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```Part II – TIME SERIES ANALYSIS
C5 ARIMA (Box-Jenkins) Models
© Angel A. Juan & Carles Serrat - UPC 2007/2008
2.5.1: Introduction to ARIMA models

Recall that stationary processes
The Autoregressive Integrated Moving Average (ARIMA)
vary about a fixed level, and
nonstationary processes have no
models, or Box-Jenkins methodology, are a class of linear
natural constant mean level.
models that is capable of representing stationary as well as
nonstationary time series.
The ACF and PACF associated to the TS

ARIMA models rely heavily on autocorrelation patterns in
data  both ACF and PACF are used to select an initial
model.

The Box-Jenkins methodology uses an iterative approach:
are matched with the theoretical
autocorrelation pattern associated with a
particular ARIMA model.
1.
An initial model is selected, from a general class of ARIMA
models, based on an examination of the TS and an
examination of its autocorrelations for several time lags
2.
The chosen model is then checked against the historical data
to see whether it accurately describes the series: the model
fits well if the residuals are generally small, randomly
distributed, and contain no useful information.
3.
If the specified model is not satisfactory, the process is
repeated using a new model designed to improve on the
original one.
4.
Once a satisfactory model is found, it can be used for
forecasting.
2.5.2: Autoregressive Models AR(p)

A pth-order autoregressive model, or AR(p), takes the form:
Yt  0  1Yt 1  2Yt 2  ...  pYt  p   t
An AR(p) model is a regression model with lagged
values of the dependent variable in the independent
variable positions, hence the name autoregressive model.
Yt  response variable at time t
Yt  k  observation (predictor variable) at time t  k
i  regression coefficients to be estimated
 t  error term at time t

Autoregressive models are appropriate for stationary time series, and the
coefficient Ф0 is related to the constant level of the series.

Theoretical behavior of the ACF and PACF for AR(1) and AR(2) models:
AR(1)
AR(2)
ACF  0
ACF  0
PACF = 0 for lag > 1
PACF = 0 for lag > 2
2.5.3: Moving Average Models MA(q)

A qth-order moving average model, or MA(q), takes the form:
Yt     t  1 t 1  2 t 2  ...  q t q
An MA(q) model is a regression model with the dependent
variable, Yt, depending on previous values of the errors
rather than on the variable itself.
Yt  response variable at time t
  constant mean of the process
i  regression coefficients to be estimated
 t k  error in time period t - k
The term Moving
Average is
historical and
should not be
confused with the
moving average
smoothing
procedures.

MA models are appropriate for stationary time series. The weights ωi do not
necessarily sum to 1 and may be positive or negative.

Theoretical behavior of the ACF and PACF for MA(1) and MA(2) models:
MA(1)
ACF = 0 for lag > 1; PACF  0
MA(2)
ACF = 0 for lag > 2; PACF  0
2.5.4: ARMA(p,q) Models

A model with autoregressive terms can be combined with a model
having moving average terms to get an ARMA(p,q) model:
Note that:
• ARMA(p,0) = AR(p)
• ARMA(0,q) = MA(q)
Yt  0  1Yt 1  2Yt 2  ...  pYt  p   t  1 t 1  2 t 2  ...  q t q

ARMA(p,q) models can describe a wide variety of behaviors for
stationary time series.

Theoretical behavior of the ACF and PACF for autoregressivemoving average processes:
ACF
PACF
AR(p)
Die out
Cut off after the order
p of the process
MA(q)
Cut off after the order
q of the process
Die out
Die out
Die out
ARMA(p,q)
In practice, the values
of p and q each rarely
exceed 2.
In this context…
• “Die out” means “tend
• “Cut off” means
“disappear” or “is zero”
2.5.5: Building an ARIMA model (1/2)

The first step in model identification is to determine whether the
series is stationary. It is useful to look at a plot of the series
along with the sample ACF.

If the series is not stationary, it can often be converted to a
stationary series by differencing: the original series is replaced
by a series of differences and an ARMA model is then specified
for the differenced series (in effect, the analyst is modeling
changes rather than levels)

A nonstationary TS is
indicated if the series
appears to grow or decline
over time and the sample
ACF fail to die out rapidly.
In some cases, it may be
necessary to difference
the differences before
stationary data are
obtained.
Models for nonstationary series are called Autoregressive
Integrated Moving Average models, or ARIMA(p,d,q), where d
indicates the amount of differencing.
Note that:

Once a stationary series has been obtained, the analyst must
identify the form of the model to be used by comparing the
sample ACF and PACF to the theoretical ACF and PACF for the
various ARIMA models.
By counting the number of
significant sample
autocorrelations and partial
autocorrelations, the orders
of the AR and MA parts can
be determined.

Principle of parsimony: “all things being equal, simple models
are preferred to complex models”

Once a tentative model has been selected, the parameters for
that model are estimated using least squares estimates.
ARIMA(p,0,q) = ARMA(p,q)
containing few rather than
many parameters. The
parameters will be evident
from an examination of the
residual ACF and PACF.
2.5.5: Building an ARIMA model (2/2)

Before using the model for forecasting, it must be checked for
adequacy. Basically, a model is adequate if the residuals cannot be
used to improve the forecasts, i.e.,

The residuals should be random and normally distributed

The individual residual autocorrelations should be small. Significant
residual autocorrelations at low lags or seasonal lags suggest the

After an adequate model has been found, forecasts can be made.
Prediction intervals based on the forecasts can also be constructed.

As more data become available, it is a good idea to monitor the
forecast errors, since the model must need to be reevaluated if:


The magnitudes of the most recent errors tend to be consistently
larger than previous errors, or

The recent forecast errors tend to be consistently positive or negative
Seasonal ARIMA (SARIMA) models contain:

Regular AR and MA terms that account for the correlation at low lags

Seasonal AR and MA terms that account for the correlation at the
seasonal lags
Many of the same
residual plots that are
useful in regression
analysis can be
developed for the
residuals from an
ARIMA model
(histogram, normal
probability plot, time
sequence plot, etc.)
In general, the
longer the forecast
lead time, the larger
the prediction
interval (due to
greater uncertainty)
nonstationary
seasonal series,
seasonal
difference is often
required
2.5.6: ARIMA with Minitab – Ex. 1 (1/4)

File: PORTFOLIO_INVESTMENT.MTW

Stat > Time Series > …
A consulting corporation wants
to try the Box-Jenkins
technique for forecasting the
Transportation Index of the
Dow Jones.
Time Series Plot of Index
290
The series show an
upward trend.
280
270
250
240
230
Autocorrelation Function for Index
220
(with 5% significance limits for the autocorrelations)
210
1
6
12
18
24
30
36
Index
42
48
54
1,0
60
0,8
0,6
The first several autocorrelations are persistently
large and trailed off to zero rather slowly  a trend
exists and this time series is nonstationary (it does
not vary about a fixed level)
Autocorrelation
Index
260
0,4
0,2
0,0
-0,2
-0,4
-0,6
-0,8
-1,0
Idea: to difference the data to see if we could
eliminate the trend and create a stationary series.
1
2
3
4
5
6
7
8
9
Lag
10
11
12
13
14
15
16
2.5.6: ARIMA with Minitab – Ex. 1 (2/4)
Time Series Plot of Diff1
First order differences.
5
4
A plot of the
differenced data
appears to vary about
a fixed level.
3
Diff1
2
1
0
-1
-2
-3
-4
1
Comparing the autocorrelations with their error limits, the only significant
autocorrelation is at lag 1. Similarly, only the lag 1 partial autocorrelation is
significant. The PACF appears to cut off after lag 1, indicating AR(1) behavior.
The ACF appears to cut off after lag 1, indicating MA(1) behavior  we will
try: ARIMA(1,1,0) and ARIMA(0,1,1)
0,8
0,8
0,6
0,6
Partial Autocorrelation
Autocorrelation
1,0
0,4
0,2
0,0
-0,2
-0,4
-0,6
0,0
-0,4
-0,6
-1,0
7
8
9
Lag
10
11
12
48
-0,2
-1,0
6
42
0,2
-0,8
5
30
36
Index
0,4
-0,8
4
24
(with 5% significance limits for the partial autocorrelations)
1,0
3
18
Partial Autocorrelation Function for Diff1
Autocorrelation Function for Diff1
2
12
13
14
15
16
54
60
A constant term in each model will be included to allow for the fact that
the series of differences appears to vary about a level greater than zero.
(with 5% significance limits for the autocorrelations)
1
6
1
2
3
4
5
6
7
8
9
Lag
10
11
12
13
14
15
16
2.5.6: ARIMA with Minitab – Ex. 1 (3/4)
ARIMA(1,1,0)
The LBQ statistics are not significant as indicated by the large pvalues for either model.
ARIMA(0,1,1)
2.5.6: ARIMA with Minitab – Ex. 1 (4/4)
Autocorrelation Function for RESI1
(with 5% significance limits for the autocorrelations)
1,0
0,8
0,4
0,2
0,0
-0,2
-0,4
-0,6
-0,8
-1,0
1
2
3
4
5
6
7
8
9
Lag
10
11
12
13
14
15
16
Autocorrelation Function for RESI2
(with 5% significance limits for the autocorrelations)
1,0
0,8
Finally, there is no significant residual
autocorrelation for the ARIMA(1,1,0) model.
The results for the ARIMA(0,1,1) are similar.
0,6
Autocorrelation
Autocorrelation
0,6
0,4
0,2
0,0
-0,2
-0,4
-0,6
Therefore, either model is adequate and provide
nearly the same one-step-ahead forecasts.
-0,8
-1,0
1
2
3
4
5
6
7
8
9
Lag
10
11
12
13
14
15
16
2.5.7: ARIMA with Minitab – Ex. 2 (1/3)


Stat > Time Series > …
A consulting corporation wants
to try the Box-Jenkins
technique for forecasting a
process.
Time Series Plot of Readings
110
The time series of readings appears to vary
about a fixed level of around 80, and the
autocorrelations die out rapidly toward zero
 the time series seems to be stationary.
100
90
80
70
The first sample ACF coefficient is significantly different form zero. The
autocorrelation at lag 2 is close to significant and opposite in sign from the lag 1
autocorrelation. The remaining autocorrelations are small. This suggests either
an AR(1) model or an MA(2) model.
60
50
40
30
20
1
7
14
21
28
35
42
Index
49
56
63
70
The first PACF coefficient is significantly different from zero, but none of the other
partial autocorrelations approaches significance, This suggests an AR(1) or
ARIMA(1,0,0)
Partial Autocorrelation Function for Readings
Autocorrelation Function for Readings
(with 5% significance limits for the partial autocorrelations)
1,0
1,0
0,8
0,8
0,6
0,6
Partial Autocorrelation
Autocorrelation
(with 5% significance limits for the autocorrelations)
0,4
0,2
0,0
-0,2
-0,4
-0,6
0,4
0,2
0,0
-0,2
-0,4
-0,6
-0,8
-0,8
-1,0
-1,0
2
4
6
8
10
Lag
12
14
16
18
2
4
6
8
10
Lag
12
14
16
18
2.5.7: ARIMA with Minitab – Ex. 2 (2/3)
AR(1) =
ARIMA(1,0,0)
A constant term is
included in both
models to allow for
the fact that the
level other than zero.
Both models appear
to fit the data well.
The estimated
coefficients are
significantly different
from zero and the
mean square (MS)
errors are similar.
Let’s take a look at the residuals ACF…
MA(2) =
ARIMA(0,0,2)
2.5.7: ARIMA with Minitab – Ex. 2 (3/3)
Autocorrelation Function for RESI1
(with 5% significance limits for the autocorrelations)
1,0
0,8
Autocorrelation
0,6
0,4
0,2
0,0
-0,2
-0,4
-0,6
-0,8
-1,0
2
4
6
8
10
Lag
12
14
16
18
Autocorrelation Function for RESI2
(with 5% significance limits for the autocorrelations)
1,0
Therefore, either model is adequate and provide nearly the same threestep-ahead forecasts. Since the AR(1) model has two parameters
(including the constant term) and the MA(2) model has three parameters,
applying the principle of parsimony we would use the simpler AR(1)
model to forecast future readings.
0,8
0,6
Autocorrelation
Finally, there is no significant residual
autocorrelation for the ARIMA(1,0,0) model.
The results for the ARIMA(0,0,2) are similar.
0,4
0,2
0,0
-0,2
-0,4
-0,6
-0,8
-1,0
2
4
6
8
10
Lag
12
14
16
18
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