Espesamiento de concentrados

3.1. Introduction

This chapter intends to apply time-series non-linear framework to daily returns of four leading stock indices, namely FTSE 100, S&P, DAX and Nikkei, with the purpose of recursive out-of-sample forecasting. In view of the fact that the main purpose of econometric modelling appears to be application of these models to forecasting, this paper is concentrating on fulfilling this objective and extending research into non-linear model forecasting.

Given that a number of studies highlight the importance of forecasts in general (Brooks, 2002), in planning and operations of companies (Holden et al., 1990), political science (Granato and Suzuki, 1996) and for economic policy-makers (Montgomery et al., 1998), there is no doubt that forecasts are required in a wide range of disciplines. The main focus of this chapter is daily stock return forecasts which are also required by a broad spectrum of market practitioners. Moreover, the degree of sensitivity of non-linearity to the frequency of data is still not entirely clear. For instance, according to Abhyankar et al. (1995) who suggest the use of high-frequency data, microstructural dynamics in the financial time-series are more apparent at higher frequencies. In addition, high- frequency data provides a large sample size for empirical investigation.

Daily stock returns predictability has been much debated over the years. The reason for this is inconsistency of returns stock predictability with efficient market hypothesis (EMH), which states that the stock price incorporates all publicly available information. Cuthbertson and Nitzsche (2004) describe efficient markets being driven by simple supply and demand mechanism of a competitive market where rational traders react and consequently adjust the stock prices according to the available information relevant to the determination of fundamental asset prices. According to the theory, due to any relevant information being costless and publicly available while new information, such as news, being unpredictable by definition, there is no opportunity to accumulate excess profit in a perfectly efficient market. Thus, Abhyankar et al. (1995; 1997) point out that returns stock predictability is inconsistent with the theory of efficient markets, however, find evidence of predictability and non-linear dependence in high-frequency FTSE returns and daily returns of S&P 500, DAX, Nikkei 225 and FTSE 100. Attempts to explain the stock market predictability suggested market inefficiency or time-varying expected returns (Brock et al., 1992; Pesaran and Timmermann, 1995). Furthermore, the presence of cyclical behaviour and asymmetric adjustments in economic and financial series implied the presence of non-linear predictability (Tong 1990; De Gooijer et al., 1992; Abhyankar et al., 1997; McMillan, 2001; Sarantis, 2001; McMillan, 2002; Bali et al., 2008; Hartmann et al., 2008; Guidolin et al., 2008). The presence of these non-linearities could be attributed to the presence of market frictions, including transaction costs, borrowing and short selling constraints, limit to arbitrage (He and Modest, 1995; Kilian and Taylor, 2003; McMillan, 2005), as well as the presence of speculative bubbles (Evans, 1991; Froot and Obstfeld, 1991; Bohl, 2003; Psaradakis et al., 2004) and interaction between noise traders and informed arbitrageurs

(Kirman, 1991, 1993; Shleifer, 2000; McMillan, 2002, 2005; McMillan and Speight, 2006).

Section 3.2 of this chapter contains a brief reminder of the methodology discussed in Section 2.3 of Chapter 2, with further discussion of the STAR-type model estimation procedure in more technical detail. Empirical results in Section 3.3 contain plots and diagrams with descriptive statistics for each time-series considered in this chapter, as well as the results of the non-linearity tests. The estimated models then are tested for goodness of fit and the results are presented in the view of the forecasting exercise. Linear and non-linear forecasts are compared in terms of forecasting performance using a number of tests of forecasting accuracy, including the tests of forecasting error magnitude, the Diebold and Mariano test of equal forecasting accuracy, forecast encompassing and trade rule tests. Moreover, the same tests are then applied to the combinations of linear and non-linear forecasts. Section 3.4 summarises the results and concludes.

3.2. Methodology

A simple random walk model and linear ARIMA models will be estimated as benchmarks for the STAR-type models. Forecasting abilities of all linear and non-linear models will then be compared using forecasting accuracy tests. A random walk model with a drift () is applied in this chapter:

= + + (3.1)

where is the price returns level at time t, is the price returns level at time t-1 and t

ε

The autoregressive integrated moving average process, ARIMA (p, d, q), is a combination of an autoregressive process of order p, AR (p), and a moving average of order q, MA (q), where d is the order of integration, or in other words, the number of times the series has to be differenced in order to achieve stationarity. For stationary series d equals zero, thus ARIMA (p, d, q) becomes ARMA (p, q). The general form for the ARIMA (p, d,q) process is as follows:

 = + + + … + + − − − … − +

(3.2)

where are the coefficients of the AR process component and are coefficients of the MA process component, and is an error term. ARIMA models are estimated using the Box and Jenkins approach introduced by Box and Jenkins (1976) and involves three stages of model building: identification, estimation and diagnostic checking. The first stage of model identification involves determining the order of the model, i.e. the values of p and q. The value of integrating order, d, is determined following the results of the stationarity test. After parameters estimation, the adequacy of the estimates is

tested with diagnostic checking of the model using an information criteria approach. Akaike’s information criteria (AIC) and Schwarz’s Bayesian information criteria (SBIC) are the most commonly used procedures in ARIMA modelling. In addition, residual tests such as tests for remaining autocorrelation and ARCH-LM test are performed as model misspecification tests.

Further to linear alternatives this paper will estimate smooth transition-type models for price returns series for the data considered. The formulae for a standard smooth transition (STR) model is as follows:

= ′K+ ′Kš F, G, ,# + , ~00> 0, # (3.3) š F, G, ,# = A1 + E) œ−F  ,− Gž# ž ž9 ŸB _{, ~00> 0, # (3.4)}

where is a parameter of the linear part of the equation and ′ _{is a parameter of the} non-linear part. š F, G, ,# is the transition function which depends on the transition variable, ,, the slope parameter, F, and the vector of location parameters, G. The transition variable, s_t, can be either part of z_t, which in the case of SETAR (self- exciting threshold autoregressive) will be the dependent variable itself, y_t, or the transition variable can be represented by another variable, such a trend, for instance. The term   can be set either to unity (  = 1) to attain an LSTAR (logistic STAR)

model, or it can be set to be equal to two (  = 2) for an ESTAR (exponential STAR) model.

There are three stages in smooth transition modelling, which include specification, estimation and evaluation. The initial stage of specification involves testing the time- series for the presence of STAR-type non-linearity and choosing the transition variable. The results will suggest whether LSTAR, ESTAR or a linear model should best fit the data. Furthermore, the estimation phase involves finding the starting values for non- linear estimation through a grid search and estimating the model based on those starting values. Results are then evaluated using a number of tests, such as misspecification tests, autocorrelation of the disturbance term, test for remaining non-linearity, ARCH test and test of non-normality. There are also graphical tests that might give an indication of whether the model was estimated correctly.

Once the significant non-linearity is reported and either ESTAR or LSTAR models are chosen, a non-linear optimisation routine known as a grid search is applied in order to estimate the starting values of STAR model parameters. The grid search requires the transition variable, ,, to be known, which is accomplished in the first stage of the specification. The procedure involves creating a linear grid within a vector of location parameters, G, and a long-linear grid in the slope parameter, F, and calculating the residual sum of squares for each of those values. The values that offer the minimum residual sum of squares are chosen as starting values for model estimation.

After the starting values have been established the Newton-Raphson algorithm is applied to maximise the likelihood function which estimates the remaining parameters of the model. Further misspecification tests are carried out on the estimated model

including a test for remaining residual autocorrelation, a test of parameter constancy, the ARCH-LM test and the Jarque-Bera normality test. In addition, the model can be tested for any remaining additive STAR-type non-linearity. The parameter constancy test, in its turn, tests whether parameters are constant or continuously change. In addition, graphical analysis may serve as a good indication tool. Thus, the tests that allow to determine validity and the goodness of fit of the estimated models used in this chapter include a test of no error autocorrelation, a test of no remaining non-linearity, and the ARCH-LM test.

The test of no error autocorrelation used in this study is based on the test commonly known as the Breusch-Godfrey test. In the case of STAR modelling this particular test is preferred over the more popular Durbin-Watson autocorrelation test. The reason for this is that the Durbin-Watson test is constructed in a way that tests relationship only between an error and its immediate previous value. In other words, it is only valid if autocorrelation is present in the first lag. The Breusch-Godfrey test, on the other hand, examines the relationship between an error and several lagged error values at the same time. Another reason for not choosing the Durbin-Watson test is that for the test to be valid there are certain conditions that have to be fulfilled, including a constant term in the regression and non-stochastic regressors. In addition, the regression must not contain lags of dependent variable. In other words, the regression should be static in nature, as opposed to dynamic. These conditions defy the very essence of the STAR- type modelling and thus a different approach is required.

However, Brooks (2002) points out that the Breusch-Godfrey test presents some difficulty in its conduct in terms of determining the appropriate value of the number of lags of residuals, +. As there is no particular rule or procedure for choosing the correct

value, it is usually down to a researcher to employ a trial-and-error approach. The frequency of data might give an initial idea about the number of lags. The test is a joint hypothesis test with a critical value following a Chi-squared distribution. The Breusch- Godfrey test for autocorrelation of rth order involves regressing residuals estimated using OLS:

= )+ )+ )aa+ ⋯ + )¡¡+ • , •~¢ 0, !£# (3.5)

where the error term follows normal distribution, •~¢ 0, !_£#. The test statistic following Chi-square distribution is: & − +#‚~T_¡, where & is the number of observations and ‚ is obtained from the above regression (3.5). The null hypothesis of no serial correlation to the order of + is tested against the alternative of autocorrelation.

]: ) = 0 `> ) = 0 `> … )¡ = 0 ]: ) ≠ 0 _+ )¡ ≠ 0 _+ … )¡ ≠ 0

] of no serial correlation is rejected if the test statistic is greater than the value of the critical value from the Chi-squared statistical tables.

Another test considered here is a test of no remaining non-linearity, which is based on the account that in the case of a correctly fitted model the residuals should contain no

remaining non-linear structure. The test naturally assumes that the remaining non- linearity is a STAR-type non-linearity.

= ′K+ ′Kš F, G, ,# + \′K] F, G, ,# + (3.6)

where ~00> 0, !# and ] is a transition function for that regression, i.e. different from the one used in the main model. The alternative hypothesis is defined as:

= M′K+ ′Kš F, G, ,# + 4 M5′K̃,5 + ∗ a

59

(3.7)

The following auxiliary model is used to test the above model, where " is regressed on =K̂′,

, K̂′, , K̂′,a ?′ and the partial derivatives of the log-likelihood function with respect to the parameters of the alternative model. The null hypothesis for this test of no remaining non-linearity is that M = M = M_a = 0. The test statistic follows F- distribution and is treated in the same fashion as a standard non-linearity test.

The ARCH-LM test is used to test for presence of ARCH in the residuals (Engle, 1982). The residuals, ", of a regression in question are squared and regressed on their own lags. The number of lags signifies the order of ARCH the test is run for. Hence, the regression for the ARCH test of order * will be as follows:

" = F+ F" + F" + ⋯ + F" + • (3.8)

where * is number of lags and ¦ is an error term. The value of ‚ obtained from this regression forms the test statistic, &‚, where & is the number of observations. &‚ is compared to a critical value obtained from the Chi-squared distribution table T *# to test the following hypotheses:

]: F = 0 `> F− 0 `> Fa = 0 `> … `> F = 0 ]: F ≠ 0 _+ F ≠ 0 _+ Fa≠ 0 _+ … _+ F ≠ 0

If test statistic is greater than the critical value, the null hypothesis of no ARCH is rejected.

3.3. Empirical results

This study will analyse daily time-series data over a twenty year period from 1st January 1988 to 31st December 2007, which consists of 5217 observations. The data consists of four price indices of major world economies. These include FTSE 100 for UK; S&P

500 Composite for US; DAX 30 Performance for Germany; and Nikkei 225 Stock Average for Japan.5