Using these 3 rough drafts, combine them into the single best version:

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.