4. Heavy tailed residuals

The ends of the normal scores plot have greater slopes than the reference line because the observations in the tails are spreading out more than the normal theory predicts.

One reason for heavy tails. The residuals come from TWO groups with different variances. Always leads to heavy tails. Interpretation: two different volatility regimes, low and high.

The volatility regime model, in which trades take place at different rates over the course of a day, relies on knowing the key facts that

The sum of normal random variables is again normal

The variance of a sum is the sum of the variances when the observations are independent

The return over a fixed period (say a day) is approximately equal to the sum of returns over contained shorter periods (say hours)

Example:

 

Table 1: Comparing hourly returns and the daily return.

HOUR	0	1	2	3	4	5	6	7
Value	100.0000	99.4819	99.6610	100.6495	101.5622	102.7503	100.9752	101.2928
Hourly return	***	-0.0052	0.0018	0.0099	0.0091	0.0117	-0.0173	0.0031

Daily return = 0.0129.
Sum of hourly returns = 0.0131

The sum of the hourly returns approximates the daily return.

If the hourly returns are approximately normal, then so is their sum, but this is just the daily return. The daily return is a sum, and the variance of a sum is the sum of the variances (provided the returns are independent), so if you buy into the model then daily returns have more variance than the hourly returns.