Question 2

1. Outline thet hinking underlying the Holt-Winters forecasting method. How would you go about determining suitable values for the parameters alpha, beta and gamma

   [8 marks]

2. What do you understand by the terms autocorrelation function and partial autocorrelation function? Outline very briefly how these are used to identify an ARIMA model for a time series using suitable diagrams.

   [5 marks]

3. Show that forecasts produced by an ARIMA (0,1,1) model: Yt Yt-1 = et 0.6et-1 are equivalent to those produced by single exponential smoothing: Ft+1 = 0.4Yt + 0.6Ft.

   [5 marks]

4. Express ARIMA (1,1,1) in terms of the backshift operator B, theta1 and 1.

   Simplify into a form suitable for forecasting.

   [3 marks]

5. An analyst is building a model to forecast the hourly number of calls to the 24/7 phoneline of a national bank. A database of calls logged in each hour stretching back several years is available. One of the models being considered is of the form:

yt = a + b1h1 + b2h2 + ... + b23h23 + b24d1 + b25d2 + ... + b29d6 + twhere:yt is the number of calls logged in hour t;

h1 to h23 are dummy variables representing the hours: 01:00 to 02:00, 02:00 to 03:00, ..., 23:00 to 24:00;

d1 to d6 are dummy variables representing the days: Monday, Tuesday, ..., Saturday;

Ordinary Least Squares Regression will be used to estimate values for a and the coefficients b1 to b29 so as to minimise the sum of squared error terms, t2.

1. i) Outline the thinking underlying the construction of the above model and in particular explain why there are no dummy variables for the hour 00:00 to 01:00 and for Sunday.

   [2 marks]

2. ii) What would be the predicted number of calls for the hour 11:00 to 12:00 on a Tuesday?