Second-order moving-average lag correction

Why a simple average stays behind

A moving average is a first-order smoother that averages a fixed lookback of ordered prices and therefore reports only the local level, leaving a systematic lag when the series is trending. Because a simple average uses only one total of the lookback prices, it can represent the local level but cannot estimate the slope of the same window.

A simple two-period average of a unit ramp from 1 to 10 and back to 1 stays a half-unit behind the series on every bar, so the first-order smoother never catches a constant rise or fall.

Building the two-sum correction

Linear regression is the algebraic construction that estimates both local level and slope from two explicit sums over the same N observations, then adds the slope term back to the simple average. Exponential smoothing is a second-order correction, drawn from inventory-forecast practice, that adds a recency-weighted slope sum so the estimate can track rising or falling prices instead of remaining one period behind.

The second-order construction adds a recency-weighted sum S, with older observations receiving smaller or even negative weights, then forms the corrected estimate as the simple average plus 6S divided by N times N plus one.

For a two-period window the uncorrected average is T/2, while the corrected average is that same level plus 6S over 6, using T as the sum of today and yesterday and S as half of today minus half of yesterday.

What the corrected lookback recovers

On the same unit-ramp series, two-, three-, and four-period corrected averages recover the current price with zero tabulated error once the window is full, whereas the corresponding simple averages remain lagged.

After startup, both totals can be updated recursively: T replaces the observation that drops out of the window, and S is refreshed from its prior value, the prior T, the newest price, and the price leaving the window.

On a standard cyclical price series and on a Treasury-bond series, the four-period and ten-period corrected averages track the underlying path more tightly than simple averages of the same length, which retain their usual lag.

Because the corrected average responds faster, crossover-style decision rules that assume simple-average delay need to be rechecked rather than transplanted unchanged.