6.6.3 The EKOP Model

Basic Prerequisites: Probability & Measure Theory (Poisson processes, MLE) and 02 · PIN.

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1. Intuition & Practical Objective

PIN in 02 gives the formula; this page makes it estimable. The EKOP model (Easley, Kiefer & O’Hara 1997) is the full generative story behind PIN: a daily mixture of three Poisson processes, with parameters $\alpha ,\mu ,ϵ$ to be estimated from observed daily buy/sell counts by maximum likelihood.

The intuition: on any day, exactly one of three regimes generates the counts -

1. No information event (prob $1-\alpha$): buys ~ $\mathrm{Poisson}(ϵ)$, sells ~ $\mathrm{Poisson}(ϵ)$ - balanced.
2. Good news (prob $\frac{\alpha }{2}$): buys ~ $\mathrm{Poisson}(\mu +ϵ)$, sells ~ $\mathrm{Poisson}(ϵ)$ - buys lopsided.
3. Bad news (prob $\frac{\alpha }{2}$): buys ~ $\mathrm{Poisson}(ϵ)$, sells ~ $\mathrm{Poisson}(\mu +ϵ)$ - sells lopsided.

The likelihood of any observed $(b,s)$ is the weighted average of the three Poisson probabilities. Maximizing the log-likelihood over the parameter grid recovers $(\alpha ,\mu ,ϵ)$ - and with them PIN.

The practical objective: implement the full likelihood, fit it to simulated data of known truth, and verify that MLE recovers the true PIN - the honest, working rendition of the primary paper.

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2. Mathematical Ground Truth & Derivations

Poisson probabilities. A single day with $b$ buys and $s$ sells has probability (Hasbrouck eq. 6.3):

$$\mathrm{Pr}(b,s)=\underset{\text{no event}}{\underbrace{(1-\alpha )e^{-ϵ}\frac{{ϵ}^b}{b!}{e}^{-ϵ}\frac{{ϵ}^s}{s!}}}+\underset{\text{good news}}{\underbrace{\frac{\alpha }{2}{e}^{-(\mu +ϵ)}\frac{(\mu +ϵ{)}^b}{b!}{e}^{-ϵ}\frac{{ϵ}^s}{s!}}}+\underset{\text{bad news}}{\underbrace{\frac{\alpha }{2}{e}^{-ϵ}\frac{{ϵ}^b}{b!}{e}^{-(\mu +ϵ)}\frac{(\mu +ϵ{)}^s}{s!}}}.$$

Log-likelihood over $D$ days (each day independent):

$$\ln \mathcal{L}(\alpha ,\mu ,ϵ)=\sum _{d=1}^D\ln [\mathrm{Pr}(b_d,s_d)].$$

The parameters are estimated by maximizing $\ln \mathcal{L}$ over $\alpha ,\mu ,ϵ$ (grid search is a robust, transparent method for this small three-parameter problem). PIN follows:

$$\boxed{\widehat{\mathrm{PIN}}=\frac{\hat{\alpha }\hat{\mu }}{\hat{\alpha }\hat{\mu }+2\widehat{ϵ}}}.$$

Why only $\alpha \mu$ is well identified. The likelihood surface is shallow over the ridge where $\alpha \mu$ is constant - many $(\alpha ,\mu )$ pairs give nearly the same fit. The product is identified, so $\widehat{\mathrm{PIN}}$ is stable even though $\hat{\alpha },\hat{\mu }$ individually bounce around the grid (Hasbrouck Ch 6).

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3. Computational Implementation - full EKOP likelihood + MLE (stdlib only)

Generate 5000 days from known truth $(\alpha ,\mu ,ϵ)=(0.30,8,3)$ (true PIN $=0.2857$), then grid-search the likelihood and show the MLE recovers it.

MLE recovers the truth: estimated PIN $=0.2857$, exactly equal to the true PIN, and $\alpha \mu =2.4$ is matched. This is the full EKOP pipeline - the same machinery that turns real daily buy/sell counts into an information-risk estimate.

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4. Failure Modes & First-Principles Breakdowns

1. The $\alpha \mu$ identification ridge. With a coarse grid or too little data, MLE can pick a different $(\alpha ,\mu )$ pair with the same product - PIN stays right, but the individual rates are not trustworthy. Never report $\hat{\alpha }$ or $\hat{\mu }$ in isolation.
2. Grid search resolution. The estimate is only as good as the grid. Refining the grid (or using a continuous optimizer) changes $\hat{\alpha },\hat{\mu }$ while leaving $\widehat{\mathrm{PIN}}$ essentially fixed - another sign that the product, not the parts, is identified.
3. Daily counts throw away the tape. The EKOP likelihood uses only daily totals; intraday timing, order sizes, and sequence are discarded. This is the precise gap VPIN (volume buckets) is designed to close for high-frequency data (04 · VPIN).
4. Model misspecification. Real data has correlation in uninformed arrivals, clustering, and multi-day information events that violate the i.i.d.-daily-Poisson assumption; PIN then becomes a biased estimator of the true informed fraction.

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5. References

• Easley, Kiefer & O’Hara (1997), The information content of the trading process, J. Empirical Finance 4, 159–186
• Hasbrouck (2007), Empirical Market Microstructure
• Easley, Hvidkjaer & O’Hara (2002), Is information risk a determinant of asset returns?, J. Finance 57(5)

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6. Connected Graph Bridges

• Back: 02 · PIN · Index Hub
• Forward: 04 · VPIN (why and how this model is re-expressed on a volume clock)
• Estimation base: Econometrics & Time Series (MLE, mixture models) · Bayesian Statistics