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This article demonstrates the use of self-excitation threshold autoregressive SETAR for the analysis of sunspot data sets frequently studied by customers. Specifically, study the estimation and prediction of the SETAR model(Click “Read the original text” at the end of the article to obtain the completecode data< /strong>).
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We consider here raw sunspot sequences to fit ARMA examples, although many sources in the literature transform the sequences before modeling.
import numpy as np import pandas as pd ......
dta.index = pd.Index(sm.....m_range('1700', '2008'))
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R language time series TAR threshold autoregressive model
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First, we will use ARMA to fit the AR(3) process to the data.
arma_mod30 = sm.tsa.ARMA(dta, (3,0)).fit()......0.hqic
To test for nonlinearity, a BDS test can be performed using the residuals of a linear AR(3) model.
bds.bds(arm......sid, 3)
This suggests that there may be an underlying nonlinear structure. To try to capture this structure, we will fit a SETAR(2) model to the data, allowing two regimes and each regime being an AR(3) process. Here, we do not specify delay or threshold values, so they will be optimally selected from the model.
Note: In the summary, the \gamma parameter is the threshold value.
ser_23 = star......).fit()
AIC and BIC guidelines prefer the SETAR model over the AR model.
Note: This is a bootstrap test, so it may be slow until it is improved.
f_sat, pae,bsf_tas = stoest()
The null hypothesis is SETAR(1), so we can reject it and choose SETAR(2) as the alternative hypothesis.
However, it should be noted that the default assumption of order_test() is the existence of heteroscedasticity, which may be unreasonable here.
f_stat_h, pvalue_h, bs_f_stats_h = ......g') print pvalue
It should be noted that when considering the time series residuals, the BDS test still rejects the null hypothesis, albeit to a smaller extent than the AR(3) model. We can look at the residual plot to see if the errors have zero mean, but may not have homoskedasticity.
print bds.bds(setar_mod23.resid, 3) setar_mod23.resid.plot(figsize=(10,5));
Relative to the ARMA model, two new types of parameters are estimated here. Delay parameters and thresholds. The Lag parameter selects the process lag to use as the threshold variable, and Threshold indicates the threshold variable that separates the data points into (here two) states.
Alternative thresholds for which the likelihood ratio statistic is less than the critical value are included in the confidence set, and the lower and upper bounds of the confidence interval are the smallest and largest thresholds in the confidence set, respectively.
setarmd23.plo......ght=5);
We can look at in-sample dynamic forecasts and out-of-sample forecasts.
predit_am_mo30 = arma_mod30.predict('1990', '2012', dynamic=True)
ax = dta.ix['1950':].pot(igsze=(12,8))......
It appears that the dynamic predictions of the SETAR model are able to track the observed data points slightly better than the AR(3) model. We can compare by root mean square prediction error and see that SETAR performs slightly better.
def rsf(y, yhat):
return (y.sb(yhat**2).man()
prnt 'AR(3): ', rse(dta.SNACTIVI......od23)
However, if the prediction window is expanded, it becomes apparent that the SETAR model is the only one that fits the shape of the data since the data is cyclic.
preict_am_mo30_long = ama_md30.predict('1960', '2012', dynamic=True)......
ax.legend();
print 'AR(3): ', rmsfe(dta.S......ong)
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This article is selected from “Python Self-Excitation Threshold Autoregression (SETAR), ARMA, BDS Test, Predictive Analysis of Sunspot Time Series Data”.
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