Computer-intensive time-varying model approach to the systematic risk of Australian industrial stock returns.

Abstract:

This paper aims to investigate the form of systematic risk of Australian industrial stock returns. We suggest using four stochastic state-space models for the analysis. The stochastic properties of systematic risk are studied by examining four classes of state-space models: random walk model, random coefficient model, ARMA(1,1) model and mean reverting model (or moving mean model). We have found that the industrial portfolio betas are unstable. The variation of industrial portfolio beta is either random or mean-reverting. Among the nineteen industrial groups, ten of them have the mean-reverting process betas but six of them seem to have a moving long-term mean. Five of the industrial groups have the random process betas, more specifically; the betas of three of them are the random walk processes while the betas of the other two are just the random coefficients. We have also identified that the betas of five industrial groups seem to follow an ARMR(1,1) process.

Keywords:

KALMAN FILTER; MAXIMUM LIKELIHOOD; RISK ANALYSIS; TIME-VARYING MODEL.

1. Introduction

Time-varying beta models have been investigated by many authors. There is an extensive literature on testing stability of beta in the market model. Blume (1971) was among the first to consider the time-varying beta market model. He revealed that the estimated beta tended to regress toward the mean. Blume (1975) suggested that the source of this mean reversion of beta was related to the idea that initially a company could choose relatively high risk projects but over time the risk of these projects declines, thus leading to a decline in the company's equity beta. His finding was supported by Brenner and Smidt (1977), and Francis (1979). Other studies have attempted to describe the Hildreth-Houck random coefficient model to beta, and their results were in favour of the random coefficient model for individual stocks over a period of six years. Alternatively, Sunder (1980) and Simonds, LaMotte and McWhorter (1986) suggested that a random-walk coefficient model was most suitable for modelling the U.S. data over a long time period. Ohlson and Rosenberg (1982) proposed an ARMR(1,1) model for the beta coefficient, which was supported by Collins, Ledolter and Rayburn (1987).

The systematic risk of asset in finance literature is normally estimated by the market model, in which the returns of asset is regressed against market return and the regression coefficient beta thus offers an estimate of systematic risk. However, recent literature has widely recognized that the systematic risk of asset change over time due to both the microeconomic factors in the level of the firm and the macroeconomic factors (see Fabozzi & Francis 1978; Bos & Newbold 1984). Empirically, considerable evidences have suggested that beta stability assumption is invalid. The evidence of beta's time-varying property can be found in Kim (1993), Bos and Ferson (1992, 1995), Wells (1994), Bos, Ferson, Martikainen and Perttunen (1995), Brooks, Faff and Lee (1992) and Cheng (1997). In Australia, Brooks, Faff and Lee (1992), and Faff, Lee and Fry (1992) were among the first to investigate the time-varying beta models. Faff, Lee and Fry (1992) employed a locally best invariant test to study the hypothesis of stationary beta. They found the evidence of nonstarionarity across all of their analysis. Brooks, Faff and Lee (1994) further suggested the use of the random coefficient model as the preferred model to best describe the systematic risk of both individual shares and portfolios. However, Pope and Warrington (1996) re-estimated the market model by using a modified random coefficients model on 191 individual companies, and found that random coefficient model was appropriate for only about 23% of these stocks.

Although most of these previous studies have explored the stochastic behaviour of betas against stationarity, very few of them have concentrated on the industrial beta's time-varying property. Faff, Lee and Fry (1992) investigated the links between beta's nonstationarity and the three firm characteristics--riskiness, size and industrial sector. They did not find the strong pattern between firm size or industry sector and nonstationarity. In Faff and Brooks (1998), industrial betas were modelled successfully by different regimes, market returns and volatility of the risk-free interest rate. However, their univariate and multivariate tests provide mixed evidence concerning the applicability of a time-varying beta model, which incorporates these variables. Groenewold and Fraser (1999) argued that the industrial sectors could be divided into two groups: one of them has volatile and non-stationary betas and the other group has relatively constant and generally stationary beta. Other recent studies include Gangemi, Brooks and Faff (2001), Josev, Brooks and Faff (2001), and others.

One prominent problem associated with beta estimation is the 'errors in variable' problem when an individual asset is used. The estimation of individual company beta normally contains a large sampling error; therefore, over- or underestimation of beta is unavoidable. Collins, Ledolter and Rayburn (1987) conducted tests on portfolios. They argue that the analysis of beta instability at the portfolio level is particularly important since, empirically, the risk-return examination has been conducted at the portfolio level. It would be interesting to see how aggregation affects the nature of beta variation through time. In addition, levels of background noise could be reduced to some degree, which will enable a better detection of sequential versus random variation in equity betas. Ohlson and Rosenberg (1982) argue that it is virtually impossible to derive the stochastic behaviour of beta at the portfolio level as a function of the stochastic behaviour of individual security betas. Therefore, a thorough investigation of time-varying portfolio betas is needed.

Practically, beta estimates for portfolios are more valuable for portfolio management than the individual betas, especially at the industry level, in the process of securities analysis, the macroeconomic and industry analysis are two major aspects. The macroeconomic condition is translated to the security market through impacts on corporate profits. The investment advice is usually tied to macroeconomic forecasts. Portfolio managers will recommend special industries when the macroeconomic condition changes according to the sensitivity of the industry. For example, the portfolio manager might recommend investment into financial stocks in a low-rate environment. However, not all industries are equally sensitive to the business cycle. Firms in the sensitive industries will have high-beta stocks and are therefore riskier. Once a financial analyst has forecast for the state of the macroeconomy, it is necessary to determine the implication of the forecast to specific industries by using industry beta information.

Industry groups usually show more dispersion in their stock market performance. Industry beta is especially useful to fund managers as many of the mutual funds are focusing on industry sectors. For example, Fidelity offers about 40 Select Funds, each of which is invested in a particular industry (O'Neal 2000). In Australia, Resources funds, Challenger Gold Trust, Lowell Australian Resources, Colonial First State Technology and Commerce funds, and many more are typical industry sector funds. One prominent feature of sector funds is the high volatility of funds' returns relative to the broad market. Even small investors can easily take positions in industry performance using mutual funds with an industry focus. Morningstar offers investors an online education course on investing sector funds using analysis of R-square and beta (Teresa 2000).

Although previous studies have argued that the stability of betas and the findings on stochastic properties of betas are not conclusive, the form of betas and estimation of industrial betas are yet to be provided. Furthermore, since industrial betas are now widely used by financial practitioners and industrial analysts in practice, a clear understanding of stochastic properties of time-varying industrial betas would be very important. The results of the paper will have important implications to the portfolio management and securities analysis. Industry betas normally serve as the 'prior' to individual beta estimation (Vasicek 1973, p. 1237). The identification of industry beta stability and the identified best stochastic model will help to determine whether the forecasting method for beta is optimal since the industry adopts different estimation methods.

The primary objective of this study is to investigate the problem of choosing a best possible time-varying beta for each individual industrial index using the state-space framework. Since it is not appropriate to assume each beta function to be a constant, the point estimation of each beta obtained by regressing the asset return over the market portfolio is therefore not justifiable. More sophisticated estimation techniques are therefore needed. State-space modelling typically deals with dynamic time series models that involve unobserved variables, and the time-varying parameters can be estimated nicely. Since beta's variability is related to the change of micro- or macro-economic factors, we suspect that industries with unique characteristics might have different stochastic properties. The earlier work of Ball and Brown (1980) discovered that Australian resources and industry sectors have different characteristics in both risk and returns. Therefore, by exploring different stochastic time-varying betas, it is likely to find the most appropriate and accurate model for each industrial index. As the choice of relatively accurate models for the nineteen industrial stock portfolio returns is very important to both financial academics and practitioners, we hope that this study would be used as a detailed model selection procedure for the choice of a best possible model for each industrial index. Several time-varying beta models, including the random walk models, random coefficient models, and mean reverting models, are examined in detail using the Kalman filter approach. By comparing models' performance with the ordinary linear model, the best possible time-varying betas are then suggested.

The paper is organised as follows. Section 2 states the methodology of our research. Our numerical studies and empirical comparisons are given in section 3. We then conclude the paper with some discussion and extensions.

2. Research Method

2.1 Time-Varying Beta and the Kalman Filter

The state-space representation of a system is a fundamental technique in modern control theory. It is now widely used for expressing dynamic systems. A state space system consists of two equations: a transition equation (or state equation) and a measurement equation. The measurement equation describes the relation between observed variables (data) and unobserved state variables. And the transition equation describes the dynamics of the state variables based on a minimum set of information from the present and past such that the future behaviour of the system can be completely described by the knowledge of the present state and the future input.

The time-varying market model would be expressed as:

[R.sub.it] = [[alpha].sub.it] + [[beta].sub.it] [R.sub.mt] + [v.sub.it] [v.sub.it] ~ (0, [[sigma].sup.2]),

where: [R.sub.t] = the industry index portfolio return; and [R.sub.mt] the market portfolio return.

As this paper considers only the case where the industrial indices are mutually independent, we estimate the parameters individually for each index. Therefore, one can denote [R.sub.it] by [R.sub.t], [[alpha].sub.it] by [[alpha].sub.t] and [[beta].sub.it] by [[beta].sub.t] for each discussed index. If both the risk-free rate [alpha] and the regression coefficient [beta] are assumed constant, the model can be estimated by ordinary least squares.

The state-space form of the time-varying market model above can be rewritten as

Measurement equation: [R.sub.t] = [X.sub.t][B.sub.t] + [v.sub.t], [v.sub.t] ~ N(0, [[sigma].sup.2]),

Transition equation: [B.sub.t] = [PHI] [B.sub.t-1] + [[omega].sub.t], [[omega].sub.t] ~ [N.sub.2](0, [XI]),

where: each [X.sub.t] - [(1, [R.sub.mt]).sup.[tau]] is a vector;{[R.sub.t]} is the asset return and {[R.sub.mt]} is the market portfolio return at time t, each [B.sub.t] = [([[alpha].sub.t], [[beta].sub.it]).sup.[tau]] is also a parameter vector, and both [v.sub.t] and [[omega].sub.t] are Gaussian and mutually independent. By setting different values to [PHI], one can derive random walk, random coefficient or mean reverting processes. The covariance matrix [XI] and any elements of the transition matrix [PHI] are known as the hyperparameters of the system.

Although a sequence of generalized least squares regressions can achieve the inferences about state vector conditional on information available up to time t, it is extremely inefficient in terms of computational burden (see Kim & Nelson 1999, p. 20). The method of Kalman filter is originally developed by Kalman (1960) within the context of linear systems, and the method now serves as the basic tool to deal with the standard state-space model. Due to the ease of implementation of the algorithm on digital computers, the Kalman filter has now become well known and widely used in many areas of applications, especially in the state-space form of varying-parameter regression.

The Kalman filter approach has been used by some financial economists to explore the dynamics of financial series. Brooks, Faff and McKenzie (1998) investigated three techniques for the time-varying beta risk of Australian industry portfolios. After comparing the three techniques: a multivariate generalized ARCH model, a time-varying beta approach suggested by Schwert and Seguin (1990), and the Kalman filter approach, the paper showed that the Kalman filter approach performed better in both in-sample and out-of-sample forecasts. Wells (1994) employed the Kalman filter approach to estimate beta models for a small sample of the Swedish stocks. Black, Fraser and Power (1992) used a similar method to estimate random walk betas for a sample of U.K. unit trusts. Groenewold and Fraser (1999) estimated time-varying beta models using the Kalman filter approach as well as the rolling regression and recursive regression method. They found that the Kalman filter betas were not totally consistent with betas obtained by the other two approaches, although much of the variability of beta parameters could be explained by a time trend.

Using the Kalman filter to estimate the time-varying parameters has two benefits. First, the calculation is recursive. Although the current estimates are based on the whole past history of measurement, there is no need for expanding memory and the extra observations available for the regression. Second, the Kalman filter converges quickly, no matter whether the underlying model is. Meinhold and Singpurwalla (1983) suggested that Kalman filter could actually be viewed as an updating procedure, which consists of forming a preliminary guess about the state of nature and then adding a correction to this guess, and the correction is determined by how well the guess has performed in predicting the next observation.

The Kalman filter (basic filter) consists of the prediction and updating two steps.

2.1.1 Prediction Treating period t-1 as the initial period, the estimate of the state and its covariance at time t, conditional on information available at t-1, are:

[B.sub.t|t-1]] = [PHI] [B.sub.t-1|t-1]],

[[SIGMA].sub.t|t-1]] = [PHI] [[SIGMA].sub.t-|t-1]] [[PHI].sup.[tau]] + [XI], respectively.

When the new observation and corresponding [X.sub.t] are available, the one-step-ahead prediction error, [v.sub.t], and its variance, [f.sub.t], can be obtained by:

[v.sub.t|t-1]] = [R.sub.t] - [R.sub.t|t-1]] = [R.sub.t] [??] [X.sub.t][B.sub.t|t-1]],

[f.sub.t|t-1]] = [X.sub.t] [[SIGMA].sub.t|t-1]] [X.sup.[tau].sub.t] + [[sigma].sup.2].

2.1.2 Updating The prediction error contains new information about [B.sub.t] beyond that contained in [B.sub.t|t-1]]. Thus, after observing [R.sub.t], a more accurate inference can be made of [B.sub.t]. [B.sub.t|t]], an inference of Bt based on information up to time t, would be of the following form:

[B.sub.t|t]] - [B.sub.t|t-1]] + [K.sub.t][v.sub.t|t-1]],

where {[K.sub.t]} is the weight assigned to new information about [B.sub.t] contained in the prediction error. Similarly,

[[SIGMA].sub.t|t]] = [[SIGMA].sub.t|t-1]] - [K.sub.t][X.sub.t][[SIGMA].sub.t|t-1]],

where each [K.sub.t] = [[SIGMA].sub.t|t-1]] [X.sup.[tau].sub.t] [f.sup.-1.sub.t|t-1] is the Kalman gain, which determines the weight assigned to information about [B.sub.t] contained in the prediction error.

When the shocks to the model and the initial unobserved variables are normally distributed, the Kalman filter enables the likelihood function to be calculated via prediction-error decomposition. The unknown hyperparameters of the system, such as elements of [PHI] and variances of the error terms, are estimated by numerical optimisation over the likelihood function given by (see Harvey 1989)

L = -T/2log(2[pi])-1/2[T.summation over t=1]log([absolute value of [f.sub.t|t-1])]-1/2[T.summation over t=1][v.sup.2.sub.t|t-1]/[f.sub.t|t-1],

where T is the number of observations.

2.2 The Models

This section considers five different classes of models: the random walk models, the random coefficient models, the ARMR(1,1) models, the mean-reverting models, and the moving mean models.

2.2.1 The Random Walk Model Setting the transition matrix [PHI] equal to the identity matrix, we can derive the following random-walk model:

[R.sub.t] = [[alpha].sub.t] + [R.sub.mt] [[beta].sub.t] + [v.sub.t] [v.sub.t] ~ (0, [[sigma].sup.2])

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

where each [([w.sub.1t], [w.sub.2t]).sup.[tau]] is also assumed to be normally distributed with zero mean and a constant covariance matrix [XI]. The state noise vector [([w.sub.1t], [w.sub.1t]).sup.[tau]] is also assumed to be serially uncorrelated. Thus, [w.sub.1t] ~ N(0, [[sigma].sup.2.sub.a]) and [w.sub.2t] ~ N(0, [[sigma].sup.2.sub.b]). This implies that [XI] has just the diagonal elements [[sigma].sup.2.sub.a] and [[sigma].sup.2.sub.B]. The hyperparameters that must be estimated are the three parameters: [[sigma].sup.2.sub.a], [[sigma].sup.2.sub.B] and [R.sub.it] [[sigma].sup.2].

2.2.2 The Random Coefficient Model Setting the transition matrix [PHI] equal to zero, we can derive a random-coefficient model. We then obtain the excess-return-version model given by:

[R.sub.et] = [R.sub.emt][[beta].sub.t] + [v.sub.t], [v.sub.t] ~ (0, [[sigma].sup.2]),

[[beta].sub.t] = [beta] + [n.sub.t],

[n.sub.t] ~ (0, [[sigma].sup.2.sub.n]),

where [R.sub.et] and [R.sub.emt] are excess returns of individual industry and the market index portfolio respectively. Thus, the three parameters need to be estimated: [bar][beta], [[sigma].sup.2] and [[sigma].sup.2.sub.n] (1).

2.2.3 The ARMR(1,1) Model To model the ARMR(1,1) process of [beta], we also use the excess version of the market model. Therefore, the model is set up as:

[R.sub.et] = [R.sub.emt][[beta].sub.t] + [v.sub.t], [v.sub.t] ~ N(0, [[sigma].sup.2]),

[[beta].sub.t] = [phi][[beta].sub.t-1] + [n.sub.t] - [theta][n.sub.t-1], [n.sub.t] ~ N(0, [[sigma].sup.2.sub.n]),

where [R.sub.et] and [R.sub.emt] are as defined above, the parameter [phi] and [theta] are chosen to ensure that {[[beta].sub.t]} is stationary. Thus, one needs to estimate the four parameters: [phi], [theta], [[sigma].sup.2.sub.n] and [[sigma].sup.2].

2.2.4 The Moving-Reverting Model." We use the excess return version model again for the mean-reverting model. Our mean-reverting model is defined as follows:

[R.sub.et] = [R.sub.emt][[beta].sub.t] + [v.sub.t], [v.sub.t] ~ N(0, [[sigma].sup.2]),

[[beta].sub.t] - [beta] = [phi]([[beta].sub.t-1] - [bar][beta]) + [n.sub.t], [n.sub.t] ~ N(0, [[sigma].sup.2.sub.n]),

where [R.sub.et] and [R.sub.emt] are the excess returns of the individual industry and the market return, respectively. The parameters needed to be estimated are [[sigma].sup.2], [beta], [[sigma].sup.2.sub.n] and [phi].

2.2.5 The Moving Mean Model Wells (1994) extended the mean-reverting model to allow the mean of [[beta].sub.t] to vary, and suggested the following moving mean model

[a.sub.t] = [[phi].sub.11][a.sub.t-1] + [[delta].sub.t], [[delta].sub.t] ~ N(0, [[sigma].sup.2.sub.[delta]],

[[beta].sub.t] - [[beta].sub.t] = [[phi].sub.22]([[beta].sub.t-1] - [[bar][beta].sub.t-1]) + [u.sub.t], [u.sub.t] ~ N(0, [[sigma].sup.2.sub.u],

[[beta].sub.t] = [[beta].sub.t-1] + [[gamma].sub.t], [[gamma].sub.t] ~ N(0, [[sigma].sup.2.sub.[gamma]].

The state-space model can then be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where [[delta].sub.t] [[gamma].sub.t] and y, are all normally distributed and mutually independent residual series. Therefore, the parameters needed to be estimated here are [[sigma].sup.2], [[sigma].sup.2.sub.[delta]], [[sigma].sup.2.sub.u], [[sigma].sup.2.sub.[gamma]], [[phi].sub.11], and [[phi].sub.22]. We estimate each industry using both the moving mean model and the mean reverting model. If the variance of [[gamma].sub.t], is close to zero, we will show only the results for the mean reverting model, as this suggests that {[[beta].sub.t]} behaves like a constant.

To implement the Kalman filter to the above models, one needs to set two different sets of initial values. One of these is the initial value for the state vector and its covariance in the Kalman filter. Wells (1996) has suggested that for the mean reverting and random coefficient models, the initial states are set to be zero and the initial covariance is set to be a large number. For the random walk model, the initial states are equal to the ordinary least squares (OLS) estimates obtained from the first ten observations; the initial covariance of the states is the covariance matrix of these OLS estimates (see Wells 1996). The second set of the initial values is for the hyper parameters to be estimated by maximizing the likelihood function. The means in the mean reverting and the random coefficient models are simply set by using the OLS estimates from the entire sample. The choice of the variance of the observation equation has no problem because it is concentrated out of the likelihood function. The coefficients [phi] involved in both the ARMR(1,1) and the mean reverting model and [[phi].sub.11] and [[phi].sub.22] involved in the moving mean model are all set to be 0.5 as the experiment has shown that the final estimates are not at all sensitive to this value.

3. Summary and Analysis of the Results

3.1 Summary Statistics

The data used in this study are nineteen monthly ASX industrial stock return indexes from December 1979 to March 2000. The risk-free rate of return is computed from the Australian three-month Treasury bill rate. (2) The data set were sourced from Datastream. The summary statistics of the stock returns data are provided in table 1 below. The means are monthly proportional rates of return and vary from a high of 2% for the media sector, to a low of 0.4% for the gold sector. The gold sector also has the highest variance while the property trusts sector has the lowest variance. The Skewness, Kurtosis and Jarque-Bera are the tests of normality on returns. The null of normality have been widely rejected. Most of return series are left skewed and leptokurtic. The ADF column shows the results for augmented Dicky Fuller unit root test. All the returns series are shown to be stationary. The last column presents the ARCH test, and six out of nineteen indexes have shown the evidence of heterosckedasticity. (3)

3.2 Empirical Comparisons

Table 2 provides the summary of the results for the best available model for each industrial index. Column three to ten of the table shows the estimated parameters of models: the moving mean model (MMM) or the mean-reverting model (MRM), the ARMA(1,1) model (ARMA), the random coefficient model (RCM), the random walk model (RWM) and ordinary least square model (OLS), respectively. Columns eleven to eighteen provides the diagnostic test results and four criteria used to measure models' performances. The diagnostic tests of the models are Box-Ljung statistics for higher order serial correlation Q(12), Goldfeld-Quandt test (G-Q) test for heteroscadaesticity, and the classical ARCH test. The cumulated periodogram test (C-P test) reports the maximum gap between the distribution function of residual series and white noises. If the residual series are white noise, the cumulated periodogram should differ only slightly from the theoretical spectral distribution function of the white noise.

To compare the models' performances, for each industrial index we use four criteria to find out the best possible model, which describes the particular industrial beta model. Apart from the normal R square, which measures the proportion of the variability of industrial returns that is explained by the model, the Akaike Information Criterion (AIC) was used to weight the reduction in the likelihood function against the increase in the number of parameters necessary to achieve this reduction. Harvey (1989, p. 245) and Wells (1996, p. 100) expressed the AIC as follows:

AIC = [[sigma].sup.2] exp(2(k + d)/T),

where [[sigma].sup.2] is the estimated residual variance of the model, d is the dimension of the state vector, k is the number of the hyper-parameters, and both are to be estimated. The 'best' possible models should have the lowest AIC values. The other two criteria used here are the mean absolute forecasting error (MAE) and the mean square error (MSE) of the estimates. The mean absolute error and the mean square error of the estimates are defined by

MA[E.sub.i] = 1/T [T.summation over t=1] [absolute value of [R.sub.it] - [R.sub.it]/T] and MS[E.sub.i] = 1/T [T.summation over t=1] [([R.sub.it] - [R.sub.it]).sup.2]/T, respectively, where Rit represents the estimated return of industry i. The best possible models should have the lowest errors of the estimates.

One problem here is the identification of the 'best' model for each industry group. Neither theory nor econometric procedures provide guidance in terms of how to estimate the betas for industry portfolio groups. Though the empirical work has provided more consistent evidence on individual stock's beta, for example, it is more widely accepted that individual betas are 'mean-reverting' process, the results of industry portfolios are mixing. The approach here we adopt is a really 'data-driven' approach. We believe that the stochastic parameter approach is both feasible and practical. It is expected that the statistical identification and estimation on different models will uncover the real pattern of the stochastic process of industry beta. Though the different criteria adopted might give us different rankings of the models, and there might not be 'perfect' stochastic model for each industry, we attempt to identify the most suitable model by balancing the four criteria here, especially, considering the R-square and the forecasting errors of the estimation.

3.3 Summary and Discussion of Results

Table 2 shows that across all the industries, there is at least one time-varying beta model, which performs better than the Ordinary Least Square model in either R-square value, low predication error or better diagnostic results. Thus the efforts spent on the estimation of time-varying beta models are quite worthwhile. Although there is no perfect model for every industry, we can certainly suggest using the most appropriate time-varying beta model balanced by the four criteria. One problem here is that the different criteria give different rankings of models. Here we try to balance four criteria to find the most suitable model for each industrial index by first considering the R-square and then forecasting errors and AIC.

For the alcohol and tobacco industry, five models have close values of R-square statistics, but the random coefficient model (RCM) has both the lowest MAE and MSE errors. It strongly suggests that the beta for alcohol and tobacco industry is close to a random process, which implies that there is no predictability in the beta. For the banks and finance industry, moving mean model (MMM) has much higher R-square numbers, however, the random coefficient model (RCM) predicts better despite a low R-square. It is also noted that OLS almost has similar performance as the time-varying beta model. Therefore, the instability of banks and finance beta is not convincing though the MMM has been chosen here. For the building materials industry, ARMA model and RCM provide mixing results on stochastic process of beta as ARMR has higher R-square but RCM has lower prediction error. The result here indicates that the mean-reverting property of this industry beta is not obvious and ARMR (1,1) process looks to be the better description of beta. In the chemicals industry, MMM is convinced as the best description for beta due to its low prediction errors than other models. The chemical industry beta is more likely a mean-reverting process but the long-term state of the beta is not constant over the time.

In the develop contractor industry, MRM and RCM actually provide the mixing results as their R-square and prediction errors are all similar though MRM could be identified as the slightly better one. However, for the diversified industrial, diversified resources and energy industry, it is quite convincing that MMM is the best to describe the beta variation process as the R-squares are higher than other models and prediction errors are also lower than other models. The results strongly imply that betas of these industry portfolios are indeed reverting to an unstable long-term mean. In the engineering industry, the MRM outperforms the RCM slightly in the R-square statistics. Therefore, it is less convincing that the beta of engineering industry is mean-reverting process rather than random. For the food and household industry, the RCM has much lower prediction errors than MMM and ARMA model. However, the R-square of RCM is also low. Thus, ARMR model has been identified as the better model for food and household industry beta, which might indicate that food and household beta is more close to an ARMR (1,1) process.

The gold industry is some kind of special case among all the industry groups that all models tested have relatively high prediction errors including the OLS model. Interestingly, the RWM looks to be the better one as it has much higher R-square and relatively lower prediction error. For the same reason, the RWM is also identified as the better model for insurance industry beta. The results here show that in these two industries the beta is more like a random walk process. The random walk process simply implies that no predictability exists in their betas. In the investment and financial services industry, MRM and RCM actually have very similar R-squares, but RCM has slightly lower forecasting errors, which indicates that the investment and financial services beta is close to a random process. For the media industry, RWM has a much higher R-square statistics despite the prediction errors are slightly higher than other three time-varying models. Thus, the media beta is identified to be close to a random walk process. In paper and packaging industry, the result is mixing. ARMR model performs slightly better than OLS. However, it is less convincing that ARMR outperforms other time-varying models. On the contrary, ARMR has much higher R square statistics than other time-varying models in property trust industry despite all models including OLS have similar prediction errors. Therefore, it might be appropriate to describe the property trust beta as an ARMR (1,1) process.

For the retail industry, MMM has highest R-square, however, the AIC number is large. It should be kept in mind that the extra parameter of this model has reduced the degrees of freedom largely. The RCM shows both a lower prediction error and lower R-square statistics. Therefore, it is not clear that the retail industry beta is either random process or mean-reverting. For the transport industry, MRM outperforms RCM in terms of a higher R-square value despite a similar level of prediction errors. Therefore, MRM has been chosen for this industry. We believe the beta of this industry has the mean-reverting property. Lastly, for the other metals industry, ARMA has achieved the highest R-square value though the prediction errors are similar to other stochastic models. The result here indicate that the other metals industry beta might follow an ARMA (1,1) process.

Therefore, in summary, the most popular model for our industrial portfolios is the moving mean model (MMM), which shows the convincing performance in chemicals, diversified industrial, diversified resources and energy industries and relatively better performance in the banks and finance and retail industries. The second popular model is the ARMA(1,1) model which well describes the betas for the building material, food and household, paper and packaging, property trusts and other metals industries. The random walk models (RWM) best fits the gold industry, insurance and media industry betas. Additionally, the mean reverting model (MRM) has been chosen as the best to model the developer contractor, engineering and transport industry beta functions. Surprisingly, the random coefficient model (RCM), which was favoured by Brooks, Faff and Lee (1992) and Wells (1994) only performs well in two of the nineteen industries, the alcohol and tobacco and the investment and financial services industries.

Overall, it is obvious that the industry portfolio betas are also best described as a time-varying stochastic process. However, the evidence on what 'type' of stochastic evolution the industry portfolio beta follows is inconclusive. Based on the diagnostic and our analysis, we can answer the question what kind of stochastic process each industry portfolio beta is 'most likely' to follow. Our results are consistent with previous findings that the industry portfolio betas have either random or sequential stochastic variation just like an individual asset beta. The industrial indexes that possess the mean-reverting type (4) of betas are banks and finance, chemicals, developer contractor, diversified industrial, diversified resources, energy, engineering, retail, and transport industries. Other industries that are believed to have stochastic betas are alcohol and tobacco, gold, insurance, investment and financial services, media and property trusts industries. We have also identified that building material, food and household, paper, packaging, and other metals industry have an ARMR beta process, which indicates that the current value of beta series depends linearly on its own pervious values plus a combination of current and previous values of a white noise error term.

The identification of mean-reverting or random property of beta has important implications to the portfolio performance evaluation, asset valuation, capital budgeting decision and tests of asset pricing models. If beta is close to the random coefficient, random walk process, the beta would not be predictable as the fluctuations in beta are purely random, the transitory does not carry over from period to period. However, a mean-reverting process or ARMA. process of beta implies that deviation in beta from its mean is serially correlated or the current beta is partially related to previous beta--thus would be at least partially predictable. Figures 1 to 5 present some example plots of the time-varying beta series and the corresponding fitted returns. Due to the limitation of space, we only display five industrial beta models and returns here as the illustration for each stochastic model tested in this study. The full detailed results are available in Yao and Gao (2002). Figure 1 displays the chemicals industry beta modelled by the moving mean model. Panel A presents the stochastic behaviour of beta and the beta series varies around its long-term moving mean. The long-term state of beta varies from 0.7 to 0.9. The corresponding fitted return is shown in panel B of the figure. To assess the performance of the random walk model, we present the gold industry result here. Figure 2 depicts the random walk process of gold industry beta and corresponding fitted returns. In figure 3, the stochastic behaviour of food and household industry beta and fitted returns are displayed, in which the beta function is modelled by an ARMR(1,1) process. The usefulness and applicability of the random coefficient beta model is demonstrated in figure 4 by the investment and financial services industry. For this industry, the beta function varies randomly around its zero mean but tends to be more volatile in some period of time, for example the late 80's than other period of time like early 90's. Figure 5 provides the time-varying beta for the developer contractor industry in which the beta is modelled as a mean reverting process. The developer contractor beta fluctuates unevenly around its steady long-term state. Overall, when comparing the actual returns with their corresponding fitted values, one will find that the time variances of the returns have been well captured by the corresponding stochastic beta model. It should be noted that, however, given the current beta estimate [B.sub.t|t], the effect of the observed market return [X.sub.t] has a lag effect on the predicted return [R.sub.t+1|t]. Therefore, in the case of market crash, the effect of such crash can only be fitted afterwards, that is, one month later. (5)

[FIGURES 3-4 OMITTED]

Overall, the stochastic parameter models fit reasonably well in modelling the timevarying systematic risk of the industry portfolios. Moreover, the results of different stochastic models discovered for each industry have presented some more accurate description of the time paths of beta series, which provides a good understanding of the risk characteristics in different industries.

4. Conclusion and Future Research

For similar industrial stock returns, Brooks, Faff and McKenzie (1998) considered using the GARCH, Schwert and Seguin, and Kalman approaches to estimating time-varying beta models. Their research overwhelmingly supported the Kalman filter approach. This paper further considered the use of the Kalman filter approach to the estimation of time-varying industrial beta models. Our study has suggested that the industry portfolio does not have a stable beta. Similar to the individual asset beta, the variation of an industry portfolio beta is either a mean reverting or random process. However, for some industry groups, the long-term mean of beta is also time-varying.

In our detailed studies, the time-varying market model performed better than the ordinary least square market model in explaining industrial returns. The industrial betas for six industries: the banks and finance, the chemicals, the diversified industrial, the diversified resources, the energy and the retail industry are best modelled as moving-mean processes while the industrial betas for the developer contractor, the engineering and the transport industries are best modelled as mean reverting processes. These results clearly show that the industrial beta does vary according to their long-term moving or stable state. Two industries--the investment and financial services and the alcohol and tobacco industries--modelled by the random coefficients models have the random betas, which wander randomly around their zero means. All of those facts have confirmed the meanreversion findings by Blume (1971), Brenner and Smidt (1977) and Francis (1979). Betas in the gold, the insurance and the media industries are best described by random-walk processes, while the building materials, the food and household, the paper and packaging, the property trusts and the other metals industries are best described by ARMA(1,1) models.

Additionally, caution needs to be taken in the use of 'best fitted' or 'best possible' models, as the 'best' model is just chosen according to one or two of the different criteria. We should also realize that the diagnostic statistics and forecasting change as the model changes, so does the time path of each beta. This might indicate that more complicated modelling techniques are desirable. This paper provides an empirical and practical comparison procedure for the selection of a best possible model for each industrial stock return. The general conclusion here is that industrial risk, which summarized by the beta, does vary randomly around its steady or moving-mean state.

The results given in this paper can be extended in a number of directions. First, it would be of interest to identify some causal variables which are primary sources or causes of the stochastic variation in beta. Second, it is possible to consider the case where the measurement error process and the transition error process are correlated and non-Gaussian. For the correlated and non-Gaussian case, a number of theoretical problems need to be worked out before considering the case for the industrial indices. Third, it would be important in both theory and practice to discuss the case where there is some kind of relationship between each industrial index and some exogenous factors. Our preliminary studies suggest that there is an explicit relationship between each industrial index and a couple of significant factors. Fourth, it would be important in both theory and practice to consider a model specification problem before choosing an appropriate model for each of the industrial indices. The approach of Josev, Brooks and Faff (2001) (see also Brooks & King 1994) can be generalized to the polynomial market model approach and the nonparametric and semiparametric econometric model approach (see Gao & King 2001). Finally, it would be wise to examine whether the industrial stock returns are of some kind of long-memory property before modelling them, as existing studies suggest that long-memory property is a key feature of some stock returns (see Ding, Granger & Engle 1993; Ding & Granger 1996; Pagan 1996; Trivedi & Brooks 1999; Brooks, Faff, McKenzie & Mitchell 2000). Some of these issues are left for possible future research.

Table 1

Summary Statistics of Returns

ASX Industry group     Mean       Variance    Skewness    Kurtosis

Alcohol and Tobacco    0.01684    0.003273    -2.21701    17.84161
Banks and Finance      0.01563    0.003439    -0.92291     6.38387
Building Mats          0.01034    0.003395    -1.6052     10.25257
Chemicals              0.01084    0.004047    -0.85512     5.68166
Devl. Contractor       0.01498    0.005001    -3.66785    34.67106
Divs. Industrial       0.01302    0.004548    -2.65531    20.92053
Divs. Resources        0.01012    0.006170    -1.06379     7.00093
Energy                 0.00522    0.007807    -0.89381     5.94637
Engineering            0.00818    0.003885    -0.86189     4.00144
Food & Household       0.01164    0.003597    -1.40992     7.50228
Gold                   0.00365    0.016585    -0.23904     3.86923
Insurance              0.01372    0.004726    -1.73007    13.68822
Inv and Fin Servs.     0.01181    0.003002    -3.81393    37.20543
Media                  0.02038    0.007967    -1.30875     6.95052
Other Metals           0.00521    0.009089    -1.84619    14.85819
Paper and Packaging    0.00892    0.003297    -1.0424      5.08132
Property Trusts        0.01067    0.001359    -1.50725    11.85543
Retail                 0.01273    0.003619    -2.07127    18.34832
Transport              0.01365    0.005303    -2.4888     20.01891
ASX Market Index       0.01106    0.00363     -3.34294    29.76842

ASX Industry group     Jarque-Bera      ADF       ARCH(6)

Alcohol and Tobacco     3422.08517    -14.0903     2.9555
Banks and Finance        447.12857    -14.6562    12.8012
Building Mats           1168.64617    -14.9155    16.7806
Chemicals                356.46287    -15.2663     5.2769
Devl. Contractor        12715.9387    -15.3434     6.249
Divs. Industrial        4716.94689    -15.2885     6.0251
Divs. Resources          542.08882    -14.2725    28.6421
Energy                   390.36768    -12.7318     8.0455
Engineering              192.20228    -14.0959     5.444
Food & Household         650.38702    -14.9562    12.4265
Gold                     153.89477    -14.4181    10.3992
Insurance               2018.31662    -15.0139    13.6265
Inv and Fin Servs.      14604.5841    -14.0324    11.0091
Media                    558.50529    -12.942     33.8767
Other Metals            2373.29603    -16.5707     5.6134
Paper and Packaging      305.43282    -13.8708    19.0422
Property Trusts         1515.08872    -15.1447     9.1044
Retail                  3582.44153    -13.9833     5.7955
Transport               4308.52451    -15.168      6.6803
ASX Market Index        9424.95718    -15.361

Note: Significance levels (5%): Skewness and Kurtosis 1.96 , Normality
5.99, ARCH(6) 12.59, ADF-3,4297.

Table 2

Stochastic Modelling and the Comparison with OLS

                                       [[sigma].sup.2.sub.a],
group    Model    [[sigma].sup.2]    [[sigma].sup.2.sub.[delta]]

A&T      RCM           0.0057
         MRM           0.0014
         ARMA          0.0014
         RWM           0.0047                  0.0000
         OLS
B&F      MMM           0.187                   0.0183
         ARMA          0.0014
         RCM           0.0063
         RWM           0.0129                  0.0000
         OLS
BM       ARMA          0.0009
         MMM           8.2319                  0.0034
         RCM           0.0034
         RWM           0.0566                  0.0001
         OLS
CHE      MMM           0.2351                  0.8048
         ARMA          0.0018
         RCM           0.0054
         RWM           0.0068                  0.0000
         OLS
DC       MRM           0.0011
         ARMA          0.0011
         RCM           0.0036
         RWM           0.0081                  0.0000
         OLS
DI       MMM           2.932                   5.7287
         ARMA          0.0012
         RCM           0.0040
         RWM           0.0089                  0.0000
         OLS
DR       MMM           0.049                   0.1006
         ARMA          0.0016
         RCM           0.0043
         RWM           0.0026                  0.0000
         OLS
ENE      MMM           0.0443                  0.2393
         ARMA          0.0026
         RCM           0.0052
         RWM           0.0012                  0.0000
         OLS
ENG      MRM           0.0014
         ARMA          0.0015
         RCM           0.0036
         RWM           0.0037                  0.0000
         OLS
F&H      ARMA          0.0015
         MMM           0.2683                  0.0001
         RCM           0.0092
         RWM           0.0014                  0.0000
         OLS
GOL      RWM           0.0088                  0.00
         MRM           0.0096
         ARMA          0.0090
         RCM           0.0183
         OLS
INS      RWM           0.0017                  0.00
         MMM           2.8053                  0.0608
         ARMA          0.0020
         RCM           0.0124
         OLS
I&F      RCM           0.0052
         MRM           0.0009
         ARMA          0.0009
         RWM           0.0056                  0.0000
         OLS
MED      RWM           0.004                   0.00
         MMM           1.1041                  0.485
         ARMA          0.0043
         RCM           0.0061
         OLS
P&P      ARMA          0.0013
         MMM           0.0238                  0.0001
         RCM           0.0072
         RWM           0.0001                  1.2E-06
         OLS
PT       ARMA          0.0036
         MMM          21.774                   0.0251
         RCM           0.0121
         RWM           0.0028                  0.0000
         OLS
RET      MMM          34.092                   0.00
         ARMA          0.0014
         RCM           0.0053
         RWM           0.0049                  0.0000
         OLS
TRA      MRM           0.0016
         ARMA          0.0017
         RCM           0.0047
         RWM           0.0618                  0.0000
         OLS
OM       ARMA          0.0013
         MRM           0.0029
         RCM           0.0049
         RWM           0.1468                  0.0000
         OLS

                  [[sigma].sup.2.sub.b],
                  [[sigma].sup.2.sub.n],
group    Model    [[sigma].sup.2.sub.u]     [[sigma].sup.2.sub.y]

A&T      RCM               3.0659
         MRM               0.0002
         ARMA              0.0099
         RWM               0.0059
         OLS
B&F      MMM               4.9603                   0.0281
         ARMA              0.0122
         RCM               3.0244
         RWM               0.0065
         OLS
BM       ARMA              0.0064
         MMM             105.90                     0.3625
         RCM               2.2723
         RWM               0.1283
         OLS
CHE      MMM              16.162                    0.0311
         ARMA              0.0020
         RCM               2.2569
         RWM               0.0010
         OLS
DC       MRM               0.0020
         ARMA              0.0225
         RCM               2.9930
         RWM               0.0077
         OLS
DI       MMM             158.38                     0.1038
         ARMA              0.0001
         RCM               2.2616
         RWM               0.0028
         OLS
DR       MMM               5.1883                   0.0017
         ARMA              0.0294
         RCM               2.2751
         RWM               0.009
         OLS
ENE      MMM              17.150                    0.6822
         ARMA              0.1002
         RCM               2.2609
         RWM               0.0098
         OLS
ENG      MRM               0.0042
         ARMA              0.0143
         RCM               2.2731
         RWM               0.0104
         OLS
F&H      ARMA              0.0047
         MMM               1.8002                   0.0157
         RCM               3.0157
         RWM               0.0075
         OLS
GOL      RWM               0.0084
         MRM               0.0000
         ARMA              0.1035
         RCM               3.0385
         OLS
INS      RWM               0.0024
         MMM              14.59                    13.092
         ARMA              0.0009
         RCM               2.2673
         OLS
I&F      RCM               3.0042
         MRM               0.5319
         ARMA              0.0021
         RWM               0.0101
         OLS
MED      RWM               0.0131
         MMM               0.0000                   3.9322
         ARMA              0.0112
         RCM               2.2622
         OLS
P&P      ARMA              0.0219
         MMM               0                        0.0724
         RCM               3.043
         RWM               2E-05
         OLS
PT       ARMA              0.000
         MMM               2.1915                   0.0000
         RCM               2.9683
         RWM               0.0025
         OLS                                        0.3725
RET      MMM             155.72                     8.7982
         ARMA              0.0004
         RCM               2.2606
         RWM               0.0059
         OLS
TRA      MRM               0.0011
         ARMA              0.0012
         RCM               2.2771
         RWM               0.0053
         OLS
OM       ARMA              0.0219
         MRM               0.0008
         RCM               2.2624
         RWM               0.0000
         OLS

                                 [[PHI].sub.11]
group    Model    [bar][beta]       ([phi])        [[PHI].sub.22]

A&T      RCM        0.2865
         MRM        0.7773           1.001
         ARMA                        0.6977
         RWM
         OLS        0.6233
B&F      MMM                         0.7622            0.000
         ARMA                        0.8889
         RCM        0.3086
         RWM
         OLS        0.7666
BM       ARMA                        0.8188
         MMM                         0.9883            0.0000
         RCM        0.7942
         RWM
         OLS        0.8608
CHE      MMM                        -1.90E-07          0.0000
         ARMA                        0.9038
         RCM        0.7397
         RWM
         OLS        0.8079
DC       MRM        0.9240           1.0001
         ARMA                        0.7959
         RCM        0.4364
         RWM
         OLS        0.8994
DI       MMM                         0.06              0.000
         ARMA                        0.8903
         RCM        0.9285
         RWM
         OLS        0.9011
DR       MMM                         0.3093            0.023
         ARMA                        0.6587
         RCM        1.0576
         RWM
         OLS        1.2187
ENE      MMM                         0.3213            0.174
         ARMA                        0.7639
         RCM        1.0873
         RWM
         OLS        1.2402
ENG      MRM        0.957                              0.9988
         ARMA                                          0.8668
         RCM        0.7579
         RWM
         OLS        0.8553
F&H      ARMA                        0.9575
         MMM                         0.988            -0.95
         RCM        0.2782
         RWM
         OLS        0.7152
GOL      RWM
         MRM        1.2218                             1.0000
         ARMA                        0.7753
         RCM        0.7537
         OLS        1.5085
INS      RWM
         MMM                         0.947            -0.92
         ARMA                        0.9947
         RCM        0.7519
         OLS        0.6624
I&F      RCM        0.3028
         MRM        0.7662           2E-04
         ARMA                        0.9400
         RWM
         OLS        0.5832
MED      RWM
         MMM                         0.4480           -0.8700
         ARMA                        0.9489
         RCM        0.8573
         OLS        0.9268
P&P      ARMA                        0.8491
         MMM                         0.9800            0.3000
         RCM        0.3075
         RWM
         OLS        0.782
PT       ARMA                        0.8678
         MMM                         0.9907           -0.9700
         RCM        0.2077
         RWM
         OLS        0.3725
RET      MMM                        -0.122            -0.94
         ARMA                        0.9385
         RCM        0.6907
         RWM
         OLS        0.6637
TRA      MRM        0.9431                             0.999
         ARMA                        0.7975
         RCM        0.9724
         RWM
         OLS        0.933
OM       ARMA                        0.8491
         MRM        0.1907                             1.0000
         RCM        1.2701
         RWM
         OLS        1.3128

group    Model    [theta]      Q(12)        C-PTest     Arch(6)

A&T      RCM                 10.892        0.0769        7.830
         MRM                 21.163 **     0.1491 **     3.8037
         ARMA                19.741        0.1440 **     2.7307
         RWM                 20.629 *      0.1642 **     4.3746
         OLS                 29.692 **     0.1662 **     3.621
B&F      MMM                 27.070 **     0.1537 **     5.970
         ARMA       1.0       7.2988       0.0601        4.848
         RCM                 11.675        0.0983        3.665
         RWM                 37.108 **     0.1659 **     6.846
         OLS                 16.283        0.0922        8.045
BM       ARMA       1.0      21.392 **     0.0568       22.202 **
         MMM                 19.992 *      0.0541       19.509 **
         RCM                 16.317        0.0793       26.731 **
         RWM                 20.956 *      0.1528 **     6.854
         OLS                 16.003        0.0465       15.540 **
CHE      MMM                 17.756        0.0713        6.015
         ARMA       1.0      20.8695 *     0.0762        6.5948
         RCM                 14.8742       0.1315 **     5.2993
         RWM                 55.0927 **    0.1998 **     5.1782
         OLS                 18.975 *      0.0662        5.397
DC       MRM                  7.645        0.0534        6.325
         ARMA       1.0       6.5304       0.0530        2.1331
         RCM                 15.6733       0.0473        7.0467
         RWM                 10.8835       0.1179 *      2.4426
         OLS                  6.950        0.0397        9.122
DI       MMM                 16.572        0.0569        5.698
         ARMA       1.0      13.8364       0.0548       12.3197 *
         RCM                 13.444        0.0576       12.1652 *
         RWM                 16.6512       0.1336 *      7.8648
         OLS                 16.554        0.0667        6.683
DR       MMM                 11.275        0.0423       21.431 **
         ARMA       1.0      12.1166       0.0955       37.048 **
         RCM                 18.0685       0.074        42.973 **
         RWM                 29.5625 **    0.1688 **     7.5516
         OLS                 13.979        0.1042       25.757 **
ENE      MMM                 19.145 *      0.0871        2.963
         ARMA       1.0      13.9942       0.1496 **     5.9862
         RCM                 18.5492       0.1573 **     9.2276
         RWM                 42.6317 **    0.2439 **    11.2807 *
         OLS                 23.053 **     0.1820 **    11.436 *
ENG      MRM                 30.599 **     0.1329 **     9.159
         ARMA       1.0      26.6067 **    0.1331 **     9.8570
         RCM                 20.3047 *     0.1904 **    13.488 **
         RWM                 20.5489 *     0.1507 **     6.3188
         OLS                 18.151        0.1218 *      5.105
F&H      ARMA       0.5      21.199 **     0.0956       11.255*
         MMM                 11.6195       0.074         7.5259
         RCM                 15.5107       0.1034 **    13.216 **
         RWM                 18.5145       0.1320 *      1.7169
         OLS                 20.744 *      0.0808       12.954 **
GOL      RWM                 15.706        0.1224 **     7.024
         MRM                 16.1011       0.1440 *      7.19728
         ARMA       1.0      13.9762       0.1343 **    10.2558
         RCM                 11.8228       0.0992 **     4.4034
         OLS                 17.407        0.1349 *     11.593 *
INS      RWM                 21.682 **     0.1158 *      3.503
         MMM                  7.1418       0.0599        2.07892
         ARMA       1.0      14.3683       0.1099 **     5.699
         RCM                 23.6557 **    0.1388 **     4.9015
         OLS                 24.354 **     0.0814       13.583 **
I&F      RCM                 17.507        0.1489 **     3.769
         MRM                 17.5051       0.1488 **     3.7686
         ARMA       0.8      16.8398       0.0944        5.8254 *
         RWM                 23.9437 **    0.1511 **     2.3241
         OLS                 23.102 **     0.0881       19.006 **
MED      RWM                 28.984 **     0.1749 **    21.994 **
         MMM                  7.4818       0.0437       28.530 **
         ARMA       0.5      15.1681       0.1485 *     24.153 **
         RCM                 15.5596       0.1264 **    25.994 **
         OLS                 17.169        0.1338 *     35.140 **
P&P      ARMA       1.0       9.105        0.1162 *     39.271 **
         MMM                 10.2503       0.0631       34.634 **
         RCM                  9.4904       0.1560 **    20.099 **
         RWM                 31.5228 **    0.1754 **     2.3658
         OLS                 11.328        0.0954       14.870 **
PT       ARMA       1.0      10.849        0.2021 **    17.575 **
         MMM                 13.7699       0.0818        8.0639
         RCM                 23.4458 **    0.3320 **     5.5857
         RWM                 10.4695       0.0717        6.1524
         OLS                 10.048        0.0688        6.638
RET      MMM                  5.861        0.0501        9.198
         ARMA       0.6       5.2754       0.0606        1.4817
         RCM                 13.2108       0.1188 **     2.6954
         RWM                 18.7931 *     0.1404 **     6.2669
         OLS                  8.924        0.0768        7.363
TRA      MRM                 14.153        0.1114 *      5.961
         ARMA       1.0      15.0978       0.1092 *      6.1264
         RCM                  6.0132       6.0132       21.051 **
         RWM                 12.9061       0.1174 *      2.6254
         OLS                 12.386        0.0872        6.327
OM       ARMA       1.0      17.673        0.1102 *      4.149
         MRM                 16.8012       0.1029        7.0744
         RCM                 13.4531       0.0579        4.4183
         RWM                 16.099        0.1745 **     1.5046
         OLS                 16.014        0.1055        5.591

group    Model     G-QTest      R-sq       AIC       MAE       MSE

A&T      RCM      1.232        0.5586     0.006     0.0964    0.0029
         MRM      0.888        0.5573     0.002     0.1465    0.0065
         ARMA     1.362        0.5080     0.002     0.1670    0.0068
         RWM      0.839        0.5910     0.005     0.2025    0.0094
         OLS      0.762        0.5369     0.002     0.2190    0.0122
B&F      MMM      0.883        0.6770     0.201     0.2122    0.0116
         ARMA     1.623 *      0.5521     0.002     0.2425    0.0165
         RCM      1.177        0.4923     0.007     0.1148    0.0039
         RWM      0.901        0.5466     0.013     0.1991    0.0101
         OLS      0.632        0.5681     0.002     0.1741    0.0091
BM       ARMA     1.0282       0.7621     0.002     0.1670    0.0078
         MMM      2.465 **     0.7161     8.864     0.1551    0.0056
         RCM      3.3211 **    0.6371     0.004     0.0866    0.0022
         RWM      0.7773       0.5311     0.059     0.2336    0.0137
         OLS      2.313 **     0.7204     0.001     0.1735    0.0073
CHE      MMM      0.983        0.8204     0.253     0.0450    0.0005
         ARMA     0.8510       0.5737     0.002     0.2607    0.0176
         RCM      1.4543 *     0.4540     0.006     0.1505    0.0068
         RWM      0.8640       0.8879     0.002     0.2218    0.0124
         OLS      0.8984       0.5300     0.002     0.2223    0.0133
DC       MRM      0.867        0.7640     0.001     0.1371    0.0061
         ARMA     0.8281       0.7209     0.001     0.2117    0.0123
         RCM      2.2208 **    0.7183     0.004     0.1051    0.0032
         RWM      0.6994       0.6353     0.009     0.2635    0.0179
         OLS      0.6392       0.7453     0.001     0.2229    O.012
DI       MMM      1.1323       0.8585     3.157     0.0709    0.0013
         ARMA     1.3428       0.6892     0.117     0.1379    0.0054
         RCM      2.2129 **    0.6281     0.004     0.095     0.0028
         RWM      0.8189       0.6333     0.009     0.2457    0.0152
         OLS      1.0039       0.716      0.001     0.2534    0.017
DR       MMM      2.367 **     0.8585     0.053     0.0466    0.0006
         ARMA     0.6073       0.7015     0.002     0.2745    0.0205
         RCM      3.2636 **    0.5660     0.005     0.1599    0.0059
         RWM      0.5523       0.6372     0.003     0.2963    0.0211
         OLS      2.408 **     0.725      0.002     0.1670    0.0076
ENE      MMM      0.998        0.9148     0.036     0.0359    0.0004
         ARMA     0.5895       0.6885     0.003     0.3460    0.0325
         RCM      0.9204       0.5540     0.005     0.2540    0.0208
         RWM      0.2082       0.5887     0.001     0.3843    0.0384
         OLS      0.525        0.5898     0.003     0.3316    0.0349
ENG      MRM      1.754 **     0.6766     0.002     0.1421    0.0061
         ARMA     1.0894 **    0.6594     0.002     0.2459    0.0181
         RCM      2.6913 **    0.5895     0.004     0.1149    0.0039
         RWM      0.772        0.5655     0.004     0.2355    0.0144
         OLS      1.392        0.5927     0.002     0.1901    0.0113
F&H      ARMA     1.847 **     0.5859     0.002     0.1507    0.0065
         MMM      2.6125 **    0.5566     0.289     0.1403    0.0059
         RCM      3.5782 **    0.4767     0.010     0.0009    0.0029
         RWM      1.22827      0.538      0.002     0.2133    0.0114
         OLS      2.823 **     0.4996     0.002     0.2107    0.0115
GOL      RWM      0.456        0.8282     0.009     0.3041    0.0226
         MRM      0.55834      0.4520     0.010     0.4193    0.0692
         ARMA     0.5796       0.3954     0.009     0.6232    0.1067
         RCM      0.6055       0.2999     0.019     0.4649    0.0615
         OLS      0.497        0.4426     0.009     0.4590    0.0653
INS      RWM      0.637        0.658      0.002     0.2086    0.0102
         MMM      0.86867      0.6065     3.021     0.2219    0.012
         ARMA     0.941        0.5307     2.978     0.1534    0.0085
         RCM      1.3666       0.3719     0.012     0.1466    0.0071
         OLS      0.583        0.4471     0.003     0.3536    0.0344
I&F      RCM      0.809        0.6795     0.005     0.1084    0.0039
         MRM      0.809        0.6796     0.010     0.1184    0.0039
         ARMA     0.4599       0.6859     0.001     0.1812    0.0076
         RWM      0.83463      0.4821     0.006     0.2054    0.0097
         OLS      0.316        0.6232     0.001     0.2588    0.0171
MED      RWM      0.836        0.7386     0.004     0.2089    0.0098
         MMM      2.90487 **   0.5565     1.189     0.1387    0.0053
         ARMA     5.8904 **    0.4005     0.005     0.1967    0.0126
         RCM      3.1508 **    0.4121     0.006     0.1329    0.0065
         OLS      1.692 **     0.4032     0.005     0.3566    0.0334
P&P      ARMA     2.011 **     0.6434     0.001     0.1482    0.0055
         MMM      1.8536 **    0.6204     0.026     0.1674    0.0064
         RCM      2.9364 **    0.5266     0.008     0.1192    0.0037
         RWM      0.9616       0.5048     0.0001    0.2254    0.0126
         OLS      2.035 **     0.561      0.002     0.1758    0.0073
PT       ARMA     1.014        0.7003     0.004     0.1544    0.0061
         MMM      0.8693       0.4832    23.446     0.1437    0.0533
         RCM      1.1471       0.5404     0.013     0.1101    0.0037
         RWM      0.8304       0.4795     0.003     0.1351    0.005
         OLS      0.840        0.4609     0.0001    0.1479    0.0056
RET      MMM      0.855        0.6519    36.710     0.1624    0.0074
         ARMA     0.8328       0.6014     0.081     0.1736    0.0082
         RCM      1.1743       0.5550     0.006     0.1099    0.0037
         RWM      0.7287       0.5916     0.005     0.1766    0.0085
         OLS      0.782        0.5557     0.002     0.2861    0.0207
TRA      MRM      0.773        0.6741     0.002     0.1470    0.0072
         ARMA     0.8098       0.6727     0.002     0.2545    0.0017
         RCM      0.8815       0.5947     0.005     0.1236    0.0045
         RWM      0.4918       0.6101     0.064     0.3008    0.0295
         OLS      0.737        0.6743     0.002     0.2269    0.0132
OM       ARMA     1.372        0.7468     0.121     0.2056    0.0119
         MRM      1.3624       0.6964     0.003     0.1944    0.012
         RCM      1.8280 **    0.579      0.005     0.1955    0.0091
         RWM      0.4025       0.6536     0.153     0.4118    0.044
         OLS      1.236        0.694      0.003     0.2860    0.0192

Note: MRM stands for mean reverting model, ARMA stands for ARMR(1,1)
model RCM stands for random coefficient model, RWM stands for random
walk model and OLS stands for ordinary least square model.
[[sigma].sup.2], [[sigma].sup.2.sub.a], [[sigma].sup.2.sub.[delta]]
[[sigma].sup.2.sub.b], [[sigma].sup.2.sub.n], [[sigma].sup.2.sub.u]
[[sigma].sup.2.sub.y] [beta] [[PHI].sub.11]([phi])
[[PHI].sub.22] [theta] are estimated hyper-parameters for each model
respectively. Q(12) is Box-Ljung statistics for serial correlation,
C-P test is the cumulated periodogram test for residual series to be
white noise, G-Q tests is the Goldfeld-Quandt test for
heteroscedasticity follows the classic ARCH test. Four criteria here
are R-square, Akaike Information Criterion (AIC), Mean Absolute Error
(MAE) and Mean Square Error (MSE) of the estimates. ** means 5% level
significance, * means 10% level significance , The best performed
models are in bold.

Note: The best performed models indicated with #.

(1.) For the simplicity of producing tables in this chapter, [v.sub.t] is used to denote the measurement equation noise with [[sigma].sup.2] as its variance. [n.sub.t] is used to denote the error process of the transition equation, with [[sigma].sup.2.sub.n] as it is variance for the three models: the random coefficient, ARMR(1,1) and mean-reverting models.

(2.) The formula to convert the three-month interest rate to monthly is taken from Knox, Zima and Brown (1996); that is. [r.sub.m] - [(1 + [r.sub.q]).sup.1/3]-1 where [r.sub.m] is the monthly rate and [r.sub.q] is the quarterly rate.

(3.) If the coefficients of the regression are time varying but are estimated as constant, the resulted residual series will be heteroscedastic.

(4.) The mean-reverting process discussed here includes the moving mean model tested in the paper as the moving mean model is indeed a mean reverting process but the mean of beta is time-varying too.

(5.) [R.sub.t+1|t] = [X.sub.t][B.sub.t+1|t], where the predicted return at time t for t+l is a product of observed market return X, and the updated state [B.sub.t+1] at time t. We thank the referee for pointing out this.

(Date of receipt of final transcript: November, 2003. Accepted by Doug Foster & Garry Twite, Area Editors.)

References

Ball, R. & Brown, P. 1980, 'Risk and return from equity investments in the Australian mining industry: January 1958-February 1979', Australian Journal of Management, vol. 5, pp. 45-66.

Black, A., Fraser, P. & Power, D. 1992, 'U.K. unit trust performance 1980-1989: A passive time-varying approach', Journal of Banking and Finance, vol. 16, pp. 1015-33.

Blume, M.E. 1971, 'On the assessment of risk', Journal of Finance, vol. 26, no. 4, pp. 275-88.

Blume, M.E. 1975, 'Betas and the regression tendencies', Journal of Finance, vol. 30, no. 3, pp. 785-95.

Bos, T. & Ferson, T.A. 1992, 'Market model nonstationarity in the Korean