Create ARIMA or ARIMAX time series model
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.
creates
an ARIMA model of degrees zero.Mdl
= arima
creates
a nonseasonal linear time series model using autoregressive degree Mdl
=
arima(p
,D
,q
)p
,
differencing degree D
, and moving average degree q
.
Mdl = arima(
creates
a linear time series model using additional options specified by one
or more Name,Value
)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 namevalue pair arguments in any order as Name1,Value1,...,NameN,ValueN
.
Note: You can only use these arguments for nonseasonal models. For seasonal models, use the namevalue syntax. 

Positive integer indicating the degree of the nonseasonal autoregressive polynomial. 

Nonnegative integer indicating the degree of nonseasonal integration in the linear time series. 

Positive integer indicating the degree of the nonseasonal moving average polynomial. 
Specify optional commaseparated pairs of Name,Value
arguments.
Name
is the argument
name and Value
is the corresponding
value. Name
must appear
inside single quotes (' '
).
You can specify several name and value pair
arguments in any order as Name1,Value1,...,NameN,ValueN
.

Cell vector of nonseasonal autoregressive coefficients corresponding
to a stable polynomial. When specified without Default: Cell vector of 

Vector of positive integer lags associated with the Default: Vector of integers 1,2,... to the degree of the nonseasonal autoregressive polynomial. 

Real vector of coefficients corresponding to the regression component in an ARIMAX conditional mean model. Default: 

Scalar constant in the linear time series. Default: 

Nonnegative integer indicating the degree of the nonseasonal differencing lag operator polynomial (the degree of nonseasonal integration) in the linear time series. Default: 

Conditional probability distribution of the innovation process. Default: 

Cell vector of nonseasonal moving average coefficients corresponding
to an invertible polynomial. When specified without Default: Cell vector of 

Vector of positive integer lags associated with the Default: Vector of integers 1,2,... to the degree of the nonseasonal moving average polynomial. 

Cell vector of seasonal autoregressive coefficients corresponding
to a stable polynomial. When specified without Default: Cell vector of 

Vector of positive integer lags associated with the Default: Vector of integers 1,2,... to the degree of the seasonal autoregressive polynomial. 

Cell vector of seasonal moving average coefficients corresponding
to an invertible polynomial. When specified without Default: Cell vector of 

Vector of positive integer lags associated with the Default: Vector of integers 1,2,... to the degree of the seasonal moving average polynomial. 

Nonnegative integer indicating the degree of the seasonal differencing lag operator polynomial in the linear time series model. Default: 

Positive scalar variance of the model innovations, or a supported
conditional variance model object (e.g., a Default: 
Notes


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 

Real vector of regression coefficients corresponding to a regression component. 

Scalar constant in the linear time series model. 

Nonnegative integer indicating the degree of nonseasonal integration in the linear time series. 

Data structure for the conditional probability distribution
of the innovation process. The field 

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 

Degree of the compound autoregressive polynomial.
The property 

Degree of the compound moving average polynomial. The property 

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 

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 

Nonnegative integer indicating the seasonal differencing polynomial degree in the linear time series model. 

Positive scalar variance of the model innovations, or a supported
conditional variance model (e.g., a 
estimate  Estimate ARIMA or ARIMAX model parameters 
filter  Filter disturbances using ARIMA or ARIMAX model 
forecast  Forecast ARIMA or ARIMAX process 
impulse  Impulse response function 
infer  Infer ARIMA or ARIMAX model residuals or conditional variances 
Display parameter estimation results for ARIMA or ARIMAX models  
simulate  Monte Carlo simulation of ARIMA or ARIMAX models 
The lag operator L is defined as $${L}^{i}{y}_{t}={y}_{ti}.$$ 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 p_{1} = s, p_{2} = 2s,...,p_{s} = c_{p}s nor q_{1} = s, q_{2} = 2s,...,q_{s} = c_{q}s 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. 
A linear time series model for response process y_{t} and innovations ε_{t} is a stochastic process that has the form
$${y}_{t}=c+{\varphi}_{1}{y}_{t1}+\dots +{\varphi}_{p}{y}_{tp}+{\epsilon}_{t}+{\theta}_{1}{\epsilon}_{t1}+\dots +\theta q{\epsilon}_{tq}.$$
In lag operator notation, this model is
$$\varphi (L){y}_{t}=c+\theta (L){\epsilon}_{t}.$$
The general times series model, which includes differencing, multiplicative seasonality, and seasonal differencing, is
$$\varphi (L){(1L)}^{D}\Phi (L){(1{L}^{s})}^{{D}_{s}}{y}_{t}=c+\theta (L)\Theta (L){\epsilon}_{t}.$$
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 $${(1L)}^{D}$$ 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 firstorder seasonal differencing if Seasonality
≥
1.
The model property P
is equal to p + D + p_{s} + D_{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.
The ARMA(p,q) model,
$$\varphi (L){y}_{t}=c+\theta (L){\epsilon}_{t},$$
where ε_{t} has mean 0, variance σ^{2}, 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
$$Cov({y}_{t},{y}_{ts})=\{\begin{array}{l}{\sigma}^{2}({\theta}_{s}+{\theta}_{1}{\theta}_{s1}+{\theta}_{2}{\theta}_{s2}+\mathrm{...}+{\theta}_{q}{\theta}_{sq})\text{if}s\ge q\\ 0\text{otherwise}.\end{array}$$
are free of t for all time points [1].
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.
[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.
Value. To learn how value classes affect copy operations, see Copying Objects in the MATLAB^{®} documentation.
estimate
 filter
 forecast
 impulse
 infer
 print
 simulate