Specify Conditional Mean Models

Default ARIMA Model

The default ARIMA(p,D,q) model in Econometrics Toolbox™ is the nonseasonal model of the form

$Δ^{D} y_{t} = c + ϕ_{1} Δ^{D} y_{t - 1} + \dots + ϕ_{p} Δ^{D} y_{t - p} + θ_{1} ε_{t - 1} + \dots + θ_{q} ε_{t - q} + ε_{t} 。$

You can write this equation in condensed form usinglag operator notation:

$ϕ (L) {(1 - L)}^{D} y_{t} = c + θ (L) ε_{t}$

In either equation, the default innovation distribution is Gaussian with mean zero and constant variance.

At the command line, you can specify a model of this form using the shorthand syntaxarima(p,D,q)。For the input argumentsp,D, andq, enter the number of nonseasonal AR terms (p), the order of nonseasonal integration (D), and the number of nonseasonal MA terms (q), respectively.

When you use this shorthand syntax,arimacreates anarimamodel with these default property values.

Property Name	Property Data Type
`AR`	Cell vector of`NaN`s
`Beta`	Empty vector`[]`of regression coefficients corresponding to exogenous covariates
`Constant`	`NaN`
`D`	程度的季节性integration,D
`Distribution`	`"Gaussian"`
`MA`	Cell vector of`NaN`s
`P`	Number of AR terms plus degree of integration,p+D
`Q`	Number of MA terms,q
`SAR`	Cell vector of`NaN`s
`SMA`	Cell vector of`NaN`s
`Variance`	`NaN`

To assign nondefault values to any properties, you can modify the created model object using dot notation.

Notice that the inputsDandqare the valuesarimaassigns to propertiesDandQ。However, the input argumentpis not necessarily the valuearimaassigns to the model propertyP。Pstores the number of presample observations needed to initialize the AR component of the model. For nonseasonal models, the required number of presample observations isp+D。

To illustrate, consider specifying the ARIMA(2,1,1) model

$(1 - ϕ_{1} L - ϕ_{2} L^{2}) {(1 - L)}^{1} y_{t} = c + (1 + θ_{1} L) ε_{t},$

where the innovation process is Gaussian with (unknown) constant variance.

Mdl = arima(2,1,1)

Mdl = arima with properties: Description: "ARIMA(2,1,1) Model (Gaussian Distribution)" Distribution: Name = "Gaussian" P: 3 D: 1 Q: 1 Constant: NaN AR: {NaN NaN} at lags [1 2] SAR: {} MA: {NaN} at lag [1] SMA: {} Seasonality: 0 Beta: [1×0] Variance: NaN

Notice that the model propertyPdoes not have value 2 (the AR degree). With the integration, a total ofp+D(here, 2 + 1 = 3) presample observations are needed to initialize the AR component of the model.

The created model,Mdl, hasNaNs for all parameters. ANaNvalue signals that a parameter needs to be estimated or otherwise specified by the user. All parameters must be specified to forecast or simulate the model.

To estimate parameters, input the model object (along with data) toestimate。This returns a new fittedarimamodel object. The fitted model object has parameter estimates for each inputNaNvalue.

Callingarimawithout any input arguments returns an ARIMA(0,0,0) model specification with default property values:

DefaultMdl = arima

DefaultMdl = arima with properties: Description: "ARIMA(0,0,0) Model (Gaussian Distribution)" Distribution: Name = "Gaussian" P: 0 D: 0 Q: 0 Constant: NaN AR: {} SAR: {} MA: {} SMA: {} Seasonality: 0 Beta: [1×0] Variance: NaN

Specify Nonseasonal Models Using Name-Value Pairs

The best way to specify models toarimais using name-value pair arguments. You do not need, nor are you able, to specify a value for every model object property.arimaassigns default values to any properties you do not (or cannot) specify.

In condensed, lag operator notation, nonseasonal ARIMA(p,D,q) models are of the form

ϕ (L) {(1 - L)}^{D} y_{t} = c + θ (L) ε_{t} 。

(1)

You can extend this model to an ARIMAX(p,D,q) model with the linear inclusion of exogenous variables. This model has the form

ϕ (L) y_{t} = c^{*} + x_{t}^{'} β + θ^{*} (L) ε_{t},

(2)

wherec^*=c/(1–L)^Dandθ^*(L)=θ(L)/(1–L)^D。

Tip

If you specify a nonzeroD, then Econometrics Toolbox differences the response seriesy_tbeforethe predictors enter the model. You should preprocess the exogenous covariatesx_tby testing for stationarity and differencing if any are unit root nonstationary. If any nonstationary exogenous covariate enters the model, then the false negative rate for significance tests ofβcan increase.

For the distribution of the innovations,ε_t, there are two choices:

Independent and identically distributed (iid) Gaussian or Student’stwith a constant variance, $σ_{ε}^{2}$ 。
Dependent Gaussian or Student’stwith a conditional variance process, $σ_{t}^{2}$ 。Specify the conditional variance model using agarch,egarch, orgjrmodel.

Thearimadefault for the innovations is an iid Gaussian process with constant (scalar) variance.

In order to estimate, forecast, or simulate a model, you must specify the parametric form of the model (e.g., which lags correspond to nonzero coefficients, the innovation distribution) and any known parameter values. You can set any unknown parameters equal toNaN, and then input the model toestimate(along with data) to get estimated parameter values.

arima(andestimate) returns a model corresponding to the model specification. You can modify models to change or update the specification. Input models (with noNaNvalues) toforecastorsimulatefor forecasting and simulation, respectively. Here are some example specifications using name-value arguments.

Model	Specification
$y_{t} = c + ϕ_{1} y_{t - 1} + ε_{t}$ $ε_{t} = σ_{ε} z_{t}$ z_tGaussian	`arima('AR',NaN)`or`arima(1,0,0)`
$y_{t} = ε_{t} + θ_{1} ε_{t - 1} + θ_{2} ε_{t - 2}$ $ε_{t} = σ_{ε} z_{t}$ z_tStudent’stwith unknown degrees of freedom	`arima('Constant',0,'MA',{NaN,NaN},... 'Distribution','t')`
$(1 - 0.8 L) (1 - L) y_{t} = 0.2 + (1 + 0.6 L) ε_{t}$ $ε_{t} = 0.1 z_{t}$ z_tStudent’stwith eight degrees of freedom	`arima('Constant',0.2,'AR',0.8,'MA',0.6,'D',1,... 'Variance',0.1^2,'Distribution',struct('Name','t','DoF',8))`
$(1 + 0.5 L) {(1 - L)}^{1} Δ y_{t} = x_{t}^{'} [\begin{matrix} - 5 \\ 2 \end{matrix}] + ε_{t}$ $ε_{t} ~ N (0, 1)$	`arima('Constant',0,'AR',-0.5,'D',1,'Beta',[-5 2])`

You can specify the following name-value arguments to create nonseasonalarimamodels.

Name-Value Arguments for Nonseasonal ARIMA Models

Name	Corresponding Model Term(s) inEquation 1	When to Specify
`AR`	Nonseasonal AR coefficients, $ϕ_{1}, \dots, ϕ_{p}$	To set equality constraints for the AR coefficients. For example, to specify the AR coefficients in the model $y_{t} = 0.8 y_{t - 1} - 0.2 y_{t - 2} + ε_{t},$ specify`'AR',{0.8,-0.2}`。 You only need to specify the nonzero elements of`AR`。If the nonzero coefficients are at nonconsecutive lags, specify the corresponding lags using`ARLags`。 Any coefficients you specify must correspond to a stable AR operator polynomial.
`ARLags`	Lags corresponding to nonzero, nonseasonal AR coefficients	`ARLags`is not a model property. Use this argument as a shortcut for specifying`AR`when the nonzero AR coefficients correspond to nonconsecutive lags. For example, to specify nonzero AR coefficients at lags 1 and 12, e.g., $y_{t} = ϕ_{1} y_{t - 1} + ϕ_{12} y_{t - 12} + ε_{t},$ specify`'ARLags',[1,12]`。 Use`AR`and`ARLags`together to specify known nonzero AR coefficients at nonconsecutive lags. For example, if in the given AR(12) model $ϕ_{1} = 0.6$ and $ϕ_{12} = - 0.3,$ specify`'AR',{0.6,-0.3},'ARLags',[1,12]`。
`Beta`	Values of the coefficients of the exogenous covariates	Use this argument to specify the values of the coefficients of the exogenous variables. For example, use`'Beta',[0.5 7 -2]`to specify $β = {[\begin{matrix} 0.5 & 7 & - 2 \end{matrix}]}^{'} 。$ By default,`Beta`is an empty vector.
`Constant`	Constant term,c	To set equality constraints forc。对于example, for a model with no constant term, specify`'Constant',0`。 By default,`Constant`has value`NaN`。
`D`	Degree of nonseasonal differencing,D	To specify a degree of nonseasonal differencing greater than zero. For example, to specify one degree of differencing, specify`'D',1`。 By default,`D`has value`0`(meaning no nonseasonal integration).
`Distribution`	Distribution of the innovation process	Use this argument to specify a Student’stinnovation distribution. By default, the innovation distribution is Gaussian. 对于example, to specify atdistribution with unknown degrees of freedom, specify`'Distribution','t'`。 To specify atinnovation distribution with known degrees of freedom, assign`Distribution`a data structure with fields`Name`and`DoF`。对于example, for atdistribution with nine degrees of freedom, specify`'Distribution',struct('Name','t','DoF',9)`。
`MA`	Nonseasonal MA coefficients, $θ_{1}, \dots, θ_{q}$	To set equality constraints for the MA coefficients. For example, to specify the MA coefficients in the model $y_{t} = ε_{t} + 0.5 ε_{t - 1} + 0.2 ε_{t - 2},$ specify`'MA',{0.5,0.2}`。 You only need to specify the nonzero elements of`MA`。If the nonzero coefficients are at nonconsecutive lags, specify the corresponding lags using`MALags`。 Any coefficients you specify must correspond to an invertible MA polynomial.
`MALags`	Lags corresponding to nonzero, nonseasonal MA coefficients	`MALags`is not a model property. Use this argument as a shortcut for specifying`MA`when the nonzero MA coefficients correspond to nonconsecutive lags. For example, to specify nonzero MA coefficients at lags 1 and 4, e.g., $y_{t} = ε_{t} + θ_{1} ε_{t - 1} + θ_{4} ε_{t - 4},$ specify`'MALags',[1,4]`。 Use`MA`and`MALags`together to specify known nonzero MA coefficients at nonconsecutive lags. For example, if in the given MA(4) model $θ_{1} = 0.5$ and $θ_{4} = 0.2,$ specify`'MA',{0.4,0.2},'MALags',[1,4]`。
`Variance`	Scalar variance of the innovation process, $σ_{ε}^{2}$ Conditional variance process, $σ_{t}^{2}$	To set equality constraints for $σ_{ε}^{2}$ 。对于example, for a model with known variance 0.1, specify`'Variance',0.1`。By default,`Variance`has value`NaN`。 To specify a conditional variance model, $σ_{t}^{2}$ 。Set`'Variance'`equal to a conditional variance model object, e.g., a`garch`model object.

Note

You cannot assign values to the propertiesPandQ。For nonseasonal models,

arimasetsPequal top+D
arimasetsQequal toq

Specify Multiplicative Models Using Name-Value Pairs

For a time series with periodicitys, define the degreep_sseasonal AR operator polynomial, $Φ (L) = (1 - Φ_{1} L^{p_{1}} - \dots - Φ_{p_{s}} L^{p_{s}})$ , and the degreeq_sseasonal MA operator polynomial, $Θ (L) = (1 + Θ_{1} L^{q_{1}} + \dots + Θ_{q_{s}} L^{q_{s}})$ 。Similarly, define the degreepnonseasonal AR operator polynomial, $ϕ (L) = (1 - ϕ_{1} L - \dots - ϕ_{p} L^{p})$ , and the degreeqnonseasonal MA operator polynomial,

θ (L) = (1 + θ_{1} L + \dots + θ_{q} L^{q}) 。

(3)

A multiplicative ARIMA model with degreeDnonseasonal integration and degreesseasonality is given by

ϕ (L) Φ (L) {(1 - L)}^{D} (1 - L^{s}) y_{t} = c + θ (L) Θ (L) ε_{t} 。

(4)

The innovation series can be an independent or dependent Gaussian or Student’stprocess. Thearimadefault for the innovation distribution is an iid Gaussian process with constant (scalar) variance.

In addition to the arguments for specifying nonseasonal models (described inName-Value Arguments for Nonseasonal ARIMA Models), you can specify these name-value arguments to create a multiplicativearimamodel. You can extend an ARIMAX model similarly to include seasonal effects.

Name-Value Arguments for Seasonal ARIMA Models

Argument	Corresponding Model Term(s) inEquation 4	When to Specify
`SAR`	Seasonal AR coefficients, $Φ_{1}, \dots, Φ_{p_{s}}$	To set equality constraints for the seasonal AR coefficients. When specifying AR coefficients, use the sign opposite to what appears inEquation 4(that is, use the sign of the coefficient as it would appear on the right side of the equation). Use`SARLags`to specify the lags of the nonzero seasonal AR coefficients. Specify the lags associated with the seasonal polynomials in the periodicity of the observed data (e.g., 4, 8,... for quarterly data, or 12, 24,... for monthly data), and not as multiples of the seasonality (e.g., 1, 2,...). 对于example, to specify the model $(1 - 0.8 L) (1 - 0.2 L^{12}) y_{t} = ε_{t},$ specify`'AR',0.8,'SAR',0.2,'SARLags',12`。 Any coefficient values you enter must correspond to a stable seasonal AR polynomial.
`SARLags`	Lags corresponding to nonzero seasonal AR coefficients, in the periodicity of the observed series	`SARLags`is not a model property. Use this argument when specifying`SAR`to indicate the lags of the nonzero seasonal AR coefficients. 对于example, to specify the model $(1 - ϕ L) (1 - Φ_{12} L^{12}) y_{t} = ε_{t},$ specify`'ARLags',1,'SARLags',12`。
`SMA`	Seasonal MA coefficients, $Θ_{1}, \dots, Θ_{q_{s}}$	To set equality constraints for the seasonal MA coefficients. Use`SMALags`to specify the lags of the nonzero seasonal MA coefficients. Specify the lags associated with the seasonal polynomials in the periodicity of the observed data (e.g., 4, 8,... for quarterly data, or 12, 24,... for monthly data), and not as multiples of the seasonality (e.g., 1, 2,...). 对于example, to specify the model $y_{t} = (1 + 0.6 L) (1 + 0.2 L^{12}) ε_{t},$ specify`'MA',0.6,'SMA',0.2,'SMALags',12`。 Any coefficient values you enter must correspond to an invertible seasonal MA polynomial.
`SMALags`	Lags corresponding to the nonzero seasonal MA coefficients, in the periodicity of the observed series	`SMALags`is not a model property. Use this argument when specifying`SMA`to indicate the lags of the nonzero seasonal MA coefficients. 对于example, to specify the model $y_{t} = (1 + θ_{1} L) (1 + Θ_{4} L^{4}) ε_{t},$ specify`'MALags',1,'SMALags',4`。
`Seasonality`	Seasonal periodicity,s	To specify the degree of seasonal integrationsin the seasonal differencing polynomialΔ_s= 1 –L^s。对于example, to specify the periodicity for seasonal integration of monthly data, specify`“季节性”12`。 If you specify nonzero`Seasonality`, then the degree of the whole seasonal differencing polynomial is one. By default,`Seasonality`has value`0`(meaning periodicity and no seasonal integration).

Note

You cannot assign values to the propertiesPandQ。For multiplicative ARIMA models,

arimasetsPequal top+D+p_s+s
arimasetsQequal toq+q_s

Specify Conditional Mean Model Using Econometric Modeler App

You can specify the lag structure and innovation distribution of seasonal and nonseasonal conditional mean models using theEconometric Modelerapp. The app treats all coefficients as unknown and estimable, including the degrees of freedom parameter for atinnovation distribution.

At the command line, open theEconometric Modelerapp.

econometricModeler

Alternatively, open the app from the apps gallery (seeEconometric Modeler).

In the app, you can see all supported models by selecting a time series variable for the response in theData Browser。Then, on theEconometric Modelertab, in theModelssection, click the arrow to display the models gallery.

TheARMA/ARIMA Modelssection contains supported conditional mean models.

For conditional mean model estimation, SARIMA and SARIMAX are the most flexible models. You can create any conditional mean model that excludes exogenous predictors by clickingSARIMA, or you can create any conditional mean model that includes at least one exogenous predictor by clickingSARIMAX。

After you select a model, the app displays theTypeModel Parametersdialog box, whereTypeis the model type. This figure shows theSARIMAX Model Parametersdialog box.

Adjustable parameters in the dialog box depend onType。In general, adjustable parameters include:

A model constant and linear regression coefficients corresponding to predictor variables
Time series component parameters, which include seasonal and nonseasonal lags and degrees of integration
The innovation distribution

As you adjust parameter values, the equation in theModel Equation部分改变以匹配您的规范。Adjustable parameters correspond to input and name-value pair arguments described in the previous sections and in thearimareference page.

For more details on specifying models using the app, seeFitting Models to DataandSpecifying Lag Operator Polynomials Interactively。