% For submission to the Journal of Futures Markets (JFM)

\section{Empirical Results} \label{ch2:sec:results}

This section reports the results of our empirical analysis of volatility transmission between commodity ETFs and their underlying assets. We organize the presentation of results around the four hypotheses stated in Section~\ref{ch2:sec:methodolog}, analyzing them in order: namely, the direction of transmission and its variation across commodity types (H1), the role of sampling frequency (H2), the relative importance of jump and continuous components (H3), and the stability of transmission over time (H4).

\subsection{Direction of Transmission and Commodity-Specific Asymmetries}

Our first hypothesis concerns whether transmission between an ETF and its iNAV is unidirectional or bidirectional, and whether this varies across commodities (H1). As discussed in Section~\ref{ch2:sec:methodolog}, theory predicts a flow only from iNAV to ETF, but a more liquid ETF can reverse the direction of the flow. In the case of commodities, the storability and settlement mechanism is also expected to matter. We therefore expect physically-backed precious metals to show unidirectional iNAV-to-ETF transmission, and futures-based energy ETFs more balanced bidirectional transmission.

The HAR-X estimates in Tables~\ref{ch2:tab:HAR_5min} through~\ref{ch2:tab:HAR_30min} reject the null of uniform bidirectional transmission across all commodities, extending the heterogeneity documented by \citet{gorton2006facts} and \citet{buyuksahin2014speculation} to the ETF setting and consistent with \citet{basak2016model} on how financialization effects vary across commodity types.

The results for precious metals show a strongly unidirectional transmission from iNAV to ETF, with little evidence of reverse transmission. Consider the iNAV-to-ETF coefficient, which is the coefficient for lagged NAV volatility in the ETF equation. For gold at a 1-minute frequency, this coefficient is 0.632 (Table~\ref{ch2:tab:HAR_1min}), which is among the largest across commodities, while the ETF-to-iNAV effect (i.e., lagged ETF volatility in the NAV equation) is not statistically different from zero ($-0.001$, $p=0.96$). The results for silver show a similar but weaker pattern: the NAV-to-ETF coefficient is 0.418 while the reverse effect is 0.004 and not significant. On the other hand, our results for energy commodities show bidirectional transmission. For crude oil at a 1-minute frequency, the iNAV-to-ETF coefficient is 0.378 and the ETF-to-iNAV coefficient is 0.089. Both are significant at the 1\% level, and the coefficient for iNAV-to-ETF is roughly four times larger. The results for natural gas are the most directionally balanced: the effects in both directions are significant and of similar magnitude (iNAV-to-ETF 0.079; ETF-to-iNAV 0.127). This contrast between the strongly unidirectional results for metals and the bidirectional results for energy supports our hypothesis H1. Indeed, the evidence confirms that physical versus futures-based arbitrage, settlement mechanism, and liquidity shape the transmission of volatility between the commodity underlying and the ETF.

These patterns are confirmed by the results for the formal tests described in Section~\ref{ch2:sec:methodolog}. The ETF-to-iNAV restriction $H_0:\alpha_1=0$ in equation~\eqref{ch2:eq:har_x_nav} is not rejected for gold or silver---their reverse coefficients are insignificant at every frequency---but is rejected at the 1\% level for crude oil and natural gas. The iNAV-to-ETF restriction $H_0:\delta_1=0$ in equation~\eqref{ch2:eq:har_x_etf} is rejected at the 1\% level for all four commodities. Transmission is therefore unidirectional (iNAV-to-ETF) for precious metals and bidirectional for energy. Moreover, the iNAV-to-ETF channel dominates for crude oil (with a magnitude of about four to one), while for natural gas the effects for the two directions are comparable.

The economic interpretation is consistent across models. In precious metals, arbitrage requires physical delivery against LBMA bullion, which is costly and slow. Authorized participants readily create or redeem ETF shares in response to underlying price moves, but cannot easily push ETF-specific shocks back into the tightly arbitraged spot market, so volatility flows essentially one way. The reverse coefficients for both metals hover near zero at all frequencies, confirming that ETF activity does not transmit volatility back to the spot market. In energy markets, by contrast, both the ETF and the underlying futures settle electronically and trade with comparable liquidity, so shocks propagate in both directions, even though fundamental supply-and-demand information still enters first through the futures-based iNAV.

Within precious metals, gold shows stronger unidirectional effects than silver at every frequency. The iNAV-to-ETF coefficients are 0.632 at a 1-minute and 0.387 at a 30-minute frequency for gold, versus 0.418 to 0.277 for silver. These results are consistent with the hypothesis that gold is a financial store of value, while silver carries additional industrial demand-related volatility. The reverse (ETF-to-iNAV) coefficients for both metals stay close to zero across frequencies: gold between $-0.015$ and 0.021, silver between $-0.034$ and 0.004, and neither is economically meaningful. In the category of energy commodities, crude oil shows a clear iNAV-to-ETF dominance across frequencies. However, the effects are more balanced for natural gas. For iNAV-to-ETF and ETF-to-iNAV, respectively, the coefficients are 0.105 and 0.126 at the 5-minute, and 0.081 and 0.074 at the 30-minute frequency. These results reflect the illiquidity, storage limits, and contango that impede arbitrage in the ETF and futures markets.

As the HAR-X specification includes cross-market terms only at the daily lag, directional transmission is identified at the daily horizon. The weekly and monthly coefficients measure each series' own persistence rather than spillovers. This own-persistence is high and, for precious metals, stable across horizons and frequencies: for instance, gold's weekly own-volatility coefficient stays near 0.33, confirming the presence of long memory in realized volatility.

\subsection{The Relevance of Using High-Frequency Data}

Our second hypothesis (H2) is that higher-frequency sampling reveals transmission that is obscured in daily data. Tables~\ref{ch2:tab:HAR_5min}, \ref{ch2:tab:HAR_1min}, and~\ref{ch2:tab:HAR_30min} report the HAR-X estimates at 5-, 1-, and 30-minute frequencies. The daily iNAV-to-ETF transmission is markedly larger at a finer sampling frequency. For crude oil, it is measured at 0.378 at a 1-minute, 0.311 at a 5-minute, and 0.20 at a 30-minute frequency. The 1-minute estimate represents a 22\% increase over the 5-minute estimate and nearly double the 30-minute estimate. Thus, much of the same-day arbitrage transmission occurs within minutes, and measurements of this mechanism would be understated at a coarser sampling, as is traditionally used in the literature.

The sensitivity of estimates according to frequency also varies by commodity. The iNAV-to-ETF transmission decreases substantially from 1-minute to 30-minute sampling in the case of crude oil (0.378 to 0.20, i.e., a 47\% drop), gold (0.632 to 0.387, or 39\% smaller), and silver (0.418 to 0.277, or 34\% smaller). It is, however, essentially flat for natural gas (0.079 versus 0.081). The economic interpretation is that transmission in the actively and continuously arbitraged crude oil and bullion markets clears within minutes, while for natural gas this adjusts over longer horizons because storage and pipeline constraints slow down the arbitrage activities that would otherwise help equalize ETF and underlying volatility.

The reverse channel, ETF-to-iNAV transmission, is empirically weaker. Moreover, unlike the iNAV-to-ETF channel, the estimated effects do not strengthen monotonically with the sampling frequency. For crude oil, the daily ETF-to-iNAV coefficient is 0.089 at a 1-minute, 0.111 at a 5-minute, and 0.103 at a 30-minute frequency, peaking at the intermediate horizon rather than rising with the frequency. The contrast between the strongly frequency-dependent iNAV-to-ETF channel and this flatter reverse channel reinforces the directional dominance finding which we describe under hypothesis H1.

Overall, we reject the null hypothesis of frequency-invariant transmission. A Wald test of $H_0:\delta_1^{(1\text{m})}=\delta_1^{(5\text{m})}=\delta_1^{(30\text{m})}$ on the daily iNAV-to-ETF coefficient rejects equality at the 1\% level for crude oil, gold, and silver; for natural gas, whose spillover is small and flat across frequencies, the difference is not significant. Daily data thus understate short-horizon transmission by up to roughly a factor of two for the most affected commodities, which would lead to incorrect conclusions about arbitrage effectiveness, addressing the sampling-frequency question raised by \citet{hansen2005realized}. For practitioners, this finding supports the relevance of high-frequency market monitoring to detect transmission patterns that are understated in daily analysis.

\subsection{Jump Components and Discontinuous Volatility Transmission}

Our third hypothesis (H3) concerns whether transmission is driven by the continuous (diffusion) component or by jumps. Tables~\ref{ch2:tab:HAR_CJ_5min}, \ref{ch2:tab:HAR_CJ_1min}, and~\ref{ch2:tab:HAR_CJ_30min} report the HAR-CJ-X model estimates. Across the four commodities, the daily cross-market jump coefficient dwarfs the continuous one: discrete price moves transmit volatility, while smooth price moves essentially do not. In the dominant iNAV-to-ETF  direction at a 1-minute frequency, the jump transmission (lagged NAV jump in the ETF equation) is large and positive---0.892 for gold, 0.578 for silver, 0.432 for crude oil, 0.09 for natural gas---while the corresponding continuous transmission is near zero or negative for every commodity (gold $-0.123$, silver $-0.065$, crude oil $-0.019$, natural gas $-0.002$). The effect is strongest for precious metals, where the jump component is the largest in the sample.

The same pattern holds in the reverse (ETF-to-iNAV) direction but with smaller economic magnitudes: the daily jump coefficient is 0.096 for crude oil and 0.128 for natural gas, while the continuous coefficients are near $-0.01$. Natural gas shows the weakest iNAV-to-ETF jump coefficient (0.090), which is consistent with natural gas volatility spikes arising from idiosyncratic, localized events---hurricanes, pipeline failures, extreme weather---that do not propagate systematically. In contrast, there is a larger value for the \emph{own} continuous effect: i.e., the weekly own-continuous coefficient reaches 0.116 at 1-minute frequency. This result suggests a separate, within-market phenomenon rather than a cross-market transmission pattern. Testing $H_0:\alpha_2=\alpha_1$ (daily jump versus continuous effect) against $H_a:\alpha_2>\alpha_1$, we find that the dominance of the jump component is confirmed at the 1\% level for all four commodities. This result supports H3: transmission mainly occurs through discrete jumps rather than continuous diffusion, which has implications for tail-risk hedging. Indeed, jump-driven volatility is harder to hedge with strategies that are usually built for the assumption of continuous price processes and volatility diffusions.

\subsection{Bayesian VAR Analysis and Stability Over Time}

Our fourth hypothesis (H4) concerns whether transmission is stable over time. The BVAR results shown in Tables~\ref{ch2:tab:VAR_USO}--\ref{ch2:tab:VAR_UNG} characterize the joint dynamics of ETF and iNAV volatility and, through sub-period estimation, the stability of the coefficient estimates. Since the BVAR treats the two volatilities as endogenous, it further provides a check on the HAR-X spillover coefficients without imposing a cascade structure. Across the four commodities, the results for the BVAR model confirm the asymmetries documented using the HAR-X model. Moreover, the cross-market coefficients remain stable across the 2010--2014, 2015--2019, and 2020--2023 sub-periods. Thus, we do not find evidence that the transmission mechanisms meaningfully change over time. The structural break test due to \citet{bai2003computation} indicates breaks in the volatility time series during the 2014 crude oil price collapse, the 2016 Brexit referendum, and the 2020 Covid-19 pandemic. However, reestimating the BVAR model for each regime period does not change the sign and relative magnitude of the cross-market coefficients. Therefore, we fail to reject the null hypothesis (H4) of a volatility transmission model that is stable over the full sample period.

For crude oil (Table~\ref{ch2:tab:VAR_USO}), both volatilities are strongly persistent (iNAV first-lag coefficient 0.5704, 95\% credible interval [0.5237, 0.6171]; second lag 0.2606 [0.2206, 0.3005]). The ETF has only a weak effect on the iNAV (first lag 0.0735 [0.0343, 0.1135]), whereas the iNAV strongly drives the ETF: its first-lag effect (0.2861 [0.2315, 0.3433]) is comparable to ETF self-persistence (0.2941 [0.2460, 0.3418]) and its second-lag effect remains economically large (0.1934 [0.1443, 0.2410]). That the iNAV's first-lag effect rivals the ETF's own self-persistence is striking: for crude oil, underlying volatility is about as important as the ETF's recent volatility in explaining today's ETF volatility---the BVAR counterpart of the strong iNAV-to-ETF spillover found in the HAR-X estimates.

Gold (Table~\ref{ch2:tab:VAR_GLD}) shows the most asymmetric configuration. The ETF has essentially no effect on the iNAV (first lag $-0.0101$ [$-0.0547$, 0.0336], with only a marginal second-lag effect of 0.0383 [0.0001, 0.0759]), while the iNAV dominates the ETF: its first-lag effect (0.3487 [0.2848, 0.4126]) exceeds ETF self-persistence (0.1549 [0.1004, 0.2083]) by more than twofold, with a large second lag (0.2758 [0.2221, 0.3307]). Silver (Table~\ref{ch2:tab:VAR_SLV}) is intermediate: iNAV self-persistence is the highest in the sample (first lag 0.6192 [0.5631, 0.6738]), the ETF-to-iNAV effect remains negligible ($-0.0446$ [$-0.0932$, 0.0051], turning weakly positive but still economically trivial at the second lag, 0.0584 [0.0176, 0.0996]), and the iNAV-to-ETF effect is strong (first lag 0.3760 [0.3102, 0.4404] versus ETF self-persistence 0.1790 [0.1226, 0.2362]) and remains economically large at the second lag (0.2016 [0.1471, 0.2563]). Natural gas (Table~\ref{ch2:tab:VAR_UNG}) shows the clearest bidirectionality. Its iNAV is moderately persistent (first lag 0.3919 [0.3488, 0.4366], second lag 0.2313 [0.1915, 0.2715]) and, unlike the other commodities, the ETF affects the iNAV through both lags, with a larger second-lag effect (0.0799 [0.0381, 0.1214] then 0.1567 [0.1189, 0.1938]), indicating that ETF activity feeds back to the underlying market through delayed channels tied to natural-gas storage operations. The forecast error variance decompositions reinforce this interpretation. Indeed, iNAV innovations explain a large and increasing share of ETF volatility forecast errors at longer horizons, while the share of iNAV forecast-error variance attributable to ETF innovations stays small for every commodity and is near zero for gold and silver. The impulse responses examined below confirm the same asymmetry.

\subsection{Graphical Evidence on Volatility Patterns and Dynamic Responses}

The realized volatility series shown in Figures~\ref{ch2:fig:rv_uso}--\ref{ch2:fig:rv_ung} corroborate the findings reported above. For crude oil (Figure~\ref{ch2:fig:rv_uso}), ETF and iNAV volatility co-move closely with synchronized peaks during stress periods, consistent with bidirectional transmission. For gold (Figure~\ref{ch2:fig:rv_gld}), iNAV volatility consistently precedes ETF volatility in periods of significant spikes, which is the visual counterpart of unidirectional iNAV-to-ETF transmission. Silver (Figure~\ref{ch2:fig:rv_slv}) shows the same pattern, but with occasional divergence, while natural gas (Figure~\ref{ch2:fig:rv_ung}) shows the most frequent ETF--iNAV divergences.

The impulse responses shown in Figures~\ref{ch2:fig:irf1}--\ref{ch2:fig:irf4} tell the same story but add evidence on how the shocks evolve over time. For crude oil (Figure~\ref{ch2:fig:irf1}), iNAV shocks produce large, persistent responses in ETF volatility while ETF shocks produce small, transitory responses in the iNAV. Gold (Figure~\ref{ch2:fig:irf2}) shows the most pronounced asymmetry, with negligible responses of the iNAV to ETF shocks. Silver (Figure~\ref{ch2:fig:irf3}) is similar but noisier, while natural gas (Figure~\ref{ch2:fig:irf4}) shows sizable responses in both directions with delayed peaks, confirming bidirectional transmission.

\subsection{Summary of the Evidence for the Hypotheses}

Taken together, the estimates from the HAR-X, HAR-CJ-X and BVAR models support our four research hypotheses, and the formal tests defined in Section~\ref{ch2:sec:methodolog} reject each null hypothesis. We can summarize as follows.

\textbf{H1.} We find that transmission is heterogeneous: the reverse (ETF-to-iNAV) coefficient is not significant for gold and silver but it is significant for crude oil and natural gas, while the forward (iNAV-to-ETF) coefficient is significant everywhere. Transmission is therefore unidirectional from iNAV to ETF for precious metals (a magnitude about four to one greater) and bidirectional for energy commodities (the coefficients are similar in magnitude). Thus, we reject the null hypothesis of uniform bidirectional transmission and we extend the evidence from \citet{gorton2006facts} to setting of commodity volatility. \textbf{H2.} Transmission is frequency-dependent: the daily iNAV-to-ETF spillover is up to roughly twice as large at the 1-minute as the 30-minute frequency. It is flat only for natural gas. These results confirm that daily data understate these volatility dynamics \citep{hansen2005realized, liu2015does}. \textbf{H3.} The daily cross-market jump component of transmission dwarfs the continuous part for all four commodities---most sharply for precious metals (i.e., the jump coefficient for gold iNAV-to-ETF jump  is 0.892 versus $-0.123$ for the continuous component)---so transmission occurs mainly through discrete jumps. \textbf{H4.} The cross-market coefficients are stable across sub-periods and are robust to astructural-break test, so the documented mechanisms hold in general and are not specific to sub-periods.

These findings build around a single mechanism, namely that the market structure and the arbitrage mechanism that links an ETF to its underlying asset---physical delivery for precious metals versus electronic futures settlement for energy---strongly shapes the characteristics of volatility transmission (e.g., direction, magnitude, horizon, and importance of jump vs diffusion parts), consistent with the arguments of \citet{petajisto2017inefficiencies} and \citet{basak2016model}. Finally, we explainthe implications for investors, market makers, and regulators in the conclusion.
