Hasbrouck's Pricing Error Variance

Every trade prints a price, and every printed price is a mix of two things: where the stock is really worth trading (the efficient price, reflecting all available information) and a temporary wobble caused purely by market mechanics — whether the last trade happened to hit the bid or the ask, inventory pressure on a market maker, or a large order temporarily pushing the tape away from fair value. If you want to know how good a market is at reflecting true value — how "efficient" the price discovery is — you need to separate these two pieces, because the visible price bounces around by more than the true value ever does.

Think of it like a scale that's slightly loose on its mounting: weigh the same object five times in a row and you'll get five slightly different readings, none of them wrong exactly, just jittering around the true weight because of mechanical play in the scale itself. The true weight didn't change between readings — only the noise did. A market's transaction price is that jittery scale reading; the efficient price is the true weight.

Hasbrouck's framework models the observed log price $p_t$ as the sum of a random-walk efficient price $m_t$ (which moves only on genuine information) and a stationary, mean-reverting pricing error $s_t$ (the noise):

$p_t = m_t + s_t$

In words: what you observe equals the true evolving value plus a wobble that doesn't persist. Because $m_t$ is a random walk, its variance grows without bound over time — that's the permanent component. Because $s_t$ is stationary, it has a fixed, finite variance that doesn't grow — that's the transient component. The key statistic is the pricing error variance, $\text{Var}(s_t)$, estimated using a vector autoregression (VAR) fit to trades and quote revisions, which decomposes the total variance of observed price changes into how much comes from genuine information ($m_t$ moving) versus how much comes from noise ($s_t$ jittering).

Worked example, simplified. Suppose over a trading day the total variance of the transaction price series is measured at $\text{Var}(p_t) = 0.00012$ (in log-price units squared, aggregated appropriately). A VAR-based decomposition attributes $\text{Var}(m_t)$'s contribution — the permanent, information-driven component — at $0.00008$, and the pricing error variance at $\text{Var}(s_t) = 0.00004$. The ratio $\text{Var}(s_t) / \text{Var}(p_t) = 0.00004/0.00012 = 0.33$ says roughly a third of observed price variance is noise, not information — a third of every tick's "movement" is mechanical wobble that will mean-revert rather than persist.

Second example, comparing two names. Stock A (a liquid, tight-spread large cap) has pricing error variance equal to 8% of total observed price variance. Stock B (a thinly traded small cap with a wide spread) has pricing error variance equal to 45% of total observed price variance. This tells you Stock B's tape is nearly half noise: almost half of every price wiggle you see is bid-ask bounce and temporary liquidity effects rather than the market actually learning something new, which means signals built directly off Stock B's tick-by-tick prices need much heavier smoothing or larger sample sizes before a real edge would be distinguishable from the noise floor.

What this means in practice

Pricing error variance is a market-quality metric: exchanges, regulators, and execution desks use it to compare how "clean" price discovery is across venues or across time for the same name (e.g., before and after a tick-size change or a change in market-making obligations). A rising pricing-error share over time on a given venue is evidence that liquidity or price discovery quality is deteriorating, independent of whether volatility itself went up or down. It's also a direct input to how much weight a signal-generation process should put on raw tick prices versus a smoothed or microprice-adjusted series.

The classic confusion

This decomposition is the formal cousin of Intraday Variance Ratios — both separate genuine price discovery from mechanical noise, using different statistical machinery — and its output directly informs Hasbrouck's Information Share, which asks which of several linked markets contributes more to the permanent, information-driven component $m_t$.