Homework#5 (1)

Homework #5 Problem 1. Please match the models listed below with the ACF and PACF plot of the time series generated from the model. Models: (1) AR(2) (2) MA(3) (3) ARIMA(2,0,1) (4) ARIMA(1,1,1) Plots: (a) MA(3) (b) ARIMA(1,1,1) ARIMA(2,0,1)

(c) AR(2) (d) ARIMA(2,0,1) AR(2) Problem 2. Consider the following models (1) y t = 0.7 y t − 1 + 0.6 ε t − 1 + ε t (2) ( 1 − B ) y t = 0.7 y t − 1 − 0.7 y t − 2 + 0.6 ε t − 1 + ε t (3) ( 1 − B 4 ) y t = 0.7 y t − 1 − 0.7 y t − 5 + 0.6 ε t − 1 + ε t (a) For each model, re-write the model equation into the form as shown slide #76 using backward shift operator B. (b) If we use notation ARIMA(p,d,q)(P,D,Q) m to represent an ARIMA model, what are the values of p, d, q, P, D, Q, m for the three models?

3 6.8 Derive the prediction at t = 3 , 4 , 5. Problem 5. (computer based problem) Consider the SouvenirSales.csv data. It contains monthly sales for a souvenir shop at a beach resort town in Queensland, Australia, between 1995 and 2001. Please partition the data into training and validation periods, with the validation set containing the last 12 months of data (year 2001). Using the training period data, please (a) Run a regression model with Sales as the output variable and with a linear trend and monthly seasonality. (b) Create an ACF and PACF plot until lag-10 for the residuals of model in (a) (c) Establish an AR(2) model for the residuals in (b) (d) Create an ACF and PACF plot until lag-10 for the residuals of the AR(2) model in (c) (e) Use auto.arima function to automatically find a best ARIMA model for the training period data. Write the model in ARIMA(p,d,q)(P,D,Q) m format. (f) Fit the identified model in (e) and plot the fitted values of the model and the observed training period data. (g) Create an ACF and PACF plot until lag-10 for the residuals of model in (f) Using the validation period data, please (h) Using the model in (a) to forecast for the validation period and obtain the forecasting accuracy (i) Combining the model in (a) and (c) to forecast for the validation period and obtain the forecasting accuracy (j) Using the model in (f) to forecast for the validation period and obtain the forecasting accuracy