Covariates Hiding in the Tails
A popular measure to assess the likelihood of extreme and rare events—such as economic crises—is the tail index. Because of the small sample of extreme events, measuring the tail index precisely is difficult. Measuring changes to this likelihood over time is even more difficult.
Under certain assumptions, using the variation across the population—a cross-section—at a given time offers a way to alleviate the problem posed by a small sample. For instance, to measure the tail index for a stock market, you could use the returns of the individual firms to imply the likelihood of a tail event for the market at a given time. Repeating this exercise for each time period produces a measure of the tail index over time.
We show that these measurements contain a bias that fluctuates over time. Our theoretical results reveal that the bias:
- results from trivial fluctuations that are common among the cross-section observations
- moves the tail index estimates in the opposite direction for events that are extremely negative or extremely positive
- is increasing in the number of extreme observations used in the measurement because the added observations are less extreme
We provide two simple remedies. First, we devise a new estimator that averages the estimate for the extreme negative and positive events. Averaging the two estimates should cancel the bias present in both tails. Second, we subtract the cross-sectional average from each observation and repeat this at each point in time.
To test for the presence, direction and size of the bias, we use monthly US stock returns and annual US Census county population data. We find that the bias plays an important role for stock returns, especially during crisis periods. However, the bias plays a small role for county population data due to the lack of common fluctuations in the population across counties.