Create Regression Models with SARIMA Errors

SARMA Error Model Without an Intercept

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This example shows how to specify a regression model with SARMA errors without a regression intercept.

Specify the default regression model with $S A R M A (1, 1) \times (2, 1, 1)_{4}$ errors:

$\begin{array}{c} y_{t} = X_{t} β + u_{t} \\ (1 - a_{1} L) (1 - A_{4} L^{4} - A_{8} L^{8}) (1 - L^{4}) u_{t} = (1 + b_{1} L) (1 + B_{4} L^{4}) ε_{t} ． \end{array}$

Mdl = regARIMA('ARLags',1,'SARLags',[4, 8],．..'Seasonality',4,'MALags',1,'SMALags',4,'Intercept',0)

Mdl = regARIMA with properties: Description: "ARMA(1,1) Error Model Seasonally Integrated with Seasonal AR(8) and MA(4) (Gaussian Distribution)" Distribution: Name = "Gaussian" Intercept: 0 Beta: [1×0] P: 13 Q: 5 AR: {NaN} at lag [1] SAR: {NaN NaN} at lags [4 8] MA: {NaN} at lag [1] SMA: {NaN} at lag [4] Seasonality: 4 Variance: NaN

The name-value pair argument:

'ARLags',1specifies which lags have nonzero coefficients in the nonseasonal autoregressive polynomial, so $a (L) = (1 - a_{1} L)$ ．
'SARLags',[4 8]specifies which lags have nonzero coefficients in the seasonal autoregressive polynomial, so $A (L) = (1 - A_{4} L^{4} - A_{8} L^{8})$ ．
'MALags',1specifies which lags have nonzero coefficients in the nonseasonal moving average polynomial, so $b (L) = (1 + b_{1} L)$ ．
'SMALags',4specifies which lags have nonzero coefficients in the seasonal moving average polynomial, so $B (L) = (1 + B_{4} L^{4})$ ．
'Seasonality',4specifies the degree of seasonal integration and corresponds to $(1 - L^{4})$ ．

The software setsInterceptto 0, but all other parameters inMdlareNaNvalues by default.

PropertyP=p+D+ $p_{s}$ + s = 1 + 0 + 8 + 4 = 13, and propertyQ=q+ $q_{s}$ = 1 + 4 = 5. Therefore, the software requires at least 13 presample observation to initializeMdl．

SinceInterceptis not aNaN, it is an equality constraint during estimation. In other words, if you passMdland data intoestimate, thenestimatesetsInterceptto 0 during estimation.

You can modify the properties ofMdlusing dot notation.

Be aware that the regression model intercept (Intercept) is not identifiable in regression models with ARIMA errors. If you want to estimateMdl, then you must setInterceptto a value using, for example, dot notation. Otherwise,estimatemight return a spurious estimate ofIntercept．

Known Parameter Values for a Regression Model with SARIMA Errors

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This example shows how to specify values for all parameters of a regression model with SARIMA errors.

Specify the regression model with $S A R I M A (1, 1, 1) \times (1, 1, 0)_{12}$ errors:

$\begin{array}{c} y_{t} = X_{t} β + u_{t} \\ (1 - 0 ． 2 L) (1 - L) (1 - 0 ． 25 L^{12} - 0 ． 1 L^{24}) (1 - L^{12}) u_{t} = (1 + 0 ． 15 L) ε_{t}, \end{array}$

where $ε_{t}$ 与单位方差高斯。

Mdl = regARIMA('AR',0.2,'SAR',{0.25, 0.1},'SARLags',[12 24],．..'D',1,'Seasonality',12,'MA',0.15,'Intercept',0,'Variance',1)

Mdl = regARIMA with properties: Description: "ARIMA(1,1,1) Error Model Seasonally Integrated with Seasonal AR(24) (Gaussian Distribution)" Distribution: Name = "Gaussian" Intercept: 0 Beta: [1×0] P: 38 D: 1 Q: 1 AR: {0.2} at lag [1] SAR: {0.25 0.1} at lags [12 24] MA: {0.15} at lag [1] SMA: {} Seasonality: 12 Variance: 1

The parameters inMdldo not containNaNvalues, and therefore there is no need to estimateMdl．然而,您可以模拟或预测反应by passingMdltosimulateorforecast．

Regression Model with SARIMA Errors and t Innovations

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This example shows how to set the innovation distribution of a regression model with SARIMA errors to atdistribution.

Specify the regression model with $S A R I M A (1, 1, 1) \times (1, 1, 0)_{12}$ errors:

$\begin{array}{c} y_{t} = X_{t} β + u_{t} \\ (1 - 0 ． 2 L) (1 - L) (1 - 0 ． 25 L^{12} - 0 ． 1 L^{24}) (1 - L^{12}) u_{t} = (1 + 0 ． 15 L) ε_{t}, \end{array}$

where $ε_{t}$ has atdistribution with the default degrees of freedom and unit variance.

Mdl = regARIMA('AR',0.2,'SAR',{0.25, 0.1},'SARLags',[12 24],．..'D',1,'Seasonality',12,'MA',0.15,'Intercept',0,．..'Variance',1,'Distribution','t')

Mdl = regARIMA with properties: Description: "ARIMA(1,1,1) Error Model Seasonally Integrated with Seasonal AR(24) (t Distribution)" Distribution: Name = "t", DoF = NaN Intercept: 0 Beta: [1×0] P: 38 D: 1 Q: 1 AR: {0.2} at lag [1] SAR: {0.25 0.1} at lags [12 24] MA: {0.15} at lag [1] SMA: {} Seasonality: 12 Variance: 1

The default degrees of freedom isNaN．If you don't know the degrees of freedom, then you can estimate it by passingMdland the data toestimate．

Specify a $t_{10}$ distribution.

Mdl.Distribution = struct('Name','t','DoF',10)

Mdl = regARIMA with properties: Description: "ARIMA(1,1,1) Error Model Seasonally Integrated with Seasonal AR(24) (t Distribution)" Distribution: Name = "t", DoF = 10 Intercept: 0 Beta: [1×0] P: 38 D: 1 Q: 1 AR: {0.2} at lag [1] SAR: {0.25 0.1} at lags [12 24] MA: {0.15} at lag [1] SMA: {} Seasonality: 12 Variance: 1

You can simulate or forecast responses by passingMdltosimulateorforecastbecauseMdlis completely specified.

In applications, such as simulation, the software normalizes the randomtinnovations. In other words,Varianceoverrides the theoretical variance of thetrandom variable (which isDoF/(DoF- 2)), but preserves the kurtosis of the distribution.

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