Bitcoin Market Efficiency Analysis Pre- and Post-COVID-19 Pandemic: An Interrupted Time Series and ARIMAX Approach

Abstract

The COVID-19 pandemic constitutes one of the most significant exogenous shocks to global financial markets in recent history, raising questions about the robustness of market efficiency under extreme uncertainty. This study examines whether the pandemic affected the weak-form efficiency of the Bitcoin market or merely heightened volatility without introducing return predictability. Using daily Bitcoin log returns from January 2013 to February 2026, the analysis first evaluates weak-form market efficiency through the Variance Ratio (VR) test. The VR statistics remain close to unity across multiple holding horizons, and the null hypothesis of a random walk cannot be rejected, indicating that daily Bitcoin returns are consistent with weak-form efficiency. Building on this baseline, an Interrupted Time Series (ITS) framework is employed to assess whether the onset of the COVID-19 pandemic in March 2020 led to structural changes in Bitcoin return dynamics. The ITS results reveal no statistically significant changes in level or slope following the outbreak. To further account for autoregressive and moving-average dynamics while explicitly modelling the intervention, an ARIMAX (0, 0, 7) model with COVID-19 intervention variables is estimated. Both the pandemic dummy and its interaction term are statistically insignificant, indicating no material change in the return-generating process after controlling for serial dependence. The moving-average structure indicates that shocks dissipate over approximately one trading week, consistent with weekly trading cycles and liquidity patterns in cryptocurrency markets rather than persistent return predictability. Diagnostic checks, including the Ljung–Box and Shapiro–Wilk tests, confirm the absence of residual autocorrelation and support the model’s white-noise properties. Although volatility increased during the pandemic period, daily Bitcoin returns continued to align with weak-form market efficiency. The evidence, therefore, suggests that COVID-19 served as a stressor without generating persistent inefficiencies. These findings reinforce the distinction between volatility and predictability, demonstrating that heightened uncertainty does not necessarily undermine informational efficiency.

1. Introduction

The Bitcoin market has emerged as a significant component of the global financial system, capturing the interest of retail and institutional investors alike. Since its launch in 2008 by the pseudonymous Satoshi Nakamoto, Bitcoin has evolved from a niche digital currency into the dominant cryptocurrency by market capitalisation, holding over 50% of the market share as of early 2021 (Statista, 2024). Its growing adoption, decentralised nature, and independence from central banks have positioned it as both a speculative investment and an alternative store of value.

Bitcoin operates on a decentralised blockchain network and is not backed by physical assets or government guarantees. Its security is ensured by cryptographic protocols, in which users rely on public and private keys to validate transactions (De Filippi & Loveluck, 2016). Unlike traditional currencies, Bitcoin is not subject to government control, regulation, or oversight, contributing to its pronounced price volatility. Its peer-to-peer structure enables low-cost global transactions and is independent of traditional banking systems. However, this operational freedom also introduces risks, as seen in early exchange failures such as Mt. Gox (Brito et al., 2014) and the lack of consumer protection on unregulated trading platforms (Bryans, 2014). Bitcoin exhibits several well-documented stylised facts, including high volatility, clustering volatility, and heavy-tailed return distributions (Baur et al., 2018; Corbet et al., 2018). These characteristics set cryptocurrency markets apart from traditional financial assets and underscore their distinct risk-return profile (Phillip et al., 2018).

Bitcoin’s high volatility, global adoption, and technological foundation set it apart from traditional financial assets. These attributes make it particularly sensitive to macroeconomic shocks, such as the COVID-19 pandemic. Beginning in early 2020, the pandemic disrupted economies worldwide, prompting government interventions through economic lockdowns and other measures, fiscal stimulus packages, and shifts in investor sentiment. These developments impacted the informational efficiency of cryptocurrency markets. Thus, understanding how Bitcoin returns were affected during and after this period is crucial.

This study first employs the Variance Ratio (VR) test to formally assess weak-form market efficiency by examining whether Bitcoin returns follow a random walk. The VR test directly evaluates return independence and serves as the primary statistical tool for determining whether past price information can predict future returns. To complement this efficiency assessment, the study further employs Interrupted Time Series (ITS) analysis, a technique commonly used in epidemiology and public policy but less frequently applied in financial research, to investigate potential structural shifts in daily Bitcoin returns around the COVID-19 pandemic. ITS is particularly useful for evaluating the effects of global, time-specific interventions by detecting both immediate level changes and longer-term trend adjustments (Chaudhuri & Carillo, 2023). In this context, the COVID-19 pandemic is treated as an exogenous intervention that may have influenced return behaviour in the Bitcoin market, thereby dividing the time series into pre- and post-intervention periods for structural comparison.

In modelling the underlying time-series dynamics and evaluating changes in return patterns, the ITS framework is applied within the Box and Jenkins (1970) methodology. This approach enables modelling of autocorrelation structures and improves inference robustness by accounting for serial dependencies. The ITS, in the context of the Box-Jenkins methods, offers a rigorous basis for identifying shifts in return behaviour and assessing market responses to the pandemic.

This research aims to determine whether the Efficient Market Hypothesis (EMH) applies to daily Bitcoin returns and to examine if the market’s informational efficiency was affected during the pre- and post-COVID-19 periods. The study tests weak-form efficiency using a time-series framework to identify structural and behavioural changes induced by a major global shock. Testing the semi-strong and strong-form efficiency presents challenges because isolating the market’s response to public announcements and events (semi-strong form) is difficult, and accessing private or insider information necessary to assess strong-form efficiency is often unavailable.

Unlike traditional econometric approaches such as GARCH or purely fundamental time-series analysis, the Interrupted Time Series (ITS) framework allows simultaneous estimation of level and trend changes following a clearly defined exogenous shock, while constructing a counterfactual trajectory based on pre-intervention dynamics (Bernal et al., 2017). Its application in economic and financial crisis contexts has demonstrated strong potential for identifying dynamic policy effects and market responses, offering a more transparent and interpretable framework than volatility-focused models such as GARCH (Nakatani & Teräsvirta, 2009). However, modelling conditional variance dynamics (e.g., GARCH) is not the primary objective of this study; instead, the analysis focuses on return predictability and structural breaks in the conditional mean process rather than volatility forecasting.

The integration of VR and ITS with the Box–Jenkins methodology further strengthens the analytical framework. ITS facilitates the identification of structural changes associated with external interventions, while Box–Jenkins techniques, implemented through ARIMA models, provide a rigorous treatment of autocorrelation and time-series dependence. Together, these methods offer a coherent and statistically robust approach to examining market efficiency and return behaviour during periods of disruption, including the COVID-19 pandemic.

Despite a growing literature on Bitcoin market efficiency, there remains limited consensus on whether major global shocks fundamentally alter weak-form efficiency or instead induce temporary deviations driven by heightened volatility. While prior studies frequently document increased volatility and abnormal returns during crisis periods, evidence regarding the persistence and structural nature of efficiency changes remains inconclusive.

This study examines the informational efficiency of daily Bitcoin returns and the impact of the COVID-19 pandemic on market behaviour. The analysis focuses on two main objectives: assessing weak-form market efficiency over the full sample period and evaluating the transitory versus persistent effects of major shocks on return dynamics.

Research Objectives

• To assess whether daily Bitcoin returns exhibit weak-form market efficiency, with particular emphasis on the random walk hypothesis over the full sample period (January 2013–February 2026).
• To determine whether the COVID-19 pandemic caused persistent structural changes in Bitcoin return dynamics, or whether any observed effects were transitory disturbances, using Interrupted Time Series (ITS) and ARIMAX intervention models.

Research Questions

• Do daily Bitcoin returns follow a random walk, consistent with weak-form market efficiency, over the full sample period?
• Did the COVID-19 pandemic induce structural shifts in Bitcoin return dynamics, or did the market absorb the shock without generating persistent inefficiencies?

Research Hypotheses

H1: 

Daily Bitcoin returns follow a random walk and therefore exhibit weak-form market efficiency.

H2: 

The COVID-19 pandemic did not generate persistent structural inefficiencies in daily Bitcoin returns.

These objectives and hypotheses guide subsequent analyses using the Variance Ratio (VR) test and ITS, combined with ARIMA-based modelling, allowing for a robust evaluation of market efficiency and the structural effects of the COVID-19 pandemic on Bitcoin returns.

The central premise of the paper is that large exogenous shocks may amplify volatility without necessarily violating market efficiency. By employing a combined ITS and ARIMA-based random walk framework, the study aims to distinguish transitory market disturbances from structural inefficiencies, while explicitly linking pandemic-related shocks to the theoretical implications of the Efficient Market Hypothesis (EMH).

This study contributes to the literature in three ways. First, it extends the application of ITS to cryptocurrency markets, providing a structured framework for evaluating the effects of global shocks on market efficiency rather than focusing solely on volatility. Second, it integrates ITS with ARIMA-based random-walk tests, thereby strengthening the empirical connection between structural interventions and the theoretical foundations of the EMH. Third, it offers a systematic examination of Bitcoin market behaviour during an unprecedented global disruption, thereby contributing to ongoing debates on the maturity and informational efficiency of cryptocurrency markets.

2. Literature Review

2.1. Variance Ratio (VR) Test

Yi et al. (2023) examine weak-form efficiency in the Bitcoin market using a quantum harmonic oscillator framework alongside the Variance Ratio (VR) test. While the VR results show mixed evidence regarding return predictability, the quantum-based analysis indicates that Bitcoin operates close to weak-form efficiency. Their findings suggest that as the market matures, abnormal returns based on historical prices become harder to exploit, and that regulators should monitor market efficiency as an indicator of potential price distortions. This study provides a foundation for assessing Bitcoin’s efficiency under periods of heightened uncertainty, such as the COVID-19 pandemic.

Kim and Shamsuddin (2008) apply multiple VR tests, including wild bootstrap and sign-based procedures, to evaluate weak-form efficiency in Asian stock markets. They find that efficiency varies across countries, with Hong Kong, Japan, Korea, and Taiwan exhibiting weak-form efficiency, while Indonesia, Malaysia, and the Philippines remain inefficient. Their results highlight the role of market development and governance in shaping efficiency. These insights motivate the current study’s approach, which combines VR testing with Interrupted Time Series (ITS) and ARIMAX models to examine whether the COVID-19 pandemic affected Bitcoin’s weak-form efficiency at a daily frequency.

2.2. Interrupted Time Series (ITS) Analysis

Interrupted time series (ITS) has been applied in the health sector to evaluate the effectiveness of interventions that occur at a clearly defined point in time (Bernal et al., 2017). Studies on ITS carried out to date have primarily focused on health issues, and it has also been applied in other fields, such as the social sciences (Chaudhuri & Carillo, 2023). However, its application in financial markets, particularly for assessing the impact of crises such as COVID-19 on digital assets, remains limited, if at all.

Studies have also examined financial aspects, including stock price movements. In this study, the COVID-19 pandemic is treated as an exogenous intervention in the Bitcoin market, as it triggered a global economic lockdown that disrupted in-person day-to-day trading and prompted an increase in online activity (Sarkodie et al., 2022). Bitcoin is a digital currency, and its movement from one person to another was not negatively affected; it gained popularity. This prompts an investigation into the effect of the COVID-19 pandemic on the Bitcoin market (Bloomberg, 2021), an effect that has not been explicitly explored in previous studies. The current study applies the ITS and Box and Jenkins (1970) methodology to assess Bitcoin price dynamics pre- and post-COVID.

Karavias et al. (2021) examined stock market returns across 61 countries from January to September 2020 using a structural break detection method tailored for panel data with interactive effects. Their analysis identified a structural break in early April 2020, with stock returns showing a significant negative response to COVID-19 before the break, and no measurable impact afterwards. This indicates a short-lived market reaction, possibly due to the financial quantitative easing programs introduced by central banks globally in late March 2020. Their approach focuses on detecting unknown structural breakpoints in panel data without assuming when changes occur. In contrast, this study employs ITS analysis, which is more appropriate when the timing of an intervention is known in advance, allowing for the modelling of both immediate level shifts and gradual slope changes to assess causal impacts over time directly. The two approaches (the structural break detection method and ITS) share the goal of identifying shifts in time series in response to significant events, which is crucial for understanding the dynamics of financial markets during crises.

Urquhart (2016) examined the efficiency of the Bitcoin market using the automatic VAR, Ljung–Box, Bartel’s, AVR, BDS, and Hurst exponent tests. The study concluded that while Bitcoin returns were significantly inefficient in the full sample, some evidence on efficiency emerged in the later sub-period. This study suggested that the Bitcoin market is evolving towards efficiency. The study did not explore the role of external shocks or interventions, such as the COVID-19 pandemic, in shaping these trends. The current study seeks to address this gap by assessing the impact of COVID-19 as an exogenous intervention that potentially shifted Bitcoin price behaviour.

Nadarajah and Chu (2017) re-examined Urquhart’s (2016) findings by applying power transformations to Bitcoin returns and conducting eight tests. Their results demonstrated that Bitcoin, by and large, satisfied the weak form of the EMH, except for tests of independence. This study highlights that Bitcoin exhibits weak market efficiency when appropriately modelled, providing a clearer understanding of Bitcoin’s market efficiency. The current study builds on Nadarajah and Chu’s (2017) findings by applying more advanced methodologies that combine the ITS and Box-Jenkins approaches, thereby offering insights into the nature of Bitcoin’s weak-form efficiency.

Vidal-Tomás and Ibañez (2018) investigated Bitcoin’s semi-strong market efficiency on the Bitstamp exchange and Mt. Gox exchange using event studies and GARCH-type models. Their findings indicated that Bitcoin became more efficient over time, particularly in response to market-specific events. The current study employs an integrated approach combining ITS analysis and Box-Jenkins modelling to detect structural and behavioural shifts in Bitcoin returns. This methodology enables an assessment of whether the COVID-19 pandemic disrupted the market’s ability to incorporate past information into prices, offering a complementary perspective on market efficiency during crises.

Caporale and Plastun (2019) investigated price overreactions and weekly patterns in Bitcoin and other major cryptocurrencies. Although statistical tests confirmed the existence of price anomalies, the study concluded that these patterns do not lead to exploitable profit opportunities due to the associated high transaction costs. These findings suggest that such anomalies do not constitute evidence against market efficiency. However, the study does not account for external market shocks, such as the COVID-19 pandemic, which may have significantly influenced Bitcoin’s return dynamics and market efficiency. To address this gap, this research applies an ITS framework within the Box-Jenkins methodology, enabling the detection of structural shifts and the accurate modelling of time-series dependencies before and after the pandemic intervention.

Radovanov et al. (2018) conducted a traditional time-series analysis of the daily returns of four major cryptocurrencies (Bitcoin, Ethereum, Ripple, and Litecoin), revealing persistent volatility and slight asymmetry in their distributions. In contrast, the current study adopts an ITS approach, focusing exclusively on Bitcoin and using daily returns to assess structural changes induced by an exogenous global shock, the COVID-19 pandemic. This methodology provides a novel lens for evaluating how external interventions affect market efficiency, offering insights that conventional volatility-focused models miss.

2.3. Efficiency Market Hypothesis (EMH)

The EMH, introduced by Fama (1995), is one of the fundamental principles in financial economics, stating that market prices reflect all available information. When markets are efficient, no investor can consistently outperform the market by exploiting available information, making active management less effective compared to passive strategies. In the context of Bitcoin, there is an ongoing debate about whether the cryptocurrency market adheres to the EMH, with some studies suggesting inefficiency (Brauneis & Mestel, 2018; Bundi & Wildi, 2019; Cheah et al., 2018), while others argue that Bitcoin follows the EMH (Kumar et al., 2020; Tiwari et al., 2018; Bariviera, 2017). Given this mixed evidence, further empirical investigation into Bitcoin’s compliance with at least the weak-form EMH remains essential, particularly given recent market disruptions and ever-evolving investor behaviour.

In analysing Bitcoin price movements, this study focuses on key characteristics such as predictability, randomness, volatility, seasonality, and anomalies, factors central to assessing weak-form market efficiency. Weak-form EMH asserts that current prices fully reflect all past price information, implying that returns should follow a random walk. To test this, the study employs methods such as autocorrelation tests, random walk analysis, and Box-Jenkins (ARIMA) modelling to identify underlying patterns and dependencies in the price and return series. While the EMH includes three forms: weak, semi-strong, and strong, each reflecting different levels of informational efficiency, this research concentrates on the weak form. In doing so, it addresses a gap in the literature on how exogenous shocks, such as the COVID-19 pandemic, may temporarily disrupt or alter weak-form efficiency in emerging digital asset markets, such as Bitcoin. It offers insights into the predictability of Bitcoin returns over time, particularly in the context of the COVID-19 pandemic as an exogenous intervention.

2.4. Bitcoin Market Efficiency in the Pre-COVID-19 Period

Prior to the COVID-19 pandemic, empirical evidence on Bitcoin market efficiency was mixed but was gradually converging toward weak-form efficiency. Early studies, such as Urquhart (2016), documented significant inefficiencies in Bitcoin returns, particularly during the market’s formative years. However, subsequent research found that these inefficiencies diminished over time as liquidity improved and market participation broadened.

Nadarajah and Chu (2017) demonstrated that once appropriate statistical transformations were applied, Bitcoin returns largely satisfied weak-form efficiency conditions. Similarly, Bariviera (2017) and Vidal-Tomás and Ibañez (2018) reported increasing efficiency consistent with market maturation and improved information diffusion. These findings suggest that, by the late 2010s, Bitcoin increasingly resembled traditional financial assets in its return-generating process, particularly at lower frequencies.

However, most pre-COVID studies implicitly assume stable market conditions and do not consider how a global systemic shock might disrupt this evolution. This gap motivates the present study’s focus on COVID-19 as a natural experiment to test Bitcoin’s weak-form efficiency.

2.5. Weak Form of the EMH

The weak form of the EMH deals with historical data. Charts, graphs, and anything under technical analysis are used to analyse this historical pricing data. Technical analysis seeks to identify trends and predict prices. The weak form states that the price of an asset or security incorporates all the historical pricing data (Fama, 1991). Technical analysis cannot consistently enable an investor to make excess risk-adjusted returns if the market is a weak-form market-efficient. If the market is efficient, simple buy-and-hold strategies (such as index tracking) can be adopted for stock investments. Assessing the weak form of EMH requires using historical price data to make predictions within a market, which is done through chart or technical analysis. Gupta et al. (2008) conducted hypothetical tests comparing mutual fund returns with those of a stock index portfolio, and the results indicated that technical analysis is no better than simple buy-and-hold strategies for stock investments. In other words, if the market is weak-form efficient, historical price data analysis is useless to a trader, as it does not provide any advantage over a random trader without such information. Any investor with fundamental or insider trading information can outperform a weak-form efficient market (Butler & Malaikah, 1992).

2.6. Semi-Strong Form of EMH

The semi-strong form of the EMH posits that all publicly available information (including historical data) is immediately reflected in asset prices, preventing investors from earning abnormal returns based on such information (Rohit et al., 2016; Theckanathukaduppil, 2021). Studies such as Goyal and Gupta (2019) have analysed semi-strong efficiency by investigating market reactions to dividend announcements on the BSE Sensex. Their findings demonstrated that the market quickly absorbed the information, suggesting that investors could not gain abnormal returns. Applying these insights to Bitcoin during the COVID-19 pandemic, significant public events such as institutional investments, economic stimulus announcements, or significant cryptocurrency policy changes can be analysed to assess Bitcoin’s market efficiency. If Bitcoin prices adjusted rapidly and unpredictably following such announcements during the pandemic, it would indicate semi-strong efficiency. Such announcements are difficult to collect globally.

2.7. Strong Form of EMH

The strong form of the EMH suggests that all information, including historical, public, and insider knowledge, is fully integrated into market prices, leaving no opportunity for consistent abnormal returns (Manzoor, 2015). Shanthaamani and Usha (2019) tested the strong form of the market efficiency by analysing the impact of dividend announcements on stock returns and market efficiency for selected companies in the BSE Sensex. It was concluded that the market was efficient in the strong form, with no exploitable advantages for investors with insider information. Drawing parallels to Bitcoin, insider-like events during the COVID-19 pandemic, such as large-scale transactions by institutional investors or significant blockchain developments, would need to be examined to determine if Bitcoin markets exhibit strong-form efficiency. It would demonstrate strong market efficiency if prices adjusted immediately and unpredictably to such events, even amid heightened uncertainty, such as during the pandemic. However, such insider information would be difficult to collect and quantify.

3. Methodology

This study adopts a multi-stage empirical strategy to assess both weak-form market efficiency and potential structural changes associated with the COVID-19 pandemic. The analysis begins with the Variance Ratio (VR) test proposed by Lo and MacKinlay (1988) to directly evaluate the random walk hypothesis, before proceeding to structural break analysis using Interrupted Time Series (ITS) and ARIMAX modelling.

An Interrupted Time Series (ITS) analysis is employed to investigate structural changes in daily Bitcoin returns surrounding the COVID-19 pandemic. ITS is well-suited for assessing the effects of clearly defined, time-bound interventions on financial time series, making it appropriate for capturing potential shifts in return behaviour associated with the pandemic.

The COVID-19 pandemic represents a clearly identifiable, exogenous shock that simultaneously affected global financial markets, thereby satisfying a key requirement of ITS analysis (Bernal et al., 2017). Unlike simple pre-post comparisons or volatility-based approaches, the ITS framework explicitly models both immediate level changes and longer-term trend adjustments, allowing for a more nuanced assessment of how markets respond and adapt over time.

In the context of market efficiency, ITS facilitates constructing a counterfactual return path based on pre-intervention dynamics. This enables an evaluation of whether post-pandemic return behaviour deviates meaningfully from what would have been expected in the absence of the COVID-19 shock. Importantly, ITS is used to detect structural breaks in return dynamics and does not, by itself, constitute a test of market efficiency.

To model the underlying time series dynamics within the ITS framework, the Box and Jenkins (1970) methodology, a well-established approach for identifying and estimating suitable ARIMA models, is employed. An ARIMAX specification incorporating COVID-19 intervention variables is then estimated to control for autoregressive and moving average components while testing the statistical significance of pandemic-related effects on Bitcoin returns.

3.1. Tests for Weak-Form Market Efficiency

To complement the structural break analysis, weak-form market efficiency is examined using established random walk tests. According to Fama (1970), a market is weak-form efficient if current asset prices fully reflect all information contained in past prices. Under this framework, returns should follow a martingale difference sequence, implying that price changes are unpredictable based on historical information. The weak-form efficiency condition can be expressed as

$$E(r_t\mid {\mathcal{F}}_{t-1}) = 0,$$

(1)

where $r_t$ denotes the return at time $t$, and ${\mathcal{F}}_{t-1}$ represents the information set available at the time $t-1$. The martingale difference condition does not require returns to follow a strict random walk with independent and identically distributed increments. Weak-form efficiency allows for time-varying volatility, conditional heteroskedasticity, and predictable risk premiums, provided that such predictability does not produce abnormal risk-adjusted profits. Thus, a random walk is sufficient but not necessary for weak-form efficiency. This study uses the random walk hypothesis as a practical benchmark for testing return independence, not as a full definition of market efficiency.

Under the strict random walk model, returns are independently and identically distributed with constant variance. However, within the broader martingale framework consistent with weak-form efficiency, returns may exhibit changing conditional variance (e.g., volatility clustering) without violating efficiency, as long as the conditional mean remains unpredictable given past price information.

To test the random walk hypothesis, the Variance Ratio (VR) test proposed by Lo and MacKinlay (1988) is used. The test is based on the property that, under a random walk, the variance of $k$-period returns should be equal to $k$ times the variance of one-period returns. The variance ratio is defined as

$$VR\left(k\right)= {\displaystyle \frac{\mathrm{Var}\left(r_t+r_{t-1}+\cdots +r_{t-k+1}\right)}{k·\mathrm{Var}\left(r_t\right)}}.$$

(2)

Under the null hypothesis of a random walk, $VR\left(k\right) = 1$. If $VR\left(k\right)\ne 1$, returns exhibit serial correlation. A rejection of the random walk hypothesis indicates linear serial dependence in returns, which may be inconsistent with weak-form efficiency if economically exploitable. However, failure to reject the random-walk null is interpreted as evidence consistent with weak-form efficiency in its return-independence dimension, while recognising that efficiency is ultimately a joint hypothesis involving both return dynamics and equilibrium asset-pricing considerations.

The null and alternative hypotheses are, therefore,

H₀: 

$VR\left(k\right) = 1$ (Random walk with constant expected returns),

H₁: 

$VR\left(k\right)\ne 1$ (Serial dependence; potential deviation from weak-form efficiency).

3.2. Segmented Regression Under Interrupted Time Series (ITS) Analysis

To formally evaluate the impact of the COVID-19 pandemic on Bitcoin return dynamics, we implement a segmented regression within an Interrupted Time Series (ITS) framework. ITS allows for the estimation of both immediate (level) and gradual (trend) structural changes following an exogenous intervention.

The model specification is

$$r_t = {\beta }_0+{\beta }_1{T}_t+{\beta }_2{D}_t+{\beta }_3\left(T_t{D}_t\right)+{\epsilon }_t,$$

(3)

where $r_t$ denotes the daily Bitcoin returns at time $t$; $T_t$ represents a continuous time trend from the beginning of the sample; $D_t$ is a dummy variable equal to 0 for the pre-COVID period and 1 for the post-COVID period; $T_t{D}_t$ captures the post-intervention slope change; ${\epsilon }_t$ is the error term.

To account for potential heteroskedasticity and serial correlation in daily financial returns, Newey–West heteroskedasticity-and autocorrelation-consistent (HAC) standard errors are used in statistical inference.

3.3. Structural Break Timing and Causality

Following standard interrupted time-series (ITS) methodology, the structural break is specified a priori on 31 March 2020, corresponding to the onset of the global COVID-19 pandemic (Bernal et al., 2017; Wagner et al., 2002).

The break date is fixed ex ante to avoid data-driven selection and preserve causal interpretation. Daily Bitcoin returns are divided into pre- and post-intervention periods relative to this fixed break. While subsequent events (e.g., the 2021 bull market and regulatory developments) also affected Bitcoin prices, the ITS framework isolates structural changes specifically linked to the onset of the pandemic.

3.4. ARIMAX Modelling Within the ITS Framework

To account for potential serial dependence in Bitcoin returns while simultaneously estimating the impact of the COVID-19 intervention, the segmented ITS regression is extended to an autoregressive integrated moving average (ARIMA) model with exogenous regressors (ARIMAX). This specification allows the deterministic intervention components to enter the conditional mean equation while modelling the stochastic error structure dynamically. Let $r_t$ denote daily Bitcoin log returns. The ARIMAX ($p,0,q$) model is specified as

$$r_t = \mu +{\displaystyle \sum _{i = 1}^p}{\varphi }_i{r}_{t-i}+{\displaystyle \sum _{j = 1}^q}{\theta }_j{\epsilon }_{t-j}+{\gamma }_1{D}_t+{\gamma }_2\left(T_t{D}_t\right)+{\epsilon }_t,$$

(4)

where $\mu$ is the unconditional mean return, ${\varphi }_i$ are autoregressive parameters, ${\theta }_j$ are moving-average parameters, $D_t$ is a post-COVID dummy variable, $T_t{D}_t$ captures the change in slope after the intervention, ${\gamma }_1$ measures the immediate level shift, ${\gamma }_2$ measures the post-intervention trend change, ${\epsilon }_t$ is a white noise innovation. Because the analysis is conducted on log returns rather than price levels, the series is stationary by construction; therefore, the differencing parameter $d = 0$, and the model reduces to an ARMA process with exogenous regressors. Model orders $p$ and $q$ are selected using inspection of ACF and PACF plots. The AIC is used for model selection. Residual diagnostics, including the Ljung–Box test, are applied to ensure that remaining serial correlation is adequately filtered. The ARIMAX model nests the baseline ITS regression as the restricted case with $p = q = 0$. The coefficients ${\gamma }_1$ and ${\gamma }_2$ therefore retain their interpretation as level and slope shifts but are estimated while controlling for autoregressive dynamics.

4. Results

4.1. Data

Daily Bitcoin data from https://za.investing.com/crypto/Bitcoin/historical-data (accessed on 1 March 2026), covering 1 January 2013 to 27 February 2026, are used. The analysis is conducted exclusively at the daily frequency to preserve short-run return dynamics and volatility clustering that are central to financial time-series modelling and tests of weak-form efficiency. Bitcoin is employed as a representative large-cap cryptocurrency; however, the findings are explicitly limited to Bitcoin and are not generalised to the entire cryptocurrency market.

The use of daily data is particularly appropriate in cryptocurrency markets, where return dependence and volatility clustering are most pronounced at higher frequencies. Prior studies document that Bitcoin returns exhibit substantial volatility, structural breaks, and speculative dynamics (Baur et al., 2018). Volatility clustering and conditional heteroskedasticity are well-established features of Bitcoin returns (Katsiampa, 2017), and weak-form efficiency has primarily been examined using daily data (Urquhart, 2016; Bariviera, 2017). Aggregating to lower frequencies may attenuate economically meaningful autocorrelation patterns and obscure short-term market dynamics.

Bitcoin prices exhibit non-stationarity, as is typical for financial asset prices. Following standard practice in financial econometrics, prices are transformed into continuously compounded (log) returns to remove stochastic trends and stabilise variance (Campbell et al., 1997; Tsay, 2010). Log returns are more statistically tractable than raw prices and are consistent with the empirical framework used in tests of weak-form market efficiency (Fama, 1970).

The formula for calculating log returns is

$$r_t = \log \left({\displaystyle \frac{P_t}{P_{t-1}}}\right) = \log \left(P_t\right)-\log \left(P_{t-1}\right),$$

(5)

where $P_t$ is the Bitcoin price at the time $t$ and $P_{t-1}$ represents the Bitcoin price at the time $t-1$.

4.2. Descriptive Statistics

The summary statistics for the daily Bitcoin prices ${(P}_t)$ and the daily Bitcoin returns ${(z}_t)$ are shown in

Table 1. Descriptive statistics of $P_t$ and $r_t$ (1 January 2013 to 27 February 2026).

Variable	Minimum	Maximum	Mean	Std. Deviation	Skewness	Kurtosis
$P_t$	13.30	12,4725.10	24,046.98	31,290.64	1.44	1.08
$r_t$	−0.85	1.47	0.00	0.05	4.63	199.66

Source: Authors’ calculations based on daily Bitcoin data obtained from Investing.com.

.

Bitcoin price data ($P_t$) exhibits substantial growth and dispersion over the sample period, ranging from $13.30 to $124,725.10. The average price of $24,046.98, alongside a high standard deviation of $31,290.64, indicates pronounced variability around the mean. Positive skewness (1.44) reflects occasional extreme price surges, while the relatively low kurtosis (1.08) suggests the price distribution is moderately flatter than a normal distribution.

Bitcoin returns ($r_t$) capture the high-frequency fluctuations typical of cryptocurrency markets, ranging from −0.85 to 1.47. The mean return of 0.00 indicates a roughly neutral average return over the entire period. Returns exhibit pronounced volatility (standard deviation = 0.05), strong positive skewness (4.63), and extremely high kurtosis (199.66), consistent with leptokurtic behaviour. This distribution implies a heavy tail and a heightened likelihood of extreme positive or negative returns, reflecting the well-known episodic surges and crashes in the Bitcoin market.

4.3. Bitcoin Returns Time Series Plot

Figure 1 illustrates the historical daily Bitcoin price and return movements from 1 January 2013 to 27 February 2026, before transforming Bitcoin prices into log returns.

Bitcoin prices show a long-term upward trajectory with pronounced volatility throughout the sample period. Major price surges occurred during 2017–2018, 2020–2021, and 2023–2025, reflecting the asset’s cyclical boom–bust behaviour. The upward trend post-March 2020 aligns with the global onset of the COVID-19 pandemic. It may be linked to heightened investor demand for alternative assets, concerns over fiat currency devaluation, expansionary monetary policies, and increased institutional adoption of Bitcoin as a store of value.

Daily returns exhibit substantial variability, with several notable spikes visible in the series. February 2014 shows both a significant upward and downward spike, reflecting market reactions to the collapse of Mt. Gox, which suspended withdrawals and later declared bankruptcy, generating severe market uncertainty (Baur et al., 2018). Another pronounced downward spike occurs in March 2020, coinciding with the COVID-19 market shock and widespread financial turmoil. These spikes illustrate episodic structural shocks superimposed on an otherwise highly volatile, largely unpredictable return series.

4.4. Weak-Form Market Efficiency Results

The Variance Ratio (VR) Test, as proposed by Lo and MacKinlay (1988), was employed to test the random walk hypothesis for Bitcoin returns. This test evaluates whether the variance of k-period returns equals k times the variance of one-period returns. If the returns follow a random walk, the variance ratio should be close to 1. Deviations from 1 suggest serial correlation, and thus inefficiency in the market. The M1 and M2 statistics were used for this analysis as they are robust to heteroskedasticity.

Table 2. Variance Ratio Test Results for Bitcoin Returns.

K	VR(k)	M1 Stat	M2 Stat
k = 2	1.0024	0.1535	0.0500
k = 5	0.8686	−4.1791	−0.8680
k = 10	0.9318	−1.4362	−0.2776
k = 20	0.9556	−0.6713	−0.1456
k = 50	1.0140	0.0316	0.0092
k = 100	1.0640	0.2577	0.0967
k = 150	0.9982	−0.1537	−0.0652
k = 200	0.9602	−0.3296	−0.1524
k = 500	0.8423	−0.6140	−0.3697

Source: Authors’ calculations based on daily Bitcoin data obtained from Investing.com.

presents the findings.

The results in Table 2 suggest that daily Bitcoin returns generally follow a random walk across various holding periods. The VR(k) values are close to 1 for most periods, indicating weak-form market efficiency, in which returns are unpredictable and exhibit no significant serial correlation. Although some deviations from 1 are observed at certain lags (e.g., k = 5, k = 50, and k = 100), these are not substantial enough to reject the random walk hypothesis. The M1 and M2 statistics, which are robust to heteroskedasticity, show no significant values beyond the critical threshold of ±1.96, supporting the conclusion that daily Bitcoin returns do not exhibit predictable trends across the tested aggregation horizons. The results indicate that Bitcoin returns are consistent with weak-form efficiency, in which historical returns do not provide reliable information for predicting future movements.

4.5. Scatter Plots

To visually examine potential changes in daily Bitcoin return behaviour, scatter plots were generated for two distinct periods: 1 January 2013 to 31 March 2020 (pre-COVID-19 period) and 1 April 2020 to 27 February 2026 (post-COVID-19 period). The post-COVID period encompasses the entire COVID-19 era, including immediate market reactions, subsequent recovery, and later pandemic-related developments in the cryptocurrency market. This division, based on the structural break identified in March 2020, facilitates an ITS analysis to assess whether the pandemic had a measurable impact on Bitcoin returns. Figure 2 presents the results.

Figure 2 displays daily Bitcoin log returns from January 2013 to February 2026, with individual returns represented by black dots and linear trends for the pre- and post-COVID-19 periods indicated by blue lines. A vertical dashed line marks the structural break on 31 March 2020, separating the two periods. While the pre-pandemic phase shows higher volatility with pronounced positive and negative returns, the post-pandemic period appears more stable, though volatility remains inherent. The lack of an abrupt level shift at the breakpoint and the gradual change in trend slope suggest that Bitcoin returns did not experience a sudden disruption but rather a gradual adjustment following the onset of the pandemic. The figure highlights the enduring volatility of Bitcoin returns while suggesting a subtle stabilisation post-COVID-19.

4.6. Interrupted Time Series (ITS) Estimation

To formally assess the potential impact of the COVID-19 pandemic on daily Bitcoin returns, the segmented regression ITS model was estimated. The model allows for the evaluation of both an immediate level change and a change in slope following the pandemic breakpoint in March 2020. Newey–West (HAC) robust standard errors were used to account for potential autocorrelation and heteroskedasticity in the daily returns.

Table 3. ITS Regression Results for Bitcoin Daily Returns (HAC Robust Standard Errors).

Variable	Coefficient	HAC Std. Error	t-Value	p-Value
β0	0.00532	0.00308	1.73	0.084
β1	−0.0000023	0.0000018	−1.28	0.200
β2	0.00205	0.00517	0.40	0.691
β3	0.00000057	0.00204	0.28	0.781

Source: Authors’ calculations based on daily Bitcoin data obtained from Investing.com.

presents the ITS parameters.

The results indicate no statistically significant immediate change in daily returns following the onset of the pandemic (β₂, p = 0.691), nor in the slope of returns thereafter (β₃, p = 0.781). The pre-pandemic time trend (β₁) is also not significant (p = 0.200), reflecting Bitcoin’s volatile yet largely unpredictable return dynamics. The joint test for structural break significance (β₂ = β₃ = 0) fails to reject the null hypothesis (F = 1.04, p = 0.354), further supporting the absence of a detectable structural change in daily returns due to COVID-19.

Residual diagnostics reveal significant autocorrelation (Ljung–Box test, p < 0.001) and strong deviations from normality (Shapiro–Wilk test, p < 0.001), indicating model residuals are not white noise and display non-normality. Moreover, volatility clustering is confirmed by significant ARCH effects (as documented in earlier volatility analyses), justifying the use of ARIMAX modelling to appropriately account for autocorrelation and conditional heteroskedasticity when estimating intervention effects.

The ITS analysis suggests that daily Bitcoin returns did not experience a statistically significant structural shift attributable to the COVID-19 pandemic, consistent with the asset’s inherently volatile and complex market behaviour. The ITS equation is expressed as

$$r_t = 0.00532-0.0000023T_t+0.00205D_t+0.00000057\left(T_t{D}_t\right)+{\epsilon }_t.$$

(6)

4.7. ARIMAX Model with Intervention Variables

Following the ITS analysis, which failed to detect a statistically significant structural break in daily Bitcoin returns due to the COVID-19 pandemic, the analysis is extended by fitting an ARIMAX model. The ARIMAX model explicitly accounts for time-series autocorrelation and volatility clustering, which were identified as significant in the ITS model’s residual diagnostics.

The Augmented Dickey–Fuller (ADF) test results support the assumption that Bitcoin daily returns are stationary and do not exhibit a unit root (p < 0.001). The ACF and PACF plots are shown in Figure 3 to determine the proposed model for daily Bitcoin returns.

The ACF and PACF plots suggest an ARIMA (0, 0, 0) model as the tentative model for the daily returns. The EACF plot is presented in

Table 4. The EACF results.

AR/MA
0 1 2 3 4 5 6 7 8 9 10 11 12 13 0 o x o x x x x o o o o   o   o   o 1 o x o x x x x o o o o   o   o   o 2 o x x x x x o o o o o   o   o   o 3 o x x o x x o x o o o   o   o   o 4 x x x o x x x x o o o   o   o   o 5 x x x x x x o x o o o   o   o   o 6 x x x x x x x x o x o   o   o   o 7 x x x x x x x o o o o   o   o   o

Source: Authors’ calculations based on daily Bitcoin data obtained from Investing.com.

to confirm the suggested models.

The EACF analysis indicated that an ARIMA (0, 0, 7) model is appropriate for modelling Bitcoin daily returns. To ensure robustness, this model is compared with alternative ARIMA models to assess the risk of overfitting. Model selection criteria, specifically the AIC, were employed to identify the most suitable model. The AIC values for various fitted models are presented in

Table 5. AIC of fitted models.

ARIMA Models with Intervention Variables	AIC
ARIMA (0, 0, 0) with non-zero mean and with intervention variables	−15,170.42
ARIMA (0, 0, 0) with zero mean and with intervention variables	−15,166.53
ARIMA (1, 0, 0) with non-zero mean and with intervention variables	−15,168.43
ARIMA (1, 0, 0) with zero mean and with intervention variables	−15,164.58
ARIMA (0, 0, 1) with non-zero mean and with intervention variables	−15,168.44
ARIMA (0, 0, 1) with zero mean and with intervention variables	−15,164.60
ARIMA (1, 0, 1) with non-zero mean and with intervention variables	−15,188.53
ARIMA (1, 0, 1) with zero mean and with intervention variables	−15,184.99
ARIMA (0, 0, 7) with non-zero mean and with intervention variables	−15,481.79
ARIMA (0, 0, 7) with zero mean and with intervention variables	−15,477.27

Source: Authors’ calculations based on daily Bitcoin data obtained from Investing.com.

.

Among the baseline models, the ARIMA (0, 0, 7) model with a non-zero mean showed better performance on daily returns, as evidenced by its lower AIC values.

Table 6. ARIMA (0, 0, 7) with non-zero mean and with intervention variables.

Parameter	Parameter Estimate	Standard Error (SE)	Test Statistic	p-Value
$\mu$	0.0023565	0.00092	2.5593	0.01049
${\theta }_1$	−0.011153	0.01442	−0.7736	0.4392
${\theta }_2$	−0.13429	0.01429	−9.3972	<0.001
${\theta }_3$	0.02062	0.01450	1.4225	0.1549
${\theta }_4$	0.17611	0.01398	12.5955	<0.0001
${\theta }_5$	0.081304	0.01445	5.6285	<0.0001
${\theta }_6$	−0.10245	0.01453	−7.0495	<0.0001
${\theta }_7$	−0.049067	0.01436	−3.4172	0.0007
${\gamma }_1$	0.0046898	0.00621	0.7552	0.4501
${\gamma }_2$	−0.0000016	0.00001	−0.1106	0.9119

Source: Authors’ calculations based on daily Bitcoin data obtained from Investing.com.

provides detailed model parameters estimated using maximum likelihood.

The ARIMAX (0, 0, 7) model results indicate that the COVID-19 pandemic’s intervention did not have a statistically significant impact on daily Bitcoin returns. The coefficient for the COVID dummy (${\gamma }_1$) is 0.0046898 (p = 0.4501), and the interaction term (${\gamma }_2$) has a coefficient of −0.0000016 (p = 0.9119), both of which are not significant. This suggests that, after accounting for autoregressive dynamics, the pandemic did not cause a measurable shift in the returns.

In contrast, the MA terms at lags 2, 4, 5, 6, and 7 were significant (p < 0.001), indicating that past error terms at these lags significantly influence current returns. Significant MA coefficients indicate short-term dependencies in shocks, reflecting temporary reversals or corrections rather than exploitable trends. The 7-day lag likely captures weekly trading cycles in cryptocurrency markets: although Bitcoin trades continuously, investor behaviour often follows calendar regularities linked to settlement cycles, liquidity patterns, and weekend–weekday differences in trading volume. This structure suggests that shocks dissipate over approximately one trading week, influencing immediate price adjustments without creating systematic predictability over longer horizons. Thus, the significant MA terms reflect transient shock propagation and gradual information absorption, consistent with short-term market dynamics rather than a violation of weak-form efficiency.

These results are consistent with the ITS analysis, where the coefficients for the level shift (β₂ = 0.00205, p = 0.691) and slope change (β₃ = 0.00000057, p = 0.781) were also not significant, further supporting the finding that there was no structural shift in Bitcoin’s return dynamics following the pandemic. The time trend (β₁ = −0.0000023, p = 0.200) was also insignificant, confirming the volatile but largely unpredictable nature of Bitcoin returns. The ARIMAX residual diagnostics revealed insignificant autocorrelation (Ljung–Box test, p = 0.9961), and the Shapiro–Wilk test indicated that model residuals were white noise.

From the results of the ARIMAX (0, 0, 7) model, we have the following equation:

$$\begin{array}{ll}{r}_t = 0.00236 +& \left(-0.011153{\epsilon }_{t-1}\right)+\left(-0.13429{\epsilon }_{t-2}\right)+\left(0.17611{\epsilon }_{t-3}\right)\\ & +\left(0.081304{\epsilon }_{t-4}\right)+\left(-0.10245{\epsilon }_{t-5}\right)+\left(-0.049067{\epsilon }_{t-6}\right)\\ & +0.00469D_t+\left(-1.6084\times {10}^{-6}\right)\left(T_t{D}_t\right)+{\epsilon }_t.\end{array}$$

(7)

This expression represents the logarithmic return dynamics of Bitcoin, accounting for intervention variables, moving-average processes, and the dependence of current returns on past shocks. The significant MA terms capture short-term shock propagation and potential weekly cyclical effects without contradicting the overall unpredictability of returns over longer horizons.

5. Discussion

A key distinction in financial economics is that heightened volatility does not automatically imply market inefficiency. Market efficiency, as outlined by Fama (1991), concerns the predictability of asset prices and the ability to generate abnormal returns using past information, whereas volatility reflects the magnitude of price fluctuations. During periods of high volatility, such as the COVID-19 pandemic, markets may experience significant price swings amid heightened uncertainty. However, the critical question remains whether these fluctuations introduce systematic predictability into returns.

The findings of this study suggest that, at the daily frequency, Bitcoin returns behave as a white noise process, meaning there is no detectable pattern or predictability in returns. This result indicates that while COVID-19 significantly increased uncertainty and trading activity, it did not create discernible linear predictability in Bitcoin’s returns. This observation is consistent with Fama’s (1991) definition of weak-form market efficiency, which posits that asset prices already incorporate all available past information, making it impossible to predict future returns from historical price data.

The results from the Variance Ratio (VR) test further support this conclusion. The test indicates that Bitcoin returns follow a random walk, meaning that price movements over time are independent and exhibit no serial correlation. This is consistent with the concept of weak-form efficiency, where the presence of a random walk implies that past price information is already reflected in the current price, and therefore no predictable patterns can be extracted (Lo, 2004; Lo & MacKinlay, 1988). The lack of significant serial correlations suggests that, even during the COVID-19 crisis, Bitcoin’s price movements remained unpredictable at the daily level.

Further support comes from the Interrupted Time Series (ITS) analysis, where the level shift and slope change parameters also failed to show statistically significant changes. These findings reinforce the notion that the pandemic did not introduce lasting changes in the underlying dynamics of Bitcoin’s return series. Notably, the significant MA (Moving Average) terms at lags 2, 4, 5, 6, and 7 suggest that past error terms continue to exert a substantial influence on current returns. This indicates that while Bitcoin’s returns are not predictable at the daily frequency, past errors (or shocks) continue to affect its price. However, this influence does not introduce a clear, exploitable pattern.

Additionally, the ARIMAX (0, 0, 7) analysis found no statistically significant impact of the COVID-19 pandemic on Bitcoin returns. Both the COVID dummy variable and the interaction term were found to be statistically insignificant, implying that after controlling for autoregressive dynamics, there was no meaningful structural break in Bitcoin’s return patterns during the pandemic. This finding is consistent with Takaishi (2025), who shows that the COVID-19 crisis did not fundamentally alter the efficiency or return behaviour of Bitcoin even as market volatility increased. It was observed that, despite heightened volatility, Bitcoin’s return process remained consistent with weak-form efficiency.

While the COVID-19 pandemic undeniably led to increased volatility and uncertainty in financial markets, it did not induce predictable patterns in Bitcoin returns. The results suggest that, even amidst these turbulent conditions, Bitcoin’s market continued to exhibit weak-form efficiency at the daily level. As the literature has shown (e.g., Fama, 1991; Lo, 2004; Takaishi, 2025), volatility does not necessarily imply inefficiency, and this study’s findings further contribute to the understanding of how Bitcoin’s market reacts under stress.

6. Conclusions

This study investigated whether the COVID-19 pandemic affected Bitcoin’s market efficiency using multiple complementary methodologies, including Interrupted Time Series (ITS) analysis, ARIMAX modelling with intervention variables, and Variance Ratio (VR) tests. With these approaches, the study assessed both structural shifts and return predictability within the weak-form efficiency framework.

The VR test results indicated that Bitcoin returns follow a random walk process. The VR statistics were close to unity across multiple holding periods, supporting the weak-form efficiency hypothesis. Furthermore, the M2 test statistics did not exceed the critical bounds of ±1.96, meaning the null hypothesis of a random walk could not be rejected. This suggests that Bitcoin returns at the daily frequency are not predictable using historical price information. These findings are consistent with the theoretical expectation that in weak-form efficient markets, successive price changes are independent and identically distributed.

The ITS analysis found no statistically significant structural shift in daily Bitcoin returns following the outbreak of COVID-19. Both the COVID dummy variable and the interaction (slope change) term were statistically insignificant. This indicates that the pandemic did not introduce a persistent change in the level or trend of Bitcoin returns. The ARIMAX (0, 0, 7) model confirmed these results, showing no measurable impact of COVID-19 on return dynamics after controlling for autoregressive and moving average components. The Ljung–Box Q-statistic indicated no remaining autocorrelation in the residuals, while the Shapiro–Wilk test results suggested that residuals behaved as white noise. Together, these diagnostic checks confirm the adequacy and robustness of the specified model.

Importantly, although the pandemic period was characterised by heightened volatility and uncertainty, increased volatility alone does not imply market inefficiency. Efficiency concerns predictability rather than the magnitude of price movements. The findings therefore suggest that while COVID-19 amplified volatility in Bitcoin markets, it did not generate exploitable return predictability at the daily frequency. Thus, the market rapidly absorbed pandemic-related information, maintaining weak-form efficiency despite turbulent conditions.

The results contribute to the growing body of literature suggesting that Bitcoin markets exhibit weak-form efficiency, particularly at higher frequencies. The absence of predictable return patterns during an unprecedented global shock strengthens the argument that Bitcoin markets, despite being relatively young and decentralised, may process information efficiently under stress.

Future research could extend this analysis by employing volatility-focused models, such as GARCH-type frameworks, to model time-varying volatility explicitly, or by employing machine learning techniques that detect nonlinear dependencies and regime shifts. Such approaches may uncover higher-order dynamics or conditional inefficiencies not observable within linear daily return models. Additionally, examining intraday data could provide further insights into whether efficiency holds at higher sampling frequencies during crisis periods.