Fig. 1. Time series of prices for the four cases in Table 2.

Fig. 2. Return series and their density distributions for Case-01, 10 and 11.

Fig. 3. ACs of returns (the left column), the absolute returns (the middle column), and the squared returns (the right column) for Case-01, 10 and 11.

Fig. 4. The price P_t and the fundamental price (top left); the geometric moving average u_t (top right); the differences (middle left) and P_t-u_t (middle right) and the corresponding distributions (bottom).

Fig. 5. The geometric volatility (v_t, top) and the demands of the trend followers (z_2t, middle) and the fundamentalists (z_1t, bottom).

Fig. 6. The phase plots of the geometric moving variance (v_t) and the demands of the trend followers (z_2t, left) and of the fundamentalists (z_1t, right).

Fig. 7. Autocorrelations of r_t (bottom), (middle) and |r_t| (top) for the MF model.

Fig. 8. The ACs of the returns, the squared returns and the absolute returns for the MF model and the DAX 30 (a), the FTSE 100 (b), the NIKKEI 225 (c), and the S&P 500 (d). The smooth lines refer to the MF model while the confidence intervals are those for the actual data.

Fig. A1. Autocorrelations of returns, the squared returns and the absolute returns for the DAX 30 (a), the FTSE 100 (b), the NIKKEI 225 (c), and the S&P 500 (d). The lines from the bottom to the top are the autocorrelations for returns, the squared returns, and the absolute returns, respectively.

Parameter settings and initial values

Four cases of the noisy effect

Summary statistics of r_t

The estimates of d for the DAX 30 with m=50,100,150,200,250

The estimates of d for the FTSE 100 with m=50,100,150,200,250

The estimates of d for the NIKKEI 225 with m=50,100,150,200,250

The estimates of d for the S&P 500 with m=50,100,150,200,250

Engle's test statistics for the presence of ARCH/GARCH effects

GARCH (1,1) parameter estimates

Note: The numbers in parentheses are standard errors. This also holds for Table A8.

FIGARCH (1,d,1) parameter estimates

The estimates of d for the MF model with m=50,100,150,200,250

The GARCH (1,1) parameter estimates for the MF model

Note: The numbers in parentheses are the standard errors, and the numbers in the last row are the percentages that the test statistics are significant at 5% level over 1000 independent simulations. This also holds for Table A11.

The FIGARCH (1,d,1) parameter estimates for the MF model

The Wald test of d with m=50,100,150,200,250