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Lecture 7:

Forecasting: Putting it ALLtogether

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The full model

The model with seasonality, quadratic trend, and ARMA

components can be written:

Ummmm, say what???? The autoregressive components allow us to control for the

fact that data is directly related to itself over time.

The moving average components, which are often less

important, can be used in instances where past errors are

expected to be useful in forecasting.

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y t b1D1,t ... bsDs,ta1ta2 t2ut,

ut 1ut1 2ut2 ... putp ...

et 1et1 ...qetq

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Model selection

Autocorrelation (AC) can be used to choose a model.

The autocorrelations measure any correlation or

persistence. For ARMA(p,q) models, autocorrelations

begin behaving like an AR(p) process after lag q.

Partial autocorrelations (PAC) only analyze direct

correlations. For ARMA(p,q) processes, PACs begin

behaving like an MA(q) process after lag p.

For AR(p) process, the autocorrelation is nevertheoretically zero, but PAC cuts off after lag p.

For MA(q) process, the PAC is never theoretically zero,

but AC cuts off after lag q.

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Important commands in EViews

ar(1): Includes a single autoregressive lag

ar(2): Includes a second autoregressive lag

Note, if you include only ar(2), EViews will not include a

first order autoregressive lag

ma(1): Includes a first order moving average term. This

is not the same as forecasting using an average of

recent values

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Selecting an appropriate time

series model, concluded

Determine if trend/seasonality is important

If it is, include it in your model

Estimate the model with necessary trend/seasonal

components. Look at the correlogram of the residuals:

From the equation dialogue box:

View => Residual Tests => Correlogram Q-statistics

If ACs decay slowly with abrupt cutoff in PAC, this is indicative

of AR components. If the PAC doesnt cutoff, you may need toinclude MA components as well.

Re-estimate the full model with trend/seasonality

included with necessary ARMA components. You will

likely have several models to choose from.

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Selecting an appropriate model,

cont.

After you estimate each model, record SIC/AIC values

Use the SIC/AIC values to select an appropriate model.

Finally, investigate the final set of residuals. Thereshould be no correlation in your residuals.

Evidenced by individual correlation coefficients within 95%

confidence intervals about zero.

Ljung-Box Q-statistics should be small with probability

values typically in excess of 0.05.

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Exponential smoothing

Very useful when we have only a handful of observations

Exponential smoothing can be modified to account for

trend and seasonality.

If you suspect your data does not contain

trend/seasonality, simply use single exponential

smoothing:

The forecast h periods into the future is constant and

given by:

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yty

t (1)y

t1 .... (1)t1y1

yTh T yT

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Exponential smoothing with trend

Obtain the smoothed level series, Lt:

Lt=yt+(1-)(Lt-1+Tt-1)

The trend series, Ttis formed as:

Tt=b(Lt-Lt-1)+(1-b)(Tt-1)

The forecasted series:

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yTh|TLT

hTT

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Exponential smoothing: Trend

and seasonality

Eviews also allows for exponential smoothing with trend

and seasonality.

With seasonality we use a smoothed series along with

estimates based on trends and seasonality.

Which option should I select?

If you believe your series lacks either seasonality or trend,

single smoothing works perfectly.

From visual inspection of your series, if only trend appears to

be present, you will need either double smoothing or Holt-Winters with no seasonal.

If seasonality and trend are expected, you will need to use

Holt-Winters with the allowance of multiplicative or additive

seasonality.

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HUH?

Eviews provides five options when you ask it, no tell it,

to provide exponential smoothing:

Single: (no seasonality/no trend)

Double: (trend value of =b).

Holt-Winters No seasonal (Trend, and bare not equal,

but are estimated in the data).

Holt-Winters Additive (Trend and Seasonality. The

seasonal component is estimated with an additive filter).

Holt-Winters - Multiplicative

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Breaks?

Uh oh? My data appears to have a break.

The developed time series methods assume the

black box generating the data is constant.

Not necessarily true:

Learning curves may cause cost curves to

decrease

Acquisition of companies or new technologiesmay alter sales/costs

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Dealing with breaks?

Solutions:

Limit the sample to the post break period

Sometimes taking logs and/or differencing can help

mitigate the effects of breaks/outliers. Include variables that help identify the breaks

Model the breaks directly:

The most obvious way is to include a break in mean and/or a

break in trend.

We should make sure that the included break is modeled in asensible way

A negative linear trend, for example, will imply the data

may eventually turn negative.

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Break in mean

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Break in trend

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0 20 40 60 80 100 120 140 160 180 200

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Statistics useful in comparing theout of sample forecasting accuracy

Mean squared error: For an h-step extrapolation

forecast:

Root mean squared error is the square root of this

number.

Mean absolute error

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( YT1|TYT1)

2 ( Y

T2|TYT2)2 .. . ( Y

Th|TYTh )2

h

| YT1|TYT1||

YT2|TYT2|.. .|

YTh|TYTh |

h

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In Eviews:

If you have a forecasted series, say xf, and an original

series x, you can calculate the mean squared error as:

genr mse=@sum((x-xf)^2)/h

To calculate the moving average forecasts:

Suppose you use the most recent four periods

Limit your data set to include only the last four observations

A variable called maf_4 is calculated by:

genr maf_4=@mean(x)

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