1.0-1

Figure 13.4. Autocorrelation function of levels.

Figure 13.5. Autocorrelation function of first differences.

ogram does not damp, thus indicating nonstationarity. The peaks at 5, 10, 15, . . . indicate a strong 5-day weekly effect. Figure 13.5 shows the correlogram of first differences. It still shows peaks at 5, 10, 15, and so on, and it does not show any sign of damping. Since the peaks do not damp, it suggests a fifth-order MA component as well. We next try fifth differences, that is X, - X,y The correlogram, shown in Figure 13.6, damps. But the initial decline and oscillation suggests the use of an ARMA model rather than a pure AR or MA model. Leuthold et al. finally arrive at the model

(I - £)(1 - ct,F - azFK = (1 - eF)(l 4- p,F 4- fiL,

1.0-\

14 16

Lag-

Figure 13.6. Autocorrelation function of fifth differences.

There are three parameters in the MA part. Hence they use the grid-search procedure on the three parameters 6, and p2- The estimated parameters were

«2 = -0.47 p2 = 0.66

0.70L)(] - 1.52L -1- 0.66L2)e,

= 0.90 a, = 1.44 e = 0.70 , = -1.52 The model therefore is

(1 - 0.90L)(1 - 1.44L + 0.47L)x, = (1 -This gives

X, = l.44x, i - 0.47x, 2 + 0.90 :, 5 -(0.90 x 1.44) :,

+ (0.90 X 0.47K„7 + e, - 1.52e,„, -1- 0.66e, 2 - 0.70e, 5 + (0.70 X 1.52K 6 - (0.70 X 0.66k, 7

This is the equation we use for forecasting purposes, using the methods outlined earlier. Note that , and 2 satisfy the conditions in (13.1) and a, and a2 satisfy the conditions in (13.4). There were no diagnostic tests and comparison with alternative models. But the example is presented here as an illustration of how first differences were not appropriate but fifth differences were.

Trend Elimination: The Traditional Method

The Box-Jenkins method eliminates trend by differencing the series. A more traditional method adopts the ratio to moving-average metfiod. An example of

0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -

3 0.2 H

0.1 0

-0.1 --0.2 -

-0.3 I I I I I I I I I I I I I I I I M I I I I I I I I M I.......I M I I I I M I I I I I I I I I I I I I I I

10 20 30 40 50 60

Lag ik)

Figure 13.7. Correlogram for the Beveridge trend-free price index.

this is the Beveridge price index." This is a historically interesting time series of wheat prices in western and centra! Europe from 1500 to 1869 (370 years). The index is made up from prices in nearly 50 places in various countries (with the mean for 1700-1745 = 100). Beveridge gets a trend-free index by expressing the index for a given year as a percentage of the mean of the 31 years for which it is the center. For instance, for the year 1600, we divide the index for 1600 by the average of the index over the years 1585-1615 and convert this to a percentage. This is the ratio to moving-average method. Note that you lose 15 observations at the beginning and 15 observations at the end. The last ones are often critical.

Table 13.2 at the end of the chapter gives the Beveridge index and Figure 13.7 gives the correlogram of the trend-free index, which shows that the series is stationary. Calculation of the correlogram of the raw index and the first difference of the series is left as an exercise.

A Summary Assessment

The Box-Jenkins models were very popular in the early 1970s because of their better forecasting performance compared to econometric models. The main drawback of econometric models at that time was that inadequate attention

"W. H. Beveridge, "Weather and Harvest Cycles," Economic Journal, Vol. 31, 1921, pp. 429-452.