LINKÖPINGS UNIVERSITET732G06 TIME SERIES ANALYSIS

Institutionen för datavetenskapFall semester 2008

Statistik, CL, ANdAssignment

Assignmentweek 39: Estimation of autocorrelations and forecasting by autoregressive processes

This assignment consists of two tasks: The first task deals with estimating autocorrelations and the other with making predictions by applying autoregressive (AR) models.

Solutions should be submitted by Sunday 12 October!

Estimating autocorrelations

Data sets: The Dollar Exchange rates; Hjalmaren month

A. Time series without seasonal variation

Use the data in the file 'dollar.txt'.Construct a time series graph of the fluctuations of the dollar exchange rate, yt, for the period 1994-1998. Construct also a point plot for all pairs (yt-1, yt) and try to visually estimate how strong the correlation between two consecutive observations is (=autocorrelation). How do the estimated autocorrelations change with increasing timelags between observations.

Construct a time series graph of the changes zt=yt - yt-1 of the dollar exchange rate. Then try to judge upon how the estimated autocorrelations for the series zt change with the time lag between observations and check your judgement by estimating the autocorrelations.

B. Time series with seasonal variation

Use the time series of monthly discharge in the lakeHjälmaren (‘Hjalmarenmonth.txt’), which you have used in the assignment for week 36. Compute the autocorrelation function (Minitab: StatTime seriesAutocorrelation…) for the variable Discharge.m. Deseasonalise the time series and make a new graph of the seasonally adjusted values. Try to visually estimate how the autocorrelations look like and check your judgement by computing the autocorrelation function.

To deseasonalise the time series, you can use in Minitab 'StatTime seriesDecomposition…', use Seasonal only and Storage Seasonally adjusted data.

C. Forecasting with autoregressive models

Data set: The Dollar Exchange rates

Consider again the time series of dollar exchange rates for the period 1994-1998. Then use the Minitab time series module ARIMA (see further below) to estimate the parameters in an AR(1)-model (1 nonseasonal autoregressive parameter) and plot the observed values together with forecasts for a period of 20 days after the last observed time-point. Investigate also if the forecasts can improve by instead using an AR(2)-model.

Minitab’s module for ARIMA-modelling (Stat Time Series ARIMA…) is a general model for times series with or without trend and seasonal effects. In the dialog box you should choose different number of Autoregressive and Moving average components and no Difference components. Also you should not mark the square for seasonal modelling. Then an ARMA-model (AR-, MA- or ARMA-) will be estimated. In the dialog box you can also as usual customize the output from the analysis. Click the different buttons (Graphs, Forecast, Results, Storage) and make suitable choices. In the output in the Session window the estimated model will appear and you can examine which model components are significant etc.

With MINITAB’s ARIMA-module you can in one sequence perform the following three operations:

  • calculate one-lag differences in the current time series
  • fit an AR(p)-model to the series of differences
  • calculate forecasts and predictions intervals for the original series

This is done by requesting 1 “difference” and p “autoregressive parameters” in the MINITAB dialog box, keeping the box for seasonal modelling unchecked. Try different values of p (start with p =1) in order to find a suitable forecasting model and draw a graph for observed values and forecasts. Finally perform a residual analysis of the errors in the one-step-ahead forecasts (can be asked for under the “Graph” button in the dialog box. By residuals we mean here the errors in the one-step-ahead forecasts).

Are there any signs of serial correlations in the residuals?

D. ARIMA models and differentiation

In this task you will first have to judge upon whether you need to differentiate the current time series ( zt = y t  yt-1 ) before forecasting with an ARMA-model can be applied. Then you shall try different models with a number of parameters to find the model that gives the last one-step-ahead prediction errors on the average. Finally you shall make some residual plots to investigate if the selected model of forecasting can be improved.

Forecasting monthly dollar exchange rates in Danish crowns (DKK)

Data set: The Dollar-Danish Crowns Exchange rates

D.1. The need for differentiation

Construct a time series graph for the monthly means of dollar exchange rates in Danish crowns (file ‘DKK.txt’). Then estimate the autocorrelations and display them in a graph. Does the time series show any obvious upward or downward trend?

Are there any signs of long-time oscillations in the time series (that can be seen from the time series graph)? Is there a fast cancel-out in the autocorrelations? Is there need for differentiation to get a time series suitable for ARMA-modelling?

In Minitab use StatTime SeriesDifferences…

D.2 Fitting different ARMA-models

Calculate the estimated autocorrelations possibly after differentiation of the original series and display these estimates in a graph. Then try to predict the dollar exchange rate by combining differentiation with ARMA-models of different orders.

How many AR- and MA-terms respectively does it pay to bring into the model (in terms of gain in predictability)? What happens if one tries to fit a very complex model with a lot of parameters to the observations?

D.3. Residual analysis

Construct a graph for the residuals (the one-step-ahead prediction errors) and examine visually if anything points to a possible improvement of the model.