Bid-ask bounce

A Gaussian Roll model: negatively autocorrelated errors are anti-redundant evidence.

In Roll's model of the effective spread, trades bounce between bid and ask. So observed prices carry negatively autocorrelated errors around the efficient price $\mu$. Here is a Gaussian stand-in: $y_t = \mu + e_t$ with AR(1) errors $e_t = a\,e_{t-1} + \sqrt{1-a^2}\,\eta_t$ of unit marginal variance. Bounce is $a < 0$. Stale, smoothed quotes are $a > 0$. The case $a = 0$ is iid.

Correlated errors couple the observations to each other. But for estimating a single $\mu$ the coupling collapses into a re-weighting of the masses. The GLS weights are the row sums of the error precision matrix. For AR(1) they come out in closed form. The first and last trade get $\tfrac{1}{1+a}$ and every interior trade gets $\tfrac{1-a}{1+a}$. Slide $a$ negative and watch every body on the rail grow. Bouncing errors cancel. The evidence is worth more than its naive iid weight and the masses more than add. Slide $a$ positive and the interior trades shrivel to near nothing. That is redundant evidence. The particle settles at the GLS estimate. The grey line is the naive sample mean, which ignores the bounce. Drag any trade to see the two estimators disagree.

autocorrelation $a$ trades $n$

GLS estimate naive mean total mass vs naive an interior trade is worth iid trades

The same trades, weighed twice. With $a=-0.6$ the total mass is more than triple the naive count $n$. The posterior is far tighter than the iid analysis admits. With $a=0.9$ twenty-four trades carry the mass of about two. Roll's estimator recovers the spread from this same signature, the negative first-order autocovariance of price changes: $s = 2\sqrt{-\mathrm{cov}(\Delta p_t, \Delta p_{t-1})}$. Roll published it in 1984 in the Journal of Finance. See the entry in the reading list, and the dictionary for where correlated residuals sit in the mechanics.

The usual explanation the GLS derivation, in full

Model the trades as $y = \mu\mathbf{1} + e$ with $e \sim \mathcal{N}(0, \Sigma)$, where $\Sigma$ is the AR(1) error covariance. The generalised least squares estimate of the level minimises the Mahalanobis distance:

$$ \hat\mu = \arg\min_{\mu}\, (y - \mu\mathbf{1})^{\top}\,\Sigma^{-1}\,(y - \mu\mathbf{1}). $$

Differentiate with respect to $\mu$ and set to zero:

$$ -2\,\mathbf{1}^{\top}\Sigma^{-1}(y - \mu\mathbf{1}) = 0. $$

Solve for $\mu$. This is the GLS estimator, an inverse-variance weighted mean:

$$ \hat\mu = \frac{\mathbf{1}^{\top}\Sigma^{-1} y}{\mathbf{1}^{\top}\Sigma^{-1}\mathbf{1}}. $$

The weights are the row sums of the precision $\Sigma^{-1}$. For an AR(1) with correlation $a$ that precision is tridiagonal, shown here for four trades:

$$ \Sigma^{-1} = \frac{1}{1-a^2} \begin{pmatrix} 1 & -a & 0 & 0 \\ -a & 1+a^2 & -a & 0 \\ 0 & -a & 1+a^2 & -a \\ 0 & 0 & -a & 1 \end{pmatrix}. $$

Sum each row. An interior row gives $\tfrac{1 - 2a + a^2}{1-a^2} = \tfrac{(1-a)^2}{(1-a)(1+a)} = \tfrac{1-a}{1+a}$, and an end row gives $\tfrac{1-a}{1-a^2} = \tfrac{1}{1+a}$. These are the weights on the rail:

$$ w_{\text{end}} = \frac{1}{1+a}, \qquad w_{\text{int}} = \frac{1-a}{1+a}. $$

The physics proof read off the springs

This is the fusion rule with the masses re-weighted. Each trade is a spring to the rail, and the estimate is their centre of mass.

$$ \hat\mu = \frac{\sum_t w_t\, y_t}{\sum_t w_t}, \qquad w_t = \big(\Sigma^{-1}\mathbf{1}\big)_t. $$

Correlated errors do not disturb that picture. They only reset each mass to the trade's row sum in the precision, which is how much independent evidence it carries once its neighbours are counted. Bouncing errors are negatively correlated, so a trade and its neighbour cancel part of their shared noise and both masses grow. Stale quotes repeat, so the interior masses shrink toward nothing. The particle finds the centre of mass either way. Weigh the trades right and the average weighs itself.