Market Microstructure & Econometrics

Madhavan-Richardson-Roompack (1997) MRR Spread Decomposition Lab

Deconstruct price formation, trade surprises, and bid-ask spreads into adverse selection, transitory order processing friction, and exogenous public news shocks using Ananth Madhavan, Matthew Richardson, and Mark Roompack's (1997) structural framework.

Paper: Review of Financial Studies 10(4) • Mechanism: AR(1) Trade Surprises & Public Shocks • Estimation: Bivariate OLS / GMM

Preset Market Scenarios:

The Structural Model of Security Price Determination

In Madhavan, Richardson, and Roompack (1997), security prices evolve in discrete trade time $t$. Let $x_t \in \{-1, +1\}$ denote the signed trade indicator ($+1$ for buyer-initiated, $-1$ for seller-initiated). Order flow follows an AR(1) Markov chain with persistence parameter $\rho$:

$$\mathbb{E}[x_t \mid x_{t-1}] = \rho x_{t-1}, \quad \text{where } \Pr(x_t = x_{t-1}) = \frac{1 + \rho}{2}$$ $$\xi_t = x_t - \rho x_{t-1} \quad \text{(Trade Order Flow Surprise)}$$

Because market participants anticipate persistent order flow, only the unexpected order flow surprise $\xi_t$ imparts new private information into fundamental value $V_t$.

Fundamental unobserved value $V_t$ and transaction price $P_t$ satisfy:

$$V_t = V_{t-1} + \alpha(x_t - \rho x_{t-1}) + u_t$$ $$P_t = V_t + \phi x_t + \epsilon_t$$ $$\Delta P_t = (\phi + \alpha)x_t - (\phi + \alpha\rho)x_{t-1} + u_t + \Delta\epsilon_t$$

Here, $\alpha$ is the adverse selection belief revision, $\phi$ is the transitory order processing half-spread, $u_t \sim \mathcal{N}(0, \sigma_u^2)$ is the public news shock, and $\epsilon_t$ is transitory noise.

Simulation Parameters

Initial Price ($P_0$):

Order Processing Half-Spread ($\phi$): $0.020

Adverse Selection / Asymmetry ($\alpha$): $0.010

Trade Autocorrelation ($\rho$): 0.40

Public News Volatility ($\sigma_u$): $0.040

Transitory Noise Volatility ($\sigma_\epsilon$): $0.005

Trade Sequence Length ($N$): 120

Implied Full Spread ($S$)

$0.060

Half: $0.030

Adverse Selection $\alpha / (\alpha + \phi)$

Order Processing $\phi / (\alpha + \phi)$

Total Return Var

$0.0028

$\mathrm{Var}(\Delta P)$

Estimated $\hat{\alpha}$ (OLS)

$0.012

True: $\alpha$

Estimated $\hat{\phi}$ (OLS)

$0.019

True: $\phi$

Estimated $\hat{\rho}$ (OLS)

0.38

True: $\rho$

Estimated $\hat{S}$ (OLS)

$0.062

$2(\hat{\alpha} + \hat{\phi})$

Price Change Return Variance Decomposition

Private Information (Trade Surprise) 12.5%

$\alpha^2(1 - \rho^2)$

Transitory Friction & Covariance 28.4%

$2(1-\rho)[\phi^2 + \alpha\phi]$

Public Information & Noise 59.1%

$\sigma_u^2 + 2\sigma_\epsilon^2$

Figure 1: Sequential transactions $P_t$ (solid cyan) vs. unobserved fundamental value $V_t$ (dashed amber). Circular markers denote buyer-initiated trades ($+1$, green) and seller-initiated trades ($-1$, red). Notice how persistent trades build up inventory friction, but only unexpected sign reversals or volume spikes alter the fundamental value trajectory.

Figure 2: True structural parameters $(\alpha, \phi, \rho, s, S)$ compared against empirical econometric estimates obtained via bivariate OLS regression on $\Delta P_t = \beta_1 x_t + \beta_2 x_{t-1} + e_t$. As trade sample size $N$ increases, OLS estimates converge asymptotically to true structural values.

Figure 3: Scatter distribution of realized trade surprises $\xi_t = x_t - \rho x_{t-1}$ versus subsequent price changes $\Delta P_t$. The dashed green line illustrates the theoretical information sensitivity slope $\alpha$. Dispersions around the slope reflect transitory spread bouncing $\phi(x_t - x_{t-1})$ and public news shocks $u_t$.

Sensitivity Grid 1: Trade Surprise Price Impact

Expected permanent price impact $\alpha(1 - \rho)$ across varying levels of information asymmetry $\alpha$ and order flow autocorrelation $\rho$.

Sensitivity Grid 2: Adverse Selection Share (%)

Fraction of total quoted spread attributable to asymmetric information $\frac{\alpha}{\alpha + \phi} \times 100\%$ across order processing costs $\phi$ and asymmetry $\alpha$.

Institutional Microstructure & Econometric Derivation

1. Why Trade Surprises Drive Permanent Value

Classical Roll (1984) models assume trades are independently distributed noise ($P(x_t = x_{t-1}) = 0.5$). However, institutional reality exhibits heavy trade clustering and algorithmic order slicing (e.g. TWAP/VWAP executions), producing positive serial correlation ($\rho > 0$).

Because a market maker expects buyer-initiated flow to follow another buy, an anticipated buy contains zero marginal surprise. In MRR (1997), the rational Bayesian market maker only adjusts expectations of true value $V_t$ by the innovation:

$$\xi_t = x_t - \mathbb{E}[x_t \mid x_{t-1}] = x_t - \rho x_{t-1}$$

When $\rho > 0$, an unexpected sell order after consecutive buys generates a surprise of $-1 - \rho(1) = -(1 + \rho)$, triggering an outsized permanent valuation markdown of $-\alpha(1 + \rho)$.

2. The GMM / OLS Identification Strategy

The empirical brilliance of the MRR specification is that both $\alpha$ and $\phi$ can be estimated without observing unobserved fundamental values $V_t$. Regressing observed price changes on contemporaneous and lagged trade signs:

$$\Delta P_t = \beta_1 x_t + \beta_2 x_{t-1} + e_t$$

Matching coefficients to theoretical moments yields:

$\beta_1 = \phi + \alpha$ (immediate gross price impact)
$\beta_2 = -(\phi + \alpha\rho)$ (lagged mean-reverting bounce)
$\beta_1 + \beta_2 = \alpha(1 - \rho) \implies \hat{\alpha} = \frac{\hat{\beta}_1 + \hat{\beta}_2}{1 - \hat{\rho}}$
$\hat{\phi} = \hat{\beta}_1 - \hat{\alpha}$ (transitory order processing margin)

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Frequently Asked Questions

The Madhavan, Richardson, and Roompack (1997) model is a seminal structural market microstructure model published in the Review of Financial Studies. It formulates security price changes as a function of order flow surprises, transitory order processing friction, and exogenous public news shocks.

MRR models trade order flow as an AR(1) Markov process with serial correlation rho. Because market participants anticipate continuing order flow, only the unexpected surprise component xi_t = x_t - rho * x_{t-1} carries new private information and causes a permanent revision in fundamental unobserved value.

While Huang and Stoll (1997) decompose spreads into three components (adverse selection, inventory holding risk, and order processing), MRR explicitly separates the permanent belief revision driven by trade surprises from transitory dealer compensation and directly models public information innovations.

Public information variance sigma_u^2 represents market-wide macroeconomic or firm-specific news arrivals that alter the security's fundamental equilibrium price independently of trading activity, allowing researchers to measure what fraction of return volatility is caused by trading versus public disclosure.

Algorithmic market makers use estimated alpha to quantify adverse selection risk against informed traders and set quote skews, phi to calculate required minimum profit margins above exchange fees, and rho to adjust quote depth following sequential buy or sell strings.