Time Series Analysis and Forecasting I. Introduction. A time series is a set of observations generated sequentially in time Continuous vs. discrete time series
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Zt = Ft + at
Economic and business planning
Inventory and production control
Control and optimization of industrial processes
Lead time of the forecasts
is the period over which forecasts are needed
Degree of sophistication
Simple ideas
Moving averages
Simple regression techniques
Complex statistical concepts
BoxJenkins methodology
ForecastingAdvantages
Quickly and easily applied
A minimum of data is required
Reasonably shortto mediumterm forecasts
They provide a basis by which forecasts developed through other models can be measured against
Disadvantages
Not useful for forecasting into the far future
Do not take into account external factors
Causeandeffect approach
Advantages
Bring more information
More accurate mediumto longterm forecasts
Disadvantages
Forecasts of the explanatory time series are required
Approaches to forecasting (cont.)(B) = 0 + 1B + 2B2 + …..
BmXt = Xt  m
Better control
Improved design
Methods for estimating transfer function models
Classical methods
Based on deterministic perturbations
Uncontrollable disturbances (“noise”) are not accounted for, and hence, these methods have not always been successful
Statistical methods
Make allowance for “noise”
The BoxJenkins methodology
Transfer function modeling (cont.)Feedback control
Deviation from target output
P
P
N
N
t
t
Deviation from
target output

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1
f
1


+
d
w
1
b
1
f
1
L
(
B
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L
(
B
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B
(
B
)
(
B
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B
L
(
B
)
L
(
B
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B
1
2
1
2
Compensating
Compen
sating
variable X
variable X
t+
t+
Control equation
Control equation
z
t
Process controlModel identification
Model estimation
Is model adequate ?
No
Modify model
Yes
Forecasts
(1st order) zt = (1 – B)zt = zt – zt1
(2nd order) 2zt = (1 – B)2zt = zt – 2zt1 + zt2
“B” is the backward shift operator
How can we determine the number of regular differencing ?
(
B
)
White noise
z
t
Linear filter
a
t
The linear filter modelIf the current observation zt depends on past observations with weights which decrease as we go back in time, the series is called invertible
For a linear process to be invertible,
Stationarity and invertibility conditions for a linear filterStationarity and invertibility conditions
Theoretical ACs and PACs
Model identificationFor a linear process to be invertible,
Stationarity and invertibility conditionsi.e., the roots of the characteristic equation 1  1B = 0 lie outside the unit circle
k = 1k k > 0
i.e., for a stationary AR(1) model, the theoretical autocorrelation function decays exponentially to zero, however, the theoretical partialautocorrelation function has a cut off after the 1st lag
i.e., the roots of the characteristic equation 1  1B = 0 lie outside the unit circle