**Assignment Task**

**Questions**

1. This question uses data from the Australian Bureau of Statistics (ABS) on the Australian labour force, from ABS Catalogue 6202.0, Labour Force, Plot your original (non-seasonally adjusted) data and seasonally adjusted data together using the following code, putting an appropriate label for your series on the y-axis.

**Tasks**

1. a. Explore the original (non-seasonally adjusted) data series using the following graphics functions, being sure to discuss what you find from each plot: gg_season(), gg_subseries(), gg_lag() and ACF() %>% autopilot

b. Consider the last ten years (i.e. the last 120 months) of your data. Use an STL decomposition of the original data (using the default settings) and produce a standard decomposition plot showing the original data and its trend-cycle, seasonal and remainder components. Discuss what you find from the decomposition plot.

c. Plot your seasonally adjusted data from the STL decomposition together with the official seasonally adjusted data for the last ten years. What observations can you make about the respective series?

d. Apply Holt-Winters’ multiplicative method with multiplicative errors to the seasonally adjusted data from the STL decomposition. Then try the method making the trend damped. Plot and compare the 24-month ahead point forecasts and prediction intervals for both methods. Using accuracy(), compare the RMSE of the of the two methods; this comparison is based on one-step-ahead in-sample forecasts. Which method do you prefer and why?

2. This question uses data from the Australian Bureau of Statistics (ABS) on Consumption and Income, from 5206.0 Australian National Accounts: National Income, Expenditure and Product, Table 1. Key National Accounts Aggregates

- Our Consumption variable is “Household Final Consumption Expenditure: Chain volume measures – Percentage changes
- Our Income variable is “Real net national disposable income: Chain volume measures – Percentage changes

2. a. Plot both Consumption and Income in the same figure. Using this single figure, discuss the characteristics of the two data series individually and relative to each other.

b. Create four dummy variables (dum1, dum2, dum3 and dum4) as follows:

- Dum1 is equal to 1 only in quarter 2 of 2020, and is otherwise equal to zero.
- Dum2 is equal to 1 only for quarters 3 and 4 of 2020, and is otherwise equal to zero.
- Dum3 is equal to 1 only for quarter 3 of 2021, and is otherwise equal to zero.
- Dum4 is equal to 1 only for quarter 4 of 2021, and is otherwise equal to zero. Explain why these dummy variables may help us model the relationship between Consumption and Income.

c. Fit a time series regression model for Consumption, using Income and seasonal dummy variables as predictor variables. Call this model1. Then add the dummy variables created in Question 2 task b to the list of predictor variables. Call this model2. Report and discuss the regression coefficients from both estimated models. For each model, plot the Consumption data series and the fitted values on the same figure and discuss. That is, there should be 3 one figure for each of the two regression models. For each model, comment on the residuals, using the standard plots (i.e. gg_tsresiduals()). Report the AICc for each model. Explain which model you prefer and why.

d. Estimate model 2 using dynamic regression (instead of time series regression), using no seasonal ARIMA components. Explain why using dynamic regression might be an appropriate thing to do in this case. Report and comment on the coefficient estimates, and the standard residual plots.

e. Using the dynamic regression model from Question 2 task d, forecast 20 quarters ahead assuming that income will continue to grow at 8% per quarter, the same rate as in the last observation for Income in the data (2023 Q4). Call this the “High” scenario. Now assume that income will grow at its mean rate of 1% per quarter. Call this the “Mean” scenario. On the same figure as the actual Consumption data from 2015, plot the point forecasts and prediction intervals generated under these two alternative assumptions. Discuss the relative forecasts from the two different scenarios.