of interest. A causal relation linking the variables is indicated if the latter forecasting rule
is outperformed by some of the former.
Each forecasting scheme is applied recursively by increasing the actual sample size from
t* = 88 to t* = T — 1 such that 120 forecasts are computed for most data sets. Let ykt*+1
denote a one step ahead forecast for ykt*+1 conditional on knowledge of explanatory variables
in time t* + 1 and some model estimate obtained from the first t* observations. Then, one
step ahead forecast errors are defined as
ukt*+1 = ykt* + ι - yJkt*+1-
To assess the accuracy of a particular model in forecasting different criteria could be consid-
ered.
In the first place, forecasting schemes could be evaluated according to randomness of the
respective one step ahead forecast errors. A sensible forecasting model should deliver serially
uncorrelated sequences Ukt*+1 which are easily diagnosed by means of a suitable LM-test.
Secondly, to measure forecasting accuracy ykt*+1 and ykt*+1 could be regarded as dichoto-
mous random variables. Along these lines a forecasting model is accurate if the distributional
properties of the forecasts ykt*+1 come close to the corresponding features of the actual quan-
tities ykt*+1. Intuitively appealing to formalize the latter idea, contingency tables are often
used in applied statistics. As a formal criterion summarizing the information content of a
contingency table we consider the so-called Henrikkson Merton statistic.38 Initially proposed
to evaluate investment performance this statistic (hm) aggregates the conditional probabili-
ties of forecasting a positive or negative value of the dependent variable, whenever the actual
realization in t* + 1 is positive or negative,
hm = Prob(ykt*+ι ≥ 0 ∧ ykt*+ι ≥ 0∣ykt*+ι ≥ 0)
+Prob(ykt*+ι < 0 ∧ ykt*+ι < 0∣ykt*+ι < 0). (17)
A successful forecasting scheme should deliver hm-statistics larger than unity. Critical values
for this test statistic depend on the number of available forecasts and can be obtained from
simulation.
In the third place one may rank competing models according to the mean absolute forecast
error (MAFE). For the empirical analysis of trade growth it turned out that for most data sets
this measure is largely affected by only a few outlying observations ykt*+1 such that the MAFE
is hardly informative for both, causality linking volatility and trade and model comparison.
Since four forecasts are available for each observation a scale invariant measure of relative
performance is the average rank of absolute forecast errors. Although rank statistics will not
be informative for the risk of (singular) large forecast errors average ranks are informative
for causality analysis as well as model comparison. In case of no causality ”zero” forecasts
should show an average rank at least smaller than 2.5 since imposing a valid restriction
is supposed to improve forecasting precision. Similarly, if there is some linear relation the
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