Chapter 15
Time Series: Descriptive
Analyses, Models, and
Forecasting
Descriptive Analysis: Index Numbers
Index Number – a number that measures the change in a
variable over time relative to the value of the variable
during a specific base period
Simple Index Number – index based on the relative
changes (over time) in the price or quantity of a single
commodity
0
1 0 0
1 0 0i
t
T im e s e r ie s v a lu e a t t im e tIn d e x n u m b e r a t t im e t
T im e s e r ie s v a lu e a t b a s e p e r io d
YI
Y
Descriptive Analysis: Index Numbers
Laspeyres and Paasche Indexes compared•The Laspeyres Index weights by the purchase quantities
of the baseline period
•The Paasche Index weights by the purchase quantities of
the period the index value represents.
•Laspeyres Index is most appropriate when baseline
purchase quantities are reasonable approximations of
purchases in subsequent periods.
•Paasche Index is most appropriate when you want to
compare current to baseline prices at current purchase
levels
Descriptive Analysis: Index Numbers
Calculating a Laspeyres Index
Collect price info for the k price series to be used, denote
as P1t, P2t…Pkt
Select a base period t0
Collect purchase quantity info for base period, denote as
Q1t0, Q2t0…..Qkt0
Calculate weighted totals for each time period using the
formula
Calculate the index using the formula
0
1
k
i t i t
i
Q P
0
1
0 0
1
1 0 0
k
i t i t
i
t k
i t i t
i
Q P
I
Q P
Descriptive Analysis: Index Numbers
Calculating a Paasche Index
Collect price info for the k price series to be used, denote
as P1t, P2t…Pkt
Select a base period t0
Collect purchase quantity info for every period, denote as
Q1t, Q2t…..Qkt
Calculate the index for time t using the formula
1
0
1
1 0 0
k
i t i t
i
t k
i t i t
i
Q P
I
Q P
Descriptive Analysis: Exponential
Smoothing
Exponential smoothing is a type of weighted
average, that applies a weight w to past and
current values of the time series.
Exponential smoothing constant (w) lies between 0
and 1, and smoothed series Et is calculated as:
1 1
2 2 1
3 3 2
1
(1 )
(1 )
(1 )t t t
E Y
E w Y w E
E w Y w E
E w Y w E
Descriptive Analysis: Exponential
Smoothing
Selection of smoothing constant w is made by
researcher.
Small values of w give less weight to current value,
yield a smoother series
Large values of w give more weight to current
value, yield a more variable series
Time Series Components
4 components of time series models:
Tt – secular or long-term trend
Ct – cyclical trend
St – seasonal effect
Rt – residual effect
These 4 components are from a widely used model called the additive model:
Yt = Tt + Ct + St + Rt
Forecasting: Exponential
Smoothing
Calculation of Exponentially Smoothed Forecasts
Calculate exponentially smoothed values E1, E2,…Et for
observed time series Y1, Y2,…Yt.
Used last smoothed value to forecast the next time series
value
Assuming that Yt is relatively free of trend and seasonal
components, use the same forecast for all future values of
Yt:
1t t t t tF E F w Y F
2 1
3 1
t t
t t
F F
F F
Forecasting Trends: The Holt-
Winters Model
The Holt-Winters model adds a trend component to the forecast.
Calculating Components of the Holt-Winters Model
1. Select exponential smoothing constant w
2. Select trend smoothing constant v
3. Calculate the two components Et and Tt from time series Ytbeginning at time t=2
E2=Y2
T2=Y2-Y1
E3=wY3+(1-w)(E2+T2)
T3=v(E3-E2)+(1-v)T2
…
Et=wYt+(1-w)(Et-1+Tt-1)
Tt=v(Et-Et-1)+(1-v)Tt-1
Forecasting Trends: The Holt-
Winters Model
Holt-Winters Forecasting
1. Calculate the exponentially smoothed and
trend components Et and Tt for each observed
value of Yt (t >2)
2. Calculate the one-step-ahead forecast using
Ft+1=Et+Tt
3. Calculate the k-step-ahead forecast using
Ft+k=Et+kTt
Measuring Forecast Accuracy:
MAD and RMSE
Measures of Forecast Accuracy for m Forecasts
Mean absolute Deviation (MAD)
Mean absolute percentage error (MAPE)
Root mean squared error (RMSE)
1
m
t t
t
Y F
M A Dm
1
1 0 0
m
t t
t t
Y F
YM A P E
m
2
1
m
t t
t
Y F
R M S Em
Forecasting Trends: Simple Linear
Regression
Simple Linear Regression is the simplest inferential forecasting model
After fitting the regression line to existing data, the least squares model can be used to forecast future values of the dependent variable
Two problems are associated with using a LSM to forecast time series:
1. Forecasting falls outside of the experimental region, increases width of prediction intervals
2. Cyclical effects are not built in to the model, and introduce the problem of correlated error
Seasonal Regression Models
Use of multiple regression model with
dummy variables to describe seasonal
differences is common
In the following example, dummy variables
are set up for Quarters 1, 2 and 3
Model that reflects seasonal component and
expected growth in usage is:
0 1 2 1 3 2 4 3( )
tE Y t Q Q Q
Seasonal Regression Models
Data to be used is in the following table:Quarterly Power Loads (megawatts) for a Southern Utility Company, 1992-2003
Year Quarter Power Load Year Quarter Power Load
1992 1 68.8 1998 1 130.6
2 65.0 2 116.8
3 88.4 3 144.2
4 69.0 4 123.3
1993 1 83.6 1999 1 142.3
2 69.7 2 124.0
3 90.2 3 146.1
4 72.5 4 135.5
1994 1 106.8 2000 1 147.1
2 89.2 2 119.3
3 110.7 3 138.2
4 91.7 4 127.6
1995 1 108.6 2001 1 143.4
2 98.9 2 134.0
3 120.1 3 159.6
4 102.1 4 135.1
1996 1 113.1 2002 1 149.5
2 94.2 2 123.3
3 120.5 3 154.4
4 107.4 4 139.4
1997 1 116.2 2003 1 151.6
2 104.4 2 133.7
3 131.7 3 154.5
4 117.9 4 135.1
Seasonal Regression Models
Result of regression analysis is:
Seasonal Regression Models
Forecast results and actual values for 2004
are:
Seasonal Regression Models
Use of multiplicative models often provides a better
forecasting model when the time series is changing at an
increasing rate over time.
Multiplicative model for Power Load problem would be:
Taking antilogarithm of both sides shows multiplicative
nature:
0 1 2 1 3 2 4 3ln ( )
tE Y t Q Q Q
0 1 2 1 3 2 4 3e x p e x p e x p e x p
tY t Q Q Q
Constant Secular
trend
Seasonal Component Residual
Component
Autocorrelation and the Durbin-
Watson Test
A residual pattern as illustrated here suggests that autocorrelationmay be an issue.
Autocorrelation is the correlation between time series residuals at different points in time. Correlation between neighboring residuals is called first-order autocorrelation
Autocorrelation and the Durbin-
Watson Test
Durbin-Watson d statistic is calculated to test for the presence of first-order autocorrelation
If residuals are uncorrelated, then d ≈ 2.
If residuals are positively autocorrelated, then d<2, and if autocorrelation is very strong, d ≈ 0.
If residuals are negatively autocorrelated, then d>2, and if autocorrelation is very strong, d ≈ 4.
2
1
2
2
1
: 0 4
n
t t
t
n
t
t
R R
d R a n g e o f d d
R
Autocorrelation and the Durbin-
Watson Test
One-Tailed Test Two-Tailed Test
H0:No first-order autocorrelation H0:No first-order autocorrelation Ha:Positive first-order autocorrelation (or Ha:Negative first-order autocorrelation)
Ha:Positive or negative first-order autocorrelation
Test Statistic
2
1
2
2
1
n
t t
t
n
t
t
R R
d
R
Rejection region: d< dL, Rejection region: d< dL,/2 or (4-d)< dL,/2
[or (4-d)< dL, if Ha:Negative first-order autocorrelation]
Where dL, is the lower tabled value corresponding to k independent variables and n observations.
The corresponding upper value dU, defines a
“possibly significant” region between dL, and dU,
Where dL,/2 is the lower tabled value corresponding to k independent variables and n observations. The corresponding upper value
dU,/2 defines a “possibly significant” region
between dL,/2 and dU,/2