**Forecasting**

**3. Basic Forecasting Methods**

In this chapter we suppose that we are currently at time *t* and
that we wish to use the data up to this time, ie *Y*_{1}, *Y*_{2},
..., *Y _{t}*

The methods to be considered are conventionally regarded as being
divided in two groups: (i) *averaging methods* and (ii) *exponential
smoothing methods*.

Though it is convenient to follow this convention it is important to
realise at the outset that this distinction is artificial in that *all*
the methods in this chapter are methods based on averages. They are thus all
similar to the moving averages considered in the last chapter. The difference
is that the averages are used for *forecasting* rather than for *describing*
past data.

This point of potential confusion is made worse by the use of the name
'exponential smoothing' for the second group. These methods are also based on
weighted averages, where the weights decay in an exponential way from the most
recent to the most distant data point. The term smoothing is being used simply
to indicate that this weighted average smoothes the data irregularities. Thus,
though the term smoothing here is used in the same sense as previously, the
smoothing is being carried out in a different context from that used in the
previous chapter.

**3.1 Averaging Methods**

The *moving average forecast of order k ,* which we write as MA(*k*),
is defined as

_{}.

This forecast is only useful if the data does not contain a trend-cycle
or a seasonal component. In other words the data must be *stationary*.
Data is said to be stationary if *Y _{t}*,
which is a random variable, has a probability distribution that does not depend
on

A convenient way of implementing this forecast is to note that

_{}

This is known as an *updating formula* as it allows a forecast
value to be obtained from the previous forecast value by a simpler calculation
than using the defining expression.

The only point of note is that moving average forecasts give a
progressively smoother forecast as the order increases, but a moving average of
large order will be slow to respond to real but rapid changes. Thus, in
choosing *k*, a balance has to be drawn between smoothness and ensuring
that this *lag* is not unacceptably large.

**3.2 Single Exponential Smoothing**

The *single exponential forecast* or *single exponential
smoothing* (SES) is defined as

*F _{t+}*

where *α* is a given *weight*
value to be selected subject to 0 < *α* < 1. Thus *F _{t+}*

Repeated application of the formula yields

_{}

showing that the dependence of the
current forecast on *Y _{t}*,

SES needs to be initialized. A simple choice is to use

*F*_{2} = *Y*_{1}.

Other values are possible, but we shall not agonise over this too much
as we are more concerned with the behaviour of the forecast once it has been in
use for a while.

**Exercise 3.1:** Employ SES for the Shampoo Data.

1. Set up *α* in a named cell.

2. Plot the *Y _{t}* and
the SES.

3. Calculate also the MSE of the SES from *Y*.

4. Try different values of *α*. Use Solver to
minimize the MSE by varying *α*.

[Web: Shampoo Data
].

**3.3 Holt's Linear
Exponential Smoothing (LES)**

This is an extension of exponential smoothing to take into account a
possible linear trend. There are two smoothing constants *α* and *β*.
The equations are:

_{}

Here *L _{t}* and

Initial estimates are needed for *L*_{1} and *b*_{1}.
Simple choices are

*L*_{1} = *Y*_{1}*
* and *b*_{1} = 0.

If however zero is atypical of the initial slope then a more careful
estimate of the slope may be needed to ensure that the initial forecasts are
not badly out.

**Exercise 3.2:** Employ LES for
the Shampoo Data.

1. Set up *α* and* β* in a named cells.

2. Plot the *Y _{t}* and
the LES.

3. Calculate also the MSE of the LES from *Y*.

4. Try different values of *α* and* β*.
Use Solver to minimize the MSE by varying *α* and* β*.

5. Make sure that you arrange the calculations in a neat
block of cells with the *Y _{t}* in a
column on the left. This will allow you to use LES on different data sets. This
will be useful later on in the Unit.

[Web: Shampoo Data
].

**3.4 Holt-Winter's Method**

This is an extension of Holt's LES to take into account seasonality.
There are two versions, with the multiplicative the more widely used.

**3.4.1 Holt-Winter's Method, Multiplicative Seasonality**

The equations are

_{}

where *s* is the number of
periods in one cycle of seasons e.g. number of months or quarters in a year.

To initialize we need one complete cycle of data, i.e. *s *values.
Then set

_{}

To initialize trend we use *s + k *time periods.

_{}.

If the series is long enough then a good choice is to make *k = s*
so that two complete cycles are used. However we can, at a pinch, use *k = *1.

Initial seasonal indices can be taken as

_{}

The parameters *α*, *β*, *γ* should lie
in the interval (0, 1), and can be selected by minimising MAD, MSE or MAPE.

**3.4.2 Holt-Winter's Method, Additive Seasonality**

The equations are

_{}

where *s* is the number of
periods in one cycle.

The initial values of *L _{s} *and

_{}.

The parameters *α*,* β*,* γ* should lie
in the interval (0, 1), and can again be selected by minimising MAD, MSE or
MAPE.

**Exercise 3.3:** Apply Holt-Winter's forecasting to
the Airline Passengers Data.

1. Follow the same arrangement as in Exercise 3.2.

2. Compare your results for Holt-Winter's methods with those
for SES and for LES.

[Web: Airline Passengers
Data ]