Forecasting with Time Series- Types of Datao Cross-sectional data – data collected at the same, or approximately the same, point in time, often for multiple variableso Time-series data – data collected at more than one period of “time” sequentially over several “periods” at regular intervals for one variable- Naïve Forecasting Methodso One Period Lag Self explanatory, simply move the actual from period before to the forecast of current periodo K-Period Moving Average Self explanatory, take the number of periods from k and average the actual from periods and use average as forecast for current periodo K-Period Weighted Moving Average Assign weights to the k most recent observations. The weights are chosen by the user, and usually assign increasing weight to more recent observations Total weight*actual and divide by total weight- Evaluating Accuracy of the Forecasting Methodo Forecast Error Forecast error for a given time period, t: et= yt−^yt Otherwise known as observed – expectedo Forecast Accuracy Measures Mean Square Error Mean Absolute Deviation Mean Absolute Percentage Error- Less than 8% signifies a good model In all three measures, n is the number of errors used to obtain the numerator The MSE and MAD are units affected while MAPE is not. This allows us to attach a rule of thumb to the use of MAPE. A forecasting method is generally considered effective if the MAPE is < 8%- Exponential Smoothingo Simple Exponential Smoothing A type of weighted average that includes all previous observations in the time series A weighted average of the most recent observation in the time series, yt −1 and the forecast of that observation, ^yt −1 = w yt −1+(1−w)^yt −1 w, called omega, is the “smoothing constant”- 0 < w < 1 Forecast for week #1 and week #2 will always be the same w above 0.5- focus on most current observation w below 0.5 w- focus on the “history”o Time Series with “components” Trend Component- Persistent, overall upward or downward pattern in the response variable over many periods- Contributors include changes in size and geographic distribution of population, technological improvements, gradual shifts in habits Cyclical Component- Measured in years- Shifts in the direction of the response variable from time to time with varying intensity and duration- We tend to view these as expansion or contraction periods in the economy, lasting 2 to several years- Contributors include buildups and depletions of inventories, shifts in rates of capital investments, changes in governmental monetary policies, weather patterns, wars etc.  Seasonal Components- Regular and repeating pattern of fluctuations- Due to weather customs etc.- Completes itself within 1 year Irregular Component- Erratic unsystematic fluctuations- Due to random variation or unforeseen eventso Union Strikeo Tornado- Short duration & nonrepeatingo Smoothing Reduce Random Component Smoothing techniques- If we can determine which components actually exist in a time series, we can develop better forecasts- We can reduce random variation by smoothing the time series- Methods to smooth the data are:o Moving averages Used to look at trends, not forecast Centered moving average- With an even number of observation included in the moving average, the average is placed between the two period in the middle- To place the moving average in an actual time period, we need to center it- Two consecutive moving averages are centered by taking their average and placing it in the middle between them. o Exponential smoothing There are two drawbacks with the moving average method of smoothing:- No moving averages for the first and last sets of time periods- The moving average “forgets” most of the previous time-series values (i.e. only looks at those around it). Exponential smoothing addresses these issues Use as for