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

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.

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)*

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)*

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)*

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)*

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)*

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)*

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)*

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)*

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`

.

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)*

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)*

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)*

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)*

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)*

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)*

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)*

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)*

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)*

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

*(1.0 point)*

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.