Time Series Analysis

　　ACCA P3考試：Time Series Analysis

　　1. Uses

　　The use of time series models is twofold：

　　To obtain an understanding of the underlying forces and structure that produced the observed data point；

　　To fit a model and proceed to forecasting, monitoring or even feedback and feed-forward control.

　　Time series analysis is used for many applications, such as：

　　Economic forecasting；               Stock market analysis；

　　Sales forecasting；                   Process and quality control；

　　Budgetary analysis；                 Inventory studies and workload projections.

　　2. Components

　　Most time series patterns can be described in terms of two basic components： trend and seasonal variation （“seasonality”）.

　　Trend describes a direction of change in the data that tends to occur at a similar rate over the short run. Within a long-term trend, data may change at varying rates or even reverse direction for short periods before continuing at the trend rate. After introducing a new product, for example, a firm may see sales grow slowly, followed by exponential growth as the new product catches on.

　　Seasonality describes calendar-related effects such as sales preceding certain holidays, air miles flown during vacation seasons, etc.

　　Cyclicality is another component. It arises when data plots in a repeating pattern around the trend line over a period lasting more than one year （e.g. economic expansion and contraction）. Business cycles are notoriously difficult to forecast, so they are often combined with trend effects into “trend-cycle” analysis.

　　Trend and seasonality components may coexist in real-life data. For example, sales of a company can grow over years but still follow consistent seasonal patterns （e.g. 25％ of sales each year are made in December and only 4％ in August）. In many cases it is necessary to establish a trend for a series and then adjust each new data point for seasonality. This may be done monthly, quarterly or semi-annually （depending on the review period over which management will compare actual results to forecast）. Various methods exist for removing seasonal and cyclical noise from data and to make forecasts：

　　Random walk： Next period's prediction is based on the latest actual. Because the data move up and down due to non-trend factors, however, this method may place too much emphasis on the latest actual result.

　　Simple moving average： Next period's prediction is based on the latest moving average of n values for the series.

　　Weighted moving average：* Weights are assigned to observations, such that more recent results may be given more weight than older results.

　　This still suffers from other problems with the simple moving average method, but can be another improvement over the random walk.

　　It may be better to select equal weights for highly variable series.

　　Exponential smoothing： Weights are assigned to last period's actual result using the “smoothing constant” and to last period's forecast amount （1 minus the smoothing constant）. Because the forecast value for the current period is the weighted actual value plus weighted forecast value of the prior period, this method implicitly gives weight to all actual values in determining the next period forecast.