Herding versus splitting decomposition: Definition & Examples

The herding versus splitting decomposition partitions order-sign autocorrelation into contributions from pairs attributed to the same agent and pairs attributed to different agents. It helps a quant investigate whether persistent order flow is concentrated within identities or across them. The attribution is statistical; identifying the economic mechanism needs additional assumptions.

Start with the pairs

Take twelve fictional aggressive orders in arrival order. A plus sign means buyer-initiated and a minus sign means seller-initiated. Assume the true investor identity and aggressor sign are known; each event receives equal weight, regardless of size.

Order event	1	2	3	4	5	6	7	8	9	10	11	12
Sign	+	+	+	−	−	−	+	+	+	−	−	−
Investor	A	A	A	B	B	B	B	B	B	A	A	A

Both investors have three buys and three sells. Their full-sample sign means, and the global mean, are zero; global sign variance is one.

For lag $\tau$ , let $M=N-\tau$ , $I_t$ be identity and $\epsilon_t\in\{-1,+1\}$ be sign. Define the globally centered product

z_t=(\epsilon_t-\bar\epsilon)(\epsilon_{t+\tau}-\bar\epsilon), \qquad \hat\sigma^2=\frac1N\sum_{t=1}^N(\epsilon_t-\bar\epsilon)^2>0.

An exact finite-sample partition is

\begin{aligned} \hat C_{\mathrm{same}}(\tau) &=\frac{1}{M\hat\sigma^2}\sum_{t=1}^{M}z_t\mathbf{1}_{\{I_t=I_{t+\tau}\}},\\ \hat C_{\mathrm{other}}(\tau) &=\frac{1}{M\hat\sigma^2}\sum_{t=1}^{M}z_t\mathbf{1}_{\{I_t\ne I_{t+\tau}\}},\\ \hat C(\tau)&=\hat C_{\mathrm{same}}(\tau)+\hat C_{\mathrm{other}}(\tau). \end{aligned}

The denominator uses all lag pairs in both components. Do not recenter or renormalize each identity subset and then add its correlations: that changes the statistic. This estimator also differs from conventions dividing the lag-product sum by

N

At lag one there are eleven overlapping pairs: $(1,2),(2,3),\ldots,(11,12)$ . Pair products in order are

+1,\ +1,\ -1,\ +1,\ +1,\ -1,\ +1,\ +1,\ -1,\ +1,\ +1.

The identity changes only at $(3,4)$ and $(9,10)$ , both with product $-1$ . The middle sign reversal, $(6,7)$ , belongs to B on both sides.

Pair class	Pair count	Product sum	Contribution using all 11 pairs
Same investor	9	$7$	$7/11\approx0.636$
Different investors	2	$-2$	$-2/11\approx-0.182$
All pairs	11	$5$	$5/11\approx0.455$

Eleven fictional lag-one pairs contain nine same-investor pairs with product sum seven and two different-investor pairs with product sum minus two. Contributions are seven elevenths and minus two elevenths, summing to five elevenths.

Original fictional lag-one decomposition. Every component uses the same eleven-pair denominator; bars are dimensionless contributions, not conditional probabilities.

Here $\hat C_{\mathrm{same}}/\hat C=7/5=1.4$ . A contribution can exceed the total because another contribution offsets it. “140% same” is an attribution ratio, not a probability; if the total is zero this ratio is undefined, and near zero it is unstable.

The conditional product average among same-investor pairs is $7/9$ , not $7/11$ . Its pair-occurrence weight is $9/11$ , giving $(9/11)(7/9)=7/11$ . The different-investor term is $(2/11)(-1)=-2/11$ . Omitting these weights overrepresents rare pair types.

Nonzero means need another term

Tóth, Palit, Lillo and Farmer, §3, retain both sign dependence and activity dependence. In stationary population notation, let

p_i=\mathbb P(I_t=i)

p_{ij}(\tau)=\mathbb P(I_t=i,I_{t+\tau}=j)

and

\mu_i=\mathbb E[\epsilon_t\mid I_t=i]

. Define

k_{ij}(\tau)=\mathbb E[\epsilon_t\epsilon_{t+\tau}\mid I_t=i,I_{t+\tau}=j]-\mu_i\mu_j.

The covariance decomposition is

\Gamma(\tau)=\sum_{i,j}p_{ij}(\tau)k_{ij}(\tau) +\sum_{i,j}\bigl(p_{ij}(\tau)-p_ip_j\bigr)\mu_i\mu_j.

Split both sums into

i=j

and

i\ne j

; divide by global sign variance for normalized contributions. The second sum is the activity/mean correction. Zero global mean alone does not remove it. Also,

k_{ij}

is not a Pearson correlation; its centering uses agent means, which need not equal means conditional on that particular pair occurrence.

Our finite example avoids this complication because every $\mu_i$ is zero. Its endpoint convention is explicit; the stationary identity should not be substituted into a short sample without treating boundary counts consistently.

What the identity does—and does not—identify

Same-investor persistence is consistent with splitting a large intended trade into child orders. Autocorrelated information received by one investor can produce it too. Positive cross-investor dependence may reflect shared news rather than imitation. The paper discusses these identification limits in §§4.1 and 4.4.

A broker code is not an investor code: a broker pools clients, and an investor may use several brokers. Consequently, same-broker pairs can contain different investors; cross-broker pairs can contain one investor's split order. §5's brokerage mapping makes this observation problem explicit.

For a reproducible study, record the identity level, event unit, lag, sign-classification method, centering, pair denominator and treatment of session boundaries. This twelve-event illustration establishes arithmetic, not a population estimate or statistical significance. Overlapping pair observations should not be treated as independent evidence.

Order Flow and Adverse Selection develops aggressor-sign estimation and explains why persistent flow does not establish informed trading. Its preview is public; the full lesson requires membership.

Sources

Tóth, Palit, Lillo and Farmer, Why is equity order flow so persistent?, arXiv v2 (2014): §3, equations (1)–(7), for identity and activity decompositions; §§4.1 and 4.4 for identification limits; §5 for brokerage maps. The worked tape and figure above are original fictional examples.
Monash University publication record, Journal of Economic Dynamics and Control 51 (2015), pp. 218–239: publication record for the research.

Start with the pairs

Nonzero means need another term

What the identity does—and does not—identify

Sources

Related terms

Broker

Order flow