In this exercise you should fit models to two observed time series. In your report you should step by step give arguments for the choices you do with respect to transforming the data, differencing, seasonality and identification. You should discuss the various models you have tried, including your reasons for trying them and your reasons for preferring one model instead of others. Thereby you also need to include in your report plots on which your reasoning is based. The final models have to be verified and written down with standard deviations for the estimated parameters and the residuals. You should use the chosen model(s) to make forecasts (with associated prediction errors) at least 15 time steps into the future, and present the prediction and prediction intervals in a plot. Try to give a natural interpretation of the fitted models, even though this may be difficult. Base your final report on plots and output from R.

When you use R commands you should specify these in your report. Please use complete sentences in your report, just keywords or half sentences are not acceptable. The report should be handed in as one (and only one) pdf file if you send it electronically. Please avoid the use of appendices in your report, as this makes it much harder to evaluate the report. In the evaluation of the report the focus will be on the reasoning and your arguments for your choices, more than on the final results.

You are allowed to work (and hand in your report) alone or in groups of two persons. You may work with the same person as you did in Project 1, or you may work with another person. Write your name(s) on the report, not student or candidate numbers.

To solve the problems you may use the computers in the student computer lab of the Department of Mathematical Sciences. If you don't already have a user and physical access to these computer labs you must send a mail to the lecturer and ask for this.

Useful R commands include the following:

• ts.plot
• acf
• pacf
• diff (to compute differences and seasonal differences)
• arima (to fit ARIMA and seasonal ARIMA models by ML)
• arima.sim
• tsdiag
• predict (or predict.Arima) (this computes forecasts)
• dev.copy2eps (or dev.copy2pdf)
• sink
• residuals

You should analyse one data set in each of the following two groups:

• Group I:
  • swedishFertility.txt, for information about the data set see http://data.is/11VdWk9.
  • treeRings.txt, for information about the data set see http://data.is/11VfM4G.
• Group II:
  • unemploymentNorway.txt, for information about the data set see http://data.is/1xV7rs7.
  • unemploymentFemale.txt, for information about the data set see http://data.is/1xV9PPs.
  • fraser.txt, for information about the data set see [http://data.is/1xVb29s]].
  • houseprices.txt. Quartely houseprices (per square meter relative to the level in year 2000) in Oslo from the first quarter of 1992 until the third quarter of 2015.

All the above data sets are from https://datamarket.com/topic/list/countries/. You may download the data in "swedishFertility.txt" into R by the R command "x = scan("http://www.math.ntnu.no/~jarlet/tidsrekker/swedishFertility.txt")' and corresponding for the other data sets.