Lesson 15: Building ARMA models. Examples

43 Pages·2014·0.45 MB·English

by  Umberto Triacca

Lesson 15: Building ARMA models.
Examples
Umberto Triacca
DipartimentodiIngegneriaeScienzedell’InformazioneeMatematica
Universit`adell’Aquila,
[email protected]
UmbertoTriacca Lesson15:BuildingARMAmodels.Examples
Examples
In this lesson, in order to illustrate the time series modelling
methodology we have presented so far, we analyze some time
series.
UmbertoTriacca Lesson15:BuildingARMAmodels.Examples
Example 1
By using a computer program we have generated a time series
x.The graph of the series is presented in the following figure
Figure : A simulated time series
UmbertoTriacca Lesson15:BuildingARMAmodels.Examples
Example 1
The objective is to build an ARMA model this time series.
The first step in developing a model is to determine if the
series is stationary.
UmbertoTriacca Lesson15:BuildingARMAmodels.Examples
Example 1
Our time series seems the realization of a stationary process
with zero mean, thus we can look at sample autocorrelation
and partial autocorrelation function to establish the orders p
and q of the ARMA model.
Figure : Sample autocorrelation and sample partial autocorrelation
functions of time serieUsmxbertoTriacca Lesson15:BuildingARMAmodels.Examples
Example 1
Since the SACF cuts off after lag 2 and the SPCAF follows a
damped cycle,
an MA(2) model
x = u +θ u +θ u , u ∼ WN(0,σ2)
t t 1 t−1 2 t−2 t
seems appropriate for the sample data.
UmbertoTriacca Lesson15:BuildingARMAmodels.Examples
Example 1
Table reports the result of the ML estimation.
300 observations, Dependent Variable x
Variabile Coefficient St. error t statistic p-value
θ 1,68559 0,0456203 36,9481 0,0000
1
θ 0,883683 0,0492842 17,9303 0,0000
2
Variance of innovations 0.941107
UmbertoTriacca Lesson15:BuildingARMAmodels.Examples
Example 1
Now, we consider the graph of the residuals
Figure : Residuals from MA(2) model
UmbertoTriacca Lesson15:BuildingARMAmodels.Examples
Example 1
Figure : SACF and SPACF of residuals from MA(2) model
UmbertoTriacca Lesson15:BuildingARMAmodels.Examples
Example 1
By analysing the SACF and SPACF of residuals presented in
figure, we note that any term isn’t significant and
Q = 16.4450 do not indicate any autocorrelation in the
25
residuals. They can be assimilate to a white noise process.
UmbertoTriacca Lesson15:BuildingARMAmodels.Examples