Most asymptotic distribution theory used in econometric research relies on moment conditions which carefully control outlier occurrences. It is not unusual in time series analysis to see conditions of the type let all required moments exist. However, in financial and commodity market time series the extent of outlier activity casts doubt on the suitability of such generic moment conditions. Mandelbrot (1963) provided suggestive evidence that even second moments may not exist for this type of data, and he proposed stable distributions with infinite variance as an alternative to finite-variance statistical models. Subsequent research has generally reached the conclusion that second moments of most datasets appear to be finite.
In many empirical works in finance, the values of kurtosis and skewness were statistically tested even though the existence of higher-order, especially fourth or sixth, moments has been studied less extensively.
This paper investigates a statistical testing method for the existence of the k-th moment for dependent, heterogeneous data using the tail index of the distribution function.
Tail index estimation depends for its accuracy on a precise choice of the sample fraction, i.e., the number of extreme order statistics on which the estimation is based.
Our test procedure has two steps. On the first step, we estimate optimal sample fraction that minimizes the mean squared error of Hill's estimator. Then, we test the hypothesis that the k-th moment is exist based on the Hill's estimator.
Results of Monte Carlo simulations show that optimal sample fractions are chosen in average (except for heavily dependent data), size of the test is a slightly higher than a nominal rate and the test has good power for light or moderately dependent data, but the power decreases in heavily dependent case.