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Time Series and Stochastic Differential Equations
Find Conditions for Stationarity and Invertibility of Time Series Processes
Find conditions for an
ARMAProcess
[2, 2]
to be weakly stationary using
WeakStationarity
.
In[1]:=
X
arma = ARMAProcess[{Subscript[a, 1], Subscript[a, 2]}, {Subscript[b, 1], Subscript[b, 2]}, \[Sigma]]; ws = WeakStationarity[arma]
Out[1]=
In[2]:=
X
RegionPlot[ws, {Subscript[a, 1], 2, 2}, {Subscript[a, 2], 2, 2}, GridLines > Automatic]
Out[2]=
Find an instance of a weakly stationary
ARMAProcess
[2, 2]
.
In[3]:=
X
aspt = ProcessParameterAssumptions[arma]; rest = Subscript[a, 1] != 0 && Subscript[a, 2] != 0 && Subscript[b, 1] != 0 && Subscript[b, 2] != 0 && \[Sigma] != 0; wsARMA = arma /. FindInstance[ ws && rest && aspt, {Subscript[a, 1], Subscript[a, 2], Subscript[ b, 1], Subscript[b, 2], \[Sigma]}][[1]]
Out[3]=
Check that the process is weakly stationary.
In[4]:=
X
WeakStationarity[wsARMA]
Out[4]=
Find conditions for an
ARMAProcess
[2, 2]
to be invertible using
TimeSeriesInvertibility
.
In[5]:=
X
in = TimeSeriesInvertibility[arma]
Out[5]=
In[6]:=
X
RegionPlot[in, {Subscript[b, 1], 2, 2}, {Subscript[b, 2], 2, 2}, GridLines > Automatic]
Out[6]=
Find an instance of an invertible
ARMAProcess
[2, 2]
.
In[7]:=
X
aspt = ProcessParameterAssumptions[arma]; rest = Subscript[a, 1] != 0 && Subscript[a, 2] != 0 && Subscript[b, 1] != 0 && Subscript[b, 2] != 0 && \[Sigma] != 0; invARMA = arma /. FindInstance[ in && rest && aspt, {Subscript[a, 1], Subscript[a, 2], Subscript[ b, 1], Subscript[b, 2], \[Sigma]}][[1]]
Out[7]=
Check that the process is invertible.
In[8]:=
X
TimeSeriesInvertibility[invARMA]
Out[8]=