JB) statistic fails to support normality at 5% significance level for Japan females, USA, Taiwan and UK Other than the visual plots of revealing their stationary behavior, the ADF and PP unit root tests performed on also rejected the null hypothesis of non-stationary at 5% level of significance for all countries Given that are stationary, we will be replacing

in equation (1.4) for all the countries with the use of

1945 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005

UK Males

Equation (1.4) is hence re-written as

Table 1.2 Descriptive Statistics & Unit Root Tests for Time-Varying Mortality

Index After First Differencing

-2.98 -3.19 1.91 -10.4 2.91 -0.55 2.98 2.60

-2.13 -3.06 9.71 -9.57 4.14 0.64 3.06 4.22

-1.41 -1.36 8.60 -7.95 3.70 0.58 3.06 3.47

-3.44 -2.92 2.54 -14.8 4.21 -0.73 3.21 3.39

-2.39 -3.13 9.95 -11.6 4.22 0.67 4.02 4.38

Unit root test

Augmented

DF

PP

-5.95 -5.92

-6.31 -7.05

-9.47 -9.74

-7.39 -8.82

-6.24 -15.3

-7.61 -8.72

-1.61 -1.88 7.59 -7.61 2.74 1.04 5.36 15.6

-1.89 -1.88 5.63 -10.2 2.48 -0.24 5.20 13.7

-1.25 -1.55 3.77 -4.74 2.01 0.77 3.23 6.49

-2.24 -2.02 3.60 -14.1 3.19 -0.81 4.80 15.0

-1.64 -1.81 10.7 -8.66 3.04 0.90 6.54 40.0

Unit root test

Augmented

DF

PP

-6.20 -6.31

-5.42 -5.34

-10.4 -10.3

-6.87 -6.90

-10.8 -10.7

-8.90 -14.5

Figure 1.2 presents the plots of the mortality indexes after first differencing

We observed across the countries that the amplitude of the series tend to change over time and the phenomenon of volatility clustering (large changes followed by large changes and small changes followed by small changes) is evident

Figure 1.2 Plots of Time-Varying Mortality Index After First Differencing

1946 1956 1966 1976 1986 1996 2006

Australia-Males

-10 -8 -6 -4 -2 0 2 4 6 8 10

-8 -6 -4 -2 0 2 4 6 8

UK-Males

-10 -8 -6 -4 -2 0 2 4 6 8 10 12

1.4.3 Estimation Results

The parameters of the conditional mean and conditional variance equations

in (1.7) and (1.5) respectively are estimated jointly using maximum likelihood technique Based on log-likelihood values, we find that either an AR(0) or AR(1) model for the conditional mean equation is adequate among all the countries

The estimation results of (1.7) for contradicted the use of the random walk with drift model in the LC method Diebold and Nerlove (1989) have argued against the use of a random walk model to safeguard from any specification error They pointed that the use of a low order AR model can account for any potential non-captured weakly serial correlation in the series

Table 1.3 displays the estimation results4 of LC model with GARCH effect and the Ljung-Box Q statistics (Ljung & Box, 1978) as diagnostic checks against the presence of serial correlation in the standardized residuals and squared standardized residuals

In the LC model, indexes the time variation of mortality rates Hence, an increase in will raise death rates for each age group, while death rates across ages will fall if declines Based on equation (1.7), negative values obtained for

μ and φ implies that is declining Since is defined as the difference of and , this is equivalent to a fall in mortality index and death rates over time

4 It is found that the conditional mean equation follows either an AR(0) or AR(1) model In addition, the conditional variance equation mostly follows the GARCH (0,1) structure We skip reporting estimates of the ARCH parameter as they are statistically insignificant at the 5% level

Due to the use of logarithm in the LC model, the death rates for each age rate will change at an exponential rate as mortality rate changes As a result, even if mortality index declines and become negative, there is no concern that negative death rates will occur

The intercept, , of the GARCH structure are positive for all the countries examined, which is consistent with the non-negativity of the conditional variance The estimated parameters for are also positive and significant at the 5% level of significance, which supports our case for modeling conditional volatility in mortality series Since is less than unity for all the samples, the variance is mean reverting, although different countries require varying timeframes The size

of is largest for Taiwan, indicating that the degree of volatility persistence for Taiwan is the greatest

Table 1.3 Estimated Parameters of LC-Univariate GARCH Model

-3.35 (0.45) -0.36 (0.12) 3.74 (1.10) 0.47 (0.24)

-2.19 (0.45) -0.28 (0.13) 12.74 (2.80) 0.35 (0.18)

-1.57 (0.33)

-0.07 (0.18)

9.51 (2.11) 0.35 (0.16)

-4.11 (0.86)

-0.17 (0.15)

9.53 (2.68) 0.57 (0.27)

-3.03 (0.68)

-0.21 (0.13)

7.81 (2.14) 0.49 (0.25)

Log-Likelihood -123.5 -120.8 -244.6 -229.6 -101.5 -98.14 Diagnostic

checks

Q(4)

Q2(4)

0.14 3.20

4.49 3.42

2.27 1.31

14.0 3.69

2.60 3.02

3.77 1.76

Country Taiwan USA UK

-2.44 (0.27)

-0.13 (0.05)

2.66 (0.95) 0.91 (0.46)

-1.30 (0.40)

0.04 (0.07)

3.57 (0.92) 0.54 (0.27)

-0.92 (0.23)

0.17 (0.11)

2.55 (0.62) 0.38 (0.19)

-3.17 (0.40) -0.52 (0.09) 3.90 (0.96) 0.62 (0.28)

-1.61 (0.36) -0.37 (0.18) 3.35 (1.13) 0.45 (0.23)

Log-Likelihood -79.3 -86.2 -146.3 -132.2 -145.2 -137.3 Diagnostic

0.89 4.98

6.43 5.37

8.52 2.37

Notes:

Figures in bold are statistically significant at 5% level

Q(4) stands for the Ljung-Box statistics of order 4 of the standardized residuals, whereas

Q2(4) is the Ljung-Box statistics of order 4 of the squared standardized residuals

The figures in parenthesis are the Bollerslev and Wooldridge (1992) consistent standard errors of the parameter estimates

heteroskedastic-We mention in passing that the diagnostic tests carried out on the standardized residuals confirmed model adequacy, which is necessary to ensure the robustness of our estimation results The standardized residuals for all samples were found to be normal and both excess kurtosis and serial correlation were absent Ljung-Box Q statistics for the standardized residuals and squared residuals also indicated no serial correlation for all countries Hence, the proposed LC-GARCH model is reasonably adequate for our data series

The use of ARCH/GARCH models, however, does not provide any insight for understanding and explaining the sources and causes of conditional variations

in time series; it only offers a mechanical way of describing the behavior of conditional variance In financial and economic markets, the arrival of any new information contributes towards improving forecasts, and ARCH/GARCH models have been interpreted as measuring the intensity of the news process In the literature, some possible reasons used to explain the existence of ARCH/GARCH effects include serially correlated information/news arriving in clusters, time lags between information arrivals, as well as information processing by market participants

Unlike financial series, the causes of volatility clustering and persistence in mortality context are unclear At the risk of over simplification, we conjecture that conditional volatility in mortality series could be related to the volatility of GDP growth rates and changes in the prevailing medical conditions of each country Indeed, a recent study by Hanewald (2009) finds support of significant correlation between real GDP growth rates in some OECD countries with the mortality index

in the LC model We postulate that the degree of shock persistence in mortality rates could also be related to the prevailing policies, living cultures and habits, as well as the unique demographic history of each country

1.4.4 LC-EGARCH Model

Besides LC-GARCH model, we include in Table 1.4 the estimation results

of LC-Exponential GARCH (EGARCH) model Under the univariate EGARCH approach, we retain the conditional mean equation in (1.7), but expressed the conditional variance in (1.5) as an asymmetric function of lagged disturbances

Table 1.4 Estimated Parameters of LC-EGARCH Model

-0.03 (0.80)

1.49 (0.30) 0.95 (0.25)

0.26

(0.18)

-3.06 (0.50)

-0.16 (0.12)

1.40 (0.26) 0.68 (0.29)

-0.17 (0.19)

-2.78 (0.37) -2.92 (0.19) 2.92 (0.19) -0.49 (0.19)

-0.04 (0.16)

-2.11 (0.36)

-0.04 (0.07)

2.53 (0.21) -0.41 (0.21)

-0.01 (0.17)

-3.81 (0.85)

-0.20 (0.14)

2.11 (0.32) 0.78 (0.25)

0.34 (0.23)

-4.03 (0.28) -0.27 (0.06) 3.62 (0.62) -1.84 (0.75)

-0.55 (0.38)

0.31 (0.27)

1.32 (0.30)

0.44 (0.31)

-1.64 (0.50)

0.08 (0.10)

2.26 (0.47) -1.44 (0.55)

0.70

(0.37)

-1.51 (0.31)

-0.13 (0.12)

0.92 (0.22) 1.03 (0.30) -0.29 (0.12)

-1.00 (0.22)

0.12 (0.12)

0.65 (0.24) 0.87 (0.22)

-0.16 (0.11)

-2.62 (0.32) -0.49 (0.07) 0.87 (0.23) 1.21 (0.19) 0.35 (0.17)

-2.19 (0.38)

-0.28 (0.16)

1.13 (0.31) 0.82 (0.32)

0.10 (0.13)

1.5 MULTIVARIATE CONDITIONAL VOLATILITY MODELS

MGARCH models are capable of accounting for time-varying correlations

at the same time when volatilities are being examined Conditional covariance is modeled using conditional variance and correlations and this approach is popular because conditional correlation models are easier to estimate and their parameters have a natural interpretation

1.5.1 VEC, BEKK and Factor-GARCH Models

There are various MGARCH-type models for estimating time-varying conditional covariance matrices (see Bauwens et al., 2006 for a survey) The key important objectives in MGARCH models are parsimony, flexibility and a positive definite conditional covariance matrix The first MGARCH model for modeling conditional covariance is the VEC model introduced by Bollerslev, Engle and Wooldridge (1988), where univariate GARCH representation is extended into a vectorized conditional-variance matrix Under the general VEC model, each conditional variance and covariance term is expressed as a linear function of all the lagged conditional variances and covariances, lagged squared errors and the cross-products of errors

The VEC model is flexible, but may not be feasible for estimation due to the problem of dimensionality which makes estimation computationally demanding

in many empirical applications Moreover, there is a need to impose further restrictions on the parameters to ensure that the conditional-variance matrix remains positive The difficulties with the VEC model leads to its simplified

version, the diagonal VEC model where diagonal matrices are used It reduces the full VEC model by assuming that the conditional covariances of variables and depend only on its lagged values and past realizations of the product Based upon the information set available at time t-1, i.e , the conditional covariance for and (i, j = 1,…n) at time t is expressed as

to converge (Ding and Engle, 1994)

The Baba-Engle-Kraft-Kroner (BEKK) model introduced by Engle and Kroner (1995) is another class of the MGARCH models Although it is a restricted

version of the VEC model, its conditional covariance matrix is positive definite by construction Unfortunately, like the VEC models, the BEKK models are not useful for systems containing more than 4 or 5 variables as there will be optimization problems resulting in unstable and inconsistent parameter estimates Similar to the case for VEC model, the difficulty associated with the large number

of unknown parameters under the BEKK model led to the simplified diagonal BEKK model

In the diagonal BEKK model, the conditional covariance matrix H at time t

is given by

where represents the column vector of residuals at time t, D is a triangular matrix and E and F are diagonal matrices The main problem associated with the use of diagonal BEKK model is that the constraints imposed on parameters can be easily violated by certain types of data

Following the VEC and BEKK models, modeling of conditional variances utilized factor models originated from economic theory Factor models are motivated under the idea that co-movements of a series may be driven by a number

of common underlying factors While the dimension of the estimation process is reduced with the use of factor models, the parameters under such models cannot be easily interpreted

1.5.2 CCC-GARCH, VC-GARCH and DCC-GARCH Models

Given the various limitations of the VEC, BEKK and factor models, we next turn our attention to another group of semiparametric MGARCH models, the CCC-GARCH model by Bollerslev (1990) and time-varying conditional correlation GARCH models (VC-GARCH and DCC-GARCH models by Tse and Tsui (2002) and Engle (2002) respectively) The CCC-GARCH model is one where the conditional correlations are assumed to be constant As a result, the conditional covariances are proportional to the product of the corresponding conditional volatility This simplification has significantly reduced the number of parameters to be estimated

For the CCC-GARCH, VC-GARCH and DCC-GARCH models, the conditional variances and correlations are modeled instead of the conditional covariance matrix The conditional variance and conditional correlation matrix

is specified and estimated in two stages In stage 1, univariate GARCH processes are constructed for each series, before using the conditional volatility results from stage 1 to compute the conditional correlation matrix in stage 2 With such a two-step estimation procedure, issues relating to the positivity of the covariance matrix and dimensionality problem do not pose a concern

The CCC-GARCH model is computationally simple, hence popular for empirical work For CCC-GARCH applications, one can refer to Kroner and Sultan (1993) who determined the minimum-variance hedge ratios of foreign currency futures; Park and Switzer (1995) in estimating the minimum-variance

hedge ratios of stock index futures; as well as Varga-Haszonits and Kondor (2007) who examined the efficiency of minimum variance portfolio optimization for stock price movements

Under the CCC-GARCH model, the conditional covariance matrix is decomposed into conditional variances and correlations The conditional correlation matrix represented by R is a constant and is independent of time The conditional covariance matrix is expressed as

where R is the constant unconditional correlation matrix with for all i = N

is a diagonal matrix containing the conditional volatility output from the N individual univariate GARCH models constructed in stage

1. The conditional covariance matrix is positive definite if and only if all the N conditional variances are positive and R is positive definite

Although the assumption of constant conditional correlations offers an attractive parameterization, it is too restrictive and may not be realistic for empirical applications (see Tusi and Yu, 1999 and Tse, 2000) There is thus a need

to extend the model to account for time-varying correlations, but at the same time

retaining the positive definite optimisation condition for the conditional correlation matrix

Both Tse and Tsui (2002) and Engle (2002) have generalized the GARCH model to VC-GARCH and DCC-GARCH models respectively Their models are able to concurrently capture the stylized features of time-varying volatility and correlations These models have clear computational advantages since the number of parameters to be estimated is independent on the number of series, hence allowing for potentially large correlation matrices to be computed The parameter estimates are also easy to interpret since univariate GARCH-type structure are used Moreover, with convergence, the conditional variance-covariance matrix is always guaranteed to be positive definite

CCC-Under these two dynamic models, the conditional correlation matrix represented by is time-dependent and an autoregressive moving average process

is applied to the matrix By imposing simple conditions on the parameters, these dynamic correlation models ensure that the conditional correlation matrix is positive definite for each point in time during the optimization process Similar to the CCC-GARCH models, they have been widely used, since including Duffee (2005) who examined the conditional correlation between stock returns and consumption growth, Yang (2005) who studied the international stock market correlations between Japan and four other Asian stock markets, as well as Higgs‟ (2009) recent application in electricity markets

In Tse and Tsui (2002) VC-GARCH model, the previous decomposition under the CCC-GARCH model is retained except that the conditional correlation matrix represented by is now time-variant It is a closely related model which models correlations directly as a weighted average of three correlations matrices The conditional correlations are expressed as functions of the unconditional correlations (R), the conditional correlations of the previous period ( and a set

In Engle (2002) DCC-GARCH model, in equation (1.14) is instead specified as

in (1.11 & 1.12) The null hypothesis of is often used to determine if the assumption of imposing constant correlations is appropriate The unconditional correlation matrix (R and Q in (1.14) and (1.18) respectively) can be computed using

of the correlation dynamics can also be extended to allow for asymmetries, which have been found to be important for financial applications

Overall, the VC-GARCH and DCC-GARCH models have been successfully applied in many areas The main disadvantage of this group of models

is that are scalars, which are necessary to ensure that remains positive definite for all t This requirement implies that there is no interdependence among the respective variances, correlations, as well as between the variances and correlations As such, the conditional correlations represent the same dynamics

1.5.3 Further Generalizations of VC/DCC-GARCH models

The VC-GARCH and the DCC-GARCH models have been further generalized since then, in view of the mentioned limitations Engle (2002) generalized the variable in (1.18) to resolve the constraint of equal dynamics for all correlations Under his generalization, is represented as

1.22

where represents the Hadamard product (element-wise matrix multiplication), A and B are positive definite square matrices The positive definiteness of A and B help to ensure that is also positive definite Although this generalization aids in resolving the problem of constant dynamics across the correlations, it significantly increases the number of parameters to be estimated and makes it unattractive for empirical work

Further attempts to extend the DCC models included the Asymmetric Generalized DCC (AG-DCC-) GARCH model by Cappiello, Engle and Sheppard (2006), who allowed for asymmetric effects in the correlation dynamics and translated the model into a quadratic form

1.23

where is the standardized residual, and A, B and G are diagonal parameter matrices Q is as defined earlier in (1.19) and F is the sample covariance matrix of However, this model not only has the mentioned system dimensional problem, there is also the challenge of ensuring that the matrix ( remains positive definite

Other extensions included the Threshold GARCH model with time-varying correlations and Quadratic Flexible DCC-GARCH models For these groups of models, the dynamic structure of the time-varying correlations is attributed to its past returns

There is another class of models which allows the dynamic correlations to

be affected by exogenous variables, see the Smooth Transition Conditional Correlation (STCC-) GARCH model by Silvennoinen and Terasvirta (2005), where the conditional correlation matrix is allowed to vary smoothly between two extreme states according to a transition variable A more recent extension, the Double Smooth Transition Correlation (DSTCC-) GARCH model was developed

in 2007 Pelletier (2006) also introduced the Regime Switching Dynamic Correlation (RSDC) model where constancy of correlations is imposed within a regime while dynamics are allowed to enter through switching regimes