Using these 5 rough drafts, combine them into the single best version paper with paragraphs:

Paper 1:
Introduction

This paper examines the dynamics between the United States 12-Month Oil Fund (USL) and futures oil prices (FUT) using advanced time series econometric techniques. By focusing on volatility modeling and causal relationships, we aim to provide insights into how oil price movements impact market conditions and inform policy decisions. Given oil's central role in economic stability and geopolitical strategies, understanding these dynamics is valuable for traders, policymakers, and risk managers seeking to comprehend market behavior and enhance price discovery processes.

Literature Review

Oil price modeling has been extensively studied due to its significant economic implications. Traditional approaches, such as Autoregressive Integrated Moving Average (ARIMA) and Vector Autoregression (VAR) models, have been used to forecast oil price movements and explore their linkages to macroeconomic variables. Variants of the Generalized Autoregressive Conditional Heteroskedasticity (GARCH) model have effectively captured volatility clustering in financial time series, a prominent feature in oil markets.

Despite extensive research, limited focus exists 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

We utilized daily closing prices for USL (an ETF representing a mix of futures oil contracts) and FUT (futures oil prices) from December 6, 2007, to December 30, 2019, resulting in 3,148 observations after data preprocessing. To stabilize variance, we computed the natural logarithm of both USL and FUT prices. We then calculated the first differences of the log-transformed data to obtain returns, ensuring stationarity suitable for time series modeling.

Econometric Techniques

Stationarity Testing: Augmented Dickey-Fuller (ADF) tests confirmed that the log-transformed price series were non-stationary, while their first differences (returns) were stationary.
Vector Autoregression (VAR): A VAR model with six lags was estimated based on statistical criteria, capturing the dynamic interdependencies between USL and FUT returns.
Hypothesis Testing: We conducted Wald tests to assess the joint significance of lagged terms in the VAR model for both variables. The tests confirmed that lagged terms of both USL and FUT were jointly significant in explaining current returns of each other, indicating bidirectional influence.
Forecast Error Variance Decomposition (FEVD): FEVD was used to quantify the proportion of the forecast error variance of each variable that could be attributed to shocks from the other variable.
Impulse Response Functions (IRF): IRFs were generated to visualize the dynamic response of USL and FUT returns to shocks in each other over time.
GARCH Modeling: GARCH(1,1) models were applied to the residuals of the VAR model to capture volatility clustering and persistence in the returns.
Empirical Results

Stationarity Analysis: ADF tests indicated that FUT and USL return series are stationary, appropriate for VAR modeling.
VAR Model Findings:
Significant Lagged Relationships: The VAR(6) model revealed that both FUT and USL returns are significantly influenced by their own past values and the past values of the other variable.
Bidirectional Influence: Hypothesis tests confirmed that lagged terms of FUT are significant predictors of USL returns and vice versa, indicating a bidirectional relationship.
FEVD Results:
When FUT is ordered first, its forecast error variance is almost entirely explained by its own shocks, while USL's variance is significantly influenced by FUT shocks.
When USL is ordered first, its variance is largely driven by its own shocks, and FUT's variance is also influenced by USL shocks, albeit to a lesser extent.
IRF Analysis:
Impact of FUT Shocks on USL: A one-standard-deviation shock to FUT results in an immediate and significant positive response in USL returns, peaking at the first time lag and gradually dissipating over subsequent periods.
Impact of USL Shocks on FUT: A shock to USL has a smaller and less persistent effect on FUT returns, underscoring FUT's leading role in the price discovery process.
GARCH Model Results:
Volatility Clustering: GARCH(1,1) models captured significant volatility clustering in both FUT and USL return series.
Persistent Volatility: High values of the GARCH parameters indicate that shocks to volatility have a persistent effect over time.
Volatility During Crises: Significant spikes in conditional variance were observed during periods of market stress, such as the 2008 financial crisis.
Discussion

The empirical results demonstrate a strong interdependency between FUT and USL prices, with futures markets playing a dominant role in price discovery. The VAR model and hypothesis tests confirm bidirectional relationships, but the influence of FUT on USL is more pronounced. FEVD results highlight that shocks to FUT have a substantial impact on the variance of USL returns.

IRF analysis corroborates these findings by showing that FUT shocks lead to immediate and significant responses in USL, emphasizing the futures market's forward-looking nature. The weaker and short-lived impact of USL shocks on FUT suggests that the ETF market reacts to information embedded in futures prices rather than driving it.

The GARCH modeling reveals persistent volatility clustering in both markets, reflecting the oil market's sensitivity to macroeconomic and geopolitical events. The significant spikes during the 2008 financial crisis underscore the importance of accounting for volatility in risk management and policy formulation.

Conclusion

This study contributes to the understanding of the dynamic relationship between futures oil prices and an ETF representing medium-term oil price trends. The findings confirm that:

FUT significantly drives USL: The futures market leads in the price discovery process, with shocks to FUT having immediate and lasting effects on USL returns.
Bidirectional Dynamics Exist: While both markets influence each other, the impact from USL to FUT is weaker and less persistent.
Volatility is Highly Persistent: GARCH models indicate that volatility shocks have long-lasting effects, highlighting the need for effective risk management strategies.
These insights are valuable for traders seeking to develop arbitrage strategies, for policymakers aiming to understand market transmission mechanisms, and for risk managers focused on hedging against oil price volatility.

Future Research

Further research could explore these dynamics using higher-frequency intraday data to capture more granular market movements. Incorporating macroeconomic variables, such as interest rates or geopolitical risk indices, could provide a more comprehensive view of the factors influencing oil markets. Additionally, employing alternative econometric models, such as Vector Error Correction Models (VECM) for cointegrated series or regime-switching GARCH models, might uncover nonlinear dynamics and structural breaks in the relationship between FUT and USL.

Appendix: Econometric Output Summary

ADF Test Results:
Levels: Non-stationary for both log_FUT and log_USL.
First Differences: Stationary for both diff_log_FUT and diff_log_USL.
VAR Model Estimation (6 Lags):
diff_log_FUT Equation:
Significant coefficients for own lags and lags of diff_log_USL.
High F-statistic (p-value < 0.001), indicating model significance.
diff_log_USL Equation:
Significant coefficients for own lags and lags of diff_log_FUT.
Lower F-statistic compared to diff_log_FUT equation but still significant (p-value < 0.05).
FEVD:
For diff_log_FUT: Variance mostly explained by own shocks (~98%), minimal influence from diff_log_USL.
For diff_log_USL: Substantial portion of variance explained by shocks from diff_log_FUT (~28%).
IRF Results:
Shock to diff_log_FUT:
Immediate positive response in diff_log_USL.
Effect dissipates over time but remains significant for several periods.
Shock to diff_log_USL:
Smaller impact on diff_log_FUT.
Effect is less persistent.
GARCH(1,1) Model Parameters:
For FUT Returns:
ω (omega): 0.000001 (constant term in variance equation).
α1 (alpha1): 0.07 (lagged squared residuals).
β1 (beta1): 0.90 (lagged conditional variance).
For USL Returns:
ω (omega): 0.000000 (constant term in variance equation).
α1 (alpha1): 0.07 (lagged squared residuals).
β1 (beta1): 0.90 (lagged conditional variance).
Interpretation: High β1 values indicate strong persistence of volatility shocks.
Residual Diagnostics:
Ljung-Box Tests: Fail to reject the null hypothesis of no autocorrelation in standardized residuals, suggesting a good fit.

Paper 2: 
Empirical Results
Data Visualization
Figure 1 presents the time series plots of the United States 12-Month Oil Fund (USL) and futures oil prices (FUT) from December 6, 2007, to December 30, 2019. Both series exhibit significant volatility, particularly during periods of economic turbulence such as the 2008 financial crisis. The FUT series demonstrates higher volatility compared to USL, reflecting its sensitivity to immediate market expectations. The logarithmic transformation applied to both series stabilizes the variance, facilitating the interpretation of percentage changes in returns.

Figure 1: Time Series Plots of USL and FUT (2007-2019)

Stationarity Analysis
Augmented Dickey-Fuller (ADF) tests were conducted to assess the stationarity of the log-transformed and differenced log data. The results confirm that the log levels of both FUT and USL are non-stationary (p-values < 0.05), while their first differences (representing daily returns) are stationary (p-values > 0.05). This stationarity of returns aligns with financial theory, ensuring the suitability of the subsequent time series models.

VAR Model
A Vector Autoregression (VAR) model with six lags was estimated to explore the dynamic interdependencies between diff_log_Fut and diff_log_USL. The VAR(6) model revealed significant bidirectional relationships between the returns of FUT and USL. Specifically, lagged values of FUT returns significantly impact current USL returns and vice versa, although the magnitude and persistence of these effects differ.

Key Findings:

Lagged Effects: Lagged FUT returns (diff_log_Fut.l1 to diff_log_Fut.l6) significantly influence both FUT and USL returns, indicating lingering impacts of past shocks.
Interdependencies: Lagged USL returns also significantly affect FUT returns, though to a lesser extent compared to the influence of FUT on USL.
Model Fit: The model exhibits low R-squared values (0.02475 for FUT and 0.01117 for USL), suggesting that while significant, the lagged terms explain a modest portion of the variability in returns.
Forecast Error Variance Decomposition (FEVD)
FEVD analysis quantifies the proportion of the forecast error variance of each variable attributable to shocks in itself and the other variable over a 30-day horizon.

Findings:

FUT Variance: The variance of diff_log_Fut is predominantly explained by its own shocks, with minimal contribution from USL shocks.
USL Variance: A substantial portion of diff_log_USL's variance is driven by FUT shocks, underscoring the dominant role of FUT in influencing USL returns.
Impulse Response Functions (IRF)
IRF analysis was conducted to trace the effect of a one-standard-deviation shock to one variable on the future values of both variables.

Key Observations:

Shock to FUT: A shock to FUT returns induces a significant and immediate positive response in USL returns, which diminishes over time.
Shock to USL: Conversely, a shock to USL has a smaller and transient impact on FUT returns, highlighting the futures market's leading role in price discovery.
Figure 2: Impulse Response Functions

Volatility Modeling Using GARCH
Generalized Autoregressive Conditional Heteroskedasticity (GARCH)(1,1) models were fitted to capture volatility clustering in the residuals of both FUT and USL returns derived from the VAR model.

Results for FUT:

Parameter Estimates: The GARCH model for FUT returns indicates significant persistence in volatility with α₁ = 0.069975 and β₁ = 0.897534 (both p-values < 0.001). This suggests that past shocks and volatility have a strong influence on current volatility levels.
Model Diagnostics: The Ljung-Box tests on standardized residuals and squared residuals indicate no remaining autocorrelation or ARCH effects, affirming the adequacy of the GARCH model.
Results for USL:

Parameter Estimates: Similarly, the GARCH model for USL returns shows significant persistence with α₁ = 0.065192 and β₁ = 0.904753 (both p-values < 0.001).
Model Diagnostics: Diagnostic tests corroborate the absence of serial correlation and ARCH effects in the standardized residuals, validating the model fit.
Figure 3: GARCH(1,1) Conditional Variance of FUT and USL

Residual Analysis
Residuals from the VAR model were examined to ensure model adequacy. The Portmanteau test yielded a Chi-squared statistic of 213.34 (p-value < 2.2e-16), indicating significant residual autocorrelation. However, the standardized residuals from the GARCH models do not exhibit significant autocorrelation, as confirmed by the Ljung-Box tests (p-values > 0.05).

Summary of Key Results
Bidirectional Influence: The VAR(6) model uncovers bidirectional influence between FUT and USL returns, with FUT exerting a more substantial and persistent effect on USL.
Dominant Price Discovery: FEVD and IRF analyses collectively emphasize the dominance of the futures market (FUT) in driving medium-term price proxies like USL.
Persistent Volatility: GARCH(1,1) models demonstrate persistent volatility clustering in both FUT and USL returns, underscoring the markets' sensitivity to economic and geopolitical shocks.
Model Adequacy: While initial VAR residuals exhibit autocorrelation, the incorporation of GARCH models effectively captures volatility dynamics, ensuring robust model specifications.
These empirical findings provide valuable insights into the intricate dynamics between immediate futures prices and medium-term price proxies, reinforcing the pivotal role of futures markets in oil price discovery and volatility transmission.

Paper 3:
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. Focusing on volatility modeling and causal relationships, we aim to provide insights into how oil price movements impact market conditions and policy decisions. Given oil's central role in economic stability and geopolitical strategies, this analysis is valuable for traders, policymakers, and risk managers seeking to understand market behavior and improve price discovery processes. 

We apply 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, forecast oil price movements and their linkages to macroeconomic variables. Structural models analyze long-run equilibria between oil prices and global economic indicators. 

VAR Models in Financial and Commodity Markets 

VAR models are well-established in capturing interdependencies among financial and commodity market variables. They are particularly useful for analyzing 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 capture 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 relevant for analyzing oil price dynamics. 

Gaps in Literature 

While oil price forecasting and volatility modeling are well-researched, limited focus exists 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 

We used daily prices for USL (a mix of futures oil contracts ETF) and FUT (futures oil prices) from December 6, 2007, to December 30, 2019, sourced from www.investing.com. After merging by date, the dataset contains 3,148 observations. We computed the natural logarithm of both USL and FUT prices to stabilize variance and then calculated first differences of the log-transformed data to obtain returns, ensuring stationarity. 

Visual inspection revealed substantial volatility and significant spikes during periods of economic uncertainty, such as the 2008 financial crisis. The differenced log data, representing returns, appeared stationary and suitable for time series modeling. 

Econometric Methods 

To analyze the dynamic relationships between USL and FUT prices, we employed several econometric techniques.  

 

Stationarity was tested using Dickey-Fuller tests, confirming that the log-transformed data were non-stationary, while the differenced log data were stationary. 

 

A VAR model was estimated to examine interdependencies between USL and FUT returns, with six lags selected based on statistical criteria. The VAR results revealed significant interrelationships between the two variables.  

 

The Hypothesis Testing was conducted Wald tests to assess the joint significance of lagged terms in the VAR model for both variables. The tests confirmed that lagged terms of both USL and FUT were jointly significant in explaining current returns of each other, indicating bidirectional influence. 

 

 

FEVD quantified the extent to which each variable's variance was attributable to shocks in the other, highlighting the strong influence of FUT on USL. 

 

Impulse Response Functions (IRF) analyzed the effects of shocks to FUT and USL over time, showing that shocks to FUT had a significant and immediate impact on USL, reinforcing the futures market's role in leading price discovery.  

 

Lastly, GARCH(1,1) models were applied to capture persistent volatility and clustering effects in both FUT and USL. 

 

 

Empirical Results 

 

Data Visualization 

The raw data for USL and FUT from 2009 to 2020 show overall trends and volatility of oil prices, highlighting key economic events such as the 2008 financial crisis. The strong co-movement between the series reflects their economic linkage. FUT appears slightly more volatile than USL due to its sensitivity to market expectations. 

The logarithmic transformation stabilizes variance and allows interpreting results in terms of percentage changes. The first differences of the logarithmic data, representing daily returns, appear stationary and are suitable for econometric modeling like VAR and GARCH. 

 

 

Stationarity Analysis 

Augmented Dickey-Fuller (ADF) tests indicate that the log-transformed levels of FUT and USL are non-stationary, while the first-differenced log data (returns) are stationary. This confirms that returns are mean-reverting and appropriate for time series modeling. 

 

VAR Model 

The optimal lag length for the VAR model was determined to be six, capturing immediate interdependencies between FUT and USL. The VAR(6) model shows significant lagged relationships between the returns of FUT and USL, indicating bi-directional influence. 

Hypothesis tests confirm that the coefficients of lagged variables are jointly significant for both FUT and USL, reinforcing the necessity of including lagged terms to adequately model dynamic interrelations. 

Residual diagnostics confirm that the model is stable, with roots of the characteristic polynomial inside the unit circle. 

 

Forecast Error Variance Decomposition (FEVD) 

FEVD results highlight the dynamics between FUT and USL. When diff_log_FUT is ordered first, its variance is almost entirely explained by its own shocks, while diff_log_USL's variance is largely explained by shocks from diff_log_FUT. Conversely, when diff_log_USL is ordered first, its variance is driven by its own shocks, and a significant portion of diff_log_FUT's variance is explained by shocks to diff_log_USL. 

 

Impulse Response Function (IRF) Results 

IRF analysis shows that shocks to FUT result in a strong and immediate positive response in USL, demonstrating the significant influence of futures prices on the ETF. Shocks to USL have a smaller and short-lived impact on FUT, highlighting the futures market's superior ability to incorporate forward-looking information. 

 

 

GARCH Model 

The GARCH(1,1) model captures volatility clustering in the residuals of FUT and USL. Significant spikes in volatility occur during periods of market stress, such as the 2008 financial crisis. High values of the GARCH parameters indicate that volatility is highly persistent. 

Standardized residuals are centered around zero, indicating that the GARCH(1,1) model effectively captures conditional heteroskedasticity in the data. 

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 lead to immediate and significant impacts on USL, reflecting the forward-looking nature of futures prices. While the VAR model reveals bidirectional relationships, IRFs show that the influence of USL on FUT is weaker and short-lived. 

 

Stationarity testing confirmed that returns are stationary, aligning with financial theory. 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. 

 

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. 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. 

Paper 4:

Title: 
Dynamic Interactions Between the United States 12-Month Oil Fund (USL) and Futures Oil Prices (FUT): A Time Series Econometric Analysis 

 

Abstract 

This study investigates the dynamic relationship between the United States 12-Month Oil Fund (USL) and futures oil prices (FUT) utilizing advanced time series econometric techniques. By employing Vector Autoregression (VAR), Forecast Error Variance Decomposition (FEVD), and Generalized Autoregressive Conditional Heteroskedasticity (GARCH) models, we explore the bidirectional influences and volatility clustering inherent in these markets. The findings reveal that FUT plays a dominant role in price discovery, significantly impacting USL returns, while the reverse influence is weaker and more transient. Additionally, persistent volatility clustering underscores the oil market's sensitivity to economic and geopolitical shocks. These insights are valuable for traders, policymakers, and risk managers aiming to understand and navigate the complexities of oil price dynamics. 

 

1. Introduction 

Oil holds a pivotal position in global economic stability and geopolitical strategies, making the understanding of oil price dynamics crucial for various market participants. This paper examines the interplay between the United States 12-Month Oil Fund (USL) and futures oil prices (FUT) using sophisticated time series econometric techniques. By focusing on volatility modeling and causal relationships, we aim to elucidate how fluctuations in oil prices influence market conditions and inform policy decisions. The insights derived from this analysis are intended to aid traders, policymakers, and risk managers in comprehending market behavior and enhancing price discovery processes. 

 

2. Literature Review 

2.1. Oil Price Modeling 

Oil price modeling has garnered extensive attention due to its profound economic implications. Traditional methodologies, such as Autoregressive Integrated Moving Average (ARIMA) and Vector Autoregression (VAR) models, have been employed to forecast oil price movements and investigate their connections with macroeconomic variables. Additionally, structural models have been utilized to analyze long-term equilibria between oil prices and global economic indicators. 

2.2. VAR Models in Financial and Commodity Markets 

VAR models are instrumental in capturing interdependencies among financial and commodity market variables. They are particularly effective in analyzing dynamic relationships between different forms of oil prices, such as futures (FUT) and exchange-traded funds (ETFs) like USL, alongside broader macroeconomic variables including exchange rates and inflation. 

2.3. Volatility Modeling Using GARCH 

The Generalized Autoregressive Conditional Heteroskedasticity (GARCH) model is adept at capturing volatility clustering—a prevalent feature in financial time series. GARCH models elucidate the persistence of volatility and the influence of external shocks, making them indispensable for analyzing oil price dynamics. 

2.4. Gaps in Literature 

Despite the extensive research on oil price forecasting and volatility modeling, there is a paucity of studies focusing on the interplay between futures oil prices (FUT) and medium-term price proxies like USL, especially through the combined application of VAR and GARCH models. This paper addresses this lacuna by examining these relationships and their economic implications, thereby contributing novel insights to the field. 

 

3. Methods and Data 

3.1. Data Description and Transformation 

The analysis employs daily closing prices for USL (an ETF representing a mix of futures oil contracts) and FUT (futures oil prices) covering the period from December 6, 2007, to December 30, 2019. The dataset comprises 3,148 observations post data preprocessing, sourced from www.investing.com. 

To stabilize variance, the natural logarithm of both USL and FUT prices was computed. Subsequently, the first differences of the log-transformed data were calculated to obtain returns, ensuring stationarity suitable for time series modeling. 

3.2. Econometric Techniques 

Stationarity Testing: Augmented Dickey-Fuller (ADF) tests were conducted to assess the stationarity of the log-transformed and differenced log data. The tests confirmed that the log levels of both FUT and USL are non-stationary (p-values < 0.05), whereas their first differences (returns) are stationary (p-values > 0.05), aligning with financial theories. 

Vector Autoregression (VAR): A VAR model with six lags was estimated based on statistical criteria, capturing the dynamic interdependencies between USL and FUT returns. The optimal lag length was determined to balance model complexity and fit. 

Hypothesis Testing: Wald tests were performed to evaluate the joint significance of lagged terms in the VAR model for both variables. These tests assess whether the lagged values of one variable significantly predict the current values of the other. 

Forecast Error Variance Decomposition (FEVD): FEVD was utilized to quantify the proportion of the forecast error variance of each variable attributable to shocks from the other variable over a 30-day horizon. 

Impulse Response Functions (IRF): IRFs were generated to visualize the dynamic response of USL and FUT returns to shocks in each other over time, providing insights into the propagation of shocks through the system. 

GARCH Modeling: GARCH(1,1) models were applied to the residuals of the VAR model to capture volatility clustering and persistence in the returns, essential for understanding the volatility dynamics within the markets. 

 

4. Empirical Results 

4.1. Data Visualization 

Figure 1 depicts the time series plots of USL and FUT from December 6, 2007, to December 30, 2019. Both series exhibit substantial volatility, particularly during periods of economic turbulence such as the 2008 financial crisis. The FUT series demonstrates higher volatility compared to USL, reflecting its sensitivity to immediate market expectations. The logarithmic transformation stabilizes variance, facilitating the interpretation of percentage changes in returns. 

Figure 1: Time Series Plots of USL and FUT (2007-2019) 

4.2. Stationarity Analysis 

Augmented Dickey-Fuller (ADF) tests confirm that the log levels of FUT and USL are non-stationary (p-values < 0.05), while their first differences (representing daily returns) are stationary (p-values > 0.05). This stationarity of returns aligns with financial theory, ensuring the suitability of the subsequent time series models. 

4.3. VAR Model 

A Vector Autoregression (VAR) model with six lags was estimated to explore the dynamic interdependencies between USL and FUT returns. The VAR(6) model revealed significant bidirectional relationships between the returns of FUT and USL, indicating that lagged values of FUT significantly impact current USL returns and vice versa. 

Lagged Effects: Lagged FUT returns (diff_log_FUT.l1 to diff_log_FUT.l6) significantly influence both FUT and USL returns, indicating lingering impacts of past shocks. 

Interdependencies: Lagged USL returns also significantly affect FUT returns, though to a lesser extent compared to the influence of FUT on USL. 

Model Fit: The model exhibits modest R-squared values (0.02475 for FUT and 0.01117 for USL), suggesting that while significant, the lagged terms explain a limited portion of the variability in returns. 

4.4. Forecast Error Variance Decomposition (FEVD) 

FEVD analysis quantifies the proportion of the forecast error variance of each variable attributable to shocks in itself and the other variable over a 30-day horizon. 

FUT Variance: The variance of diff_log_FUT is predominantly explained by its own shocks (~98%), with minimal contribution from USL shocks. 

USL Variance: A substantial portion of diff_log_USL's variance (~28%) is driven by FUT shocks, underscoring the dominant role of FUT in influencing USL returns. 

4.5. Impulse Response Functions (IRF) 

IRF analysis traces the effect of a one-standard-deviation shock to one variable on the future values of both variables. 

Shock to FUT: A shock to FUT returns induces a significant and immediate positive response in USL returns, which diminishes over time but remains significant for several periods. 

Shock to USL: Conversely, a shock to USL has a smaller and transient impact on FUT returns, highlighting the futures market's superior role in price discovery. 

Figure 2: Impulse Response Functions 

4.6. Volatility Modeling Using GARCH 

GARCH(1,1) models were fitted to capture volatility clustering in the residuals of both FUT and USL returns derived from the VAR model. 

Results for FUT: 

Parameter Estimates: ω = 0.000001, α₁ = 0.07, β₁ = 0.90 (all p-values < 0.001), indicating significant persistence in volatility. 

Model Diagnostics: Ljung-Box tests on standardized residuals and squared residuals indicate no remaining autocorrelation or ARCH effects, affirming model adequacy. 

Results for USL: 

Parameter Estimates: ω = 0.000000, α₁ = 0.07, β₁ = 0.90 (all p-values < 0.001), similarly indicating persistent volatility. 

Model Diagnostics: Diagnostic tests corroborate the absence of serial correlation and ARCH effects in the standardized residuals. 

Figure 3: GARCH(1,1) Conditional Variance of FUT and USL 

4.7. Residual Analysis 

Residuals from the VAR model exhibited significant autocorrelation, evidenced by a Portmanteau test yielding a Chi-squared statistic of 213.34 (p-value < 2.2e-16). However, the standardized residuals from the GARCH models did not show significant autocorrelation, as confirmed by the Ljung-Box tests (p-values > 0.05), indicating that the GARCH models effectively captured the volatility dynamics. 

 

5. Discussion 

The empirical findings illustrate a robust interdependency between FUT and USL prices, with the futures market (FUT) playing a dominant role in price discovery. The VAR model and subsequent hypothesis tests confirm bidirectional relationships; however, the influence of FUT on USL is significantly more pronounced. 

FEVD results reinforce this dominance, showing that a substantial portion of USL's forecast error variance is attributable to FUT shocks. The IRF analysis further corroborates these findings by demonstrating that shocks to FUT elicit immediate and significant responses in USL returns, highlighting the futures market's forward-looking nature. In contrast, shocks to USL exert a weaker and more transient effect on FUT returns, suggesting that the ETF market primarily reacts to information embedded in futures prices rather than driving it. 

The GARCH modeling reveals persistent volatility clustering in both markets, reflecting the oil market's inherent sensitivity to macroeconomic and geopolitical events. Notably, significant spikes in conditional variance were observed during periods of market stress, such as the 2008 financial crisis, underscoring the importance of volatility modeling in risk management and policy formulation. 

 

6. Conclusion 

This study enhances the understanding of the dynamic relationship between futures oil prices (FUT) and an ETF representing medium-term oil price trends (USL). The key findings are as follows: 

Dominant Price Discovery by FUT: The futures market (FUT) significantly drives USL, with shocks to FUT having immediate and lasting effects on USL returns. 

Bidirectional Influence: While both markets influence each other, the impact from USL to FUT is weaker and short-lived compared to the influence of FUT on USL. 

Persistent Volatility Clustering: GARCH models indicate that volatility shocks have enduring effects, highlighting the necessity for effective risk management strategies in the face of persistent volatility. 

These insights are invaluable for traders developing arbitrage strategies, policymakers seeking to understand market transmission mechanisms, and risk managers focused on hedging against oil price volatility. 

 

7. Future Research 

Future studies could expand on these results by analyzing higher-frequency intraday data to capture more granular market movements. Incorporating macroeconomic variables, such as interest rates or geopolitical risk indices, could provide a more comprehensive understanding of the factors influencing oil markets. Additionally, employing alternative econometric models, such as Vector Error Correction Models (VECM) for cointegrated series or regime-switching GARCH models, may uncover nonlinear dynamics and structural breaks in the relationship between FUT and USL. 

 

 

 

 

 

 

 

Appendix: Econometric Output Summary 

ADF Test Results: 

Levels: Non-stationary for both log_FUT and log_USL. 

First Differences: Stationary for both diff_log_FUT and diff_log_USL. 

VAR Model Estimation (6 Lags): 

diff_log_FUT Equation: 

Significant coefficients for own lags and lags of diff_log_USL. 

High F-statistic (p-value < 0.001), indicating model significance. 

diff_log_USL Equation: 

Significant coefficients for own lags and lags of diff_log_FUT. 

Lower F-statistic compared to diff_log_FUT equation but still significant (p-value < 0.05). 

FEVD: 

diff_log_FUT: Variance mostly explained by its own shocks (~98%), minimal influence from diff_log_USL. 

diff_log_USL: Substantial portion of variance explained by shocks from diff_log_FUT (~28%). 

IRF Results: 

Shock to diff_log_FUT: 

Immediate positive response in diff_log_USL. 

Effect dissipates over time but remains significant for several periods. 

Shock to diff_log_USL: 

Smaller impact on diff_log_FUT. 

Effect is less persistent. 

GARCH(1,1) Model Parameters: 

FUT Returns: 

ω (omega): 0.000001 

α₁ (alpha1): 0.07 

β₁ (beta1): 0.90 

USL Returns: 

ω (omega): 0.000000 

α₁ (alpha1): 0.07 

β₁ (beta1): 0.90 

Interpretation: High β₁ values indicate strong persistence of volatility shocks. 

Residual Diagnostics: 

Ljung-Box Tests: Fail to reject the null hypothesis of no autocorrelation in standardized residuals, suggesting a good fit. 

 
Paper 5:

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.