This model fits the data the best9. Table 2.12 contains the estimation results of the autoregressive model in the period January 5, 1983 until February 4, 1985 and of the GARCH model in the period February 5, 1985 until December 31, 1987 with the t-ratios within parenthesis.
The returns in the first period show significantly positive 16-th order autocorrelation, while the returns in the second period show significantly negative second order autocorrelation. The constant is in the first period significantly positive at the 10% significance level, while in the second period it is significantly negative at the 1% significance level. This is an indication that the drift is first positive and then negative. With this final bootstrap procedure we can test whether the good results of the technical trading rules can be explained by the trend structure in the data series and the strong autocorrelation in returns.
Table 2.12: Coefficient estimates structural break in trend model
Estimates of an autoregressive model on the daily return series of the LIFFE cocoa futures prices in the period January 5, 1983 until February 4, 1985 and of a GARCH model in the period February 5, 1985 until December 31, 1987. The autoregressive model is estimated using OLS and White's (1980) heteroskedasticity-consistent standard errors. The GARCH model is estimated using maximum likelihood using the Marquardt iterative algorithm and Bollerslev-Wooldridge (1992) heteroskedasticity-consistent standard errors and covariance. The numbers within parenthesis are t-ratios.
The autoregressive model coefficients estimates 1/5/1983 - 2/4/1985 α φ16
0.001213 0.161887 (1.74) (3.67)
The GARCH-model coefficients estimates 2/5/1985 - 12/31/1987 α φ2 α0 α1 β1
-0.001511 -0.113115 3.85E-06 0.064247 0.905622 (-3.95) (-2.85) (1.48) (1.68) (18.6)
2.6.2 Bootstrap tests: empirical results
Random walk process
In table 2.13 we display the bootstrap results under the null of a random walk, an autoregressive model, an exponential GARCH model and the structural break in trend model63