OR I G I N A L A R T I C L E Improving momentum returns using generalized linear models Hui Zeng1 | Ben R. Marshall1 | Nhut H. Nguyen2 | Nuttawat Visaltanachoti1 1School of Accounting, Economics and Finance, Massey University, North Shore Auckland, New Zealand 2Department of Economics and Finance, Auckland University of Technology, Auckland, New Zealand Correspondence Hui Zeng, School of Accounting, Economics and Finance, Massey University, Private Bag, North Shore Auckland 102,904, New Zealand. Email: h.zeng@massey.ac.nz Abstract We estimate the enduring momentum probabilities of past winners and losers continuing as future winners and losers by incorporating a comprehensive set of firm characteris- tics. Our results reveal that combining the price momentum signals and enduring momentum probabilities generates returns double those of the traditional price momentum strategy. Furthermore, the robust performance of the enduring momentum strategy cannot be fully attributed to factors such as seasonality, limits to arbitrage, and transac- tion costs. K E YWORD S enduring momentum probability, firm characteristics, momentum J E L C L A S S I F I C A T I ON C13, C31, C53, G17 1 | INTRODUCTION The price momentum strategy, where past winning firms continue to outperform while past losing firms persist in underperformance, remains one of the most puzzling anomalies in asset-pricing research (Fama, 1998). Researchers have identified that specific firm attributes and fundamental factors can enhance traditional price momentum returns by providing additional predictive information (e.g., Huang et al., 2019; Sagi & Seasholes, 2007). In this study, we employ a Cox Proportional Hazards (Cox PH) model (Cox, 1972), and five other generalized linear models (GLM), to We thank our editor Prof Tang, and two anonymous referees for comments that have substantially improved the paper. All errors are our own. Received: 12 April 2022 Revised: 10 March 2025 Accepted: 11 March 2025 DOI: 10.1111/irfi.70014 This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited. © 2025 The Author(s). International Review of Finance published by John Wiley & Sons Australia, Ltd on behalf of International Review of Finance Ltd. International Review of Finance. 2025;25:e70014. wileyonlinelibrary.com/journal/irfi 1 of 35 https://doi.org/10.1111/irfi.70014 mailto:h.zeng@massey.ac.nz http://creativecommons.org/licenses/by/4.0/ http://wileyonlinelibrary.com/journal/irfi https://doi.org/10.1111/irfi.70014 http://crossmark.crossref.org/dialog/?doi=10.1111%2Firfi.70014&domain=pdf&date_stamp=2025-04-11 identify stocks that deliver superior momentum returns. Our models integrate information from past momentum sur- vival time and 37 firm characteristics to estimate the probability that a winner (loser) will continue being a winner (loser) over a 6-month investment period. This probability is referred to as the “enduring momentum probability”. Specifically, in the first stage of our estimation process, we identify winners and losers at the end of each month t based on their cumulative returns over the preceding 6-month period (t � 5 to t). We then compute the momentum survival time for each winner and loser over the subsequent 6-month period (t + 1 to t + 6) by counting the number of months a winner or loser continues in its respective category.1 We refer to this momentum survival time as ‘enduring time’. We also calculate a large set of firm characteristics for each firm at the end of each month t. This process is repeated every month in our sample period to generate a time series of enduring time and firm characteristics. In the second stage of our estimation process, we fit the enduring time and firm characteristics from month t � 60 to month t � 6 to the Cox PH model to estimate the model parameters.2 The enduring momentum probability for a winner (loser) to remain in the winning (losing) group in the next six months (t + 1 to t + 6) is calculated by applying these parameters to the winner's (loser's) firm characteristics in month t. Finally, we establish our enduring momen- tum strategy by buying 10 winners and selling 10 losers in month t with the highest enduring momentum probabili- ties. We hold this long-short portfolio for six months and rebalance it on a monthly basis. Our analysis reveals that the enduring momentum probabilities estimated by the Cox PH model for winners and losers are significantly associated with the persistence of stock returns. Firms with higher enduring momentum prob- abilities exhibit stronger return persistence compared to those with lower probabilities. In addition, we show that these probabilities play a critical role in predicting cross-sectional stock returns. By conditioning on a comprehensive set of 37 firm characteristics, the enduring momentum probabilities enhance traditional momentum returns, exten- ding beyond the conventionally examined variables such as price momentum signals, book-to-market ratio, size, illi- quidity ratio, and volatility. Our results show that the enduring momentum long-short portfolio achieves an average monthly return of 2.19%, nearly double the 1.12% return of the traditional price momentum portfolio (e.g., Jegadeesh & Titman, 1993). In addition to the Cox PH model, we employ five other GLMs, that is, Poisson, logit, fractional logit, linear probability, and OLS models, to estimate the enduring momentum probability and form long- short portfolios in a similar manner. We find that these portfolios also outperform the price momentum portfolio, although their performances are still inferior to the Cox PH portfolio's performance. Since the superior performance of the enduring momentum strategy to the price momentum strategy is exhibited in both average portfolio returns and risk-adjusted returns, with the loser portfolio component apparently playing a central role in driving this outperformance, we investigate whether the returns are due to limits to arbi- trage. Li et al. (2009) argue that trading winners is generally less costly than trading losers. Investors face fewer bar- riers when buying stocks than short selling, which is more complex and costly due to borrowing requirements, fees, regulatory constraints, and the risk of unlimited losses. Consistent with this perspective, studies by Shleifer and Vishny (1997) and Lamont and Thaler (2003) highlight how these barriers prevent arbitrageurs from effectively cor- recting market mispricing. Arena et al. (2008) suggest that momentum profits are associated with limits to arbitrage; however, our results show that the enduring momentum returns are unlikely attributable to limits to arbitrage. They prevail after we remove small stocks, volatile stocks, and illiquid stocks from the sample. We also conduct numerous robustness checks on the performance of our enduring momentum strategy based on previous literature on price momentum. These include excluding the top 10% of microcap stocks (Asness et al., 2013), varying the portfolio holding period of up to 12 months (Jegadeesh & Titman, 1993), addressing the sea- sonal effect or long-term reversal of momentum performance (De Bondt & Thaler, 1985; Jegadeesh & Titman, 1993), extending the number of winners and losers in the portfolio, applying different weighting schemes (Korajczyk & Sadka, 2004), and adjusting for different risk factors (e.g., Barroso & Santa-Clara, 2015; Da et al., 2014). Overall, we show that the enduring momentum continues to show consistent results and outperforms the price momentum returns in most tests. Finally, our results indicate that the enduring momentum strategy performs well during market crashes and that it bears a higher break-even transaction cost than the price momentum strategy. 2 of 35 ZENG ET AL. 14682443, 2025, 2, D ow nloaded from https://onlinelibrary.w iley.com /doi/10.1111/irfi.70014 by M assey U niversity L ibrary, W iley O nline L ibrary on [27/04/2025]. See the T erm s and C onditions (https://onlinelibrary.w iley.com /term s-and-conditions) on W iley O nline L ibrary for rules of use; O A articles are governed by the applicable C reative C om m ons L icense Our paper contributes to existing research on firm characteristics that moderate momentum returns. A growing body of literature highlights that momentum is intricately driven by various firm characteristics rather than solely by price signals (e.g., Daniel & Titman, 1999; Guo et al., 2022; Hong et al., 2000). We identify age, bid-ask spread, trad- ing volume, leverage, maximum daily return, volatility, secured debt, and the sales-to-price ratio as the most critical characteristics in selecting stocks in enduring momentum portfolios. Several studies support the importance of these characteristics. For example, Lee and Swaminathan (2000) demonstrate that past trading volume predicts both the magnitude and persistence of price momentum. Forner et al. (2018) show that incorporating leverage on the short side of momentum portfolios can yield higher abnormal returns. Barroso and Santa-Clara (2015) find that the realized variance of daily returns is highly predictable, and scaling the long-short portfolio by its realized volatility results in substantial economic gains. Avramov et al. (2007) document a strong relationship between momentum and credit rating, where momentum profitability is higher for low-grade firms but non-existent among high-grade firms. Moti- vated by Daniel and Moskowitz (2016), this paper contributes to the literature on conditional momentum by compre- hensively incorporating firm characteristics to refine momentum signals and develop an enhanced trading strategy, namely the enduring momentum strategy. The remainder of this paper is organized as follows. Section 2 provides the data and methodology for estimating the enduring momentum probability and constructing an enduring momentum strategy. Section 3 presents the empirical results. Section 4 reports robustness check results, and Section 5 concludes the paper. 2 | DATA AND METHODOLOGY 2.1 | Data We collect daily returns for all common stocks on the New York Stock Exchange (NYSE), the American Stock Exchange (AMEX), and Nasdaq from the Center for Research in Security Prices (CRSP) database. The monthly returns of all the individual firms used to construct the 6-month/6-month price momentum strategy are obtained by com- pounding their daily returns, and we exclude stocks with share prices less than $1 at the beginning of the holding month. In addition, we obtain data from the Compustat database and follow Green et al. (2017)3 to construct 102 firm characteristics. Green et al. (2017) argue that data from January 1980 become robustly available. Thus, our analysis starts in January 1980 and ends in December 2018, covering 38 years. To maximize the predictive ability of firm characteristics and ensure sufficient data for the Cox PH model and other GLM models to calculate the enduring momentum probability, we only retain firm characteristics with missing data of less than 5% over the entire sample period. In addition, we exclude all momentum variables to avoid the momentum effect in the construction of the enduring momentum strategy. After these screening criteria, 37 firm characteristics remain, and we provide their descriptions in Table A1. Finally, we collect the four Fama–French and Carhart factors (MKT, SMB, HML, and UMD)4 from Kenneth French's data library.5 2.2 | Methodology 2.2.1 | Enduring momentum probability and enduring momentum strategy This section introduces the terminology common to enduring momentum analysis and discusses the estimation of enduring momentum probability by applying the Cox PH model. It also introduces the difference between the endur- ing momentum and survival probability and describes how we use the estimated enduring momentum probability to construct the enduring momentum strategy. Figure 1 illustrates the steps and timeline for constructing the enduring momentum strategy. This estimation process includes two stages. In the first stage, we identify momentum winners ZENG ET AL. 3 of 35 14682443, 2025, 2, D ow nloaded from https://onlinelibrary.w iley.com /doi/10.1111/irfi.70014 by M assey U niversity L ibrary, W iley O nline L ibrary on [27/04/2025]. See the T erm s and C onditions (https://onlinelibrary.w iley.com /term s-and-conditions) on W iley O nline L ibrary for rules of use; O A articles are governed by the applicable C reative C om m ons L icense and losers, count their enduring time, and compute their 37 firm characteristics. Specifically, at the end of each month t, we identify momentum winners and losers based on their cumulative returns over the preceding 6-month period (t � 5 to t). Next, we determine the survival time of each winner (loser) firm, calculated as the number of months it remains in its respective category during the subsequent 6-month period (i.e., t + 1 to t + 6) and denote this enduring time as NW i (NL i ). The 37 firm characteristics associated with each winner (loser) are also calculated or observed at the end of month t. These steps provide us with a time series of winners and losers, their enduring time, and firm characteristics throughout our sample period. In the second stage of estimation, for each month t, we use the time series data from month t � 60 to month t � 6 (i.e., 54 months) and fit the enduring time and firm characteristics into the Cox PH model to estimate the model parameters. We then compute the enduring momentum probability for a winner (loser) in month t to stay in the win- ning (losing) group in the following six months (t + 1 to t + 6) by applying these parameters to the winner's (loser's) firm characteristics in that month. Finally, we rank these enduring momentum probabilities and form our enduring momentum portfolio by buying 10 winners and selling 10 losers with the highest estimated enduring momentum probabilities. We hold this long-short portfolio for six months and rebalance it monthly. In the traditional Cox PH models, survival analysis is typically conducted over a continuous interval from t = 1 to t = T, with each subject i assigned a survival time Ti upon failure. However, in our analysis, the winners or losers formed at the end of month t to keep performing as winners or losers can only occur at discrete points over the 6-month holding period (i.e., from t + 1 to t + 6). Therefore, our enduring time variables, NW i for winners and NL i for losers, represent discrete survival time. For example, if firm i is identified as a winner at the end of formation month t and continues to be a winner in the second (t + 2), fourth (t + 4), and sixth months (t + 6) of the holding period, the enduring time NW i for this winner firm is 3. F IGURE 1 Steps and timeline to estimate the enduring momentum probability. First, at the end of each month, we identify momentum winners (W) and losers (L) based on their cumulative returns over the preceding 6-month period. Second, we determine the enduring time of each winner (NW) or loser firm (NL), calculated as the number of months it remains in its respective category during the subsequent 6-month period. The 37 firm characteristics associated with each winner (FCW) or loser (FCL) are also calculated or observed at the end of the month. These steps provide us with a time series of winners and losers, their enduring time, and firm characteristics throughout our sample period. Third, we use the time series data from month t � 60 to month t � 6 (i.e., 54 months) and fit the enduring time and firm characteristics into the Cox PH model to estimate the model parameters. Fourth, we compute the enduring momentum probability for a winner (loser) in month t to stay in the winning (losing) group in the following six months (t + 1 to t + 6) by applying these parameters to the winner's (loser's) firm characteristics in that month. Finally, we rank these enduring momentum probabilities and form our enduring momentum portfolio by buying 10 winners and selling 10 losers with the highest estimated enduring momentum probabilities. We hold this long-short portfolio for six months and rebalance it monthly. 4 of 35 ZENG ET AL. 14682443, 2025, 2, D ow nloaded from https://onlinelibrary.w iley.com /doi/10.1111/irfi.70014 by M assey U niversity L ibrary, W iley O nline L ibrary on [27/04/2025]. See the T erm s and C onditions (https://onlinelibrary.w iley.com /term s-and-conditions) on W iley O nline L ibrary for rules of use; O A articles are governed by the applicable C reative C om m ons L icense The below hazard function describes the relative likelihood of a winner or loser exiting the top or bottom decile group based on τ times (0≤ τ ≤ 6) that it has survived in the top or bottom group: λ τjXð Þ¼ lim Δτ!0 P τ ≤ T ≤ τþΔτjT ≥ τ,X,βð Þ Δτ ¼�dlogS τjXð Þ dτ ¼ f τjXð Þ S τjXð Þ¼ λ0 τð Þexp β0Xð Þ ð1Þ where f τjXð Þ is the density function associated with the distribution of survival duration, λ0 τð Þ is the baseline hazard function with all covariates equal to zero, β is a vector of the parameters, and X represents a set of 37 firm character- istics. The survival function S τjXð Þ¼Prob T ≥ τjXð Þ is related to the hazard by: S τjXð Þ¼ exp � ðτ 0 λ ujXð Þdu � � ¼ exp � ðτ 0 λ0 uð Þexpβ0Xdu � � , ¼ exp �Λ0 τð Þexp β0Xð Þð Þ ¼ S0 τð Þexp β0Xð Þ, ð2Þ where Λ0 τð Þ¼ Ð τ0λ0 uð Þdu represents the baseline cumulative hazard function at time τ, and S0 τð Þ¼ exp �Λ0 τð Þð Þ. Let x� denote a particular value of X at the end of rolling window t. The estimated enduring momentum probability is defined by substituting estimators for the unknown quantities as: bS τjx�ð Þ¼ exp �bΛ0 τð Þexp bβ0X� �� � , ð3Þ At the end of each rolling window in month t, we obtain seven estimated enduring momentum probabilities6 for each winner or loser to stay in the corresponding group during the 6-month holding period (t + 1 to t + 6), as τ ranges from one to six. For example, bSW=L 6ð Þ (τ¼6) represents the estimated enduring probability for a winner (W) or loser (L) to remain so throughout the 6-month holding period. Unlike the price momentum strategy to hold and sell all past winners and losers, our enduring momentum portfolio only includes the top 10 winners and top 10 losers with the highest estimated enduring probabilities, bSW 6ð Þ and bSL 6ð Þ, respectively. 2.2.2 | Enduring momentum with general linear model (GLM) estimation In this section, we introduce a series of generalized linear models, including the Poisson, logit, fractional logit, linear probability, and OLS models, to construct enduring momentum portfolios and compare their performances with the Cox PH. Although all five GLMs follow the same estimation procedure to facilitate a fair comparison with the Cox PH model, variations exist in the definition of the dependent variable due to the unique characteristics of each model. We use the same enduring time variables, NW i and NL i , as the dependent variable in the Poisson and OLS models. Since in the logit model the dependent variable needs to be a binary, we define this variable, yW Lð Þ i , as follows: yW Lð Þ i ¼ 1, NW i NL i � � 6 ≥0:5 0, NW i NL i � � 6 <0:5 8>><>>: : ð4Þ ZENG ET AL. 5 of 35 14682443, 2025, 2, D ow nloaded from https://onlinelibrary.w iley.com /doi/10.1111/irfi.70014 by M assey U niversity L ibrary, W iley O nline L ibrary on [27/04/2025]. See the T erm s and C onditions (https://onlinelibrary.w iley.com /term s-and-conditions) on W iley O nline L ibrary for rules of use; O A articles are governed by the applicable C reative C om m ons L icense In the fractional logit and linear probability models, the dependent variable is defined as in Equation (5) below: D_ratioWi D_ratioLi � � ¼NW i NL i � � 6 , ð5Þ where D_ratioWi D_ratioLi � � represents the enduring ratio for winner (loser) firm i, calculated by dividing its enduring time NW i NL i � � by six. Using these dependent variables, we obtain the model parameters, estimate enduring momen- tum probabilities, and then construct enduring momentum portfolios in a similar manner as described above. Compared with the Cox PH model, the above five GLMs are conventional methods commonly used in estimating the enduring momentum probabilities. However, there are some restrictions and limitations for these models. For exam- ple, the OLS model assumes that the effects are linear and additive. Moreover, the estimated enduring time fit in the Poisson, linear probability, and OLS models will cause the out-of-bound prediction problem that the estimated enduring probabilities of the Poisson model may exceed the total investment months (6) or the estimated enduring probabilities of the OLS and linear probability models may be less than 0. Besides, the Poisson model is required to adhere to the assumption E(Y) = V(Y), where the expected value of enduring time is equivalent to its variance. The dependent variable in the logit model is binary, which will compress the total number of enduring time information to 1 and 0. Since the Cox PH model is more flexible, as it is semi-parametric, it does not make strong assumptions about the shape of the baseline hazard function. Moreover, the Cox PH model produces greater statistical power than a logistic regression model with the same available follow-up data (Annesi et al., 1989; Cuzick, 1982). Given the above limitations of the five other GLMs compared to the Cox PH, our analysis mainly focuses on the enduring momentum probability (EMP) and enduring momentum portfolio/returns (EM) based on the Cox PH model unless otherwise specified. 2.2.3 | Autocorrelation regression Sagi and Seasholes (2007) claim that some specific firm attributes (revenues, costs, and real options) restrict winners and losers, and that restricted winners and losers have more persistent expected returns. We test whether the enduring momentum probability is related to the persistence of stock returns. We first run the autoregression with a one-month lag (AR (1)) for each firm based on a 60-month rolling window: ri,t�60:t ¼ ai,tþρi,tri,t�61:t�1þei,t, ð6Þ where ri,t�60:t (ri,t�61:t�1) is the stock return of firm i from month t � 60 to t (t � 61 to t � 1). After obtaining the slope ρi,t (the proxy for stock return persistency), we determine the relationship between the enduring momentum proba- bility and the persistence of stock returns using the multiple cross-sectional regression over time as follows: ρh,t ¼ αh,tþβEMP h,t EMPh,t�1þεh,t, ð7Þ where EMPh,t�1 represents the enduring momentum probabilities for the winner (h = W) or loser (h = L) firms, esti- mated at the end of month t � 1. Subsequently, we calculate the average value of all estimated coefficients βh,t and its corresponding t value. 2.2.4 | Logit regression We compare the performance of the estimated enduring momentum probability and the price momentum signal (i.e., past 6-month cumulative returns) in determining future winners or losers over the 6-month holding period using the following panel logit regression: 6 of 35 ZENG ET AL. 14682443, 2025, 2, D ow nloaded from https://onlinelibrary.w iley.com /doi/10.1111/irfi.70014 by M assey U niversity L ibrary, W iley O nline L ibrary on [27/04/2025]. See the T erm s and C onditions (https://onlinelibrary.w iley.com /term s-and-conditions) on W iley O nline L ibrary for rules of use; O A articles are governed by the applicable C reative C om m ons L icense Yh,t ¼ akhþ γkhx k h,tþϵkh,t, ð8Þ where Yh,t is a dummy variable equal to one for a winner (h = W) or loser (h = L) firm of price momentum identified at the end of month t to keep performing as a winner or loser in any month of the subsequent 6-month holding period and zero otherwise. xkh,t is the estimated enduring momentum probability (k = EMP) or the cumulative returns over the preceding 6-month period (k = MOM) associated with that winner or loser. We also explore the explanatory ability of the enduring momentum probability after controlling for the price momentum signal as depicted in Equation (9) below. Yh,t ¼ ahþ γEMP h EMPh,tþ γMOM h MOMh,tþϵh,t, ð9Þ where Yh,t is the same dummy variable as in Equation (8). MOMh,t is the price momentum signal for a winner (h = W) or loser (h = L). EMPh,t is the estimated enduring momentum probability for the same winner or loser.7 2.2.5 | Decomposition analysis We compare the predictive ability of the enduring momentum probability and momentum signal in forecasting future returns using the following cross-sectional regressions: rh,tþ1 ¼ akhþϕk hx k h,tþϵkh,t, ð10Þ where rh,tþ1 is the stock return for a winner (h = W) or loser (h = L) firm in month t + 1. xkh,t is the estimated enduring momentum probability (k = EMP) or the price momentum signal (k = MOM) for that winner or loser. We also include both EMP and MOM in the same predictive regression: rh,tþ1 ¼ akhþϕEMP h EMPh,tþϕMOM h MOMh,tþ εh,tþ1, ð11Þ However, Hou and Loh (2016) argue that the traditional method of adding competing variables cannot quantify the explanatory fraction of candidate variables. They introduce a decomposition method to comprehensively evaluate the explained fraction of the candidate variables. We follow their method to detect whether the momentum signal can explain the return captured by the enduring momentum probability. We first explore whether enduring momentum probability affects the predictability of stock returns using the following cross- sectional regression: ri,tþ1 ¼ αh,tþηh,tEMPh,i,tþεh,i,t, ð12Þ where ri,tþ1 is the stock return of firm i in month t + 1. EMPh,i,t is the enduring momentum probability for a winner (h = W) or loser (h = L) firm estimated in month t. We consider the price momentum return, MOMh,i,t, as a candidate variable and examine the relationship between the EMPh,i,t and MOMh,i,t as follows: EMPh,i,t ¼ ah,tþ ζh,tMOMh,i,tþϵh,i,t, ð13Þ Subsequently, we decompose EMPh,i,t into two orthogonal components based on the estimated coefficients ηh,t in Equation (12): ZENG ET AL. 7 of 35 14682443, 2025, 2, D ow nloaded from https://onlinelibrary.w iley.com /doi/10.1111/irfi.70014 by M assey U niversity L ibrary, W iley O nline L ibrary on [27/04/2025]. See the T erm s and C onditions (https://onlinelibrary.w iley.com /term s-and-conditions) on W iley O nline L ibrary for rules of use; O A articles are governed by the applicable C reative C om m ons L icense ηh,t ¼ Cov rh,i,tþ1, EMPh,i,tð Þ Var EMPh,i,tð Þ , ¼Cov rh,i,tþ1, ζh,tMOMh,i,t � � Var xi,j,t�1 � � þCov rh,i,tþ1, ah,tþϵh,i,tð Þ Var xi,j,t�1 � � , ¼ ηCh,tþηRh,t, ð14Þ where ζh,tMOMh,i,t (ah,tþϵh,i,t) is the related (residual) component of EMPh,i,t. Finally, we use ηCh,t ηh,t ( ηRh,t ηh,t ) to calculate the explained (residual) fractions from the price momentum signal of winners (h = W) or losers (h = L) by estimating the mean and variance of the fractions over the whole regression period as follows: bE ηCh,t ηh,t ! ≈ ηCh,t ηh,t ,bE ηRh,t ηh,t ! ≈ ηRh,t ηh,t , ð15Þ dVar ηCh,t ηh,t ! ≈ 1 T ηCh,t ηh,t !2 σ2 ηC h,t ηCh,t 2 þ σ2ηh,t ηh,t 2 �2 bρη C h,t ,ηh,t σ2 ηC h,t σ2ηh,t ηCh,tηh,t 0@ 1A, ð16Þ dVar ηRh,t ηh,t ! ≈ 1 T ηRh,t ηh,t !2 σ2 ηR h,t ηRh,t 2 þ σ2ηh,t ηh,t 2 �2 bρη R h,t ,ηh,t σ2 ηR h,t σ2ηh,t ηRh,tηh,t 0@ 1A, ð17Þ and, tηC h,t ηh,t ¼ ηCh,t ηh,t σ ηCh,t ηh,t � � ,tηR h,t ηh,t ¼ ηRh,t ηh,t σ ηRh,t ηh,t � � , ð18Þ where tηC h,t ηh,t tηR h,t ηh,t 0@ 1A is the t value used to examine whether the explained (residual) fraction is significantly different from zero. 3 | EMPIRICAL RESULTS 3.1 | Summary statistics of firm characteristics Table 1 summarizes the 37 firm characteristics from January 1980 to December 2018, presenting the mean, median, standard deviation (std), and the fifth (P5), twenty-fifth (P25), seventy-fifth (P75), and ninety-fifth (P95) percentile values. We can see that most firm characteristics are positive; for example, the average monthly return volatility (retvol) and idiosyncratic volatility (idiovol) are 3.0% and 7.0%, with standard deviations of 0.03 and 0.04, respec- tively, suggesting that stock returns are volatile over January 1980 to December 2018. Guo et al. (2022) show that the average monthly return volatility and idiosyncratic volatility are 2.5% and 2.3%, with standard deviations of 0.014 and 0.013 from July 1963 to December 2016. Table 2 presents the Pearson correlations between the 37 firm characteristics and the estimated enduring momentum probabilities from January 1985 to December 2018. 25 of them display positive and statistically signifi- cant relationships with the estimated enduring momentum probabilities. For example, the results for baspread, beta, bm, retvol, idiovol indicate that winner and loser firms with higher bid-ask spread, beta, book-to-market ratio, return 8 of 35 ZENG ET AL. 14682443, 2025, 2, D ow nloaded from https://onlinelibrary.w iley.com /doi/10.1111/irfi.70014 by M assey U niversity L ibrary, W iley O nline L ibrary on [27/04/2025]. See the T erm s and C onditions (https://onlinelibrary.w iley.com /term s-and-conditions) on W iley O nline L ibrary for rules of use; O A articles are governed by the applicable C reative C om m ons L icense TABLE 1 Summary statistics of firm characteristics. N Variables Mean Median Std P5 P25 P75 P95 1 age 11.32 9.00 9.18 1.00 4.00 16.00 31.00 2 baspread 0.06 0.04 0.07 0.01 0.02 0.06 0.15 3 beta 1.09 1.02 0.67 0.14 0.61 1.48 2.34 4 betasq 1.65 1.04 1.90 0.03 0.38 2.19 5.46 5 bm 0.69 0.55 0.64 0.06 0.30 0.91 1.85 6 bm_ia 26.86 0.05 772.36 �42.80 �0.36 0.65 6.74 7 cashdebt �0.04 0.11 1.34 �1.27 0.01 0.26 0.82 8 cashpr �0.95 �0.26 57.47 �53.92 �8.64 5.34 49.39 9 convind 0.12 0.00 0.33 0.00 0.00 0.00 1.00 10 currat 3.33 1.94 5.05 0.68 1.18 3.26 10.23 11 depr 0.29 0.17 0.42 0.05 0.11 0.31 0.88 12 dolvol 11.37 11.32 3.06 6.47 9.17 13.64 16.40 13 dy 0.02 0.00 0.03 0.00 0.00 0.02 0.07 14 ep �0.04 0.04 0.36 �0.49 �0.02 0.08 0.17 15 herf 0.08 0.05 0.09 0.02 0.03 0.09 0.23 16 idiovol 0.07 0.06 0.04 0.02 0.04 0.09 0.14 17 ill 5.16 0.07 27.23 0.00 0.00 1.06 20.69 18 indmom 0.14 0.11 0.29 �0.26 �0.03 0.27 0.65 19 IPO 0.07 0.00 0.26 0.00 0.00 0.00 1.00 20 lev 2.23 0.64 4.77 0.04 0.22 1.88 10.03 21 maxret 0.08 0.05 0.08 0.01 0.03 0.09 0.22 22 mve 11.91 11.80 2.26 8.38 10.28 13.47 15.81 23 mve_ia �201.51 �486.39 6479.43 �4861.02 �1723.79 �104.34 4157.84 24 pricedelay 0.16 0.07 1.05 �0.74 �0.05 0.33 1.24 25 quick 2.64 1.31 4.48 0.41 0.88 2.41 9.00 26 retvol 0.03 0.03 0.03 0.01 0.02 0.04 0.09 27 roic �0.13 0.06 1.08 �1.01 �0.01 0.13 0.30 28 salecash 56.10 8.78 173.34 0.28 2.26 36.80 229.12 29 salerec 11.79 5.82 58.78 0.10 3.59 8.81 45.83 30 securedind 0.47 0.00 0.50 0.00 0.00 1.00 1.00 31 sin 0.01 0.00 0.09 0.00 0.00 0.00 0.00 32 sp 2.03 0.97 3.27 0.05 0.41 2.22 7.59 33 std_dolvol 0.87 0.80 0.42 0.32 0.53 1.15 1.65 34 std_turn 4.31 2.17 7.25 0.35 1.04 4.59 15.15 35 tang 0.54 0.54 0.16 0.25 0.46 0.62 0.82 36 turn 1.14 0.62 1.83 0.07 0.27 1.39 3.83 37 zerotrade 1.36 0.00 3.38 0.00 0.00 0.00 9.55 Note: This table reports the summary statistics of the 37 firm characteristics from January 1980 to December 2018. The definitions of these characteristic variables are presented in Table A1. ZENG ET AL. 9 of 35 14682443, 2025, 2, D ow nloaded from https://onlinelibrary.w iley.com /doi/10.1111/irfi.70014 by M assey U niversity L ibrary, W iley O nline L ibrary on [27/04/2025]. See the T erm s and C onditions (https://onlinelibrary.w iley.com /term s-and-conditions) on W iley O nline L ibrary for rules of use; O A articles are governed by the applicable C reative C om m ons L icense TABLE 2 Correlations with enduring momentum probability. N Variables Corr p-value 1 age �0.13 <0.0001 2 baspread 0.25 <0.0001 3 beta 0.14 <0.0001 4 betasq 0.14 <0.0001 5 bm 0.05 <0.0001 6 bm_ia 0.29 <0.0001 7 cashdebt �0.08 <0.0001 8 cashpr 0.00 0.1227 9 convind 0.02 <0.0001 10 currat 0.02 <0.0001 11 depr 0.04 <0.0001 12 dolvol 0.02 <0.0001 13 dy �0.04 <0.0001 14 ep �0.29 <0.0001 15 herf 0.01 <0.0001 16 idiovol 0.22 <0.0001 17 ill �0.03 <0.0001 18 indmom 0.12 <0.0001 19 IPO 0.07 <0.0001 20 lev 0.10 <0.0001 21 maxret 0.32 <0.0001 22 mve �0.07 <0.0001 23 mve_ia �0.07 <0.0001 24 pricedelay 0.02 <0.0001 25 quick 0.02 <0.0001 26 retvol 0.34 <0.0001 27 roic �0.10 <0.0001 28 salecash �0.02 <0.0001 29 salerec 0.01 <0.0001 30 securedind 0.06 <0.0001 31 sin 0.00 0.4162 32 sp 0.16 <0.0001 33 std_dolvol 0.06 <0.0001 34 std_turn 0.33 <0.0001 35 tang 0.08 <0.0001 36 turn 0.26 <0.0001 37 zerotrade �0.08 <0.0001 Note: This table reports the correlation between enduring momentum probability and firm characteristics from January 1985 to December 2018. The definitions of these characteristic variables are presented in Table A1. 10 of 35 ZENG ET AL. 14682443, 2025, 2, D ow nloaded from https://onlinelibrary.w iley.com /doi/10.1111/irfi.70014 by M assey U niversity L ibrary, W iley O nline L ibrary on [27/04/2025]. See the T erm s and C onditions (https://onlinelibrary.w iley.com /term s-and-conditions) on W iley O nline L ibrary for rules of use; O A articles are governed by the applicable C reative C om m ons L icense volatility, and idiosyncratic volatility have higher probabilities to continue to be winners and losers over the following 6-month investment period, respectively. The results are consistent with the findings of Guo et al. (2022) which show that the stocks with high past returns tend to be those with high past idiosyncratic volatility and return volatility. The Diff column in Panel A of Table A2 reports the average difference of each of the 37 firm characteristics cal- culated by averaging the monthly firm characteristics difference between the enduring momentum selected top ten winner (loser) firms and the non-selected firms within the same winner (loser) group from January 1985 to December 2018. Most firm characteristics are significantly higher for the enduring momentum selected winners and losers. For example, the monthly average bid-ask spread, baspread, is 0.06 and 0.01 higher for the enduring momentum's top ten winners and losers, respectively. The construction of the traditional price momentum strategy assumes that past winners or losers will continue to perform as winners or losers. However, this assumption does not hold for all winners and losers. Therefore, we summarize the average enduring months and the proportion of past winners and losers to keep performing as future winners and losers during the 6-month holding period in Panel B of Table A2. The N column represents the total number of months for all past winners (losers) to keep performing as winners (losers) over the 6-month holding period. We report the corresponding marginal survival and cumulative fail proportions for winners (losers) in the sec- ond (fourth) and third (fifth) columns of Panel B. In the second (fourth) column, we can see that only 7% (6%) of past winners (losers) continue to be future winners (losers) throughout the 6-month holding period, whereas 74% (75%) of past winners (losers) will no longer appear in the top (bottom) decile group over the entire investment period. The last two rows in Panel B show that the average number of enduring months for winners and losers is approximately two. Table 3 reports the partial likelihood estimates of enduring momentum portfolio returns and marginal differ- ences in R2 when each firm characteristic is dropped. We estimate the impact of each firm characteristic on the enduring momentum probability using the Cox PH specification in Equation (1). Each beta estimate reflects the par- tial impact of a firm characteristic on the probabilities of winners (losers) surviving as winners (losers) over the invest- ment period, holding the estimated enduring time constant. Besides, a positive (negative) coefficient estimate implies a shorter (longer) enduring time because the survival time is inversely related to the hazard rate. The results in Table 3 suggest that out of the 37 characteristics, 9 (8) characteristics are statistically significantly associated with a longer enduring time for winners (losers) while 20 (21) other characteristics exhibit significantly shorter effects on the enduring time. Interestingly, many characteristics share positive/shorter impacts on the enduring time of both winners and losers such as idiosyncratic volatility, idiovol, and size, mve, whereas only two statistically significant characteristics, systematic risk, beta, and secured debt indicator, securedind, exhibit common longer effects on the enduring time of both groups. In addition, many other characteristics show opposite effects, including bid-ask spread, baspread, return volatility, retvol, and maximum daily return in the formation month, maxret. Both winners and losers have eight firm characteristics that do not show a statistical significance at the 10% level on their respective endur- ing time. However, only the current ratio, currat, quick ratio, quick, price delay, pricedelay, are insignificant for both groups. The other five characteristics are statistically significant in the winner regression and insignificant in the loser regression and vice versa. The two ΔR2 columns show the R2 difference between the Cox PH model that incorporates all 37 firm charac- teristics in the regression and the model obtained by iteratively excluding each characteristic. Therefore, ΔR2 indi- cates the marginal impact of an individual firm characteristic on the enduring time. The results show that the magnitude of maximum daily return in the formation month, maxret, has the largest contribution to the enduring time of winners. Its ΔR2 of 0.22% is equivalent to 8% of the full regression R2 of 2.77%. Other characteristics such as baspread, sales to price ratio, sp, the number of zero trading days in the formation month, zerotrade, also have rela- tively large impact on winners' enduring time. For the impacts on losers' enduring time, maxret also has the largest contribution. Its ΔR2 of 0.36% is equivalent to 11% of the full regression R2 of 3.38% for losers. Following maxret are return volatility, retvol, leverage, lev, dollar trading volume, dolvol, and firm age, age, in exerting their impacts on ZENG ET AL. 11 of 35 14682443, 2025, 2, D ow nloaded from https://onlinelibrary.w iley.com /doi/10.1111/irfi.70014 by M assey U niversity L ibrary, W iley O nline L ibrary on [27/04/2025]. See the T erm s and C onditions (https://onlinelibrary.w iley.com /term s-and-conditions) on W iley O nline L ibrary for rules of use; O A articles are governed by the applicable C reative C om m ons L icense TABLE 3 Partial likelihood estimates of enduring momentum portfolio returns. N Variable β Winners ΔR2 β Losers ΔR2p-value p-value 1 age 0.00 0.0002 0.01% 0.01 <0.0001 0.14% 2 baspread �1.96 <0.0001 0.14% 0.75 <0.0001 0.03% 3 beta �0.09 <0.0001 0.03% �0.05 0.0002 0.00% 4 betasq 0.02 <0.0001 0.01% 0.01 0.0774 0.00% 5 bm 0.01 0.0057 0.01% 0.06 <0.0001 0.09% 6 bm_ia 0.00 <0.0001 0.02% 0.00 0.0685 0.00% 7 cashdebt �0.01 0.0006 0.00% 0.01 <0.0001 0.01% 8 cashpr 0.00 0.1084 0.00% 0.00 0.0416 0.00% 9 convind 0.04 <0.0001 0.02% �0.08 <0.0001 0.05% 10 currat 0.00 0.998 0.00% 0.00 0.9157 0.00% 11 depr �0.03 <0.0001 0.02% 0.03 <0.0001 0.00% 12 dolvol 0.01 0.0124 0.00% �0.07 <0.0001 0.15% 13 dy 1.19 <0.0001 0.06% 0.02 0.8945 0.00% 14 ep 0.04 <0.0001 0.02% �0.01 0.6340 0.00% 15 herf 0.07 0.0123 0.00% �0.35 <0.0001 0.04% 16 idiovol 0.22 0.0297 0.01% 0.41 0.0008 0.01% 17 ill 0.00 0.295 0.00% 0.00 <0.0001 0.01% 18 indmom 0.02 0.0034 0.00% 0.01 0.1872 0.00% 19 IPO �0.01 0.5865 0.00% �0.12 <0.0001 0.09% 20 Lev 0.00 0.0998 0.00% �0.01 <0.0001 0.15% 21 maxret �1.65 <0.0001 0.22% 1.51 <0.0001 0.36% 22 mve 0.03 <0.0001 0.02% 0.04 <0.0001 0.07% 23 mve_ia 0.00 <0.0001 0.02% 0.00 <0.0001 0.03% 24 pricedelay 0.00 0.5654 0.00% 0.00 0.2684 0.00% 25 quick 0.00 0.4073 0.00% 0.00 0.3334 0.00% 26 retvol 3.46 <0.0001 0.05% �4.94 <0.0001 0.29% 27 roic 0.00 0.0917 0.00% 0.01 0.0031 0.01% 28 salecash 0.00 <0.0001 0.02% 0.00 0.1007 0.00% 29 salerec 0.00 0.0206 0.00% 0.00 <0.0001 0.01% 30 securedind �0.05 <0.0001 0.08% �0.04 <0.0001 0.02% 31 sin �0.08 0.0023 0.00% 0.08 0.0345 0.00% 32 sp �0.01 <0.0001 0.13% 0.01 <0.0001 0.04% 33 std_dolvol �0.01 0.5408 0.00% 0.06 <0.0001 0.00% 34 std_turn 0.00 0.0005 0.01% 0.00 <0.0001 0.03% 35 tang �0.21 <0.0001 0.08% 0.05 0.0454 0.00% 36 turn 0.00 0.1749 0.00% 0.00 0.0934 0.04% 37 zerotrade 0.02 <0.0001 0.11% 0.00 0.8696 0.00% R2 2.77% 3.38% Note: This table reports the partial likelihood estimates of enduring momentum portfolio returns from January 1985 to December 2018. The β and p-value columns indicate the partial likelihood estimated coefficients from Equation (1) and corresponding p-value for all 37 firm characteristics. The R2 in the last row represents the coefficient of determination for the Cox PH model utilizing all 37 firm characteristics for winner and loser firms, respectively. Moreover, the marginal difference in R2 when each firm characteristic is dropped is presented in the ΔR2 column, highlighting the impact of each characteristic on the model's explanatory power. 12 of 35 ZENG ET AL. 14682443, 2025, 2, D ow nloaded from https://onlinelibrary.w iley.com /doi/10.1111/irfi.70014 by M assey U niversity L ibrary, W iley O nline L ibrary on [27/04/2025]. See the T erm s and C onditions (https://onlinelibrary.w iley.com /term s-and-conditions) on W iley O nline L ibrary for rules of use; O A articles are governed by the applicable C reative C om m ons L icense the enduring time of losers. Overall, while many characteristics exhibit statistically significant effects on the enduring time of winners and losers, the above characteristics are particularly important for investors in selecting future win- ners and losers for their long-short portfolios. Table A3 reports the summary statistics of the estimated enduring momentum probabilities for all winners and losers, which include the average, standard deviation, median, maximum, and minimum estimated probabilities of all enduring months. At the end of each month t, we estimate the probability of each winner or loser firm continuing as a winner or loser for one (two, three, four, five, and six) month (months) over the entire 6-month holding period. We then calculate the cross-sectional average, median, standard deviation, maximum, and minimum values of all esti- mated enduring momentum probabilities. Finally, we consider the time series average of each statistic across all months. Panel B of Table A3 shows that the average estimated enduring momentum probability for winners to appear for one (six) month(s) over the 6-month holding period is 58% (5%), and these statistics are similar for losers. 3.2 | Returns of price momentum and enduring momentum strategies Table 4 reports the monthly average return of the traditional price momentum (PM) and that of the enduring momentum portfolio constructed based on the Cox PH model (EM) and five other GLMs: Poisson, Logit, fractional logit (Frac Logit), linear probability model (LPM), and ordinary least squares (OLS). We describe the formation and investment process in Sections 2.2.1 and 2.2.2. The results in Panel A suggest that all five GLMs consistently yield statistically significant positive returns. Moreover, among the five GLM estimations, the Poisson (OLS) model pro- duces the highest (lowest) monthly average enduring momentum return of 1.97% (1.69%). All these GLM models outperform the traditional price momentum's monthly average return of 1.12% displayed in the PM column in Panel B. However, their performances are inferior to the EM, which boasts a monthly average return of 2.19%. Apart from PM and EM results, we also report the monthly average return for the long-short portfolio of win- ners and losers that are not picked for the enduring momentum portfolio. We denote this portfolio as non-EM and show that it earns an average return of 0.63% a month. The return difference between EM and PM in the EM-PM column demonstrates that our enduring momentum strategy based on the Cox PH model produces an average of 1.07% higher monthly return than the traditional price momentum strategy. Moreover, the loser firms in EM gener- ate a 1.01% lower monthly return compared to their corresponding counterparts in PM, suggesting that the out- performance of EM primarily stems from the loser portfolio. Furthermore, the last column shows that compared to the portfolio constructed based on the non-enduring momentum strategy, the enduring momentum strategy gener- ates an average of 1.56% higher monthly return, and this is also driven by loser stocks. Panel C reports the value-weighted and probability-weighted EM returns, and the portfolio return of a combined strategy that is constructed based on the most popular momentum stocks selected by all six models. Returns in the value-weighted EM use firms' market capitalizations as weights, while those in the probability-weighted EM employ the estimated enduring momentum probabilities at the beginning of each month as weights. The results show that all three long-short portfolios yield significant monthly average returns, with the probability-weighted EM achieving the highest monthly return of 2.13%, followed by the combined portfolio of 2.06% a month. Figures 2 and 3 present the time series patterns of the EM and non-EM portfolio returns over January 1985 to December 2018. In Figure 2, the return shown in a particular year is the average of monthly returns in that year. In Figure 3, the annual return is cumulative across all months in a year. The results show that EM outperforms non-EM in most years of our sample period and that EM's cumulative return is approximately five times more than that of non-EM. 3.3 | Enduring momentum predictive ability This subsection investigates the relationship between stock returns and winner/loser indicators based on the Fama- Macbeth (FM) regression. Concurrently, we consider the impact of 37 firm characteristics and past returns. Utilizing ZENG ET AL. 13 of 35 14682443, 2025, 2, D ow nloaded from https://onlinelibrary.w iley.com /doi/10.1111/irfi.70014 by M assey U niversity L ibrary, W iley O nline L ibrary on [27/04/2025]. See the T erm s and C onditions (https://onlinelibrary.w iley.com /term s-and-conditions) on W iley O nline L ibrary for rules of use; O A articles are governed by the applicable C reative C om m ons L icense Fama-MacBeth regression, Guo et al. (2022) successfully identified firm fundamentals as the most promising factor among established momentum explanations. In each month, we calculate stocks' average monthly returns over the holding period. We then regress these returns on an EM indicator that is equal to one for enduring momentum stocks, and zero otherwise. We also include the past returns and the 37 firm characteristics as additional control variables. We report the average values for the intercept, EM indicator, adjusted R2, and number of observations in Panels A, B, and C of Table 5, respectively. We do not report the coefficients for control variables to save space. The results in Panel A without any control variables show that enduring momentum winners do not earn significantly higher returns in the holding period than TABLE 4 Momentum returns of price momentum and generalized regression models. Panel A: Enduring momentum returns of five GLMs Poisson Logit Frac logit LPM OLS Winners 1.22%*** 1.27%*** 1.22%*** 1.22%*** 1.16%*** [2.50] [2.60] [2.47] [2.48] [2.34] Losers �0.74% �0.54% �0.69% �0.57% �0.53% [�1.17] [�0.87] [�1.10] [�0.92] [�0.83] WML 1.97%*** 1.80%*** 1.92%*** 1.79%*** 1.69%*** [3.91] [3.73] [3.80] [3.69] [3.30] Panel B: Returns of price momentum and Cox PH enduring momentum portfolios PM EM Non-EM EM-PM EM-NonEM Winners 1.32%*** 1.38%*** 1.40%*** 0.06% �0.02% [3.96] [2.77] [4.21] [0.22] [�0.08] Losers 0.19% �0.81% 0.78% �1.01%*** �1.59%*** [0.39] [�1.29] [1.58] [�3.39] [5.14] WML 1.12%*** 2.19%*** 0.63%* 1.07%*** 1.56%*** [3.20] [4.24] [1.79] [2.58] [3.69] Panel C: Value- and probability-weighted Cox PH enduring momentum returns Value-weighted EM Probability-weighted EM Combined Winners 1.40%*** 1.30%*** 1.26%*** [2.63] [2.58] [2.45] Losers �0.04% �0.83% �0.80% [�0.07] [�1.31] [�1.25] WML 1.45%*** 2.13%*** 2.06%*** [2.55] [4.04] [3.86] Note: This table reports the momentum returns generated by various generalized regression models from January 1985 to December 2018. Panel A presents the monthly average enduring portfolio returns by applying the five popular GLMs, including the Poisson, Logit, fractional logit (Frac Logit), linear probability model (LPM), and ordinary least squares (OLS) models. Panel B reports the monthly average returns of the price momentum (PM), Cox PH enduring momentum (EM), and non-enduring momentum (Non-EM) portfolios. The return differences between EM, PM, and Non-EM are also reported in Panel B. Panel C reports the value-weighted and probability-weighted EM returns, and the portfolio return of a combined strategy that is constructed based on the most popular momentum stocks selected by all six GLMs. Returns in the value- weighted EM use firms' market capitalization as weights, while those in the probability-weighted EM employ the estimated enduring momentum probabilities at the beginning of each month as weights. t-statistics are in brackets. * and *** indicate statistical significance at the 10% and 1% levels, respectively. 14 of 35 ZENG ET AL. 14682443, 2025, 2, D ow nloaded from https://onlinelibrary.w iley.com /doi/10.1111/irfi.70014 by M assey U niversity L ibrary, W iley O nline L ibrary on [27/04/2025]. See the T erm s and C onditions (https://onlinelibrary.w iley.com /term s-and-conditions) on W iley O nline L ibrary for rules of use; O A articles are governed by the applicable C reative C om m ons L icense non-enduring winners. However, enduring momentum losers exhibit significantly lower holding returns than non- enduring losers. These results are consistent with those in Table 4 and suggest that the outperformance of the enduring momentum portfolio is driven mainly by the loser stocks. In Panels B and C, we find consistent patterns even after controlling for past month returns and firm characteristics. 3.4 | Risk-adjusted returns Table 6 presents the risk-adjusted returns for the price momentum and enduring momentum strategies based on the Capital Asset Pricing Model (Sharpe, 1964), Fama and French's (1992) three-factor model, and Carhart's (1997) four- factor model. The results show that EM excess returns, that is, alpha, remain positive and statistically significant across the three factor models. For example, in the Carhart four-factor model, alpha is 0.0143, suggesting that the EM earns an average return of 1.43% a month during our sample period. The results for PM in the last three columns in Table 6 show markedly smaller values for alpha compared to those for EM. PM's alpha even becomes insignificant after controlling for the four popular risk factors. These results confirm our previous findings regarding the superior- ity of the enduring momentum strategy to the traditional price momentum strategy. Annual Averaged Portfolio Returns: Enduring vs. Non-Enduring Momentum 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 Year -0.10 -0.05 0.00 0.05 A nn ua l R et ur ns Non_Enduring_momentum_returnsEnduring_momentum_returns F IGURE 2 Enduring and non-enduring momentum portfolio returns. This figure presents the performance of enduring momentum and non-enduring momentum portfolios from January 1985 to December 2018. The enduring momentum portfolio is constructed by buying the top ten winner firms and selling the top ten loser firms with the highest enduring momentum probabilities estimated by the Cox PH model. The non-enduring momentum portfolio is constructed by buying winners and selling losers that are not selected for the enduring momentum portfolio. The return shown in a particular year is the average of monthly returns in that year. ZENG ET AL. 15 of 35 14682443, 2025, 2, D ow nloaded from https://onlinelibrary.w iley.com /doi/10.1111/irfi.70014 by M assey U niversity L ibrary, W iley O nline L ibrary on [27/04/2025]. See the T erm s and C onditions (https://onlinelibrary.w iley.com /term s-and-conditions) on W iley O nline L ibrary for rules of use; O A articles are governed by the applicable C reative C om m ons L icense 4 | ADDITIONAL ROBUSTNESS CHECKS This section examines the validity of our empirical findings through a comprehensive series of robustness tests. The initial robustness test set focuses on validating the enduring momentum probability. We then explore enduring momentum returns by excluding micro stocks, varying sample sizes, and investment horizons. In addition, we investi- gate the seasonal effects, detect the limits of arbitrage effects, probe long-term reversal effects, evaluate the risk- managed momentum factor and the information discreteness (ID) index, and assess returns during market crashes, turnover ratio, and break-even costs. 4.1 | Predictive ability of enduring momentum probability In this section, we examine the predictive ability of the Cox PH model's enduring momentum probability (EMP) in explaining the persistence of stock returns and detecting future winner and loser firms. Sections 2.2.3 and 2.2.4 describe the methodologies, and Table A4 shows the regression results. Panel A shows positive and significant values Annual Cumulative Portfolio Returns: Enduring vs. Non-Enduring Momentum 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 Year 0.0 0.2 0.4 0.6 0.8 1.0 C um ul at iv e R et ur ns NEM_cumulative_returnsEM_cumulative_returns F IGURE 3 Enduring and non-enduring momentum cumulative portfolio returns. This figure presents the cumulative returns of enduring momentum and non-enduring momentum portfolios from January 1985 to December 2018. The enduring momentum portfolio is constructed by buying the top ten winner firms and selling the top ten loser firms with the highest enduring momentum probabilities estimated by the Cox PH model. The non- enduring momentum portfolio is constructed by buying winners and selling losers that are not selected for the enduring momentum portfolio. The annual return is cumulative across all months in a year. 16 of 35 ZENG ET AL. 14682443, 2025, 2, D ow nloaded from https://onlinelibrary.w iley.com /doi/10.1111/irfi.70014 by M assey U niversity L ibrary, W iley O nline L ibrary on [27/04/2025]. See the T erm s and C onditions (https://onlinelibrary.w iley.com /term s-and-conditions) on W iley O nline L ibrary for rules of use; O A articles are governed by the applicable C reative C om m ons L icense TABLE 5 Fama-Macbeth regression results. Intercept EM indicator R2 N Panel A: Enduring momentum winner/loser indicator only Winners 0.0135*** �0.0111 0.0067 385 [8.85] [�0.62] Losers 0.0242*** �0.0348*** 0.0078 272 [3.30] [�5.16] Panel B: Enduring momentum winner/loser indicator with past returns Winners 0.0136*** �0.0008 0.0148 385 [8.93] [�0.45] Losers 0.0138** �0.0322*** 0.0184 272 [1.99] [�4.76] Panel C: Enduring momentum winner/loser indicators with past returns and 37 firm characteristics Winners 0.0265*** �0.0019 0.1893 385 [5.24] [�1.00] Losers �0.0119*** �0.0059*** 0.1897 272 [�3.75] [�2.32] Note: This table reports the Fama-Macbeth (FM) regression results. In each month, we regress the average monthly holding returns on an EM indicator that is equal to one for enduring momentum stocks and zero otherwise. We include the past- month return and the 37 firm characteristics as additional control variables in Panels B and C, respectively. We report the average values for the intercept, EM indicator, adjusted R2, and number of observations in this table and do not show the coefficients for control variables to save space. The t-statistics in brackets are calculated using Newey and West (1987) standard errors. ** and *** indicate statistical significance at the 5% and 1% levels, respectively. TABLE 6 Risk-adjusted portfolio returns. Panel A: Enduring momentum Panel B: Price momentum CAPM FF3 Carhart4 CAPM FF3 Carhart4 alpha 0.2570*** 0.0270*** 0.0143*** 0.0143*** 0.0148*** 0.0034 [4.97] [5.22] [3.38] [4.03] [4.15] [1.49] MKT �0.0035*** �0.0044*** �0.0006 �0.0033*** �0.0033*** 0.0001 [�3.00] [�3.65] [�0.57] [�4.09] [�3.99] [0.22] SMB 0.0021 0.0025* �0.0018 �0.0015* [1.22] [1.75] [�1.48] [�1.95] HML �0.0040** 0.0014 �0.0021* 0.0026*** [�2.11] [0.87] [�1.65] [3.12] UMD 0.0084*** 0.0075*** [14.87] [24.61] R2 0.0217 0.0398 0.3801 0.0396 0.0489 0.6199 N 408 408 408 408 408 408 Note: This table presents the risk-adjusted returns for enduring momentum and price momentum portfolios from the CAPM, Fama–French three-factor, and Carhart four-factor models. The last row reports the total number of regression observations. t-statistics are in brackets. ***, **, and * indicate statistical significance at the 1%, 5%, and 10% levels, respectively. ZENG ET AL. 17 of 35 14682443, 2025, 2, D ow nloaded from https://onlinelibrary.w iley.com /doi/10.1111/irfi.70014 by M assey U niversity L ibrary, W iley O nline L ibrary on [27/04/2025]. See the T erm s and C onditions (https://onlinelibrary.w iley.com /term s-and-conditions) on W iley O nline L ibrary for rules of use; O A articles are governed by the applicable C reative C om m ons L icense for βEMP from Equation (7), indicating that EMP is significantly related to the persistence of stock returns for both winner and loser firms. Panel B presents the results of Equations (8) and (9). For the winner portfolio, both γEMP W and γMOM W are positive and highly statistically significant, suggesting that past returns, MOM, and EMP can individually predict future winners. More importantly, the predictive ability of EMP remains strong after controlling the effect of MOM. We observe a similar pattern for the loser portfolio. The positive coefficient of EMP indicates that a higher EMP is associated with a higher likelihood that the current loser stock is also a future loser stock. The negative coef- ficient of MOM indicates that lower past returns increase the likelihood of being a loser in the future. The predictive power of EMP survives the inclusion of MOM in the regression, as shown in the last column in Panel B. We also esti- mate Equation (9) using OLS regression as an alternative to logit regression and report the results in Panel C. We still observe a significant predictive power of EMP across the winner and loser portfolios. We next examine the predictive ability of EMP in forecasting future stock returns. We estimate Equations (10) and (11) and report their results in Panel A of Table A5. The results for EMP alone indicate that EMP has a significant power to forecast future stock returns. The coefficients, ϕEMP W and ϕEMP L are positive and negative for winner and loser portfolios, respectively. This suggests that higher enduring momentum probabilities lead to higher (lower) future returns for winners (losers), which is consistent with the portfolio results in Table 4. The results for past returns, however, show that the effect of MOM effect on future returns is only significant for loser firms and that its effect does not explain away the forecasting power of EMP, as shown in the last column. In Panel B, we decompose the effect of EMP into 2 components: an explained component representing how much MOM shares its predictive impact with EMP and a residual component representing the magnitude of EMP predictive power that is not shared with MOM. We present this methodology in Section 2.2.5 following Hou and Loh (2016). We find that the momen- tum signal fails to share the predictive information that the enduring momentum probability provides in forecasting stock returns for winner and loser firms. The magnitude of the explained fraction is small and statistically insignificant for winner firms and significantly negative for loser firms. Overall, in this section we show that the enduring momentum probability is strongly associated with the persis- tence of returns, able to detect future winners and losers and forecast their returns. More importantly, its predictive information is not shared by the price momentum signal based on past returns. 4.2 | Excluding micro stocks, varying investment firm numbers, different investment horizons, and weighting schemes This subsection examines how the results change by investigating four separate sample adjustments. First, we follow Asness et al. (2013) to rank all the stocks into ten groups based on their market capitalization in descending order at the beginning of each month and then exclude the group with the smallest market capitalization. Second, we expand the number of stocks to be included in the enduring momentum portfolio to 20 and 30 instead of 10 as in the main analysis. Third, we include the 3, 9, and 12 months as alternative investment periods and show the portfolio returns using equal weighting and value weighting. The results reported in Tables A6–A8 generally show that the enduring momentum returns remain robust and mostly larger than the price momentum returns. 4.3 | Seasonal effect Jegadeesh and Titman (1993) document a seasonal behavior of the price momentum strategy: loser firms generate significantly higher returns than winner firms in January. McLean and Pontiff (2016) find that anomalies decay over time. Thus, we examine the performance of the enduring momentum and price momentum strategies in the January and non-January periods, report the results in Panel A of Table A9, and explore the performance of the two strate- gies during the earlier and later periods of our sample in Panel B. 18 of 35 ZENG ET AL. 14682443, 2025, 2, D ow nloaded from https://onlinelibrary.w iley.com /doi/10.1111/irfi.70014 by M assey U niversity L ibrary, W iley O nline L ibrary on [27/04/2025]. See the T erm s and C onditions (https://onlinelibrary.w iley.com /term s-and-conditions) on W iley O nline L ibrary for rules of use; O A articles are governed by the applicable C reative C om m ons L icense The results show that the enduring momentum strategy is less affected by seasonality and shows consistent long-short returns over time than the price momentum strategy. Consistent with Jegadeesh and Titman (1993), we find that the PM return is negative for January and positive in other months. The EM return, however, remains posi- tive in both January and other months. The results for subsample periods indicate that both EM and PM strategies generate lower long-short portfolio returns during the 2000.01–2018.12 period than during the 1985.01–1999.12 period, consistent with the findings of McLean and Pontiff (2016). However, both subperiod returns are significantly higher for EM than for PM portfolios. 4.4 | Limits of arbitrage Our results in Table 4 and other Tables A6–A8 apparently indicate that the outperformance of the EM over the PM is mainly driven by the performance of loser firms. In this section, we investigate whether this is due to limits to arbi- trage. Researchers have argued that limits to arbitrage prevent arbitrageurs from correcting market mispricing effec- tively (e.g., Shleifer and Vishny, 1997; Lamont and Thaler, 2003) and these limits tend to be more constrained when investors sell short stocks than when they buy them (e.g., Li et al., 2009). Stringent borrowing requirements, fees, regulatory constraints, and the risk of unlimited losses are among the limits to arbitrage. Arena et al. (2008) suggest that momentum profits are associated with limits to arbitrage. We use popular proxies for limits to arbitrage: size, illiquidity, and return volatility, and reconstruct the enduring momentum and price momentum portfolios by excluding from our sample 20% of firms with the smallest market capitalization, highest illiquidity, and highest return volatility. For the sake of comparison, we alternatively exclude 20 firms that are largest in size, lowest in illiquidity, and lowest in return volatility. We present the portfolio returns in Table A10. The results in the 2nd column show that the long-short portfolio returns remain positive and significant in each instance of excluding stocks with high limits to arbitrage. The return difference between EM and PM portfo- lios is still positive and significant, except when volatility is used in the screening. The exclusion of firms with the least limits to arbitrage does not materially alter the portfolio returns of the two strategies and their differences, as compared to the results in Table 4. 4.5 | Long-term reversal effects The long-term reversal effect is a widely recognized anomaly in asset pricing (e.g., Balvers & Wu, 2006; De Bondt & Thaler, 1985; Spierdijk et al., 2012). Stocks that have performed poorly over a long time tend to have higher returns in the future, while stocks that have performed well over a long period tend to have lower returns. This section examines whether the winners and losers from month t � 12 to t � 24 tend to have reversal performance after time t in enduring momentum. We report the performance of the enduring momentum portfolio in Table A11 for both equal-weighted and value-weighted returns. The results show that the average long-short portfolio return is 1.62% (1.58%) a month, statistically significant at the 1% level, suggesting that long-term reversal does not have a significant impact on the enduring momentum performance. 4.6 | Risk-managed momentum factor Barroso and Santa-Clara (2015) find that the performance of price momentum could be enhanced by effectively managing its risk by using realized volatility to scale the long-short portfolio return. This section compares the risk- managed momentum strategy with the enduring momentum strategy. We first follow Barroso and Santa-Clara (2015) to construct the risk-managed momentum strategy with the same sample period as the enduring momentum ZENG ET AL. 19 of 35 14682443, 2025, 2, D ow nloaded from https://onlinelibrary.w iley.com /doi/10.1111/irfi.70014 by M assey U niversity L ibrary, W iley O nline L ibrary on [27/04/2025]. See the T erm s and C onditions (https://onlinelibrary.w iley.com /term s-and-conditions) on W iley O nline L ibrary for rules of use; O A articles are governed by the applicable C reative C om m ons L icense strategy. We then regress the enduring momentum portfolio returns on the four popular risk factors plus this risk- managed momentum return factor, denoted by WML*, and report the results in Table A12. The results in Panel A show that the EM portfolio continues generating significant returns after controlling for all five risk factors. The value of Alpha ranges from 1.24% to 1.53% across the four regression specifications. How- ever, the PM portfolio returns turn insignificant or even negative as soon as WML* is included in the regression. These results indicate that the risk-managed momentum strategy fails to explain away the enduring momentum strategy's returns. 4.7 | Information discreteness index Da et al. (2014) argue that investors' limited attention to information that arrives continuously in small amounts cau- ses stronger momentum observed in stocks with more continuous information. They create the information discrete- ness (ID) index based on past returns and the proportion difference between positive and negative prices to measure a stock's ID. Following Da et al. (2014), we construct the ID for each stock using daily returns over the past 12 months, rank them into quintiles, and buy the stocks in the top group of continuous information and sell those in the bottom group of discrete information. The long-short portfolio is held for six months and rebalanced monthly. We include these ID-based monthly returns in the factor model regression. The results reported in Table A13 sug- gest that information discreteness does not have an impact on the risk-adjusted returns for the EM portfolio. All alpha coefficients are relatively large and statistically significant at the 1% level. The results for the PM portfolio are relatively similar to those in Panel B of Table 6. 4.8 | Returns when market crashes Table A14 reports the enduring momentum returns when financial market crashes occur. The results suggest that the enduring momentum strategy successfully avoids market crashes. Specifically, the EM portfolio yields positive returns during the top eight crashes in the aggregate financial market while its return is significantly less negative for the other two market crashes. 4.9 | Turnover ratio and breakeven transaction cost In many studies, a price momentum portfolio usually has a higher turnover ratio than a market portfolio. This section examines whether high turnover-related transaction costs offset the overperformance of the price momentum and enduring momentum strategies. Following Brandt et al. (2009), we calculate the turnover ratio in month t as the summation of the absolute values of the weight changes of all securities in the corresponding portfolio between months t � 1 and t. We report the turnover ratio and breakeven cost for the price momentum and enduring momentum in Table A15. We find that the turnover ratio for winners (losers) of the enduring momentum strategy is 1.3101 (1.138), and this is higher than the turnover ratio of 0.7398 (0.7587) for winners (losers) of the price momentum strategy.8 However, the breakeven cost for the EM portfolio is 0.95%, higher than 0.81% for the PM portfolio. This result suggests that while it takes 81 basis points for the price momentum portfolio to break even, a transaction cost of 95 basis points will make the enduring momentum portfolio achieve zero returns. For comparison's sake, both these break-even costs are higher than real-world transaction costs of 8.37 basis points for long-short trades and 9.61 basis points for implementation shortfall (Frazzini et al., 2018). 20 of 35 ZENG ET AL. 14682443, 2025, 2, D ow nloaded from https://onlinelibrary.w iley.com /doi/10.1111/irfi.70014 by M assey U niversity L ibrary, W iley O nline L ibrary on [27/04/2025]. See the T erm s and C onditions (https://onlinelibrary.w iley.com /term s-and-conditions) on W iley O nline L ibrary for rules of use; O A articles are governed by the applicable C reative C om m ons L icense 5 | CONCLUSION This study utilizes information from 37 firm characteristics and a Cox PH model to estimate the enduring momentum probabilities that winner and loser firms continue to be winners and losers over the next investment period. Instead of trading on all past winners and losers, we construct the enduring momentum strategy by buy- ing the top ten past winners and selling the top ten past losers with the highest estimated enduring momentum probabilities. Our results indicate that the enduring momentum strategy generates significantly higher returns than the tradi- tional price momentum strategy. This finding remains robust when the enduring momentum probabilities are esti- mated by the five alternative GLMs models: Poisson, Logit, fractional logit, linear probability model, and ordinary least squares. Moreover, we show that the enduring momentum indicator plays an important role in predicting cross-sectional stock returns, even after controlling for past returns and various firm-specific characteristics. Besides, traditional models, including the CAPM, Fama–French three-factor, and Carhart four-factor model, fail to explain the enduring momentum anomaly even after we augment them with a risk-managed momentum factor or an information discrete index. Enduring momentum returns withstand a series of robustness tests. Firstly, we find that the enduring momen- tum probabilities for winner and loser firms are significantly related to the persistence of stock returns, successfully detect future winners and losers, and show significant ability in forecasting future returns. Secondly, enduring momentum returns are held after excluding microcap stocks, varying the number of selected firms, and applying vari- ous weighting schemes and investment horizons. Thirdly, positive returns are consistently observed with the endur- ing momentum strategy in January and non-January periods and surpassing the price momentum strategy across all subsamples. Fourthly, the enduring momentum returns prove resilient against arbitrage limits and long-term reversal effects. Finally, the enduring momentum portfolio performs well during market crashes and has a higher breakeven transaction cost than the price momentum portfolio. ACKNOWLEDGMENT Open access publishing facilitated by Massey University, as part of the Wiley - Massey University agreement via the Council of Australian University Librarians. ENDNOTES 1 Given our monthly data, the survival time in our study is discrete rather than continuous as in the traditional survival analysis. 2 Note that the data from months t � 5 to t are used to compute the enduring time for winners and losers in month t � 6. 3 We thank Jeremiah Green for providing the SAS code on his webpage: https://sites.google.com/site/jeremiahrgreenacctg/home. 4 MKT: market excess return; SMB is the return difference between small-capitalization firms and big-capitalization firms; HML is the return difference between high book-to-market and low book-to-market firms; UMD is the return difference between momentum winner and loser stocks. 5 We thank Kenneth French for providing the data on his website: https://mba.tuck.dartmouth.edu/pages/faculty/ken. french/index.html. 6 The seven estimated enduring momentum probabilities illustrate the seven different occurrences for past winners (losers) to continue to be winners (losers) in the following 6-month investment period, ranging from 0 to 6 times. We only focus on the probability of six occurrences (τ¼6) in our analysis. 7 We also perform the OLS regression as an alternative to the logit regression. 8 Compared with the price momentum strategy, which selects 385 winners and 272 losers each month, the enduring momentum strategy in the last row of Table A15 selects only 10 winners and 10 losers per month. This smaller pool can explain why the enduring momentum strategy's turnover ratio exceeds 100%. ZENG ET AL. 21 of 35 14682443, 2025, 2, D ow nloaded from https://onlinelibrary.w iley.com /doi/10.1111/irfi.70014 by M assey U niversity L ibrary, W iley O nline L ibrary on [27/04/2025]. 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See the T erm s and C onditions (https://onlinelibrary.w iley.com /term s-and-conditions) on W iley O nline L ibrary for rules of use; O A articles are governed by the applicable C reative C om m ons L icense https://doi.org/10.1111/irfi.70014 https://doi.org/10.1111/irfi.70014 APPENDIX A TABLE A1 Definitions of firm characteristics. N Acronym Variable Reference Definitions of characteristics 1 age Firm age Jiang et al. (2005) Number of years since the company's IPO year 2 baspread Bid-ask spread Amihud and Mendelson (1989) Monthly averaged daily bid-ask spread divided by averaged daily spread 3 beta Beta Fama and MacBeth (1973) Estimated market beta based on weekly returns and equal- weighted market returns for 3 years ending month t � 1 with at least 52 weeks of returns 4 betasq Beta squared Fama and MacBeth (1973) Market beta squared 5 bm Book-to-market equity Rosenberg, Reid, and Lanstein (1985) Book value of equity divided by fiscal year-end market capitalization 6 bm_ia Industry- adjusted book to market Asness et al. (2000) Industry adjusted book-to-market ratio 7 cashdebt Cash-flow-to- debt Ou and Penman (1989) Earnings before depreciation and extraordinary items divided by the average of total liabilities 8 cashpr Cash productivity Chandrashekar and Rao (2009) Fiscal year-end market capitalization plus long-term liabilities minus total assets scaled by cash and cash equivalents 9 convind Convertible debt indicator Valta (2016) A dummy variable equal to 1 for a company that has convertible debt obligations, zero otherwise 10 currat Current ratio Ou and Penman (1989) Current assets divided by current liabilities 11 depr Depreciation / PP&E Holthausen and Larcker (1992) Depreciation divided by PP&E 12 dolvol Dollar trading volume Chordia et al. (2001) Stock price times natural log of trading volume stock price from month t � 2 13 dy Dividend to price Litzenberger and Ramaswamy (1982) Annual total dividends divided by fiscal year-end market capitalization 14 ep Earnings to price Basu (1977) Annual income before extraordinary items divided by fiscal year-end market cap 15 herf Industry sales concentration Hou and Robinson (2006) 2-digit SIC-fiscal-year sales concentration (sum of the squared percentages of sales in the industry for each company). 16 idiovol Idiosyncratic return volatility Ali et al. (2003) Standard deviation of residuals of weekly returns on weekly equal-weighted market returns for 3 years before month-end 17 ill Illiquidity Amihud (2002) Average of absolute daily return divided by daily dollar volume 18 indmom Industry momentum Moskowitz and Grinblatt (1999) Equal-weighted average industry 12-month returns 19 IPO New equity issue Loughran and Ritter (1995) An indicator equals 1 if the first year available on CRSP monthly stock file 24 of 35 ZENG ET AL. 14682443, 2025, 2, D ow nloaded from https://onlinelibrary.w iley.com /doi/10.1111/irfi.70014 by M assey U niversity L ibrary, W iley O nline L ibrary on [27/04/2025]. See the T erm s and C onditions (https://onlinelibrary.w iley.com /term s-and-conditions) on W iley O nline L ibrary for rules of use; O A articles are governed by the applicable C reative C om m ons L icense TABLE A1 (Continued) N Acronym Variable Reference Definitions of characteristics 20 lev Leverage Bhandari (1988) Total liabilities divided by fiscal year-end market capitalization 21 maxret Maximum daily return Bali et al. (2011) Maximum daily return during calendar month t � 1 22 mve Size Banz (1981) Natural logarithm of market capitalization at the end of month t � 1 23 mve_ia Industry- adjusted size Asness et al. (2000) 2-digit SIC industry-adjusted fiscal year-end market capitalization 24 pricedelay Price delay Hou and Moskowitz (2005) The proportion of variation in weekly returns for 36 months ending in month t � 1 explained by 4 lags of weekly market returns incremental to contemporaneous market return 25 quick Quick ratio Ou and Penman (1989) Value of current assets minus inventory divided by current liabilities 26 retvol Return volatility Ang et al. (2006) The standard deviation of daily returns from month t � 1 27 roic Return on invested capital Brown and Rowe (2007) Annual earnings before interest and taxes minus non- operating income divided by non-cash enterprise value 28 salecash Sales to cash Ou and Penman (1989) Annual sales divided by cash and cash equivalents 29 salerec Sales to receivables Ou and Penman (1989) Annual sales divided by accounts receivable 30 securedind Secured debt indicator Valta (2016) An indicator equal to 1 if the company has secured debt obligations 31 sin Sin stocks Hong and Kacperczyk (2009) An indicator variable equal to 1 if a company's primary industry classification is in beer or alcohol, smoke or tobacco, or gaming 32 sp Sales to price Barbee et al. (1996) Annual operating revenue divided by fiscal year-end market capitalization 33 std_dolvol Volatility of liquidity Chordia et al. (2001) The monthly standard deviation of daily dollar trading volume 34 std_turn Volatility of liquidity Chordia et al. (2001) The monthly standard deviation of daily share turnover 35 tang Debt capacity/ firm tangibility Almeida and Campello (2007) Cash holdings +0.715 � receivables +0.547 � inventory +0.535 � PPE/total assets 36 turn Share turnover Datar et al. (1998) Average monthly trading volume for the most recent three months scaled by the number of shares outstanding in the current month 37 zerotrade Zero trading days Liu (2006) Turnover weighted number of zero trading days for most recent 1 month Note: This table presents detailed definitions of the 37 firm characteristics applied in this study to estimate the enduring momentum probability. ZENG ET AL. 25 of 35 14682443, 2025, 2, D ow nloaded from https://onlinelibrary.w iley.com /doi/10.1111/irfi.70014 by M assey U niversity L ibrary, W iley O nline L ibrary on [27/04/2025]. See the T erm s and C onditions (https://onlinelibrary.w iley.com /term s-and-conditions) on W iley O nline L ibrary for rules of use; O A articles are governed by the applicable C reative C om m ons L icense TABLE A2 Firm characteristics differences and enduring proportions. Panel A: Firm characteristics difference between EM and non-EM stocks N Variables Winners Losers Diff t-value Diff t-value 1 age �2.41 �17.55 �3.42 �25.26 2 baspread 0.06 33.82 0.01 10.76 3 beta 0.09 4.76 0.25 14.79 4 betasq 0.39 7.02 0.93 14.26 5 bm 0.23 10.36 �0.13 �6.68 6 bm_ia 270.59 2.70 272.87 3.07 7 cashdebt �0.38 �12.18 �0.44 �6.74 8 cashpr �9.12 �10.63 7.72 6.16 9 convind �0.02 �4.20 0.11 13.10 10 currat 0.02 0.21 0.91 6.77 11 depr 0.12 9.33 0.01 0.49 12 dolvol �2.44 �44.36 2.27 34.09 13 dy �0.01 �4.43 0.00 3.28 14 ep �0.50 �10.28 �0.08 �5.45 15 herf 0.01 3.00 0.01 3.82 16 idiovol 0.03 36.41 0.01 8.98 17 ill 7.40 8.97 �4.69 �14.84 18 indmom 0.01 0.95 0.02 2.93 19 IPO 0.01 1.41 0.11 14.66 20 lev 1.75 9.53 1.81 4.84 21 maxret 0.19 40.40 �3.42 2.37 22 mve �1.69 �55.78 0.67 13.75 23 mve_ia �841.04 �16.84 278.49 2.79 24 pricedelay 0.10 5.15 0.02 1.33 25 quick 0.04 0.41 0.87 7.74 26 retvol 0.05 39.81 0.02 34.54 27 roic �0.61 �10.10 �0.40 �8.44 28 salecash 3.04 1.02 �11.03 �4.18 29 salerec 5.50 8.00 1.57 2.70 30 securedind 0.11 12.03 0.00 0.53 31 sin 0.01 3.52 0.00 �1.10 32 sp 3.65 12.50 �0.58 �10.92 33 std_dolvol 0.41 41.75 �0.06 �8.01 34 std_turn 11.42 17.44 20.44 31.00 35 tang 0.04 10.00 0.03 7.51 36 turn 0.55 7.04 3.49 30.60 37 zerotrade 0.55 8.27 �0.89 �17.28 26 of 35 ZENG ET AL. 14682443, 2025, 2, D ow nloaded from https://onlinelibrary.w iley.com /doi/10.1111/irfi.70014 by M assey U niversity L ibrary, W iley O nline L ibrary on [27/04/2025]. See the T erm s and C onditions (https://onlinelibrary.w iley.com /term s-and-conditions) on W iley O nline L ibrary for rules of use; O A articles are governed by the applicable C reative C om m ons L icense TABLE A3 Summary statistics of the enduring momentum probability. N Mean Std Median Max Min Panel A: Winners 1 0.58 0.07 0.58 0.78 0.31 2 0.43 0.80 0.42 0.68 0.17 3 0.30 0. 80 0.30 0.58 0.08 4 0.20 0.07 0.19 0.48 0.04 5 0.12 0.06 0.11 0.38 0.01 6 0.05 0.04 0.04 0.25 0.00 N Mean Std Median Max Min Panel B: Losers 1 0.58 0.09 0.59 0.79 0.24 2 0.43 0.10 0.43 0.69 0.12 3 0.30 0.10 0.29 0.59 0.05 4 0.19 0.08 0.19 0.48 0.02 5 0.11 0.06 0.10 0.37 0.01 6 0.05 0.04 0.03 0.24 0.00 Note: This table reports the summary statistics of the enduring momentum probability from the Cox PH model for winner and loser firms. TABLE A2 (Continued) Panel B: Proportion of past winners (losers) to be future winners (losers) N Winners Losers Marginal Cumulative Marginal Cumulative 0 0.26 0.74 0.25 0.75 1 0.21 0.52 0.22 0.53 2 0.17 0.36 0.17 0.36 3 0.13 0.23 0.13 0.23 4 0.10 0.14 0.10 0.13 5 0.07 0.06 0.08 0.06 6 0.07 0.00 0.06 0.00 Mean 2.06 2.06 Median 2.00 2.00 Note: The Diff column in Panel A reports the average difference of each of the 37 firm characteristics calculated by averaging the monthly firm characteristic differences between the enduring momentum selected top ten winner (loser) firms, EM, and the non-selected firms within the same winner (loser) group, non-EM, from January 1985 to December 2018. We also show the corresponding t values. The N column in Panel B represents the total number of months for all past winners (losers) to keep performing as winners (losers) over the 6-month holding period. Marginal and Cumulative columns denote the corresponding marginal survival and cumulative fail proportions for winners (losers). ZENG ET AL. 27 of 35 14682443, 2025, 2, D ow nloaded from https://onlinelibrary.w iley.com /doi/10.1111/irfi.70014 by M assey U niversity L ibrary, W iley O nline L ibrary on [27/04/2025]. See the T erm s and C onditions (https://onlinelibrary.w iley.com /term s-and-conditions) on W iley O nline L ibrary for rules of use; O A articles are governed by the applicable C reative C om m ons L icense TABLE A4 Detecting future winners/losers with enduring momentum probability. Panel A: Enduring momentum probability and persistency of stock returns Winners Losers βEMP 0.1033*** 0.3767*** t-value [5.07] [15.76] R2 0.0080 0.0181 Panel B: Enduring momentum probability detecting future winners/losers (Logit) Winners γEMP W 2.8596*** 1.0711*** p-value [<0.0001] [<0.0001] γMOM W 0.5088*** 0.4882*** p-value [<0.0001] [<0.0001] R2 0.0119 0.0679 0.0694 Losers γEMP L 4.7710*** 3.6054*** p-value [<0.0001] [<0.0001] γMOM L �1.5889*** �1.4265*** p-value [<0.0001] [<0.0001] R2 0.0260 0.0511 0.0645 Panel C: Enduring momentum probability detecting future winners/losers (OLS) Winners γEMP W 0.7268*** 0.4049*** t-value [30.52] [15.16] γMOM W 0.0659*** 0.0608*** t-value [20.93] [19.57] R2 0.0069 0.0240 0.0260 Losers γEMP L 1.2584*** 0.8838*** t-value [24.83] [19.63] γMOM L �0.4962*** �0.4415*** t-value [�63.53] [�53.63] R2 0.0153 0.0351 0.0416 Note: This table assesses the predictive ability of the enduring momentum probability (EMP) in explaining the persistence of stock returns and detecting future winner and loser firms. Sections 2.2.3 and 2.2.4 in the main text describe the methodologies. Panel A reports the results of Equation (7) while Panel B shows the logit regression results of Equations (8) and (9). Panel C contains the results of Equations (8) and (9) using OLS regression. We report the regression observations in parentheses in the first column. t-statistics (p-value) associated with each coefficient are in brackets (parentheses) for Panels A and C (B). *** indicates statistical significance at the 1% level. 28 of 35 ZENG ET AL. 14682443, 2025, 2, D ow nloaded from https://onlinelibrary.w iley.com /doi/10.1111/irfi.70014 by M assey U niversity L ibrary, W iley O nline L ibrary on [27/04/2025]. See the T erm s and C onditions (https://onlinelibrary.w iley.com /term s-and-conditions) on W iley O nline L ibrary for rules of use; O A articles are governed by the applicable C reative C om m ons L icense TABLE A5 Forecasting future returns with enduring momentum probability. Panel A: OLS regression Winners ϕEMP W 0.0696** 0.0717*** [2.23] [2.41] ϕMOM W �0.0004 �0.0018 [�0.22] [�1.11] Losers ϕEMP L �0.2687*** �0.3230*** [�8.31] [�10.17] ϕMOM L �0.0948*** �0.1220*** [�5.73] [�7.23] R2 0.0108 0.0106 0.0207 Panel B: Decomposition Explained fraction Residual fraction Winners 0.0324 0.9676*** [0.32] [9.61] Losers �0.1322*** 1.1322*** [�3.13] [26.81] Note: Panel A presents the regression results of equations (10) and (11) to examine the predictive ability of EMP in forecasting future stock returns. In Panel B, we decompose the effect of EMP into 2 components: an explained component representing how much MOM shares its predictive impact with EMP and a residual component representing the magnitude of EMP predictive power that is not shared with MOM. We present this methodology in Section 2.2.5 following Hou and Loh (2016). t-statistics are in brackets. *** and ** indicate statistical significance at the 1% and 5% levels, respectively. TABLE A6 Portfolio returns after excluding micro stocks. Price momentum Enduring momentum Difference Winners 1.31%*** 1.45%*** 0.13% [3.90] [2.88] [0.44] Losers 0.06% �0.31% �0.51%* [0.11] [�0.50] [�1.89] WML 1.26%*** 1.76%*** 0.63% [3.49] [3.45] [1.61] Note: This table shows the average monthly portfolio returns for price and enduring momentum and their difference after excluding the 10% micro stocks at the beginning of each month. t-statistics are in brackets. *** and * indicate statistical significance at the 1% and 10% levels, respectively. ZENG ET AL. 29 of 35 14682443, 2025, 2, D ow nloaded from https://onlinelibrary.w iley.com /doi/10.1111/irfi.70014 by M assey U niversity L ibrary, W iley O nline L ibrary on [27/04/2025]. See the