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Forecast horizons, EWMA, and model diagnostics

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Start with the idea

A recursive forecast balances adaptation with stability. Fast updates react to shocks but may force costly position changes; slow updates can leave the book exposed to volatility that has already risen.

Symbols, units & horizon

σ̂ₜ²: variance forecast made before innovation εₜ
λ: EWMA weight on old variance, between 0 and 1
εₜ: observed residual return
μ: AR process mean
φ: AR coefficient
h: integer forecast horizon in periods
E[xₜ₊ₕ|xₜ]: expected future level given the current level
Hat: estimated quantity

When and why to use this

Use EWMA as a transparent volatility baseline and multi-step forecasts to align the model horizon with the actual holding period.

{\hat{σ}}_{t + 1}^{2} = λ {\hat{σ}}_{t}^{2} + (1 - λ) ε_{t}^{2}, 0 < λ < 1

Algebra and arithmetic

Unroll an EWMA variance forecast

Substitute the previous recursion repeatedly: $v_{t + 1} = λ^{k} v_{t + 1 - k} + (1 - λ) \sum_{j = 0}^{k - 1} λ^{j} ε_{t - j}^{2}$ .
The weights sum to $λ^{k} + \frac{(1 - λ) (1 - λ^{k})}{(1 - λ)} = 1$ . A shock’s weight halves at $h = \frac{\ln (.5)}{\ln λ}$ .

Work it by hand

λ=.94, old variance=.0001, shock=.02 gives .94(.0001)+.06(.0004)=.000118; next SD≈1.086%.

EWMA updates the variance forecast using the last innovation. Larger λ means slower adaptation; the weight half-life is ln(1/2)/ln(λ). Unlike stationary GARCH with an intercept, this form has no fixed long-run variance toward which the forecast reverts.

𝔼 [x_{t + h} | x_{t}] = μ + ϕ^{h} (x_{t} - μ)

Algebra and arithmetic

Iterate the conditional AR forecast

Write $x_{t + 1} - μ = ϕ (x_{t} - μ) + ε_{t + 1}$ . Conditional mean-zero future innovations disappear under expectation.
At two steps multiply the current deviation by φ²; induction gives φʰ. Add μ back.

Work it by hand

Current spread 5, mean 3, φ=.8, horizon 2: forecast=3+.64(2)=4.28, an expected decline of .72.

For a stable AR(1), the conditional forecast converges toward μ as h grows. A negative φ alternates the sign; a half-life for absolute deviation is not a monotone price-reversion forecast. Validate residual autocorrelation, changing variance and forecast errors at the actual trading horizon.

Python implementation

Self-contained teaching example. Python 3.10+; dependencies and input conventions are shown in the code and notation. Run in your own Python environment.

def ewma_variance(previous_forecast, innovation, decay=.94):
    if not 0 < decay < 1 or previous_forecast < 0:
        raise ValueError("Require variance >= 0 and decay in (0,1)")
    return decay*previous_forecast+(1-decay)*innovation**2

def ar_forecast(current, mean_level, phi, horizon):
    if horizon < 0 or int(horizon) != horizon:
        raise ValueError("Nonnegative integer horizon required")
    return mean_level+phi**horizon*(current-mean_level)

print(ewma_variance(.0001, .02), ar_forecast(12, 10, .8, 2))

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