Documentation 
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.
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 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. 
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.
'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 equivalentlength 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 a string you specify as '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 equivalentlength cell vector of coefficients associated with the lags in MALags. Default: Cell vector of NaNs. 
'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 equivalentlength 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 equivalentlength 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 (the degree of seasonal integration) 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 
Notes

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.