0.15

-0.05

-0.1

0.05

0.1

-0.15

Figure 11.6 Correlogram of WTI crude oil futures returns.

Figure 11.6 shows the empirical correlogram of a typical return process, the WTI crude oil near futures returns that were analysed in §6.2.3. The largest autocorrelations occur at lag 1 and lag 2, and there are signs of negative second-order correlation. But it is not clear that they are significantly different from zero because the values here are little more than 0.1. Indeed, some of the autocorrelations at higher-order lags are almost as great as they are at lags 1 and 2. If an AR(2) model were appropriate, the estimated model would be:10

However, it is unlikely that this model will have much predictive ability, even though the next section shows that the second-order autocorrelation is statistically significant. The forecastability of crude oil prices using ARMA models will be returned to later in this and the following chapter.

At the opposite extreme, the correlogram of the gas daily futures prices, which were found to be (weakly) stationary in §11.1.5, is shown in Figure 11.7. This has the characteristic shape of an AR(1) correlogram, and in §11.3.3 it will be shown that the AR(1) specification is more appropriate than others that are considered.

11.3.2 Autocorrelation Tests

Even in the highly efficient FX markets there is autocorrelation in returns, but not normally at the daily frequency. Like skewness and kurtosis, autocorrelation tends to increase with sampling frequency, and there is much evidence that FX markets exhibit autocorrelated returns at the intra-day frequencies (§13.1.3).

Like skewness and kurtosis, autocorrelation tends to increase with sampling frequency

= 0.1334! - 0.1373 , 2 + 8,.

(8.47) (-3.58)

luThe term uj5 insignificant and therefore omitted.

0.9 0.8 0.7 0.6 0.5 j 0.4 0.3 0.2 0.1 0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Figure 11.7 Correlogram of log futures prices for natural gas.

Table 11.2: Box-Pierce tests for autocorrelation in daily returns to WTI crude oil

futures

There are a number of tests for the significance of autocorrelation, one of the most common being the Box-Pierce test. This test for autocorrelation up to order p is based on the statistic

e = r£cp(«)2, (11.27)

where T is the sample size and cp(«) is the wth-order sample autocorrelation

y<y<-"

, n t=n+\

9(«) =-j-

The Box-Pierce statistic is a form of Lagrange multiplier test so it is asymptotically distributed as chi-squared with p degrees of freedom (§A.2.5).

Applying Box-Pierce tests to the daily returns to the WTI crude oil near futures data shows that autocorrelations are highly significant even though they are small. The values of Q for each lag along with the 1% chi-squared

Table 11.3: Box-Pierce tests for autocorrelation in US equity returns

critical value are shown in Table 11.2. As expected from the correlogram in Figure 11.6, the autocorrelations at lags 1 and 2 are significant at 1%, but beyond that there is little increase in the value of the -statistic, indicating that the series is dominated by first- and second-order autocorrelation.

Since the second-order autocorrelation is very highly significant the series could have an AR(2) representation, a suggestion that is reinforced by the shape of the correlogram, although the oscillating behaviour of the correlogram could also be in line with an ARMA( 1,1) representation.

Equity returns can exhibit significant autocorrelation even at a daily frequency, although this is the exception rather than the rule. Table 11.3 shows the Box-Pierce g-statistics for lags 1 and 2 for the twenty US stocks that were discussed in Chapters 4 and 6. Only American International, Ford and perhaps Exxon show signs of autocorrelation. Readers may wish to replicate these results using the PcGive package on the CD.

11.3.3 Testing Down

A standard approach to ARMA model identification is to estimate the model (11.23) with a variety of lags of both autoregressive and moving average terms and then to use standard hypothesis tests on the coefficients. At the first stage