Madhavan–Richardson–Roomans model

For a quant estimating spread decomposition, this matters immediately: the coefficient on today's trade sign combines information and liquidity costs. Reading that coefficient as a pure adverse-selection cost overstates what the regression identifies.

Prices respond to the surprise in the trade

Here $\theta\geq0$ measures the value revision per unit of trade-direction surprise, and $\varepsilon_t$ is a public-information innovation. Transaction prices also compensate liquidity providers:

In the manuscript's Markov specification, $\mathbb{E}[x_t\mid x_{t-1}]=\rho x_{t-1}$, where $\rho$ is the first-order autocorrelation of trade direction. Substituting the price equation at consecutive observations yields

The zero indicator is an actual transaction

Under the manuscript's symmetric Markov transition specification, define

The probability of a midpoint transaction following either a buy or a sell is taken to be $\lambda$. After a buy, the probabilities of buy, midpoint and sell are therefore $\gamma$, $\lambda$ and $1-\gamma-\lambda$. Hence

Here $\gamma$ conditions only on the previous sign; it does not also condition on the current transaction being nonzero. Thus $\gamma$ and $\rho$ are different objects. A midpoint transaction following a buy has surprise $0-\rho$, rather than automatically having zero information content. Arbitrarily inserting zeros during periods with no trades would change the model and its estimation problem.

One buy history, two price changes

with monetary coefficients in USD per share per unit of sign. Under a symmetric two-state Markov law, $\rho=0.5$ corresponds to a 75% chance of repeating the previous sign. Suppose the previous trade was a buy, $x_{t-1}=+1$.

What estimation must identify

Write the price regression as $\Delta p_t=b_0x_t+b_1x_{t-1}+u_t$. For the toy,

Once $\rho$ is identified from trade-direction behaviour, the structural coefficients can be recovered, provided $\rho\ne1$:

Here that gives $\theta=0.01/0.5=0.02$ and $\phi=0.01$. The two price-regression coefficients alone are insufficient: the same $b_0,b_1$ with $\rho=0$ instead imply $\theta=0.01$ and $\phi=0.02$. As $\rho$ approaches one, division by $1-\rho$ makes the recovery sensitive to estimation error.

The authors use generalised method of moments to combine the price-regression restrictions with moments identifying trade-sign autocorrelation and the crossing probability. The differenced rounding disturbance $\xi_t-\xi_{t-1}$ also creates serial dependence even when rounding-error levels are independent; ordinary independent-error standard errors are inappropriate. A systematic tendency to round buys upward and sells downward can be absorbed into $\phi$, further limiting a literal processing-cost interpretation.

Persistent buy signs do not prove informed trading. Order splitting, reporting conventions and other trading mechanisms can produce persistence; $\rho$ describes that predictability. Interpreting $\theta$ as an information component relies on the model's value-update and orthogonality assumptions, rather than establishing a causal motive for each trade.

Sources

• Madhavan, A., Richardson, M. and Roomans, M. (1997). Why Do Security Prices Change? A Transaction-Level Analysis of NYSE Stocks. The Review of Financial Studies, 10(4), 1035–1064. Publication details and abstract.
• The authors' November 1996 manuscript, FIN-96-034, especially §§2.1–2.2, printed pp.4–9, equations (1)–(3) and estimation moments; §3.1 on trade signing and quote timing. This is the accessible primary source for the detailed model discussed here.