2.1.2. Price Impact Benchmarks

In this study, we employ three high-frequency price impact measures: Hasbrouck's (2009) λ coefficient (*LAMBDA*), Goyenko et al.'s (2009) five-minute price impact (*IMP*), and Huang and Stoll's (1996) adverse selection costs (*ASC*). Hasbrouck (2009) estimates that, while using a regression of returns, the slope of the price function is measured over five-minute time intervals while considering the aggregate signed square root of the dollar volume during the same intervals. It is measured as the coefficient *λ* in the following regression model:

$$r\_n = \lambda \left[ \sum\_t \text{sign}(\text{volume}\_{t,n}) \sqrt{\text{volume}\_{t,n}} \right] + \varepsilon\_{n,n}$$

where *rn* is the return over the *n*th five-minute interval, *volumet*,*<sup>n</sup>* is the dollar volume of the *t*th trade during the *n*th interval, and *sign*(*volumet*,*<sup>n</sup>*) takes the value +1 if the *t*th transaction is a buy order, and −1 if it is a sell order.

Our next price impact benchmark is the five-minute price impact that was introduced by Goyenko et al. (2009). This measure captures the permanent price change over a five-minute window subsequent to a trade. It measures the change in quote midpoints from the time of the trade to five minutes after the trade.

$$IMP\_{\mathfrak{l}} = 2D\_{\mathfrak{l}}[\ln(m\_{\mathfrak{l}+\mathfrak{s}}) - \log(m\_{\mathfrak{l}})]\_{\mathfrak{l}}$$

In the above specification, *mt* and *mt*+5 are the quote midpoints at *t* and five minutes after *t*, respectively.

Our last price impact benchmark is adverse selection costs, as developed by Huang and Stoll (1996). Huang and Stoll (1996) calculate adverse selection costs by subtracting the realized spread from the effective spread. This measure captures the portion of investors' transaction costs attributable to the permanent price change, as follows:

$$ASC\_t = ES\_t - RS\_t = 2D\_t(m\_{t+\tau} - m\_t) / m\_{t-\tau}$$

#### *2.2. Liquidity Proxies from Low-Frequency Data*
