A Shock Is Still Half Remembered Thirty-Four Days Later

Two weights that add to 0.98 give the variance forecast a half-life of 34.31 days; delete the second one and the half-life is 0.2744 days, gone before the next open. The same recursion turns strictly normal daily draws into a year with kurtosis exactly 297/67, and one shuffle of those same numbers separates the fat tail from the clustering. It also has a condition nobody quotes: stationarity is not enough for that kurtosis to be finite.

Take a year of daily returns on one share and fit a single variance to the whole series. The number you get is not wrong so much as it is a description of nothing that happened. Quiet stretches ran quieter than it and violent stretches ran more violent, and the model has no way of saying so, because it was told in advance that every day is drawn from the same distribution.

The repair is two extra terms and one extra idea: today's variance is allowed to depend on yesterday. The consequences are sharper than the description suggests. Two weights that add to 0.98 give a shock a half-life of 34.31 days in the forecast, and the same recursion turns strictly normal daily draws into a year whose kurtosis is exactly $297/67 = 4.4328$ .

What a single variance gets wrong

The assumption in ordinary least squares is that the residuals all have the same variance. When they do not, the coefficient estimates are still unbiased, so nothing looks broken. What breaks is everything built on top of them. The usual standard-error formula assumes the constant variance it was given, so the reported errors on estimated coefficients are wrong, and every confidence interval and every significance test inherits the error. A regression can report a comfortable result on data that does not support it.

The clustering is easy to see and hard to write down with a single number. In a simulated 400,000-day series from the recursion below, the calm stretches run at 0.665 percent a day and the violent ones at 1.700 percent, a factor of 2.56 between them. A single fitted value sits in the middle and describes neither.

Fig. 1 — The series and the variance behind it, from the same recursion. The lower curve is not fitted to the upper one; it generated it.

Two weights and a floor

GARCH(1,1)

Returns satisfy $u_t \mid \mathcal{F}_{t-1} \sim N(0, h_t)$ with $\;h_t = \gamma_0 + \gamma_1 u_{t-1}^2 + \gamma_2 h_{t-1}$ and $\gamma_0 > 0$ , $\gamma_1, \gamma_2 \ge 0$ . Today's variance is a floor, plus a weight on yesterday's squared return, plus a weight on yesterday's variance. Setting $\gamma_2 = 0$ leaves ARCH(1).

The figures below use $\gamma_1 = 0.08$ , $\gamma_2 = 0.90$ and $\gamma_0 = 2 \times 10^{-6}$ , which is a canonical shape rather than a fit to any real series. Every number here comes from the recursion itself, and the series in Fig. 1 was generated by it.

Notice what each term buys. The squared return is the only channel through which news enters, and its weight of 0.08 is small, so a single day never moves the variance far on its own. The lagged variance is the memory, and it is what carries a shock forward long after the day that produced it has passed.

The level the forecast returns to

Write $p = \gamma_1 + \gamma_2$ for the persistence. Taking unconditional expectations in the recursion and using $\mathbb{E}[u_{t-1}^2] = \mathbb{E}[h_{t-1}]$ gives one equation for the long-run variance:

v \;=\; \frac{\gamma_0}{1 - p} \;=\; \frac{2\times 10^{-6}}{0.02} \;=\; 10^{-4}

(1)

A variance of $10^{-4}$ is a standard deviation of exactly one percent a day, and the floor was chosen to make it come out round. It is a genuine fixed point of the recursion: a variance sitting there stays there, since $2\times 10^{-6} + 0.98 \times 10^{-4} = 10^{-4}$ .

The decay, in one line

The reason the fixed point matters is that the forecast approaches it geometrically, and the proof is a substitution. Replace $\gamma_0$ by $v(1-p)$ from (1) and the recursion rearranges into a statement about distances rather than levels:

h_{t+1} - v \;=\; \gamma_0 + p\,h_t - v \;=\; p\,(h_t - v) \quad\Longrightarrow\quad \mathbb{E}_t[h_{t+k}] - v = p^{\,k}\,(h_t - v)

(2)

The distance from the long-run level is multiplied by $p$ every day, no matter how far away it started. So the half-life is a division:

k_{1/2} \;=\; \frac{\ln(1/2)}{\ln p} \;=\; \frac{\ln(1/2)}{\ln 0.98} \;=\; 34.31 \text{ days}

(3)

Now delete the second weight. ARCH(1) has the same equation with $p = \gamma_1 = 0.08$ , so its half-life is $\ln(1/2)/\ln 0.08 = 0.2744$ days. Rather more than half of any shock is forgotten before the next open. That difference between 34.31 and 0.2744 is the entire case for the second term, and it is a factor of 125 in memory bought by one extra parameter.

Fig. 2 — Both curves are the same geometric decay with a different rate. The one-weight version has essentially no memory at the resolution of a trading day.

One caution about reading equation (3). It is a statement about the forecast, meaning the expected variance $k$ days out conditional on today. It is not a promise about any single path, which will wander around the forecast rather than tracing it.

Normal days, fat-tailed year, and two facts that are not one

Every single day in this model is a normal draw, and yet a year of it is not normal at all. The fourth moment can be computed in closed form, and for normal innovations it comes out as a clean rational number:

\kappa \;=\; \frac{3\,(1 - p^2)}{1 - p^2 - 2\gamma_1^2} \;=\; \frac{3 \cdot 99/2500}{67/2500} \;=\; \frac{297}{67} \;=\; 4.4328

(4)

Against the 3 of a normal, that is a real excess, and nothing fat-tailed was assumed anywhere. Mixing variances is enough on its own: a distribution built by drawing a variance and then drawing a normal has more mass far from the centre than any single normal with the same total variance. A 400,000-day simulation gives a sample kurtosis of 4.371 against the exact 4.4328, and days beyond three times the typical size arrive 0.804 percent of the time against the 0.270 percent a single fixed normal allows.

Clustering and fat tails get conflated constantly, and one measurement pulls them apart. In that same simulation a day 2.5 times the typical size happens 1.96 percent of the time unconditionally, and 11.70 percent of the time immediately after another one, six times as often. Now shuffle the identical numbers into a random order. The fat tail is unchanged, since shuffling cannot alter a histogram. The clustering disappears completely: the rate of a big day following a big day falls back to the unconditional 1.96 percent. So the fat tail is a fact about the distribution and the clustering is a fact about the ordering, and only the second one is what equation (2) is about.

Where the half-life stops existing

Equation (1) needs $p < 1$ , and everything after it inherits that. At $p = 1$ the denominator vanishes: there is no long-run variance for the forecast to return to, shocks are permanent rather than merely persistent, and equation (3) returns infinity, correctly. Above 1 the forecast diverges and the model is unusable as written. With $\;p = 0.98$ the process sits close enough to that boundary that the half-life is sensitive to the third decimal of the estimate: 0.99 would give 69 days, and 0.95 would give 13.5.

Equation (4) needs strictly more than stationarity, and this is the constraint most often skipped. The kurtosis is finite only when the denominator is positive:

p^2 + 2\gamma_1^2 < 1 \quad\Longleftrightarrow\quad 3\gamma_1^2 + 2\gamma_1\gamma_2 + \gamma_2^2 < 1

(5)

With these weights the left side is 0.9732, so the fourth moment exists with room to spare. Hold $p$ at 0.98 and the left form of (5) collapses to a bound on the news weight alone, $\gamma_1 < \sqrt{(1-p^2)/2} = 0.1407$ . So raise $\gamma_1$ to 0.15 while lowering $\gamma_2$ to 0.83. The persistence is untouched, the long-run variance in (1) is untouched, the half-life in (3) is still 34.31 days, and the left side of (5) is now 1.0054. The kurtosis has become infinite. A model can therefore be stationary, have a published half-life, and have no finite fourth moment to speak of.

Sources and further reading

Wikipedia: Autoregressive conditional heteroskedasticity, and Volatility clustering for the effect the recursion is built to reproduce.
Wikipedia: Kurtosis, for the moment computed in equation (4).
Robert Engle, “Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation”, Econometrica50 (1982), 987–1007, and Tim Bollerslev, “Generalized Autoregressive Conditional Heteroskedasticity”, Journal of Econometrics31 (1986), 307–327.

The half-life was verified exactly rather than by simulation: the forecast was iterated 120 steps in rational arithmetic and matched $p^{\,k}$ times today's distance to the last bit. The simulated channel reproduces the long-run variance of equation (1) to within 0.3 percent over 400,000 days, and the clustering measurement is reported alongside its shuffled control so that the two effects cannot be confused. No real market data appears anywhere in this article.