ssSARIMA#
Purpose#
Estimates SARIMA models using a state space representation, the Kalman filter, and maximum likelihood.
Format#
- sOut = ssSARIMA(y, p, d, q, p_s, d_s, s[, const, trend])#
- Parameters:
y (Nx1 vector) – data.
p (Scalar) – the autoregressive order.
d (Scalar) – Optional argument, the order of differencing. Default = 0.
q (Scalar) – Optional argument, the moving average order. Default = 0.
const (Scalar) – Optional argument, an indicator variable to include a constant in the model. Set to 1 to include trend, 0 otherwise. Default = 0.
trend (Scalar) – Optional argument, an indicator variable to include a trend in the model. Set to 1 to include trend, 0 otherwise. Default = 0.
- Returns:
sOut (Struct) –
an instance of an
ssOutstructure. For an instance named sOut, the members are:sOut.final_params
String array, final parameter estimates.
sOut.resid
Vector, residuals.
sOut.fitted
Vector, the fitted y values based on final parameter estimates.
sOut.df_model
Vector, degrees of freedom of the model.
sOut.df_resid
Vector, degrees of freedom of the residuals.
sOut.numObs
Vector, number of observations.
sOut.mleResults
Instance of a
cmlmtResultsstructure. Further information provided in thecmlmtdocumentation.sOut.kfResults
Instance of a
kalmanOutstructure, contains the results from thekalmanFilter().kfResults.filtered_state
Matrix, k_endog x numObs, filtered states.
kfResults.filtered_state_cov
Array, numObs x k_endog x k_endog, filtered state covariances.
kfResults.predicted_state
Matrix, k_endog x (numObs+1), predicted states.
kfResults.predicted_state_cov
Array, numObs x k_endog x k_endog, predicted state covariances.
kfResults.forecast
Matrix, k_endog x numObs, forecasts.
kfResults.forecast_error
Matrix, k_endog x numObs, forecast error.
kfResults.forecast_error_cov
Array, numObs x k_endog x k_endog, forecast error covariances.
kfResults.loglikelihood
Matrix, k_endog x (numObs+1), computed loglikelihood.
sOut.ssmFinal
Instance of a
ssModelstructure, contains the final state space system matrices used in thekalmanFilter(). Contains the following members:ssmFinal.Z
k_endog x k_states, transition matrix.
ssmFinal.d
k_endog x 1, observation intercept.
ssmFinal.H
k_endog x k_endog, observation disturbance covariance.
ssmFinal.T
k_states x k_states, design matrix.
ssmFinal.c
k_states x k_states, state intercept.
ssmFinal.R
k_states x k_posdef, selection matrix.
ssmFinal.Q
k_states x k_posdef, state disturbance covariance.
ssmFinal.a_0
k_states x 1, initial prior state mean.
ssmFinal.p_0
k_states x k_states, initial prior state covariance.
sOut.aic
Scalar, model Akaike’s information criterion.
sOut.aicc
Scalar, model corrected Akaike’s information criterion.
sOut.bic
Scalar, model Schwarz’ Bayesian information criterion.
sOut.hqic
Scalar, model Hannan–Quinn information criterion.
sOut.ssy
Scalar, sum of squares total (Deviations of y from mean of y).
sOut.sse
Scalar, sum of squared errors.
sOut.mse
Scalar, means squared errors.
sOut.rsquared
Scalar, model r-squared.
sOut.ljung_box
Scalar, Ljung-Box Q-test for autocorrelation.
sOut.ljung_box_pval
Scalar, p-value of the Ljung-Box Q-test for autocorrelation.
sOut.hetero_test
Scalar, tests for the null hypothesis of no heteroskedasticity.
sOut.hetero_test_pval
Scalar, p-value of the test for the null hypothesis of no heteroskedasticity.
sOut.jb_stat
Scalar, the Jarque-Bera goodness-of-fit test on model residuals.
sOut.jb_stat_pval
Scalar, p-value ofthe Jarque-Bera goodness-of-fit test.
sOut.standardized_forecast_errors
Scalar, standardized forecast errors used in all residual diagnostics.
sOut.skew
Scalar, sample skewness of the standardized forecast errors.
sOut.kurtosis
Scalar, sample kurtosis of the standardized forecast errors.
sOut.irf
Scalar, model impulse response functions.
sOut.forecasts
Scalar, forecasts.
Example#
new;
library tsmt, cmlmt, sslib;
// Create file name with full path
fname = __FILE_DIR $+ "wpi1.dat";
// Load variable 'wpi' from 'wpi1.dat'
y = loadd(fname, "wpi");
// Model settings
p = 1;
d = 1;
q = 1;
trend = 0;
const = 1;
// Declare 'sOut' to be an ssOut structure
// to hold the estimation results and then
// estimate the model
struct ssOut sOut;
sOut = ssARIMA(y, p, d, q, const, trend);
The example above prints the following results
Return Code: 0
Log-likelihood: -4.603e+04
Number of Cases: 130
AIC: 9.207e+04
AICC: 9.207e+04
BIC: 9.208e+04
HQIC: 9.207e+04
Covariance Method: ML covariance matrix
==========================================================================
Parameters Estimates Std. Err. T-stat Prob. Gradient
-------------------------------------------------------------------------------------------------------------
theta : e[t-1] -0.4073 0.0753 -5.4103 0.0000 .
theta : e[t-12] -0.5506 0.0891 -6.1764 0.0000 .
Sigma2 0.0014 0.0002 9.2080 0.0000 .
Correlation matrix of the parameters
--------------------------------------------------------------------------
1.0000 -0.2894 -0.0638
-0.2894 1.0000 -0.0066
-0.0638 -0.0066 1.0000
Model and residual diagnostics:
==========================================================================
Ljung-Box (Q): 0.0381
Prob(Q): 0.845
Heteroskedasticity (H): 0.544
Prob(H): 0.049
Jarque-Bera (JB): 2.02
Prob(JB): 0.363
Skew: 0.0671
Kurtosis: 3.59
==========================================================================
Source#
ssarima.src