CHARQUI

Interval forecasts provide a prediction of a range of values in which the future value of the variable is expected to lie. This study will apply a technique based on a study by Christoffersen (1998) in order to carry out interval forecasts on the linear and non-linear models considered in this chapter. The methodology involves setting interval prediction barriers in the form of upper and lower limits each with assigned certain probability, with further evaluation of goodness of fit of the forecast using a success ratio approach.

The upper and lower limits are set as a time-series of forecasted values plus or minus respectively the standard error term at the 95% level of confidence assuming normal distribution (Figures 3.8 – 3.11). The goodness of fit test will determine the success rate of the forecast value falling inside the set limits.

6

The main objective of this thesis is an investigation of point forecasting with non-linear models and does not include a thorough examination into interval forecasts. The subject of interval forecasts is an important area of time-series research that lacks extensive empirical examination in the literature. I would like to thank my examiners for their valuable comments and recommendations for further research within the field of forecasting.

Figure 3.8. FTSE interval random walk.

Figure 3.9. FTSE interval linear forecast.

-.06 -.04 -.02 .00 .02 .04 .06 2001 2002 2003 2004 2005 2006 2007 FTSE returns lower limit upper limit

-.06 -.04 -.02 .00 .02 .04 .06 2001 2002 2003 2004 2005 2006 2007 FTSE returns lower limit upper limit

Figure 3.10. FTSE interval LSTAR forecast.

Figure 3.11. FTSE interval LSTAR trend forecast.

-.06 -.04 -.02 .00 .02 .04 .06 .08 2001 2002 2003 2004 2005 2006 2007 FTSE returns lower limit upper limit

-.06 -.04 -.02 .00 .02 .04 .06 2001 2002 2003 2004 2005 2006 2007 FTSE returns lower limit upper limit

The success rate of the interval forecast can be easily seen on the graphical representation, where the actual returns will be either within or outside the set limit barriers, thus indicating success or failure of the forecast respectively. Naturally, the upper and lower limits for the random walk interval forecast are characterised by a somewhat less volatile line as opposed to limit barriers of the linear and non-linear interval forecasts which mimic the movements of the actual returns series. Moreover, all the interval forecasts share a characteristic of a common trend level. As expected, while the most of actual returns series values lie within the interval forecast, the outliers and extreme points rest outside the prediction barriers. The most successful forecast based on graphical representation in terms of following outliers is the LSTAR forecast (Figure 3.10), where the model attempts to correct for extreme value in the beginning of the sample characterised with high volatility.

The out-of-sample goodness of fit evaluation of interval forecast applied in this chapter is based on assessing the success ratio of the indicator variable, , for a given interval forecast, Z¶_| )#, „_| )#[ for time t, made at time t-1, with the coverage probability, p, for a time-series of a random variable, , which is defined as follows:

= ¸1,_0, 01 ₀₁ ∈ ‰¶| )#, „| )#Š ∉ ‰¶| )#, „| )#Š

/ (3.19)

Where, ¶_| )# and „_| )# are lower and upper limits respectively. In other words, zero value is assigned to every forecasted value outside the prediction barriers, while forecasts within the range are assigned a value of unity. The mean of the indicator

variable is the success ratio of the interval forecast. The results (Table 3.18) suggest that none of the interval forecasts performed in this section surpassed the limit required by the 95% confidence interval.

Table 3.18. Interval forecast success ratio results. Success Ratio FTSE Random walk 0.9243 ARMA (1, 3) 0.9183 LSTAR 0.9178 LSTAR-trend 0.9210 S&P Random walk 0.9287 ARMA (2, 1) 0.9276 ESTAR 0.9265 DAX Random walk 0.9216 ARMA (3, 2) 0.9205 ESTAR 0.9227 Nikkei Random walk 0.9468 ARMA (0, 3) 0.9484 ESTAR 0.9490 ESTAR-trend 0.9473

These results could be explained by the fact that the goodness of fit evaluation procedure was based on the assumption of normal t distribution. Generally the distribution of financial daily data is characterised with fat tails due to daily data being very noisy and containing extreme values. Similarly to the results of the point forecast

in this chapter it seems that the argument of the daily data lacking defined patterns still holds when applying interval forecasting techniques. Therefore, the suggestion that the less noisy monthly data will demonstrate more clearly defined forecasting performance of non-linear as well as linear models is also applicable to interval forecast.

3.4. Conclusion

This chapter intended to assess the forecasting abilities of non-linear STAR-type models using daily stock price data over the period of twenty years between 1988 and 2007 using four price indices of four major world economies, including FTSE 100, S&P 500 Composite, DAX 30 Performance and Nikkei 225 Stock Average.

Results of the empirical investigation suggest the presence of stock returns predictability and presence of STAR-type non-linearity. These results are consistent with extensive literature on the issue of forecastability of stock returns and successful use of STAR-type models in forecasting these dynamics (Abhyankar et al., 1995; Clements and Smith, 1999; Clements and Smith, 2001; McMillan, 2001; Lekkos and Milas, 2004; McMillan, 2004; Teräsvirta et al., 2005). Moreover, in parallel with notion of traders interaction in financial markets suggested by McMillan (2001) and the presence of market frictions including transaction costs, limits to arbitrage, short selling and borrowing constraints (Martens et al., 1998; Kapetanious et al., 2003; McMillan, 2005b), it can be argued that small changes in pricing equilibria can be foregone and not

corrected immediately, thus, displaying non-linear dynamics within the series. STAR models produce reasonably accurate results in comparison with linear alternatives, however, any additional gains achieved by non-linear framework are only marginal to the results of a random walk and ARMA models. Hence, drawing from the results of the forecasting accuracy tests and taking into account specific behaviour and characteristics of daily data, and combining aspects of forecasting accuracy and ease of implementation, it can be concluded that the random walk model seems to be the most superior model for the purpose of forecasting daily stock returns. It has to be noted, however, that there is no clear evidence of exceeding superiority of the random walk model compared to other linear and non-linear approaches. Nevertheless, it is assumed that for forecasting high-frequency data on a daily level it is vital that the model is fast and easy to apply in addition to clear interpretation of results, which the random walk models appears to provide. The conclusion of these results is similar to those of an empirical study of high-frequency stock returns by Abhyankar et al. (1995), where the researchers confirmed the presence of non-linearity, however found the time-series to be adequately explained by a simple alternative, namely the GARCH (generalised autoregressive conditional heteroskedasticity) model process. Thus, while Abhyankar et al. (1995) encourage the use of high-frequency data due to the fact that it allows for a larger sample and thus increases the likelihood of better understanding the underlying process, there is also a possibility that small changes in high-frequency time-series returns might be too noisy and would not fully reflect the long-run dynamics.

Moreover, Fair and Shiller (1990) pointed out the fact that a specific model displays either good or poor forecasting abilities for one sample period might not necessarily mean it will have the same results for a different forecasting period. One of the reasons

for this could be a change in economic structure or other events that will change the behavioural dynamics of the data. Furthermore, Montgomery et al. (1998) found that in their study of US unemployment rate with the aid of non-linear forecasting models, the quarterly series is much smoother comparing to a more frequent monthly series. Both series shared similar cyclical and trend characteristics, however it is evident that there is a strong possibility that the long-horizon data might utilise the benefits of the non-linear forecasting much more efficiently than data sets with much higher frequency. Hence, this chapter will be concluded on the notion that the results obtained here suggest the use of the random walk model as the best forecasting model for daily stock returns in terms of the ease of implementation and relative forecasting accuracy it provides. However, it is not to suggest that the non-linearity should be disregarded and that researchers should consider its presence. This study further anticipates that an investigation of non-linear forecasting models should be extended to long-horizon data, as a non-linear approach seems to be more appropriate in this case.

Chapter 4