n Chapter 1
, we have learned that our objective is to construct forecasts based on time series models. These are representations (equations) that link past information with the present, and by doing so, they summarize the time dependence in the data. Time dependence is the key to predicting future values of the variable of interest. However, before constructing a time series model, the forecaster needs to consider three basic elements that will guide the production of the forecast. These are:
1. The information set.
2. The forecast horizon.
3. XThe loss function.
These elements are a priori choices that the forecaster must make. In this sense, we call them tools. If you are ready to build a dining table, you need to choose nails, hammers, wood, glues, machine saws, and so on. In the same fashion, if you are ready to build a time series model, you need to choose at least the three basic elements just listed. For instance, suppose that you wish to forecast the number of new homes in Riverside County. You will need to collect information related to the construction sector in the area, the state of the local economy, the population inflows, the actual supply of houses, and so on. You are constructing the information set by gathering relevant and up-to-date information for the problem at hand. This information will be fed into the time series models. Because different models will process information differently, it may happen that some information is more important than others or that some information may be irrelevant for the forecast of interest.
The forecaster needs to choose how far into the future she wishes to predict. Do we want a 1-month-ahead, a 1-day-ahead, or a 1-year-ahead prediction? It depends on the use of the forecast. For instance, think about policy makers who plan to design or revamp the transportation services of the area or any other infrastructure. It is likely that they will be more interested in long-term predictions of new housing (i.e., forecasts for 1 year, 2 years, 5 years) than in short-term predictions (i.e., forecasts for 1 day, 1 month, 1 quarter). The forecast horizon
influences the choice of the frequency of the time series data. If our interest is a 1-month-ahead prediction, we may wish to collect monthly data, or if our interest is a 1-day-ahead forecast, we may collect daily data. Of course, it is possible to forecast 1 month ahead with daily observations but, in some instances, this may not be desirable.
The forecaster must deal with uncertainty, which is inherent in any exercise involving the future. Only when time passes and the future becomes a reality does the forecaster know whether her prediction was right or wrong, and if it is wrong, by how much. In other words, forecast errors will happen but more importantly, they will be costly. The loss function is a representation of the penalties associated with forecast errors. Suppose that based on the forecast of new housing construction, some policy makers decide to invest in a system of new highways. If the forecast happens to overestimate the construction of new housing and, as a result, new construction is less than expected, the highways likely will be underutilized. On the contrary, if the forecast underestimates the construction of new housing and construction is more than expected, the highways will be overcrowded and congested. Either case has a cost. In the first case, more was invested than needed; thus, resources were wasted. In the second case, there were costs associated with traffic congestion, air pollution, longer commuting times, and so on. The costs of underestimation and of overestimation may be of different magnitude. It is sensible to assume that the forecaster may want to avoid forecast errors that are costly and to choose a forecast that minimizes the forecaster’s losses. This is deeme...