What Explains Bitcoin Volatility? Evidence from an Extended HAR Framework

Abstract

This study investigates the dynamics of Bitcoin’s realized volatility by extending the Heterogeneous Autoregressive (HAR) framework to incorporate external shocks from major financial and commodity markets, namely the NASDAQ-100, Brent crude oil, and gold. To capture potential asymmetries, external market returns are decomposed into positive and negative components. In addition, structural changes in volatility dynamics are examined using structural break tests. The empirical results reveal strong volatility persistence at the daily and weekly horizons, consistent with the HAR structure. Shocks associated with the NASDAQ and gold markets are significantly related to Bitcoin’s realized volatility, whereas the association with crude oil prices is limited. Moreover, both negative and positive gold-market shocks display stronger linkages in the post-2022 period, suggesting time variation in the volatility relationship between Bitcoin and gold.

1. Introduction

Understanding and forecasting financial market volatility are central to risk management, asset pricing, and portfolio allocation. This challenge has become particularly salient in the context of cryptocurrencies—Bitcoin in particular—which has rapidly evolved into a globally traded asset with a market capitalization exceeding USD 2.2 trillion as of October 2025. Despite its growing market size and increasing participation by institutional investors, Bitcoin continues to exhibit substantially higher volatility than traditional financial assets. Unlike equities or commodities, Bitcoin lacks widely accepted fundamental valuation anchors such as earnings or cash flows, rendering its price formation process less transparent and amplifying uncertainty surrounding its volatility dynamics (Cheah & Fry, 2015; Corbet et al., 2018).

In this context, identifying the factors associated with Bitcoin’s volatility has emerged as an important research question with implications for both investors and researchers. While early studies primarily emphasized internal persistence and market-specific characteristics, a growing body of evidence suggests that Bitcoin’s volatility is increasingly influenced by developments in traditional financial markets (Bouri et al., 2018; Uzonwanne, 2021). In particular, spillovers from equity markets and selected commodity markets appear to play a non-negligible role, reflecting Bitcoin’s gradual integration into the broader financial system (Bouri et al., 2018; Corbet et al., 2018). However, the magnitude and transmission channels of these spillovers remain subject to debate, especially in light of recent changes in global financial conditions.

This study examines the empirical patterns of Bitcoin’s realized volatility within an extended Heterogeneous Autoregressive (HAR) framework. The HAR model, originally proposed by Corsi (2009), is well suited for capturing volatility persistence across heterogeneous time horizons and has demonstrated strong empirical performance in modeling and forecasting realized volatility. Building on this framework, we extend the baseline HAR specification in several dimensions. First, we incorporate day-of-the-week effects to account for pronounced intra-week seasonality documented in cryptocurrency markets. Second, we introduce external market shocks originating from the NASDAQ-100 index, Brent crude oil, and gold to examine their respective roles in shaping Bitcoin’s volatility. Third, to capture potential asymmetries in volatility transmission, external market returns are decomposed into positive and negative components, allowing Bitcoin’s volatility to respond differently to upside and downside shocks.

Importantly, the framework accommodates potential time variation in the relationships underlying volatility dynamics. The rapid maturation of cryptocurrency markets, shifts in global monetary policy, and major events within the crypto ecosystem suggest that volatility transmission mechanisms may not remain constant across market regimes. To address this issue, we incorporate structural break tests to identify potential regime shifts and to assess whether the relationship between Bitcoin volatility and its explanatory variables changes over time.

The empirical findings reveal several key results. Bitcoin’s realized volatility exhibits strong persistence at daily and weekly horizons, consistent with the HAR structure. Volatility spillovers from the NASDAQ-100 and gold markets are statistically significant, whereas the impact of crude oil prices remains limited. Moreover, the influence of gold becomes markedly stronger in the post-2022 period, with both positive and negative shocks exerting a more pronounced effect on Bitcoin volatility, indicating an intensification of volatility linkages between Bitcoin and traditional safe-haven assets.

This study advances the literature by addressing three main respects. First, it introduces a framework that jointly accounts for volatility persistence, weekday seasonality, and external market spillovers within an extended HAR model. Second, by modeling asymmetric responses to positive and negative external shocks, it offers a more nuanced understanding of volatility transmission mechanisms between Bitcoin and traditional financial markets. Third, by incorporating structural break analysis, the study shows new evidence on the time-varying nature of Bitcoin’s volatility linkages, highlighting how its exposure to equity and commodity markets has evolved over time. While existing studies acknowledge that Bitcoin volatility is shaped by both internal persistence and external market conditions, relatively little attention has been paid to jointly modeling weekday seasonality, asymmetric spillovers, and structural changes within a unified HAR framework. This study contributes to the literature by offering such an integrated approach.

The rest of the paper is organized as follows. Section 2 reviews the related literature. Section 3 presents the data employed in the analysis. Section 4 describes the empirical methodology. Section 5 presents the estimation results, and Section 6 concludes.

2. Literature Review

The volatility of financial asset returns is well known to be time-varying, exhibiting volatility clustering, persistence, and asymmetric responses to shocks. Early studies modeled these features using the ARCH framework introduced by Engle (1982) and the GARCH model developed by Engle and Bollerslev (1986). Numerous extensions such as EGARCH (Nelson, 1991) and GJR-GARCH (Glosten et al., 1993) capture asymmetric responses, while FIGARCH models (Baillie et al., 1996) account for long-memory behavior. In parallel, stochastic volatility (SV) models provided an alternative framework by treating volatility as an unobserved stochastic process, with Bayesian estimation methods developed by Jacquier et al. (2002).

While GARCH-type and SV models have been widely applied, these approaches treat volatility as a latent variable inferred from low-frequency return data, typically constructed from daily observations. This reliance often results in a substantial loss of information embedded in high-frequency intraday price fluctuations. To address this limitation, Andersen and Bollerslev (1998) introduced the concept of Realized Volatility (RV). Unlike latent volatility estimates, RV utilizes high-frequency intraday data to compute the sum of squared returns, thereby providing precise information regarding volatility. Corsi (2009) subsequently proposed the Heterogeneous Autoregressive (HAR) model. Based on the Heterogeneous Market Hypothesis, the HAR framework suggests that volatility is driven by constructing realized volatility as a function of lagged daily, weekly, and monthly volatility components. Andersen et al. (2007) demonstrated that HAR models significantly outperform traditional GARCH and SV models in forecasting the volatility of the S&P 500 and foreign exchange rates. Patton and Sheppard (2015) further extended the framework to separately account for “good” and “bad” volatility, confirming the asymmetric impact of negative shocks on equity volatility.

A growing body of literature employs these models to Bitcoin and other cryptocurrencies. Early studies on Bitcoin volatility focused on documenting its statistical properties, including fat tails, and volatility clustering, typically using GARCH-type and SV models and daily data (Cheah & Fry, 2015; Katsiampa, 2017; Ma & Tanizaki, 2019). As the market matured, subsequent research moved toward modeling internal volatility dynamics using high-frequency data and realized volatility measures, emphasizing volatility persistence, jump behavior, and microstructure effects (Bergsli et al., 2022; Shen et al., 2020). In addition, intra-week seasonality has been documented in cryptocurrency markets. Ma and Tanizaki (2019) and Aharon and Qadan (2019) reported statistically significant day-of-the-week effects in Bitcoin returns and volatility.

Beyond internal dynamics, the expanding literature highlights the importance of external market spillovers in shaping Bitcoin’s volatility. Spillovers from equity markets have been widely documented, reflecting the increasing integration of cryptocurrencies with traditional financial markets (Bouri et al., 2018; Chen, 2024; Conrad et al., 2018; Uzonwanne, 2021). In particular, the NASDAQ is especially relevant for Bitcoin volatility, as it is heavily weighted toward technology-oriented firms. Several studies documented a strong co-movement and volatility transmission between cryptocurrency and technology-focused equity indices (Bonelli, 2020; Matkovskyy & Jalan, 2019).

In addition, the relationship between Bitcoin and commodity markets—specifically crude oil and gold—has attracted substantial attention. Energy prices are often examined due to the electricity-intensive nature of Bitcoin mining. Because of limitations in obtaining electricity price data, several studies proxy electricity prices using crude oil or natural gas prices when examining their impacts on Bitcoin price dynamics and volatility. Ciaian et al. (2016) documented a significant impact of crude oil price movements on Bitcoin returns, while Omura et al. (2024) provided evidence that natural gas influences Bitcoin volatility. By contrast, Baur et al. (2018) found no statistically significant relationship between crude oil prices and Bitcoin returns. Moreover, gold has been increasingly examined in the literature in relation to Bitcoin, particularly within the context of the “digital gold” narrative. Dyhrberg (2016) pioneered this line of research by suggesting that Bitcoin exhibits hedging properties similar to those of gold. However, subsequent studies have produced mixed evidence, examining whether Bitcoin and gold function as substitutes or whether their relationship weakens during periods of market turbulence (Baur et al., 2018; Klein et al., 2018).

Despite these advances, several gaps remain in the existing literature. While prior studies have examined volatility persistence, weekday seasonality, external market spillovers, or structural breaks in isolation, relatively little attention has been paid to jointly modeling these features within a unified framework. In particular, the asymmetric transmission of external shocks and the potential time variation in volatility spillovers across market regimes have not been explored. This paper addresses these gaps by developing an extended HAR framework that simultaneously accounts for volatility persistence, intra-week seasonality, asymmetric spillovers from equity and commodity markets, and structural breaks.

3. Data

This study constructs Bitcoin realized volatility using high-frequency intraday transaction data. Bitcoin price data are obtained from Binance. Following Andersen and Bollerslev (1998), daily realized volatility is computed as

$$R{V}_t=ln\left(\sum _{j=1}^k{r}_{t,j}^2\right),$$

(1)

where $r_{t,j}$ denotes the intraday return for interval j within day t, calculated as the natural logarithmic difference between consecutive prices. A key issue in constructing realized volatility is the influence of microstructure noise in high-frequency prices. As noted by Liu et al. (2015), using overly fine sampling intervals increases the proportion of noise relative to the true price signal, leading to biased volatility estimates. They show that realized volatility based on 5-min return intervals generally outperforms other sampling intervals with respect to in-sample fit and forecasting accuracy. In line with their findings, this study constructs realized volatility using 5-min intraday returns.

Daily price data for the NASDAQ-100 index, Brent crude oil, and gold are obtained from Yahoo Finance (https://finance.yahoo.com) for the period 1 January 2012 to 30 August 2025. Daily returns are computed as the first differences of the natural logarithms of the closing prices.

Table 1. Descriptive Statistics.

Variable	Mean	SD	Min	Max	Skewness	Kurtosis
Bitcoin realized volatility (RV)	−7.2982	1.1138	−14.1981	−2.2022	0.0497	4.0037
Monday	−7.0961	1.0081	−10.1279	−3.5673	0.4151	3.4371
Tuesday	−7.1084	0.9866	−9.6078	−3.2904	0.4338	3.7206
Wednesday	−7.0437	0.9875	−9.3093	−2.6530	0.4356	3.4920
Thursday	−7.0878	0.9938	−9.9140	−3.0154	0.4162	3.6691
Friday	−7.1280	1.0121	−9.6728	−2.2022	0.4504	4.3587
Saturday	−7.8450	1.2609	−11.7001	−4.2242	0.0824	2.8773
Sunday	−7.6211	1.1356	−10.9638	−4.0738	0.3269	3.1868
NASDAQ-100 return	0.0440	1.2940	−13.0032	11.3528	−0.3649	14.4885
Brent crude oil return	−0.0005	2.1276	−27.9761	19.0774	−1.6031	32.0841
Gold return	0.0345	0.8071	−5.1069	5.7775	−0.2436	9.4907

Notes: This table reports descriptive statistics for Bitcoin realized volatility (RV) and for the daily returns of the NASDAQ-100 index, Brent crude oil, and gold. RV is presented in natural logarithmic form, and all return series are expressed in daily percentage terms. The rows labeled Monday through Sunday summarize Bitcoin realized volatility by day of the week. The sample period covers 31 January 2018 to 29 August 2025.

summarizes the descriptive statistics for Bitcoin’s realized volatility—both overall and by day of the week—along with the daily returns of the NASDAQ-100, Brent crude oil, and gold. A key observation is the distinct day-of-the-week effect, justifying the inclusion of daily dummy variables to control for this feature. Regarding the exogenous variables, all three return series—NASDAQ, crude oil, and gold—show significant deviations from normality. They are characterized by negative skewness and excessively high kurtosis, particularly in the crude oil market.

4. Methodology

This section presents the empirical model used to analyze the determinants of Bitcoin’s realized volatility. The baseline specification is based on the HAR model introduced by Corsi (2009), which is designed to capture persistence of realized volatility across heterogeneous time horizons. The model is defined as follows:

$$\begin{array}{c}\hfill R{V}_t=\alpha +{\beta }_{d}R{V}_t^{\left(d\right)}+{\beta }_{w}R{V}_t^{\left(w\right)}+{\beta }_{m}R{V}_t^{\left(m\right)}+{\epsilon }_t,\end{array}$$

(2)

where $R{V}_t^{\left(d\right)}$, $R{V}_t^{\left(w\right)}$, and $R{V}_t^{\left(m\right)}$ correspond to the realized volatility components over daily, weekly, and monthly horizons, respectively. Specifically, the daily component is the lagged realized volatility, $R{V}_t^{\left(d\right)}=R{V}_{t-1}$. Consistent with the standard HAR framework adapted for the cryptocurrency market, the weekly and monthly components are constructed as the averages of past realized volatilities:

$$\begin{array}{c}\hfill R{V}_t^{\left(w\right)}={\displaystyle \frac{1}{7}}\sum _{i=1}^{7}R{V}_{t-i},\mathrm{and}R{V}_t^{\left(m\right)}={\displaystyle \frac{1}{30}}\sum _{i=1}^{30}R{V}_{t-i}.\end{array}$$

The coefficients ${\beta }_d$, ${\beta }_w$, and ${\beta }_m$ capture the persistence of volatility attributable to short-, medium-, and long-term market activities, respectively. The error term ${\epsilon }_t$ is assumed to have mean zero and variance ${\sigma }^2$.

Taking into account the presence of intra-week seasonality in Bitcoin’s realized volatility, as reported in Table 1, the baseline HAR specification is augmented to include day-of-the-week dummy variables as follows:

$$R{V}_t=\alpha +{\beta }_{d}R{V}_t^{\left(d\right)}+{\beta }_{w}R{V}_t^{\left(w\right)}+{\beta }_{m}R{V}_t^{\left(m\right)}+\sum _{s\in \mathcal{D}}{\delta }_s{D}_{s,t}+{\epsilon }_t,$$

(3)

where $\mathcal{D}=\{\mathrm{Mon},\dots ,\mathrm{Sat}\}$ represents the set of day-of-the-week dummy variables, with Sunday omitted. For instance, $D_{\mathrm{Mon},t}=1$ if day t corresponds to a Monday and 0 otherwise. Including these indicators allows the model to account for weekday variations in Bitcoin’s volatility.

To identify volatility spillovers from traditional financial markets, we further augment the above specification as follows:

$$\begin{array}{c}\hfill R{V}_t=\alpha +{\beta }_{d}R{V}_t^{\left(d\right)}+{\beta }_{w}R{V}_t^{\left(w\right)}+{\beta }_{m}R{V}_t^{\left(m\right)}+\sum _{s\in \mathcal{D}}{\delta }_s{D}_{s,t}+\sum _{k\in \mathcal{K}}{\beta }_k\left|R_{k,t}\right|+{\epsilon }_t,\end{array}$$

(4)

where the set $\mathcal{K}=\{\mathrm{Nas},\mathrm{Oil},\mathrm{Gold}\}$ denotes the set of external markets corresponding to the NASDAQ index, Brent crude oil, and gold, respectively. The variable $R_{k,t}$ represents the daily return of asset $k\in \mathcal{K}$, computed as the first difference of the natural logarithm of daily closing prices1. The absolute return $|R_{k,t}|$ captures the magnitude of external market shocks, abstracting from their direction, and is used to proxy the intensity of volatility spillovers from these markets. This specification emphasizes risk transmission through shock magnitude instead of contemporaneous price correlation.

In the following equation, to examine potential asymmetries in volatility spillovers from external markets, we further decompose external shocks into their negative and positive components as follows:

$$\begin{array}{c}\hfill R{V}_t=\alpha +{\beta }_{d}R{V}_t^{\left(d\right)}+{\beta }_{w}R{V}_t^{\left(w\right)}+{\beta }_{m}R{V}_t^{\left(m\right)}+\sum _{s\in \mathcal{D}}{\delta }_s{D}_{s,t}+\sum _{k\in \mathcal{K}}\left({\beta }_k^{-}{R}_{k,t}^{-}+{\beta }_k^{+}{R}_{k,t}^{+}\right)+{\epsilon }_t,\end{array}$$

(5)

where $R_{k,t}^{-}=\left|R_{k,t}\right|I(R_{k,t}<0)$ and $R_{k,t}^{+}=R_{k,t}I(R_{k,t}>0)$ represent the negative and positive returns for each variable $k\in \mathcal{K}$, respectively, capturing negative or positive shocks exert influence on the Bitcoin’s volatility. For instance, when the NASDAQ return is negative, $R_{Nas,t}^{-}$ takes the absolute value of the return, while $R_{Nas,t}^{+}$ equals zero.

Considering that the relationship between Bitcoin volatility and its explanatory variables may evolve over time, we further augment the HAR specification to allow for coefficient instability over time. To detect such changes, we apply the multiple structural break procedure developed by Bai and Perron (1998, 2003), which identifies break points by searching for the partition that minimizes the overall sum of squared residuals. Specifically, the break dates $({\widehat{T}}_1,\dots ,{\widehat{T}}_n)$ are obtained as

$$\begin{array}{cc}\hfill ({\widehat{T}}_1,\dots ,{\widehat{T}}_n)& =arg\underset{T_1,\dots ,T_n}{min}\sum _{j=1}^{n+1}\sum _{t=T_{j-1}+1}^{T_j}(R{V}_t-{\alpha }_j-{\beta }_{d,j}R{V}_t^{\left(d\right)}-{\beta }_{w,j}R{V}_t^{\left(w\right)}-{\beta }_{m,j}R{V}_t^{\left(m\right)}\hfill \\ & -\sum _{k\in \mathcal{K}}\left({\beta }_{k,j}^{-}{R}_{k,t}^{-}+{\beta }_{k,j}^{+}{R}_{k,t}^{+}\right)-\sum _{s\in \mathcal{D}}{\delta }_{s,j}{D}_{s,t}{)}^2,\hfill \end{array}$$

(6)

where $j=1,2,\dots ,n+1$ denotes the regimes such that $T_{j-1}<t\le T_j$, with $T_0=0$ and $T_{n+1}=T$. The optimal number of structural breaks is selected using a combination of the sequential $\mathrm{supF}(l+1\mid l)$ test, which compares a model with l breaks against one with $l+1$ breaks, and the Bayesian Information Criterion (BIC), which selects the model that balances goodness of fit and simplicity.

5. Empirical Results

Table 2. HAR Model Estimates for Bitcoin Realized Volatility.

	Model 1	Model 2	Model 3
Intercept	$-0.3322$ **	$-0.5754$ ***	$-0.6128$ ***
	$\left(0.1196\right)$	$\left(0.1138\right)$	$\left(0.1134\right)$
$R{V}^{\left(d\right)}$	$0.2553$ ***	$0.2395$ ***	$0.2395$ ***
	$\left(0.0198\right)$	$\left(0.0203\right)$	$\left(0.0202\right)$
$R{V}^{\left(w\right)}$	$0.7278$ ***	$0.7549$ ***	$0.7457$ ***
	$\left(0.0328\right)$	$\left(0.0316\right)$	$\left(0.0315\right)$
$R{V}^{\left(m\right)}$	$-0.0286$	$-0.0444$ *	$-0.0403$
	$\left(0.0287\right)$	$\left(0.0261\right)$	$\left(0.0260\right)$
$D_{\mathrm{Mon}}$		$0.4683$ ***	$0.4139$ ***
		$\left(0.0434\right)$	$\left(0.0444\right)$
$D_{\mathrm{Tue}}$		$0.3357$ ***	$0.2845$ ***
		$\left(0.0457\right)$	$\left(0.0465\right)$
$D_{\mathrm{Wed}}$		$0.4014$ ***	$0.3502$ ***
		$\left(0.0457\right)$	$\left(0.0464\right)$
$D_{\mathrm{Thu}}$		$0.3416$ ***	$0.2824$ ***
		$\left(0.0461\right)$	$\left(0.0470\right)$
$D_{\mathrm{Fri}}$		$0.3156$ ***	$0.2583$ ***
		$\left(0.0458\right)$	$\left(0.0467\right)$
$D_{\mathrm{Sat}}$		$-0.3881$ ***	$-0.3881$ ***
		$\left(0.0456\right)$	$\left(0.0453\right)$
$|R_{\mathrm{Nas}}|$			$0.0691$ ***
			$\left(0.0146\right)$
$|R_{\mathrm{Oil}}|$			$0.0013$
			$\left(0.0082\right)$
$|R_{\mathrm{Gold}}|$			$0.0615$ ***
			$\left(0.0232\right)$
Observations	2768	2768	2767
Residual Std. Error	$0.6677$	$0.6068$	$0.6033$
$R^2$/Adjusted $R^2$	$0.6312$/$0.6308$	$0.6961$/$0.6951$	$0.6996$/$0.6983$
F-statistic (df)	$1578.0$ $(3,2764)$	$702.1$ $(9,2758)$	$534.7$ $(12,2754)$

Notes: The dependent variable is the realized volatility of Bitcoin. Model 1 includes the standard HAR components with daily, weekly, and monthly lags (1, 7, and 30 days). Model 2 augments this specification with weekday dummy variables (Monday through Saturday). Model 3 further incorporates external market variables, here measured by the negative absolute daily returns of the NASDAQ-100 index, Brent crude oil, and gold. Estimated coefficients are reported with standard errors in parentheses. ***, **, and * denote significance at the 1%, 5%, and 10% levels, respectively.

presents the estimation results for the stepwise extensions of the Heterogeneous Autoregressive (HAR) model. The results are reported in three columns, corresponding to the model specifications defined in Section 3. Model 1 corresponds to the baseline specification, designed to identify the persistence of realized volatility components over daily, weekly, and monthly horizons. Model 2 extends the baseline by incorporating day-of-the-week dummy variables to account for seasonality in Bitcoin volatility. Model 3 presents the specification of Equation (4), which further includes the NASDAQ-100, Brent crude oil, and gold to capture volatility spillovers from external markets.

The coefficients for the daily and weekly volatility components are positive and statistically significant at the 1% level across all specifications, confirming strong short- and medium-term persistence in Bitcoin volatility. However, the monthly component is statistically insignificant at the 5% level, suggesting that long-term volatility dynamics have a limited impact relative to short- and medium-term horizons. Model 2 provides evidence of a pronounced day-of-the-week effect; that is, volatility is significantly higher on weekdays compared to on Sundays, while Saturday exhibits a significantly lower level of volatility; among weekdays, Monday exhibits the highest level of volatility. These findings are consistent with the empirical evidence provided by Ma and Tanizaki (2019), Aharon and Qadan (2019), who similarly documented distinct weekly seasonality in cryptocurrency markets. Moreover, compared with Model 1, Model 2 exhibits a lower residual standard error and a higher adjusted R-squared, indicating that accounting for day-of-the-week effects improves the overall model fit. In Model 3, the coefficients for the NASDAQ and gold are positive and significant at the 1% level, implying that shocks in these markets spill over to the Bitcoin volatility. In contrast, Brent crude oil does not exert statistically significant effects on Bitcoin.

Considering that the dynamics of Bitcoin’s volatility may change over time, we extend the HAR framework by testing for structural breaks. Following the methodology of Bai and Perron (1998, 2003), the break dates are estimated by minimizing the sum of squared residuals over all possible partitions of the sample. Estimating the same model across different regimes allows us to assess whether the volatility transmission mechanism remains stable over time. The model specification remains unchanged, and the subsample estimates are used to capture time variation rather than to introduce additional complexity. Applying this procedure to Equations (4) and (5), we identify a single break occurring on 11 September 2022. Both the sequential supF tests and the BIC criterion consistently select the single break date, indicating a robust regime shift in Bitcoin volatility dynamics. This period coincides with heightened uncertainty in global financial markets, associated with rising monetary tightening expectations following unexpected inflation surprises in the United States. At the same time, cryptocurrency markets experienced heightened volatility surrounding Ethereum’s Merge upgrade in mid-September 2022. The vulnerable market environment was further strained by subsequent events, including the collapse of the FTX exchange in November 2022, contributing to a prolonged phase of heightened uncertainty in cryptocurrency markets.

Based on the structural breaks test,

Table 3. HAR Model Estimates for Structural Break Subsamples.

	Period I	Period II
	(2018–2022)	(2022–2025)
Intercept	$-0.6725$ ***	$-1.1828$ ***
	$\left(0.1464\right)$	$\left(0.2790\right)$
$R{V}^{\left(d\right)}$	$0.2258$ ***	$0.2448$ ***
	$\left(0.0262\right)$	$\left(0.0323\right)$
$R{V}^{\left(w\right)}$	$0.7721$ ***	$0.7100$ ***
	$\left(0.0398\right)$	$\left(0.0502\right)$
$R{V}^{\left(m\right)}$	$-0.0724$ ***	$-0.0676$
	$\left(0.0325\right)$	$\left(0.0483\right)$
$D_{\mathrm{Mon}}$	$0.1991$ ***	$0.6073$ ***
	$\left(0.0594\right)$	$\left(0.0710\right)$
$D_{\mathrm{Tue}}$	$0.1234$ ***	$0.3890$ ***
	$\left(0.0599\right)$	$\left(0.0819\right)$
$D_{\mathrm{Wed}}$	$0.1957$ ***	$0.4453$ ***
	$\left(0.0597\right)$	$\left(0.0821\right)$
$D_{\mathrm{Thu}}$	$0.1678$ ***	$0.3186$ ***
	$\left(0.0604\right)$	$\left(0.0834\right)$
$D_{\mathrm{Fri}}$	$0.1418$ ***	$0.3007$ ***
	$\left(0.0597\right)$	$\left(0.0822\right)$
$D_{\mathrm{Sat}}$	$-0.1691$ ***	$-0.7274$ ***
	$\left(0.0557\right)$	$\left(0.0743\right)$
$|R_{Nas}|$	$0.0539$ ***	$0.0609$ ***
	$\left(0.0160\right)$	$\left(0.0213\right)$
$|R_{Oil}|$	$0.0072$	$0.0235$
	$\left(0.0082\right)$	$\left(0.0167\right)$
$|R_{Gold}|$	$0.0202$	$0.0877$ ***
	$\left(0.0264\right)$	$\left(0.0315\right)$
Observations	1685	1083
Residual Std. Error	$0.6010$	$0.5549$
$R^2$/Adjusted $R^2$	$0.6687$/$0.6663$	$0.6753$/$0.6716$
F-statistic (df)	$281.4$ $(12,1672)$	$185.2$ $(12,1070)$

Notes: Table 3 reports the estimation results for Model 3, in which the Bai–Perron structural break test is applied to identify regime changes, yielding two subsample periods: Period I (31 January 2018–11 September 2022) and Period II (12 September 2022–29 August 2025). Each specification includes the standard HAR components with daily, weekly, and monthly lags (1, 7, and 30 days), day-of-the-week dummy variables for Monday through Saturday (with Sunday omitted), and the absolute daily returns of the NASDAQ-100 index, Brent crude oil, and gold. Estimated coefficients are reported with standard errors in parentheses. *** denotes statistical significance at the 1% level.

presents the estimation results for two sub-periods: Period I (31 January 2018 to 11 September 2022) and Period II (12 September 2022 to 29 August 2025). For volatility persistence, the HAR structure remains robust in both periods. The daily and weekly components are consistently positive and statistically significant at the 1% level. The monthly component is statistically negative, reducing the Bitcoin volatility in Period I; however, it becomes insignificant in Period II, indicating that long-horizon information plays a limited role in shaping Bitcoin’s realized volatility. In addition, the result reflects an intra-weekly seasonality in both regimes. Interestingly, the day-of-the-week becomes substantially more pronounced in Period II. The coefficients for Monday through Friday increase sharply in the later period, while Saturday continues to exhibit significantly lower volatility relative to Sunday. Regarding external spillovers, the role of traditional assets changes over time. The NASDAQ index ($|R_{\mathrm{Nas}}|$) maintains a significant positive impact on Bitcoin volatility in both periods, confirming the integration between the technology equity market and cryptocurrency. In contrast, the influence of gold ($|R_{\mathrm{Gold}}|$) shows a structural evolution. In Period I, gold exerts limited influence on Bitcoin volatility. However, in Period II, the coefficient for gold increases and becomes significant at the 1% level, indicating that shocks in the gold market have begun to spill over to Bitcoin, likely driven by common exposure to macroeconomic risks in the post-2022 period. In contrast, Brent crude oil ($|R_{\mathrm{Oil}}|$) remains statistically insignificant throughout both periods. Our results confirm that the volatility transmission mechanism is not time-invariant; thus, a segmented analysis is essential to avoid model misspecification and to capture the evolving linkages between Bitcoin and traditional assets.

Table 4. HAR Model with Sign-Split External Shocks for Structural Break Subsamples.

	Period I	Period II
	(2018–2022)	(2022–2025)
Intercept	$-0.6637$ ***	$-1.1793$ ***
	$\left(0.1462\right)$	$\left(0.2795\right)$
$R{V}^{\left(d\right)}$	$0.2289$ ***	$0.2449$ ***
	$\left(0.0262\right)$	$\left(0.0323\right)$
$R{V}^{\left(w\right)}$	$0.7663$ ***	$0.7124$ ***
	$\left(0.0397\right)$	$\left(0.0503\right)$
$R{V}^{\left(m\right)}$	$-0.0685$ ***	$-0.0696$
	$\left(0.0325\right)$	$\left(0.0484\right)$
$D_{\mathrm{Mon}}$	$0.2054$ ***	$0.6088$ ***
	$\left(0.0594\right)$	$\left(0.0711\right)$
$D_{\mathrm{Tue}}$	$0.1342$ ***	$0.3909$ ***
	$\left(0.0600\right)$	$\left(0.0820\right)$
$D_{\mathrm{Wed}}$	$0.2050$ ***	$0.4450$ ***
	$\left(0.0598\right)$	$\left(0.0822\right)$
$D_{\mathrm{Thu}}$	$0.1746$ ***	$0.3170$ ***
	$\left(0.0605\right)$	$\left(0.0836\right)$
$D_{\mathrm{Fri}}$	$0.1438$ ***	$0.3001$ ***
	$\left(0.0598\right)$	$\left(0.0824\right)$
$D_{\mathrm{Sat}}$	$-0.1704$ ***	$-0.7275$ ***
	$\left(0.0556\right)$	$\left(0.0743\right)$
$R_{\mathrm{Nas}}^{-}$	$0.0831$ ***	$0.0767$ ***
	$\left(0.0185\right)$	$\left(0.0273\right)$
$R_{\mathrm{Nas}}^{+}$	$0.0150$	$0.0486$ **
	$\left(0.0207\right)$	$\left(0.0249\right)$
$R_{\mathrm{Oil}}^{-}$	$0.0027$	$0.0189$
	$\left(0.0095\right)$	$\left(0.0188\right)$
$R_{\mathrm{Oil}}^{+}$	$0.0108$	$0.0288$
	$\left(0.0114\right)$	$\left(0.0217\right)$
$R_{\mathrm{Gold}}^{-}$	$0.0452$	$0.1013$ ***
	$\left(0.0311\right)$	$\left(0.0383\right)$
$R_{\mathrm{Gold}}^{+}$	$-0.0052$	$0.0748$ **
	$\left(0.0333\right)$	$\left(0.0375\right)$
Observations	1685	1083
Residual Std. Error	$0.5996$	$0.5550$
$R^2$/Adjusted $R^2$	$0.6710$/$0.6680$	$0.6760$/$0.6715$
F-statistic (df)	$226.9$ $(15,1669)$	$148.4$ $(15,1067)$

Notes: This table reports the estimation results for Equation (5), in which external market shocks are decomposed into positive and negative components across two subsample periods: Period I (31 January 2018–11 September 2022) and Period II (12 September 2022–29 August 2025). Each specification includes the standard HAR components with daily, weekly, and monthly lags (1, 7, and 30 days), day-of-the-week dummy variables for Monday through Saturday (with Sunday omitted), and the daily returns of the NASDAQ-100 index, Brent crude oil, and gold. Estimated coefficients are reported with standard errors in parentheses. ***, and ** denote statistical significance at the 1% and 5%, respectively.

reports the estimation results of Equation (5) for Period I and Period II, where external shocks are decomposed into negative and positive components. A key finding is the pronounced asymmetry in volatility spillovers from the NASDAQ market. In Period I, volatility spillovers are primarily driven by downside risk. Negative shocks from the NASDAQ exert a strong and statistically significant effect on Bitcoin’s volatility, whereas positive shocks remain insignificant. This pattern indicates that, prior to 2022, Bitcoin markets responded to bad news in the tech sector while ignoring positive news. The dynamics change noticeably in Period II. Although negative shocks continue to play a dominant role, positive shocks also begin to exert a marginal influence on Bitcoin’s volatility. This shift suggests that sharp upward movements in the NASDAQ index have become an additional source of volatility transmission. As integration between the equity and cryptocurrency markets deepens, risk-on rallies in the technology sector increasingly spill over into the Bitcoin market, amplifying volatility even during market upswings.

For the relationship of gold with Bitcoin, neither negative nor positive gold shocks significantly affect Bitcoin volatility, suggesting a decoupling between the digital and traditional safe havens in Period I. However, Period II marks a regime shift where gold emerges as a significant predictor. Negative gold shocks become statistically significant, and positive shocks also have marginal effects on Bitcoin volatility.

The strong reaction to negative gold shocks in the later period suggests that declines in gold prices—often associated with a strengthening U.S. dollar or rising yields—serve as a potent signal for increased turbulence in the Bitcoin market. On the other hand, the impacts of positive shocks in the gold market tend to be associated with bad news in global financial environment, which leads to an increase in the volatility of Bitcoin as well.

In contrast, crude oil shocks exhibit no statistically significant impact on Bitcoin’s volatility. Across both positive and negative shocks, as well as across subsample periods, oil returns fail to explain the dynamics of Bitcoin’s realized volatility. This finding supports the view that Bitcoin’s price discovery process remains largely decoupled from fluctuations in energy markets, despite the energy-intensive nature of its mining activity.

Table 5. Out-of-Sample Forecasting Performance of HAR-Based Models.

Model	Overall		Period I		Period II
	MSE	MAE	MSE	MAE	MSE	MAE
Model 1 (HAR)	0.4491	0.4970	0.3142	0.4163	0.5346	0.5481
Model 2 (HAR with weekdays)	0.3149	0.4176	0.2766	0.3901	0.3392	0.4350
Model 3 (HAR with weekdays and external shocks)	0.3131	0.4155	0.2786	0.3890	0.3350	0.4322
Model 4 (HAR with weekdays and sign-split external shocks)	0.3135	0.4153	0.2756	0.3864	0.3374	0.4335

Notes: This table reports the out-of-sample forecasting performance of alternative HAR-based specifications. Out-of-sample forecasts are generated over the full evaluation period using a rolling-window scheme, and the resulting forecast errors are subsequently partitioned into two subsample periods—Period I (31 January 2018–11 September 2022) and Period II (12 September 2022–29 August 2025)—as defined by the structural break analysis. Model 1 includes the standard HAR components with daily, weekly, and monthly lags (1, 7, and 30 days). Model 2 augments the baseline specification with day-of-the-week dummy variables (Monday through Saturday, with Sunday omitted). Model 3 further incorporates external market variables, measured by the absolute daily returns of the NASDAQ-100 index and gold. Model 4 decomposes these external shocks into negative and positive components to allow for asymmetric spillover effects. Forecast accuracy is evaluated using the mean squared error (MSE) and mean absolute error (MAE). Bold values denote the minimum MSE or MAE within each column.

reports the out-of-sample forecasting performance of the four HAR-based specifications. Forecasts are generated using a fixed-length rolling window, where the models are re-estimated each day using the most recent 1000 observations. Out-of-sample evaluation therefore begins at t = 1001, and the resulting forecast errors are partitioned into Period I (31 January 2018–11 September 2022) and Period II (12 September 2022–29 August 2025), as identified by the structural-break analysis. Model 1 includes the baseline HAR components. Model 2 adds day-of-the-week dummies; Model 3 further incorporates external shocks from the NASDAQ-100, Brent crude oil and gold; and Model 4 decomposes these shocks from external markets into negative and positive components. Forecast accuracy is evaluated using the mean-squared error (MSE) and mean-absolute error (MAE). The baseline HAR model reveals the weakest forecasting performance, while the inclusion of weekday effects in Model 2 yields a marked improvement. Specifically, extending the baseline specification to include day-of-the-week effects reduces the out-of-sample MSE by approximately 30% and the MAE by about 16%, highlighting the economic relevance of weekly seasonality in Bitcoin volatility dynamics. External shocks further enhance predictive accuracy in Model 3, and allowing for asymmetric spillovers in Model 4 produces the lowest MSE and MAE overall. These results indicate that weekday seasonality has meaningful predictive power and that external shocks—especially when modeled asymmetrically—offer further refinement to volatility forecasting.

6. Conclusions

This study examines Bitcoin’s realized volatility by extending the Heterogeneous Autoregressive (HAR) framework to incorporate asymmetric spillovers from major financial and commodity markets. Using high-frequency intraday data to construct realized volatility, and decomposing external market returns into positive and negative components, the empirical analysis reveals strong volatility persistence at daily and weekly horizons, consistent with the HAR structure. Among the external factors considered, shocks originating from the NASDAQ and gold markets play a significant role in shaping Bitcoin’s volatility, whereas the influence of Brent crude oil prices remains limited across specifications.

To capture potential time variation in these relationships, we further allow for structural breaks in volatility dynamics. The analysis identifies a regime shift in September 2022, marking a change in how external shocks are transmitted to Bitcoin volatility. After this break, Bitcoin exhibits emerging volatility linkages with gold, suggesting that its interaction with gold evolves over time and that volatility spillovers are inherently regime-dependent. These findings also have practical implications for hedging and risk management. In particular, the strengthening volatility linkage between Bitcoin and gold after 2022 implies that shocks to gold may spill over to Bitcoin volatility, potentially weakening its role as a volatility hedge and highlighting the importance of accounting for regime-dependent spillovers when designing diversification strategies.

Furthermore, the out-of-sample forecasting results indicate that the extended HAR specifications—particularly those incorporating day-of-the-week seasonality and asymmetric external shocks—outperform the baseline HAR model. These findings highlight the value of accounting for both intra-week patterns and asymmetric spillover effects, thereby contributing to a more comprehensive understanding of the evolving nature of Bitcoin’s volatility.

Several directions for future research remain. One direction is to extend the analysis to other cryptocurrencies in order to assess whether similar volatility transmission patterns arise. Another is to incorporate broader macroeconomic or sentiment-related variables that may interact with external market shocks. While the present analysis focuses on reduced-form transmissions, future work may also complement this approach with identification-based models to further explore underlying mechanisms.

Finally, the flexibility of the HAR framework makes it particularly well suited for these extensions. Its structure allows additional explanatory variables to be incorporated without increasing estimation complexity while preserving the interpretability of volatility persistence across heterogeneous horizons. This tractability makes the HAR approach a useful framework for studying volatility transmission and forecasting in rapidly evolving financial markets.