Learning Objectives: Explain how forecasts are generated from ARMA models. Describe the role of mean reversion in long-horizon forecasts. Explain how seasonality is modeled in a covariance-stationary ARMA.
Questions:
25.9.1. An analyst at Derivatech Hedge Fund is tasked with assessing the behavior of forecasted S&P 500 Volatility Index (VIX) data derived using an Autoregressive Integrated Moving Average (ARIMA) model, specifically ARIMA(1, 0, 1).
The model summary or regression output below provides detailed statistical results and diagnostics for the ARIMA(1, 0, 1) model fitted to VIX data. The results are as follows:
Based on the above, which of the following is the most accurate answer?
a. The ADF statistic and its p-value suggest that the data has stationarity; therefore, he should proceed with the ARMA model.
b. The Auto-Regressive coefficient (ar.L1 aka the AR1) suggests that differencing might not be necessary to achieve stationarity.
c. The skewness and kurtosis confirm that residuals have heavy tails and are not symmetric, suggesting that the model is mis-specified and needs further transformation of data.
d. Heteroskedasticity test indicates heteroskedasticity is present, suggesting that the variance of residuals will change significantly over time.
25.9.2. An analyst at Smart Puts Capital LLP is forecasting the long-term price of the Breakwave Dry Bulk Shipping ETF, which is believed to revert to a specific price level due to the cyclical nature of the shipping industry. The ETF's price is modeled using an AR(1) process with a long-run mean (mean reversion level) of $25, while the current price is $30. The time frame for the analysis spans from January 1, 2019, to December 31, 2024.
He has conducted the following tests and specifications:
Which of the following is the most accurate?
a. The initial ADF test value indicates the presence of a unit root, therefore justifying the belief in stationarity.
b. The high AR coefficient is an indicator of drift and, therefore, not of mean reversion.
c. The ADF test after adjustment indicates the series is stationary, therefore justifying the belief in stationarity.
d. This cannot be ascertained because five years may not be sufficient for this analysis.
25.9.3. Edward Moore, an analyst at Moore Africa Investment Partners LLP, is reviewing a dataset of Natural Gas Futures prices from January 20X1 to December 20X4. He uses a seasonal ARMA model to forecast prices for the upcoming period. His model incorporates a restricted AR(13) with significant coefficients at lags 1, 12, and 13 to capture the pronounced annual seasonal pattern, including winter demand spikes and subsequent corrections.
Model Parameters:
Based on this seasonal ARMA model specification (AR(1) with additional lags at 12 and 13), which of the following is the most accurate description of the expected price forecast for January 2025?
a. The forecast will likely increase, reflecting the continuation of the December seasonal effect carried over into January by lag 12.
b. The forecast will likely decline, as lag 12 reflects December’s seasonal peak, and lag 13 adjusts for the typical drop in prices observed in January after the winter spike.
c. The model will forecast prices to revert directly to the unconditional mean due to stationarity and mean reversion in the AR(1) component.
d. The model will produce a constant forecast since, beyond lag 12, seasonal effects are fully differenced out in stationary seasonal ARMA processes.
Answers here:
Questions:
25.9.1. An analyst at Derivatech Hedge Fund is tasked with assessing the behavior of forecasted S&P 500 Volatility Index (VIX) data derived using an Autoregressive Integrated Moving Average (ARIMA) model, specifically ARIMA(1, 0, 1).
The model summary or regression output below provides detailed statistical results and diagnostics for the ARIMA(1, 0, 1) model fitted to VIX data. The results are as follows:
Y_t = 0.9762Y_(t - 1) + 0.2527ϵ_(t - 1) + ϵ_t
Based on the above, which of the following is the most accurate answer?
a. The ADF statistic and its p-value suggest that the data has stationarity; therefore, he should proceed with the ARMA model.
b. The Auto-Regressive coefficient (ar.L1 aka the AR1) suggests that differencing might not be necessary to achieve stationarity.
c. The skewness and kurtosis confirm that residuals have heavy tails and are not symmetric, suggesting that the model is mis-specified and needs further transformation of data.
d. Heteroskedasticity test indicates heteroskedasticity is present, suggesting that the variance of residuals will change significantly over time.
25.9.2. An analyst at Smart Puts Capital LLP is forecasting the long-term price of the Breakwave Dry Bulk Shipping ETF, which is believed to revert to a specific price level due to the cyclical nature of the shipping industry. The ETF's price is modeled using an AR(1) process with a long-run mean (mean reversion level) of $25, while the current price is $30. The time frame for the analysis spans from January 1, 2019, to December 31, 2024.
He has conducted the following tests and specifications:
- Augmented Dickey-Fuller (ADF) Test: p-value > 0.05.
- He applied first differencing, which led to a subsequent ADF test result with a p-value < 0.05.
- The fitted ARMA model had an AR coefficient of 0.94.
Which of the following is the most accurate?
a. The initial ADF test value indicates the presence of a unit root, therefore justifying the belief in stationarity.
b. The high AR coefficient is an indicator of drift and, therefore, not of mean reversion.
c. The ADF test after adjustment indicates the series is stationary, therefore justifying the belief in stationarity.
d. This cannot be ascertained because five years may not be sufficient for this analysis.
25.9.3. Edward Moore, an analyst at Moore Africa Investment Partners LLP, is reviewing a dataset of Natural Gas Futures prices from January 20X1 to December 20X4. He uses a seasonal ARMA model to forecast prices for the upcoming period. His model incorporates a restricted AR(13) with significant coefficients at lags 1, 12, and 13 to capture the pronounced annual seasonal pattern, including winter demand spikes and subsequent corrections.
Model Parameters:
Based on this seasonal ARMA model specification (AR(1) with additional lags at 12 and 13), which of the following is the most accurate description of the expected price forecast for January 2025?
a. The forecast will likely increase, reflecting the continuation of the December seasonal effect carried over into January by lag 12.
b. The forecast will likely decline, as lag 12 reflects December’s seasonal peak, and lag 13 adjusts for the typical drop in prices observed in January after the winter spike.
c. The model will forecast prices to revert directly to the unconditional mean due to stationarity and mean reversion in the AR(1) component.
d. The model will produce a constant forecast since, beyond lag 12, seasonal effects are fully differenced out in stationary seasonal ARMA processes.
Answers here:
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