تدابیر جدید گسترش فرکانس پایین

I develop new spread proxies that pick up on three attributes of the low-frequency (daily) data: (1) price clustering, (2) serial price covariance accounting for midpoint prices on no-trade days, and (3) the quoted spread that is available on no-trade days. I develop and empirically test two different approaches: an integrated model and combined models. I test both new and existing low-frequency spread measures relative to two high-frequency benchmarks (percent effective spread and percent quoted spread) on three performance dimensions: (1) higher individual firm correlation with the benchmarks, (2) higher portfolio correlation with the benchmarks, and (3) lower distance relative to the benchmarks. I find that on all three performance dimensions the new integrated model and the new combined model do significantly better than existing low-frequency spread proxies.

In a classic and influential paper, Roll (1984) develops a simple proxy for the effective spread using price data only. Lesmond et al. (1999) and Hasbrouck (2004) develop additional proxies for the effective spread using low-frequency (daily) data. Amihud (2002) and Pastor and Stambaugh (2003) develop low-frequency liquidity measures that perhaps might be viewed as proxies for price impact, more than for the effective spread. Collectively, these low-frequency spread proxies allow the study of liquidity over relatively long periods of time and across countries. This is helpful to the asset pricing literature, because recent studies suggest that liquidity is a priced risk factor. This is also helpful to recent studies in the market efficiency and corporate finance literatures, which utilize spread proxies for the cost of trade by stock, by time period, and across countries. Is it possible to create new low-frequency spread proxies that perform better than the existing low-frequency spread proxies? In this paper, better performance is primarily evaluated relative to two high-frequency benchmarks (percent effective spread and percent quoted spread) and on three dimensions: (1) higher individual firm correlation with the benchmarks, (2) higher portfolio correlation with the benchmarks, and (3) lower distance (tracking error) relative to the benchmarks. I find that the answer is “yes” on all three dimensions. Spread proxies can be constructed from daily data going back more than 80 years in the United States and for various time spans in countries around the world. For U.S. equity markets, the Center for Research in Security Prices (CRSP) provides five key daily stock variables: prices, returns adjusted for splits and dividends, volume, high/ask, and low/bid.1 These five variables are available for all NYSE/AMEX firms from December 31, 1925 to the present and for all NASDAQ firms from December 14, 1972 to the present. High-performing low-frequency spread measures would be very helpful to the asset pricing literature. Chordia et al. (2000), Sadka (2003), Acharya and Pedersen (2005), Fujimoto (2004), Hasbrouck (2009), and others show that in recent U.S. experience various liquidity measures vary systematically and are priced; Bekaert et al. (2007) provide similar evidence for emerging markets where liquidity concerns may be more pronounced. Spread proxies going back in time and/or across countries are needed to determine whether or not these asset pricing relationships hold up across time and space. High-performing low-frequency spread measures would be very helpful to the market efficiency and corporate finance literatures. De Bondt and Thaler (1985), Jegadeesh and Titman (1993), Jegadeesh and Titman (2001), Chan et al. (1996), Rouwenhorst (1998), and many others have found trading strategies that appear to generate significant abnormal returns. Correctly scaled spread proxies over time and/or across countries are needed to determine if these trading strategies are truly profitable net of a relatively precise measure of cost of trading. Similarly, Dennis and Strickland (2003), Kalev et al. (2003), Cao et al. (2004), Lipson and Mortal (2004a), Schrand and Verrecchia (2004), Lesmond et al. (2005), and many others examine the impact of corporate finance events on stock liquidity. Helfin and Shaw (2000), Lipson and Mortal (2004b), Lerner and Schoar (2004), and many others examine the influence of liquidity on capital structure, security issuance form, and other corporate finance decisions. Spread proxies over time would expand the potential sample size of this literature. Spread proxies across countries would greatly extend the potential diversity of international corporate finance environments that this literature could analyze. This paper develops new, low-frequency spread measures that pick up on three attributes of the daily data. One attribute is price clustering—the higher likelihood for trade prices to be on rounder increments. One can directly observe the frequency of various price clusters (odd eighths, odd quarters, etc. on a fractional price grid and off-pennies, off-nickels, off-dimes, etc.2 on a decimal price grid) and use this information to infer the effective spread. The second attribute is serial covariance of price changes. I extend the Roll framework of serial covariance to account for no-trade days in which the reported price is the closing midpoint. The third attribute is the high/ask and low/bid variables available in the CRSP stock data. These variables directly supply the quoted spread on no-trade days. I develop and empirically test two different approaches to incorporating these three attributes. First, I develop an integrated model that directly includes these attributes. The base integrated model, which I call “Holden,” combines two attributes: price clustering and serial correlation. The expanded integrated model, which I call “Holden2,” combines all three attributes. Second, I develop combined models, which I call Multi-Factor models, that are linear combinations of simpler one-attribute or two-attribute models. 3 I show theoretically that Multi-Factor models have the potential to diversify away some imperfectly-correlated error terms. Then, I test the new, low-frequency spread measures against the existing low-frequency spread measures (Hasbrouck Gibbs, LOT Mixed, LOT Y-split, Pastor and Stambaugh, Roll, and Zeros).4 All proxies are compared to two high-frequency benchmarks: (1) percent effective spread and (2) percent quoted spread. Both of these benchmarks are computed from the NYSE's Trade and Quote (TAQ) dataset for 400 firms from 1993 to 2005. Percent effective spread is a volume-weighted average based on every trade and corresponding BBO5 quote in the stock-month. Percent quoted spread is a time-weighted average based on every BBO quote in the stock-month. I test on three performance dimensions. First, I compute the correlation of each spread proxy with each benchmark based on individual firms. Second, I create an aggregate spread measure for each proxy and benchmark based on an equally weighted portfolio across all 400 firms. Then, I compute the pure time-series correlation of each aggregate spread proxy with each aggregate benchmark. Third, I compute the average root mean squared error for each spread proxy compared to each benchmark. I find that on all three performance dimensions with regard to both benchmarks the new integrated model Holden2 does significantly better than existing low-frequency spread proxies. I find that on all three performance dimensions with regard to both benchmarks the new combined model Multi-Factor2 does significantly better than existing low-frequency spread proxies, except for one tie. Summarizing six tests (three performance dimensions X two benchmarks), the combined model Multi-Factor2 does significantly better than the integrated model Holden2 on four out of six tests. I also find that these new proxies are robust by size quintiles, price quintiles, and tick size regime. Consistently, Holden2 is the best integrated model and Multi-Factor2 is the best combined model. Across all size, price, and tick size regime comparisons, Multi-Factor2 is the most frequent winner and Holden2 is the second most frequent winner. Finally, I compare the proxies to low-frequency spread benchmarks and find that new proxies consistently outperform existing proxies. An empirical companion paper, Goyenko et al. (2009), performs extensive testing of monthly and annual liquidity proxies. It tests earlier-developed proxies, Holden, Effective Tick, and Effective Tick2, and existing proxies from the prior literature. In their monthly effective spread comparisons, they found that Holden, Effective Tick, and LOT Y-split significantly outperformed all other spread proxies then in existence. In this paper, later-developed proxies, Holden2, Multi-Factor1, Multi-Factor2, and other newly developed proxies are tested for the first time. I find that Holden2 and Multi-Factor2 outperform the earlier-developed proxies, Holden, Effective Tick, and LOT Y-split, and outperform existing proxies from the prior literature. Hasbrouck (2009) tests annual estimates of four liquidity proxies using U.S. data. Lesmond (2005) tests quarterly estimates of five liquidity proxies using data from 31 emerging countries. The vast majority of asset pricing, market efficiency, or corporate finance literature that examines the role of liquidity is based on monthly (or finer) data. Goyenko et al. (2009) and this paper contribute to our knowledge of potential spread proxies by testing monthly estimates. The paper is organized as follows. Section 2 develops the integrated Holden model. Section 3 develops both single-attribute models and combined multi-factor models and shows that multi-factor models can diversify away some imperfectly correlated measurement error. Section 4 describes the data and empirically tests both new and existing low-frequency spread measures on three performance dimensions. Section 5 concludes. Three appendices contain technical details.

I develop new spread proxies that are computed from low-frequency (daily) data. First, I develop an integrated model, the Holden model, which directly includes three attributes of the daily data: price clustering, serial covariance accounting for no-trade midpoints, and the no-trade quoted spread. Second, I develop combined models, the Multi-Factor models, that are linear combinations of simpler one-attribute or two-attribute models. I show theoretically that Multi-Factor models have the potential to diversify away some imperfectly-correlated error terms. Next, I empirically test both new and existing low-frequency spread measures on three performance dimensions: (1) higher individual firm correlation with effective or quoted spread, (2) higher portfolio correlation with effective or quoted spread, and (3) lower distance (tracking error) relative to effective or quoted spread. I find that on all three performance dimensions with regard to both high-frequency benchmarks, the new integrated model Holden2 does significantly better than existing low-frequency spread proxies. I find that on all three performance dimensions with regard to both benchmarks, the new combined model Multi-Factor2 does significantly better than existing low-frequency spread proxies, except for one tie. Summarizing six tests (three performance dimensions X two benchmarks), the combined model Multi-Factor2 does significantly better than the integrated model Holden2 on four out of six tests. I also find that these new proxies are robust by size quintiles, price quintiles, and tick size regime. Consistently, Holden2 is the best integrated model and Multi-Factor2 is the best combined model. Across all size, price, and tick size regime comparisons, Multi-Factor2 is the most frequent winner and Holden2 is the second most frequent winner. Finally, I compare the proxies to low-frequency spread benchmarks and find that new proxies consistently outperform existing proxies.