6.5.5 Failure Modes & Real-World Practice

Basic Prerequisites: 03 · The Roll Model and 04 · Spread Decomposition.

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1. Intuition & Practical Objective

The Roll estimator is beautiful and fragile. Its entire output rests on four assumptions: (1) the efficient price is a random walk, (2) $q_t$ is i.i.d. with mean zero, (3) $q_t$ is independent of the innovation $u_t$, and (4) the spread is constant. This page names precisely which assumptions fail in real markets, how the bias runs (up or down), and what to do instead. The objective is not cynicism - it is the discipline of knowing when a number is trustworthy and when it is a trap.

The four failures, in one line each:

1. Momentum/persistence in the efficient returns flips ${\gamma }_1$ positive → the estimator is undefined.
2. Serial-correlated order flow (buys-follow-buys) biases the spread downward.
3. Information correlated with trade direction biases the spread upward.
4. Non-constant spread / non-stationarity contaminates the single autocovariance.

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2. Mathematical Ground Truth & Derivations

Where the assumptions live (Hasbrouck Ch 3–4). The Roll formulas ${\gamma }_0=2c^2+{\sigma }_u^2,{\gamma }_1=-c^2$ use: (A1) $\Delta p_t=u_t+c(q_t-q_{t-1})$ with $u_t$ white noise; (A2) $q_t$ i.i.d.; (A3) $q_t\perp u_t$; (A4) $c$ constant. Each relaxation changes the moments:

Failure 1 - persistent efficient returns. If $u_t$ is itself an AR(1) with $\varphi >0$, the added positive autocovariance $\varphi {\sigma }_u^2/(1-{\varphi }^2)$ competes with $-c^2$. When the momentum term dominates, ${\gamma }_1\ge 0$ and the estimator is undefined (Harris 1990 documents positive ${\hat{\gamma }}_1$ as common).

Failure 2 - serial-correlated order flow (Hasbrouck Ex 4.2). Let $\rho =\mathrm{corr}(q_t,q_{t-1})>0$ (real data: $\approx 0.34$). Then

$${\gamma }_1=-c^2(1-2\rho ),$$

which is less negative than $-c^2$ for $0<\rho <\frac{1}{2}$, so the Roll spread $\sqrt{-{\gamma }_1}$ is biased downward (this is the lag-1-only Ex 4.2 result; a persistent Markov direction process instead gives the geometric form ${\gamma }_1=-c^2(1-\rho {)}^2$, also biased downward).

Failure 3 - information correlated with direction (Hasbrouck Ex 4.3). Let $\rho =\mathrm{corr}(q_t,u_t)>0$ (informed traders buy before good news). Then

$${\gamma }_1=-c(c+\rho {\sigma }_u),$$

which is more negative than $-c^2$, so the Roll spread is biased upward.

Failure 4 - non-stationarity. Roll assumes covariance-stationary $\Delta p_t$ with a constant spread. A time-varying spread (wider in high-vol, tighter in calm), overnight gaps, or drift in ${\sigma }_u^2$ all make a single ${\gamma }_1$ a poor description. Standard practice drops overnight changes (missing values at day breaks) and estimates conditionally.

Decomposition caveat. In the generalized-Roll decomposition, ${\sigma }_w^2={\lambda }^2+{\sigma }_u^2={\gamma }_0+2{\gamma }_1$ is identified, but $\{c,\lambda ,{\sigma }_u^2\}$ are not from autocovariances alone - under-identification is a structural feature, not just an estimation nuisance.

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3. Computational Implementation - the failures in numbers

Experiment 1 - serial-correlated orders bias the spread down. Simulate a Roll process with true spread $0.04$ but order flow that clusters ($\alpha =\mathbb{P}(q_{t+1}=q_t)=0.8$, i.e. $\rho =0.6$). The Roll estimator under-reads.

The estimator reports $0.016$ against a true spread of $0.040$ - a 60% understatement, purely from order clustering. This is the canonical Roll failure in dealer/institutional markets.

Experiment 2 - information correlated with direction biases the spread up. Informed traders buy before good news, so $\mathrm{corr}(q_t,u_t)>0$.

Now it overstates: $0.0426$ vs true $0.040$. The two bias directions are opposite, so if both are present they can partially cancel - but you cannot know by how much without modeling them.

Experiment 3 - momentum breaks the estimator. Make the efficient returns a persistent AR(1); the bounce term loses, ${\gamma }_1\ge 0$, and the spread is undefined.

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4. Failure Modes & First-Principles Breakdowns (numbered)

1. Positive autocovariance (A1 fails). Momentum or clustering in the efficient returns makes ${\gamma }_1\ge 0$; the estimator returns NaN. Check the sign of ${\hat{\gamma }}_1$ before trusting anything - a positive first autocovariance means Roll is silent, not that the spread is zero.
2. Serial-correlated order flow (A2 fails). $\rho >0$ shrinks $|{\gamma }_1|$ and the spread is underestimated - badly, as Experiment 1 shows. Real order flow clusters (institutional splitting), so plain Roll is a lower bound in practice.
3. Information–direction correlation (A3 fails). Informed flow makes ${\gamma }_1$ more negative and the spread is overestimated. This is precisely why the adverse-selection literature exists: part of the measured “spread” is really information moving the price.
4. Non-stationarity (A4 fails). Non-constant spread, overnight gaps, and drift in ${\sigma }_u^2$ corrupt ${\gamma }_1$. Drop overnight changes; estimate on intraday segments; use quote-based or regression-based estimators when quotes exist.
5. Under-identification in decomposition. Autocovariances alone can’t separate $c$, $\lambda$, and ${\sigma }_u^2$. If you want components, you need trade-direction data (the 04 · Decomposition regression) - not just prices.

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5. References

• Harris (1990), Statistical properties of the Roll serial covariance bid/ask spread estimator, Journal of Financial Economics 27(2)
• Hasbrouck (2007), Empirical Market Microstructure
• Stoll (1989), Inferring the components of the bid-ask spread
• Hasbrouck (2005), Trading costs and returns for US equities

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6. Connected Graph Bridges

• Back: 04 · Spread Decomposition · Index Hub
• Forward: 06 · Advanced Extensions (generalized Roll, VAR/price impact, random-walk decomposition)
• Sibling: Adverse Selection & Glosten–Milgrom (the A3 failure, structural) · Toxic Order Flow & VPIN (signing trades, Lee–Ready)
• Base: Econometrics & Time Series (stationarity, estimation)