Using time series models for forecasting

fTime series models may be used for forecasting purposes. As usual, we should find not only a point forecast, but also a prediction interval. Given an information set consisting of observations up to Y_t, we wish to find a forecast , at time t, with horizon h ≥ 1, that is “best” in some well specified sense. A reasonable criterion is the minimization of the mean squared error

It can be shown that this is obtained by the conditional expectation

To be concrete, assume that we have estimated the parameters of a model like

If we want to forecast with horizon h = 1, we should step forward to

To build a point forecast, we note the following:

• If all of the observations Y_t−p+1, …, Y_t are available,¹⁶ we should just plug their values into Eq. (11.42).
• Of course we do not know the future random shock ; however, if random shocks are uncorrelated, we may just plug its expected value  into the equation.
• Unfortunately, we cannot directly observe the past shocks . Simple linear regression, we faced a similar issue related to the difference between unobservable errors and observable residuals. We have to estimate  on the basis of prediction errors.

Example 11.14 Consider the model

In order to forecast Y_t+1, we write

where the prediction of the future shock is , and the past shock is estimated by the observed forecast error

If we want to forecast with horizon h > 1, we should rewrite Eq. (11.42) for the appropriate time subscript, and run a multistep forecasting procedure, whereby successive forecasts are generated and used. As the reader can imagine, when the moving-average order q is large, things are not as simple as in the example above. Indeed, forecasting based on ARIMA models is by no way a trivial business, and alternative approaches have been proposed. One possibility is to rewrite the model as an infinite-order moving average

and plug estimates of random shocks. The procedures to accomplish all of this are beyond the scope of this book;¹⁷ luckily, statistical software packages are available to carry out model estimation and forecasting. These software tools also provide the user with a prediction interval, using procedures which are not unlike those we used to find confidence intervals. The idea is surrounding the point forecast with an interval related to some standard prediction error, using quantiles from t or standard normal distributions. When using these procedures, we should keep in mind the following:

• Typically, it is assumed that random shocks are normal and uncorrected.
• Since the mathematics of time series is quite involved, the only source of uncertainty accounted for is the realization of the future random shock. However, we have seen in Section 10.4 that uncertainty in parameter estimates also plays a role. Unfortunately, simple linear regression is a simple problem with an analytical solution, which lends itself to an accurate analysis; often, to estimate parameters of time series models, numerical optimization is required, which makes a formal analysis quite difficult.

As a result, the forecast uncertainty could be underestimated. Hence, it is good practice to split available data into a fit and a test sample, and assess forecast errors out-of-sample before applying a model in a business setting.

Problems

11.1 Consider the demand data in the table below:

We want to apply exponential smoothing with trend:

• Using a fit sample of size 3, initialize the smoother using linear regression.
• Choose smoothing coefficients and evaluate MAPE and RMSE on the test sample.
• After observing demand in the last time bucket, calculate forecasts with horizons h = 2 and h = 3.

11.2 The following table shows quarterly demand data for 3 consecutive years:

Choose smoothing coefficients and apply exponential smoothing with seasonality:

• Initial parameters are estimated using a fit sample consisting of two whole cycles.
• Evaluate MAD and MPE on the test sample, with h = 1.
• What is F_5,3?

11.3 In the table below, “−” indicates missing information and “??” is a placeholder for a future and unknown demand:

Initialize a smoother with multiplicative seasonality by using a fit sample of size 7.

11.4 We want to apply the Holt–Winter method, assuming a cycle of one year and a quarterly time bucket, corresponding to ordinary seasons. We are at the beginning of summer and the current parameter estimates are

• Level 80
• Trend 10
• Seasonality factors: winter 0.8, spring 0.9, summer 1.1, autumn 1.2

On the basis of these estimates, what is your forecast for next summer? If the demand scenario (summer 88, autumn 121, winter 110) is realized, what are MAD and MAD%?

11.5 Prove that the weights in Eq. (11.18) add up to one. (Hint: Use the geometric series.)

11.6 Prove Eqs. (11.31) and (11.32).

11.7 Consider a moving-average algorithm with time window n. Assume that the observed values are i.i.d. variables. Show that the autocorrelation function for two forecasts that are k time buckets apart is