How is the rough draft of  my second report should consist of an application of the time series econometrics learnt in Modules 4-5 to your data. Full submission should consist of a document containing the introduction, literature review, methods and data and preliminary results; the file containing your data and your computer codes. Please present the results (tables, plots, etc. in your report).

Outline for Time Series Econometrics Report
1. Introduction
Objective
This paper aims to analyze the dynamics between the United States 12-Month Oil Fund (USL) and future oil prices (FUT) using time series econometrics. The study focuses on volatility modeling and examining causal relationships between these variables to gain insights into how oil price movements affect market conditions and policy decisions.
Context
Understanding oil price dynamics is crucial for a wide range of stakeholders, from traders to policymakers. Oil prices directly influence economic stability, market forecasting, and geopolitical decision-making. Therefore, analyzing these prices using econometric tools can provide valuable insights into future market behavior, risk management, and price discovery processes.
Overview
The paper employs several time series econometric techniques to model the relationships between USL and FUT, including the Vector Autoregression (VAR) model, Forecast Error Variance Decomposition (FEVD), and Generalized Autoregressive Conditional Heteroskedasticity (GARCH) model. These techniques allow for a deeper understanding of the interdependencies, volatility clustering, and dynamic responses of oil prices to market shocks.
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2. Literature Review
Key Studies on Oil Price Modeling
Many studies have focused on modeling oil prices due to their fundamental importance in the global economy. Oil price modeling has traditionally utilized autoregressive models, such as ARIMA and VAR, to forecast oil price dynamics and their linkages to other economic variables. Researchers have also used structural models to capture the long-run equilibrium between oil prices and global economic indicators.
VAR Models in Financial and Commodity Markets
The use of VAR models in financial markets is well-established. These models are instrumental in capturing the interdependencies between multiple variables, including commodities like oil. VAR models help analyze the dynamic relationships between different oil price forms (spot, future) and other macroeconomic variables, such as exchange rates and inflation.
Volatility Modeling Using GARCH
GARCH models have become the standard for capturing volatility clustering in financial time series. They account for time-varying volatility, which is a common feature of commodity markets, especially oil. Many studies have applied GARCH models to oil price returns, showing the persistence of volatility and the sensitivity of oil prices to external shocks.
Gaps in Literature
While substantial research has been conducted on oil price forecasting and volatility modeling, fewer studies focus on the interplay between spot oil prices (like USL) and futures prices (FUT), especially using the combination of VAR and GARCH models. This paper aims to fill this gap by analyzing these relationships and their economic implications.
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3. Methods and Data
3.1 Data Description
•	Data Source: The dataset used in this analysis is from OILNF_US0_NEW.xlsx, containing daily prices for USL (spot oil ETF) and FUT (futures oil prices) from 2009 to 2020.
•	Frequency and Time Period: The dataset is daily, covering a time period of approximately 11 years.
•	Variables:
o	Raw Data: The raw price series for FUT and USL.
o	Log-Transformed Data: The logarithmic transformation of FUT and USL to stabilize the variance.
o	Differenced Data: The first difference of the log-transformed series to obtain returns.
3.2 Preliminary Data Exploration
•	Summary Statistics:
o	Raw and transformed data were analyzed, and summary statistics showed volatility and large variations in both series, with significant spikes during periods of economic uncertainty (e.g., 2008 financial crisis).
•	Visualizations:
1.	Raw Data Plot: The raw data for FUT and USL revealed their general trends and key economic events.
2.	Log-Transformed Data Plot: The log-transformed data helped smooth volatility and highlighted longer-term trends.
3.	Differenced Data Plot: First-differenced data indicated stationarity, confirming their suitability for econometric modeling.
3.3 Econometric Methods
•	Stationarity Testing:
o	Dickey-Fuller tests were performed on the raw and differenced series.
o	Results: The log-transformed data for both FUT and USL were non-stationary, but the differenced series (returns) were stationary.
•	VAR Modeling:
o	Lag Selection: Optimal lag length was selected based on AIC, BIC, and HQ criteria. A lag of 6 was chosen for the VAR model.
o	Model Estimation: The VAR model revealed significant interdependencies between FUT and USL returns.
•	Impulse Response Functions (IRF):
o	IRFs were computed to understand how shocks to FUT and USL affect each other over time. The results showed a significant and immediate impact of FUT shocks on USL, reinforcing the role of futures in leading price discovery.
•	Forecast Error Variance Decomposition (FEVD):
o	FEVD decomposed the forecast error variance for FUT and USL, showing how much of each variable's variance is explained by the shocks to the other.
•	Volatility Modeling with GARCH:
o	GARCH(1,1) models were applied to residuals from the VAR model, showing persistent volatility and clustering effects in both FUT and USL.
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4. Results
4.1 Data Visualization
•	Include time series plots for:
o	Original data (FUT and USL).









 

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
 
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
 
The third graph illustrates the first differences of the logarithmic transformations of USL and FUT, effectively showing the daily or periodic percentage changes in prices (returns). 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 analysis.
Econometric Implications
These visualizations set the stage for applying advanced time series econometric techniques:
1.	Stationarity and Differencing: The differenced logarithmic series (returns) are stationary, as suggested by their mean-reverting patterns.
2.	Volatility Analysis: The clear periods of high volatility in the differenced data could be explored further using GARCH models to understand volatility clustering.
3.	Co-movement and Causality: The co-movement between USL and FUT in both raw and log-transformed data can be analyzed using Granger causality tests or cointegration analysis.
4.	Forecasting: With the stabilized log-transformed series, predictive models such as ARIMA or VAR could be applied to forecast future trends in oil prices.



4.2 Stationarity Analysis
•	Summarize Dickey-Fuller test results for raw and differenced series.
•	Highlight implications for model choice (e.g., need for differencing).
Interpretation of Augmented Dickey-Fuller (ADF) Test Results
Log-Transformed Levels of FUT and USL
The ADF tests on the log-transformed levels of FUT (Future Oil Prices) and USL (United States 12-Month Oil Fund LP) were performed under three specifications:
1.	Trend: Tests with a deterministic trend.
2.	Drift: Tests with a constant but no trend.
3.	None: Tests without drift or trend.
For both log_FUT and log_USL, the test statistics across all specifications 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 higher than the 5% critical value (-3.41). This indicates that the log-transformed levels are non-stationary, with trends that are likely stochastic.
First-Differenced Log-Transformed Data
The ADF tests on first-differenced log-transformed data (diff_log_FUT and diff_log_USL), representing returns, show strong evidence of stationarity across all specifications:
•	In the trend specification, test statistics are -42.6185 for diff_log_FUT and -40.1064 for diff_log_USL, both well below the critical threshold, confirming that the returns are stationary.
Key Implications
•	Non-Stationarity in Levels: The log-transformed levels are non-stationary, meaning shocks to prices have persistent effects.
•	Stationarity in Returns: The first differences (returns) are stationary, suitable for time series models like ARIMA or GARCH.
In summary, FUT and USL levels are integrated of order 1 (I(1)I(1)I(1)), while their returns are stationary. This allows us to use stationary modeling on returns and explore cointegration in levels to analyze potential long-term relationships.
4.3 VAR Model Results
•	Present the summary statistics for the fitted VAR models.
•	Include hypothesis test results on coefficients.
•	Plot and interpret IRFs showing responses of FUT and USL to shocks.
•	Include FEVD plots and discuss the proportion of variance explained by each variable. 
Interpretation of VAR Model Results with Economic Intuition
Optimal Lag Selection
The optimal lag length for the VAR model was determined using multiple criteria:
•	AIC selected 80 lags, favoring more complex models to minimize forecast error variance.
•	BIC selected 6 lags, favoring simplicity to avoid overfitting.
•	HQ selected 22 lags, balancing between 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.
VAR Model Estimation Results
The VAR(6) model estimates show significant lagged relationships between the first differences of FUT and USL (returns):
•	For the diff_log_FUT equation:
o	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.
o	The negative coefficients for further lags suggest a partial mean-reverting effect, reflecting market corrections after initial shocks.
•	For the diff_log_USL equation:
o	Lagged terms of diff_log_FUT are significant at several lags (e.g., lag 3, p<0.001p < 0.001p<0.001), showing that shocks to future oil prices propagate to longer-term oil price structures.
o	The economic intuition behind this relationship is that futures prices often incorporate expectations of oil supply/demand, which subsequently influence the broader oil market (captured by USL).
The interdependencies observed align with financial theory: futures prices lead immediate changes in the spot price (via USL), while spot market adjustments also influence expectations embedded in futures.
Hypothesis Testing
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<0.001p < 0.001p<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 and spot prices, where short-term oil price changes are influenced by both past expectations (via FUT) and market conditions (via USL).
Residual Diagnostics and Stability
The residual diagnostics indicate:
•	Serial Correlation: The Portmanteau test (p<0.001p < 0.001p<0.001) shows some serial correlation in the residuals, suggesting potential room for refinement in the model. This could imply that certain dynamics or higher-order dependencies are not fully captured.
•	Stability: The roots of the characteristic polynomial lie within the unit circle, confirming that the VAR(6) model is stable and well-specified for analyzing the relationships between FUT and USL.
Economic Implications
1.	Short-Term Interactions: The significant lagged relationships suggest that oil market participants rely heavily on past trends in both futures and spot markets to form expectations, reflecting the recursive nature of price adjustments.
2.	Predictive Power: The model captures how shocks to one market propagate to the other, demonstrating the integration of futures and spot markets. For instance, unexpected futures market changes (e.g., due to geopolitical tensions) affect longer-term spot prices, which eventually feed back into futures markets.
3.	Policy and Market Decisions: These results are crucial for policymakers and traders, as they highlight the need to monitor futures markets for signals of broader price trends and to consider feedback effects when formulating strategies.
The VAR(6) model provides a robust framework for understanding and forecasting the interconnected dynamics between futures and spot oil prices, offering insights into the mechanics of market expectations and real-time adjustments.














 
Interpretation of Forecast Error Variance Decomposition (FEVD) with Economic Intuition
Overview of Results
The FEVD results from the two VAR models highlight the sensitivity of the decomposition to variable ordering, revealing interesting dynamics between the future oil prices (diff_log_FUT) and the spot oil ETF prices (diff_log_USL). This highlights how the underlying structure of the model and ordering of variables can influence the interpretation of lead-lag relationships.
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VAR Model 1: diff_log_FUT Ordered First
 
1.	diff_log_FUT (Futures Prices):
o	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 shows that futures price changes are largely self-driven, reflecting the futures market's role as the primary mechanism for incorporating information about future supply, demand, and geopolitical factors.
2.	diff_log_USL (Spot Prices):
o	The variance of diff_log_USL is dominated by shocks from diff_log_FUT, with approximately 72% explained by diff_log_FUT across all forecast horizons. This result underscores that spot prices (represented by the ETF) are influenced by the futures market, as spot prices adjust to information already priced into futures contracts. Economically, this reflects the well-known role of futures in leading the price discovery process.
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VAR Model 2: diff_log_USL Ordered First
 
1.	diff_log_USL (Spot Prices): 
o	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 spot prices dominate. From an economic standpoint, this might represent the idea that spot prices react primarily to immediate market conditions and inventory adjustments.
2.	diff_log_FUT (Futures Prices):
o	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 respond to information reflected in spot prices when the latter is ordered first. This is less consistent with conventional financial theory but can occur due to feedback loops where spot prices reflect unobservable factors (e.g., immediate physical market conditions) that later influence futures pricing.
 
Interpretation of Impulse Response Function (IRF) Results with Economic Intuition
The impulse response function (IRF) analysis visualizes how shocks to one variable (diff_log_FUT or diff_log_USL) affect the dynamic responses of both variables in the VAR model over time. The results help to understand the causal relationships and interdependencies between the two variables in the oil market.
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Responses of diff_log_FUT (Future Oil Prices)
 
1.	Shock to diff_log_FUT:
o	A positive shock to diff_log_FUT leads to a strong immediate positive response in itself, reflecting the autoregressive nature of futures prices.
o	The effect dissipates rapidly after a few periods, with minor fluctuations around zero. This suggests that the impact of a futures price shock is short-lived, consistent with the efficient market hypothesis, where price changes incorporate all available information quickly.
2.	Shock to diff_log_USL (Spot Prices):
o	A shock to diff_log_USL causes a relatively smaller, immediate, and positive response in diff_log_FUT. However, this effect diminishes quickly, suggesting that while spot price movements influence futures prices, the relationship is not as dominant as the reverse.
o	This reflects the idea that futures markets incorporate forward-looking information more effectively than spot prices, which are more reactive to real-time market conditions.
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Responses of diff_log_USL (Spot Prices)
1.	Shock to diff_log_FUT:
o	A positive shock to diff_log_FUT leads to a significant and immediate positive response in diff_log_USL, demonstrating the strong influence of futures prices on spot prices. This highlights the futures market's role in price discovery and how spot prices adjust to incorporate the information contained in futures prices.
o	The response stabilizes after a few periods, indicating that the adjustment process is efficient but short-lived.
2.	Shock to diff_log_USL:
o	A shock to diff_log_USL results in a strong initial response in itself, which is expected given the autoregressive nature of the spot price series. However, the effect decays rapidly, indicating that spot price shocks do not have a prolonged impact.
o	This aligns with the notion that spot prices react to immediate physical market conditions and supply-demand factors, which are quickly absorbed into the market.
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Key Economic Insights
1.	Futures Lead Spot Prices:
o	The IRF results clearly indicate that shocks to futures prices (diff_log_FUT) have a significant impact on spot prices (diff_log_USL). This reflects the role of futures markets as the primary venue for price discovery, incorporating forward-looking information that guides spot price adjustments.
o	Shocks to spot prices have a weaker and less persistent effect on futures prices, suggesting that futures markets are less reactive to real-time market conditions reflected in spot prices.
2.	Market Efficiency:
o	The rapid dissipation of shocks in both series demonstrates market efficiency, where new information is quickly reflected in prices, and the effects of shocks are not prolonged.
3.	Policy and Trading Implications:
o	For market participants, the strong influence of futures prices on spot prices implies that monitoring futures markets is critical for understanding and anticipating spot price movements.
o	Policymakers can use this dynamic to gauge market expectations about future supply and demand, as reflected in futures prices.
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Conclusion
The IRF results confirm the dominant role of futures prices in the oil market's pricing dynamics, consistent with financial theory. While spot prices respond strongly to futures price shocks, the reverse relationship is weaker, highlighting the predictive power of futures markets. These findings reinforce the importance of futures as a leading indicator of oil market trends and underscore their significance for both traders and policymakers.




4.4 Volatility Modeling Results
•	Present GARCH (1,1) model output for FUT and USL residuals.
•	Include plots of conditional variances and standardized residuals.
•	Discuss the behavior and implications of volatility dynamics.

Interpretation of GARCH(1,1) Results and Economic Intuition
Conditional Variance from GARCH(1,1) Models
The conditional variances for both FUT (future oil prices) and USL (spot oil ETF) are modeled using a GARCH(1,1) process, which captures time-varying volatility dynamics in the returns. These results provide insights into the volatility clustering and persistence of oil price changes over time.  
1.	Conditional Variance Dynamics:
o	The GARCH(1,1) model captures volatility clustering, where periods of high volatility are followed by high volatility, and periods of low volatility are followed by low volatility.
o	The top panel shows the conditional variance of FUT:
	There are significant spikes in volatility during periods of market stress, particularly around major economic or geopolitical events (e.g., 2008 financial crisis and oil price shocks around 2014–2015).
	The variance stabilizes in more recent periods but remains subject to occasional spikes.
o	The bottom panel depicts the conditional variance of USL:
	The pattern is similar to FUT, with periods of heightened volatility aligning closely with those in FUT. This similarity reflects the interlinked nature of the futures and spot oil markets.
2.	Key GARCH Parameters:
o	For both series, the high values of alpha1 (short-term shock persistence) and beta1 (long-term volatility persistence) indicate that volatility is highly persistent. This implies that shocks to volatility have long-lasting effects, a common feature in financial time series.
o	The sum of alpha1 and beta1 for both series is close to 1, suggesting that volatility decays slowly over time and periods of high uncertainty persist.
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Standardized Residuals from GARCH Models
 
1.	Behavior of Residuals:
o	The standardized residuals for both FUT and USL appear approximately centered around zero, with no significant patterns, indicating that the GARCH(1,1) model successfully captures the conditional heteroskedasticity in the data.
o	However, there are occasional large deviations, reflecting extreme market events or outliers in the data.
2.	Distribution of Residuals:
o	The summary statistics for standardized residuals show that the mean is close to zero, and the maximum and minimum values reveal a few extreme observations. These outliers are likely related to significant market events that drive unusual price movements.
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Economic Intuition
1.	Volatility Clustering:
o	The results confirm that oil price changes exhibit volatility clustering. This is consistent with economic intuition, as periods of heightened uncertainty (e.g., due to geopolitical tensions, changes in OPEC policy, or global recessions) lead to persistent volatility in oil markets.
2.	Market Dynamics and Feedback:
o	The similarity in conditional variance patterns between FUT and USL reflects the integrated nature of the futures and spot markets. Futures markets, being forward-looking, often lead in absorbing new information, but the spot market adjusts quickly, leading to aligned volatility dynamics.
o	The persistence in volatility suggests that traders and market participants adjust slowly to new information, leading to extended periods of uncertainty.
3.	Policy and Trading Implications:
o	Policymakers can use the volatility persistence observed in these markets to anticipate prolonged periods of instability following major shocks. Similarly, traders can exploit volatility patterns to implement hedging strategies, knowing that high volatility periods are likely to persist.
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Conclusion
The GARCH(1,1) results reveal critical insights into the volatility dynamics of FUT and USL. Both series exhibit high volatility persistence and clustering, aligning with the nature of financial markets under uncertainty. These findings emphasize the importance of monitoring market conditions and developing strategies to manage risks associated with prolonged volatility.
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Discussion and Conclusion for Time Series Econometrics Report
Discussion
1.	Dynamic Interplay Between Futures (FUT) and Spot (USL) Oil Prices
o	The results demonstrate the strong interdependency between FUT and USL prices, with futures markets playing a dominant role in price discovery. Shocks to FUT consistently led to significant immediate impacts on USL, supporting the hypothesis that futures prices incorporate forward-looking market expectations.
o	The bidirectional relationship observed in the VAR model underlines the recursive feedback between the futures and spot markets. However, the impulse response functions (IRFs) reveal that the influence of USL on FUT is relatively weaker and dissipates faster, suggesting that the spot market adjusts primarily to information already embedded in futures prices.
2.	Stationarity and Econometric Model Choices
o	Stationarity testing confirmed that the log-transformed levels of both FUT and USL are non-stationary, while their returns (first-differenced log series) are stationary. This aligns with financial theory, which suggests that price levels in markets like oil exhibit random walk properties.
o	The VAR model's choice of lag length (6 lags based on BIC) was validated by its ability to capture short-term dynamics without overfitting. The results provided a statistically significant framework to explore interdependencies, dynamic responses, and variance decomposition.
3.	Volatility Dynamics and GARCH Results
o	GARCH(1,1) models effectively captured the persistent volatility and clustering present in both FUT and USL returns. This finding reflects the inherent uncertainty and sensitivity of oil markets to macroeconomic and geopolitical shocks.
o	The high persistence of volatility (sum of alpha and beta parameters close to 1) indicates that periods of market stress are likely to have prolonged effects. This persistence is consistent across both FUT and USL, reinforcing the interconnected nature of these markets.
4.	Economic and Policy Implications
o	The dominance of FUT in explaining variance and influencing USL dynamics highlights the importance of futures markets for market participants and policymakers. Futures prices act as leading indicators of market sentiment, supply-demand expectations, and geopolitical risk assessments.
o	For risk management, the volatility clustering observed in GARCH results underscores the necessity for traders and firms to implement robust hedging strategies, especially during periods of heightened market uncertainty.
Conclusion
1.	Summary of Key Findings
o	The study confirms that futures oil prices (FUT) lead and significantly influence spot prices (USL), establishing their role in price discovery.
o	The VAR model and IRFs illustrate a strong dynamic relationship between the two variables, where FUT shocks propagate quickly and efficiently to USL, but the reverse effect is less pronounced.
o	GARCH(1,1) modeling of residuals highlights persistent volatility clustering, consistent with financial market behavior during periods of uncertainty.
2.	Contributions to Literature
o	This paper bridges a gap in existing literature by jointly applying VAR and GARCH models to analyze the interplay between spot (USL) and futures (FUT) oil prices. The findings contribute to understanding the volatility dynamics and causal relationships in oil markets, offering practical insights for market participants and policymakers.
3.	Implications for Stakeholders
o	Policymakers: The results emphasize the need to monitor futures markets for signals of economic and geopolitical risks, as these markets often reflect collective expectations about future conditions.
o	Traders and Investors: The findings highlight the critical role of volatility management and the importance of futures markets in predicting spot price movements. Developing strategies based on the observed lead-lag relationship can enhance decision-making.
4.	Limitations and Future Research
o	While this study focused on daily data from 2009 to 2020, future research could extend the analysis to incorporate higher-frequency data or explore the impact of external macroeconomic variables on these relationships.
o	Additionally, exploring alternative models, such as structural VAR or regime-switching GARCH, could provide deeper insights into the complex dynamics of oil markets under different economic conditions.
This report offers a comprehensive econometric analysis of oil price dynamics, highlighting the pivotal role of futures markets in shaping spot prices and the persistent volatility inherent in oil markets. The findings serve as a foundation for future research and practical applications in trading and policy formulation.
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7. References
•	List all references cited in the literature review and methods.
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8. Appendix
•	Include:
o	R code for reproducibility.
o	Additional plots or test results not included in the main text.
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Accompanying Files
•	Data File: OILNF_US0_NEW.xlsx (with FUT and USL data).
•	R Code: Full code for preprocessing, modeling, and visualization.