T3 Moving Average

Tim Tilson's smoothing filter built from six nested EMAs blended by a 'volume factor' coefficient set.

What it is

The T3 Moving Average was published by Tim Tilson in 1998 as a smoother, lower-lag alternative to standard moving averages. Despite often being described vaguely as 'a better EMA', it has an exact, fully deterministic construction: six exponential moving averages applied in sequence (an EMA of an EMA, six layers deep), combined with four fixed coefficients derived from a single parameter called the volume factor (usually written v or b, default 0.7). The 'volume factor' has nothing to do with traded volume — it is just a weighting constant that controls how aggressively the filter compensates for lag. The result is a very smooth line that reacts faster than its smoothness would suggest, at the cost of some overshoot around sharp turns. Like every moving average, T3 is a summary of past prices: it describes what price has done, not what it will do. This entry is educational only, not financial advice, and all trading involves risk of loss.

How it works

The full expansion — which many sites hand-wave — is as follows. First compute six nested EMAs of the source, all with the same length N: e1 = EMA(price, N), e2 = EMA(e1, N), e3 = EMA(e2, N), e4 = EMA(e3, N), e5 = EMA(e4, N), e6 = EMA(e5, N). Then, with volume factor v (default 0.7), compute four coefficients: c1 = −v³, c2 = 3v² + 3v³, c3 = −6v² − 3v − 3v³, and c4 = 1 + 3v + v³ + 3v². The output is T3 = c1·e6 + c2·e5 + c3·e4 + c4·e3. Conceptually, T3 is a 'generalized DEMA' applied three times: where DEMA uses 2·EMA − EMA(EMA) to cancel lag, Tilson's version uses (1+v)·EMA − v·EMA(EMA), and composing that operation three times produces the coefficient set above. At v = 0 the formula collapses to a plain triple-smoothed EMA (maximum smoothness, maximum lag); at v = 1 it becomes a triple DEMA (least lag, most overshoot). The default 0.7 sits between those extremes. Note the coefficients sum to 1, so T3 stays on price scale.

How traders read it

• Slope as a smoothed trend read: because six smoothing passes remove most bar-to-bar noise, traders often read the T3's slope as a cleaner description of recent direction than a single EMA — still a lagging description, not a forecast.
• Side of the line: price persistently above a rising T3 is read as recent upward drift, below a falling T3 as downward drift, with the usual caveat that price can cross through it at any time.
• Colour or slope flips: some implementations colour the line by slope; a flip marks the point where the heavily smoothed average changed direction — which, given the nesting, happens well after the price turn began.
• Compare with an EMA of the same length: the T3 is visibly smoother yet turns at a comparable time; that comparison shows what the lag-compensation coefficients are doing.
• Watch turns for overshoot: with higher volume factors the T3 can bend past price after sharp reversals — a structural artifact of lag compensation, not information about the market.

Common settings

Common defaults are length 14 (some platforms use 5 or 8 for faster variants) with volume factor 0.7, applied to the close. Raising the volume factor toward 1 reduces lag but increases overshoot; lowering it toward 0 gives a very smooth, very slow triple-EMA. Note that because six EMAs are nested, the effective smoothing window is much longer than the nominal length — a T3(14) is far slower than an EMA(14), so lengths are not comparable one-to-one across average types.

Strengths

• Exceptionally smooth output for its response speed, because the coefficient blend cancels much of the lag the six-fold smoothing would otherwise add.
• Fully deterministic and reproducible: six EMAs and four published coefficients — nothing proprietary, despite its black-box reputation.
• One intuitive tuning knob: the volume factor moves the filter continuously between a smooth triple EMA and a fast triple DEMA.
• Coefficients sum to one, so the line stays on price scale with no drift or normalization issues.
• Useful as a noise-reduced input for other calculations, since it is on the same scale as price.

Pitfalls to watch

• Substantial effective lag: six nested smoothing passes mean a T3 responds much more slowly than its nominal length suggests; comparing T3(14) to EMA(14) is comparing different speeds.
• Overshoot from lag compensation: higher volume factors make the line swing past price at reversals — smoothness and responsiveness are borrowed against accuracy at turns.
• Misleading name: the 'volume factor' does not use volume data at all; it is a coefficient. Descriptions that imply volume-awareness are wrong.
• Warm-up sensitivity: with six recursive EMAs, early-chart values take a long time to converge, so readings near the start of the data are unreliable.
• Settings sensitivity: length and volume factor interact; a combination that looks clean on one chart can behave very differently on another instrument or timeframe.
• No predictive power: T3 is arithmetic on past prices. Smoother lines look more convincing but forecast nothing, and no setting removes the risk of loss.

Pine v6 example

//@version=6
indicator("T3 Moving Average", overlay = true)

length = input.int(14, "Length", minval = 1)
vf     = input.float(0.7, "Volume Factor", minval = 0.0, maxval = 1.0, step = 0.05)
src    = input.source(close, "Source")

// Six nested EMAs
e1 = ta.ema(src, length)
e2 = ta.ema(e1, length)
e3 = ta.ema(e2, length)
e4 = ta.ema(e3, length)
e5 = ta.ema(e4, length)
e6 = ta.ema(e5, length)

// Tilson coefficients (sum to 1)
c1 = -math.pow(vf, 3)
c2 = 3 * math.pow(vf, 2) + 3 * math.pow(vf, 3)
c3 = -6 * math.pow(vf, 2) - 3 * vf - 3 * math.pow(vf, 3)
c4 = 1 + 3 * vf + math.pow(vf, 3) + 3 * math.pow(vf, 2)

t3 = c1 * e6 + c2 * e5 + c3 * e4 + c4 * e3

plot(t3, "T3", color = color.purple, linewidth = 2)

Pro tip: Set the volume factor to 0 and then to 1 and watch the line transform from a slow triple-EMA into a fast, overshooting triple-DEMA — that single experiment teaches you exactly what the coefficient blend does and where your chosen setting sits on the smoothness-versus-overshoot spectrum. Remember the effective window is far longer than the nominal length when comparing T3 to other averages. Educational information only, not financial advice; no moving average predicts price, and all trading carries risk of loss.

Educational only — not financial advice, not a recommendation to trade. No indicator is predictive; trading involves substantial risk of loss.