Chapter 3 – Moving Average and Exponential Smoothing

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## Chapter 3 – Moving Average and Exponential Smoothing

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**Installing ForecastX**• Here we go • Link to Software 1. Install 2. Activate the plug-in 3. Activate Analysis ToolPak, if it’s not already Note: on lab machines, you may have to install every day you use it… Deep freeze!!!**The naïve model – assumes most recent values are the best**predictors. Alternatively: • Moving average (MA) – assumes the best predictor is some average of past values. • Exponential smoothing models – best predictors are both present and past values, but more distant values are of decreasing usefulness. • You could think of both MA and ES as “smoothing models,” just with different weighting schemes, with different emphasis placed on the past.**MA & Smoothing Models**• Advantages • -minimal data requirements • -can produce accurate short-range forecasts • -some models allow for a trend or seasonality • -widely used and accepted • Disadvantages • -don’t allow for a cyclical pattern in the data • -add little understanding with regard to why the • variable to be forecast changes over time**Moving Averages-an “MA” process**• An MA assumes there is some underlying “smooth” curve that represents the path the data is actually taking. • Sharp up and down movements are assumed to be random departures from this underlying curve. • Estimating the MA helps “smooth out” short-term fluctuations to reveal that curve. • A moving average is calculated just exactly like it sounds.**Forecasting with MA’s**MA(2) or a two-period Moving average • Ft = (Xt-1 + Xt-2) / 2 MA(3) or a two-period Moving average • Ft = (Xt-1 + Xt-2 + Xt-3) / 3 …and, so on Question…what does an MA(1) look like???**Why use the MA in this instance?**• May supply clearer signaling of direction and momentum. • Suppose you are following exchange rates… You can use it to develop rules of thumb to determine which currency to hold. If you see a few periods of decline in the MA, it might be a signal of the market momentum downward…sell off yen. If you see a few periods of increases in the MA, it might signal an upward momentum…buy yen.**Two Real-World Applications Using Moving Averages**• Initial Unemployment Claims to forecast changes in the economy • Buy and Sell Signals for the Stock Market**Initial Unemployment Claims**• Compute 4-week moving average of initial unemployment insurance claims for the U.S. • When 4-week moving average < 400,000 • Indicates improving U.S. labor market conditions • When 4-week moving average > 400,000 • Indicates deteriorating U. S. labor market conditions**Buy and Sell Signals for Stock Market**• Compute 24-month moving average of stock market index • When index > than 24-month moving average, • A buy signal is triggered • When index < than 24-month moving average, • A sell signal is triggered**Time Magazine Handout**Some folks use a rule of thumb to determine their stock buys and sells. Handout**Chapter 3: Un-graded Homework**• P 151-152 Problems 6 and 7 (to be covered next week on Tue) • I’ll post the questions just in case.**Smoothing Models**-all smoothing model forecast are simply based on weighted averages of past values… The weights are just different for different models**Smoothing Models**• Advantages • -minimal data requirements • -can produce accurate short-range forecasts • -some models allow for a trend or seasonality • -widely used and accepted • Disadvantages • -don’t allow for a cyclical pattern in the data, no trends or seasonality. • -add little understanding with regard to why the • variable to be forecast changes over time**A Specific Problem with theMA Model**• The MA model treats each of the observations used in the average equally. • Intuitively, past data should affect the present, but the effect should decline over time or with distance.**Exponential Smoothing Models**• forecast is weighted average of values from all available previous periods where more recent periods are given more weight. • Simple Exponential Smoothing Model - • - the simplest model assumes no trend or seasonality in data**Simple Exponential Smoothing (SES) Model**• Ft +1 = αXt + (1 - α ) Ft where • Ft +1 = Forecast value for period t + 1 • α = Smoothing constant ( 0 < α < 1) • Xt = Actual value in present (period t) • Ft = Forecast for present (period t) i.e., the smoothed value**Re-writing the model to see one of the neat things about the**SES Model Ft +1 = αXt + (1 –α ) Ft Ft +1 = αXt + Ft–αFt Ft +1 = Ft + α (Xt – Ft) It’s forecasting error for period t…hmmm. Each forecast “learns” from past error. What is this???**The model “learns” from its past mistakes!!!**• The forecast value at period t+1 is increased if the actual value for period t is greater than it was forecasted to be, and vice versa. Thus, all past forecasting error is used to adjust the model as it goes forward.**Simple Example**5 periods of data + 1 period forecast Every past forecast value is used to produce each current forecast. This is a Recursive forecast. The entire history is used to make the current forecast…at least to some degree. Each value of F affects all subsequent values. • Ft -3 = αXt-4 + (1 - α ) Ft-4 • Ft -2 = αXt-3 + (1 - α ) Ft-3 • Ft -1 = αXt-2 + (1 - α ) Ft-2 • Ft = αXt-1 + (1 - α ) Ft-1 • Ft +1 = αXt + (1 - α ) Ft The forecast**Recursive?!…how is this different than earlier forecasts?**• Each period’s forecast contains all of the previous forecasts. So, anything that changes past forecasts (a.k.a., fitted values) will affect all subsequent forecasts. • Previous forecasting methods only use the actual values of X in forecasts…not, past “fitted values” (those estimated by the model where data already exists for X). • See page 105, equation (3.3) for more insight.**Simple Exponential Smoothing Model**• Ft +1 = weighted average of actual values in past • It can be shown (trust me on this) that • Ft +1 = αXt + (1 - α ) α Xt-1 + (1 - α )2α Xt-2 + . . . • where • Xt = value in present period • Xt-1 = value in previous period • Xt-2 = value in period before previous period**How is this accomplished in practice???**The forecast value for the first period is chosen somewhat arbitrarily. $F$1 is where I put a. This called Initializing the model. Notice the use of the Previous period Forecast (column C) in the formula.**How is this accomplished in practice???**Using different initialization values affects all the periods’ forecasts.**Common Initialization methods**• Using the first X • Averaging the first three X’s • Using the average X for the series • Square root of mother’s SSN… well, maybe not that one….**Here is a Little Excel Hint…**• To show the formulas instead of the values they create hold down <CTRL> and the <`> The <`> is usually on the same key as the <~>. Do the same thing to toggle back to the values.**Excel hint # 2, Checking Formulas**• You can check your formulas by using the [tools] • >[formula auditing] • >formula auditing toolbar • You can show the flow of the formula so you can quickly check to see if you are doing it right and using the correct cells.**This button allows you to check the**Components of your formula. These arrows Indicate what cells are being used.**Why is it called exponential?**• It’s base on the weights and how they decline in importance with time. • (geometrically or exponentially, rather than arithmetically or linearly) • Small values of a (like 0.1) result in a slow decay. Past values are weighted almost as much as more recent ones. • Large values of a (like 0.9) result in a fast decay. Recent values are weighted more than past ones.**Simple Exponential Smoothing Model**• the closerα is to 0, the more evenly the weights are spread out over more periods • α = . 1 • Ft +1 = αXt + (1 - α ) α Xt-1 + (1 - α )2α Xt-2 + . . . • = .1Xt + (1 - .1) .1 Xt-1 + (1 - .1 )2 .1 Xt-2 + . . . • = .1Xt + (.9) .1 Xt-1 + (.9)2 .1 Xt-2 + . . . • = .1Xt + .09 Xt-1 + (.81) .1 Xt-2 + . . . • = .1Xt + .09 Xt-1 + .081 Xt-2 + . . . • 19% of weight is given to two most recent periods**Simple Exponential Smoothing Model**• the closerα is to 1, the more weight is given to the most recent periods • α = . 9 • Ft +1 = αXt + (1 - α ) α Xt-1 + (1 - α )2α Xt-2 + . . . • = .9Xt + (1 - .9) .9 Xt-1 + (1 - .9 )2 .9 Xt-2 + . . . • = .9Xt + (.1) .9 Xt-1 + (.1)2 .9 Xt-2 + . . . • = .9Xt + .09 Xt-1 + (.01) .9 Xt-2 + . . . • = .9Xt + .09 Xt-1 + .009 Xt-2 + . . . • 99% of weight is given to two most recent periods**How does the choice ofa affect the present weight of past**values 0.3044 of the total weight in the first 4 periods. 0.9999 of the total weight in the first 4 periods. So, your choice of a has a substantial effect on the forecast.**Simple Exponential Smoothing Model(degenerate SES model)**• whenα = 1, model becomes simple naïve model • α = 1 • Ft +1 = αXt + (1 - α ) α Xt-1 + (1 - α )2α Xt-2 + . . . • = 1Xt + (1 - 1) 1 Xt-1 + (1 - 1)2 1 Xt-2 + . . . • = 1Xt + (0) 1 Xt-1 + (0)2 1 Xt-2 + . . . • = 1Xt + 0 Xt-1 + 0 Xt-2 + . . . • = Xt • All of weight is given to the most recent period**Simple Exponential Smoothing Model(degenerate SES model)**• asα approaches 0, simple exponential smoothing model becomes a moving average with all previous periods given equal weight…but, this is just aaaaaa What???? Horizontal straight line! Or, an MA(N) where N is the total number of periods.**Choosing a**• Select values close to 0 if the series has a lot of “random variation.” More distant past values remain important predictors • Select values close to 1 if there appears to be a recent shift (or some other structural change) in the series. Recent values are more important than those in the more distant past • Use RMSE to optimize.**E.g., The UMICH Index of Consumer Sentiment**• The purpose is to measure changes in consumer attitudes and expectations. • Questions about the decisions to save, borrow, or make discretionary purchases, and to forecast changes in aggregate consumer behavior. • Attempts to measure consumer “confidence.” • Started in the late 1940s (quarterly). During 1977 and thereafter, (monthly). • Current data location: http://www.icpsr.umich.edu**Remember: SES Model Summary**• Uses past X’s and F’s. • Learns from errors. • assumes no trend or seasonality in data. • produces biased forecasts when there is a trend in the data. • With a positive trend, forecasts are too low • With a negative trend, forecasts are too high