4.1. Data Availability
A time series is a sequence of similar measurements taken at regular time intervals. Time series analysis, our topic in Chapters 4, 5, 6, 7, and 9, means examining the history of the time series itself to gather information about the future. An inherent assumption in time series analysis is that the past of a series contains information about the future of the same series. Regression modeling, which is our topic in Chapter 8, means using the information contained in another data series to make predictions about the future of a focal time series.
One essential aspect of time series forecasting is that time needs to be “bucketed” into time periods. Many demand forecasts are made on a monthly basis (“how much product will our customers demand next month?”), and thus the time series requires aggregating data into monthly buckets. Note that “a month” is not an entirely regular time interval, since some months have more days than others, but for most applications, this nonregularity is inconsequential enough to be ignored. Financial forecasting works on a quarterly or yearly basis, whereas some operational forecasting requires weekly, daily, and sometimes even quarter-hourly time buckets, for example, in the case of call center traffic forecasting (Minnucci 2006). This temporal dimension of aggregation does, as we shall explain later in this chapter, imply different degrees of statistical aggregation as well, making forecasting more or less challenging. It also raises the question of temporal hierarchies (see Chapter 14), that is, at what level should the organization forecast, and how does the organization aggregate or disaggregate to longer or shorter segments of time?
Another important aspect to understand about your forecast is the availability of relevant historical data. Most methods discussed in the next chapters assume that some demand history for a time series exists. For example, if exponential smoothing is used (see Chapter 6) and the model includes seasonality, then at least nine past quarters are needed for quarterly time series models, and 17 past months are needed for monthly time series models (Hyndman and Kostenko 2007)—and these are minimum requirements; if the time series is very noisy, much more data is needed to get reliable estimates for the method’s parameters and thus obtain dependable forecasts.1 This data requirement may be excessive in many business contexts where the lifecycle of products is only 3 years. If lifecycles are short, changes to the product portfolio need to be carefully assessed as to whether they represent true new product introductions, that is, the introduction of a novel good or service that is incomparable to any existing product in the firm’s portfolio, or a semi-new product introduction, that is, a modified version of a product the firm has sold before (Tonetti 2006). In the former case, the methods we will discuss here do not hold; forecasts will require modeling the product lifecycle, which requires good market research and extensive conjoint analysis to have a chance of being successful. Interested readers are referred to a different book by the same publisher for further information on new product forecasting (Berry 2010). In the latter case, forecasting can proceed as we discuss here, as long as some existing data can be deemed representative of the semi-new product. For example, if the semi-new product is a simple engineering change of a previous version of the product, the history of the previous version of the product should apply and can be used to initialize the forecasting method for the semi-new product. If the change is a change in packaging or style, the level of the time series may change, but other components of the series, such as the trend, seasonality, and possibly even the uncertainty in demand, may remain constant. Thus, the estimates of these components from the past can be used for the new model as initial estimates, greatly reducing the need for a data history to be available. Similarly, if the semi-new product is simply a new variant within a category, then the trend and seasonality that exists at the category level may apply to the new variant as well; in other words, smart top-down forecasting in a hierarchy (see Chapter 14) can allow forecasters to learn about these time series components by looking at the collection of other, similar variants within the same category.
A first step in time series analysis is to understand what data underlies the series. Demand forecasting means making statements about future demand; the data that is stored in company databases often only shows actual sales. The difference between demand and sales comes into play during stock-outs. If inventory runs out, customers may still demand a product, so sales may be lower than the actual demand. In such a case, customers may turn to a competitor, delay their purchase, or buy a substitute product. In the latter case, the substitute’s sales are actually higher than the raw demand for it. If sales are used as an input for demand forecasting, both point forecasts and their associated prediction intervals will be wrong. Modern forecasting software can adjust sales data accordingly if stock-out information is recorded. The mathematics of such adjustments are beyond the scope of this book. Interested readers are referred to Nahmias (1994) for further details.
Adjusting sales to estimate demand requires clearly understanding whether data represents sales or demand. Demand can be very difficult to observe in business-to-consumer contexts. If a product is not on the shelf, it is hard to tell whether a customer walking through the store demanded the product or not. In online retail contexts, demand can be clearly observed if inventory availability is not shown to the customers before they place an item into their shopping basket. However, if this information is presented to customers before they click on purchase, demand is again difficult to observe. Demand is generally easier to observe in business-to-business settings, since customer requests are usually recorded. In modern ERP software, salespeople usually work with an “available-to-promise” number. Running out of “available-to-promise” means that some customer requests are not converted into orders; if these requests are not recorded by the salespeople, databases again only show sales and not demand.
4.2. Stationarity
One key attribute of a time series is referred to as stationarity. Stationarity means that the mean of demand is constant over time, that the variance of demand remains constant, and that the correlation between current and most recent demand observations (and other parameters of the demand distribution) remains constant. Stationarity in essence requires that the time series has constant properties when looked at over time. Many time series violate these criteria; for example, a time series with a trend (see Chapter 5) is not stationary, since the mean demand is persistently increasing or decreasing. Similarly, a simple random walk (see Chapter 5) is not stationary since mean demand randomly increases or decreases in every time period. In essence, nonstationary series imply that demand conditions for a product change over time, whereas stationary series imply that demand conditions are very stable. Some forecasting methods, such as the ARIMA methods discussed in Chapter 7, work well only if the underlying time series is stationary.
Time series are often transformed to become stationary before they are analyzed. Typical data transformations include first differencing, that is, examining only the changes of demand between consecutive time periods; calculating growth rates, that is, examining the normalized first difference; or taking the natural logarithm of the data. Suppose, for example, one observes the following four observations of a time series: 100, 120, 160, and 150. The corresponding three observations of the first difference series become 20, 40, and -10. Expressed as growth rates, this series of first differences becomes 20, 33, and -6 percent.
Essential to these transformations is that they are reversible. While estimations are made on the transformed data, the resulting forecasts can be easily transformed back to apply to the untransformed time series. The benefit of such transformations usually lies in the reduction of variability and in filtering out the unstable portions of the data. There are several statistical tests for stationarity that will usually be reported in statistical software, such as the Dickey–Fuller test. It is useful to apply these tests to examine whether a first-differenced time series has achieved stationarity or not.
A common mistake in managerial thinking is to assume that using “old” data (i.e., 4–5 years ago) for forecasting is bad, since obviously so much has changed since then. Modern forecasting techniques will deal with this change without excluding the data; in fact, they need data that shows how much has changed over time, otherwise the methods may underestimate how much the future can change from the present. Excluding data is rarely a good practice in forecasting. A long history of data allows the forecaster and his/her methods to more clearly assess market volatility and change.
4.3. Forecastability and Scale
Another aspect to understand about a time series is the forecastability of the series. As discussed in Chapter 1, some time series contain more noise than others, making the task of predicting their future realizations more challenging. The less forecastable a time series is, the wider the prediction interval associated with the forecast will be. Understanding the forecastability of a series not only helps in terms of setting expectations among decision makers, but is also important when examining appropriate benchmarks for forecasting performance. A competitor may be more accurate at forecasting if they have a better forecasting process or if their time series are more forecastable. The latter may simply be a fact of them operating at a larger scale, with less variety, or their products being less influenced by current fashion and changing consumer trends.
One metric that is used to measure the forecastability of a time series is to calculate the ratio of the standard deviation of the time series data itself to the standard deviation of forecast errors using a benchmark method (Hill, Zhang, and Burch 2015). The logic behind this ratio is that the standard deviation of demand is in some sense a lower bound on forecasting performance since it generally corresponds to using a simple, long-run average as your forecasting method for the future. Any useful forecasting method should not lead to more uncertainty than the uncertainty inherent in demand. Thus, if this ratio is >1, forecasting in a time series can benefit from more complex methods than using a long-run average. If this ratio is close to 1 (or even <1), the time series currently cannot be forecast any better than using a long-run average.
This conceptualization is very similar to what some researchers call “Forecast Value Added” (Gilliland 2013). In this concept, one defines a base accuracy for a time series by calculating the forecast accuracy achieved (see Chapter 11) by the best simple method—either using a long-run average or the most recent demand—to predict the future. Every step in the forecasting process, whether it is the output of a statistical forecasting model, the consensus forecast from a group, or the judgmental adjustment to a forecast by a higher level executive, is then benchmarked in terms of their long-run error against this base accuracy; if a method requires effort from the organization but does not lead to better forecast accuracy compared to a method that requires less effort, it can be eliminated from future forecasting processes. Results from such comparisons are often sobering—some estimates suggest that in almost 50 percent of time series, the existing toolset available for forecasting does not improve upon simple forecasting methods (Morlidge 2014). In other words, demand averaging or simple demand chasing may sometimes be the best a forecaster can do to create predictions.
Some studies examine what drives the forecastability of a time series (Schubert 2012). Key factors here include the overall volume of sales (larger volume means more aggregation of demand, thus less observed noise), the coefficient of variation of the series (more variability relative to mean demand), and the intermittency of data (data with only few customers that place large orders is more difficult to predict than data with many customers that place small orders). In a nutshell, the forecastability of a time series can be explained by characteristics of the product as well as characteristics of the firm within its industry. There are economies of scale in forecasting, with forecasting at higher volumes being generally easier than forecasting for very low volumes.
The source of these economies of scale lies in the principle of statistical aggregation. Imagine trying to forecast who among all the people living in your street will buy a sweater this week. You would end up with a forecast for each person living in the street that is highly uncertain for each individual. However, if you just want to forecast how many people living in your street buy a sweater in total, the task becomes much easier. At the individual level, you can make many errors, but at the aggregate level, these errors cancel out. This effect will increase the more you aggregate—that is, predicting at the neighborhood, city, county, state, region, or country level. Thus, the forecastability of a series is often a question of the level of aggregation that a time series is focused on. Very disaggregate series can become intermittent and thus very challenging to forecast (see Chapter 9 for details). Very aggregate series are easier to forecast, but if the level of aggregation is too high, these forecasts become less useful for planning purposes as the information they contain is not detailed enough.
It is important in this context to highlight the difference between relative and absolute comparisons in forecast accuracy. In absolute terms, a higher level of aggregation will have more uncertainty than each individual series, but in relative terms, the uncertainty at the aggregate level will be less than the sum of the uncertainties at the lower level. If you predict whether a person buys a sweater or not, your absolute error is at most 1, whereas the maximum error of predicting how many people in your street buy a sweater or not depends on how many people live in your street; nevertheless, the sum of the errors you make at the individual level will be less than the error you make in the sum. For example, suppose five people live in your street, and we can order them by how far into the street (i.e. first house, second house, etc.) they live. You predict that the first two residents buy a sweater, whereas the last three do not. Your aggregate prediction is just that two residents buy a sweater. Suppose now, actually only the last two residents buy a sweater. Your forecast is 100 percent accurate at the aggregate level, but only 20 percent accurate at the disaggregate level. In general, the standard deviation of forecast errors at the aggregate level will be less than the sum of the standard deviations of forecast errors made at the disaggregate level.
The ability to use more aggregate forecasts in planning can also be achieved through product and supply chain design, and the benefits of aggregation here are not limited to better forecasting performance but also include reduced inventory costs. For example, the concept of postponement in supply chain design favors postponing the differentiation of products until later in the process. This enables forecasting and planning at higher levels of aggregation for longer within the supply chain. Paint companies were early adopters of this idea by producing generic colors that are then mixed into the final product at the retail level. This allows forecasting (and stocking) at much higher levels of aggregation. Similarly, Hewlett-Packard demonstrated how to use distribution centers for the localization of their products in order to produce and ship generic printers into the distribution centers. A product design strategy that aims for better aggregation is component commonality, or so-called platform strategies. Here, components across SKUs are kept in common, enabling production and procurement to operate with forecasts and plans at a higher level of aggregation. Volkswagen is famous for pushing the boundaries of this approach with its MQB platform, which allows component sharing and final assembly on the same line across such diverse cars as the Audi A3 and the Volkswagen Touran. Additive manufacturing may become a technology that allows planning at very aggregate levels (e.g., printing raw materials and flexible printing capacity), thereby allowing companies to deliver a variety of products without losing economies of scale in forecasting and inventory planning.
4.4. Key Takeaways
• Understanding your data is the first step to a good forecast.
• The objective of most forecasts is to predict demand, yet the data available to prepare these forecasts often reflects sales; if stock-outs occur, sales are less than demand.
• Many forecasting methods require time series to be stationary, that is, to have constant parameters over time. Stationarity can often be achieved by suitable transformations of the original time series such as differencing the series.
• A key attribute of a time series is its forecastability. Your competitors may have more accurate forecasts because their forecasting process is better or because their time series are more forecastable.
• There are economies of scale in forecasting; predicting at a larger scale tends to be easier due to statistical aggregation effects.
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1If too little data is available, the risk of detecting seasonality where none exists is much higher than the risk of failing to detect seasonality if it exists. Shrinkage methods to better deal with seasonality in such settings are available (Miller and Williams 2003).