Chapter 5 Time Series Analysis Time series are analyzed to discover past patterns of variability that can be used to forecast future values. Decomposition - identify components that influence the series. Trend Cyclical Seasonal Irregular

Decomposition Additive model: Multiplicative model: Yt = T + S + I

Decomposition An annual series is a product of trend and cyclical fluctuations: Y = TC This is a multiplicative model where trend is in original units and the cyclical is an index. Series that is measured in less than a year (monthly and quarterly data): Y = TSCI

Trend Basic forces in trend: population change, price change, technological change, productivity change, product life cycles Two basic purposes: project the trend and to eliminate it from the original data. Trend analysis: independent variable (X) is time Method most widely used to describe straight line trends is least squares method. Computes the line that best fits a group of points mathematically. Assumes that the correct trend curve is selected and that the curve that fits the past is indicative of the future.

Trend Curves Life cycle curves: introduction, growth, maturity, decline. Linear models assume that a variable is increasing by a constant amount each period. Life cycle curves assume increases at an increasing rate. Exponential curves fit data that is growing at a constant rate instead of a constant amount. Growth curves (Gompertz) represent industries and products that grow at a declining rate. Project management life cycles. Refer to articles on forecasting product life cycles

Seasonal Variation Trend is determined directly from all available data. Seasonal component is determined by eliminating all the other components. Trend is represented by one equation. A separate seasonal value has to be calculated each period, usually in the form of an index number. An index number is a percentage that represents changes over time. Most common calculation is ratio-to-moving average for the multiplicative decomposition model. Seasonal index represents the extent of seasonal influence for a particular segment of the year. The calculation involves a comparison of the expected values of that period to the overall average.

Seasonal Variation A seasonal index of 100 for a particular month indicates that the expected value of that month is 1/12 of the total for the annual period. A seasonal index of 125 indicates that the expected value for that month s 25% greater than 1/12 of the annual total A seasonal index of 80 indicates that the expected value for that month is 20% less than 1/12 of the total activity for the year. Monthly index indicates the expected ups and downs in monthly (quarterly) activity with effects due to trend, cyclical, and irregular components REMOVED.

Ratio-to-Moving Average Centered moving average is used for comparison of values at different points in time. Moving average values are placed at the period in which they are calculated. For example, for a moving average length of 3, the first numeric moving average value is placed at period 3, the next at period 4, and so on. When you center the moving averages, they are placed at the center of the range rather than the end of it. This is done to position the moving average values at their central positions in time. See new car registrations for example

Car Registrations For monthly data use a 12-month centered moving average, quarterly data uses a 4-month CMA. This removes seasonal effects leaving only long-term trend, cyclical, and irregular components. CMA smoothes short-run fluctuations. Median is less sensitive to outliers

Seasonally Adjusted Data Allows reliable comparison of values at different points in time Easier to understand the relationships among economic/business variables once seasonal effects are removed Helpful for short-term forecasts Simplify data for easy interpretation without significant loss of information Deseasonalized - original values are divided by their corresponding seasonal index. TCI = Y/S

Cyclical Variation Residual Method - cyclical component of time series data is identified by eliminating or averaging out trend effects. If the data is an annual series, trend components are removed. If the data are monthly/quarterly, trend and seasonal effects are removed. Multiplied by 100 for percentage

Cyclical index shows the position of each Y value relative to the trend line. New registrations were about 18% below what was expected from the trend line

Cyclical Variation Plot the cyclical index over time The trend line is the 100% base line. Once plotted, it is very easy to see the cyclical patterns. Does the series cycle? If so, how extreme is the cycle? Does the series follow the general economy/business cycle? (Do peaks occur when the economy is strong and bottom out when the economy is weak? Business indicator - business related time series that are used to help assess the general state of the economy.

Economic Indicators Certain statistical time series may be useful as direct indicators of cyclical expansions and contractions in business activity. National Bureau of Economic Research has 22 business indicators: Leading (11 of 22) - anticipate turning points up or down Coincident (4 of 22) - indicate economy’s current performance Lagging (7 of 22) - lag behind the general upswing/downswing of the economy.

Cyclical Cautions Difficult to identify cyclical turning points near the time they occur - because the series also contains short term irregular components No uniformity occurs in the length of time by which a given leading indicator precedes cyclical turns in the economy. For example, leading indicators may signal a recession or recovery some time in the future, but they provide less help in establishing the timing of the turn. False signals - a turning point does not materialize. Should be used together with other data - but analysts should beware of limitations.

Long Term Forecasts Most important aspect is to predict the direction! If the trend is the dynamic value, the model can be used to forecast long term. This is determined if the trend equation does a good job at fitting past data. If the cyclical is the most important, the model should only be used to forecast one period ahead. The equation estimates T and we use subjective data to estimate the cyclical effect for Y-hat = T x C 7.988 + .0687(34) = 10.324 (this is the value for T) Based on coincident and leading indicators, estimate an upswing. C is estimated to be 83. Forecast for period 34 = 10.324*.83 = 8.569

Cyclical and Irregular Effects CI = Y/TS I = CI/C The irregular component measures the variability of the time series after the other components have been removed. Outboard Sales Example: (key the data on Page 329) Use a combination of Minitab and Excel for the data analysis.

Outboard Sales Example T column is based on time series regression. Calculate the fitted values for each period SCI column is Y/T S column is from Minitab output - seasonal index per period TCI column is Y/S CI column is Y/TS C column is a 3 month moving average of CI. This is the cyclical index for every quarter I column is CI/C Need to use Excel to calculate CI, C, and I

Seasonal Forecasting Forecast for trend with the equation Use the adjusted seasonal index for the appropriate month/quarter Estimate the cyclical component using indicators - remember the important aspect is to get the direction correct Commonly use 1 to indicate the irregular component since irregular effects are usually random noise