arima class

Superclasses:

Create ARIMA or ARIMAX time series model

Description

arima creates model objects for stationary or unit root nonstationary linear time series model. This includes moving average (MA), autoregressive (AR), mixed autoregressive and moving average (ARMA), integrated (ARIMA), multiplicative seasonal, and linear time series models that include a regression component (ARIMAX).

Specify models with known coefficients, estimate coefficients with data using estimate, or simulate models with simulate. By default, the variance of the innovations is a positive scalar, but you can specify any supported conditional variance model, such as a GARCH model.

Construction

Mdl = arima creates an ARIMA model of degrees zero.

Mdl = arima(p,D,q) creates a nonseasonal linear time series model using autoregressive degree p, differencing degree D, and moving average degree q.

Mdl = arima(Name,Value) creates a linear time series model using additional options specified by one or more Name,Value pair arguments. Name is the property name and Value is the corresponding value. Name must appear inside single quotes (''). You can specify several name-value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

Input Arguments

Note

You can only use these arguments for nonseasonal models. For seasonal models, use the name-value syntax.

p	Positive integer indicating the degree of the nonseasonal autoregressive polynomial.
D	Nonnegative integer indicating the degree of nonseasonal integration in the linear time series.
q	Positive integer indicating the degree of the nonseasonal moving average polynomial.

Name-Value Pair Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside quotes. You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

'AR'	Cell vector of nonseasonal autoregressive coefficients corresponding to a stable polynomial. When specified without ARLags, AR is a cell vector of coefficients at lags 1,2,... to the degree of the nonseasonal autoregressive polynomial. When specified with ARLags, AR is an equivalent-length cell vector of coefficients associated with the lags in ARLags. Default: Cell vector of NaNs.
'ARLags'	Vector of positive integer lags associated with the AR coefficients. Default: Vector of integers 1,2,... to the degree of the nonseasonal autoregressive polynomial.
'Beta'	Real vector of coefficients corresponding to the regression component in an ARIMAX conditional mean model. Default: [] (no regression coefficients corresponding to a regression component)
'Constant'	Scalar constant in the linear time series. Default: NaN
'D'	Nonnegative integer indicating the degree of the nonseasonal differencing lag operator polynomial (the degree of nonseasonal integration) in the linear time series. Default: 0 (no nonseasonal integration)
'Distribution'	Conditional probability distribution of the innovation process. Distribution is 'Gaussian' or 't'. Alternatively, specify it as a data structure with the field Name to store the distribution 'Gaussian' or 't'. If the distribution is 't', then the structure also needs the field DoF to store the degrees of freedom. Default: 'Gaussian'
'MA'	Cell vector of nonseasonal moving average coefficients corresponding to an invertible polynomial. When specified without MALags, MA is a cell vector of coefficients at lags 1,2,... to the degree of the nonseasonal moving average polynomial. When specified with MALags, MA is an equivalent-length cell vector of coefficients associated with the lags in MALags. Default: Cell vector of NaN values.
'MALags'	Vector of positive integer lags associated with the MA coefficients. Default: Vector of integers 1,2,... to the degree of the nonseasonal moving average polynomial.
'SAR'	Cell vector of seasonal autoregressive coefficients corresponding to a stable polynomial. When specified without SARLags, SAR is a cell vector of coefficients at lags 1,2,... to the degree of the seasonal autoregressive polynomial. When specified with SARLags, SAR is an equivalent-length cell vector of coefficients associated with the lags in SARLags. Default: Cell vector of NaNs.
'SARLags'	Vector of positive integer lags associated with the SAR coefficients. Default: Vector of integers 1,2,... to the degree of the seasonal autoregressive polynomial.
'SMA'	Cell vector of seasonal moving average coefficients corresponding to an invertible polynomial. When specified without SMALags, SMA is a cell vector of coefficients at lags 1,2,... to the degree of the seasonal moving average polynomial. When specified with SMALags, SMA is an equivalent-length cell vector of coefficients associated with the lags in SMALags. Default: Cell vector of NaNs.
'SMALags'	Vector of positive integer lags associated with the SMA coefficients. Default: Vector of integers 1,2,... to the degree of the seasonal moving average polynomial.
'Seasonality'	Nonnegative integer indicating the degree of the seasonal differencing lag operator polynomial in the linear time series model. Default: 0 (no seasonal integration)
'Variance'	Positive scalar variance of the model innovations, or a supported conditional variance model object (e.g., a garch model object). Default: NaN
'Description'	String scalar or character vector describing the model. By default, this argument describes the parametric form of the model, for example, "ARIMA(1,1,1) Model (Gaussian Distribution)".

Notes

• Each AR, SAR, MA, and SMA coefficient is associated with an underlying lag operator polynomial and is subject to a near-zero tolerance exclusion test. That is, the software compares each coefficient to the default lag operator zero tolerance, 1e-12. If the magnitude of a coefficient is greater than 1e-12, then the software includes it in the model. Otherwise, the software considers the coefficient sufficiently close to 0, and excludes it from the model. For additional details, see LagOp.

• Specify the lags associated with the seasonal polynomials SAR and SMA in the periodicity of the observed data, and not as multiples of the Seasonality parameter. This convention does not conform to standard Box and Jenkins [1] notation, but it is a more flexible approach for incorporating multiplicative seasonality.

Properties

AR	Cell vector of nonseasonal autoregressive coefficients corresponding to a stable polynomial. Associated lags are 1,2,... to the degree of the nonseasonal autoregressive polynomial, or as specified in ARLags.
Beta	Real vector of regression coefficients corresponding to a regression component.
Constant	Scalar constant in the linear time series model.
D	Nonnegative integer indicating the degree of nonseasonal integration in the linear time series.
Description	String scalar for the model description.
Distribution	Data structure for the conditional probability distribution of the innovation process. The field Name stores the distribution name "Gaussian" or "t". If the distribution is "t", then the structure also has the field DoF to store the degrees of freedom.
MA	Cell vector of nonseasonal moving average coefficients corresponding to an invertible polynomial. Associated lags are 1,2,... to the degree of the nonseasonal moving average polynomial, or as specified in MALags.
P	Degree of the compound autoregressive polynomial. P is the total number of lagged observations necessary to initialize the autoregressive component of the model. P includes the effects of nonseasonal and seasonal integration captured by the properties D and Seasonality, respectively, and the nonseasonal and seasonal autoregressive polynomials AR and SAR, respectively. The property P does not necessarily conform to standard Box and Jenkins notation. It only conforms if the model has no integration nor seasonal autoregressive component.
Q	Degree of the compound moving average polynomial. Q is the total number of lagged innovations necessary to initialize the moving average component of the model. Q includes the effects of nonseasonal and seasonal moving average polynomials MA and SMA, respectively. The property Q does not necessarily conform to standard Box and Jenkins notation. It only conforms if the model has no seasonal moving average component.
SAR	Cell vector of seasonal autoregressive coefficients corresponding to a stable polynomial. Associated lags are 1,2,... to the degree of the seasonal autoregressive polynomial, or as specified in SARLags.
SMA	Cell vector of seasonal moving average coefficients corresponding to an invertible polynomial. Associated lags are 1,2,... to the degree of the seasonal moving average polynomial, or as specified in SMALags.
Seasonality	Nonnegative integer indicating the seasonal differencing polynomial degree in the linear time series model.
Variance	Positive scalar variance of the model innovations, or a supported conditional variance model (e.g., a garch model).

Methods

estimate	Estimate ARIMA or ARIMAX model parameters
filter	Filter disturbances using ARIMA or ARIMAX model
forecast	Forecast ARIMA or ARIMAX model responses or conditional variances
impulse	Impulse response function
infer	Infer ARIMA or ARIMAX model residuals or conditional variances
print	(To be removed) Display parameter estimation results for ARIMA or ARIMAX models
simulate	Monte Carlo simulation of ARIMA or ARIMAX models
summarize	Display ARIMA model estimation results

Copy Semantics

Value. To learn how value classes affect copy operations, see Copying Objects (MATLAB).

Examples

Specify a Nonseasonal ARIMA Model

Specify an ARIMA(2,1,2) model,

$$(1-{\varphi }_{1}L-{\varphi }_2{L}^2)(1-L)y_t=(1+{\theta }_{1}L+{\theta }_2{L}^2){\epsilon }_t$$

Mdl = arima(2,1,2)

Mdl = 
  arima with properties:

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

The model is nonseasonal, so you can use shorthand syntax. The result is a model with two nonseasonal AR coefficients ($$p$$ = 2), two nonseasonal MA coefficients ($$q$$ = 2), and one degree of differencing ($$D$$ = 1). The property P is equal to $$p$$ + $$D$$ = 3. NaN values indicate estimable parameters.

Modify an ARIMA Model Object

Create, and then modify an arima model.

Specify an AR(3) model with known coefficients,

$$y_t=0.05+0.6y_{t-1}+0.2y_{t-2}-0.1y_{t-3}+{\epsilon }_t,$$

where $${\epsilon }_t$$ has a Gaussian distribution with mean 0 and variance 0.01.

Mdl = arima('Constant',0.05,'AR',{0.6,0.2,-0.1},...
	'Variance',0.01)

Mdl = 
  arima with properties:

     Description: "ARIMA(3,0,0) Model (Gaussian Distribution)"
    Distribution: Name = "Gaussian"
               P: 3
               D: 0
               Q: 0
        Constant: 0.05
              AR: {0.6 0.2 -0.1} at lags [1 2 3]
             SAR: {}
              MA: {}
             SMA: {}
     Seasonality: 0
            Beta: [1×0]
        Variance: 0.01

Modify the model object to make all the model parameters unknown (set them to NaN).

Mdl.Constant = NaN;
Mdl.AR = {NaN NaN NaN};
Mdl.Variance = NaN

Mdl = 
  arima with properties:

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

Make the innovation distribution a $$t$$ distribution with 10 degrees of freedom.

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

Mdl = 
  arima with properties:

     Description: "ARIMA(3,0,0) Model (t Distribution)"
    Distribution: Name = "t", DoF = 10
               P: 3
               D: 0
               Q: 0
        Constant: NaN
              AR: {NaN NaN NaN} at lags [1 2 3]
             SAR: {}
              MA: {}
             SMA: {}
     Seasonality: 0
            Beta: [1×0]
        Variance: NaN

Specify an Additive Seasonal ARIMA Model

Specify an MA model with no constant, and moving average terms at lags 1, 2, and 12,

$$y_t={\epsilon }_t+{\theta }_1{\epsilon }_{t-1}+{\theta }_2{\epsilon }_{t-2}+{\theta }_{12}{\epsilon }_{t-12}.$$

Mdl = arima('Constant',0,'MALags',[1,2,12])

Mdl = 
  arima with properties:

     Description: "ARIMA(0,0,12) Model (Gaussian Distribution)"
    Distribution: Name = "Gaussian"
               P: 0
               D: 0
               Q: 12
        Constant: 0
              AR: {}
             SAR: {}
              MA: {NaN NaN NaN} at lags [1 2 12]
             SMA: {}
     Seasonality: 0
            Beta: [1×0]
        Variance: NaN

Specify a Multiplicative Seasonal Model

Specify a multiplicative seasonal ARIMA model with seasonal and nonseasonal integration,

$$(1-L)(1-L^{12})y_t=(1+{\theta }_1)(1+{\theta }_{12}{L}^{12}){\epsilon }_t.$$

Mdl = arima('Constant',0,'D',1,'Seasonality',12,...
	'MALags',1,'SMALags',12)

Mdl = 
  arima with properties:

     Description: "ARIMA(0,1,1) Model Seasonally Integrated with Seasonal MA(12) (Gaussian Distribution)"
    Distribution: Name = "Gaussian"
               P: 13
               D: 1
               Q: 13
        Constant: 0
              AR: {}
             SAR: {}
              MA: {NaN} at lag [1]
             SMA: {NaN} at lag [12]
     Seasonality: 12
            Beta: [1×0]
        Variance: NaN

An ARIMA model is assumed to be multiplicative any time SMALags or SARLags are specified.

Specify an ARIMAX Model

Specify an ARIMAX model with one or more regression coefficients corresponding to predictor data.

Specify the ARIMAX(1,1,1) model,

$$(1-0.2L)(1-L{)}^1{y}_t=0.5x_t+(1+0.3L){\epsilon }_t$$

using arima.

Mdl = arima('AR',0.2,'D',1,'MA',0.3,'Beta',0.5)

Mdl = 
  arima with properties:

     Description: "ARIMAX(1,1,1) Model (Gaussian Distribution)"
    Distribution: Name = "Gaussian"
               P: 2
               D: 1
               Q: 1
        Constant: NaN
              AR: {0.2} at lag [1]
             SAR: {}
              MA: {0.3} at lag [1]
             SMA: {}
     Seasonality: 0
            Beta: [0.5]
        Variance: NaN

In the output, the property P is sum of the AR lags and degree of nonseasonal integration p + D = 2.

Modify this ARIMAX(1,1,1) model by adding two more regression coefficients,

 Mdl.Beta=[0.5,4,-0.6]

Mdl = 
  arima with properties:

     Description: "ARIMAX(1,1,1) Model (Gaussian Distribution)"
    Distribution: Name = "Gaussian"
               P: 2
               D: 1
               Q: 1
        Constant: NaN
              AR: {0.2} at lag [1]
             SAR: {}
              MA: {0.3} at lag [1]
             SMA: {}
     Seasonality: 0
            Beta: [0.5 4 -0.6]
        Variance: NaN

Specify a Composite Conditional Variance Model

Specify an ARIMA(1,0,1) conditional mean model with a GARCH(1,1) conditional variance model.

Specify the conditional mean model.

Mdl = arima(1,0,1);

Specify the conditional variance model.

Mdl.Variance = garch(1,1)

Mdl = 
  arima with properties:

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

More About

Lag Operator

The lag operator L is defined as $$L^i{y}_t=y_{t-i}.$$ You can create lag operator polynomials using them to condense the notation and solve linear difference equations. The lag operator polynomials in the linear time series model definitions are:

• $$\varphi (L)=1-\varphi L-{\varphi }^2{L}^2-\mathrm{...}-{\varphi }^p{L}^p,$$ which is the degree p autoregressive polynomial.

• $$\theta (L)=1+\theta L+{\theta }^2{L}^2+\mathrm{...}+{\theta }^q{L}^q,$$ which is the degree q moving average polynomial.

• $$\Phi (L)=1-{\Phi }_{p_1}{L}^{p_1}-{\Phi }_{p_2}{L}^{p_2}-\mathrm{...}-{\Phi }_{p_s}{L}^{p_s},$$ which is the degree p_s seasonal autoregressive polynomial.

• $$\Theta (L)=1+{\Theta }_{q_1}{L}^{q_1}+{\Theta }_{q_2}{L}^{q_2}+\mathrm{...}+{\Theta }_{q_s}{L}^{q_s},$$ which is the degree q_s seasonal moving average polynomial.

Note

The degrees of the lag operators in the seasonal polynomials Φ(L) and Θ(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 Multiplicative ARIMA Model Specifications.

Linear Time Series Model

A linear time series model for response process y_t and innovations ε_t is a stochastic process that has the form

$$y_t=c+{\varphi }_1{y}_{t-1}+\dots +{\varphi }_p{y}_{t-p}+{\epsilon }_t+{\theta }_1{\epsilon }_{t-1}+\dots +\theta q{\epsilon }_{t-q}.$$

In lag operator notation, this model is

$$\varphi (L)y_t=c+\theta (L){\epsilon }_t.$$

The general time series model, which includes differencing, multiplicative seasonality, and seasonal differencing, is

$$\varphi (L){(1-L)}^D\Phi (L){(1-L^s)}^{D_s}{y}_t=c+\theta (L)\Theta (L){\epsilon }_t.$$

• The coefficients of the nonseasonal and seasonal autoregressive polynomials $$\varphi (L)$$ and $$\Phi (L)$$ correspond to AR and SAR, respectively. The degrees of these polynomials are p and p_s. Similarly, the coefficients of polynomials $$\theta (L)$$ and$$\Theta (L)$$ correspond to MA and SMA. The degrees of these polynomials are q and q_s, respectively.

• The polynomials $${(1-L)}^D$$ and $${(1-L^s)}^{D_s}$$ have a degree of nonseasonal and seasonal integration D and D_s, respectively. Note that s corresponds to model property Seasonality. D_s is 1 if Seasonality is nonzero, and it is 0 otherwise. That is, the software applies first-order seasonal differencing if Seasonality ≥ 1.

• The model property P is equal to p + D + p_s + s.

• The model property Q is equal to q + q_s.

• You can extend this model by including a matrix of predictor data. For details, see ARIMA Model Including Exogenous Covariates.

Stationarity Requirements

The ARMA(p,q) model,

$$\varphi (L)y_t=c+\theta (L){\epsilon }_t,$$

where ε_t has mean 0, variance σ², and $$Cov({\epsilon }_t,{\epsilon }_s)=0$$ for t ≠ s, is stationary if its expected value, variance, and covariance between elements of the series are independent of time. For example, the MA(q) model, with c = 0, is stationary for any $$q<\infty$$ because

• $$E(y_t)=\theta (L)0=0,$$

• $$Var(y_t)={\sigma }^2{\displaystyle \sum _{i=1}^q{\theta }_i^2},$$ and

• $$Cov(y_t,y_{t-s})=\{\begin{array}{l}{\sigma }^2({\theta }_s+{\theta }_1{\theta }_{s-1}+{\theta }_2{\theta }_{s-2}+\mathrm{...}+{\theta }_q{\theta }_{s-q})\text{ if }s\ge q\\ 0\text{ otherwise}.\end{array}$$

are free of t for all time points [1].

Unit Root

The time series $$\{y_t;t=1,\mathrm{...},T\}$$ is a unit root process if its expected value, variance, or covariance grows with time. Subsequently, the time series is not stationary.

References

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

[2] Enders, W. Applied Econometric Time Series. Hoboken, NJ: John Wiley & Sons, Inc., 1995.