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1. Basic Forecasting Tools

1.1 Forecasting Methods and Examples

1.1.1 Examples:

The first example, [Web: Australian Monthly Electricity Production ], displays a clear trend and seasonality. Note that both the seasonal variability as well as the mean show a trend.

The data [Web: US Treasury Bill Contracts ] shows a trend, but there is less certainty as to whether thistrend will continue.

The data on [Web: Australian Clay Brick Production ] contains occasional large fluctuations which are difficult to explain, and hence predict, without knowing the underlying causes.

Exercise 1.1: Make Timeplots of each data set: Australian Monthly Electricity, US Treasury Bills, Australian Clay Brick.

1.1.2 Quantitative and Qualitative Approach:
Quantitative approachrelies on sufficient reliable quantitative information being available. Qualitative approach is an alternative if expert knowledge is available.

1.1.3 Explanatory Versus Black-Box Models:
An explanatory model is one that attempts to explain the relationship between the variable to be forecast and a number of independent variables. eg

GNP = f(monetary and tax policies, inflation,capital spending, imports, exports) + Error

A time series model is one that attempts to relate the value of a variable(s) at one time point with values of the variable(s) at previous time points. eg

GNPt+1 = f(GNPt, GNPt-1, GNPt-2, ....) + Error

A black-box model is one that simply tries to relate future values of the variable of interest to previous values, withoutattempting to explain its behaviour in terms of other variables.

Thus simple time series models, like the one above, are 'black-box'.

More complex time series models are explanatory in that they try to relate the value of the variable of interest not simply with its previous values but also with previous values of other 'explanatory' variables.

1.2 Graphical Summaries

1.2.1 Time plot.
Alwaysmake a time plot and look for patterns:
(i) A time series is said to be stationary if distribution of the fluctuations is not time dependent In particular both the variability about the mean, as well as the mean must be independent of time.
(ii) A seasonal/periodic pattern is one with a yearly, monthly or weekly period.
(iii) A cyclical pattern is one where there are rises and falls but notof regular period.
(iv) A trend is a long term increase or decrease in the variable of interest.
eg [Web: Australian beer production: Time Plot ]

1.2.2 Seasonal plot.
A seasonal plot is one where the time series is cut into regular periods and the time plots of each period are overlaid on top of one another.
eg [Web: Australian beer production: Seasonal Plot ]

Exercise 1.2: Producetime and seasonal plots for the Australian beer production data.

1.2.3 Scatterplots.
This plots the relationship between two variables, but does not necessarily have to have time as one of the variables.
eg [Web: Price/Mileage relationship for 45 cars ]

Exercise 1.3: Produce a scatterplot for the Price/Mileage relationship for 45 cars data.

1.3 Numerical Summaries

1.3.1 Statistics.A statistic is a summary quantity calculated from a data set.

1.3.2 Univariate Statistics.
Commonly used statistics are the mean, median, deviation, mean absolute deviation (MAD), variance or mean square deviation (MSD); standard deviation (SD).

These are calculated for the data:[Web: 19 Japanese Cars ]

EXCEL contains several of these statistics as Worksheet Functions, specifically:AVERAGE, MEDIAN, VAR, STDEV.
Note: VAR and STDEV now use n – 1 in the divisor. [Also they use an old fashioned version of the formula, which is not fully robust.]

Exercise 1.4: Reproduce, in a spreadsheet the calculations made in the example: 19 Japanese Cars.

1.3.3 Bivariate.
The most commonly used statistics for bivariate data is the covariance, and the correlation coefficient. If...