Make my paper more concise without losing quality:

Introduction 

This paper examines the dynamics between the United States 12-Month Oil Fund (USL) and futures oil prices (FUT) using time series econometric techniques. By focusing on volatility modeling and causal relationships, the study aims to provide insights into how oil price movements impact market conditions and policy decisions. Given oil’s central role in economic stability, market forecasting, and geopolitical strategies, such an analysis is valuable for traders, policymakers, and risk managers seeking to understand market behavior and improve price discovery processes. 

The study applies advanced econometric tools, including Vector Autoregression (VAR), Forecast Error Variance Decomposition (FEVD), and Generalized Autoregressive Conditional Heteroskedasticity (GARCH), to explore interdependencies, volatility clustering, and the dynamic responses of oil prices to market shocks. These methods enable a comprehensive analysis of the relationships between USL and FUT, contributing to both academic research and practical decision-making in volatile energy markets. 

Literature Review 

Key Studies on Oil Price Modeling 

Oil price modeling has been extensively studied due to its economic importance. Traditional approaches, such as ARIMA and VAR models, are widely used to forecast oil price movements and their linkages to macroeconomic variables. Structural models have also been employed to analyze long-run equilibria between oil prices and global economic indicators. 

VAR Models in Financial and Commodity Markets 

VAR models are well-established in financial and commodity markets for capturing interdependencies among variables. They are particularly useful for analyzing the dynamic relationships between different forms of oil prices (e.g., futures and ETFs like USL) and broader macroeconomic variables, such as exchange rates and inflation. 

Volatility Modeling Using GARCH 

GARCH models are standard for capturing volatility clustering in financial time series, a prominent feature in oil markets. These models highlight the persistence of volatility and the impact of external shocks, making them particularly relevant for analyzing oil price dynamics. 

Gaps in Literature 

While oil price forecasting and volatility modeling are well-researched, there is limited focus on the interplay between futures oil prices (FUT) and medium-term price proxies like USL, particularly through combined VAR and GARCH modeling. This paper addresses this gap by examining these relationships and their economic implications, providing a novel contribution to the field. 

Methods and Data 

Data Description and Transformation 

The dataset used for this analysis, OILNF_US0_NEW.xlsx, was sourced from www.investing.com and contains daily prices for USL (mix of futures oil contracts ETF) and FUT (futures oil prices) from December 6, 2007, to December 30, 2019, comprising 3,151 observations. To prepare the data for econometric analysis, several transformations were performed. First, the USL and FUT data were merged by date, resulting in a reduction to 3,148 observations. Next, the natural logarithm of both USL and FUT prices was computed to stabilize variance, creating a log-transformed dataset. Finally, the first differences of the log-transformed data were calculated to obtain returns, ensuring stationarity, and this transformation reduced the dataset by one additional observation. 

Visual inspection of the data provided insights into its characteristics. The raw and log-transformed data both exhibited substantial volatility and large variations, with significant spikes during periods of economic uncertainty such as the 2008 financial crisis. In contrast, the differenced log data, representing returns, appeared stationary, making it suitable for time series modeling. These transformations and initial analyses provided a robust foundation for the econometric methods applied in this study. 

Econometric Methods 

To analyze the dynamic relationships between USL and FUT prices, a series of econometric techniques were employed. First, stationarity was tested using the Dickey-Fuller tests on both the log-transformed and differenced log datasets. The results indicated that the log-transformed data were non-stationary, while the differenced log data were stationary, confirming their suitability for further modeling. 

A Vector Autoregression (VAR) model was then estimated to examine the interdependencies between USL and FUT returns. The optimal lag length was determined based on AIC, BIC, and HQ criteria, with six lags selected for the model. The VAR results revealed significant interrelationships between the two variables, capturing the dynamic interplay between futures and USL prices. Forecast Error Variance Decomposition (FEVD) was performed to decompose the forecast error variance for FUT and USL, quantifying the extent to which each variable’s variance was attributable to shocks in the other. This analysis highlighted the strong influence of FUT on USL. 

Impulse Response Functions (IRF) were computed to analyze the effects of shocks to FUT and USL over time. The results showed that shocks to FUT had a significant and immediate impact on USL, reinforcing the futures market's role in leading price discovery. Lastly, to model volatility dynamics, GARCH(1,1) models were applied to the residuals of the VAR model. These models captured persistent volatility and clustering effects in both FUT and USL, consistent with the well-known characteristics of financial time series. Together, these methods provided a comprehensive analysis of the relationships and volatility dynamics between USL and FUT prices. 

 

 

Empirical Results 

Raw Data Visualization 

 

The first graph shows the raw data for United States 12-Month Oil Fund LP (USL) and Future Oil Prices (FUT) from 2009 to 2020. The plots capture the general trends and volatility of oil prices over time, highlighting key economic events such as the sharp price drop during the 2008 financial crisis and subsequent recovery phases. The strong co-movement between the two series reflects their economic and financial linkage. FUT appears slightly more volatile than USL, potentially due to its sensitivity to market expectations about future oil supply and demand dynamics. 

 Log-Transformed Data 

A graph of a graph of a graph of a graph of a graph of a graph of a graph of a graph of a graph of a graph of a graph of a graph of a graph of

Description automatically generated 

The second graph presents the logarithmic transformation of the same series, which helps in stabilizing the variance of the data over time and interpreting the results in terms of percentage changes. By taking the logarithmic scale, the steep increases and decreases in the raw data appear smoother, making it easier to identify long-term trends and periodic fluctuations. The two series still demonstrate co-movement, reinforcing their relationship. 

First-Differenced Logarithmic Data 

A graph of a graph showing different types of sound waves

Description automatically generated with medium confidence 

The third graph illustrates the first differences of the logarithmic transformations of USL and FUT, effectively showing the daily changes in prices. These series appear stationary, which is a prerequisite for many time series models used in econometrics. Both USL and FUT show periods of heightened volatility, often corresponding to significant economic events, such as geopolitical tensions or market crises. The stationarity of the differenced series suggests they are suitable for econometric modeling such as VAR and GARCH analysis. 

The differenced logarithmic data appears stationary, as indicated by its mean-reverting patterns and periods of high volatility. This stationarity supports the application of a VAR model, while the evident volatility clustering makes a strong case for using a GARCH model to analyze volatility dynamics. Additionally, the co-movement observed between USL and FUT in both raw and log-transformed data can be explored further through Granger causality tests. 

 

 

 

 

 

 

 

Stationarity Analysis 

The results of the Augmented Dickey-Fuller (ADF) tests provide important insights into the stationarity of the data. For the log-transformed levels of FUT (Future Oil Prices) and USL (United States 12-Month Oil Fund LP), the ADF tests were conducted under three specifications: with a deterministic trend (Trend), with a constant but no trend (Drift), and without drift or trend (None). Across all specifications, the test statistics for both log_FUT and log_USL fail to reject the null hypothesis of a unit root. For example, under the trend specification, the test statistic for log_FUT is -2.1862, and for log_USL is -1.9289, both of which are higher than the 5% critical value (-3.41). These results confirm that the log-transformed levels are non-stationary, with trends likely being stochastic. 

In contrast, the ADF tests on the first-differenced log-transformed data, representing returns (diff_log_FUT and diff_log_USL), provide strong evidence of stationarity across all specifications. Under the trend specification, the test statistics are -42.6185 for diff_log_FUT and -40.1064 for diff_log_USL, both significantly below the critical threshold. This confirms that the differenced series (returns) are stationary and suitable for time series modeling. 

These findings have key implications. The non-stationarity in the log-transformed levels indicates that shocks to prices have persistent effects. Conversely, the stationarity in the first-differenced series suggests that returns are mean-reverting and appropriate for a GARCH model. Although stationarity in levels often justifies exploring cointegration, this analysis does not do so because it is already known that FUT and USL are cointegrated due to their shared underlying asset. Therefore, the focus of this study remains on understanding the short-term dynamics and volatility of the returns using stationary models. In summary, the levels of FUT and USL are integrated of order 1 (I(1)), while their returns are stationary, enabling the use of VAR and GARCH models to explore their relationships and volatility patterns. 

 

 

 

 

VAR Model 

The optimal lag length for the VAR model was determined using multiple criteria: The AIC selected 80 lags, favoring more complex models to minimize forecast error variance. The BIC selected 6 lags, favoring simplicity to avoid overfitting. The HQ selected 22 lags, balancing model complexity and fit. Given the trade-offs, the lag of 6 (selected by BIC) was chosen, aligning with economic intuition that shorter lags capture immediate interdependencies between FUT and USL while maintaining model parsimony. This lag length is appropriate for analyzing short-term dynamics without overcomplicating the model with excessive parameters, which could dilute statistical significance and economic interpretability. 

The VAR(6) model estimates show significant lagged relationships between the first differences of FUT and USL.For the diff_log_FUT equation, significant lagged terms of both FUT and USL suggest bi-directional influence. For example, a positive diff_log_USL at lag 1 significantly predicts a higher diff_log_FUT (t-value = 3.837, p < 0.001). This indicates that movements in USL, representing oil prices across 12 months, have a predictive impact on immediate future oil price changes. The negative coefficients for further lags suggest a partial mean-reverting effect, reflecting market corrections after initial shocks. 

For the diff_log_USL equation, the lagged terms of diff_log_FUT are significant at several lags (e.g., lag 3, p < 0.001), showing that shocks to futures oil prices propagate to longer-term trends captured by USL. The economic intuition behind this relationship is that futures prices incorporate expectations of oil supply and demand, which subsequently influence broader oil market trends as reflected in USL's ETF structure. 

The interdependencies observed align with financial theory: futures prices lead immediate changes in USL (a proxy for medium-term oil price movements), while adjustments in USL also influence expectations embedded in futures prices. 

Hypothesis Testing 

The Linear hypothesis tests reveal that the coefficients of lagged variables (up to 6 lags) are jointly significant for both FUT and USL. This reinforces the necessity of including lagged terms to adequately model dynamic interrelations. For FUT equation, the null hypothesis that all lagged terms are jointly zero is rejected (p<0.001), indicating that past values of both FUT and USL significantly drive current FUT returns. For USL equation, similarly, lagged terms are significant, suggesting predictive power of FUT on USL. 

These findings underscore the feedback loop between futures prices (FUT) and USL, where short-term oil price changes are influenced by both past expectations in FUT and the aggregated trends represented by USL. 

 

Residual Diagnostics and Stability 

The residual diagnostics for the VAR(6) model reveal some serial correlation in the residuals (Portmanteau test, p < 0.001), indicating potential room for refinement in capturing higher-order dependencies. However, the model is stable, as confirmed by the roots of the characteristic polynomial lying within the unit circle, ensuring its suitability for analyzing FUT and USL dynamics. 

Economically, the significant lagged relationships highlight that market participants rely on past trends in both futures markets (FUT) and the broader price movements reflected in USL. The model effectively captures how shocks in one market, such as unexpected futures price changes driven by geopolitical events, influence the other, demonstrating the integration of futures-based ETFs like USL and the underlying futures market. 

These findings are valuable for both policymakers and traders, emphasizing the importance of monitoring futures markets for broader price signals and accounting for feedback effects. The VAR(6) model provides a robust tool for understanding and forecasting oil price dynamics, aiding decision-making in volatile markets. 

 

Forecast Error Variance Decomposition (FEVD) 

The FEVD results from the two VAR models highlight the sensitivity of the decomposition to variable ordering, revealing interesting dynamics between the futures oil prices (diff_log_FUT) and the ETF prices (diff_log_USL). This demonstrates how the structure of the model and the ordering of variables can influence the interpretation of lead-lag relationships. 

VAR Model 1: diff_log_FUT Ordered First 

A comparison of a graph

Description automatically generated with medium confidence 

The diff_log_FUT (Futures Prices), the variance of diff_log_FUT is almost entirely explained by its own shocks over all forecast horizons, with contributions from diff_log_USL remaining below 2%. This indicates that futures price changes are primarily self-driven, reflecting the futures market's role as the main mechanism for incorporating information about future supply, demand, and geopolitical factors. While the diff_log_USL (ETF Prices), the variance of diff_log_USL is dominated by shocks from diff_log_FUT, with approximately 72% of its variance explained by FUT across all forecast horizons. This underscores the significant influence of futures prices on the ETF, as USL adjusts to incorporate information already priced into futures contracts. This dynamic reflects the role of futures in leading the price discovery process. 

Shape 

 

 

VAR Model 2: diff_log_USL Ordered First 

A comparison of different types of data

Description automatically generated with medium confidence 

The diff_log_USL (ETF Prices), the variance of diff_log_USL is almost entirely driven by its own shocks, with contributions from diff_log_FUT below 1%. This reflects the impact of variable ordering, where the model assumes that the ETF (USL) drives the dynamics. Economically, this might suggest that USL reacts primarily to medium-term market trends rather than directly influencing futures prices. The diff_log_FUT (Futures Prices), approximately 70% of the variance in diff_log_FUT is explained by shocks to diff_log_USL at all horizons. This contrasts with Model 1 and suggests that futures prices may respond to information embedded in the ETF (USL), particularly in cases where USL captures broader trends or unobservable factors that later influence futures pricing. 

 

Impulse Response Function (IRF) Results 

The impulse response function (IRF) analysis provides insights into how shocks to one variable, such as diff_log_FUT (futures prices) or diff_log_USL (ETF prices), affect the other over time. This analysis reveals the causal relationships and interdependencies within the oil market. 

 

A screenshot of a graph

Description automatically generated 

For diff_log_FUT, a positive shock results in a strong and immediate positive response within the same series, consistent with its autoregressive nature. The effect dissipates rapidly, suggesting that the impact of futures price shocks is short-lived, aligning with the efficient market hypothesis, where prices quickly incorporate new information. When diff_log_USL experiences a shock, the response in diff_log_FUT is smaller but immediate and positive. However, this effect diminishes quickly, indicating that while ETF price movements influence futures prices, the relationship is not as dominant as the reverse. This dynamic highlights the futures market's superior ability to incorporate forward-looking information compared to ETFs like USL, which reflect aggregated medium-term trends. 

For diff_log_USL, a positive shock to diff_log_FUT leads to a significant and immediate positive response, demonstrating the strong influence of futures prices on the ETF. This underscores the role of futures markets in price discovery, as ETF prices adjust to incorporate information embedded in futures prices. The response stabilizes after a few periods, indicating an efficient adjustment process. Conversely, a shock to diff_log_USL results in a strong initial response within its own series, reflecting the autoregressive nature of ETF prices. However, the effect decays rapidly, suggesting that shocks to USL prices have limited long-term impact, consistent with its focus on medium-term futures data. 

The IRF results confirm that futures prices significantly influence ETF prices, underscoring the role of futures markets in price discovery. Futures markets incorporate forward-looking information that drives ETF price adjustments, while ETF shocks have a weaker and shorter-lived impact on futures prices. The rapid dissipation of shocks in both series reflects market efficiency. 

These findings highlight the predictive power of futures markets as a leading indicator of oil market trends, offering critical insights for traders and policymakers in understanding and anticipating price movements. 

 

GARCH Model 

The GARCH(1,1) model was applied to the residuals of FUT (futures oil prices) and USL (a 12-month futures-based ETF) to capture time-varying volatility dynamics. The model effectively captures the volatility clustering phenomenon, where periods of high volatility are followed by more high volatility and periods of low volatility are followed by more low volatility. Significant spikes in volatility are observed for FUT during periods of market stress, such as the 2008 financial crisis and oil price shocks in 2014–2015, while more recent periods exhibit relative stability with occasional spikes. Similarly, the conditional variance of USL shows a pattern closely aligned with FUT, reflecting the interconnected nature of these markets. 

Conditional Variance from GARCH(1,1) Models 

A graph of a graph of a number of data

Description automatically generated with medium confidence 

Key GARCH parameters for both series indicate that volatility is highly persistent, with high values of alpha1 (short-term shock persistence) and beta1 (long-term volatility persistence). The sum of these parameters approaches 1, suggesting that volatility decays slowly over time and uncertainty persists during volatile periods. This persistence underscores the need for careful risk management and forecasting in these markets. 

 

Standardized Residuals from GARCH Models 

A group of black dots

Description automatically generated 

The standardized residuals for both FUT and USL are approximately centered around zero, with no discernible patterns, indicating that the GARCH(1,1) model successfully captures conditional heteroskedasticity in the data. However, occasional large deviations highlight the impact of extreme market events or outliers. Summary statistics for residuals further confirm that these deviations correspond to significant market disruptions that drive unusual price movements. 

From an economic perspective, these results validate the presence of volatility clustering in oil price returns, a common feature in financial markets. Futures markets, being forward-looking, absorb new information quickly, with USL aligning its volatility patterns as a reflection of medium-term trends. The persistent nature of volatility suggests that market participants adjust gradually to new information, leading to extended periods of uncertainty. This has practical implications for both policymakers and traders: policymakers can anticipate prolonged instability after major shocks, while traders can design hedging strategies to mitigate risks during such periods. 

In conclusion, the GARCH(1,1) model reveals critical insights into the volatility dynamics of FUT and USL, confirming their high persistence and clustering characteristics. These findings are essential for understanding the behavior of oil prices under uncertain market conditions and for developing strategies to manage associated risks effectively. 

Discussion and Conclusion 

Discussion 

The results demonstrate a strong interdependency between FUT and USL prices, with futures markets playing a dominant role in price discovery. Shocks to FUT consistently lead to immediate and significant impacts on USL, reflecting the forward-looking nature of futures prices. While the VAR model reveals bidirectional relationships, impulse response functions (IRFs) show that the influence of USL on FUT is weaker and short-lived, suggesting that USL, as an ETF tracking medium-term futures trends, primarily adjusts to changes in immediate futures prices. 

Stationarity testing confirmed that the log-transformed levels of FUT and USL are non-stationary, while their returns are stationary, aligning with financial theory about oil price behaviors. The VAR model, with an optimal lag length of six, effectively captured short-term dynamics and interdependencies without overfitting. GARCH(1,1) models revealed persistent volatility clustering in both FUT and USL returns, highlighting the sensitivity of oil markets to macroeconomic and geopolitical shocks. The high persistence of volatility underscores the likelihood of prolonged effects during periods of market stress. 

Futures markets’ dominance in explaining variance and influencing USL dynamics reinforces their role as leading indicators of market sentiment and supply-demand expectations. These findings have practical implications, particularly for policymakers monitoring futures prices for economic signals and for traders implementing hedging strategies during volatile periods. 

Conclusion 

This study confirms that FUT significantly drives USL, emphasizing the futures market's role in price discovery. VAR and IRF results reveal a strong dynamic relationship, where FUT shocks propagate efficiently to USL, while GARCH models capture persistent volatility clustering characteristic of financial markets under uncertainty. These findings contribute to understanding the interplay between immediate futures prices and ETFs representing medium-term trends, offering insights for market participants and policymakers. 

Future research could expand on these results by analyzing higher-frequency data or incorporating macroeconomic variables. Alternative models, such as structural VAR or regime-switching GARCH, may further clarify the complex dynamics of oil markets. Overall, this analysis highlights the critical role of futures markets in shaping ETF behavior and provides a foundation for more informed decision-making in trading and policy formulation.