Illiquidity and stock returns: the moderating role of investors ’ holding period in Central and Eastern European markets

Purpose – The positive illiquidity – return relationship (so-called liquidity premium) is a well-established pattern in international developed stock markets. The magnitude of liquidity premium should increase with market illiquidity. Existing studies, however, do not confirm this conjecture with regard to frontier markets. Thismayresultfromapplyingdifferentapproachestotheinvestors ’ holdingperiod.Thepaperaimstoidentify the role of the holding period in shaping the illiquidity – return relationship in emerging and frontier stock markets, which are arguably considered illiquid. Design/methodology/approach – The authors utilise the data on stocks listed on fourteen exchanges in CentralandEasternEurope.Theauthorsregressstockreturnsonliquiditymeasuresvariouslytransformedto reflecttheclienteleeffectinaliquidity – return relationship. Findings – The authors show that the investors ’ holding period moderates the illiquidity – return relationship in CEE markets and also show that the liquidity premium in these markets is statistically and economically relevant. Practical implications – The findings may be of great interest to investors, companies and regulators. Investors and companies should take liquidity into account when making decisions; regulators should employ liquidity-enhancing actionsto decreasecompanies ’ costof capital and expandfirms ’ investmentopportunities, which will improve growth perspectives for the entire economy. Originality/value – These findings enrich the understanding of the role that the investors ’ holding period playsinthe illiquidity – returnrelationship inCEEmarkets.Tothe bestknowledge,thisisthe firststudywhich investigates the effect of holding period on liquidity premium in emerging and frontier markets.


Introduction
Risk-averse investors prefer to hold less risky stocks until they are compensated for bearing additional risk. Similarly, investors prefer to allocate their capital into liquid investments until they are compensated for lower liquidity. Thus, in equilibrium, less liquid investments must offer higher expected returns to ensure efficient capital allocation in the market. In their seminal paper, Amihud and Mendelson (1986a) prove that expected capital assets returns are functions of both, liquidity and investment holding period. As up-to-date research concerns mainly the liquidity itself as a driver of expected stock returns, in this paper we aim to empirically test the role the holding period plays in a liquidity-return relationship. If the holding period significantly affects this relationship, assuming an inappropriate holding period may result in biased estimation of the liquidity premium, which may thus result in an inefficient capital allocation in the economy.
The negative liquidity-return relationship (hereafter also referred to as the liquidity premium) is quite a well-established cross-sectional pattern within international developed stock markets. Similar studies carried out on emerging or frontier capital markets still yield ambiguous results. Bekaert et al. (2007) and Amihud et al. (2015) found that the liquidity premium in emerging markets is higher than in developed ones, which is the result of the overall lower level of liquidity in these markets. One should thus expect the liquidity premium in frontier markets, which are arguably far less liquid than even emerging markets, to be higher than in emerging markets. However, Batten and Vo (2014) and Stere nczak et al. (2020) found the liquidity-return relationship to be insignificantly positive in frontier equity markets. So why does the positive illiquidityreturn relationship not hold in the least liquid (frontier and some of the emerging) markets?
The liquidity effect on stock returns depends not only on the amount of liquidity costs themselves, but also on the frequency of trading and, in turn, the frequency of incurring these costs (Amihud and Mendelson, 1986a;Anginer, 2010;Chalmers and Kadlec, 1998;Florackis et al., 2011;Naes and Ødegaard, 2009). Thus, investors with a short investment horizon hold highly-liquid assets while long-term investors prefer less liquid shares (clientele effect) (Amihud and Mendelson, 1986a;Anginer, 2010;Atkins and Dyl, 1997).
The liquidity clientele effect causes the liquidity-return relationship to be nonlinear, and stock returns to be a growing and concave function of liquidity costs. The most common approach to the concavity of the illiquidity-return relationship is the log-transformation of illiquidity measures. However, as the results of the study by Gniadkowska-Szyma nska (2016) suggest, the log-transformation of the illiquidity measure may not fully reflect the clientele effect in less liquid (emerging or frontier) markets. This could imply that simply taking the log of spread (and other liquidity measures) when analysing the liquidity-return relationship in emerging and frontier markets, that are considered illiquid, may lead to biased results.
Thus, the paper aims to identify the role of the investors' holding period in shaping the illiquidity-return relationship in arguably less liquid (emerging and frontier) stock markets. To this end, we utilise the data on stocks listed on stock exchanges in fourteen Central and Eastern European (also CEE herein) countries (Bulgaria, Croatia, Czechia, Estonia, Hungary, Latvia, Lithuania, Poland, Romania, Russia, Serbia, Slovenia, Slovakia and Ukraine). CEE markets provide a good empirical setting to study the role of the investors' holding period in the liquidity premium. While similar in their history, geographical location, investors' profile and legal framework, CEE markets differ significantly in terms of size (measured by both the number of listed companies and capitalisation), liquidity, integration with the global economy, and development. According to the MSCI classification, CEE markets are either emerging (Czechia, Hungary, Poland and Russia) or frontier (Bulgaria, Croatia, Estonia, Lithuania, Romania, Serbia, Slovenia and Ukraine). Latvia and Slovakia are not covered by any MSCI index.
All these countries share a common history, as the market economy appeared there only after 1989. The capital markets of Central and Eastern Europe may not seem attractive, as their capitalisation constitutes only a small fraction of global capitalisation and they are very illiquid. The average bid-ask spread on CEE markets equals 5.61%, as compared to an average of 1.96% in developed markets (Fong et al., 2017). CEE markets are also characterised by highly concentrated trading and ownership. In Poland and Russia the 3 largest shareholders hold about 85% of shares on average, and the percent free-float is 20% on average (De La Cruz et al., 2019). However, while being less integrated with the global economy, CEE markets may offer some diversification benefits for global investors. Thus, these markets are an important area of research, as they offer a unique opportunity for international investors and yet, surprisingly, are under-researched. In addition, their importance may be set to grow. IJOEM All the features of CEE stock markets, especially low liquidity and shareholders' structure, allow us to conjecture that the investors' holding period may play a significant role in liquidity effects in these markets. Moreover, the geographic proximity of these countries allows us to conjecture that CEE markets have common pricing factors. As the group of CEE markets consists of both emerging and frontier markets, our results could possibly be expanded to other emerging and frontier markets that are definitely less liquid than developed stock markets.
The study provides several important contributions to the literature. Our first contribution is an examination of the liquidity premium in fourteen CEE stock markets. The existing research does not cover most of the markets covered by our study and reports ambiguous results. Moreover, studies covering some of the markets included in our sample apply filters that discard a significant number of firms. For example, Stere nczak et al. (2020) dropped on average 54% of stock-month observations in the countries covered by their study and ours, while we discard only 27%. Our study suggests that the liquidity premium in CEE markets is both statistically significant and economically relevant. One standard deviation increase in the illiquidity measure causes an increase in monthly stock return by about 0.3 p.p.
Our second contribution emerges from the analysis of the role the investors' holding period plays in shaping the stock liquidity-return relationship in emerging and frontier markets. While most research is focused on various transformations of liquidity measures to reflect the clientele effect, our study shows that such a transformation is insufficient, as the relationship between the stock illiquidity investors' holding period is much more complex when markets are illiquid and the ownership concentrated. Our study shows that one should use liquidity measures amortised over the holding period to analyse the liquidity premium in emerging and frontier markets.
Furthermore, the contribution of the paper comes from the fact that the study utilises a unique methodology. We study the relationship between liquidity and returns on a single stock level, rather than on a portfolio level. As pointed out by Amihud and Noh (2021b), testing the asset pricing patterns on individual stocks has some advantages over testing on stock portfolios. The use of portfolios sorted on stock characteristics as test assets may impart a strong factor structure across them, which in turn may lead to biased results. Moreover, such results may be sensitive to a subjective choice of sorting stock characteristics. Using single stock as test assets may lead to bias due to errors-in-variables, though we address this problem by employing estimators designed for panel data analysis.
The remainder of the paper is structured as follows. The next section presents a brief literature overview. Then we move to the presentation of the methodology of the study, followed by the empirical results. The final section concludes.
2. Literature review and hypothesis development 2.1 Illiquidity-return relationship In their seminal paper, Amihud and Mendelson (1986a) developed the model of the relationship between the return expected on stock and its liquidity, and found that stock returns are an increasing and concave function of the spread. This results from the liquidity clientele effect, i.e. that short-term investors prefer more liquid shares while long-term investors allocate relatively illiquid stocks to their portfolios. This is in line with Constantinides' (1986) proposition that investors tend to rebalance their portfolios less frequently when they face high illiquidity. Both a longer holding period and less frequent portfolio rebalancing are aimed at reducing the overall liquidity costs incurred.
A positive illiquidity-return relationship (liquidity premium) is a quite well-established pattern within developed stock markets. Recent studies by Amihud (2019), Amihud et al. (2015), Amihud and Noh (2021a, b), Guo et al. (2017), Hsieh andNguyen (2021), Chulia et al. (2021), among others, confirm that lower stock liquidity results in higher stock returns in the US and Investor holding period and liquidity premium other developed markets. Studies on international markets suggest that the liquidity premium is stronger in emerging markets than in developed ones (Amihud et al., 2015;Bekaert et al., 2007;Ben-Rephael et al., 2015). A higher liquidity premium in emerging markets results from the overall lower level of liquidity in these markets. This is in line with Jensen and Moorman (2010), Ben-Rephael et al. (2015), He and Kryzanowski (2006) and Hsieh and Nguyen (2021), who prove that when market liquidity is low, the illiquidity return premium tends to be high. Generally speaking, the liquidity premium in emerging markets should be higher than in developed markets as emerging markets exhibit higher illiquidity. One should thus expect the liquidity premium in frontier markets to be higher than in emerging markets, as frontier markets are even less liquid than emerging ones. Surprisingly, Batten and Vo (2014) and Stere nczak et al. (2020) found the illiquidity-return relationship to be insignificantly negative in frontier markets. However, the studies presented above differ in the approach to the investors' holding period, which could influence the results.

Different approaches to the investors' holding period and illiquidity-return relationship
Standard procedures applied in asset pricing studies seem to neglect the liquidity clientele effect, as they assume frequent portfolio rebalancing, which means that they assume holding periods equal to all stocks. Assuming an inappropriate holding period may result in biased estimation of the liquidity premium. Neglection of the clientele effect in asset pricing studies may be the cause of the ambiguity about the liquidity premium in recent studies. As an example, a recent study by Cakici and Zaremba (2021) has proved that liquidity premium is observable only among microcap, hardly investable stocks. Since large stocks are most heavily traded (their holding periods are relatively short), assuming that their holding periods are equal to the holding periods of microcap stocks may lead to biased results.
To incorporate the holding period into the analysis of the relationship between liquidity and stock returns (liquidity premium), in several studies stock (or portfolio) returns have been regressed on stock characteristics. In such regressions, measures of stock liquidity are one of the explanatory variables and are transformed to reflect the concavity of the relationship. The most popular transformation is the logarithmic one (Amihud and Mendelson, 1986b;Eleswarapu and Reinganum, 1993;Harris and Amato, 2019), but the square-root transformation of liquidity measures is also applied (see, e.g. Chordia et al., 2009;Hasbrouck, 1999Hasbrouck, , 2009. The log or square-root transformation of liquidity measure has an additional advantage in regression analysis. Raw values of liquidity measures often reveal elevated skewness and kurtosis, and taking the logarithm makes their distributions closer to normal. Another way to include a holding period in the analysis of the relationship between stock liquidity and returns is by amortising liquidity measures. Amortised liquidity measures are simply liquidity measures over investors' holding periods. The essence of amortisation is because liquidity costs are incurred by investors only at the moment of the purchase and the moment of sales. Thus, the longer the holding period is, into more periods liquidity costs can be divided. Therefore, the required compensation for liquidity costs (liquidity premium) per one period should decrease with an increase in the investment horizon. As evidenced by He and Kryzanowski (2006), stock returns should be growing and linear functions of amortised spread, not growing and concave functions of the spread itself.
The amortisation of liquidity measure was first used by Chalmers and Kadlec (1998) who have used amortised spreads as the measure of liquidity for NYSE and AMEX stocks and found that the observed liquidity premium is stronger when one uses an amortised spread (bid-ask spread scaled by the holding period) instead of a simple spread (even log-transformed to reflect the concavity of the relationship). However, Marshall and Young (2003) have found a significantly negative relationship between spread and returns, as well as a positive (though insignificant) relationship between amortised spread and returns in the IJOEM Australian Stock Market. Concerning emerging markets, recent studies by Stere nczak (2021) have utilised amortised liquidity measures to analyse the liquidity premium in the Warsaw Stock Exchange. He found a significantly positive relationship between amortised liquidity measures and stock returns.

Hypothesis development
Amortising the liquidity measures in order to take investors' holding periods into account may have some advantages over the logarithmic transformation of liquidity measures, especially in non-developed markets. As evidenced by Anginer (2010), liquidity clientele effect is stronger among more sophisticated investors. Karolyi et al. (2020) in turn suggest that investors in emerging economies are presumably less sophisticated than those in developed economies. Combining the two studies, one may conjecture that the liquidity clientele effect is less pronounced in emerging and frontier markets, which is empirically evidenced by Gniadkowska-Szyma nska (2016), who found log spreads (together with some control variables) have explained only from 9.4% to 20% (dependent on the data frequency) of the variations in investors' holding periods in the Warsaw Stock Exchange. In a comparable study by Atkins and Dyl (1997), log spreads explained 27.2% (for NYSE stocks) and 43.4% (for Nasdaq firms) of the variation in holding periods. Thus, a relationship between liquidity and holding periods may be more complex in relatively less liquid markets. This may result in biased estimation of the liquidity premium when applying log liquidity measures instead of amortised ones.
Also, some recent theoretical studies put stress on the role that investment horizon plays in estimating the liquidity premium. Beber et al. (2021) demonstrate that the investment horizon has a considerable impact on the way liquidity affects asset prices. Pereira and Zhang (2010) and Stere nczak (2020), by calibrating their models for different investment horizons showed that, keeping liquidity costs constant, a change in investment horizon results in a significant change in liquidity premium.
Equally important, as evidenced by Daryaei and Fattahi (2022), with an increase in institutional ownership up to a certain level, the relationship between liquidity and stock returns strengthens. However, when institutional ownership exceeds a certain level, its further increase results in a weakening of the liquidity-return relationship. As ownership structure is closely related to the investors' holding period, one could conjecture that the investors' holding period impacts the strength of the illiquidity-return relationship. The above considerations lead us to the following hypothesis.
Hypothesis. The investors' holding period moderates the relationship between illiquidity and stock returns in CEE markets.

Methodology and data 3.1 Research sample
Our study covers companies listed on exchanges located in fourteen countries in Central and Eastern Europe: Bulgaria, Croatia, Czechia, Estonia, Hungary, Latvia, Lithuania, Poland, Romania, Russia, Serbia, Slovenia, Slovakia and Ukraine. Our basic source of data is S&P Capital IQ and our initial coverage encompasses CEE markets companies available in this database, both survivors and non-survivors. Our sample period runs from January 2007 to December 2020, and its length is dictated by the availability of data on the bid and ask prices.
Including an earlier period in our sample would cause a higher overrepresentation of Polish companies. Overall, our initial sample comprises 2,142 firms and 206,544 stock-month observations.

Investor holding period and liquidity premium
To be included in the research sample in month t, we require a company to have a complete set of financial data and meet our requirements for having estimated at least one of our liquidity measures for month t-1, and a return for month t. To calculate a liquidity measure, we require data on at least 10 trading days for a stock-month observation. We do not exclude micro-and small-capitalisation firms, penny stocks and least liquid stocks, as CEE exchanges are densely populated with such firms, and this would considerably reduce our research sample. Our final sample consists of 1,552 firms and 159,807 stock-month observations. The structure by country of our sample is presented in Table 1. The fraction of discarded stock-month observations (23%) is much smaller than in recent studies on frontier markets (see, e.g. Stere nczak et al., 2020; Zaremba and Maydybura, 2019), in which 61% of observations have been discarded due to the filters applied.

Variables
To encompass more than just one liquidity dimension, we utilise two liquidity measures. First, we use the Percent Quoted Closing Spread (PQCS), which is the relative bid-ask spread quoted at the end of the trading day (Chung and Zhang, 2014). Recent studies indicate this measure as the best performing proxy for transaction costs (Chung and Zhang, 2014;Fong et al., 2017). Most research suggests that the best proxy of a price impact dimension is the Amihud (2002) illiquidity measure (ILLIQ). This has been proved true for developed (Fong et al., 2017), emerging (Ahn et al., 2018) and frontier (Marshall et al., 2013) stock markets. Thus, we use Amihud's (2002) ILLIQ as our second liquidity measure; we calculate it with the daily value of shares traded expressed in millions of EUR and the return in the numerator replaced by the log of the daily price range. This helps us to capture the short-lived price pressure that disappears before the end of the trading day. As indicated in recent studies, such a modification overperforms the standard Amihud measure in CEE markets (Będowska-S ojka, 2018; Stere nczak, 2019). To account for the outliers, we cross-sectionally winsorise computed liquidity measures at 2.5 and 97.5 percentile.
For the purpose of the study, we calculate excess (over risk-free return) monthly log returns (ret), based on the closing prices denominated in EUR on the last trading day in a month. We assume the risk-free is equal to one-month EURIBOR. We also include other firm characteristics as a set of control variables in our analyses. We use the logarithm of capitalisation expressed in millions of EUR (Size) as our proxy for firm size. Growth opportunities are proxied by the book-to-market ratio (B-MV), computed as the book value of equity scaled by capitalisation. Risk is an annualised standard deviation of weekly returns from the previous twelve months. Momentum is a cumulative log return from the months from tÀ12 to tÀ2, and Reversal is a return from month tÀ1. Table 2 presents the summary statistics for all the variables used in our study as well as statistics of the Fisher-type ADF test of the unit-root for panel data. It is visible that liquidity measures exhibit elevated skewness and kurtosis, especially amortised ones. However, log-transformed liquidity measures' distributions are close to normal. Noticeably, the mean PQCS in our sample equals 5.61% and is higher than in the Stere nczak et al. (2020) sample of frontier markets (4.4%), which may be a result of the filters that excluded the least liquid stocks from their sample. The statistics of the Fisher-type ADF tests (Choi, 2001;Maddala and Wu, 1999) of the panel unit-root indicate that all variables are stationary. We are not able to apply other tests of the panel unit-root (e.g. Harris and Tzavalis, 1999;Im et al., 2003;Levin et al., 2002) as they are designed for balanced panels and in our sample the number of periods varies between panel units.  (Belsley et al., 1980) suggest no collinearity concerns in our models.

Methods
In our study, we test the effect of stock liquidity on monthly returns. To this end, we estimate the following basic regressions: where ret denotes monthly excess log return on the stock, Liquidity denotes one of our liquidity measures, and Controls are the control variables as described in the previous subsection. To detect the role of the holding period in estimating the liquidity premium in our research sample, beyond using the raw values we also use logarithms of liquidity measures and their amortised versions as well. The differences in estimated βs for differently captured holding periods allow us to indicate how it should be taken into account when estimating the liquidity premium in markets we consider. We amortise liquidity measures by dividing their values by the average holding period as proxied by the inverse of the turnover ratio in the following month: where amLIQ is the amortised value of our liquidity measure (PQCS or ILLIQ), HP is the average holding period, V denotes the volume of shares traded and NoS is the number of outstanding shares. In this way, the amortised liquidity measure is indeed averaged across investors with different investment horizons. In fact, as ownership concentration may vary between various stocks, one should take it into account. If not all shares are freely tradeable (are held by longterm investors), the average holding period as computed with the simple inverse of the turnover ratio may be biased upwards. To omit this problem, we also adjust the method of computing the holding period by adjusting the number of outstanding shares by the stock held by long-term investors. We use percent free float for that purpose: where ff it denotes the percent of free float shares.
As liquidity measures amortised over the holding period exhibit elevated skewness and kurtosis, in our regressions we also include log-transformations of these variables. Log transformations of the amortised liquidity measures also help us to indicate whether significantly positive coefficients on amortised liquidity measures are not simply the result of dividing the liquidity measure by the holding period. To this end, we could assume the equal value of PQCS/ILLIQ for all stocks (e.g. equal to 1) and regress stock returns on the reciprocal of the holding period. However, if we take into account that the logarithm of the ratio of two variables is equal to the difference in the logs of these variables, i.e.: we are able to include both the log transformed liquidity measure and the log transformed holding period in the regression.

Empirical results
In model (1) the coefficient on Liquidity (β) denotes the "per unit" liquidity premium, i.e. the liquidity premium required for the unit of liquidity costs incurred (Stere nczak, 2021). In line with the theory, the estimated β is expected to be significantly positive, as higher values of our liquidity measures denote lower liquidity. Thus, a positive coefficient results in the fact IJOEM that less liquid stocks yield higher returns than more liquid shares. As our dataset constitutes an unbalanced panel of data, we carry out a series of panel diagnostic tests to indicate the most appropriate method of estimation. Due to the results of the Welch's F-test, the Breusch-Pagano test, and the Hausman test, we utilise the fixed effects (FE) estimator with time dummies in all the estimated models. To account for the potential heteroskedasticity and autocorrelation of residuals, we use robust standard errors clustered at the firm level. The results of the estimation are presented in Table 4. In general, the coefficients on control variables seem to be consistent with the theory and previous research on stock markets. The negative coefficient on Size indicates that smaller firms yield higher returns. Though we should expect a positive coefficient on Risk, a negative relationship between risk and future returns was reported in some earlier studies (Ang et al., 2006(Ang et al., , 2009e.g. Chalmers and Kadlec, 1998;Stere nczak et al., 2020). Significantly positive coefficients on Momentum are also in line with previous findings (e.g. Jegadeesh and Titman, 1993). The coefficients on Reversal are consistent with our expectations, though statistically insignificant.
The estimated coefficients on raw (non-transformed) liquidity measures are statistically insignificant, suggesting no clear relationship between liquidity and stock returns. However, coefficients on log-transformed liquidity measures are negative and statistically significant with t-statistics equal to 3.421 (PQCS) and 2.667 (ILLIQ). Significantly negative coefficients on log-transformed PQCS and ILLIQ suggest the returns are decreasing and concave functions of illiquidity, which is contrary to the theory. All the theoretical models produce propositions stating that returns are increasing functions of stock illiquidity, some models show this function is concave (e.g. Amihud and Mendelson, 1986b), and someconvex (Jacoby et al., 2000), but no theoretical model indicates the illiquidity-return relationship is negative.
The estimated coefficients on amortised liquidity measures are positive and statistically significant at a 0.05 level. This suggests that the amortisation of liquidity measures results in an estimated per unit liquidity premium of a value which is in line with the theory. Only the coefficient on ILLIQ amortised over the holding period adjusted for free float is insignificantly positive. This is hardly surprising, as ILLIQ measures stock liquidity in terms of the price impact dimension and reflects the liquidity costs for large volume trades, which are frequently carried out by large, often long-term investors. Since amILLIQ FF reflects ILLIQ amortised over the average holding period of short-term investors, an insignificantly positive coefficient suggests short-term investors are not concerned about the price impact of their trades (that are rather of small volume).
Taken all together, we can summarise as follows. As the use of raw (non-transformed) liquidity measures neglects the liquidity clientele effect, and the coefficient on nontransformed PQCS and ILLIQ is (contrary to our expectations) insignificant, we can conjecture that we should take the liquidity clientele effect into account when analysing the illiquidity-return relationship. However, as evidenced by the (contrary to the theory) significantly negative coefficient on log-transformed PQCS and ILLIQ, log-transformation does not reflect the clientele effect in a liquidity-return relationship in CEE markets. If it did so, the estimated coefficient would be significantly positive. Significantly positive coefficients on amortised PQCS and ILLIQ seem to support this conjecture.
Not less important, the estimated coefficients on amortised liquidity measures are of economically relevant values. One standard deviation increase in amortised liquidity measure results in an increase in a monthly excess return of about 0.333% (0.119% if we adjust the holding period for free float), which is 195% (66.5%) of the mean excess return and 4.44% (1.51%) of its standard deviation.
Though we find evidence that, unlike raw or log transformations of liquidity measures, amortised liquidity measures are priced coherently with the theory, we should also take into Note(s): The table presents the estimated effect of market liquidity on stock returns with different approaches to taking the holding period into account. The dependent variable is the monthly excess (over risk-free return) return on the stock. t-statistics are given in the parentheses and asterisks denote the statistical significance at the 0.1 (*), 0.05 (**) and 0.01 (***) levels Table 4. The effects of stock market liquidity on returns IJOEM account that they reveal highly non-normal distribution, which could bias our results. As depicted in Panel A of Table 2, liquidity measures amortised over the holding period exhibit elevated skewness and kurtosis. To check whether our results are not driven mainly due to the non-normal distribution of amortised liquidity measures, we log-transform them to alleviate concerns related to skewness and kurtosis. As presented in Panel B of Table 2, log transformation vividly reduces the skewness and kurtosis of amortised liquidity measures. The results of the estimation for log transformations of amortised liquidity measures are delivered in Table 5, columns (1)-(4). Coefficients on a log of amortised PQCS and ILLIQ remain significantly positive. This effect is not only statistically significant but also economically relevant. One standard deviation increase in log-transformed amortised liquidity measures results in an increase of monthly excess return by 1.18-1.28%, which constitutes 15.64-17.05% of its standard deviation.
By log transformation of the amortised liquidity measure we are able to indicate whether significantly positive coefficients on amortised liquidity measures are not simply the result of dividing the liquidity measure by the holding period. To this end, based on equation (4), we include both the log transformed liquidity measures and the log transformed holding period in the regressions. If both variables are significant in the liquidity-return relationship, we should expect significantly positive coefficients on log(PQCS) and log(ILLIQ) and significantly negative coefficient on log(HP) and log(HP FF ). The results of the estimation are presented in Table 5, columns (5)-(8).
The estimated coefficients are in line with our expectations: in all the regressions, the coefficients on log-transformed liquidity measures are significantly positive, and the coefficients on the log-transformed holding period (also adjusted for free float) are significantly negative. This means that both the level of liquidity and the investors' holding period play a significant role in the liquidity-return relationship, as both are significantly priced. We can thus claim that investors take their expected holding period into account when forming their expectations as to compensations for holding illiquid stocks. Our results are in line with the theory, i.e. the shorter the investment horizon, the higher the compensation for illiquidity (liquidity premium).
Keeping the investors' holding period constant, a one standard deviation increase in log(PQCS) (log(ILLIQ)) results in an increase of monthly excess return by 0.32-0.35% (0.46-0.56%), which constitutes about 4.5% (6.7%) of its standard deviation. One should remember that an increase in illiquidity would result in an increase in the investors' holding period, hence the above numbers may seem overestimated. However, these numbers (at least for PQCS) are similar to respective values for the amortised illiquidity measures. This suggests that the effect of stock liquidity on returns is not only statistically relevant, but also economically.
It is worth noticing that the investors' holding period seems to moderate the relationship between the log-transformed liquidity measure and stock returns, as including the holding period in the regressions changes the value of the estimated coefficient on the log-transformed illiquidity measure from negative to positive. While the coefficient on logtransformed illiquidity measures without controlling for the holding period is significantly negative, this coefficient becomes significantly positive when we control for the investors' holding period. This directly supports our hypothesis on the moderating role of the investors' holding period in the illiquidity-return relationship.
Our results imply that, at least in the CEE stock markets, the effect of liquidity clientele in a liquidity-return relationship cannot be simply reduced to a logarithmic transformation of the liquidity measure. As evidenced, even after the log transformation of PQCS and ILLIQ, the holding period remains statistically significant in shaping the effect of illiquidity on stock returns. This in turn suggests that assuming a holding period equal for all stocks in asset pricing studies on relatively illiquid markets can lead to a biased estimation of liquidity premium.

Investor holding period and liquidity premium
Model (1) (3) Note(s): The table presents the estimated effect of market liquidity and holding period on stock returns. The dependent variable is the monthly excess (over risk-free return) return on the stock. t-statistics are given in the parentheses and asterisks denote the statistical significance at the 0.1 (*), 0.05 (**) and 0.01 (***) levels  (Chung and Zhang, 2014) and Amihud's (2002) illiquidity ratio with its original definition (Amih). The results of the estimation are presented in Table 6. As presented, the coefficients on log-transformed PECS and Amih are significantly negative in regressions without controlling for the investors' holding period and become significantly positive after controlling for the holding period. This confirms that the investors' holding period moderates the relationship between stock liquidity and returns, and our conclusions remain unchanged when liquidity is measured differently. The results presented in Section 4 may be subject to bias due to the heteroskedasticity of residuals. Though we estimate t-statistics for the coefficients using robust standard errors clustered at the firm level, we also apply a Weighted Least Squares estimation to check whether the coefficients remain statistically significant if we take the heteroskedasticity into account. To this end, we utilise two different weighting schemes. Our first approach is based on the scheme of Asparouhova et al. (2010) and we use 1 plus return from the previous month as a weighting variable.
Our second approach is the Groupwise Weighted Least Squares for panel data. WLS estimates are computed with the weights based on the inverses of the estimated variances of residuals for the respective cross-sectional units in the sample. We iterate this procedure, i.e. at each round, the residuals are re-computed using the current WLS parameter estimates, which gives a rise to a new set of estimates of the variances of residuals, and a hence a new set of weights. Iteration is stopped when the maximum difference in the coefficients estimates from one round to the next falls below 0.0001 or the number of iterations reaches 20.
The results of the estimation with WLS with both weighting schemes are presented in Table 7. The coefficients on the log transformations of our liquidity measures remain significantly positive, while the coefficients on log holding periods remain significantly negative. This suggests that our previous results are not subject to bias resulting from the heteroskedasticity of residuals, and are robust to the choice of the estimation method.

Discussion and conclusions
The paper aimed to identify the role the investment holding period plays in a liquidity-return relationship in fourteen CEE stock markets. As CEE capital markets are very illiquid and characterised by highly concentrated trading and ownership, they provide an excellent setting to study the role of investors' holding periods in a liquidity-return relationship. As the group of CEE markets consists of both emerging and frontier markets, our results could possibly be expanded to other emerging and frontier markets that are definitely less liquid than developed stock markets. We conjecture that the investors' holding period does indeed moderate the illiquidity-return relationship in our sample.
We find no convincing evidence that would force us to reject our hypothesis. Coefficients on log-transformed liquidity measures are significantly negative when we do not control for investors' holding period and become significantly positive after controlling for it. It is worth noticing that the coefficients on other control variables remain unchanged. Also, the coefficients on liquidity measures amortised over the average investors' holding period are significantly positive. All our results suggest that investors take their investment horizon into account when forming the expected compensations for stock illiquidity (liquidity premium).
Our results also suggest that one should be careful in making assumptions about investors' holding periods when estimating the liquidity premium, as assuming an inappropriate holding the monthly excess (over risk-free return) return on the stock. t-statistics are given in the parentheses and asterisks denote the statistical significance at the 0.1 (*), 0.05 (**) and 0.01 (***) levels Note(s): The table presents the estimated effect of market liquidity and holding period on stock returns using Weighted Least Squares estimation. The dependent variable is the monthly excess (over risk-free return) return on the stock. t-statistics are given in the parentheses and asterisks denote the statistical significance at the 0.1 (*), 0.05 (**) and 0.01 (***) levels Table 7. The effects of stock market liquidity on returns -WLS estimation Investor holding period and liquidity premium period may lead to a biased estimation of the liquidity premium. We show that a simple log transformation of liquidity variables is insufficient to capture the liquidity clientele effect on the illiquidity-return relationship in the CEE markets. The study has important practical implications for investors, companies, policymakers and regulators. As suggested by our results, the liquidity premium in the CEE stock markets is both statistically significant and economically relevant. The effect of illiquidity on stock returns may however be alleviated by lengthening the investment holding period. Thus, investors should consider stock liquidity when making investment decisions to assure their returns will exceed liquidity costs. This is especially important for short-term investors, especially high-frequency traders, as a short investment horizon or frequent trading strengthens the liquidity effects. Also, companies should consider the liquidity of their shares and recognise the investment horizon of their investors when making financing decisions as stock liquidity affects the cost of equity capital and the investors' holding period moderates the strength of this effect.
Since stock liquidity affects expected returns and companies' cost of equity capital as a consequence, low liquidity in CEE markets limits their development opportunities. Because the opportunity cost of capital is higher for firms with more illiquid stock, managers accept fewer investment projects as the supply of projects is a declining function of their expected rate of return. As a consequence, enhanced stock liquidity expands the set of investment opportunities (Amihud and Levi, 2018;Im et al., 2022). Thus, liquidity-enhancing actions should be employed to decrease the cost of capital of the companies listed on the CEE markets, expand the investment opportunities and, as a consequence, improve growth perspectives for the entire economy. To improve stock liquidity, managers can consider measures aimed at increasing their investor base by attracting small investors (Amihud and Mendelson, 2000). Policymakers and regulators should strive for systemic improvement in the listed companies' disclosing quality.
Our study also shows that the stock markets of Central and Eastern Europe may be attractive for foreign investors. As they are less integrated with the global economy, CEE markets offer some diversification opportunities and benefits related to it. Furthermore, since CEE markets offer a significant liquidity premium, foreign investors, especially longterm ones, can be compensated for the low liquidity of stocks listed on them. Thus, investing in the CEE stock markets would allow investors to decrease their portfolios' risks and earn a premium for holding relatively less liquid stocks. This effect may be even more pronounced in periods of global crises, such as the 2007 crisis and COVID-19 pandemic, as CEE markets are not highly integrated, and they might not be really affected by global crises.
We perceive one serious limitation of our study, which results from the proxy for the investors' holding period that was applied. As suggested by Naes and Ødegaard (2009), the inverse of the turnover ratio may be an imperfect proxy for the investment horizon as the turnover ratio is a characteristic of a stock, while the holding period is a decision made by individual investors. However, if we take the marginal investor into account, an inverse of the turnover ratio may function quite well as a proxy for her holding period as it would reflect an average holding period of all investors for a given stock. In this way, the use of the inverse of turnover ratio as a proxy for the holding period seems to be justified.
Taking the limitations of our study into account, we also perceive a need for further research on the role the investors' holding period plays in the stock liquidity-return relationship, especially in other emerging and frontier markets. Such studies could also be of importance to developed markets. The use of more sophisticated proxies for the holding period as well as the application of other estimation techniques and liquidity measures would be desired.