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A new spread estimator

Abstract

A new estimator of bid-ask spreads is presented. When the trade direction is known, any estimate of the spread is associated with a unique series of conjectural mid-prices derived by adjusting the observed transaction price by half the estimated spread. It is shown that the covariance of successive conjectural mid-price returns is maximised (or least negative) when the estimated spread is equal to the true spread. A search procedure to maximise this covariance may therefore be used to estimate the true spread. The performance of this estimator under various conditions is examined both theoretically and with Monte Carlo simulations. The simulations confirm the theoretical results. The performance of the estimator is good.

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Fig. 1

Notes

  1. A referee suggests that we also run simulations calibrated to the stock market. If we set the ratio of the spread to the standard deviation of daily returns to one (i.e. the spread is much larger than in Table 1), as is typical of the stock market, the relative performance of estimators is similar to that shown in Table 1. The results of these simulations can be found in the online Appendix.

  2. We have also done some simulations with a higher value of. For \(\kappa =0.85\), the biases are larger in every case than those shown in Table 2, but the relative performance of the different estimators is similar. The results of these simulations can be found in the Online Appendix.

  3. We have also carried out some simulations with lower values of \(\eta\) The results are intermediate between those of Tables 2 (\(\eta =0\)) and 3 (\(\eta =0.5\)). The results of these simulations can be found in the Online Appendix.

  4. One referee points out that by introducing lagged feedback trading, trade direction indicators will also exhibit autocorrelation. Furthermore, as the referee notes, the new estimator is not influenced by autocorrelated trade direction indicators. Bleaney and Li (2015) show that the HS and the CS estimators are not influenced by trade direction indicators, and the Roll estimator is influenced by them in the tick-by-tick case.

References

  • Bleaney M, Li Z (2015) The performance of bid-ask spread estimators under less than ideal conditions. Stud Econ Financ. Forthcoming

  • Choi JY, Salandro D, Shastri K (1988) On the estimation of bid-ask spreads: theory and evidence. J Financ Quant Anal 23(2):219–230

    Article  Google Scholar 

  • Corwin SA, Schultz P (2012) A simple way to estimate bid-ask spreads from daily high and low prices. J Finance 67(2):719–760

    Article  Google Scholar 

  • Daníelsson J, Love R (2006) Feedback trading. Int J Financ Econ 11(1):35–53

    Article  Google Scholar 

  • De Long JB, Shleifer A, Summers LH, Waldmann RJ (1990) Positive feedback investment strategies and destabilizing rational speculation. J Finance 45(2):379–395

    Article  Google Scholar 

  • Dean WG, Faff RW (1990) Evidence of feedback trading with Markov switching regimes. Rev Quant Financ Account 30(2):133–151

    Article  Google Scholar 

  • Evans MDD, Lyons RK (2002) Order flow and exchange rate dynamics. J Polit Econ 110(1):170–180

    Article  Google Scholar 

  • George TJ, Kaul G, Nimalendran M (1991) Estimation of the bid-ask spread and its components: a new approach. Rev Financ Stud 4(4):623–656

    Article  Google Scholar 

  • Glosten L, Milgrom P (1985) Bid ask and transaction prices in a specialist market with heterogeneously informed trades. J Financ Econ 14(1):71–100

    Article  Google Scholar 

  • Glosten LR, Harris LE (1988) Estimating the components of the bid/ask spread. J Financ Econ 21(1):123–142

    Article  Google Scholar 

  • Goyenko RY, Holden CW, Trzcinka CA (2009) Do liquidity measures measure liquidity? J Financ Econ 92(2):153–181

    Article  Google Scholar 

  • Hasbrouck J (1991) Measuring the information content of stock trades. J Finance 46(1):179–207

    Article  Google Scholar 

  • Hasbrouck J (2004) Liquidity in the futures pits: Inferring market dynamics from incomplete data. J Financ Quant Anal 39(2):305–326

    Article  Google Scholar 

  • Hasbrouck J (2009) Trading costs and returns for U.S. equities: estimating effective costs from daily data. J Finance 64(3):1445–1477

    Article  Google Scholar 

  • Holden CW (2009) New low-frequency spread measures. J Financ Mark 12(4):778–813

    Article  Google Scholar 

  • Huang RD, Stoll HR (1997) The components of the bid-ask spread: a general approach. Rev Financ Stud 10(4):995–1034

    Article  Google Scholar 

  • Kyle AS (1985) Continuous auctions and insider trading. Econometrica 53(6):1315–1335

    Article  Google Scholar 

  • Laux PA, Senchack AJ Jr (1994) Estimating the bid-ask spread in a heteroskedastic market: the case of foreign currency futures. Rev Quant Financ Account 4(3):219–237

    Article  Google Scholar 

  • Lesmond DA, Ogden JP, Trzcinka CA (1999) A new estimate of transaction costs. Rev Financ Stud 12(5):1113–1141

    Article  Google Scholar 

  • Lin CH, Hsu H, Chiang CY (2005) Trading patterns and performance of trader types in Taiwan futures market. Rev Pac Basin Financ Mark Polic 8(2):217–234

    Article  Google Scholar 

  • Nofsinger JR, Sias RW (1999) Herding and feedback trading by institutional and individual investors. J Finance 54(6):2263–2295

    Article  Google Scholar 

  • Roll R (1984) A simple implicit measure of the effective bid-ask spread in an efficient market. J Finance 39(4):1127–1139

    Article  Google Scholar 

  • Stoll HR (1989) Inferring the components of the bid-ask spread: theory and empirical tests. J Finance 44(1):115–134

    Article  Google Scholar 

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Correspondence to Michael Bleaney.

Electronic supplementary material

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Supplementary material 1 (pdf 159 KB)

Appendices

Appendix 1: Proof of Proposition 1

Definition 3

Let A be a set of all conjectures of the true spread \(A=\{\widetilde{SP}_1,\widetilde{SP}_2,\cdots ,\widetilde{SP}_n\}\)

Definition 4

Let B be a set of covariances of two adjacent conjectural mid-price returns obtained according to the conjecture of the true spread \(B=\{Cov_1,Cov_2,\cdots ,Cov_n\}\), where \(Cov_i=Cov[\widetilde{M}(\widetilde{SP_i})_t,\widetilde{M}(\widetilde{SP_i})_{t-1}]\).

One can find that sets A and B are one to one mapping.

Proposition 1

If there is no feedback trading, and no inventory control or asymmetric information components of the spread, then the spread and its conjecture, and thus the conjectural error, are serially independent or are fixed. If a conjecture of the spread \(\widetilde{SP}_i \in A\) corresponds to \(Cov_i=max(B)\), it equals the true spread i.e. \(\widetilde{SP}_i=SP\).

Proof

The covariance of two adjacent conjectures of mid-price returns is:

$$\begin{aligned}&Cov(\Delta \widetilde{M}_t,\Delta \widetilde{M}_{t-1}) \\&\quad =E\{[\Delta \widetilde{M}_t-E(\Delta \widetilde{M}_t)][\Delta \widetilde{M}_{t-1}-E(\Delta \widetilde{M}_{t-1})]\} \end{aligned}$$
(42)

Assume the conjectural errors are fixed, expectations of errors are given by:

$$\begin{aligned} E(\Omega _t)=E(\Omega _{t-1})=E(\Omega _{t-2})= \Omega \end{aligned}$$
(43)

and the exceptions of the multiplication of the conjectural errors are given by:

$$\begin{aligned} E(\Omega _t\Omega _{t-1})= E(\Omega _t\Omega _{t-2})= E(\Omega _{t-1}\Omega _{t-2})=\Omega ^2 \end{aligned}$$
(44)

Furthermore, assume the expectation of the conjectural mid-prices is zero. Thus, the covariance can be written as:

$$\begin{aligned}&Cov(\Delta \widetilde{M}_t,\Delta \widetilde{M}_{t-1}) \\&\quad =E[\Delta \widetilde{M}_t\cdot \Delta \widetilde{M}_{t-1}] \\&\quad =E\left[ \left( \Delta {M}_t+\frac{1}{2}\Omega {BS}_t-\frac{1}{2}\Omega {BS}_{t-1}\right) \left( \Delta {M}_{t-1}+\frac{1}{2}\Omega {BS}_{t-1}-\frac{1}{2}\Omega {BS}_{t-2}\right) \right] \\&\quad = E\left[ \left( \Delta M_t + \frac{1}{2} \Omega {BS}_t - \frac{1}{2} \Omega {BS}_{t - 1}\right) \Delta M_{t - 1}\right. \\&\quad \quad + \frac{1}{2} \Omega {BS}_{t - 1} \left( \Delta M_t + \frac{1}{2} \Omega {BS}_t - \frac{1}{2} \Omega {BS}_{t - 1}\right) \\&\quad \quad \left. - \frac{1}{2} \Omega {BS}_{t - 2} \left( \Delta M_t + \frac{1}{2} \Omega {BS}_t - \frac{1}{2} \Omega {BS}_{t - 1}\right) \right] \\&\quad = E\left[ \left( \Delta M_t \Delta M_{t - 1} + \frac{1}{2} \Omega {BS}_t \Delta M_{t - 1} - \frac{1}{2} \Omega {BS}_{t - 1} \Delta M_{t - 1}\right) \right. \\&\quad \quad +\left( \Delta M_t \frac{1}{2} \Omega {BS}_{t - 1} + \frac{1}{2} \Omega {BS}_t \frac{1}{2} \Omega {BS}_{t - 1} - \frac{1}{2} \Omega {BS}_{t - 1} \frac{1}{2} \Omega {BS}_{t - 1}\right) \\&\quad \quad \left. - \left( \Delta M_t \frac{1}{2} \Omega {BS}_{t - 2} + \frac{1}{2} \Omega {BS}_t \frac{1}{2} \Omega {BS}_{t - 2} - \frac{1}{2} \Omega {BS}_{t - 1} \frac{1}{2} \Omega {BS}_{t - 2} \right) \right] \end{aligned}$$
(45)

Re-arrange the equation further, we have:

$$\begin{aligned}&= E\left[ \left( \Delta M_t \Delta M_{t - 1} + \frac{1}{2} \Omega {BS}_t \Delta M_{t - 1} - \frac{1}{2} \Omega {BS}_{t - 1} \Delta M_{t - 1}\right) \right. \\&\quad +\left( \frac{1}{2}\Delta M_t \Omega {BS}_{t - 1} + \frac{1}{4} \Omega ^2 {BS}_t {BS}_{t - 1} - \frac{1}{4} \Omega ^2 { {BS}_{t - 1} }^2\right) \\&\quad \left. - \left( \Delta M_t \frac{1}{2} \Omega {BS}_{t - 2} + \frac{1}{4} \Omega ^2 {BS}_t {BS}_{t - 2} - \frac{1}{4} \Omega ^2 {BS}_{t - 1} {BS}_{t - 2}\right) \right] \\&= E(\Delta M_t \Delta M_{t - 1}) \\&\quad + E\left( \frac{1}{2} \Omega {BS}_t \Delta M_{t - 1} - \frac{1}{2} \Omega {BS}_{t - 1} \Delta M_{t - 1} + \frac{1}{2}\Delta M_t \Omega {BS}_{t - 1} - \Delta M_t \frac{1}{2} \Omega {BS}_{t - 2}\right. \\&\quad \left. + \frac{1}{4} \Omega ^2 {BS}_t {BS}_{t - 1} - \frac{1}{4} \Omega ^2 { {BS}_{t - 1} }^2 - \frac{1}{4} \Omega ^2 {BS}_t {BS}_{t - 2} + \frac{1}{4} \Omega ^2 {BS}_{t - 1} {BS}_{t - 2}\right) \end{aligned}$$
(46)

Because the variable \(BS\) is a binary variable (1 or \(-1\)), then:

$$\begin{aligned} E(BS_{t-1}^2)=1 \end{aligned}$$
(47)

Furthermore, because we assume there is no feedback trading, then:

$$\begin{aligned}&E({BS}_{t} \Delta M_{t - 1})=0 \\&E({BS}_{t - 1} \Delta M_{t - 1})=0 \end{aligned}$$
(48)

Because we assume there is no IC&AS components, then:

$$\begin{aligned}&E({BS}_{t-1} \Delta M_{t})=0 \\&E({BS}_{t - 2}\cdot \Delta M_{t})=0 \end{aligned}$$
(49)

Equation (46) can be written as:

$$\begin{aligned}&Cov(\Delta \widetilde{M}_t,\Delta \widetilde{M}_{t-1}) \\&\quad = E(\Delta M_t \Delta M_{t - 1}) \\&\quad \quad + E\left( \frac{1}{4} \Omega ^2 {BS}_t {BS}_{t - 1} - \frac{1}{4} \Omega ^2 - \frac{1}{4} \Omega ^2 {BS}_t {BS}_{t - 2} + \frac{1}{4} \Omega ^2 {BS}_{t - 1} {BS}_{t - 2}\right) \\&\quad =Cov(\Delta {M}_t\cdot \Delta {M}_{t-1})+\frac{1}{4}\Omega ^2\cdot [2E(BS_t \cdot BS_{t-1})-E(BS_t \cdot BS_{t-2})-1] \end{aligned}$$
(50)

The right hand side of the equation is a quadratic polynomial of the expectation of the error of the conjecture. For a given series, the first term on the right hand side is a constant. It is straightforward that when the expectation of the error is zero (i.e. \(E(\Omega )=0\)), the second term is zero. Furthermore, when \(E(\Omega )=0\), there is a global extreme for the right hand side polynomial, symmetrically, the left hand side of the equation \(Cov(\Delta \widetilde{M}_t,\Delta \widetilde{M}_{t-1})\) is also at the extreme value:

$$\begin{aligned} \mathop {{ arg}\; { max}} \limits _{\Omega } { Cov}(\Delta \widetilde{M}_t,\Delta \widetilde{M}_{t-1})=0 \end{aligned}$$
(51)

When the conjectural error is zero, the conjectural spread is the true spread:

$$\begin{aligned} \Omega =SP-\widetilde{SP}_i=0 \end{aligned}$$
(52)

Therefore the conjectural spread which maximises the covariance equals the true spread.

$$\begin{aligned} \mathop {{ arg} \; { max}} \limits _{\widetilde{SP}_i \in A} { Cov}(\Delta \widetilde{M}_t,\Delta \widetilde{M}_{t-1})=SP \end{aligned}$$
(53)

Q.E.D.□

Appendix 2: Simplify Equations (16) and (27)

This section shows the detail of the simplification of Eqs. (16) and (27). Feedback trading, inventory holding costs and asymmetric information costs are considered together.

Considering the inventory control and asymmetric information components of the spread, the true mid-price returns are given by:

$$\begin{aligned} \Delta {M}_t= \frac{1}{2}\varrho {SP} {BS}_{t-1}+\epsilon _t \end{aligned}$$
(54)

The covariance of the two adjacent conjectural mid-price returns is given by,

$$\begin{aligned}&{ Cov}(\Delta \widetilde{M}_t,\Delta \widetilde{M}_{t-1}) \\&\quad =E\{[\Delta \widetilde{M}_t-E(\Delta \widetilde{M}_t)][\Delta \widetilde{M}_{t-1}-E(\Delta \widetilde{M}_{t-1})]\} \end{aligned}$$
(55)

Assume the expectation of conjectural mid-price returns to be zero. Thus the equation above can be written as:

$$\begin{aligned}&{ Cov}(\Delta \widetilde{M}_t,\Delta \widetilde{M}_{t-1}) \\&\quad =E[\Delta \widetilde{M}_t\cdot \Delta \widetilde{M}_{t-1}] \\&\quad =E\left\{ \left[ \frac{1}{2}(\varrho -1)\Omega {BS}_{t-1}+\frac{1}{2}\Omega {BS}_t+\frac{1}{2}\varrho \widetilde{SP}_{t-1} {BS}_{t-1}+\epsilon _t \right] \right. \\&\quad \quad \left. \cdot \left[ \frac{1}{2}(\varrho -1)\Omega {BS}_{t-2}+\frac{1}{2}\Omega {BS}_{t-1}+\frac{1}{2}\varrho \widetilde{SP}_{t-2} {BS}_{t-2}+\epsilon _{t-1}\right] \right\} \\&\quad =E\left\{ \frac{1}{2}( - 1 + \varrho ) \Omega {BS}_{t - 2} \left[ \frac{1}{2}( - 1 + \varrho ) \Omega {BS}_{t - 1} + \epsilon _t + \frac{1}{2} \Omega {BS}_t + \frac{1}{2}\varrho \widetilde{SP}_{t-1} {BS}_{t - 1} \right] \right. \\&\quad \quad +\,\epsilon _{t - 1} \left[ \frac{1}{2}( - 1 + \varrho ) \Omega {BS}_{t - 1} + \epsilon _t + \frac{1}{2} \Omega {BS}_t + \frac{1}{2}\varrho \widetilde{SP}_{t-1} {BS}_{t - 1} \right] \\&\quad \quad + \frac{1}{2} \Omega {BS}_{t - 1} \left[ \frac{1}{2}( - 1 + \varrho ) \Omega {BS}_{t - 1} + \epsilon _t + \frac{1}{2} \Omega {BS}_t + \frac{1}{2}\varrho \widetilde{SP}_{t-1} {BS}_{t - 1} \right] \\&\quad \quad \left. + \frac{1}{2}\varrho \widetilde{SP}_{t-2} {BS}_{t - 2} \left[ \frac{1}{2}( - 1 + \varrho ) \Omega {BS}_{t - 1} + \epsilon _t + \frac{1}{2} \Omega {BS}_t + \frac{1}{2}\varrho \widetilde{SP}_{t-1} {BS}_{t - 1} \right] \right\} \\&\quad =E\left\{ \left[ \frac{1}{2}\cdot \frac{1}{2}( - 1 + \varrho ) \Omega {BS}_{t - 2} ( - 1 + \varrho ) \Omega {BS}_{t - 1} + \frac{1}{2}( - 1 + \varrho ) \Omega {BS}_{t - 2} \epsilon _t \right. \right. \\&\quad \quad \left. + \frac{1}{2}\cdot \frac{1}{2}( - 1 + \varrho ) \Omega {BS}_{t - 2} \Omega {BS}_t + \frac{1}{2}\cdot \frac{1}{2}( - 1 + \varrho ) \Omega {BS}_{t - 2} \varrho \widetilde{SP}_{t-1} {BS}_{t - 1} \right] \\&\quad \quad + \left[ \frac{1}{2}( - 1 + \varrho ) \Omega {BS}_{t - 1} \epsilon _{t - 1} + \epsilon _t \epsilon _{t - 1} + \frac{1}{2} \Omega {BS}_t \epsilon _{t - 1} + \frac{1}{2}\varrho \widetilde{SP}_{t-1} {BS}_{t - 1} \epsilon _{t - 1} \right] \\&\quad \quad + \left[ \frac{1}{2}\cdot \frac{1}{2} \Omega {BS}_{t - 1} ( - 1 + \varrho ) \Omega {BS}_{t - 1} + \frac{1}{2} \Omega {BS}_{t - 1} \epsilon _t + \frac{1}{2}\cdot \frac{1}{2} \Omega {BS}_{t - 1} \Omega {BS}_t \right. \\&\quad \quad \left. + \frac{1}{2}\cdot \frac{1}{2} \Omega {BS}_{t - 1} \varrho \widetilde{SP}_{t-1} {BS}_{t - 1} \right] + \left[ \frac{1}{2}\cdot \frac{1}{2}\varrho \widetilde{SP}_{t-2} {BS}_{t - 2} ( - 1 + \varrho ) \Omega {BS}_{t - 1} \right. \\&\quad \quad \left. \left. + \frac{1}{2}\varrho \widetilde{SP}_{t-2} {BS}_{t - 2} \epsilon _t + \frac{1}{2}\cdot \frac{1}{2}\varrho \widetilde{SP}_{t-2} {BS}_{t - 2} \Omega {BS}_t + \frac{1}{2}\cdot \frac{1}{2}\varrho \widetilde{SP}_{t-2} {BS}_{t - 2} \varrho \widetilde{SP}_{t-1} {BS}_{t - 1} \right] \right\} \end{aligned}$$
(56)

Re-arrange the equation further, we have:

$$\begin{aligned}&{ Cov}(\Delta \widetilde{M}_t,\Delta \widetilde{M}_{t-1}) \\&\quad = E\left\{ \left[ \frac{1}{4} {( - 1 + \varrho )}^2 \Omega ^2 {BS}_{t - 1} {BS}_{t - 2} + \frac{1}{2}( - 1 + \varrho ) \Omega {BS}_{t - 2} \epsilon _t \right. \right. \\&\quad \quad \left. + \frac{1}{4}( - 1 + \varrho ) \Omega ^2 {BS}_t {BS}_{t - 2} + \frac{1}{4}\varrho ( - 1 + \varrho )\widetilde{SP} {BS}_{t - 1} {BS}_{t - 2} \Omega \right] \\&\quad \quad + \left[ \frac{1}{2}( - 1 + \varrho ) \Omega {BS}_{t - 1} \epsilon _{t - 1} + \epsilon _t \epsilon _{t - 1} + \frac{1}{2} \Omega {BS}_t \epsilon _{t - 1} + \frac{1}{2}\varrho \widetilde{SP} {BS}_{t - 1} \epsilon _{t - 1} \right] \\&\quad \quad + \left[ \frac{1}{4}( - 1 + \varrho ) \Omega ^2 { {BS}_{t - 1} }^2 + \frac{1}{2} \Omega {BS}_{t - 1} \epsilon _t + \frac{1}{4} \Omega ^2 {BS}_t {BS}_{t - 1} \right. \\&\quad \quad \left. + \frac{1}{4}\varrho \widetilde{SP} { {BS}_{t - 1} }^2 \Omega \right] + \left[ \frac{1}{4}\varrho ( - 1 + \varrho ) \Omega {BS}_{t - 1} {BS}_{t - 2} \widetilde{SP} + \frac{1}{2}\varrho \widetilde{SP} {BS}_{t - 2} \epsilon _t \right] \\&\quad \quad \left. + \frac{1}{4}\varrho \widetilde{SP} \Omega {BS}_t {BS}_{t - 2} + \frac{1}{4} \varrho ^2 \widetilde{SP}^2 {BS}_{t - 2} {BS}_{t - 1} \right\} \\&\quad = E( \epsilon _t \epsilon _{t - 1} ) +E\left[ \frac{1}{4} {( - 1 + \varrho )}^2 \Omega ^2 {BS}_{t - 1} {BS}_{t - 2} + \frac{1}{4}( - 1 + \varrho ) \Omega ^2 {BS}_t {BS}_{t - 2} \right. \\&\quad \quad + \frac{1}{4}( - 1 + \varrho ) \Omega ^2 { {BS}_{t - 1} }^2 + \frac{1}{4} \Omega ^2 {BS}_t {BS}_{t - 1} + \frac{1}{2}( - 1 + \varrho )\Omega {BS}_{t - 2} \epsilon _t \\&\quad \quad + \frac{1}{2}( - 1 + \varrho )\Omega {BS}_{t - 1} \epsilon _{t - 1} + \frac{1}{2}\Omega {BS}_t \epsilon _{t - 1} + \frac{1}{2}\Omega {BS}_{t - 1} \epsilon _t \\&\quad \quad + \frac{1}{4}\varrho ( - 1 + \varrho )\widetilde{SP} {BS}_{t - 1} {BS}_{t - 2} \Omega + \frac{1}{4}\varrho \widetilde{SP} { {BS}_{t - 1} }^2 \Omega \\&\quad \quad + \frac{1}{4}\varrho ( - 1 + \varrho ) \Omega {BS}_{t - 1} {BS}_{t - 2} \widetilde{SP} + \frac{1}{4}\varrho \widetilde{SP} \Omega {BS}_t {BS}_{t - 2} \\&\quad \quad \left. + \frac{1}{2}\varrho \widetilde{SP} {BS}_{t - 1} \epsilon _{t - 1} + \frac{1}{2}\varrho \widetilde{SP} {BS}_{t - 2} \epsilon _t + \frac{1}{4} \varrho ^2 \widetilde{SP}^2 {BS}_{t - 2} {BS}_{t - 1} \right] \\&\quad = E( \epsilon _t \epsilon _{t - 1}) +E \left\{ \left[ \frac{1}{4} {( - 1 + \varrho )}^2 {BS}_{t - 1} {BS}_{t - 2} + \frac{1}{4}( - 1 + \varrho ) {BS}_t {BS}_{t - 2} \right. \right. \\&\quad \quad \left. + \frac{1}{4}( - 1 + \varrho ) { {BS}_{t - 1} }^2 + \frac{1}{4} {BS}_t {BS}_{t - 1} \right] \Omega ^2 + \left[ \frac{1}{2}( - 1 + \varrho ) {BS}_{t - 2} \epsilon _t \right. \\&\quad \quad \left. + \frac{1}{2}( - 1 + \varrho ) {BS}_{t - 1} \epsilon _{t - 1} + \frac{1}{2} {BS}_t \epsilon _{t - 1} + \frac{1}{2} {BS}_{t - 1} \epsilon _t \right] \Omega \\&\quad \quad + \left[ \frac{1}{4}( - 1 + \varrho ) {BS}_{t - 1} {BS}_{t - 2} + \frac{1}{4} { {BS}_{t - 1} }^2 \right. \\&\quad \quad \left. + \frac{1}{4}( - 1 + \varrho ) {BS}_{t - 1} {BS}_{t - 2} + \frac{1}{4} {BS}_t {BS}_{t - 2} \right] \varrho \widetilde{SP} \Omega \\&\quad \quad +\left[ \frac{1}{2} {BS}_{t - 1} \epsilon _{t - 1} + \frac{1}{2} {BS}_{t - 2} \epsilon _t\right] \varrho \widetilde{SP} \\&\quad \quad \left. + \frac{1}{4} \varrho ^2 \widetilde{SP}^2 {BS}_{t - 2} {BS}_{t - 1} \right\} \\ \end{aligned}$$
(57)

Let following symbols to represent some parts of the equation above, because we assume order flows do not influence the mid-price shocks following parts are zeros.

$$\begin{aligned} E( {BS}_{t - 2} \epsilon _t)=E( {BS}_{t - 1} \epsilon _t)=0 \end{aligned}$$
(58)

Because the variable BS is a binary variable and with a mean of zero, then:

$$\begin{aligned} E(BS_{t-1}^{2})&= 1\end{aligned}$$
(59)
$$\begin{aligned} \Pi _0&= E( {BS}_{t - 1} {BS}_{t - 2}) \\ \Pi _1&= E\left[ \frac{1}{4} {( - 1 + \varrho )}^2 {BS}_{t - 1} {BS}_{t - 2} + \frac{1}{4}( - 1 + \varrho ) {BS}_t {BS}_{t - 2} + \frac{1}{4}( - 1 + \varrho ) + \frac{1}{4} {BS}_t {BS}_{t - 1} \right] \\&= \frac{1}{4} E\left[ {( - 1 + \varrho )}^2 \Pi _0 + ( - 1 + \varrho ) {BS}_t {BS}_{t - 2} + ( - 1 + \varrho ) + \Pi _0 \right] \\ \Pi _2&= E\left[ \frac{1}{2} ( - 1 + \varrho ) {BS}_{t - 2} \epsilon _t + \frac{1}{2}( - 1 + \varrho ) {BS}_{t - 1} \epsilon _{t - 1} + \frac{1}{2} {BS}_t \epsilon _{t - 1} + \frac{1}{2} {BS}_{t - 1} \epsilon _t \right] \\&= \frac{1}{2} E\left[ ( - 1 + \varrho ) {BS}_{t - 1} \epsilon _{t - 1} + {BS}_t \epsilon _{t - 1} \right] \\ \Pi _3&= E \left[ \frac{1}{4}( - 1 + \varrho ) {BS}_{t - 1} {BS}_{t - 2} + \frac{1}{4} { {BS}_{t - 1} }^2 + \frac{1}{4}( - 1 + \varrho ) {BS}_{t - 1} {BS}_{t - 2} + \frac{1}{4} {BS}_t {BS}_{t - 2} \right] \\&= \frac{1}{4} E[2( - 1 + \varrho ) \Pi _0 + 1 + {BS}_t {BS}_{t - 2} ] \\ \Pi _4&= E\left( \frac{1}{2} {BS}_{t - 1} \epsilon _{t - 1} + \frac{1}{2} {BS}_{t - 2} \epsilon _t \right) \\&= \frac{1}{2} E({BS}_{t - 1} \epsilon _{t - 1}) \end{aligned}$$
(60)

Substitute above equations into Eq. (57), we have:

$$\begin{aligned}&{ Cov}(\Delta \widetilde{M}_t,\Delta \widetilde{M}_{t-1}) \\&\quad = E(\epsilon _t \epsilon _{t - 1} ) + \Pi _1 \Omega ^2 + \Pi _2 \Omega + \Pi _3 \varrho \widetilde{SP} \Omega \\&\quad \quad + \Pi _4 \varrho \widetilde{SP} + \frac{1}{4} \varrho ^2 \widetilde{SP}^2 \Pi _0 \end{aligned}$$
(61)

Equation (61) suggests that when there are IC&AS components and feedback trading, the covariance of the conjectural mid-price returns (\({ Cov}(\Delta \widetilde{M}_t,\Delta \widetilde{M}_{t-1})\)) is no longer a function of conjectural errors (\(\Omega\)) only but also a function of the conjecture of the spread (\(\widetilde{SP}\)). Furthermore, because the true spread (\(SP\)) is certain for a given series, the conjectural errors (\(\Omega\)) is a function of the conjecture of the spread. To investigate the relationship between the true spread and the conjecture of it, we re-arrange the equation to make \(\widetilde{SP}\) be the only variable of the equation. Replace the conjectural error (\(\Omega\)) by the true spread (\(SP\)) and the conjectural spread (\(\widetilde{SP}\)), we have:

$$\begin{aligned}&Cov(\Delta \widetilde{M}_t,\Delta \widetilde{M}_{t-1}) \\&\quad = E(\epsilon _t \epsilon _{t - 1} ) + \Pi _1 {(SP - \widetilde{SP} )}^2 + \Pi _2 \left( SP - \widetilde{SP} \right) \\&\quad \quad + \Pi _3 \left[ \varrho \widetilde{SP} (SP - \widetilde{SP} )\right] + \Pi _4 \varrho \widetilde{SP} + \frac{1}{4} \Pi _0 \varrho ^2 \widetilde{SP}^2 \\&\quad = E(\epsilon _t \epsilon _{t - 1} ) + \Pi _1 {SP}^2 - 2 \Pi _1 \widetilde{SP} SP + \Pi _1 \widetilde{SP}^2 + \Pi _2 SP \\&\quad \quad - \Pi _2 \widetilde{SP} + \Pi _3 \varrho \widetilde{SP} SP - \Pi _3 \varrho \widetilde{SP}^2 + \Pi _4 \varrho \widetilde{SP} + \frac{1}{4} \varrho ^2 \widetilde{SP}^2 \Pi _0 \\&\quad = E(\epsilon _t \epsilon _{t - 1} ) + \Pi _1 {SP}^2 + \Pi _2 SP - 2 \Pi _1 \widetilde{SP} SP - \Pi _2 \widetilde{SP} \\&\quad \quad + \Pi _3 \varrho \widetilde{SP} SP + \Pi _4 \varrho \widetilde{SP} + \Pi _1 \widetilde{SP}^2 + \frac{1}{4} \varrho ^2 \widetilde{SP}^2 \Pi _0 - \Pi _3 \varrho \widetilde{SP}^2 \\&\quad =\left( \Pi _1 + \frac{1}{4}\varrho ^2 \Pi _0 - \varrho \Pi _3\right) \cdot \widetilde{SP}^2 +[ \varrho \Pi _3 \cdot SP + \varrho \Pi _4 - 2 \Pi _1 \cdot SP - \Pi _2 ] \\&\quad \quad \cdot \widetilde{SP} +E(\epsilon _t\cdot \epsilon _{t-1}) + \Pi _1 \cdot {SP}^2 + \Pi _2\cdot SP \end{aligned}$$
(62)

We now discuss possible errors that if we still let \(\left( SP-\Omega \right)\) which maximises the covariance between two adjacent conjectures of mid-price returns \(Cov(\Delta \widetilde{M}_t,\Delta \widetilde{M}_{t-1})\) to be the estimate of the true spread. Thus, the estimate is given by:

$$\begin{aligned} \widehat{SP}= - \left[ \frac{ - 2 \Pi _1 SP - \Pi _2 + \Pi _3 SP\varrho + \Pi _4 \varrho }{2\left( \Pi _1 + \frac{1}{4} \Pi _0 \varrho ^2 - \Pi _3 \varrho \right) } \right] \end{aligned}$$
(63)

where \(\widehat{SP}\) is the value of \(\left( SP-\Omega \right)\) which maximises \(Cov(\Delta \widetilde{M}_t,\Delta \widetilde{M}_{t-1})\), and is the estimate of the true spread. When there are no IC&AS components of the spread (\(\varrho =0\)), no feedback trading (\(\Pi _2=\Pi _4=0\)), and no autocorrelated order flows (\(\Pi _o=0\) and \(\Pi _1=-1\) and \(\Pi _3=1\)), the equation above becomes:

$$\begin{aligned} \widehat{SP}= SP \end{aligned}$$
(64)

and thus,

$$\begin{aligned} \Omega = 0 \end{aligned}$$
(65)

The equations above suggest that under the ideal conditions, Eq. (63) reduces to the simple version of the estimator and in this circumstance, the estimator is unbiased.

Appendix 3: Errors of the estimator

Feedback trading

Assume there is feedback trading and there are no inventory control and asymmetric information components of the spread, thus \(\varrho =0\), then Eq. (61) becomes,

$$\begin{aligned}&Cov(\Delta \widetilde{M}_t,\Delta \widetilde{M}_{t-1}) \\&\quad =E(\Delta \widetilde{M}_t\cdot \Delta \widetilde{M}_{t-1}) \\&\quad =E(\epsilon _t\cdot \epsilon _{t-1})+\Pi _1\cdot \Omega ^2+\Pi _2\cdot \Omega \end{aligned}$$
(66)

where

$$\begin{aligned} \Pi _0&= E( {BS}_{t - 1} \cdot {BS}_{t - 2} ) \\ \Pi _1&= \frac{1}{4} E\left[ \Pi _0 - {BS}_t\cdot {BS}_{t - 2} -1 + \Pi _0 \right] \\ \Pi _2&= \frac{1}{2} E\left[ - {BS}_{t - 1}\cdot \epsilon _{t - 1}+{BS}_t \cdot \epsilon _{t - 1} \right] \end{aligned}$$
(67)

Because the covariance of order flows are usually very small compare to 1, it is safe to take approximation that let \(\Pi _1=-\frac{1}{4}\). From (67) into Eq. (63), it becomes:

$$\begin{aligned}&\widehat{SP} \\&\quad = - \frac{ - 2 \Pi _1 SP - \Pi _2 }{2\left( \Pi _1 \right) } \\&\quad = SP+\frac{\Pi _2 }{2 \Pi _1 } \\&\quad = SP+\frac{ \frac{1}{2} E\left( {BS}_t \epsilon _{t - 1} - {BS}_{t - 1} \epsilon _{t - 1} \right) }{-2\cdot \frac{1}{4}} \\&\quad = SP+E({BS}_{t - 1} \epsilon _{t - 1}) - E( {BS}_t \epsilon _{t - 1} ) \end{aligned}$$
(68)

Inventory control and asymmetric information components

Assume there are inventory control and asymmetric information components of the spread and there is no feedback trading, then Eq. (61) becomes,

$$\begin{aligned}&{ Cov}(\Delta \widetilde{M}_t,\Delta \widetilde{M}_{t-1}) \\&\quad =E(\Delta \widetilde{M}_t\cdot \Delta \widetilde{M}_{t-1}) \\&\quad =E(\epsilon _t\cdot \epsilon _{t-1})+\Pi _1\cdot \Omega ^2+\varrho \Pi _3\cdot (SP-\Omega ) \cdot \Omega \\&\quad \quad +\frac{1}{4}\varrho ^2\Pi _0\cdot [(SP-\Omega )^2] \end{aligned}$$
(69)

where

$$\begin{aligned} \Pi _0&= E( {BS}_{t - 1} \cdot {BS}_{t - 2} ) \\ \Pi _1&= \frac{1}{4} E\left[ {(\varrho -1 )}^2 \Pi _0 + (\varrho -1 ) {BS}_t\cdot {BS}_{t - 2} + (\varrho -1 ) + \Pi _0 \right] \\ \Pi _3&= \frac{1}{4}E[ 1 + 2(\varrho -1 ) \Pi _0 + {BS}_t \cdot {BS}_{t - 2} ] \end{aligned}$$
(70)

And Eq. (63) becomes:

$$\begin{aligned} \widehat{SP} = - \frac{{ - 2 \Pi _1 SP + \Pi _3 SP\varrho }}{{2( \Pi _1 + \frac{1}{4} \Pi _0 \varrho ^2 { - \Pi _3 \varrho }_{} )}} = \frac{{(2 \Pi _1 - \Pi _3 \varrho )}}{{2( \Pi _1 + \frac{1}{4} \Pi _0 \varrho ^2 { - \Pi _3 \varrho }_{} )}} \cdot SP \end{aligned}$$
(71)

The numerator of Eq. (71) can be simplified:

$$\begin{aligned}&(2 \Pi _1 - \Pi _3 \varrho ) \\&\quad = 2 {( - 1 + \varrho )}^2 \Pi _0 + 2( - 1 + \varrho ) {BS}_t {BS}_{t - 2} \\&\quad \quad + 2( - 1 + \varrho ) + 2 \Pi _0 - 2( - 1 + \varrho ) \Pi _0 \varrho - \varrho - {BS}_t {BS}_{t - 2} \varrho \\&\quad = 2 {( - 1 + \varrho )}^2 \Pi _0 + 2 \Pi _0 - 2( - 1 + \varrho ) \Pi _0 \varrho \\&\quad \quad + 2( - 1 + \varrho ) {BS}_t {BS}_{t - 2} - {BS}_t {BS}_{t - 2} \varrho \\&\quad \quad + 2( - 1 + \varrho ) - \varrho \\&\quad = 2 \Pi _0 \varrho ^2 - 4 \Pi _0 \varrho + 4 \Pi _0 - 2 \Pi _0 \varrho ^2 + 2 \Pi _0 \varrho \\&\quad \quad + 2\varrho {BS}_t {BS}_{t - 2} - 2 {BS}_t {BS}_{t - 2} - {BS}_t {BS}_{t - 2} \varrho \\&\quad \quad - 2 + \varrho \\&\quad = 4 \Pi _0 - 2 \Pi _0 \varrho + \varrho {BS}_t {BS}_{t - 2} - 2 {BS}_t {BS}_{t - 2} - 2 + \varrho \\&\quad = 4 {BS}_{t - 1} {BS}_{t - 2} - 2 {BS}_t {BS}_{t - 2} - 2 + ( {BS}_t {BS}_{t - 2} + 1 - 2 {BS}_{t - 1} {BS}_{t - 2} )\varrho \end{aligned}$$
(72)

The dominator of Eq. (71) can be simplified:

$$\begin{aligned}&2( \Pi _1 + \Pi _0 \varrho ^2 { - \Pi _3 \varrho }) \\&\quad = 2([ {( - 1 + \varrho )}^2 \Pi _0 + ( - 1 + \varrho ) {BS}_t {BS}_{t - 2} + ( - 1 + \varrho ) + \Pi _0 ] \\&\quad \quad + \Pi _0 \varrho ^2 { - [2( - 1 + \varrho ) \Pi _0 + 1 + {BS}_t {BS}_{t - 2} ]\varrho }) \\&\quad = 2( \Pi _0 \varrho ^2 - 2 \Pi _0 \varrho + \Pi _0 + ( - 1 + \varrho ) {BS}_t {BS}_{t - 2} - 1 + \varrho + \Pi _0 \\&\quad \quad +\Pi _0 \varrho ^2 + 2(1 - \varrho ) \Pi _0 \varrho - \varrho - {BS}_t {BS}_{t - 2} \varrho ) \\&\quad = 2(2 {BS}_{t - 1} {BS}_{t - 2} - {BS}_t {BS}_{t - 2} - 1) \\&\quad \quad + 2( \Pi _1 + \Pi _0 \varrho ^2 { - \Pi _3 \varrho }) \\&\quad = 2([ {( - 1 + \varrho )}^2 \Pi _0 + ( - 1 + \varrho ) {BS}_t {BS}_{t - 2} + ( - 1 + \varrho ) + \Pi _0 ] \\&\quad \quad + \Pi _0 \varrho ^2 { - [2( - 1 + \varrho ) \Pi _0 + 1 + {BS}_t {BS}_{t - 2} ]\varrho }) \\&\quad = 2( \Pi _0 \varrho ^2 - 2 \Pi _0 \varrho + \Pi _0 + ( - 1 + \varrho ) {BS}_t {BS}_{t - 2} - 1 + \varrho + \Pi _0 \\&\quad \quad + \Pi _0 \varrho ^2 + 2(1 - \varrho ) \Pi _0 \varrho - \varrho - {BS}_t {BS}_{t - 2} \varrho ) \\&\quad = 2(2 {BS}_{t - 1} {BS}_{t - 2} - {BS}_t {BS}_{t - 2} - 1) \end{aligned}$$
(73)

Substitute the results of the simplification back to Eq. (71), the equation becomes,

$$\begin{aligned}&\widehat{SP} \\&\quad =\frac{{(2 \Pi _1 - \Pi _3 \varrho )}}{{2( \Pi _1 + \frac{1}{4} \Pi _0 \varrho ^2 { - \Pi _3 \varrho })}}\cdot SP \\&\quad = \frac{{3 {BS}_{t - 1} {BS}_{t - 2} - 2 {BS}_t {BS}_{t - 2} - 2 + ( {BS}_t {BS}_{t - 2} + 1 - 2 {BS}_{t - 1} {BS}_{t - 2} )\varrho }}{{(4 {BS}_{t - 1} {BS}_{t - 2} - 2 {BS}_t {BS}_{t - 2} - 2)}}\cdot SP \\&\quad =\left[ 1 + \frac{{( {BS}_t {BS}_{t - 2} + 1 - 2 {BS}_{t - 1} {BS}_{t - 2} )\varrho }}{{(4 {BS}_{t - 1} {BS}_{t - 2} - 2 {BS}_t {BS}_{t - 2} - 2)}}\right] \cdot SP \\&\quad = \left( 1 - \frac{1}{2}\varrho \right) \cdot SP \end{aligned}$$
(74)

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Bleaney, M., Li, Z. A new spread estimator. Rev Quant Finan Acc 47, 179–211 (2016). https://doi.org/10.1007/s11156-015-0499-z

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Keywords

  • Bid-ask spread
  • Feedback trading
  • Estimation

JEL Classification

  • C15
  • G20