Improving Business Cycle Forecasts’ Accuracy - What Can We Learn from Past Errors?

Improving Business Cycle Forecasts’ Accuracy

There is a further reason why (3) may lead to wrong results when it is applied
in a way widely used. In the estimate the entire sample of forecast errors and
indicators is used. But this is not the typical situation a forecaster is in: In
period t he only knows the t-1, t - 2 .. observations. In the regression, also the
t+1, t+2... data are used. Therefore, the test for informational efficiency as de-
scribed in (3) may be misleading. To avoid this problem here, the x will be
“normalised” with respect to past data. Campbell/Ghysels (1995) propose in
this context to subtract the median of x in the most recent k years from x. The
transformation is similar to (6)

(7)                       ^xc,k = ^xt^-median(^χ_t—k^,χt—k +₁,.∙∙,^χt).

If x follows a clear upward (downward) trend, this transformation could result
in x^c being positive (negative) in most cases. Therefore the x must be trans-
formed in a way making the indicators stationary. Subsequently in the case of
non-stationary indicators, their growth rates will be used to conduct the test.

Having done this, the orthogonality test for independence of e and x can be
based on a rather simple statistic

(8)                                      zk = et ∙ ^xCk.

Defining I+ in the same way asin (5), a first test statistic can be calculated as:

(9)                            SO=∑I+ (zt^k).

t

It is cumulative binominal distributed with probability 0.5. The critical values
are the same as cited before.

The sign test requires rather weak assumptions only, but it is on the other hand
also a rather weak test. If errors are distributed symmetrically, a ranked sign
test can be applied which was proposed by Campbell/Ghysels (1995:23-24)¹.
In this test, the absolute forecast errors are ranked, and the ranks are used as
weights for the signs in (9)

(10)                      WO = ∑ I ₊ ( zk )∙ rank (| e_t |).

t

WO is Wilcoxon rank signed distributed. For small samples, the critical values
can be taken from special tables (e.g. Siegel 1956: 254). In the present case of
14 observations the hypothesis that x_t,k and e_t are independent must be re-
jected on a 10%-level when the sum of positive ranks is above 80 or below 25
(5%: 84 and 21; 1%: 92 and 13) in a two tailed test. For large samples, the

¹ For a deduction of this test see Campbell/Dufour 1995.

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