regARIMA

Create regression model with ARIMA time series errors

Description

The regARIMA function returns a regARIMA object specifying the functional form and storing the parameter values of a regression model with ARIMA time series errors for a univariate response process y_t.

$$\begin{array}{c}{y}_t=c+X_t\beta +u_t\\ a\left(L\right)A\left(L\right){\left(1-L\right)}^D\left(1-L^s\right)u_t=b\left(L\right)B\left(L\right){\epsilon }_t,\end{array}$$

Because they completely specify the model structure, the key components of a regARIMA object are the:

• Regression model coefficients c and β

• Polynomial degrees of the ARIMA disturbances u_t, for example, the AR polynomial degree p and the degree of integration D

Given only polynomial degrees, the regression model contains only a constant. All parameters, such as the model constant, and error model coefficients and innovation-distribution parameters, are unknown and estimable unless you specify their values. regARIMA determines the number of coefficients in the regression model by the number of variables in the supplied predictor data or by other specifications.

To estimate a model containing unknown parameter values, pass the model and data to the estimate object function. To work with an estimated or fully specified regARIMA object, pass it to an object function.

Alternatively, you can:

• Create and work with regARIMA model objects interactively by using Econometric Modeler.

• Create a standard ARIMA model containing exogenous predictors (ARIMAX). For more details, see the arima function and Alternative ARIMA Model Representations.

• Create a Bayesian linear regression model by using the bayeslm function.

Creation

Syntax

Mdl = regARIMA

Mdl = regARIMA(p,D,q)

Mdl = regARIMA(Name=Value)

Description

Mdl = regARIMA creates a regression model containing degree 0 ARIMA disturbances. The regression model contains an intercept; the software determines the number of regression coefficients when you fit the model to data by using estimate. The innovations are iid Gaussian random variables with a mean of 0 and unknown variance.

Mdl = regARIMA(p,D,q) creates a regression model with ARIMA(p,D,q) disturbances. The disturbance model contains nonseasonal AR polynomial lags from 1 through p, a degree D nonseasonal integration polynomial, and nonseasonal MA polynomial lags from 1 through q. The regression model contains an intercept; the software determines the number of regression coefficients when you fit the model to data by using estimate. The innovations are iid Gaussian random variables with a mean of 0 and unknown variance.

This shorthand syntax provides an easy way to create a model template in which you specify the degrees of the nonseasonal polynomials explicitly. The model template is suited for unrestricted parameter estimation. After you create a model, you can alter property values using dot notation.

Mdl = regARIMA(Name=Value) sets properties and polynomial lags using name-value arguments. For example, regARIMA(ARLags=[1 4],AR={0.5 –0.1}) creates a regression model containing an unknown model intercept and innovations variance, and AR(4) disturbances, where the lag 1 nonseasonal AR coefficient is –0.5 and the lag 4 nonseasonal AR coefficient is 0.1.

This longhand syntax allows you to create more flexible models. For example, you can create a regression model with seasonal errors by using only longhand syntax. regARIMA infers all disturbance model polynomial degrees from the properties that you set. Therefore, property values that correspond to polynomial degrees must be consistent with each other.

Input Arguments

The shorthand syntax provides an easy way for you to create model templates of regression models with nonseasonal ARIMA errors. Model templates are suitable for unrestricted parameter estimation. For example, to create a regression model with ARMA(2,1) errors containing an unknown model intercept and innovations variance, enter:

Mdl = regARIMA(2,0,1);

To impose equality constraints on parameter values during estimation, or include seasonal components, set the appropriate property values using dot notation.

p — Nonseasonal autoregressive polynomial degree

nonnegative integer

Nonseasonal autoregressive polynomial degree for the error model, specified as a nonnegative integer.

Data Types: double

D — Degree of nonseasonal integration

nonnegative integer

Degree of nonseasonal integration (the degree of the nonseasonal differencing polynomial) for the error model, specified as a nonnegative integer. The D input argument sets the property D.

Data Types: double

q — Nonseasonal moving average polynomial degree

nonnegative integer

Nonseasonal moving average polynomial degree for the error model, specified as a nonnegative integer.

Data Types: double

Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Before R2021a, use commas to separate each name and value, and enclose Name in quotes.

The longhand syntax enables you to create seasonal error models or models in which some or all coefficients are known. During estimation, estimate imposes equality constraints on any known parameters.

Example: regARIMA(ARLags=[1 4],AR={0.5 –0.1}) creates a regression model containing an unknown model intercept and innovations variance, and AR(4) disturbances, where the lag 1 nonseasonal AR coefficient is –0.5 and the lag 4 nonseasonal AR coefficient is 0.1, symbolically, $$1-0.5L^1+0.1L^4$$.

ARLags — Lags associated with nonseasonal AR polynomial coefficients

1:numel(AR) (default) | numeric vector of unique positive integers

Lags associated with the nonseasonal AR polynomial coefficients for the error model u_t, specified as a numeric vector of unique positive integers. The maximum lag is p.

AR{j} is the coefficient of lag ARLags(j), where AR is the value of the property AR.

Example: ARLags=4 specifies the nonseasonal AR polynomial $$1-{\varphi }_4{L}^4$$.

Example: ARLags=1:4 specifies the nonseasonal AR polynomial $$1-{\varphi }_1{L}^1-{\varphi }_2{L}^2-{\varphi }_3{L}^3-{\varphi }_4{L}^4$$.

Example: ARLags=[1 4] specifies the nonseasonal AR polynomial $$1-{\varphi }_1{L}^1-{\varphi }_4{L}^4.$$

Data Types: double

MALags — Lags associated with nonseasonal MA polynomial coefficients

1:numel(MA) (default) | numeric vector of unique positive integers

Lags associated with the nonseasonal MA polynomial coefficients for the error model u_t, specified as a numeric vector of unique positive integers. The maximum lag is q.

MA{j} is the coefficient of lag MALags(j), where MA is the value of the property MA.

Example: MALags=3 specifies the nonseasonal MA polynomial $$1+{\theta }_3{L}^3$$.

Example: MALags=1:3 specifies the nonseasonal MA polynomial $$1+{\theta }_1{L}^1+{\theta }_2{L}^2+{\theta }_3{L}^3.$$

Example: MALags=[1 3] specifies the nonseasonal MA polynomial $$1+{\theta }_1{L}^1+{\theta }_3{L}^3$$.

Data Types: double

SARLags — Lags associated with seasonal AR polynomial coefficients

1:numel(SAR) (default) | numeric vector of unique positive integers

Lags associated with the seasonal AR polynomial coefficients for the error model u_t, specified as a numeric vector of unique positive integers. The maximum lag is p_s.

SAR{j} is the coefficient of lag SARLags(j), where SAR is the value of the property SAR.

Specify SARLags as the periodicity of the observed data, not as multiples of the Seasonality property. This convention does not conform to standard Box and Jenkins [1] notation, but it is more flexible for incorporating multiplicative seasonality.

Example: SARLags=[4 8] specifies the seasonal AR polynomial $$1-{\Phi }_4{L}^4-{\Phi }_8{L}^8.$$

Data Types: double

SMALags — Lags associated with seasonal MA polynomial coefficients

1:numel(SMA) (default) | numeric vector of unique positive integers

Lags associated with the seasonal MA polynomial coefficients for the error model u_t, specified as a numeric vector of unique positive integers. The maximum lag is q_s.

SMA{j} is the coefficient of lag SMALags(j), where SMA is the value of the property SMA.

Specify SMALags as the periodicity of the observed data, not as multiples of the Seasonality property. This convention does not conform to standard Box and Jenkins [1] notation, but it is more flexible for incorporating multiplicative seasonality.

Example: SMALags=4 specifies the seasonal MA polynomial $$1+{\Theta }_4{L}^4.$$

Data Types: double

Note

Polynomial degrees are not estimable. If you do not specify a polynomial degree, or regARIMA cannot infer it from other specifications, regARIMA does not include the polynomial in the model.

Properties

You can set writable property values when you create the model object by using name-value argument syntax, or after you create the model object by using dot notation. For example, to create the fully specified regression model with ARMA(2,1) disturbances

$$\begin{array}{l}{y}_t=1+3x_1+5x_2+u_t\\ u_t=0.3u_{t-1}-0.15u_{t-2}+{\epsilon }_t+0.2{\epsilon }_{t-1},\end{array}$$

enter:

Mdl = regARIMA(Intercept=1,Beta=[3; 5],AR={0.3 -0.15},MA=0.2);
Mdl.Variance = 1;

Note

• NaN-valued properties indicate estimable parameters. Numeric properties indicate equality constraints on parameters during model estimation. Coefficient vectors can contain both numeric and NaN-valued elements.

• You can specify polynomial coefficients as vectors in any orientation, but regARIMA stores them as row vectors.

Regression Model Properties

Intercept — Regression model intercept c

NaN (default) | numeric scalar

Regression model intercept c, specified as a numeric scalar.

Example: Intercept=1

Data Types: double

Beta — Regression model coefficients β

empty row vector [] (default) | numeric vector

Regression component coefficients β associated with predictor variables x_t, specified as a numeric vector.

The default indicates one of the following conditions:

• estimate infers the size of Beta from the number of columns of the specified predictor data X. Therefore, if you plan to fit all regression coefficients to data, you do not need to specify Beta.

• The model does not include regression coefficients.

Example: Beta=[0.5 NaN 3] specifies three regression coefficients. During estimation, estimate fixes β₁ to 5 and β₃ to 3, and it fits β₂ to the data associated with the second predictor variable.

Data Types: double

Error Model Properties

P — Compound AR polynomial degree

Read-only: nonnegative integer

This property is read-only.

Compound AR polynomial degree of the error model, specified as a nonnegative integer.

P does not necessarily conform to standard Box and Jenkins notation [1] because P captures the degrees of the nonseasonal and seasonal AR polynomials (properties AR and SAR, respectively), nonseasonal integration (property D), and seasonality (property Seasonality). Explicitly, P = p + D + p_s + s. P conforms to Box and Jenkins notation for models without integration or a seasonal AR component (D = 0 and SAR = {}).

P specifies the number of lagged observations required to initialize the AR components of the model.

Data Types: double

Q — Compound MA polynomial degree

Read-only: nonnegative integer

This property is read-only.

Compound MA polynomial degree of the error model, specified as a nonnegative integer.

Q does not necessarily conform to standard Box and Jenkins notation [1] because Q captures the degrees of the nonseasonal and seasonal MA polynomials (properties MA and SMA, respectively). Explicitly, Q = q + q_s. Q conforms to Box and Jenkins notation for models without a seasonal MA component (SMA = {}).

Q specifies the number of lagged innovations required to initialize the MA components of the model.

Data Types: double

AR — Nonseasonal AR polynomial coefficients ϕ

cell vector | empty cell vector {}

Nonseasonal AR polynomial coefficients ϕ for the error model u_t, specified as a cell vector. Cells contain numeric scalars or NaN values. A fully specified nonseasonal AR polynomial must be stable.

Coefficient signs correspond to the model expressed in difference-equation notation. For example, for the nonseasonal AR polynomial $$\varphi \left(L\right)=1-0.5L+0.1L^2,$$ specify AR={0.5 –0.1}.

If you do not set the ARLags name-value argument, AR{j} is the coefficient of lag j, j = 1,…,p, where p = numel(AR).

Otherwise, if ARLags = arlags, with p = max(arlags), the following conditions apply:

• The lengths of AR and arlags must be equal.

• AR{j} is the coefficient of lag arlags(j), for each j.

• regARIMA stores AR as a length p cell vector. All cells that do not correspond to lags in arlags contain 0.

The default value of AR depends on other specifications:

• If you use the shorthand syntax to specify p > 0, AR is a length p cell vector, where each cell contains a NaN value.

• If you specify ARLags, AR is a length p cell vector. AR{j} = NaN for each lag arlags(j). All other cells contain 0.

• Otherwise, AR is an empty cell vector {}, meaning the model does not contain a nonseasonal AR polynomial.

The coefficients in AR correspond to coefficients in an underlying LagOp lag operator polynomial, and they are subject to a near-zero tolerance exclusion test. If a coefficient is 1e–12 or below, regARIMA excludes that coefficient and its corresponding lag in ARLags from the model.

Example: AR={0.8} sets the only AR lag coefficient associated with lag ARLags(1) to 0.8.

Example: regARIMA(AR={0.2 0 0.1}) sets the error model, in difference-equation form, to $$u_t=0.2u_{t-1}+0.1u_{t-3}+{\epsilon }_t$$.

Example: regARIMA(AR={NaN –0.1},ARLags=[4 8]) sets the AR lag polynomial to $$1-{\varphi }_4{L}^4+0.1L^8$$, where ϕ₄ is unknown and estimable.

Data Types: cell

MA — Nonseasonal MA polynomial coefficients θ

cell vector | empty cell vector {}

Nonseasonal MA polynomial coefficients θ for the error model u_t, specified as a cell vector. Cells contain numeric scalars or NaN values. A fully specified nonseasonal MA polynomial must be invertible.

If you do not set the MALags name-value pair argument, MA{j} is the coefficient of lag j, j = 1,…,q, where q = numel(MA).

Otherwise, if MALags = malags, with q = max(MALags), the following conditions apply:

• The lengths of MA and malags must be equal.

• MA{j} is the coefficient of lag malags(j), for each j.

• regARIMA stores MA as a length q cell vector. All cells that do not correspond to lags in malags contain 0.

The default value of MA depends on other specifications:

• If you use the shorthand syntax to specify q > 0, MA is a length q cell vector, where each cell contains a NaN value.

• If you specify MALags, MA is a length q cell vector. MA{j} = NaN for each lag malags(j). All other cells contain 0.

• Otherwise, MA is an empty cell vector {}, meaning the error model does not contain a nonseasonal MA polynomial.

The coefficients in MA correspond to coefficients in an underlying LagOp lag operator polynomial, and they are subject to a near-zero tolerance exclusion test. If a coefficient is 1e–12 or below, regARIMA excludes that coefficient and its corresponding lag in MALags from the model.

Example: MA=0.8 sets the only MA lag coefficient associated with lag MALags(1) to 0.8.

Example: regARIMA(MA={0.2 0.1}) sets the error model to $$u_t={\epsilon }_t+0.2{\epsilon }_{t-1}+0.1{\epsilon }_{t-2}.$$

Example: regARIMA(MA={NaN –0.1},MALags=[4 8]) sets the MA lag polynomial to $$1+{\theta }_4{L}^4-0.1L^8$$, where θ₄ is unknown and estimable.

Data Types: cell

SAR — Seasonal AR polynomial coefficients Φ

cell vector | empty cell vector {}

Seasonal AR polynomial coefficients Φ for the error model u_t, specified as a cell vector. Cells contain numeric scalars or NaN values. A fully specified seasonal AR polynomial must be stable.

Coefficient signs correspond to the model expressed in difference-equation notation. For example, for the seasonal AR polynomial $$\Phi \left(L\right)=1-0.5L^4+0.1L^8,$$ specify SAR={0.5 –0.1}.

If you do not set the SARLags name-value argument, SAR{j} is the coefficient of lag j, j = 1,…,p_s, where p_s = numel(SAR).

Otherwise, if SARLags = sarlags, with p_s = max(sarlags), the following conditions apply:

• The lengths of SAR and sarlags must be equal.

• SAR{j} is the coefficient of lag sarlags(j), for each j.

• regARIMA stores SAR as a length p_s cell vector. All cells that do not correspond to lags in sarlags contain 0.

The default value of SAR depends on the value of SARLags:

• If you specify SARLags, SAR is a length p_s cell vector. SAR{j} = NaN for each lag SARLags(j). All other cells contain 0.

• Otherwise, SAR is an empty cell vector {}, meaning the error model does not contain a seasonal AR polynomial.

The coefficients in SAR correspond to coefficients in an underlying LagOp lag operator polynomial, and they are subject to a near-zero tolerance exclusion test. If a coefficient is 1e–12 or below, regARIMA excludes that coefficient and its corresponding lag in SARLags from the model.

Example: SAR=0.8 sets the only SAR lag coefficient associated with lag SARLags(1) to 0.8.

Example: regARIMA(SAR={0.2 0.1},Seasonality=4) sets the error model to $$\left(1-0.2L^1-0.1L^2\right)\left(1-L^4\right)u_t={\epsilon }_t$$.

Example: regARIMA(SAR={NaN –0.1},SARLags=[4 8],Seasonality=4) sets the SAR lag polynomial to $$\left(1-{\Theta }_4{L}^4-0.1L^8\right)\left(1-L^4\right)$$, where Φ₄ is unknown and estimable.

Data Types: cell

SMA — Seasonal MA polynomial coefficients

cell vector | empty cell vector {}

Seasonal MA polynomial coefficients for the error model, specified as a cell vector. Cells contain numeric scalars or NaN values. A fully specified seasonal MA polynomial must be invertible.

If you do not set the SMALags name-value argument, SMA{j} is the coefficient of lag j, j = 1,…,q_s, where q_s = numel(SMA).

Otherwise, if SMALags = smalags, with q_s = max(smalags), the following conditions apply:

• The lengths of SMA and SMALags must be equal.

• SMA{j} is the coefficient of lag smalags(j), for each j.

• regARIMA stores SMA as a length q_s cell vector. All cells that do not correspond to lags in smalags contain 0.

The default value of SMA depends on other specifications:

• If you specify SMALags, MA is a length q cell vector. MA{j} = NaN for each lag MALags(j). All other cells contain 0.

• Otherwise, SMA is an empty cell vector {}, meaning the error model does not contain a seasonal MA polynomial.

The coefficients in SMA correspond to coefficients in an underlying LagOp lag operator polynomial, and they are subject to a near-zero tolerance exclusion test. If a coefficient is 1e–12 or below, regARIMA excludes that coefficient and its corresponding lag in SMALags from the model.

Example: SMA=0.8 sets the only SMA lag coefficient associated with lag SMALags(1) to 0.8.

Example: regARIMA(SMA{0.2 0.1},Seasonality=4) specifies the error model $$\left(1-L^4\right)u_t=\left(1+0.2L+0.1L^2\right){\epsilon }_t.$$

Example: regARIMA(SMALags=[1 4],SMA={0.2 0.1},Seasonality = 4) specifies the error model $$\left(1-L^4\right)u_t=\left(1+0.2L+0.1L^4\right){\epsilon }_t.$$

Data Types: cell

D — Degree of nonseasonal integration

0 (default) | nonnegative integer

Degree of nonseasonal integration, or the degree of the nonseasonal differencing polynomial, for the error model specified as a nonnegative integer.

If you use shorthand syntax to create Mdl, the input d sets D.

Example: D=1

Example: regARIMA(0,1,2) sets D to 1.

Data Types: double

Seasonality — Degree of seasonal differencing polynomial

0 (default) | nonnegative integer

Degree of the seasonal differencing polynomial s for the error model, specified as a nonnegative integer.

Example: Seasonality=12 specifies monthly periodicity.

Data Types: double

Variance — Variance σ² of model innovations process ε_t

NaN (default) | positive scalar

Variance σ² of the model innovations process ε_t, specified as a positive scalar.

NaN specifies an unknown and estimable variance, which estimate fits to data.

Example: Variance=1

Data Types: double

Other Properties

Description — Model description

string scalar | character vector

Model description, specified as a string scalar or character vector. regARIMA stores the value as a string scalar. The default value describes the parametric form of the model, for example, "Regression with ARMA(2,1) Error Model (Gaussian Distribution)".

Example: "Model 1"

Data Types: string | char

Distribution — Conditional probability distribution of innovation process ε_t

"Gaussian" (default) | "t" | structure array

Conditional probability distribution of the innovation process ε_t, specified as a string or structure array. regARIMA stores the value as a structure array.

Distribution	String	Structure Array
Gaussian	"Gaussian"	struct('Name',"Gaussian")
Student’s t	"t"	struct('Name',"t",'DoF',DoF)

The 'DoF' field specifies the t distribution degrees of freedom parameter.

• DoF > 2 or DoF = NaN.

• DoF is estimable.

• If you specify "t", DoF is NaN by default. You can change its value by using dot notation after you create the model. For example, Mdl.Distribution.DoF = 3.

• If you supply a structure array to specify the Student's t distribution, then you must specify both the 'Name' and the 'DoF' fields.

Example: Distribution=struct('Name',"t",'DoF',10)

SeriesName — Response series name

"Y" (default) | string scalar | character vector

Since R2023b

Response series name, specified as a string scalar or character vector. regARIMA stores the value as a string scalar.

Example: "StockReturn"

Data Types: string | char

Object Functions

estimate	Fit univariate regression model with ARIMA errors to data
infer	Infer residuals of univariate regression model with ARIMA time series errors
summarize	Display estimation results of regression model with ARIMA errors
simulate	Monte Carlo simulation of univariate regression model with ARIMA time series errors
filter	Filter disturbances through regression model with ARIMA errors
impulse	Generate regression model with ARIMA errors impulse response function (IRF)
forecast	Forecast responses of univariate regression model with ARIMA time series errors
arima	Convert regression model with ARIMA errors to ARIMAX model

Examples

Specify Regression Model with Nonseasonal ARIMA Errors

Specify the following regression model with ARIMA(2,1,3) errors:

$$\begin{array}{c}{y}_t=u_t\\ (1-{\varphi }_{1}L-{\varphi }_2{L}^2)(1-L)u_t=(1+{\theta }_{1}L+{\theta }_2{L}^2+{\theta }_3{L}^3){\epsilon }_t.\end{array}$$

Mdl = regARIMA(2,1,3)

Mdl = 
  regARIMA with properties:

     Description: "ARIMA(2,1,3) Error Model (Gaussian Distribution)"
      SeriesName: "Y"
    Distribution: Name = "Gaussian"
       Intercept: NaN
            Beta: [1×0]
               P: 3
               D: 1
               Q: 3
              AR: {NaN NaN} at lags [1 2]
             SAR: {}
              MA: {NaN NaN NaN} at lags [1 2 3]
             SMA: {}
        Variance: NaN

The output displays the values of the properties P, D, and Q of Mdl. The corresponding autoregressive and moving average coefficients (contained in AR and MA) are cell arrays containing the correct number of NaN values. Because P = p + D = 3, you need three presample observations to initialize the model for estimation.

Modify Regression Model with ARIMA Errors

Define the regression model with ARIMA errors:

$$\begin{array}{l}\begin{array}{c}{y}_t=2+X_t\left[\begin{array}{c}1.5\\ 0.2\end{array}\right]+u_t\\ (1-0.2L-0.3L^2)u_t=(1+0.1L){\epsilon }_t,\end{array}\end{array}$$

where $${\epsilon }_t$$ is Gaussian with variance 0.5.

Mdl = regARIMA(Intercept=2,AR={0.2 0.3},MA={0.1}, ...
    Variance=0.5,Beta=[1.5 0.2])

Mdl = 
  regARIMA with properties:

     Description: "Regression with ARMA(2,1) Error Model (Gaussian Distribution)"
      SeriesName: "Y"
    Distribution: Name = "Gaussian"
       Intercept: 2
            Beta: [1.5 0.2]
               P: 2
               Q: 1
              AR: {0.2 0.3} at lags [1 2]
             SAR: {}
              MA: {0.1} at lag [1]
             SMA: {}
        Variance: 0.5

Mdl is fully specified to, for example, simulate a series of responses given the predictor data matrix, $$X_t$$.

Modify the model to estimate the regression coefficient, the AR terms, and the variance of the innovations.

Mdl.Beta = [NaN NaN];
Mdl.AR = {NaN NaN};
Mdl.Variance = NaN;

Change the innovations distribution to a $$t$$ distribution with 15 degrees of freedom.

Mdl.Distribution = struct("Name","t","DoF",15)

Mdl = 
  regARIMA with properties:

     Description: "Regression with ARMA(2,1) Error Model (t Distribution)"
      SeriesName: "Y"
    Distribution: Name = "t", DoF = 15
       Intercept: 2
            Beta: [NaN NaN]
               P: 2
               Q: 1
              AR: {NaN NaN} at lags [1 2]
             SAR: {}
              MA: {0.1} at lag [1]
             SMA: {}
        Variance: NaN

Specify Regression Model with SARIMA Errors

Specify the following model:

$$\begin{array}{l}\begin{array}{c}{y}_t=1+6X_t+u_t\\ (1-0.2L)(1-L)(1-0.5L^4-0.2L^8)(1-L^4)u_t=(1+0.1L)(1+0.05L^4+0.01L^8){\epsilon }_t,\end{array}\end{array}$$

where $${\epsilon }_t$$ is Gaussian with variance 1.

Mdl = regARIMA(Intercept=1,Beta=6,AR=0.2,MA=0.1,D=1, ...
    SAR={0.5,0.2},SARLags=[4, 8],SMA={0.05,0.01},SMALags=[4 8], ...
    Seasonality=4,Variance=1)

Mdl = 
  regARIMA with properties:

     Description: "Regression with ARIMA(1,1,1) Error Model Seasonally Integrated with Seasonal AR(8) and MA(8) (Gaussian Distribution)"
      SeriesName: "Y"
    Distribution: Name = "Gaussian"
       Intercept: 1
            Beta: [6]
               P: 14
               D: 1
               Q: 9
              AR: {0.2} at lag [1]
             SAR: {0.5 0.2} at lags [4 8]
              MA: {0.1} at lag [1]
             SMA: {0.05 0.01} at lags [4 8]
     Seasonality: 4
        Variance: 1

If you do not specify SARLags or SMALags, then the coefficients in SAR and SMA correspond to lags 1 and 2 by default.

Mdl = regARIMA(Intercept=1,Beta=6,AR=0.2,MA=0.1,D=1, ...
    SAR={0.5,0.2},SARLags=[4, 8], ...
    Seasonality=4,Variance=1)

Mdl = 
  regARIMA with properties:

     Description: "Regression with ARIMA(1,1,1) Error Model Seasonally Integrated with Seasonal AR(8) (Gaussian Distribution)"
      SeriesName: "Y"
    Distribution: Name = "Gaussian"
       Intercept: 1
            Beta: [6]
               P: 14
               D: 1
               Q: 1
              AR: {0.2} at lag [1]
             SAR: {0.5 0.2} at lags [4 8]
              MA: {0.1} at lag [1]
             SMA: {}
     Seasonality: 4
        Variance: 1

More About

Regression Model with ARIMA Time Series Errors

A regression model with ARIMA time series errors explains the behavior of a response series by applying linear regression with predictor data, though the errors have autocorrelation indicative of an ARIMA process.

The model has the form (in lag operator notation)

$$\begin{array}{c}{y}_t=c+X_t\beta +u_t\\ a\left(L\right)A\left(L\right){\left(1-L\right)}^D\left(1-L^s\right)u_t=b\left(L\right)B\left(L\right){\epsilon }_t,\end{array}$$

where:

• t = 1,...,T.

• y_t is the response series.

• X_t is row t of X, which is the matrix of concatenated predictor data vectors. That is, X_t is observation t of each predictor series.

• c is the regression model intercept.

• β is the regression coefficient.

• u_t is the disturbance series.

• ε_t is the innovations series.

• $$L^j{y}_t=y_{t-j}.$$

• $$a\left(L\right)=\left(1-a_{1}L-\mathrm{...}-a_p{L}^p\right),$$ which is the degree p, nonseasonal autoregressive polynomial.

• $$A\left(L\right)=\left(1-A_{1}L-\mathrm{...}-A_{p_s}{L}^{p_s}\right),$$ which is the degree p_s, seasonal autoregressive polynomial.

• $${\left(1-L\right)}^D,$$ which is the degree D, nonseasonal integration polynomial.

• $$\left(1-L^s\right),$$ which is the degree s, seasonal integration polynomial.

• $$b\left(L\right)=\left(1+b_{1}L+\mathrm{...}+b_q{L}^q\right),$$ which is the degree q, nonseasonal moving average polynomial.

• $$B\left(L\right)=\left(1+B_{1}L+\mathrm{...}+B_{q_s}{L}^{q_s}\right),$$ which is the degree q_s, seasonal moving average polynomial.

Regression models with ARIMA errors contain a hierarchy of error series. The unconditional disturbance, u_t, or structural disturbance, is based on the structural regression component. The conditional error (one-step-ahead forecast or prediction error), ε_t is the innovation of u_t.

Note

The degrees of the lag operators in the seasonal polynomials A(L) and B(L) do not conform to those defined by Box and Jenkins [1]. In other words, Econometrics Toolbox™ does not treat p₁ = s, p₂ = 2s,...,p_s = c_ps nor q₁ = s, q₂ = 2s,...,q_s = c_qs, where c_p and c_q are positive integers. The software is flexible as it lets you specify the lag operator degrees. See Create Multiplicative ARIMA Models.

References

[1] Box, George E. P., Gwilym M. Jenkins, and Gregory C. Reinsel. Time Series Analysis: Forecasting and Control. 3rd ed. Englewood Cliffs, NJ: Prentice Hall, 1994.

Version History

Introduced in R2013b

R2023b: Name the response series of a regression model with ARIMA errors

Name the response series of a regression model with ARIMA errors by setting the SeriesName property to a string scalar. When you supply input response data to model object functions in a table or timetable, the functions choose the variable with name SeriesName as the response variable by default.

R2018a: Describe a response series of a regression model with ARIMA errors

Describe a response series of a regression model with ARIMA errors by setting the Description property to a string scalar.

R2018a: Use indices that are consistent with MATLAB cell array indexing

The indices of cell arrays of lag operator polynomial coefficients follow MATLAB^® cell array indexing rules. Affected model properties are AR, MA, SAR, and SMA.

• You cannot access any lag-zero coefficients by using an index of 0. For example, Mdl.AR{0} issues an error.

  Remove any instances of such zero indices from your code. The value of all lag-zero coefficients is 1, except for the lag operator polynomial corresponding to the ARCH property, which has the value 0.

• You cannot index beyond the maximal lag in the polynomial. For example, if Mdl.P is 4, then Mdl.AR{p} issues an error when p is greater than 4. For details on the maximal lags of the lag operator polynomials, see the corresponding property descriptions.

  Remove any instances of such indices beyond the maximal lag from your code. All coefficients beyond the maximal lag are 0.

R2018a: Models store innovation distribution name as a string scalar

The Name field of the Distribution property of regARIMA model objects stores the innovation distribution name as a string scalar, for example, "Gaussian" for Gaussian innovations. Before R2018a, MATLAB stored the innovation distribution name as a character vector, for example 'Gaussian' for Gaussian innovations. Although most text-data operations accept character vectors and string scalars for text-data input, the two data types have some differences. For details, see Text in String and Character Arrays.