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PDF Submissiоn Only GrаdeScоpe Submissiоn Link (10-minute submission window) Cаnvаs file upload here QUESTION 1: TREND AND STATIONARITY (25 points) Question 1a (6 points)Plot the full time series and its ACF. Evaluate the stationarity of natural gas consumption. Support your analysis with the plots and provide a detailed commentary on stationarity, including which assumptions of stationarity (if any) are violated and why. Question 1b (3 points)Why is stationarity an important assumption for ARMA-type models? What issues arise when this assumption is violated but an ARMA model is fit anyway? Question 1c (3 points)How does the presence (or absence) of trend and seasonality in a series influence the choice between an ARMA vs. an ARIMA model, and between a seasonal vs. a non-seasonal specification? Question 1d (5 points)Split the data into a training set (2005–2018) and a test set (2019, the last 12 months), as provided in the setup code above. Using the training data, fit two trend models covered in the course (e.g., moving average and a parametric quadratic trend). Overlay the fitted values on the training series, and examine the residual ACFs. Comment on how well each model captures the trend and whether either leaves behind additional structure. Question 1e (2 points)Both a trend and a seasonal component can produce an ACF that fails to decay quickly, so a slowly-decaying ACF alone does not tell you which one you're dealing with. Explain how the shape of the ACF differs between a series driven purely by trend versus one driven purely by seasonality, and describe a diagnostic you could use to tell which component (or both) is present just from the correlogram. Question 1f (3 points)Suppose two trend models fit the training data about equally well by eye. What additional, more quantitative criteria could you use to decide between them? Separately, explain why a trend model that fits the historical data very well can still forecast poorly on new data, and what role model complexity plays in this tradeoff. Question 1g (3 points)Non-parametric trend estimators (e.g., a moving average or a smoothing spline) rely on a tuning parameter, a bandwidth, a span, a number of knots, or an equivalent degrees-of-freedom setting, that controls how much the fitted trend is allowed to flex. Explain the trade-off this tuning parameter creates between underfitting and overfitting the trend, and describe how each failure mode would typically show up in the fitted trend curve and in the residual diagnostics. QUESTION 2: SEASONALITY AND DIFFERENCING (23 points) Question 2a (8 points)Using the training data, fit a trend + seasonal model that combines a non-parametric (spline) trend with ANOVA-type monthly seasonal dummies — that is, a single GAM of the form gam(train_ts ~ s(time.pts) + season_factor). Overlay the fitted values on the training series, and evaluate the residuals. Has this specification resolved the dependence left over from Question 1(d)? Question 2b (3 points)How would you formally assess whether the monthly seasonal effects in the model from 2(a) are statistically meaningful, rather than assuming seasonality is present just because the plot looks seasonal? What would it imply if they were not significant? Question 2c (6 points)Apply first-order (regular) differencing, seasonal (lag-12) differencing, and both combined to the training series. Plot each differenced series with its ACF, and compare these differencing-based approaches to the trend-seasonal fitting approach used in 2(a). Which approach — differencing or explicit trend/seasonal modeling — would you recommend for this series, and why? Question 2d (3 points)What information about the original series is removed when you take a first difference, and what information is preserved? Question 2e (3 points)For a seasonal ARMA process with period s, describe theoretically how a purely seasonal AR component versus a purely seasonal MA component would each appear in the ACF and PACF at the seasonal lags s, 2s, 3s, etc. Relate this to the residual ACF patterns you found for the trend-only models in 1(d) versus the trend+seasonal model in 2(a). QUESTION 3: ARIMA AND SARIMA MODELING (33 points) Question 3a (5 points)Using the residuals from your trend-seasonal model in 2(a), fit an ARMA(1,0,1) model to capture the remaining serial dependence. Report the fitted coefficients, and assess residual adequacy (Ljung–Box test, Shapiro–Wilk test). Question 3b (3 points)How should the coefficients of a fitted ARMA model be interpreted, and what does the statistical significance (or lack thereof) of those coefficients imply about the model's adequacy? Relate this directly to what you found in 3(a). Question 3c (8 points)Now work directly with the original (un-differenced) training series, and consider a non-seasonal ARIMA(p,d,q) model only (no seasonal terms at all). Run the grid search provided below over p, q in {0,1,2} and d in {0,1}, which selects orders by AICc. Based on the results, choose an order you believe is appropriate and justify your choice (you are not required to pick strictly the lowest-AICc model if you have good reason not to). Fit your chosen model, report coefficient significance, assess causality and invertibility using the characteristic roots, and evaluate residual diagnostics. Question 3d (3 points)Looking back at the ACF and PACF plots of the ARIMA residuals in 3(c): how are ACF and PACF plots generally used to assess whether a fitted ARMA-type model has captured the available serial dependence? What specific pattern in those plots told you the model had not succeeded here? Question 3e (5 points)Fit the following SARIMA model, already selected for you via a broader search of the seasonal order space: SARIMA(1,1,1) x (0,1,1)_12. Report the fitted coefficients and their significance, assess causality and invertibility using the characteristic roots, and evaluate residual diagnostics (Ljung–Box, Shapiro–Wilk). Question 3f (3 points)Compare the ARIMA model from 3(c) with the SARIMA model from 3(e) in terms of: (i) how each handles trend, (ii) how each represents seasonality, and (iii) overall model flexibility. Question 3g (6 points)Compare the three models fit so far — the hybrid model (3a), the non-seasonal ARIMA (3c), and the SARIMA model (3e) — based on their residual diagnostics alone. Which appears best specified in-sample, and which is clearly inadequate? Discuss the role of the near-non-invertible coefficient found in 3(e) and the non-significant coefficients found in 3(a). QUESTION 4: FORECASTING (19 points) Question 4a (8 points)Using all three models — the hybrid model (2a + 3a), the non-seasonal ARIMA (3c), and the SARIMA model (3e) — forecast the 12 months of 2019 (the test set). The hybrid forecast is given below as a worked template; write the analogous forecast code for the ARIMA and SARIMA models. Then overlay all three sets of forecasts on the actual test-set values (plotting code given). Question 4b (7 points)Compute RMSE, MAPE, PM, and 95% CI coverage for all three models' 2019 forecasts. Does the model with the cleanest in-sample residuals (per 3g) also forecast best out-of-sample? Which model would you recommend, and why? Question 4c (4 points)The hybrid forecast in 4(a) combines predictions from two separately-fit components (the GAM trend-seasonal model and the ARMA residual model). Discuss potential sources of error propagation when combining forecasts from multiple model components this way, and relate this to the hybrid model's spring 2019 under-performance found in 4(b).