setar models

8/17/2019 Setar Models

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This continues with the Sunspots section of the ARMA Notebook example.

Note: here we consider the raw Sunspot series to match the ARMA example, although many

sources in the literature apply a transformation to the series before modeling.

In [ ]:import numpy as np

import pandas as pd

from scipy import stats

import matplotlib.pyplot as plt

import statsmodels.api as sm

import statsmodels.tsa.setar_model as setar_model

In [ ]:

print sm.datasets.sunspots.NOTE

In [ ]:

dta = sm.datasets.sunspots.load_pandas().data

In [ ]:

dta.index = pd.Index(sm.tsa.datetools.dates_from_range('!""'# '$""%'))

del dta[&E&]

In [ ]:

dta.plot(figsi*e=($#%))+

First we'll fit an AR(3) process to the data as in the ARMA Notebook Example.

In [ ]:

arma_mod," = sm.tsa.-(dta# (,#")).fit()

print arma_mod,".params

print arma_mod,".ai# arma_mod,"./i# arma_mod,".01i

To try and capture nonlinearities, we'll fit a SETAR(2) model to the data to allow for two

regimes, and we let each regime be an AR(3) process. Here we're not specifying the delay or

threshold values, so they'll be optimally selected from the model.

Note: In the summary, the \gamma parameter(s) are the threshold value(s).

In [ ]:

setar_mod$, = setar_model.2ET(dta# $# ,).fit()

print setar_mod$,.summar3()

Note that the The AIC and BIC criteria prefer the SETAR model to the AR model.We can also directly test for the appropriate model, noting that an AR(3) is the same as a

SETAR(1;1,3), so the specifications are nested.

Note: this is a bootstrapped test, so it is rather slow until improvements can be made.

In [ ]:

setar_mod$, = setar_model.2ET(dta# $# ,).fit()

http://statsmodels.sourceforge.net/devel/examples/notebooks/generated/tsa_arma.htmlhttp://statsmodels.sourceforge.net/devel/examples/notebooks/generated/tsa_arma.htmlhttp://statsmodels.sourceforge.net/devel/examples/notebooks/generated/tsa_arma.html


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f_stat# p4alue# /s_f_stats = setar_mod$,.order_test() # by default tests

against SETAR(1)

print p4alue

The null hypothesis is a SETAR(1), so it looks like we can safely reject it in favor of the

SETAR(2) alternative.

One thing to note, though, is that the default assumptions oforder_test() is that there is

homoskedasticity, which may be unreasonable here. So we can force the test to allow for

heteroskedasticity of general form (in this case it doesn't look like it matters, however).

In [ ]:

f_stat_0# p4alue_0# /s_f_stats_0 =

setar_mod$,.order_test(0eteros5edastiit3='g')

print p4alue

In [ ]:

setar_mod$,.resid.plot(figsi*e=("#6))+

We have two new types of parameters estimated here compared to an ARMA model. The delayand the threshold(s). The delay parameter selects which lag of the process to use as the

threshold variable, and the thresholds indicate which values of the threshold variable separate

the datapoints into the (here two) regimes.

The confidence interval for the threshold parameter is generated (as in Hansen (1997)) by

inverting the likelihood ratio statistic created from considering the selected threshold value

against ecah alternative threshold value, and comparing against critical values for various

confidence interval levels. We can see that graphically by plotting the likelihood ratio sequence

against each alternate threshold.

Alternate thresholds that correspond to likelihood ratio statistics less than the critical value areincluded in a confidence set, and the lower and upper bounds of the confidence interval are the

smallest and largest threshold, respectively, in the confidence set.

In [ ]:

setar_mod$,.plot_t0res0old_i("# fig7idt0="# fig0eig0t=6)+

As in the ARMA Notebook Example, we can take a look at in-sample dynamic prediction and

out-of-sample forecasting.

In [ ]:

predit_arma_mod," = arma_mod,".predit('88"'# '$"$'# d3nami=True)

predit_setar_mod$, = setar_mod$,.predit('88"'# '$"$'# d3nami=True)

In [ ]:

ax = dta.ix['86"':].plot(figsi*e=($#%))

ax = predit_arma_mod,".plot(ax=ax# st3le='r99'# line7idt0=$#

la/el='(,) 3nami ;redition')+

ax = predit_setar_mod$,.plot(ax=ax# st3le='599'# line7idt0=$#

la/el='2ET($+,#,) 3nami ;redition')+

ax.legend()+


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ax.axis((9$"."# ,%."# 9N?TI@IT# predit_arma_mod,")

print '2ET($+,#,): '# rmsfe(dta.2>N?TI@IT# predit_setar_mod$,)

If we extend the forecast window, however, it is clear that the SETAR model is the only one that

even begins to fit the shape of the data, because the data is cyclic.

In [ ]:

predit_arma_mod,"_long = arma_mod,".predit('8A"'# '$"$'#

d3nami=True)

predit_setar_mod$,_long = setar_mod$,.predit('8A"'# '$"$'#

d3nami=True)

ax = dta.ix['86"':].plot(figsi*e=($#%))

ax = predit_arma_mod,"_long.plot(ax=ax# st3le='r99'# line7idt0=$#

la/el='(,) 3nami ;redition')+

ax = predit_setar_mod$,_long.plot(ax=ax# st3le='599'# line7idt0=$#

la/el='2ET($+,#,) 3nami ;redition')+

ax.legend()+

ax.axis((9$"."# ,%."# 9N?TI@IT# predit_arma_mod,"_long)

print '2ET($+,#,): '# rmsfe(dta.2>N?TI@IT# predit_setar_mod$,_long)

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