1. Time Series Analysis Autocorrelation Naive & Simple Averaging
Moving Averages
Exponential Smoothing
Regression Analysis

2. Time Series Models Trends:
linear moving average, exponential smoothing, Regression, growth curves
Seasonality:
classical decomposition, multiple regression, time series & Box-Jenkins
Cyclical:
classical decomposition, economic indicators, econometric models, multiple regression and Box-Jenkins

3. Evaluating Methods
Forecasting method is selected - many times by intuition, previous experience, or computer resource availability
Divide the data into two sections - an initialization part and a test part
Use the forecast technique to determine the fitted values for the initialization data set
Use the forecast technique to forecast the test data set and determine the forecast errors
Evaluate errors (MAD, MPE, MSE, MAPE)
Use the technique, modify, or develop new model

4. Naive Models Recent periods are the best predictors of the future
Adjustments to naive models
Trend
Rate of Change

5. Use 1990-95 as initialization Use 1996 as the test data set
Forecast the first period in 1996
Forecast error:
Forecast for the remaining 1996 quarters and calculate the error - what do you see happening?

6. Naïve Models: Trended Data
Nonstationary - data values increase over time

7. Naïve Methods: Rate of Change
Can also use Naïve models for seasonal forecasts - data indicates that Quarter 1 seems to be higher than 2,3,4.

8. Averaging Methods
Simple Averages - quick, inexpensive (should only be used on stationary data)
Moving Averages - a constant number specified at the outset and a mean computed for the most recent observations - such as a 3 or 4 period moving average.
Works best with stationary data.
The larger the order of the moving average, the greater the smoothing effect. Larger n when there are wide, infrequent fluctuations in the data.
By smoothing recent actual values, removes randomness.

9. If you suspect seasonality, with quarterly data, it makes sense to use a 4-period moving average (monthly data would use a 12 period moving average). The larger the number of periods, the smoother the fluctuations become.

11. How many periods?
To determine how many periods to use for a moving average, remember:
The smaller the number, the more weight given to recent periods.
A smaller number is desirable when there are sudden shifts in the level of the series.
The greater the number, less weight is given to more recent periods.
A larger number is desirable when there are wide or infrequent fluctuations in the data

12. Double Moving Averages
Double Moving Averages - designed to handle trending data. One set of moving averages is calculated and then a second set is calculated as a moving average of the first set.
Weighted Moving Average - place more weight on recent observations. Sum of the weights needs to equal 1.

13. More on Moving Averages
Moving averages are used with quarterly or monthly data to help examine the components within a time series.
Used as a forecast, large-order moving averages pays very little attention to fluctuations in the data series
Minitab does a great job with SMA. However, you will need to use Excel to calculate DMA.

14. Stat, Time Series, Moving Average Enter the variable name and
Data Rents
Length
NMissing 0
Moving Average
Length: 3
Accuracy Measures
MAPE:
MAD:
MSD:
Row Period Rents AVER1 Predict Error
* * *
* * *
* *
Stat, Time Series, Moving Average
Enter the variable name and
the number of periods - this example uses a 3 period moving average.
MAPE, MAD and MSE (noted as MSD for Mean Squared Deviations) is automatically calculated.
Row Period FORE1 Lower Upper

15. Formulas for DMA 1. In Excel, Calculate a SMA.
2. Calculate the DMA from the SMA with a SMA using the same number of periods
3. Compute the differences between SMA and DMA. This value is noted as a.
4. Calculate an adjustment factor (similar to the slope in regression) that measures the change over the series, noted as b.
5. Calculate the forecast p periods into the future (usually 1)
6. Calculate the error for each period

16. Weekly Rents DMA example

17. Cell Formulas
Excel has a built-in Moving Average function within the Data Analysis tool pack - but it is only a SMA.

18. Prediction Intervals
A good way to test to see if your model has good predictability to is look at the probability that the actual values will be within a 95% interval.
If n<30, use ta/2,n-1

19. Exponential Smoothing Methods
Single Exponential Smoothing (Averaging)
Tracking
Double Exponential Smoothing
Holt’s Method
Winter’s Model

20. Exponential Smoothing Methods
Continually revising a forecast in light of more recent experiences. Averaging (smoothing) past values of a series in a decreasing (exponential) manner. The observations are weighted with more weight being given to the more recent observations

21. Exponential Smoothing Methods
When looking at the formula - it is really the old forecast plus a times the error in the old forecast
To get started, we need a smoothing constant, an initial forecast, and an actual value. Can use the first actual as the forecast value or you can average the first n observations. Minitab’s default is 6.
The smoothing constant serves as the weighting factor. When a is close to 1, the new forecast will include a substantial adjustment for any error that occurred in the preceding forecast. When a is close to 0, the new forecast is very similar to the old forecast. Iterative procedure to chose a by minimizing MSE.
The smoothing constant is not an arbitrary choice - but generally falls between .1 and .5. If we want predictions to be stable and random variation smoothed, use a small a. If we want a rapid response a larger a value is required.

22. Use Minitab to generate output!!!
Row Time Sales SMOO1 Predict RESI1
Row Period Forecast Lower Upper
Outside tracking interval
Single Exponential Smoothing
Data Sales
Length
NMissing 0
Smoothing Constant
Alpha: 0.1
Accuracy Measures
MAPE:
MAD:
MSD:
Note that a is set to .1
and that the initial forecast value is the first actual value

23. The small a here smooths the data.

24. Row Time Sales Smooth Predict Error
Row Period Forecast Lower Upper
Single Exponential Smoothing
Data Sales
Length
NMissing 0
Smoothing Constant
Alpha: 0.6
Accuracy Measures
MAPE:
MAD:
MSD:
You can only forecast for 1 period - because the formula requires actual data from the current period. If you forecast more than 1 period, it will remain the same as a 1 period forecast.

25. The large a in this example responds quickly to the data.

26. Tracking
Use a tracking signal (measure of errors over time) and setting limits. For example, if we forecast 10 periods, count the number of negative and positive errors. If the number of positive errors is substantially less or greater than n/2, then the process is out of control.
Can also use 95% prediction interval (1.96 * sqrt (MSE)). If the forecast error is outside of the interval, use a new optimal a.
Looking back at the .1 single exponential smoothing:
1.96*sqrt(24261) = Observation #21 is out-of-control. We need to re-evaluate alpha level because this technique is biased.

27. Double Exponential Smoothing
Also known as Brown’s Method. Used to forecast a series with a linear trend.

28. Double Exponential Smoothing
Uses a single coefficient, alpha, for both smoothing operations. Calculates the difference between single and double smoothed values as a measure of trend (at). It then adds this value to the single smoothed value together with an adjustment for the current trend (bt).
In Minitab, use the same a value for the trend and level. Do NOT optimize for Brown’s Method!
Minitab sets the initial value for the smoothed series and trend adjustment by calculating the trend’s slope and intercept using the least square’s method.
If you use Excel, use the Actual values of period 1 to estimate the single and double exponentially smoothed values. Realize that Excel is assuming there is no trend present and will tend to underestimate!

29. Before running Brown’s method; here is single exponential for Rental data with a set to .2

30. Row Time Rents Smooth Predict Error
Row Period Forecast Lower Upper
Double Exponential Smoothing
Data Rents
Length
NMissing 0
Smoothing Constants
Alpha (level): 0.4
Gamma (trend): 0.4
Accuracy Measures
MAPE:
MAD:
MSD:
The SMOOTH value is a Minitab does NOT output the value of b you can calculate the value of b1 by taking Prediction for period 2 and subtracting the Smooth value for period 1. You can store the b in Mintab by selecting the TREND option in Results.

31. Notice that the MSE is much lower than single exponential smoothing and that the smoothed value is much closer to the data. This is due to the trending.

32. Remember, large a responds rapidly to the data.

33. The low a has smoothed the data.

34. Setting the best a
The a level should always be between 0 and 1. However, Minitab is known for violating this rule. We will generally use .1 to .5. It becomes an art and a science in picking the “correct” level - stay with the objective of minimizing MSE. This may require running different levels and comparing MSE values.

35. Holt’s Method
Extension of Brown’s double exponential smoothing but it uses two coefficients.
a is the smoothing constant for the level
b is the trend smoothing constant - used to remove random error
Using Minitab, select the OPTIMAL - but realize, Minitab will violate the a between 0 and 1.

36. Winter’s Method
Extends Holt’s Method to include an estimate for seasonality.
a is the smoothing constant for the level
b is the trend smoothing constant - used to remove random error
g smoothing constant for seasonality
This fun formula removes seasonal effects. The forecast is modified by multiplying by a seasonal index. We will calculate this seasonal index in Chapter 8.

37. Winters' multiplicative model
Data Sales
Length
NMissing 0
Smoothing Constants
Alpha (level):
Gamma (trend):
Delta (seasonal): 0.3
Accuracy Measures
MAPE:
MAD:
MSD:
Row Time Sales Smooth Predict Error
Row Period Forecast Lower Upper
Multiplicative model because we are multiplying the seasonal index by the current smoothed and trend values. Can use the Storage option to store trend and seasonal results.

39. More on Winter’s
In order to Forecast, you would need to have all three estimates. Use Minitab to forecast.
On the exam, you would be required to state the forecast from the output. For Brown’s, Holt, & Winter’s you will not need to calculate a forecast. You should be able to calculate a forecast for Moving averages and Simple exponential smoothing. You should also be able to develop Prediction intervals, track the forecast, and determine the best forecast by comparing MSEs.
Do not try to memorize formula’s but do know the differences between Models

40. For next time
We will run all models from Chapter 4 and compare with MSE and also track the errors based on 95% intervals using the data from Case Key in this data.