Introduction

Forecasting is a common task in business management. Simple linear regression models, we have met a kind of statistical model that can be used as a forecasting tool, provided that

• We are able to find potential explanatory variables
• We have enough data on all the relevant variables, in order to obtain reliable estimates of model parameters

Even though, strictly speaking, linear regression captures association and not causation, the idea behind such a model is that knowledge about explanatory variables is useful to predict the value of the explained variable. Unfortunately, there are many cases in which we are not able to find a convincing set of explanatory variables, or we lack data about them, possibly because they are too costly to collect. In some extreme cases, not only do we lack enough information about the explanatory variables, but we even lack information about the predicted variable. One common case is forecasting sales for a brand-new kind of product, with no past sales history. Then, we might have to settle for a qualitative, rather than quantitative forecasting approach. Qualitative forecasting may take advantage of qualified expert opinion; various experts may be pooled in order to obtain both a forecast and a measure of its uncertainty.¹ Actually, these two families of methods can be and, in fact, are often integrated. Even when plenty of data are available, expert opinions are a valuable commodity, since statistical models are intrinsically backward-looking, whereas we should look forward in forecasting.

We stick to quantitative approaches, such as linear regression, leaving qualitative forecasting to the specialized literature. Within the class of quantitative forecasting methods, an alternative to regression models is the family of time series models. The distinguishing feature of time series models is that they aim at forecasting a variable of interest, based only on observations of the variable itself; no explanatory variable is considered.

Example 11.1 Stock trading on financial markets is one of those human endeavors in which good forecasts would have immense value. One possible approach is based on fundamental analysis. Given a firm, its financial and industrial performance is evaluated in order to assess the prospect for the price of its shares. In this kind of analysis, it is assumed that there is some rationality in financial markets, and we try to explain stock prices, at least partially, by a set underlying factors, which may also include macroeconomic factors such as inflation or oil price. On the contrary, technical analysis is based only on patterns and trends in the stock price itself. No explanatory variable is sought. The idea is that financial markets are mostly irrational, and that psychology should be used to explain observed behavior. In the first case, some statistical model, possibly a complicated linear regression model, could be arranged. In the second case, time series approaches are used.²

(Note: Time series models can be built to forecast a wide variety of variables, such as interest rates, electric power consumption, inflation, unemployment, etc. To be concrete, in most, if not all of our examples, we will deal with demand forecasting. Yet, the approaches we outline are much more general than it might seem.)

To illustrate the nature of time series models formally, let us introduce the fundamental notation, which is based on a discrete time representation in time buckets or periods (e.g., weeks) denoted by t:

• Y_t is the realization of the variable of interest at time bucket t. If we are observing weekly demand for an item at a retail store, Y_t is the demand observed at that store during time bucket t.
• F_t,h is the forecast generated at the end of time bucket t with horizon of h time buckets; hence, F_t,h is a prediction of demand at time t+h, where h = 1, 2, 3,…. It is important to clarify the roles of subscripts t and h in our definition of forecast:
  • t indicates when the forecast is made, and it defines the information set on the basis of which the forecast is built; for instance, at time t we have information about demand during all the time buckets up to and including t.
  • h defines how many steps ahead in the future we want to forecast; the simplest case is h = 1, which implies that after observing demand in time bucket t, we are forecasting future demand in time bucket t + 1.
  We should distinguish quite clearly when and for when the forecast is calculated. One might wonder why we should forecast with a horizon h > 1. The answer is that sometimes we have to plan actions in considerable advance. Consider, for instance, ordering an item from a supplier whose lead time is three time buckets; clearly, we cannot base our decision on a demand forecast with h = 1, since what we order now will not be delivered during the next time bucket.

In time series models, the information set consists only of observations of Y_t; no explanatory variable is considered. Trivial examples of forecasting formulas could be

In the first case, we just use the last observation as a forecast for the next time bucket. In the second case, we take the average of all of the past t observations, from time bucket 1 up to time bucket t. In a sense, these are two extremes, because we either use a very tiny information set, or a set consisting of the whole past history. Maybe one observation is too prone to spikes and random shocks, which would probably add undesirable noise, rather than useful information, to our decision process; on the other hand, the choice of using all of them does not consider the fact that some observations far in the past could be hardly relevant. Furthermore, we are not considering the possibility of systematic variations due to trends or seasonality. In the following, we describe both heuristic and more formal approaches to forecasting.

There is an enormous variety of forecasting methods, and plenty of software packages implementing them. What is really important is to understand a few recurring and fundamental concepts, in order to properly evaluate competing approaches. However, there is an initial step that is even more important: framing the forecasting process within the overall business process. We insist on this in Section 11.1. The most sophisticated forecasting algorithm is utterly useless, if it is not in tune with the surrounding process. Whatever approach we take, it is imperative to monitor forecast errors; in Section 11.2 we define several error measures that can be used to choose among alternative models, to fine-tune coefficients governing their functioning, and to check and improve performance. Section 11.3 illustrates the fundamental ideas of time series decomposition, which highlights the possible presence of factors like trend and seasonality. Section 11.4 deals with a very simple approach, moving average, which has a limited domain of applicability but is quite useful in pointing out basic tradeoffs that we have to make when fine-tuning a forecasting algorithm. Section 11.5 is actually the core where we describe the widely used family of exponential smoothing methods; they are easy to use and quite flexible, as they can account for trend and seasonality in a straightforward and intuitive way. Finally, Section 11.6 takes a more formal route and deals with autoregressive and moving-average models within a proper statistical framework. This last section is aimed at more advanced readers and may be safely skipped.