Variance Ratio Test

If stock prices follow a random walk—moving up or down each period with no pattern—then variance should grow linearly with time. A four-day variance should be roughly four times a one-day variance. The Variance Ratio Test compares actual variances across timeframes to see if this holds. When it doesn’t, markets reveal memory: momentum that makes recent winners keep winning, or mean reversion that brings extreme moves back to centre.

The logic: why random walk variance scales linearly

Assume a stock’s returns follow pure randomness. Each day, it’s equally likely to jump +1% or –1% (simplistically). Over one day, variance is 1%.

Over four days, the returns compound: (1 ± 1%) × (1 ± 1%) × (1 ± 1%) × (1 ± 1%). If the changes are truly independent, the four-day return variance should be 1% × 4 = 4% (variance adds linearly under independence). The variance ratio VR(4) = 4% / (4 × 1%) = 1.0.

Now suppose day-to-day returns are not independent. If a 1% jump on Day 1 makes a 1% jump on Day 2 more likely (momentum), then the four-day return is amplified; variance inflates beyond 4%, and VR(4) > 1.0. If a 1% jump makes a –0.8% correction on Day 2 more likely (mean reversion), variance shrinks below 4%, and VR(4) < 1.0.

By testing whether VR(k) = 1 across multiple k values (2, 4, 8, 16 periods), you uncover temporal structure in prices. Markets are not random walks—they reveal preferences for mean reversion or momentum at different timescales.

Calculating the variance ratio

Take daily returns for 500 trading days (approximately two years). Denote them r₁, r₂, …, r₅₀₀.

One-day variance: Var(R₁) = variance of {r₁, r₂, …, r₅₀₀}.

Four-day variance: Construct four-day returns: {r₁ + r₂ + r₃ + r₄, r₅ + r₆ + r₇ + r₈, …}. Then Var(R₄) = variance of these cumulative returns.

Variance Ratio: VR(4) = Var(R₄) / (4 × Var(R₁)).

If VR(4) = 1.2, four-day returns are 20% more variable than random walk predicts. This hints that a move on day 1 is likely to persist into days 2–4 (momentum).

What the test reveals at different horizons

For most stocks and indices, the variance ratio varies across k:

• VR(2) to VR(4): Often slightly below 1.0 (mean reversion on very short timescales).
• VR(8) to VR(20): Often above 1.0 (momentum dominates medium-term; winners keep winning).
• VR(60+): Often drifts back toward 1.0 or slightly below (long-term reversion to fundamentals).

This pattern suggests markets are not random walks at any single timescale, but different timescales have different biases. A trader exploiting VR(8) > 1.2 might ride momentum, while a rebalancer exploiting VR(4) < 0.95 might bet on reversal.

Statistical significance: beyond the ratio

Computing VR(k) is half the battle; testing whether it’s statistically different from 1.0 requires a test statistic. Lo and MacKinlay (1988) showed that under the random walk null, VR(k) is asymptotically normal. A z-score test tells you if the deviation is real or sampling noise:

z = (VR(k) – 1) / SE(VR(k))

If |z| > 1.96 (at the 5% level), you can reject the random walk hypothesis. A VR(8) = 1.05 with small sample size might not be significant; a VR(8) = 1.15 over 20 years likely is.

Market implications and practical limits

Evidence from the variance ratio test suggests:

1. Equities show mild momentum. Over weeks and months, VR often exceeds 1.0, meaning winners keep winning. This supports momentum strategies and explains why trend-following funds exist.

2. Very long horizons (years) revert. VR(250) or VR(500) sometimes drops below 1.0, suggesting mean reversion dominates multiyear moves. Extreme valuations do eventually correct.

3. Differences by asset class. Currencies and commodities show different patterns; forex pairs often exhibit stronger mean reversion than equities.

4. Time-varying results. A market in boom phase has different VR(k) patterns than one in crisis. The test is static; it doesn’t adapt to regime change.

The test’s practical limits: VR(k) is noisy with real-world data (transaction costs, bid-ask spreads, non-synchronous trading). A significant variance ratio doesn’t guarantee a profitable trade—you still face implementation costs and the risk that the pattern breaks.

Relation to efficient markets and predictability

The random walk hypothesis is a building block of the efficient market hypothesis. If VR(k) ≠ 1 significantly, prices contain predictable patterns, and markets are not efficient at that horizon. Many academics interpret significant variance ratios as evidence that:

• Arbitrage hasn’t fully exploited the pattern, or
• The pattern is real but too small (after costs) to trade profitably, or
• The market is inefficient at that timescale.

Most economists accept that variance ratios deviate meaningfully from 1 in real data—rejecting the strict random walk—yet debate whether this creates genuine profit opportunities or is noise.

See also