Description
This project involved generating and analyzing simulated time series for ARMA and ARCH processes to study their properties, visualize their behavior, and compare theoretical and sample statistics.
ARMA(2,1) Process Simulation and Analysis
Methodology
- Manual Simulation:
- An ARMA(2,1) process of length 50 was simulated with parameters:
- A burn-in period of 3 iterations was used to stabilize the process.
- A pre-specified white-noise series was provided to ensure reproducibility.
- The process was computed iteratively using the ARMA(2,1) relationship:
- R’s
arima.simFunction: - The same ARMA(2,1) process was generated using the
arima.simfunction. - The burn-in values were set manually to match the initial conditions from step 1.
- This approach served to validate the manually computed series.
- Autocorrelation Analysis:
- Theoretical autocorrelation was computed using the
ARMAacffunction. - Sample autocorrelation functions (ACFs) were calculated for both manually simulated and
arima.simseries. - Differences between theoretical and sample ACFs were analyzed.
Interpretation
Essentially, we are comparing the theoretical expectations with what we observed in the simulated data. In both cases, theoretical and sample ACFs, we have the autocorrelation values at various lags. Based on the general patterns, we see that the autocorrelation values in the theoretical ACF tend to be larger in magnitude compared to those in the sample ACF, especially at higher lags. The signs of the autocorrelation values alternate more frequently in theoretical ACF compared to sample ACF, where they appear to be more consistent. The autocorrelation values from sample ACF deviate more from
the theoretical expectations to a greater extent compared to those in theoretical ACF. This could suggest that the model used in sample ACF might not fully capture the underlying process, leading to discrepancies between the simulated and theoretical results.
- Extended Realizations:
- Four independent realizations of length 400 were generated using
arima.sim. - Sample ACFs were plotted for each realization, and their variability was discussed.
Results
- Manual vs.
arima.simComparison:
The two methods produced series with similar trends but slight deviations due to differences in implementation. Sample ACFs showed minor discrepancies, especially at higher lags, emphasizing the stochastic nature of the white-noise component.
- Theoretical vs. Sample ACFs:
- Theoretical ACFs matched closely at lower lags but deviated slightly as the lag increased, particularly for the
arima.simseries. - Alternating signs of ACF values aligned with the negative and positive feedback introduced by and .
- Extended Realizations:
All four realizations displayed consistent patterns in ACFs, with minor variations due to random white noise. Peaks at initial lags confirmed significant short-term autocorrelation.
ARCH(1) Process Simulation and Analysis
Methodology
- Simulation of ARCH(1) Process:
- An ARCH(1) process of length 1,000 was simulated using the model:
where , and is standard normal noise.
- ACF Analysis:
- Sample ACFs of two realizations were computed and plotted.
- Weak white noise characteristics (stationarity and uncorrelated structure) were evaluated.
- Ljung-Box Test:
- Applied the Ljung-Box test to detect autocorrelation in the realizations.
- Results supported the weak white noise properties of the ARCH(1) process.
- Influence of Parameter :
- Two realizations were simulated with and .
- Sample ACFs were compared to study the influence of on persistence.
Results
- ARCH(1) Behavior:
Both realizations exhibited stationary behavior, consistent with the weak white noise definition. Sample ACFs showed minimal autocorrelation, aligning with expectations.
- Ljung-Box Test:
Results indicated significant autocorrelation at lower lags, affirming the autoregressive influence of the ARCH(1) process.
- Impact of :
Higher values led to slower decay in the ACF, reflecting stronger persistence in the process. This aligns with the theoretical behavior of ARCH models, where controls the degree of autocorrelation.
Conclusion
This project successfully demonstrated the simulation, visualization, and statistical analysis of ARMA and ARCH processes. Key takeaways include:
- Manual simulation and R’s built-in functions can complement each other for validating results.
- Theoretical expectations align closely with simulated outcomes at lower lags, with discrepancies arising from stochastic noise and model assumptions.
- ARCH processes exhibit predictable behavior based on their parameters, influencing persistence and autocorrelation.
These analyses showcase advanced time series modeling techniques and their practical implications which contributes to my expertise in statistical modeling and R programming.
