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.