6.5.6 Advanced Extensions

Basic Prerequisites: 04 · Spread Decomposition, 05 · Failure Modes, and Econometrics & Time Series (MA/VAR, cointegration).

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1. Intuition & Practical Objective

The plain Roll model is univariate: prices only. This page is the launchpad to the modern toolkit that fixes its limits - (1) the generalized Roll model (Hasbrouck Ch 8) that splits the spread into $c$ (transitory) and $\lambda$ (permanent/price-impact), (2) the random-walk / permanent-transitory decomposition (Beveridge–Nelson 1981; Watson 1986) that separates the efficient price from the pricing error, and (3) the multivariate VAR / price-impact machinery (Hasbrouck Ch 9) that measures $\lambda$ directly from the trade-innovation response.

Why these first? They are the measurement completion of Roll: Roll gives the total spread; these give its components and its permanent vs transitory split. Everything farther - Hasbrouck information shares across venues, PIN/VPIN, structural adverse-selection models - links from here.

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2. Mathematical Ground Truth & Derivations

2.1 The generalized Roll model (Hasbrouck Ch 8)

Let the efficient price be driven by both public information $u_t$ and the information content of the trade itself, $\lambda q_t$:

$$p_t=m_t+c{q}_t,m_t=m_{t-1}+\lambda q_t+u_t.$$

Here $c$ = non-informational (order-processing/inventory) cost, $\lambda$ = adverse-selection / price-impact cost. The bid/ask straddles $m_{t-1}+u_t$, and the spread is $2(c+\lambda )$. Differencing:

$$\Delta p_t=c(q_t-q_{t-1})+\lambda q_t+u_t,$$ $${\gamma }_0=c^2+(c+\lambda {)}^2+{\sigma }_u^2,{\gamma }_1=-c(c+\lambda ).$$

The central identification result: only two autocovariances are observable, but there are three structural parameters $\{c,\lambda ,{\sigma }_u^2\}$ - so the components are under-identified from prices alone. However, the random-walk innovation variance

$$\boxed{{\sigma }_w^2\equiv {\lambda }^2+{\sigma }_u^2={\gamma }_0+2{\gamma }_1}$$

is identified, because it is the permanent component and does not depend on the bounce. This is the economic reason the permanent/transitory split is recoverable while the spread components are not.

2.2 Random-walk (Beveridge–Nelson / Watson) decomposition

Any covariance-stationary $\Delta p_t$ is an MA(1) (Wold): $\Delta p_t={\epsilon }_t+\theta {\epsilon }_{t-1}$. The Beveridge–Nelson decomposition writes $p_t=m_t+s_t$ where $m_t$ is the random-walk (permanent) component and $s_t$ is the stationary pricing error:

$${\sigma }_w^2=(1+\theta {)}^2{\sigma }_{\epsilon }^2\text{(permanent innovation variance)},$$ $${\sigma }_s^2={\theta }^2{\sigma }_{\epsilon }^2\text{(lower bound on the pricing-error variance)}.$$

In the pure Roll case, ${\sigma }_w^2={\sigma }_u^2$ and the actual pricing-error variance is $\mathrm{Var}(p_t-m_t)=c^2$, attained by the bound ${\theta }^2{\sigma }_{\epsilon }^2$ only when ${\sigma }_u^2=0$ (in Roll, $c^2=\theta {\sigma }_{\epsilon }^2>{\theta }^2{\sigma }_{\epsilon }^2$). The decomposition is invariant to the identification of the MA parameters - a strong robustness property.

2.3 Variance ratio

The variance ratio compares variances over different horizons:

$$V_{M,N}=\frac{\mathrm{Var}(p_t-p_{t-M})/M}{\mathrm{Var}(p_t-p_{t-N})/N}.$$

With microstructure (bid-ask bounce) inflating the short horizon variance, for $M>N$ (longer horizon in numerator) one gets $V_{M,N}<1$, declining toward $1$ as the horizon grows and the transitory component washes out. A ratio below 1 is the signature of a transitory (pricing-error) component in the price.

2.4 Multivariate price impact (Hasbrouck Ch 9) - the empirical route to $\lambda$

Stack the price change and trade variables into $y_t=[\Delta p_t,x_t^{\prime }{]}^{\prime }$ and fit a VAR; the impulse response of $\Delta p$ to a trade-innovation $v_t$ measures $\lambda$ directly. Structural form:

$$q_t=v_t+\beta v_{t-1}(\text{MA(1) order flow}),w_t=u_t+\lambda v_t,$$ $$\Delta p_t=u_t+\lambda v_t+c[(v_t+\beta v_{t-1})-(v_{t-1}+\beta v_{t-2})].$$

The permanent/trade-driven variance splits as ${\sigma }_w^2={\sigma }_u^2+{\lambda }^2{\sigma }_v^2$, where ${\lambda }^2{\sigma }_v^2$ is the absolute information content of trade flow and ${\lambda }^2{\sigma }_v^2/{\sigma }_w^2$ the relative share (≈ $R^2$ of price changes on trades). This is the bridge to Toxic Order Flow & VPIN and to price discovery across venues.

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3. Computational Implementation - the decomposition in numbers

Recover the MA(1) parameters from the sample autocovariances, then split the price into permanent (random-walk) and transitory (pricing-error) components, and confirm the permanent variance equals ${\gamma }_0+2{\gamma }_1$. Stdlib only.

The permanent (random-walk) innovation variance $0.000088$ matches ${\gamma }_0+2{\gamma }_1$ and recovers the true efficient ${\sigma }_u^2=0.0001$; the pricing-error variance $0.000436$ approximates the bounce $c^2$. The variance ratio below 1 is the empirical signature of the transitory component.

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4. Failure Modes & First-Principles Breakdowns

1. Reading components out of prices alone. $\{c,\lambda ,{\sigma }_u^2\}$ are under-identified from autocovariances; only ${\sigma }_w^2$ is. Any claim to have separated order-processing from adverse-selection using only a price tape is either an extra assumption or an error.
2. Variance-ratio sign confusion. The ratio is less than 1 when the shorter horizon is inflated (bounce); “>1 means microstructure noise” only when the short horizon is in the numerator. State $M,N$ explicitly.
3. VAR identification is ordering-dependent. The Cholesky decomposition that splits ${\sigma }_w^2$ into public vs trade information depends on the causal ordering; report bounds or justify the ordering (the same caveat that drives Hasbrouck information shares).
4. Trade signing. $\lambda$ estimation needs correct aggressor-side signing (Lee–Ready). Mis-signed flow attenuates or reverses the price-impact estimate (see Toxic Order Flow & VPIN).

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5. References

• Hasbrouck (2007), Empirical Market Microstructure
• Beveridge & Nelson (1981), A new approach to decomposition of economic time series into permanent and transitory components, Journal of Monetary Economics 7(2)
• Watson (1986), Univariate detrending methods with stochastic trends, Journal of Monetary Economics 18(1).
• Huang & Stoll (1997), The components of the bid-ask spread: a general approach, RFS 10(4)
• Hasbrouck (1991), Measuring the information content of stock trades, Journal of Finance 46(1)

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6. Connected Graph Bridges

• Back: 04 · Spread Decomposition · 05 · Failure Modes
• Forward topic pages: Adverse Selection & Glosten–Milgrom (structural $\lambda$) · Toxic Order Flow & VPIN (signing, PIN) · Limit Order Book Mechanics & L3 Data (trade/quote data)
• Base: Econometrics & Time Series (MA/VAR, Wold, cointegration) · Probability & Measure Theory (martingales)
• Cross-pillar: Optimal Execution (price impact in execution cost)