$\left\|\mathcal{F}_{1}-\mathcal{F}_{0}\right\|\left\|\mathcal{F}_{0}\right\|\leq\psi_{\min}^{2}(\mathcal{F}_{0})/2$
and
$\left\|\mathcal{F}_{1}-\mathcal{F}_{2}\right\|\left\|\mathcal{F}_{0}\right\|\leq\psi_{\min}^{2}(\mathcal{F}_{0})/4$.
Denote

$R_{1}\coloneqq\operatorname*{arg\,min}_{O\in\mathcal{O}^{r\times
r}}\left\|\mathcal{F}_{1}O-\mathcal{F}_{0}\right\|_{F},R_{2}\coloneqq\operatorname*{arg\,min}_{O\in\mathcal{O}^{r\times
r}}\left\|\mathcal{F}_{2}O-\mathcal{F}_{0}\right\|_{F}.$

Then we have

$\left\|\mathcal{F}_{1}R_{1}-\mathcal{F}_{2}R_{2}\right\|\leq
5\frac{\psi_{\max}^{2}(\mathcal{F}_{0})}{\psi_{\min}^{2}(\mathcal{F}_{0})}\left\|\mathcal{F}_{1}-\mathcal{F}_{2}\right\|\quad\text{and}\quad\left\|\mathcal{F}_{1}R_{1}-\mathcal{F}_{2}R_{2}\right\|_{F}\leq
5\frac{\psi_{\max}^{2}(\mathcal{F}_{0})}{\psi_{\min}^{2}(\mathcal{F}_{0})}\left\|\mathcal{F}_{1}-\mathcal{F}_{2}\right\|_{F}.$

###### Proof.

This is the same as Lemma 37 in Ma et al., (2020). ∎

##  Appendix E Additional empirical findings: comparison with the two-way
fixed effect model in Chung et al., (2020)

Finally, we provide further details of the comparison between our model and
the two-way fixed effect model in Chung et al., (2020) which is omitted in the
main text to save space. Denote the quote, trade and trade-at-rule dummy
variables by $\mathcal{Q}_{i}$, $\mathcal{T}_{i}$, and
$\mathcal{T}\mathcal{A}_{i}$, respectively, and the pilot period dummy
variable by $Pilot_{t}$. Chung et al., (2020) consider the following two-way
fixed effect model:

$y_{it}=(\mathcal{Q}_{i}\times Pilot_{t})\theta^{(1)}+(\mathcal{T}_{i}\times
Pilot_{t})\theta^{(2)}+(\mathcal{T}\mathcal{A}_{i}\times
Pilot_{t})\theta^{(3)}+x_{it}^{\top}\beta+\alpha_{i}+\delta_{t}+\epsilon_{it},$

where $x_{it}$ is the set of $\left(\mathcal{Q}_{i}\times Pilot_{t}\times
TBC_{it}\right)$, $\left(Pilot_{t}\times TBC_{it}\right)$ and other control
variables like stock prices and trading volumes. Since $y_{it}=\sum_{0\leq
d\leq 3}\Upsilon_{it}^{(d)}y_{it}^{(d)}$, where $\Upsilon_{it}^{(d)}=1$ if and
only if unit $i$ receives treatment $d$ at time $t$, and zero otherwise, with
the convention that the treatment $0$ is the control, this model can be
represented as Model (4.1). On the other hand, our model can be represented
as:

$y_{it}=(\mathcal{Q}_{i}\times
Pilot_{t})\theta^{(1)}_{it}+(\mathcal{T}_{i}\times
Pilot_{t})\theta^{(2)}_{it}+(\mathcal{T}\mathcal{A}_{i}\times
Pilot_{t})\theta^{(3)}_{it}+x_{it}^{\top}\beta+\zeta_{i}^{\top}\eta_{t}^{(0)}+\epsilon_{it}.$

As noted in the main text, the above model is nested in our model and highly
likely to be misspecified.

| $\beta_{1}$ | $\beta_{2}$ | $\theta^{(1)}$ | $\theta^{(2)}$ | $\theta^{(3)}$ | $R^{2}$
---|---|---|---|---|---|---
Our model | 2.20 *** | -1.46 *** | $\hat{\theta}^{(1)}_{it}$ | $\hat{\theta}^{(2)}_{it}$ | $\hat{\theta}^{(3)}_{it}$ | 0.79
(0.10) | (0.07) | [mean: -0.40] | [mean: 0.86] | [mean: -0.98] |
Two-way | 3.68*** | -0.75 *** | -0.27 *** | 0.28 *** | -0.99 *** | 0.67
(0.09) | (0.06) | (0.05) | (0.05) | (0.05) |
Table E.1: Estimation results: ‘Two-way’ denotes the two-way fixed effect
model. Numbers in the parenthesis ( ) are standard errors. ‘mean’ denotes the
average of $\hat{\theta}^{(d)}_{it}$ over all treated stocks in the pilot
periods.

Table E.1 provides estimates for both models. $\beta_{1}$ and $\beta_{2}$ are
the coefficients for $\left(\mathcal{Q}_{i}\times Pilot_{t}\times
TBC_{it}\right)$ and $(Pilot_{t}\times TBC_{it})$, respectively. Note that the
positive $\beta_{1}$ means that a larger TBC results in a larger treatment
effect of the Q rule. It shows that as the minimum quoted spread increases
from 1 cent to 5 cents under the Q rule, the effective spread increases, and
this effect increases when the extent to which the new tick size ($0.05) is a
binding constraint on quoted spreads is larger.

It is worth noting that the treatment effect of the Q rule is
$\theta_{it}^{(1)}+\beta_{1}\cdot TBC_{it}$ since

$\mathbb{E}[y_{it}|\mathcal{Q}_{i}=1,Pilot_{t}=1]-\mathbb{E}[y_{it}|\mathcal{Q}_{i}=0,Pilot_{t}=1]=\theta_{it}^{(1)}+\beta_{1}\cdot
TBC_{it}$

while that of the T rule and the TA rule are $\theta_{it}^{(2)}$ and
$\theta_{it}^{(3)}$, respectively. Figure E.1 shows the dynamics of the cross-
sectional average of the treatment effects of the Q rule.

Figure E.1: The dynamics of the cross-sectional average of the effect of Q
rule: For the confidence band, we use the 95% uniform critical value,
$\Phi^{-1}(1-0.025/53)$. The dots denote the weekly average.

Note also that the sign of treatment effect of the Q rule is determined by the
magnitudes of the positive effect of TBC and the negative effect of
$\theta^{(1)}_{it}$, the effect of coarser quotable prices. To see why the Q
rule results in coarser quotable prices, consider, for example, if the quoted
spread is 17 cents without the Q rule. It may change to 15 cents or 20 cents
under the Q rule. This effect is different from the effect related to the
minimum quoted spread captured by $TBC$. $\theta^{(1)}_{it}$ can capture the
effect of coarser quotable prices. Most of the time, the positive effect of
TBC is greater than the negative effect of $\theta^{(1)}_{it}$, and therefore
the treatment effect of Q rule is positive. Especially, as time passes, the
negative effect of $\theta^{(1)}_{it}$ becomes weaker, and the treatment
effect of Q rule becomes more positive.

## References

  * Abadie, (2021) Abadie, A. (2021).  Using synthetic controls: Feasibility, data requirements, and methodological aspects.  Journal of Economic Literature, 59(2):391–425.
  * Abadie et al., (2010) Abadie, A., Diamond, A., and Hainmueller, J. (2010).  Synthetic control methods for comparative case studies: Estimating the effect of california’s tobacco control program.  Journal of the American statistical Association, 105(490):493–505.
  * Abadie and Gardeazabal, (2003) Abadie, A. and Gardeazabal, J. (2003).  The economic costs of conflict: A case study of the basque country.  American economic review, 93(1):113–132.
  * Agarwal et al., (2021) Agarwal, A., Dahleh, M., Shah, D., and Shen, D. (2021).  Causal matrix completion.  arXiv preprint arXiv:2109.15154.
  * Agarwal et al., (2020) Agarwal, A., Shah, D., and Shen, D. (2020).  Synthetic interventions.  arXiv preprint arXiv:2006.07691.
  * Ahn and Horenstein, (2013) Ahn, S. C. and Horenstein, A. R. (2013).  Eigenvalue ratio test for the number of factors.  Econometrica, 81(3):1203–1227.
  * Albuquerque et al., (2020) Albuquerque, R., Song, S., and Yao, C. (2020).  The price effects of liquidity shocks: A study of the sec’s tick size experiment.  Journal of Financial Economics, 138(3):700–724.
  * Athey et al., (2021) Athey, S., Bayati, M., Doudchenko, N., Imbens, G., and Khosravi, K. (2021).  Matrix completion methods for causal panel data models.  Journal of the American Statistical Association, 116(536):1716–1730.
  * Bai, (2003) Bai, J. (2003).  Inferential theory for factor models of large dimensions.  Econometrica, 71(1):135–171.
  * Bai, (2009) Bai, J. (2009).  Panel data models with interactive fixed effects.  Econometrica, 77(4):1229–1279.
  * Bai and Ng, (2021) Bai, J. and Ng, S. (2021).  Matrix completion, counterfactuals, and factor analysis of missing data.  Journal of the American Statistical Association, 116(536):1746–1763.
  * Belloni et al., (2018) Belloni, A., Chernozhukov, V., Chetverikov, D., Hansen, C., and Kato, K. (2018).  High-dimensional econometrics and regularized GMM.  arXiv preprint arXiv:1806.01888.
  * Bhattacharya and Chatterjee, (2022) Bhattacharya, S. and Chatterjee, S. (2022).  Matrix completion with data-dependent missingness probabilities.  IEEE Transactions on Information Theory, 68(10):6762–6773.
  * Cahan et al., (2023) Cahan, E., Bai, J., and Ng, S. (2023).  Factor-based imputation of missing values and covariances in panel data of large dimensions.  Journal of Econometrics, 233(1):113–131.
  * Candes and Plan, (2010) Candes, E. J. and Plan, Y. (2010).  Matrix completion with noise.  Proceedings of the IEEE, 98(6):925–936.
  * Candès and Recht, (2009) Candès, E. J. and Recht, B. (2009).  Exact matrix completion via convex optimization.  Foundations of Computational mathematics, 9(6):717.
  * (17) Chen, J., Liu, D., and Li, X. (2020a).  Nonconvex rectangular matrix completion via gradient descent without $\ell_{2,\infty}$ regularization.  IEEE Transactions on Information Theory, 66(9):5806–5841.
  * (18) Chen, Y., Chi, Y., Fan, J., Ma, C., and Yan, Y. (2020b).  Noisy matrix completion: Understanding statistical guarantees for convex relaxation via nonconvex optimization.  SIAM journal on optimization, 30(4):3098–3121.
  * (19) Chen, Y., Fan, J., Ma, C., and Yan, Y. (2019a).  Inference and uncertainty quantification for noisy matrix completion.  Proceedings of the National Academy of Sciences, 116(46):22931–22937.
  * (20) Chen, Y., Fan, J., Ma, C., and Yan, Y. (2019b).  Inference and uncertainty quantification for noisy matrix completion.  Proceedings of the National Academy of Sciences, 116(46):22931–22937.
  * Chernozhukov et al., (2021) Chernozhukov, V., Hansen, C., Liao, Y., and Zhu, Y. (2021).  Inference for low-rank models.  arXiv preprint arXiv:2107.02602.
  * Chung et al., (2020) Chung, K. H., Lee, A. J., and Rösch, D. (2020).  Tick size, liquidity for small and large orders, and price informativeness: Evidence from the tick size pilot program.  Journal of Financial Economics, 136(3):879–899.
  * Griffith and Roseman, (2019) Griffith, T. G. and Roseman, B. S. (2019).  Making cents of tick sizes: The effect of the 2016 us sec tick size pilot on limit order book liquidity.  Journal of Banking & Finance, 101:104–121.
  * Hamdi and Bayati, (2022) Hamdi, N. and Bayati, M. (2022).  On low-rank trace regression under general sampling distribution.  The Journal of Machine Learning Research, 23(1):14424–14472.
  * Jin et al., (2021) Jin, S., Miao, K., and Su, L. (2021).  On factor models with random missing: Em estimation, inference, and cross validation.  Journal of Econometrics, 222(1):745–777.
  * Klopp, (2014) Klopp, O. (2014).  Noisy low-rank matrix completion with general sampling distribution.  Bernoulli, 20(1):282–303.
  * Koltchinskii et al., (2011) Koltchinskii, V., Lounici, K., Tsybakov, A. B., et al. (2011).  Nuclear-norm penalization and optimal rates for noisy low-rank matrix completion.  The Annals of Statistics, 39(5):2302–2329.
  * Ma et al., (2020) Ma, C., Wang, K., Chi, Y., and Chen, Y. (2020).  Implicit regularization in nonconvex statistical estimation: Gradient descent converges linearly for phase retrieval, matrix completion, and blind deconvolution.  Foundations of Computational Mathematics, 20:451–632.
  * Mazumder et al., (2010) Mazumder, R., Hastie, T., and Tibshirani, R. (2010).  Spectral regularization algorithms for learning large incomplete matrices.  Journal of machine learning research, 11:2287–2322.
  * Negahban and Wainwright, (2012) Negahban, S. and Wainwright, M. J. (2012).  Restricted strong convexity and weighted matrix completion: Optimal bounds with noise.  The Journal of Machine Learning Research, 13(1):1665–1697.
  * Rindi and Werner, (2019) Rindi, B. and Werner, I. M. (2019).  Us tick size pilot.  Fisher College of Business Working Paper:2017-03-018.
  * Tropp et al., (2015) Tropp, J. A. et al. (2015).  An introduction to matrix concentration inequalities.  Foundations and Trends® in Machine Learning, 8(1-2):1–230.
  * Vershynin, (2010) Vershynin, R. (2010).  Introduction to the non-asymptotic analysis of random matrices.  arXiv preprint arXiv:1011.3027.
  * Werner et al., (2022) Werner, I. M., Rindi, B., Buti, S., and Wen, Y. (2022).  Tick size, trading strategies, and market quality.  Management Science, 69(7):3818–3837.
  * Xia and Yuan, (2021) Xia, D. and Yuan, M. (2021).  Statistical inferences of linear forms for noisy matrix completion.  Journal of the Royal Statistical Society: Series B (Statistical Methodology), 83(1):58–77.
  * Xiong and Pelger, (2023) Xiong, R. and Pelger, M. (2023).  Large dimensional latent factor modeling with missing observations and applications to causal inference.  Journal of Econometrics, 233(1):271–301.