1. Time Series and Forecasting
Session 8

2. Time Series, Forecasting & Index Numbers
Using Statistics
Trend Analysis
Seasonality and Cyclical Behavior
The Ratio-to-Moving-Average Method
Exponential Smoothing Methods
Index Numbers
Using the computer
Summary and Review of Terms

3. 8-1 Using Statistics
A time series is a set of measurements of a variable that are ordered through time. Time series analysis attempts to detect and understand regularity in the fluctuation of data over time.
Regular movement of time series data may result from a tendency to increase or decrease through time - trend- or from a tendency to follow some cyclical pattern through time - seasonality or cyclical variation.
Forecasting is the extrapolation of series values beyond the region of the estimation data. Regular variation of a time series can be forecast, while random variation cannot.

4. The Random Walk A random walk: Zt- Zt-1=at or equivalently:
The difference between Z in time t and time t-1 is a random error. 5 4 3 2 1 t Z R a n d o m W l k
There is no evident regularity in a random walk, since the difference in the series from period to period is a random error. A random walk is not forecastable.

5. A Time Series as a Superposition of Cyclical Functions
A Time Series as a Superposition of Two Wave
Functions and a Random Error (not shown) 5 4 3 2 1 - t z
In contrast with a random walk, this series exhibits obvious cyclical fluctuation. The underlying cyclical series - the regular elements of the overall fluctuation - may be analyzable and forecastable.

6. 8-2 Trend Analysis: A Linear Model
Year Loans Time
MTB > Regress 'Loans' 1 'Time'
Regression Analysis
The regression equation is
Loans = Time
Predictor Coef Stdev t-ratio p
Constant
Time
s = R-sq = 98.5% R-sq(adj) = 98.2% 7 2 - 3 8 T i m e L o a n s 1 R S q u r d = . 9 5 Y 6 + 4 X t f D E x p l
Example 12-1

7. Example 8-2 Year Line Fit Plot y = 26.23x - 52065 100 200 300 400 1988
SUMMARY OUTPUT
Regression Statistics
Multiple R
R Square
Adjusted R Square
Standard Error
Observations 9
ANOVA df SS MS F
Significance F
Regression 1
1.2256E-05
Residual 7
Total 8
Coefficients
t Stat
P-value
Lower 95%
Upper 95%
Lower 95.0%
Upper 95.0%
Intercept
E-05
X Variable 1
Year Line Fit Plot
y = 26.23x
100
200
300
400
1988
1990
1992
1994
1996
1998
Year
Price

8. Trend Analysis: An Exponential Model
MTB > let 'LOGLoan' = loge('Loans')
MTB > Name c5 = 'FITS1'
MTB > Regress 'LOGLoan' 1 'Time';
SUBC> Fits 'FITS1';
SUBC> Constant.
Regression Analysis
The regression equation is
LOGLoan = Time
8 cases used 1 cases contain missing values
Predictor Coef Stdev t-ratio p
Constant
Time
s = R-sq = 99.6% R-sq(adj) = 99.5%
MTB > let 'FITS' = expo(‘FITS1’) 1 9 6 5 4 3 2 8 7 Y e a r L o n s

9. 8-3 Seasonality and Cyclical Behavior88 87 86 1 9 8 7 6 t P a s e n g r M o h l y N u m b f A i 89 90 91 92 93 94 95 5 4 3 2 Y E G : J F D O S T
When a cyclical pattern has a period of one year, it is usually called seasonal variation. A pattern with a period of other than one year is called cyclical variation.

10. Time Series Decomposition
Types of Variation
Trend (T)
Seasonal (S)
Cyclical (C)
Random or Irregular (I)
Additive Model
Zt=Tt+ St+ Ct+ It
Multiplicative Model
Zt=(Tt )(St )(Ct )(It)

11. Estimating an Additive Model with Seasonality
An additive regression model with seasonality:
Zt=0+ 1 t+ 2 Q1+ 3 Q2 + 4 Q3 + at
where Q1=1 if the observation is in the first quarter, and 0 otherwise; Q2=1 if the observation is in the second quarter, and 0 otherwise; Q3=1 if the observation is in the third quarter, and 0 otherwise.

12. 8-4 The Ratio-to-Moving- Average Method
A moving average of a time series is an average of a fixed number of observations that moves as we progress down the series.
Time, t:
Series, Zt:
Five-period
moving average:
Time, t:
Series, Zt:
( )/5=15.4
( )/5=15.6
( )/5=16.0
( )/5=17.6

13. Comparing Original Data and Smoothed Moving Average1 5 2 t Z O r i g n a l S e s d F v - P o M A
Moving Average:
“Smoother”
Shorter
Deseasonalized
Removes seasonal and irregular components
Leaves trend and cyclical components

14. Ratio-to-Moving Average
Ratio-to-Moving Average for Quarterly Data
Compute a four-quarter moving-average series.
Center the moving averages by averaging every consecutive pair and placing the average between quarters.
Divide the original series by the corresponding moving average. Then multiply by 100.
Derive quarterly indexes by averaging all data points corresponding to each quarter. Multiply each by 400 and divide by sum.

15. Ratio-to-Moving Average: Example 8-21 5 7 6 4 3 S a l e s T i m A c t u o h d M D : P E L n g v r 2 . 9 8 F - Q
Simple Centered Ratio
Moving Moving to Moving
Quarter Sales Average Average Average
1992W 170 * * *
1992S 148 * * *
1992S 141 *
1992F
1993W
1993S
1993S
1993F
1994W
1994S
1994S
1994F
1995W
1995S
1995S * *
1995F * *

16. Seasonal Indexes: Example 8-2
Quarter
Year Winter Spring Summer Fall
Sum
Average
Sum of Averages =
Seasonal Index = (Average)(400)/400.07
Seasonal
Index

17. Deseasonalized Series: Example 8-2
Seasonal Deseasonalized
Quarter Sales Index (S) Series(Z/S)*100
1992W
1992S
1992S
1992F
1993W
1993S
1993S
1993F
1994W
1994S
1994S
1994F
1995W
1995S
1995S
1995F
Original
Deseasonalized - - - 1 9 2 F S W 7 6 5 4 3 t a l e s O r i g n d o y A j u

18. The Cyclical Component: Example 8-21 8 7 6 5 4 3 2 9 F S W t a l e s T r n d L i M o v g A
The cyclical component is the remainder after the moving averages have been detrended. In this example, a comparison of the moving averages and the estimated regression line:
illustrates that the cyclical component in this series is negligible.

19. Example 8-2 using Excel
The centered moving average, ratio to moving average, seasonal
index, and deseasonalized values were determined using the
Ratio-to-Moving-Average method (implemented with formulas).

20. Example 8-2 using Excel
Comparison of actual sales data given in Example 12-2 for versus predicted sales fit using (1) the trend model determined linear regression and (2) the seasonal index estimated from Ratio-to-Moving-Average method. The trend equation and seasonal index were used to calculate forecasts of sales for 1998 and 1999.
Quarter
Sales
Trend
Seas Indx
Fit/Forcast
1994 W
170
169.8 S
148
97.448 S
141
91.291
Model Fit/Forecast F
150
1995 W
161 S
137
148.64
97.448 S
132
91.291
180 F
158
1996 W
157
160 S
145
97.448 S
128
143.11
91.291
Sales F
134
Sales
140
1997 W
160
Fit/Forecast S
139
97.448
120 S
130
91.291 F
144
1998 W
100 S
97.448 S
91.291
122.57 4 8 12 16 20 24 F
1999 W
145.44
Period (Fall 1993 = 0) S
97.448 S
129.84
91.291

21. Forecasting a Multiplicative Series: Example 8-2

22. Multiplicative Series: Review

23. 8-5 Exponential Smoothing Methods
Smoothing is used to forecast a series by first removing sharp variation, as does the moving average.
Exponential smoothing is a forecasting method in which the forecast is based in a weighted average of current and past series values. The largest weight is given to the present observations, less weight to the immediately preceding observation, even less weight to the observation before that, and so on. The weights decline geometrically as we go back in time. - 5 1 . 4 3 2 L a g W e i h t s D c l n G o B k T m d S u
Weights Decline as we go back in Time
-10
Weight
Lag

24. The Exponential Smoothing Model

25. Example 8-3
Day Z w=.4 w=.8
16 *
Original data:
Smoothed, w=0.4:
Smoothed, w=0.8: ----- D a y 1 5 9 6 4 3 2 w = . E x p o n e t i l S m h g : d 8

26. 8-6 Index Numbers
An index number is a number that measures the relative change in a set of measurements over time. For example: the Dow Jones Industrial Average (DJIA), the Consumer Price Index (CPI), the New York Stock Exchange (NYSE) Index.

27. Index Numbers: Example 8-4
Index Index
Year Price = =100 Y e a r P i c n d I x ( 1 9 8 2 = ) o f N t u l G s 5
Original
Index (1982)
Index (1989)

28. Consumer Price Index Example 12-5: Adjusted Year Salary Salary C P I Y e a r 4 5 3 2 1 9 8 7 6
Consumer Price index (CPI): 1967=100

29. Using the Computer: Example 8-2 (1)
MTB > %Decomp 'Sales' 4;
SUBC> Start 1.
Time Series Decomposition
Data Sales
Length
NMissing 0
Trend Line Equation
Yt = *t
Seasonal Indices
Period Index

30. Using the Computer: Example 8-2 (2)

31. Using the Computer: Example 8-2 (3)