ARCH model

Take any regression's residuals, square them, and regress the squared residual on its own lag:

e_t^2 = a0 + a1 e_(t-1)^2 + u_t

If a₁ is zero, the size of yesterday's surprise tells you nothing about the size of today's, and the constant-variance assumption behind every ordinary standard error survives. If a₁ is reliably positive, variance has a memory, and Engle's 1982 paper named that condition autoregressive conditional heteroskedasticity. The test is the same t-test you already run on any slope, and Engle's Lagrange multiplier version, n × R² from that auxiliary regression against a chi-square with one degree of freedom, is the form you will meet in software output.

Worked example: a 500-day midcap sample

On the full 500-day midcap sample the coefficient is a₁ = 0.212654 with a standard error of 0.044269, a t-statistic of 4.80 and a p-value of 2.1 × 10⁻⁶. The Lagrange multiplier statistic is 22.14 against a 5% critical value of 3.84. The lag-1 autocorrelation of the raw returns is −0.0733, inside the ±0.0894 Bartlett band, while the lag-1 autocorrelation of the squared residuals is +0.2086, outside it. Returns are close to unpredictable in sign and clearly predictable in magnitude, and those two facts are not in tension.

What a positive a₁ does to the parent regression is the part most often skipped. Fit the mean equation r_t = c + φ r_(t−1) on the same 500 days and ordinary least squares reports φ = −0.0743 with a standard error of 0.044972, giving t = −1.65. The heteroskedasticity-consistent standard error for the same coefficient is 0.059416, which is 32.1% larger, and the t-statistic falls to −1.25. The coefficient estimate itself is unchanged and remains unbiased. What changes is the confidence interval, which widens by a factor of 1.3212, and the honest verdict, which moves from "borderline" to "nothing here". Unmodelled ARCH does not corrupt your point estimates; it manufactures significance, and it does so in exactly the direction that flatters whoever is presenting.

How do you forecast variance with an ARCH model?

Having tested for the effect, use it. The one-step-ahead conditional variance is the fitted value of that auxiliary regression at the last observed squared residual:

h_(T+1) = a0 + a1 e_T^2

and the conditional standard deviation is its square root. Feeding it into a normal-based value at risk gives z × √h × position, which is the arithmetic behind the two numbers in this node's opening. Iterating one more step gives h_(T+2) = a₀ + a₁ h_(T+1), and continuing forever converges on a₀/(1 − a₁), the ARCH(1) model's own unconditional variance. That last quantity is a diagnostic in disguise, because it has to be close to the sample variance you can compute directly. On the full sample it comes out at 0.00017196 against a sample variance of 0.00017198, agreeing to four significant figures. On the nineteen-pair teaching window it comes out at 0.00099084, implying a daily standard deviation of 3.1478% against a sample figure of 1.3114%, and that gap is how you know the window estimate cannot be trusted no matter how significant its t-statistic looked.