Purpose: Understand the difference between additive and multiplicative decomposition. Solidifying understanding of deterministic trend forecasts as well as in- and out-of-sample forecasts and their assessments. Deadline: Tuesday, September 26, 2017

Preliminaries

First, we have to load our Airline Passengers data. If you have questions about he data you can find more information via executing ?AirPassengers in the R console.

data("AirPassengers")
AP.ts <- AirPassengers

The data is available as a .csv-file here. Please ensure that you are able to save files such as this one locally on your computer. Make sure you know how to set a directory in R using the setwd() and check what your working directory is via getwd().

We split the \(12\times 12\) observations on Airline Passengers into three subperiods of equal length. We then split these three subperiod into two subsample, an in-sample of the first two years and an out-of-sample period of two years. So altogether there are six periods:

Period ID Period Type Start End
1.1 in-sample Jan 1949 Dec 1950
1.2 out-of-sample Jan 1951 Dec 1952
2.1 in-sample Jan 1953 Dec 1954
2.2 out-of-sample Jan 1955 Dec 1956
3.1 in-sample Jan 1957 Dec 1958
3.2 out-of-sample Jan 1959 Dec 1960

Questions 1-6 refer to forecast using the AP.ts time series.

Using a linear trend model in period 1.1, 2.1, and 3.1 to forecast values in periods 1.2, 2.2, and 3.2 respectively.

Question 1

What is the MEA for the first out-of-sample period 1.2 using the linear trend model estimated on in-sample period 1.1 for the series AP.ts. (0.5 points)

Question 2

What is the MEA for the second out-of-sample period 2.2 using the linear trend model estimated on the in-sample period 2.1 for the series AP.ts. (0.5 points)

Question 3

What is the MEA for the third out-of-sample period 3.2 using the linear trend model estimated on the in-sample period 3.1 for the series AP.ts. (0.5 points)

Question 4

What is the RMSE for the first out-of-sample period 1.2 using the linear trend model estimated on in-sample period 1.1 for the series AP.ts. (0.5 points)

Question 5

What is the RMSE for the second out-of-sample period 2.2 using the linear trend model estimated on the in-sample period 2.1 for the series AP.ts. (0.5 points)

Question 6

What is the RMSE for the third out-of-sample period 3.2 using the linear trend model estimated on the in-sample period 3.1 for the series AP.ts. (0.5 points)

Question 7

What is the MAPE for the first out-of-sample period 1.2 using the linear trend model estimated on in-sample period 1.1 for the series AP.ts. (0.5 points)

Question 8

What is the MAPE for the second out-of-sample period 2.2 using the linear trend model estimated on the in-sample period 2.1 for the series AP.ts. (0.5 points)

Question 9

What is the MAPE for the third out-of-sample period 3.2 using the linear trend model estimated on the in-sample period 3.1 for the series AP.ts. (0.5 points)

Now transform the series AP.ts into the series AP.l.ts by taking the natural log. Plot both AP.ts and AP.l.ts in a single diagram. You might to search around the internet a bit as to how to do that.

Questions 10-18 refer to the series AP.l.ts.

Question 10

What is the MEA for the first out-of-sample period 1.2 using the linear trend model estimated on in-sample period 1.1 for the series AP.l.ts. (0.5 points)

Question 11

What is the MEA for the second out-of-sample period 2.2 using the linear trend model estimated on the in-sample period 2.1 for the series AP.l.ts. (0.5 points)

Question 12

What is the MEA for the third out-of-sample period 3.2 using the linear trend model estimated on the in-sample period 3.1 for the series AP.l.ts. (0.5 points)

Question 13

What is the RMSE for the first out-of-sample period 1.2 using the linear trend model estimated on in-sample period 1.1 for the series AP.l.ts. (0.5 points)

Question 14

What is the RMSE for the second out-of-sample period 2.2 using the linear trend model estimated on the in-sample period 2.1 for the series AP.l.ts. (0.5 points)

Question 15

What is the RMSE for the third out-of-sample period 3.2 using the linear trend model estimated on the in-sample period 3.1 for the series AP.l.ts. (0.5 points)

Question 16

What is the MAPE for the first out-of-sample period 1.2 using the linear trend model estimated on in-sample period 1.1 for the series AP.l.ts. (0.5 points)

Question 17

What is the MAPE for the second out-of-sample period 2.2 using the linear trend model estimated on the in-sample period 2.1 for the series AP.l.ts. (0.5 points)

Question 18

What is the MAPE for the third out-of-sample period 3.2 using the linear trend model estimated on the in-sample period 3.1 for the series AP.l.ts. (0.5 points)

Question 19

Compare and rank the MAE, RMSE, and the MAPE for the three periods for the original series with the log transformations.

(1.0 point)

Optional

Visualize the three in-sample and out-of-sample periods, the predicted values in-sample, the predicted values out-of-sample, as well as the three sets of in-sample and the three sets of out-of-sample forecast errors.