Market Predictability Before the Closing Bell Rings

Abstract

This study examines the predictability of the last 30 min of intraday stock price movements within the US financial market. The analysis encompasses several potential explanatory variables, including returns from each 30 min intraday trading session, overnight returns, the federal reserve fund rate decision days and the subsequent three days, the US dollar index, month effects, weekday effects, and market volatilities. Market-adaptive trading strategies are developed and backtested on the basis of the study’s insights. Unlike the commonly employed multiple linear regression methods with Gaussian errors, this research utilizes a Bayesian linear regression model with Student-t error terms to more accurately capture the heavy tails characteristic of financial returns. A comparative analysis of these two approaches is conducted and the limitations inherent in the traditionally used method are discussed. Our main findings are based on data from 2007 to 2018. We observed that well-studied factors such as overnight effects and intraday momentum have diminished over time. Some other new factors were significant, such as lunchtime returns during boring days and the tug-of-war effect over the days after a federal fund rate change decision. Ultimately, we incorporate findings derived from data spanning 2022 to 2024 to provide a contemporary perspective on the examined components, followed by a discussion of the study’s limitations.

1. Introduction

The efficient markets theory (EMT) tells us that stock prices have already reflected all the information and suggests that returns are not predictable in an efficient market. However, quite a few research studies on market anomalies have recently been conducted to indicate that intraday returns1 are still predictable, especially for those returns during the last half-hour trading session of the regular trading hours in the US stock market, that is, from 3:30 p.m. to 4:00 p.m. in New York time2. Most such research involves two main themes: One is market intraday momentum, which uses the first half-hour returns to predict the last half-hour returns, and the other is to use overnight returns to predict the last half-hour returns. Gao et al. (2018) found evidence of intraday momentum and attributes it to the trading behavior of informed traders. Baltussen et al. (2021) links it with the demand for gamma hedge. Elaut et al. (2018)’s results suggest that intraday momentum in the ruble market is induced by the risk aversion to overnight holdings among liquidity providers. However, Ho et al. (2021) did not find statistically significant intraday momentum in the Australian market, possibly due to the relatively small number of daily trades. Griffin and Lim (1984) demonstrate the presence of market momentum within the global stock market, where profits generated worldwide exhibit notable economic significance and statistical reliability under various economic conditions, including favorable and unfavorable states.

According to Zhang et al. (2019), the final half-hour return can be predicted in China. In addition, second-last-to-half-hour returns have also been found to have predictive potential. In contrast, Baltussen et al. (2021) presents an alternative perspective, suggesting that the return for the remaining period of the day (spanning from the previous market closure to the last 30 min) exhibits a positive correlation with the return observed in the last half hour. Bogousslavsky (2021) attributed the last 30 min returns to institutional restrictions and overnight risks. Lou et al. (2019) studied overnight and intraday reversal effects. Other studies in this direction of research include Akbas et al. (2022); Cheema et al. (2022); Hendershott et al. (2020); Ho et al. (2021); Jin et al. (2020); Limkriangkrai et al. (2023); Shen et al. (2022).

There are mixed conclusions from the above papers. Some are consistent, and some are only consistent and/or significant under certain conditions. An issue of inquiry emerges: Is the return of the last 30 min predictable at all and, if so, to what extent is it predictable?

To answer this question, we consider dividing the time period from 3:30 p.m. on the previous trading day to 4:00 p.m. on the current trading day into 15 trading sessions, as illustrated in Figure 1. Previous research on intraday momentum has primarily examined the influence of intraday returns on returns in the final half-hour of the trading day. For example, Gao et al. (2018) employed $r_1$ and $r_{12}$ as explanatory variables to predict the last half-hour return $r_{13}$, while Baltussen et al. (2021) examined the impact of what they defined as the “rest-of-day” return on $r_{13}$. This rest-of-day return was calculated as the return from the prior day’s closing price to the closing price 30 min before the current day’s end. In contrast, our study extends this approach by incorporating the complete series of intraday returns ($r_{13lag}$, $r_{on}$, $r_1$, …, $r_{12}$) and further considers additional variables that may influence the returns in the last hour. More specifically, we also consider the days the Federal Reserve made changes to fund rates and their subsequent consecutive days, as well as the US dollar index3, month effects, weekday effects and market volatility in terms of the VIX4 as explanatory variables. Moreover, instead of using the multiple linear regression model with Gaussian error terms that have been widely used in the literature, we used a Bayesian linear regression model with Student-t errors to better capture the heavy tails of the financial returns.

To our knowledge, our study is the first to consider these many factors in studying the intraday momentum. We present significant findings regarding the predictability of the final 30 min return for two major exchange-traded funds (ETFs) in the US market: SPY and QQQ. The former tracks the S&P 500 stock market index, and the latter tracks the stocks of Nasdaq 100 companies. First, we verify some existing findings on associations between factors such as overnight returns and returns in the last 30 min, and moreover, we find that these associations have diminished over the years. Furthermore, we have identified some new significant factors, such as returns between 1:30 p.m. and 2:00 p.m. ($r_9$, see

Table A1. Bayesian linear regression model with Student-t error terms (all variables and non-scaled data). The values included in brackets represent estimates in the $2.5\%$ [$5\%$], $50\%$ [$50\%$], and $97.5\%$ [$95\%$] quantiles. The scale of these estimates is determined by the “scale” column. “*” indicates that the variable is significant. Significant factors were highlighted in bold format.

	95% Credible Intervals		90% Credible Intervals		Scale
	SPY	QQQ	SPY	QQQ
r13_lag	(−6.11, −2.36, 1.28)	(−9.84, −6.07, −2.31) *	(−5.47, −2.36, 0.71)	(−9.23, −6.07, −2.87) *	$\times {10}^{-2}$
r_on	(2.48, 4.07, 5.56) *	(0.77, 2.37, 3.96) *	(2.75, 4.07, 5.33) *	(1.03, 2.37, 3.69) *	$\times {10}^{-2}$
r1 (09:31–10:00 a.m.)	(−8.27, 23.50, 55.30)	(−1.63, 25.50, 51.50)	(−2.67, 23.50, 50)	(2.52, 25.50, 47.70) *	$\times {10}^{-3}$
r2 (10:00–10:30 a.m.)	(−1.85, 1.40, 4.68)	(−2.37, 0.63, 3.65)	(−1.39, 1.40, 4.21)	(−1.91, 0.63, 3.22)	$\times {10}^{-2}$
r3 (10:30–11:00 a.m.)	(−1.11, 2.80, 6.76)	(0.27, 3.59, 6.88) *	(−0.42, 2.80, 6.09)	(0.80, 3.59, 6.39) *	$\times {10}^{-2}$
r4 (11:00–11:30 a.m.)	(−1.32, 3.03, 7.43)	(−2.15, 1.70, 5.59)	(−0.66, 3.03, 6.79)	(−1.54, 1.70, 5.06)	$\times {10}^{-2}$
r5 (11:30–12:00 p.m.)	(1.61, 6.36, 11.30) *	(−4.88, −0.56, 3.68)	(2.38, 6.36, 10.50) *	(−4.19, −0.56, 2.94)	$\times {10}^{-2}$
r6 (12:00–12:30 p.m.)	(5.95, 59.40, 113) *	(−27.40, 21.50, 69.80)	(13.50, 59.40, 104) *	(−19.70, 21.50, 62.40)	$\times {10}^{-2}$
r7 (12:30–01:00 p.m.)	(−2.64, 3.15, 8.93)	(−3.64, 1.72, 7.06)	(−1.69, 3.15, 8.05)	(−2.70, 1.72, 6.16)	$\times {10}^{-2}$
r8 (01:00–01:30 p.m.)	(−5.62, −0.02, 5.35)	(−8.19, −3.26, 1.88)	(−4.68, −0.02, 4.43)	(−7.46, −3.26, 1.04)	$\times {10}^{-2}$
r9 (01:30–02:00 p.m.)	(2.11, 8.09, 13.90) *	(0.01, 5.09, 10.10) *	(3.06, 8.09, 13) *	(0.77, 5.09, 9.31) *	$\times {10}^{-2}$
r10 (02:00–02:30 p.m.)	(−5.85, −0.68, 4.55)	(−4.53, 0.38, 5.08)	(−5.07, −0.68, 3.75)	(−3.70, 0.38, 4.38)	$\times {10}^{-2}$
r11 (02:30–03:00 p.m.)	(1.35, 6.11, 10.80) *	(−3.71, 0.94, 5.35)	(2.15, 6.11, 10.10) *	(−2.96, 0.94, 4.68)	$\times {10}^{-2}$
r12 (03:00–03:30 p.m.)	(−4.73, −0.54, 4.33)	(−6.59, −1.56, 3.36)	(−4.05, −0.54, 3.52)	(−5.69, −1.56, 2.57)	$\times {10}^{-2}$
vixlagclose	(1.95, 16.60, 30.10) *	(−18.10, −2.33, 13.70)	(4.73, 16.60, 28) *	(−15.90, −2.33, 11.20)	$\times {10}^{-6}$
vixpctlag	(−1.15, 0.01, 1.22)	(−1.17, 0.06, 1.36)	(−0.98, 0.01, 1.02)	(−0.97, 0.06, 1.14)	$\times {10}^{-3}$
usdlagclose	(−8.68, 0.33, 9.37)	(−10.60, −0.79, 9.35)	(−7.16, 0.33, 8.05)	(−9.27, −0.79, 7.87)	$\times {10}^{-6}$
usdpctlag	(−1.41, 0.10, 1.62)	(−0.87, 0.87, 2.66)	(−1.18, 0.10, 1.38)	(−0.60, 0.87, 2.36)	$\times {10}^{-2}$
I_fomc1	( −6.21, 4.44, 15.00)	(−8.20, 5.62, 18.70)	(−4.24, 4.44, 13.20)	(−5.66, 5.62, 16.50)	$\times {10}^{-4}$
I_fomc1lag1	(−2.96, −1.39, 0.31)	(−2.66, −1.25, 0.12)	(−2.72, −1.39, 0.04)	(−2.45, −1.25, −0.11) *	$\times {10}^{-3}$
I_fomc2lag1	(4.47, 14.10, 24.90) *	(6.93, 21.50, 35.80) *	(6.07, 14.10, 22.90) *	(9.29, 21.50, 33.60) *	$\times {10}^{-4}$
I_fomc3lag1	(−2.26, −1.33, −0.35) *	(−1.63, −0.42, 0.74)	(−2.13, −1.33, −0.53) *	(−1.42, −0.42, 0.58)	$\times {10}^{-3}$
weekday2	(−3.44, −1.14, 1.21)	(−4.52, −1.99, 0.48)	(−3.04, −1.14, 0.82)	(−4.10, −1.99, 0.07)	$\times {10}^{-4}$
weekday3	(−2.31, 0.01, 2.21)	(−3.20, −0.62, 1.89)	(−1.96, 0.01, 1.90)	(−2.76, −0.62, 1.50)	$\times {10}^{-4}$
weekday4	(−2.89, −0.63, 1.60)	(−5.18, −2.57, −0.11) *	(−2.52, −0.63, 1.25)	(−4.70, −2.57, −0.46) *	$\times {10}^{-4}$
weekday5	(8.43, 31.30, 54.10) *	(−15.20, 9.77, 35.30)	(12.40, 31.30, 50.10) *	(−11.80, 9.77, 31.10)	$\times {10}^{-5}$
mon2	(−6.27, −2.79, 0.73)	(−6.27, −2.30, 1.78)	(−5.74, −2.79, 0.13)	(−5.72, −2.30, 1.09)	$\times {10}^{-4}$
mon3	(−7.89, −4.56, −1.20) *	(−6.02, −2.33, 1.52)	(−7.38, −4.56, −1.69) *	(−5.38 −2.33 0.92)	$\times {10}^{-4}$
mon4	(−5.17, −1.66, 1.72)	(−4.37, −0.57, 3.33)	(−4.53, −1.66, 1.19)	(−3.82, −0.57, 2.75)	$\times {10}^{-4}$
mon5	(−5.16, −1.83, 1.60)	(−4.92, −0.89, 3.21)	(−4.66, −1.83, 1.05)	(−4.22, −0.89, 2.47)	$\times {10}^{-4}$
mon6	( −8.30, −4.81, −1.28) *	(−7.90, −3.87, 0.20)	(−7.78, −4.81, −1.81) *	(−7.23, −3.87, −0.44) *	$\times {10}^{-4}$
mon7	(−4.52, −1.14, 2.32)	(−4.10, −0.23, 3.56)	(−3.95, −1.14, 1.66)	(−3.53, −0.23, 2.95)	$\times {10}^{-4}$
mon8	(−7.68, −4.29, −0.86) *	(−7.56, −3.76, 0.14)	(−7.08, −4.29, −1.39) *	(−6.92, −3.76, −0.43) *	$\times {10}^{-4}$
mon9	(−5.53, −2.18, 1.34)	(−4.38, −0.62, 3.29)	(−5.05, −2.18, 0.78)	(−3.78, −0.62, 2.72)	$\times {10}^{-4}$
mon10	(−4.92, −1.46, 2.04)	(−5.05, −1.15, 2.81)	(−4.38, −1.46, 1.46)	(−4.40, −1.15, 2.11)	$\times {10}^{-4}$
mon11	(−3.96, −0.44, 3.02)	(−4.13, −0.15, 3.88)	(−3.35, −0.44, 2.45)	(−3.49, −0.15, 3.24)	$\times {10}^{-4}$
mon12	(−7.09, −3.64, −0.19) *	(−6.53, −2.66, 1.28)	(−6.52, −3.64, −0.70) *	(−5.90, −2.66, 0.68)	$\times {10}^{-4}$
$\sigma$	(1.36, 1.43, 1.52) *	(1.53, 1.62, 1.71) *	(1.37, 1.43, 1.50) *	(1.55, 1.62, 1.70) *	$\times {10}^{-3}$
$\nu$	( 1.67, 1.84, 2.02) *	(1.84, 2.03, 2.24) *	(1.70, 1.84, 1.98) *	(1.87, 2.03, 2.20) *	$\times 1$

) as well as returns during lunch (11:30 a.m. to 1:00 p.m., $r_5,r_6,r_7$, see Section 4.3) when the overnight volatility of the previous day was relatively low. In addition to the returns of each intraday trading session, we find an interesting oscillating pattern after the rate change decisions were made (see Figure A4).

It showed a significant tug-of-war effect between bulls and bears after a critical change in the rate of the federal fund was announced. To harness the dynamic nature of the associations between the predictive factors and the returns of the last 30 min, we propose some trading strategies that are adaptive to market changes. The backtest of the proposed strategies is conducted and shows better performance than their benchmark strategies.

Technically, we have pointed out limitations of the Gaussian linear regression model that has been widely used in the literature. We compared our results based on the Bayesian model with those obtained from the Gaussian linear model to assess the significance of explanatory variables and to backtest the trading strategies developed based on the study.

The structure of this paper is as follows: Section 2 presents the data used in the article. The methods used including the Bayesian linear regression model with Student-t error terms are discussed in Section 3. The main results are reported in Section 4, with corresponding trading strategies being developed and backtested in Section 5. More comparisons between the Bayesian model and the Gaussian linear model are reported in Section 6. Finally, we conclude the paper in Section 7.

2. Data

The sample spans from 1 August 2007 to 31 July 2018. To obtain the values for open, high, low, close prices, and volume (OHLCV), we used 1-min intervals. Data were subjected to a screening process that ensured retention of the entire period of daily trading, which occurred between 9:30 a.m. and 4:00 p.m. Two major ETFs were studied: SPY and QQQ. We considered the days when all the trading sessions, as illustrated in Figure 1, were available in the dataset, and 2743 days were studied for SPY and 2741 days for QQQ.

To assess the reliability of intraday returns on any given trading day t, we calculated the initial half-hour return using the price at 9:31 a.m. and 10:00 a.m. Subsequently, we computed the return for each subsequent half-hour from 10:00 a.m. to 4:00 p.m. In addition to the returns for the current trading day, we also take into account the returns from the previous day. Specifically, we consider the overnight return $r_{on}$ and the return of the last 30 min of the regular trading hours of the previous day, denoted as $r_{13lag}$. This results in fourteen returns per day plus an additional return $r_{13lag}$ from the previous day. Returns were calculated by fraction of close prices at different time points, as illustrated in Figure 1.

In addition to return factors, we also take into account indicator variables that indicate the days when FOMC decided to alter the rates of the Federal Reserve fund and the following one, two, and three days, respectively.

Table 1. Dates and rate changes of the FOMC over the sampling period. This table displays the dates when the FOMC announced changes in federal fund rates, as well as the following 1, 2, and 3 trading days, respectively.

FOMC Decision Date	Rate Changes (bps)	Federal Fund Rates	Day 1	Day 2	Day 3
2007-09-18	−50	$4.75\%$	2007-09-19	2007-09-20	2007-09-21
2007-10-31	−25	$4.50\%$	2007-11-01	2007-11-02	2007-11-05
2007-12-11	−25	$4.25\%$	2007-12-12	2007-12-13	2007-12-14
2008-01-22	−75	$3.50\%$	2008-01-23	2008-01-24	2008-01-25
2008-01-30	−50	$3.00\%$	2008-01-31	2008-02-01	2008-02-04
2008-03-18	−75	$2.25\%$	2008-03-19	2008-03-20	2008-03-24
2008-04-30	−25	$2.00\%$	2008-05-01	2008-05-02	2008-05-05
2008-10-08	−50	$1.50\%$	2008-10-09	2008-10-10	2008-10-13
2008-10-29	−50	$1.00\%$	2008-10-30	2008-10-31	2008-11-03
2008-12-16	−100	$0\%$ to $0.25\%$	2008-12-17	2008-12-18	2008-12-19
2015-12-17	+25	$0.25\%$ to $0.5\%$	2015-12-18	2015-12-21	2015-12-22
2016-12-15	+25	$0.5\%$ to $0.75\%$	2016-12-16	2016-12-19	2016-12-20
2017-03-16	+25	$0.75\%$ to $1.00\%$	2017-03-17	2017-03-20	2017-03-21
2017-06-15	+25	$1.00\%$ to $1.25\%$	2017-06-16	2017-06-19	2017-06-20
2017-12-14	+25	$1.25\%$ to $1.5\%$	2017-12-15	2017-12-18	2017-12-19
2018-03-22	+25	$1.50\%$ to $1.75\%$	2018-03-23	2018-03-26	2018-03-27
2018-06-14	+25	$1.75\%$ to $2.0\%$	2018-06-15	2018-06-18	2018-06-19

shows the dates, rate changes, and rates of federal funds for FOMC indicators.

Table 2. Summary statistics for the numerical variables utilized in both the Student-t Bayesian linear regression model and the Gaussian linear regression model. The return variables, such as r13_lag, r1, usdlagclose (previous day US dollar index close price), and vixlagclose (previous day VIX close price) are expressed in basis points (bps), where 1 bps equals $0.0001$. In contrast, the variables vixpctlag (previous day VIX close price percentage change), and usdpctlag (previous day US dollar index close price percentage change) are reported without scaling in bps. Summary measures include the mean, standard deviation (std), median, maximum, minimum, skewness, and kurtosis values.

	QQQ							SPY
	Mean (bps)	Std (bps)	Median (bps)	Max (bps)	Min (bps)	Skewness	Kurtosis	Mean (bps)	Std (bps)	Median (bps)	Max (bps)	Min (bps)	Skewness	Kurtosis
r13_lag	−0.37	38	1	337	−329	−0.44	21.1	−0.11	39	1	325	−413	−0.33	26.2
r_on	2.83	78	6	543	−1372	−2.38	43.7	1.15	71	4	522	−671	−0.58	13.0
r1	1.68	49	2	1081	−446	4.07	100.1	0.16	34	0	448	−354	0.66	24.7
r2	0.38	36	1	205	−360	−0.39	10.9	0.46	30	1	193	−290	−0.15	11.6
r3	−0.33	31	1	204	−188	−0.11	8.0	−0.43	26	1	262	−205	−0.16	14.3
r4	0.06	26	1	130	−202	−0.30	7.5	−0.22	22	1	117	−154	−0.44	8.0
r5	0.47	25	1	232	−189	0.14	11.5	0.41	21	1	213	−143	0.40	13.5
r6	−0.25	23	0	158	−213	−0.53	12.6	−0.09	20	1	214	−249	−0.48	22.2
r7	0.08	22	1	180	−253	−0.36	16.2	0.27	20	1	166	−281	−0.97	27.8
r8	0.00	23	0	275	−131	0.31	15.2	0.13	21	1	235	−150	0.63	18.5
r9	−0.68	25	0	276	−230	0.07	19.8	−0.43	22	0	228	−235	−0.32	21.7
r10	0.16	27	1	304	−225	0.31	16.6	0.09	25	0	230	−213	0.40	16.3
r11	1.11	28	1	219	−185	0.63	13.8	0.93	26	1	233	−173	0.99	18.4
r12	0.57	33	1	591	−259	3.40	57.6	0.82	32	1	647	−258	4.11	79.7
r13	−0.44	38	1	337	−329	−0.46	21.2	−0.18	39	1	325	−413	−0.34	26.4
vixpctlag	28.57	803	−58	11,560	−2957	2.17	22.5	27.27	800	−58	11,560	−2957	2.17	22.7
usdpctlag	0.63	52	0	256	−268	0.02	5.1	0.61	52	0	256	−268	0.01	5.1
vixlagclose	19.97	9.65	17.26	80.86	9.14	2.33	10.3	19.96	9.64	17.25	80.86	9.14	2.33	10.3
usdlagclose	85.04	8.27	81.82	103.29	71.33	0.48	1.9	85.04	8.27	81.82	103.29	71.33	0.48	1.9

displays the summary statistics for the numerical variables utilized in our proposed model.

3. Methods

For each day, we consider the trading sessions illustrated in Figure 1. The last half-hour return $r_{13}$ is treated as the response variable and $r_{13lag},r_{on},r_1,\dots ,r_{12}$ as explanatory variables. Other explanatory variables include the VIX index, the US dollar index, and indicator variables to indicate whether there is an FOMC decision on changing the Federal Reserve fund rate on the current day and in the next 1, 2, and 3 days, respectively. We also include “month” and “weekdays” as explanatory variables, as there could be month and weekday effects that might affect the intraday market predictability of our interest.

In the literature on financial data modeling, multiple linear regression models with Gaussian error terms have been used as the standard modeling tool. We will argue that financial returns often appear to be heavy-tailed and a linear regression model with Gaussian error terms may be misleading, especially when one needs to identify those statistically significant explanatory variables. Instead, we used a Bayesian multiple linear regression model with Student-t error terms. For the data samples under study, we have observed significant heavy tails of returns (see Figure A1). The Student-t error term can well account for heavy tails of the returns, and thus our assessment on whether some explanatory variables are significant or not would be more reliable than those obtained from an ordinary multiple linear regression model with Gaussian error terms. To compare the discrepancy between the results obtained from the Bayesian linear model with Student-t errors and those from the Gaussian linear regression models, we also performed the corresponding analysis based on the latter model. However, for interpretation, we rely more on the Bayesian model. A comparison between the estimates of the two models is reported in Section 4, and comparisons in terms of trading strategies and their performance are discussed in Section 5.

3.1. Bayesian Linear Regression with Student-t Errors

We follow Zellner (1976) to illustrate the Bayesian linear regression model with the Student-t error terms. Assume that we have n observations and the model for the response variable $\mathit{y}={(y_1,y_2,\dots ,y_n)}^{\prime }$ with the covariate matrix ${\mathit{X}}_{n\times p}$ is

$$\mathit{y}=\mathit{X}\mathit{\beta }+ϵ\mathit{,}$$

(1)

where $\mathit{\beta }$ is a vector $p\times 1$ of regression parameters and $\mathit{ϵ}={({ϵ}_1,{ϵ}_2,\dots ,{ϵ}_n)}^{\prime }$ is a random error vector. The joint probability density function (pdf) for the vector $\mathit{ϵ}$ is

$$p\left(\mathit{ϵ}\right|\nu ,\sigma )={\displaystyle \frac{{\nu }^{\nu /2}(\nu +n/2)/{\pi }^{n/2}\Gamma (v/2)}{{\left({\sigma }^2\right)}^{n/2}{\{\nu +{\mathit{ϵ}}^{\prime }\mathit{ϵ}/{\sigma }^2\}}^{(n+\nu )/2}}},$$

(2)

where $\sigma ,\nu >0,-\infty <{ϵ}_i<\infty ,i=1,2,\dots ,n$. When $\nu$ is finite, ${ϵ}_i/\sigma$ follows a univariate Student-t pdf with $\nu$ degrees of freedom, and ${ϵ}_i$ will have marginal distributions with heavy tails if $\nu$ is small.

Thus, the likelihood function for the regression model is given by

$$\begin{array}{cc}\hfill p\left(\mathit{y}\right|\mathit{\beta },\nu ,\sigma )& ={\displaystyle \frac{{\nu }^{\nu /2}(\nu +n/2)/{\pi }^{n/2}\Gamma (v/2)}{{\left({\sigma }^2\right)}^{n/2}}}\hfill \\ \hfill & \times {\{\nu +{\displaystyle \frac{{(\mathit{y}-\mathit{X}\widehat{\mathit{\beta }})}^{\prime }(\mathit{y}-\mathit{X}\widehat{\mathit{\beta }})+{(\mathit{\beta }-\widehat{\mathit{\beta }})}^{\prime }{\mathit{X}}^{\prime }\mathit{X}(\mathit{\beta }-\widehat{\mathit{\beta }})}{{\sigma }^2}}\}}^{-(n+\nu )/2},\hfill \end{array}$$

(3)

where $\widehat{\mathit{\beta }}={\left({\mathit{X}}^{\prime }\mathit{X}\right)}^{-1}{\mathit{X}}^{\prime }\mathit{y}$. Equations (1)–(3) will be analyzed from a Bayesian point of view with a given prior distribution for the parameters $\mathit{\beta }$, $\nu$, and $\sigma$. Specifically, in our experiment, we assign Gaussian, Gamma, and Cauchy as prior distributions for $\mathit{\beta }$, $\nu$, and $\sigma$, respectively. The posterior distributions of the parameters were derived from the observed data using Markov chain Monte Carlo (MCMC) simulations. We use statistical software R and the package brms (Bürkner 2017) to perform the analysis. We will then interpret the results based on the model derived in Section 4.

3.2. Multiple Linear Regression with Gaussian Errors

Follow the regression model (see Equation (1)), which is equivalent to $y_i={\beta }_0+{\beta }_1{x}_{i1},\dots ,{\beta }_p{x}_{ip}+{ϵ}_i$, for $i=1,\dots ,n$. For the Gaussian error terms, we assume that ${ϵ}_i$ are independent and identically distributed (i.i.d.) with ${ϵ}_i\sim \mathcal{N}(0,{\sigma }^2)$. Least squares methods (LSE) were applied to estimate the regression coefficients $\mathit{\beta }={({\beta }_0,{\beta }_1,\dots ,{\beta }_p)}^{\prime }$, which is consistent to looking for $\widehat{\mathit{\beta }}$ minimizing (Grégoire 2014):

$$\begin{array}{cc}\hfill \mathcal{S}\left(\mathit{\beta }\right)& =\sum _{i=1}^n{(y_i-{\beta }_0-{\beta }_1{x}_{i1}-\cdots -{\beta }_p{x}_{ip})}^2\hfill \\ \hfill & ={(\mathit{y}-\mathit{X}\mathit{\beta })}^{\prime }(\mathit{y}-\mathit{X}\mathit{\beta })\hfill \end{array}$$

(4)

The estimated coefficient vector, denoted $\widehat{\mathit{\beta }}$, can be obtained by putting the first derivative of the function ${\mathcal{S}}^{\prime }\left(\mathit{\beta }\right)$ equal to zero. Mathematically, this can be expressed as $\widehat{\mathit{\beta }}={\left({\mathit{X}}^{\prime }\mathit{X}\right)}^{-1}{\mathit{X}}^{\prime }\mathit{y}$.

4. Results

In this section, we present the results obtained from the Bayesian model, and a comparison with those from the multiple linear regression model will be discussed.

4.1. Model Fitting and Diagnostics

We first performed a comprehensive analysis using the Bayesian model, involving all the relevant variables that we were initially interested in. The results are reported in Table A1 with credible intervals of $95\%$ and $90\%$. After reviewing these results, we decided to include only the variables that we were still interested in for further analysis. At this stage, the variables for the effects of month, the effects of the US dollar and the changes in VIX were removed from our further analysis, as these variables were probably not statistically significant. The results for SPY and QQQ were slightly different, but share similar patterns of the effects of covariates; see Figure A4 for a comparison. To explain the main idea and findings, we focus on SPY for the following discussion.

Panel A in

Table 3. Panels A and B represent estimates for the Bayesian linear regression model with Student-t error terms and the multiple linear regression model, respectively. “sig” in panel A indicates whether the variable is significant based on whether the value 0 is within the $2.5\%$ and $97.5\%$ quantiles. Significant factors were highlighted in bold format. The “scale” column determines the scale of these estimates (except for the t-value).

	Panel A: Bayesian Linear Model				Panel B: Gaussian Linear Model			Scale
	2.5%	50%	97.5%	Sig	Estimate	t-Value	Sig
r13_lag	−5.70	−2.18	1.36	No	−16.00	−8.55	Yes	$\times {10}^{-2}$
r_on	2.49	4.05	5.59	Yes	8.54	8.47	Yes	$\times {10}^{-2}$
r1	−1.06	2.29	5.50	No	−1.59	−0.75	No	$\times {10}^{-2}$
r2	−2.43	0.95	4.17	No	8.13	3.47	Yes	$\times {10}^{-2}$
r3	−1.19	2.67	6.52	No	12.30	4.55	Yes	$\times {10}^{-2}$
r4	−1.63	2.66	7.16	No	2.22	0.69	No	$\times {10}^{-2}$
r5	1.67	6.36	11.20	Yes	11.50	3.47	Yes	$\times {10}^{-2}$
r6	1.12	6.47	12.00	Yes	5.16	1.50	No	$\times {10}^{-2}$
r7	−2.41	3.46	9.31	No	−12.70	−3.51	Yes	$\times {10}^{-2}$
r8	−5.74	−0.32	5.20	No	0.41	0.12	No	$\times {10}^{-2}$
r9	2.32	8.17	13.80	Yes	−1.86	−0.58	No	$\times {10}^{-2}$
r10	−5.89	−0.52	4.90	No	7.79	2.75	Yes	$\times {10}^{-2}$
r11	1.68	6.83	11.60	Yes	2.31	0.84	No	$\times {10}^{-2}$
r12	−4.77	−0.59	4.21	No	16.00	7.19	Yes	$\times {10}^{-2}$
vixlagclose	2.61	15.70	28.30	Yes	6.85	0.93	No	$\times {10}^{-6}$
I_fomc1	−7.19	3.66	14.30	No	−43.70	−4.87	Yes	$\times {10}^{-4}$
I_fomc1lag1	−2.97	−1.45	0.17	No	−2.21	−2.45	Yes	$\times {10}^{-3}$
I_fomc2lag1	3.11	12.50	23.20	Yes	15.20	1.70	No	$\times {10}^{-4}$
I_fomc3lag1	−2.36	−1.44	−0.48	Yes	0.41	0.46	No	$\times {10}^{-3}$
weekday2	−3.54	−1.23	1.07	No	−1.79	−0.80	No	$\times {10}^{-4}$
weekday3	−2.17	0.11	2.42	No	−5.23	−2.35	Yes	$\times {10}^{-4}$
weekday4	−2.92	−0.63	1.63	No	−1.30	−0.58	No	$\times {10}^{-4}$
weekday5	7.56	31.10	53.40	Yes	8.14	0.36	No	$\times {10}^{-5}$
$\sigma$	1.36	1.44	1.52	Yes	−	−	−	$\times {10}^{-3}$
$\nu$	1.68	1.84	2.02	Yes	−	−	−	$\times 1$

reports the results based on the selected variables. The last row of Table 3 is the degree of freedom $\nu$ of the Student-t error terms. It is clear that the posterior median of the degree of freedom $\nu$ of the Student-t distribution was approximately $1.84$, suggesting a very heavy-tailed distribution for the error term. For comparison, in Panel B of Table 3, we also include the results obtained from the multiple linear regression model, which did not account for the heavy tails of the returns. There are quite a few discrepancies between the results of the two models, and we suggest using the results from the Bayesian model to assess whether the variables were statistically significant.

The results of the Bayesian model suggest that after controlling the variables of interest, the following explanatory variables were significant at the significance level of $5\%$: $r_{on},r_5,r_6,r_9,r_{11}$, vixlagclose, fomc2lag, fomc3lag, and the Friday effect. We have also tried including the following variables in the regression model: the US dollar index in the previous day, the changes in the US dollar index in the previous day, the changes in the VIX index in the previous day, and the month of the current day in the model. They did not show significance and, therefore, were excluded from the model reported in Panel A of Table 3. We refer the reader to Table A1 for the estimates of all the variables.

Conditional effect graphs are shown in Figure 2 and Figure 3. The effect of each variable is illustrated along its range of values, conditioning on the other continuous variables taking their corresponding mean values, and categorical variables taking their corresponding reference levels.

The model diagnostics were performed, and the results are contained in the Appendix A.1. The posterior distributions of the parameters are illustrated in Figure A3, and it is clear that the convergences are achieved satisfactorily. The PSIS diagnostic plot is shown in Figure A2, and all the values of k were less than $0.5$, also suggesting a satisfactory convergence of the Bayesian model.

4.2. Dynamic Market Predictability

To study the potential evolution of the predictability of the last 30 min returns, we consider the rolling-window method that covers the current year of interest and the two previous years. The results are reported in Figure 4. There were no federal fund rate change decisions for certain rolling windows, so relevant explanatory variables about FOMC decision days were excluded from the plots.

4.3. Overnight Effects

To further study the effect of $r_{on}$, the previous midnight returns, we split the data into two scenarios: small and large $r_{on}$, using the median overnight returns of all study years as the cut-off point. The results are reported in Figure 5. It is clear that for those days with larger previous overnight returns, the intraday momentum effect of $r_{on}$ is relatively greater. We look at those days with smaller previous overnight returns as boring days that have fewer news to generate market volatilities. For those boring days, there are interesting results in $r_5$ and $r_6$, which are returns before lunch breaks on Wall Street. Based on our study, there is a significant momentum effect right before the lunch break time period when the market is relatively boring. There was a significant positive association between returns before the midday break and those before the end of the day break on those days when the market is relatively boring. To the best of our knowledge, this suggests another momentum effect that has not been documented in the literature.

5. Trading Strategy and Backtesting

An approach to evaluating the efficacy of a predictor is to analyze its economic performance using backtesting. Based on the results obtained in Figure 4, we can develop some trading strategies. In the literature, trading strategies based on significant factors were developed from the aggregated data of many years. They are mainly based on the overnight returns ($r_{on}$) or the returns of the last 30 min of the previous day ($r_{13lag}$). We denote the two indicated strategies as $\mathring{\eta }\left(r_{on}\right)$ and $\mathring{\eta }\left(r_{lag}\right)$, respectively, in the following.

However, from the results of the three-year rolling windows (see Figure 4), the significance of trading signals has evolved over the years. We now develop some new trading strategies based on data from the past three years only and update the strategies yearly. Specifically, if the trading signal is significant at the beginning of the final half-hour of the regular trading hours, we initiate either a long or a short position and then close the position at the end of the 30 min trading window. The direction of the position is determined by the sign of the effects of the reference returns, and a summary table of the timing signals that includes the reference returns and their trading directions is

Table 4. Trading signal summary table. For the signal $r_p^{\pm }$, p and ± represent the reference return $r_p$ and its trading direction, respectively. “+” suggests initiating a long position, and “−” suggests initiating a short position. For example, (1) $r_{on}^{+}$ in 2010 indicates that we take a long position when $r_{on}$ is positive and a short position otherwise; (2) $r_{lag}^{-}$ in 2018 suggests that we take a long position when $r_{lag}$ is negative and a short position otherwise. “-” indicates that no positions are initiated. $m=3,5,n=3,12,i=3,5,7,8,12,j=3,5,7,8,12$. The economic return for trades based on $r_p^{\pm }$ in the last 30 min becomes $r_{13}$ if a long position is initialized and $-r_{13}$ if a short position is initialized. $T,G$ represents Student-t Bayesian regression and Gaussian linear regression, respectively. The two subscripts of T and G represent which trading sessions are used for 2017 and 2018, respectively.

Strategy	2010	2011	2012	2013	2014	2015	2016	2017	2018
$\mathring{\eta }\left(r_{on}\right)$	$r_{on}^{+}$	$r_{on}^{+}$	$r_{on}^{+}$	$r_{on}^{+}$	$r_{on}^{+}$	$r_{on}^{+}$	$r_{on}^{+}$	$r_{on}^{+}$	$r_{on}^{+}$
$\eta \left(r_{on}\right)$	$r_{on}^{+}$	$r_{on}^{+}$	$r_{on}^{+}$	$r_{on}^{+}$	$r_{on}^{+}$	$r_{on}^{+}$	$r_{on}^{+}$	-	-
$\mathring{\eta }\left(r_{lag}\right)$	$r_{lag}^{-}$	$r_{lag}^{-}$	$r_{lag}^{-}$	$r_{lag}^{-}$	$r_{lag}^{-}$	$r_{lag}^{-}$	$r_{lag}^{-}$	$r_{lag}^{-}$	$r_{lag}^{-}$
$\eta \left(r_{lag}\right)$	$r_{lag}^{-}$	$r_{lag}^{-}$	$r_{lag}^{-}$	-	-	$r_{lag}^{+}$	$r_{lag}^{+}$	-	-
$\eta \left(T_{m,n}\right)$	$r_{on}^{+}$	$r_{on}^{+}$	$r_{on}^{+}$	$r_{on}^{+}$	$r_{on}^{+}$	$r_{on}^{+}$	$r_{on}^{+}$	$r_m^{+}$	$r_n^{+}$
$\eta \left(G_{i,j}\right)$	$r_{on}^{+}$	$r_{on}^{+}$	$r_{on}^{+}$	$r_{on}^{+}$	$r_{on}^{+}$	$r_{on}^{+}$	$r_{on}^{+}$	$r_i^{+}$	$r_j^{+}$
$\eta \left(G_{9,j}\right)$	$r_{on}^{+}$	$r_{on}^{+}$	$r_{on}^{+}$	$r_{on}^{+}$	$r_{on}^{+}$	$r_{on}^{+}$	$r_{on}^{+}$	$r_9^{-}$	$r_j^{+}$
$\eta \left(G_{i,9}\right)$	$r_{on}^{+}$	$r_{on}^{+}$	$r_{on}^{+}$	$r_{on}^{+}$	$r_{on}^{+}$	$r_{on}^{+}$	$r_{on}^{+}$	$r_i^{+}$	$r_9^{-}$

. When the trading signal is deemed insignificant, no trades are made unless the timing signal is substituted with another significant signal.

The strategy $\eta \left(r_{on}\right)$ is an improved version of $\mathring{\eta }\left(r_{on}\right)$, following the observation of statistically significant results from the Student-t Bayesian regression results (Figure 4). The significance of $r_{on}$ disappeared in 2017 and 2018, and therefore, before 2016, inclusive, $\eta \left(r_{on}\right)$ and $\mathring{\eta }\left(r_{on}\right)$ performed identically. However, in 2017 and 2018, $\eta \left(r_{on}\right)$ takes neutral positions when the signals were no longer significant. The strategy $\eta \left(r_{lag}\right)$ is an improved version of $\mathring{\eta }\left(r_{lag}\right)$, and takes a long position when $r_{lag}$ is negative and a short position otherwise from 2010 to 2012. In years 2013, 2014, 2017, and 2018, $\eta \left(r_{lag}\right)$ does not take positions, and in 2015 and 2016, however, $\eta \left(r_{lag}\right)$ takes a long position when $r_{lag}$ is positive and a short position otherwise. The benchmark strategy “Always Long” (AL) always initializes a long position at the beginning of the last 30 min and closes the position at the end of the last 30 min.

When the trading signals $r_{on}$ and $r_{lag}$ were no longer significant for 2017 and 2018, we consider strategies based on the returns of other intraday trading sessions, such as $r_3$ and $r_5$. We use T and G to represent Student-t Bayesian regression and Gaussian linear regression, respectively. m, n, i and j are determined by observing the significance status of the return covariates in the Bayesian Student-t linear regression (Figure 4) or the Gaussian linear regression (Figure A5) of the rolling window of 3 years in 2016 and 2017. Note that we used the significance of factors from the previous three years to predict the following year; for example, if we observed $r_3$ as a significant predictor in 2016, we will use $r_3$ as a trading signal in 2017. When i or j is equal to 9, we will take a long position when $r_9$ is negative and a short position otherwise because we observed a negative association between $r_9$ and the response variable $r_{13}$.

For strategies $\eta \left(T_{m,n}\right)$ or $\eta \left(G_{i,j}\right)$, before 2016, inclusive, they performed the same as $\eta \left(r_{on}\right)$. However, unlike $\eta \left(r_{on}\right)$ and $\eta \left(r_{lag}\right)$, $\eta \left(T_{m,n}\right)$ and $\eta \left(G_{i,j}\right)$ are more aggressive in their operations in 2017 and 2018. In 2017, $\eta \left(T_{m,n}\right)$ [$\eta \left(G_{i,j}\right)$] takes a long position when $r_m$ [$r_i$] is positive and a short position otherwise; in 2018, $\eta \left(T_{m,n}\right)$ [$\eta \left(G_{i,j}\right)$] takes a long position when $r_n$ [$r_j$] is positive and a short position otherwise.

By analyzing the values of the subscripts: m, n, i and j, it becomes evident that the strategies $\eta \left(T_{m,n}\right)$ form a subset of $\eta \left(G_{i,j}\right)$. Theoretically, the Bayesian linear regression model with Student-t error is more sensitive to extreme values, such as outliers, compared to Gaussian error linear models. This increased sensitivity leads to a more conservative approach, as the Bayesian method tends to select fewer strategies based on the significance of return signals.

In summary, 30 strategies ($\eta \left(G_{i,j}\right)$) were identified using the observed significance of returns in 2016 and 2017 using the Gaussian linear model with the rolling window of three years (GLR3) method. In contrast, only four strategies ($\eta \left(T_{m,n}\right)$) were selected using the Bayesian linear regression of the Student-t with a rolling window of three years (BLRT3) method. Interestingly, these four strategies are a strict subset of the 30 proposed by the GLR3 model, underscoring the conservative nature of the Bayesian approach. A more detailed comparison between Bayesian and Gaussian error term models will be discussed in Section 6.

Table 5. Trading strategy results based on BLRT3 (Figure 4). This table shows the daily average returns, standard deviations, Sharpe ratios (SRatio), skewness, kurtosis, success rates, t-values from the Newey–West t test, and the number of three distinct positions: no trading (neutral), long position, and short position. “*” indicates that the robust Newey–West t test is significant at a significance level of $5\%$. The sample period spans from January 2010 to July 2018, with the optimal returns and Sharpe ratios highlighted for emphasis.

Strategy	AvgRet (bps)	StdDev (bps)	SRatio (pbs)	Skewness	Kurtosis	Success (%)	t-Value	Neutral	Long	Short
$\mathring{\eta }\left(r_{on}\right)$	0.92	24.70	372	−0.25	24.9	53.9	1.55	0	1155	988
$\eta \left(r_{on}\right)$	1.17	22.84	510	−0.37	29.9	62.9	2.12 *	394	928	821
$\mathring{\eta }\left(r_{lag}\right)$	−0.19	24.74	−77	1.08	24.8	48.6	−0.38	0	1005	1138
$\eta \left(r_{lag}\right)$	0.27	21.34	127	1.36	39.1	71.5	0.62	892	599	652
$\eta \left(T_{3,3}\right)$	1.06	24.71	429	−0.09	24.8	53.6	1.80	0	1136	1007
$\eta \left(T_{3,12}\right)$	1.36	24.70	551	−0.36	24.9	54.1	2.42 *	0	1131	1012
$\eta \left(T_{5,3}\right)$	1.01	24.72	409	−0.09	24.8	53.5	1.73	0	1153	990
$\eta \left(T_{5,12}\right)$	1.32	24.70	534	−0.35	24.9	54.0	2.34 *	0	1148	995
AL	−0.21	24.74	−85	0.39	24.8	53.1	−0.41	0	2143	0

presents summary statistics on the returns generated from the proposed strategies observed from the Bayesian linear regression of Student-t with a rolling window of three years (BLRT3). For example, when we use the $\eta \left(r_{on}\right)$ strategy to trade in the last half hour, the average return is $1.17$ pbs per day. The aggressive strategy $\eta \left(T_{3,12}\right)$ has optimal returns with an average return of around $1.36$ pbs, and the Sharpe ratio equals 551 pbs per day. To gauge the performance of the proposed strategies, we also calculate the returns for AL. We always take a long position on the market at the beginning of the last half-hour and close it at the close of the market. The results indicate that the daily average return of this strategy is lower, at $-0.21$ pbs per day, and statistically insignificant. Therefore, the new strategies proposed outperform the benchmark AL strategy. The main limitation of all strategies is that we only trade and hold the position for 30 min and the fund utilization rate is quite low.

Unlike their improved versions, $\mathring{\eta }\left(r_{on}\right)$ and $\mathring{\eta }\left(r_{lag}\right)$ are not strictly proposed based on significant signals from BLRT3. Therefore, it is not surprising that their performance is not as good as that of their improved versions. A comparison between the results of $\mathring{\eta }\left(r_{lag}\right)$ [$\mathring{\eta }\left(r_{on}\right)$] and $\eta \left(r_{lag}\right)$ [$\eta \left(r_{on}\right)$] demonstrates that the dynamic strategy we propose, utilizing rolling windows, outperforms the static strategy commonly used in the intraday momentum literature. This superiority is evidenced by the significantly higher returns and Sharpe ratios achieved by the dynamic approach. The aggressive strategies $\eta \left(T_{3,3}\right)$, $\eta \left(T_{3,12}\right)$, $\eta \left(T_{5,3}\right)$, and $\eta \left(T_{5,12}\right)$ are proposed based on BLRT3, and they not only maintain adaptive in trading strategies, but also achieve average daily return $1.19$5 pbs (average return 299.256 pbs on an annual basis). Certainly, we also need to take into account the risk. The average standard deviation is around 24.7 pbs per day among aggressive strategies, resulting in an average Sharpe ratio (SRatio) of 480.757 pbs. The strategy AL has the same standard deviation value, but with a negative Sharpe ratio $-85$ pbs. Meanwhile, we also measure the success rate by defining it as the proportion of trading days that produce zero or positive returns (Gao et al. 2018). The AL strategy holds $53.1\%$ of the success rate, indicating that the unconditional probability that the last 30 min return is positive is $53\%$. Other strategies, such as $\eta \left(T_{3,3}\right)$, have a slightly higher success rate, for example, of $53.6\%$ than the benchmark strategy. Furthermore, it is not surprising that $\eta \left(r_{on}\right)$ and $\eta \left(r_{lag}\right)$ have much higher success rates ($63.5\%$ and $71.5\%$, respectively) due to the fact that these strategies improved their peers by avoiding over-trade during unfavorable years.

Kurtosis and skewness are used to report the density shape of each proposed strategy. Positive values of “Kurtosis” (>1) indicate that the distribution is peaked and has heavy tails. The value of “Skewness” shows the asymmetrical state of the distributions. A negative skewness value indicates the tail towards the left side, while a positive skewness value indicates the right side. Finally, we also summarize the number of neural, long, and short positions that were operated from January 2010 to July 2018 in Table 5.

Cumulative returns are calculated by compounding the periodic returns over time. Specifically, the cumulative return $r_T$ from time 0 to T is calculated by $r_T={\prod }_{t=1}^T(1+r_t)-1$, where $r_t$’s denote the periodic returns of consecutive periods from time 0 to T. Figure 6 visually illustrates the returns of the strategies. We plot the daily cumulative performance of the strategies that we are interested in and the benchmark AL strategy. First, we observe that the strategy based on a significant factor $r_{on}$ ($\mathring{\eta }\left(r_{on}\right)$ and $\eta \left(r_{on}\right)$) and its variants $\eta \left(T_{3,3}\right)$, $\eta \left(T_{3,12}\right)$, $\eta \left(T_{5,3}\right)$, $\eta \left(T_{5,12}\right)$ outperforms the passive strategy AL, and is consistent over time. Moreover, it is clear that aggressive strategies $\eta \left(T_{3,3}\right)$, $\eta \left(T_{3,12}\right)$, $\eta \left(T_{5,3}\right)$, and $\eta \left(T_{5,12}\right)$ could be a potential way to avoid the “drop” trend in the tail. Figure 7 further supports our conclusion by providing a comparison graph of cumulative annual returns on the proposed strategy and the benchmark. As mentioned above, the strategy $\eta \left(r_{lag}\right)$ is not the one proposed based on BLRT3 methods, and therefore, it is not surprising that it does not defeat the benchmark.

As outlined in our methodology, it is essential to clarify the rationale for not pursuing strategy development in 2007, 2008, and 2009, as our approach emphasizes the development of adaptive strategies based on models trained using data from the previous three years. For example, to formulate strategies for 2010, we use models trained on data from 2007, 2008, and 2009 (denoted 2009(3) in the accompanying Figure 4). However, in the cases of 2007, 2008, and 2009, the requisite three years of rolling trading data are not available, thus the diagram does not have the performance for these three years. Moreover, our research indicates that the FOMC variable does not significantly contribute to formulating optimal trading strategies at this stage. Notably, given our use of a rolling three-year window to model and extract meaningful insights continuously, the FOMC variable remains constant across certain three-year periods. To maintain analytical consistency and avoid potential model distortions, it is, therefore, more effective to exclude the FOMC variable in the design of trading strategies.

6. Comparisons with Gaussian Linear Models

Ordinary multiple linear regression models with default Gaussian error terms have been widely used in the literature on financial econometrics and unfortunately many of such articles conducted empirical research on financial returns without model diagnostics. We argue that ignoring model diagnostics may lead to misleading results. Ignoring heavy tails of financial returns is a common issue for using multiple linear regression models with Gaussian error terms to model returns data. Financial returns often have heavy tails that cannot be well modeled by the Gaussian distribution. For example, Figure A1 is the QQ plot of the residuals obtained from the multiple linear regression model with Gaussian error terms for the SPY data under study. Clearly, the Gaussian assumption is inadequate. Using Gaussian distributions to model heavy tails tends to underestimate the variability of the error terms, thus increasing the chances of false positives. We believe that the results obtained from the Bayesian linear models are more reliable than those obtained from the multiple linear regression model with Gaussian errors.

For comparison, the results obtained from the Gaussian linear regression model are in panel B of Table 3, and the column of “sig” indicates whether the p-values are less than $5\%$. The variables “month”, “vixpctlag”, and “usdlagclose” were not significant and were excluded from the model. We kept all intraday return variables, relevant variables associated with days when FOMC made fund rate change decisions, and weekdays to have a relatively more complete picture of the variables of interest and to compare them with the corresponding results obtained from the Bayesian linear regression model with Student-t error terms.

We also used the Gaussian linear model with a rolling window of three years (GLR3) to perform a regression analysis on identical response variables and covariates as those used in the Bayesian model with Student-t errors. Similar aggressive strategies (denoted as $\eta \left(G_{i,j}\right)$) are created on the basis of the Gaussian linear model. These strategies use $r_{on}$ as trading signals before 2017, the same as the strategies proposed by BLRT3 (see Table 4), and use their corresponding significant components, such as $r_3$ and $r_5$ after 2017, inclusive. For a further comparison, Figure 8 shows the results of the comparison of aggressive strategies proposed by GLR3 (gray line) and BLRT3 (black line), respectively.

7. Conclusions and Limitations

Motivated by extensive studies of intraday market momentum in the literature, we conducted a study to find out, in addition to those commonly studied factors, what other variables may contribute to the predictability of the returns of the last 30 min and to what extent the returns of the last 30 min of the financial market are predictable. To adequately address the questions, we first recognized the limitations of using the Gaussian linear regression model that are widely used in the literature and then, instead, used a Bayesian linear regression model with Student-t error terms to better capture the heavy tails of financial returns.

Our study suggests that overnight returns were positively associated with returns from the last 30 min, but the association became weaker during the later years of the period of our study. The associations between the returns of the first 30 min and those of the last 30 min were not as significant as those with the overnight returns, and such associations also became insignificant over the years. Similar patterns of such diminishing associations were also observed between the last 30 min of the previous day and that of the current day. It seems that all the associations commonly studied became less relevant over the years, and this may reflect that the market becomes more efficient and less predicable over the years.

Some other associations that come into our view are those between lunchtime returns and returns of the last 30 min. There were significant associations between them during relatively boring days with lower market volatilities. In addition, we found a tug-of-war effect during the three days after the Federal Reserve made fund rate change decisions.

We have proposed some trading strategies that are adaptive to market evolution, and the backtesting shows that their performance is relatively better than their nonadaptive versions. Furthermore, a comparison between the performance of the trading strategies developed from the Bayesian Student-t model and the Gaussian linear model suggests that the Bayesian model is more prudent and leads to less variability in financial returns.

One limitation of our study is that, like much other research work for the financial markets, our main findings are based on the specific time period. Financial markets are complex systems that continually evolve. As a result, any factor that demonstrates predictive power at a moment can decrease in effectiveness once identified, potentially altering the direction of future predictions. For example, in

Table A2. Bayesian linear regression model with Student-t error terms (all variables and non-scaled data) for recent years including 1 January 2022 to 30 September 2024. The values included in brackets represent estimates in the $2.5\%$ [$5\%$], $50\%$ [$50\%$], and $97.5\%$ [$95\%$] quantiles. The scale of these estimates is determined by the “scale” column. “*” indicates that the variable is significant. Significant factors were highlighted in bold format.

	95% Credible Intervals		90% Credible Intervals		Scale
	SPY	QQQ	SPY	QQQ
r13_lag	(−8.06, −0.99, 5.89)	(−9.29, −1.54, 6.06)	(−6.93, −0.99, 4.87)	(−8.02, −1.54, 4.93)	$\times {10}^{-2}$
r_on	(−3.45, −0.40, 2.59)	(−3.41, −0.77, 1.88)	(−2.94, −0.40, 2.11)	(−2.95, −0.77, 1.55)	$\times {10}^{-2}$
r1 (09:31–10:00 a.m.)	(−6.91, −0.30, 5.92)	(−4.47, 0.19, 4.93)	(−5.86, −0.30, 5.00)	(−3.78, 0.19, 4.28)	$\times {10}^{-2}$
r2 (10:00–10:30 a.m.)	(−1.08, 5.31, 11.80)	(−0.41, 5.47, 11.40)	(−0.14, 5.31, 10.80)	(0.57, 5.47, 10.40) *	$\times {10}^{-2}$
r3 (10:30–11:00 a.m.)	(−10.40, −3.22, 4.03)	(−6.22, −0.19, 5.96)	(−9.18, −3.22, 2.82)	(−5.24, −0.19, 5.03)	$\times {10}^{-2}$
r4 (11:00–11:30 a.m.)	(−4.94, 2.62, 10.40)	(−3.67, 3.36, 10.70)	(−3.85, 2.62, 9.11)	(−2.36, 3.36, 9.38)	$\times {10}^{-2}$
r5 (11:30–12:00 p.m.)	(−12.80, −2.79, 7.37)	(−12.30, −3.33, 5.66)	(−11.30, −2.79, 5.63)	(−10.80, −3.33, 4.21)	$\times {10}^{-2}$
r6 (12:00–12:30 p.m.)	(−16.70, −6.98, 2.88)	(−16.80, −8.51, 0.13)	(−15.20, −6.98, 1.15)	(−15.60, −8.51, −1.28) *	$\times {10}^{-2}$
r7 (12:30–01:00 p.m.)	(−8.49, 1.23, 10.70)	(−7.35, 1.61, 10.50)	(−6.99, 1.23, 9.12)	(−5.85, 1.61, 8.88)	$\times {10}^{-2}$
r8 (01:00–01:30 p.m.)	(−6.23, 2.05, 10.50)	(−5.97, 1.52, 9.35)	(−4.79, 2.05, 9.14)	(−4.67, 1.52, 8.06)	$\times {10}^{-2}$
r9 (01:30–02:00 p.m.)	(−15.10, −5.68, 3.67)	(−13.90, −5.18, 3.85)	(−13.50, −5.68, 2.14)	(−12.40, −5.18, 2.18)	$\times {10}^{-2}$
r10 (02:00–02:30 p.m.)	(−13.70, −3.98, 5.70)	(−13.60, −4.35, 5.17)	(−12.20, −3.98, 4.02)	(−12.00, −4.35, 3.58)	$\times {10}^{-2}$
r11 (02:30–03:00 p.m.)	(−9.46, −0.21, 8.95)	(−7.65, 1.02, 9.50)	(−7.91, −0.21, 7.46)	(−6.29, 1.02, 8.11)	$\times {10}^{-2}$
r12 (03:00–03:30 p.m.)	(1.05, 10.80, 19.60) *	(−1.61, 8.49, 18.00)	(2.80, 10.80, 18.30) *	(0.24, 8.49, 16.50) *	$\times {10}^{-2}$
vixlagclose	(−6.89, −3.45, 0.00)	(−8.97, −4.67, −0.47) *	(−6.28, −3.45, −0.58) *	(−8.26, −4.67, −1.12) *	$\times {10}^{-5}$
vixpctlag	(−5.20, −2.00, 1.12)	(−7.30, −3.59, 0.08)	(−4.70, −2.00, 0.62)	(−6.72, −3.59, −0.53) *	$\times {10}^{-3}$
usdlagclose	(−6.53, 0.23, 7.19)	(−7.05, 1.44, 10.10)	(−5.38, 0.23, 6.01)	(−5.66, 1.44, 8.64)	$\times {10}^{-5}$
usdpctlag	(−5.00, −0.86, 3.17)	(−5.54, −0.63, 4.33)	(−4.29, −0.86, 2.53)	(−4.79, −0.63, 3.46)	$\times {10}^{-2}$
I_fomc1	(−3.79, −1.48, 1.10)	(−5.03, −1.73, 1.46)	(−3.43, −1.48, 0.69)	(−4.48, −1.73, 1.03)	$\times {10}^{-3}$
I_fomc1lag1	(−2.11, −0.60, 0.92)	(−2.31, −0.39, 1.45)	(−1.85, −0.60, 0.67)	(−2.01, −0.39, 1.16)	$\times {10}^{-4}$
I_fomc2lag1	(−1.58, −0.31, 0.94)	(−1.63, −0.09, 1.47)	(−1.36, −0.31, 0.76)	(−1.37, −0.09, 1.22)	$\times {10}^{-5}$
I_fomc3lag1	(−1.60, −0.27, 1.03)	(−2.03, −0.41, 1.11)	(−1.40, −0.27, 0.83)	(−1.75, −0.41, 0.88)	$\times {10}^{-6}$
weekday2	(−7.05, −1.84, 3.34)	(−8.35, −1.78, 4.77)	(−6.20, −1.84, 2.44)	(−7.34, −1.78, 3.74)	$\times {10}^{-4}$
weekday3	(−11.30, −5.76, −0.30) *	(−14.40, −7.50, −0.85) *	(−10.50, −5.76, −1.12) *	(−13.20, −7.50, −1.87) *	$\times {10}^{-4}$
weekday4	(−9.85, −4.61, 0.78)	(−12.90, −6.23, 0.65)	(−9.02, −4.61, −0.12) *	(−11.90, −6.23, −0.41) *	$\times {10}^{-4}$
weekday5	(−10.90, −5.67, −0.42) *	(−13.30, −6.61, 0.09)	(−10.10, −5.67, −1.30) *	(−12.20, −6.61, −1.09) *	$\times {10}^{-4}$
mon2	(−9.43, −1.43, 6.34)	(−9.37, 0.73, 10.60)	(−8.11, −1.43, 5.21)	(−7.64, 0.73, 8.98)	$\times {10}^{-4}$
mon3	(−3.81, 4.34, 12.90)	(−5.54, 4.55, 14.60)	(−2.69, 4.34, 11.40)	(−3.84, 4.55, 13.00)	$\times {10}^{-4}$
mon4	(−12.30, −4.00, 4.32)	(−13.10, −3.26, 6.40)	(−10.90, −4.00, 2.94)	(−11.40, −3.26, 4.82)	$\times {10}^{-4}$
mon5	(−9.97, −2.00, 6.10)	(−12.00, −2.45, 7.31)	(−8.78, −2.00, 4.66)	(−10.40, −2.45, 5.71)	$\times {10}^{-4}$
mon6	(−12.60, −4.48, 3.31)	(−13.60, −3.81, 5.80)	(−11.10, −4.48, 2.14)	(−12.10, −3.81, 4.34)	$\times {10}^{-4}$
mon7	(−11.20, −2.96, 5.12)	(−12.70, −2.96, 7.22)	(−9.88, −2.96, 3.90)	(−11.30, −2.96, 5.56)	$\times {10}^{-4}$
mon8	(−9.76, −1.93, 5.79)	(−11.00, −1.59, 7.78)	(−8.56, −1.93, 4.62)	(−9.70, −1.59, 6.24)	$\times {10}^{-4}$
mon9	(−8.01, 0.28, 8.25)	(−9.76, 0.75, 10.50)	(−6.57, 0.28, 7.06)	(−7.92, 0.75, 9.03)	$\times {10}^{-4}$
mon10	(−15.30, −4.90, 5.49)	(−16.30, −4.23, 8.34)	(−13.80, −4.90, 3.63)	(−14.40, −4.23, 6.22)	$\times {10}^{-4}$
mon11	(−11.10, −1.83, 7.58)	(−11.30, −0.14, 11.20)	(−9.60, −1.83, 5.94)	(−9.32, −0.14, 9.39)	$\times {10}^{-4}$
mon12	(−9.36, −0.63, 8.31)	(−10.00, 0.38, 11.20)	(−7.92, −0.63, 6.97)	(−8.54, 0.38, 9.35)	$\times {10}^{-4}$
$\sigma$	(1.57, 1.76, 1.97) *	(1.99, 2.21, 2.45 ) *	(1.61, 1.76, 1.94) *	(2.03, 2.21, 2.41) *	$\times {10}^{-3}$
$\nu$	(2.38, 3.11, 4.21) *	(2.70, 3.57, 4.92) *	(2.49, 3.11, 4.02) *	(2.82, 3.57, 4.66) *	$\times 1$

, we have presented the results based on the recent data from 1 January 2022 to 30 September 2024. Neither the overnight effect ($r_{on}$) nor the two days after the federal rate change days (I_fomc2lag) is significant anymore. However, there are some other factors, such as $r_{12}$ becoming significant during recent years, which indicate that a price trend established in the first 30 min of the final hour of regular trading tends to persist until the market closes during recent years.

Our research objective is to examine the predictability of the last 30 min of the US financial market, using various factors, including some well-studied in the literature as well as some new factors. We advocate Bayesian linear regression models with Student-t error as standard models to identify significant factors, not the commonly used multiple linear regression models with Gaussian error terms. We must admit that identifying long-lasting significant factors that have predictive powers in the financial market is very challenging; instead, monitoring changes in market regimes and the trend of the effects of factors may be more relevant, and results based on rolling windows, such as those in Figure 4, can be helpful in identifying potential effects trends. For market participants relying on factors to predict price movement, it is very critical that many significant factors studied for a specific time period and for a specific financial asset are likely short-lived. For the major issues that arise in the analysis of financial data and their potential solutions, we refer to Zhang and Hua (2024) for a recent survey.