1. Lecture 4 Time-Series Forecasting
Business and Economic Forecasting
Lecture 4
Time-Series Forecasting

2. Learning Objectives In this lecture, you learn:
About different time-series forecasting models: moving averages, exponential smoothing, linear trend, quadratic trend, exponential trend, autoregressive models, and least-squares models for seasonal data
To choose the most appropriate time-series forecasting model
Chap 16-2
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3. The Importance of Forecasting
DCOVA
Governments forecast unemployment rates, interest rates, and expected revenues from income taxes for policy purposes
Marketing executives forecast demand, sales, and consumer preferences for strategic planning
College administrators forecast enrollments to plan for facilities and for faculty recruitment
Retail stores forecast demand to control inventory levels, hire employees and provide training
Chap 16-3
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4. Common Approaches to Forecasting
DCOVA
Common Approaches to Forecasting
Qualitative forecasting methods
Quantitative forecasting methods
Used when historical data are unavailable
Considered highly subjective and judgmental
Time Series
Causal
Use past data to predict future values
Chap 16-4
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5. Time-Series Data Numerical data obtained at regular time intervals
DCOVA
Numerical data obtained at regular time intervals
The time intervals can be annually, quarterly, monthly, weekly, daily, hourly, etc.
Example:
Year:
Sales:
Chap 16-5
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6. A time-series plot is a two-dimensional plot of time series data
DCOVA
A time-series plot is a two-dimensional plot of time series data
the vertical axis measures the variable of interest
the horizontal axis corresponds to the time periods
Chap 16-6
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7. Time-Series Components
DCOVA
Time Series
Trend Component
Seasonal Component
Cyclical Component
Irregular Component
Regular periodic fluctuations, usually within a 12-month period
Repeating swings or movements over more than one year
Erratic or residual fluctuations
Overall, persistent, long-term movement
Chap 16-7
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8. Trend Component
DCOVA
Long-run increase or decrease over time (overall upward or downward movement)
Data taken over a long period of time
Sales
Upward trend
Time
Chap 16-8
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9. Trend Component Trend can be upward or downward
(continued)
DCOVA
Trend can be upward or downward
Trend can be linear or non-linear
Sales
Sales
Time
Time
Downward linear trend
Upward nonlinear trend
Chap 16-9
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10. Seasonal Component Short-term regular wave-like patterns
DCOVA
Short-term regular wave-like patterns
Observed within 1 year
Often monthly or quarterly
Sales
Summer
Winter
Summer
Fall
Winter
Spring
Fall
Spring
Time (Quarterly)
Chap 16-10
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11. Cyclical Component Long-term wave-like patterns
Regularly occur but may vary in length
Often measured peak to peak or trough to trough
DCOVA
1 Cycle
Sales
Year
Chap 16-11
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12. Irregular Component Unpredictable, random, “residual” fluctuations
DCOVA
Unpredictable, random, “residual” fluctuations
Due to random variations of
Nature
Accidents or unusual events
“Noise” in the time series
Chap 16-12
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13. Does Your Time Series Have A Trend Component?
DCOVA
A time series plot should help you to answer this question.
Often it helps if you “smooth” the time series data to help answer this question.
Two popular smoothing methods are moving averages and exponential smoothing.
Chap 16-13
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14. Smoothing Methods Moving Averages Exponential Smoothing DCOVA
Calculate moving averages to get an overall impression of the pattern of movement over time
Averages of consecutive time series values for a chosen period of length L
Exponential Smoothing
A weighted moving average
Chap 16-14
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15. Moving Averages Used for smoothing
DCOVA
Used for smoothing
A series of arithmetic means over time
Result dependent upon choice of L (length of period for computing means)
Examples:
For a 5 year moving average, L = 5
For a 7 year moving average, L = 7
Etc.
Chap 16-15
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16. Moving Averages Example: Five-year moving average DCOVA First average:
(continued)
Example: Five-year moving average
First average:
Second average:
etc.
DCOVA
Chap 16-16
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17. Example: Annual Data DCOVA Year Sales 1 2 3 4 5 6 7 8 9 10 11 etc… 2340 25 27 32 48 33 37 50
Chap 16-17
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18. Calculating Moving Averages
DCOVA
Average Year
5-Year Moving Average 3
29.4 4
34.4 5
33.0 6
35.4 7
37.4 8
41.0 9
39.4 …
Year
Sales 1 23 2 40 3 25 4 27 5 32 6 48 7 33 8 37 9 10 50 11
etc…
Each moving average is for a consecutive block of 5 years
Chap 16-18
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19. Annual vs. Moving Average
DCOVA
The 5-year moving average smoothes the data and makes it easier to see the underlying trend
Chap 16-19
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20. Exponential Smoothing
DCOVA
Used for smoothing and short term forecasting (one period into the future)
A weighted moving average
Weights decline exponentially
Most recent observation is given the highest weight
Chap 16-20
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21. Exponential Smoothing
(continued)
DCOVA
The weight (smoothing coefficient) is W
Subjectively chosen
Ranges from 0 to 1
Smaller W gives more smoothing, larger W gives less smoothing
The weight is:
Close to 0 for smoothing out unwanted cyclical and irregular components
Close to 1 for forecasting
Chap 16-21
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22. Exponential Smoothing Model
DCOVA
Exponential smoothing model
For i = 2, 3, 4, …
where:
Ei = exponentially smoothed value for period i
Ei-1 = exponentially smoothed value already
computed for period i - 1
Yi = observed value in period i
W = weight (smoothing coefficient), 0 < W < 1
Chap 16-22
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23. Exponential Smoothing Example
DCOVA
Suppose we use weight W = 0.2
Time Period (i)
Sales
(Yi)
Forecast from prior period (Ei-1)
Exponentially Smoothed Value for this period (Ei) 1 2 3 4 5 6 7 8 9 10
etc. 23 40 25 27 32 48 33 37 50 --
23.000
26.400
26.120
26.296
27.437
31.549
31.840
32.872
33.697
(.2)(40)+(.8)(23)=26.4
(.2)(25)+(.8)(26.4)=26.12
(.2)(27)+(.8)(26.12)=26.296
(.2)(32)+(.8)(26.296)=27.437
(.2)(48)+(.8)(27.437)=31.549
(.2)(48)+(.8)(31.549)=31.840
(.2)(33)+(.8)(31.840)=32.872
(.2)(37)+(.8)(32.872)=33.697
(.2)(50)+(.8)(33.697)=36.958
E1 = Y1 since no prior information exists
Chap 16-23
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24. Sales vs. Smoothed Sales
DCOVA
Fluctuations have been smoothed
NOTE: the smoothed value in this case is generally a little low, since the trend is upward sloping and the weighting factor is only .2
Chap 16-24
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25. Forecasting Time Period i + 1
DCOVA
The smoothed value in the current period (i) is used as the forecast value for next period (i + 1) :
Chap 16-25
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26. Exponential Smoothing in Excel
DCOVA
Use data analysis / exponential smoothing
The “damping factor” is (1 - W)
Time
Sales
Smoothed 1 23 2 40 3 25
26.4 4 27
26.12 5 32
26.296 6 48 7 33 8 37 9 10 50
Chap 16-26
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27. There Are Three Popular Methods For Trend-Based Forecasting
DCOVA
Linear Trend Forecasting
Nonlinear Trend Forecasting
Exponential Trend Forecasting
Chap 16-27
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28. Linear Trend Forecasting
DCOVA
Estimate a trend line using regression analysis
Use time (X) as the independent variable:
Year
Time Period (X)
Sales (Y)
2004
2005
2006
2007
2008
2009 1 2 3 4 5 20 40 30 50 70 65
In least squares linear, non-linear, and
exponential modeling, time periods are
numbered starting with 0 and increasing
by 1 for each time period.
Chap 16-28
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29. Linear Trend Forecasting
(continued)
DCOVA
The linear trend forecasting equation is:
Year
Time Period (X)
Sales (Y)
2004
2005
2006
2007
2008
2009 1 2 3 4 5 20 40 30 50 70 65
Chap 16-29
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30. Linear Trend Forecasting
(continued)
DCOVA
Forecast for time period 6 (2010):
Year
Time Period (X)
Sales (y)
2004
2005
2006
2007
2008
2009
2010 1 2 3 4 5 6 20 40 30 50 70 65 ??
Chap 16-30
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31. Nonlinear Trend Forecasting
DCOVA
A nonlinear regression model can be used when the time series exhibits a nonlinear trend
Quadratic form is one type of a nonlinear model:
Compare adj. r2 and standard error to that of linear model to see if this is an improvement or test significance of the quadratic term
Can try other functional forms to get best fit
Chap 16-31
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32. Exponential Trend Model
DCOVA
Another nonlinear trend model:
Transform to linear form:
Chap 16-32
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33. Exponential Trend Model
(continued)
DCOVA
Exponential trend forecasting equation:
where b0 = estimate of log(β0)
b1 = estimate of log(β1)
Interpretation:
is the estimated annual compound
growth rate (in %)
Chap 16-33
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34. Trend Model Selection Using Differences
DCOVA
Use a linear trend model if the first differences are approximately constant
Use a quadratic trend model if the second differences are approximately constant
Chap 16-34
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35. Trend Model Selection Using Differences
(continued)
DCOVA
Use an exponential trend model if the percentage differences are approximately constant
Chap 16-35
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36. Autoregressive Modeling
DCOVA
Used for forecasting
Takes advantage of autocorrelation
1st order - correlation between consecutive values
2nd order - correlation between values 2 periods apart
pth order Autoregressive model:
Random Error
Chap 16-36
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37. Autoregressive Model: Example
DCOVA
The Office Concept Corp. has acquired a number of office units (in thousands of square feet) over the last eight years. Develop the second order Autoregressive model.
Year Units
03 3
06 2
07 2
08 4
Chap 16-37
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38. Autoregressive Model: Example Solution
DCOVA
Develop the 2nd order table
Use Excel to estimate a regression model
Year Yi Yi Yi-2
Excel Output
Chap 16-38
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39. Autoregressive Model Example: Forecasting
DCOVA
Use the second-order equation to forecast number of units for 2010:
Chap 16-39
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40. Autoregressive Modeling Steps
DCOVA
1. Choose p (note that df = n – 2p – 1)
2. Form a series of “lagged predictor” variables
Yi-1 , Yi-2 , … ,Yi-p
3. Use Excel or Minitab to run regression model using all p variables
4. Test significance of Ap
If null hypothesis rejected, this model is selected
If null hypothesis not rejected, decrease p by 1 and repeat
Chap 16-40
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41. Choosing A Forecasting Model
DCOVA
Perform a residual analysis
Eliminate a model that shows a pattern or trend
Measure magnitude of residual error using squared differences and select the model with the smallest value
Measure magnitude of residual error using absolute differences and select the model with the smallest value
Use simplest model
Principle of parsimony
Chap 16-41
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42. Cyclical effects not accounted for
Residual Analysis
DCOVA e e T T
Random errors
Cyclical effects not accounted for e e T T
Trend not accounted for
Seasonal effects not accounted for
Chap 16-42
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43. Measuring Errors
DCOVA
Choose the model that gives the smallest measuring errors
Sum of squared errors (SSE)
Sensitive to outliers
Mean Absolute Deviation (MAD)
Less sensitive to extreme observations
Chap 16-43
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44. Principal of Parsimony
DCOVA
Suppose two or more models provide a good fit for the data
Select the simplest model
Simplest model types:
Least-squares linear
Least-squares quadratic
1st order autoregressive
More complex types:
2nd and 3rd order autoregressive
Least-squares exponential
Chap 16-44
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45. Forecasting With Seasonal Data
DCOVA
Time series are often collected monthly or quarterly
These time series often contain a trend component, a seasonal component, and the irregular component
Suppose the seasonality is quarterly
Define three new dummy variables for quarters:
Q1 = 1 if first quarter, 0 otherwise
Q2 = 1 if second quarter, 0 otherwise
Q3 = 1 if third quarter, 0 otherwise
(Quarter 4 is the default if Q1 = Q2 = Q3 = 0)
Chap 16-45
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46. Exponential Model with Quarterly Data
DCOVA
Transform to linear form:
(β1–1)x100% is the quarterly compound growth rate
βi provides the multiplier for the ith quarter relative to the 4th quarter (i = 2, 3, 4)
Chap 16-46
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47. Estimating the Quarterly Model
DCOVA
Exponential forecasting equation:
where b0 = estimate of log(β0), so
b1 = estimate of log(β1), so
etc…
Interpretation:
= estimated quarterly compound growth rate (in %)
= estimated multiplier for first quarter relative to fourth quarter
= estimated multiplier for second quarter rel. to fourth quarter
= estimated multiplier for third quarter relative to fourth quarter
Chap 16-47
Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall

48. Quarterly Model Example
DCOVA
Suppose the forecasting equation is:
b0 = 3.43, so
b1 = .017, so
b2 = -.082, so
b3 = -.073, so
b4 = .022, so
Chap 16-48
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49. Quarterly Model Example
(continued)
DCOVA
Value:
Interpretation:
Unadjusted trend value for first quarter of first year
4.0% = estimated quarterly compound growth rate
Average sales in Q1 are 82.7% of average 4th quarter sales, after adjusting for the 4% quarterly growth rate
Average sales in Q2 are 84.5% of average 4th quarter sales, after adjusting for the 4% quarterly growth rate
Average sales in Q3 are 105.2% of average 4th quarter sales, after adjusting for the 4% quarterly growth rate
Chap 16-49
Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall

50. Lecture Summary Discussed the importance of forecasting
Addressed component factors of the time-series model
Performed smoothing of data series
Moving averages
Exponential smoothing
Described least square trend fitting and forecasting
Linear, quadratic and exponential models
Chap 16-50
Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall

51. Lecture Summary Addressed autoregressive models
(continued)
Addressed autoregressive models
Described procedure for choosing appropriate models
Addressed time-series forecasting of monthly or quarterly data (use of dummy variables)
Chap 16-51
Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall