regARIMA class

Superclasses:

Create regression model with ARIMA time series errors

Description

regARIMA creates a regression model with ARIMA time series errors to maintain the sensitivity interpretation of regression coefficients. To create an ARIMA model containing a linear regression component for exogenous predictors (ARIMAX), see arima.

By default, the time series errors (also called unconditional disturbances) are independent, identically distributed, mean 0 Gaussian random variables. If the errors have an autocorrelation structure, then you can specify models for them. The models include:

• moving average (MA)

• autoregressive (AR)

• mixed autoregressive and moving average (ARMA)

• integrated (ARIMA)

• multiplicative seasonal (SARIMA)

Specify error models containing known coefficients to:

• Simulate responses using simulate.

• Explore impulse responses using impulse.

• Forecast future observations using forecast.

• Estimate unknown coefficients with data using estimate.

Construction

Mdl = regARIMA creates a regression model with degree 0 ARIMA errors and no regression coefficient.

Mdl = regARIMA(p,D,q) creates a regression model with errors modeled by a nonseasonal, linear time series with autoregressive degree p, differencing degree D, and moving average degree q.

Mdl = regARIMA(Name,Value) creates a regression model with ARIMA errors using additional options specified by one or more Name,Value pair arguments. Name can also be a 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

For regression models with nonseasonal ARIMA errors, use p, D, and q. For regression models with seasonal ARIMA errors, use Name,Value pair arguments.

p	Nonseasonal, autoregressive polynomial degree for the error model, specified as a positive integer.
D	Nonseasonal integration degree for the error model, specified as a nonnegative integer.
q	Nonseasonal, moving average polynomial degree for the error model, specified as a positive integer.

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.

Intercept	Regression model intercept, specified as the comma-separated pair consisting of 'Intercept' and a scalar. Default: NaN
Beta	Regression model coefficients associated with the predictor data, specified as the comma-separated pair consisting of 'Beta' and a vector. Default: [] (no regression coefficients corresponding to predictor data)
AR	Nonseasonal, autoregressive coefficients for the error model, specified as the comma-separated pair consisting of 'AR' and a cell vector. The coefficients must yield a stable polynomial. If you specify ARLags, then AR is an equivalent-length cell vector of coefficients associated with the lags in ARLags. For example, if ARLags = [1, 4] and AR = {0.2, 0.1}, then, ignoring all other specifications, the error model is $$u_t=0.2u_{t-1}+0.1u_{t-4}+{\epsilon }_t.$$ If you do not specify ARLags, then AR is a cell vector of coefficients at lags 1,2,...,p, which is the nonseasonal, autoregressive polynomial degree. For example, if AR = {0.2, 0.1} and you do not specify ARLags, then, ignoring all other specifications, the error model is $$u_t=0.2u_{t-1}+0.1u_{t-2}+{\epsilon }_t.$$ The coefficients in AR correspond to coefficients in an underlying LagOp lag operator polynomial, and are subject to a near-zero tolerance exclusion test. If you set a coefficient to 1e–12 or below, regARIMA excludes that coefficient and its corresponding lag in ARLags from the model. Default: Cell vector of NaNs with the same length as ARLags.
MA	Nonseasonal, moving average coefficients for the error model, specified as the comma-separated pair consisting of 'MA' and a cell vector. The coefficients must yield an invertible polynomial. If you specify MALags, then MA is an equivalent-length cell vector of coefficients associated with the lags in MALags. For example, if MALags = [1, 4] and MA = {0.2, 0.1}, then, ignoring all other specifications, the error model is $$u_t={\epsilon }_t+0.2{\epsilon }_{t-1}+0.1{\epsilon }_{t-4}.$$ If you do not specify MALags, then MA is a cell vector of coefficients at lags 1,2,...,q, which is the nonseasonal, moving average polynomial degree. For example, if MA = {0.2, 0.1} and you do not specify MALags, then, ignoring all other specifications, the error model is $$u_t={\epsilon }_t+0.2{\epsilon }_{t-1}+0.1{\epsilon }_{t-2}.$$ The coefficients in MA correspond to coefficients in an underlying LagOp lag operator polynomial, and are subject to a near-zero tolerance exclusion test. If you set a coefficient to 1e–12 or below, regARIMA excludes that coefficient and its corresponding lag in MALags from the model. Default: Cell vector of NaNs with the same length as MALags.
ARLags	Lags associated with the AR coefficients in the error model, specified as the comma-separated pair consisting of 'ARLags' and a vector of positive integers. Default: Vector of integers 1,2,...,p, the nonseasonal, autoregressive polynomial degree.
MALags	Lags associated with the MA coefficients in the error model, specified as the comma-separated pair consisting of 'MALags' and a vector of positive integers. Default: Vector of integers 1,2,...,q, the nonseasonal moving average polynomial degree.
SAR	Seasonal, autoregressive coefficients for the error model, specified as the comma-separated pair consisting of 'SAR' and a cell vector. The coefficient must yield a stable polynomial. If you specify SARLags, then SAR is an equivalent-length cell vector of coefficients associated with the lags in SARLags. For example, if SARLags = [1, 4], SAR = {0.2, 0.1}, and Seasonality = 4, then, ignoring all other specifications, the error model is $$\left(1-0.2L-0.1L^4\right)\left(1-L^4\right)u_t={\epsilon }_t.$$ If you do not specify SARLags, then SAR is a cell vector of coefficients at lags 1,2,...,ps, which is the seasonal, autoregressive polynomial degree. For example, if SAR = {0.2, 0.1} and Seasonality = 4, and you do not specify SARLags, then, ignoring all other specifications, the error model is $$\left(1-0.2L-0.1L^2\right)\left(1-L^4\right)u_t={\epsilon }_t.$$ The coefficients in SAR correspond to coefficients in an underlying LagOp lag operator polynomial, and are subject to a near-zero tolerance exclusion test. If you set a coefficient to 1e–12 or below, regARIMA excludes that coefficient and its corresponding lag in SARLags from the model. Default: Cell vector of NaNs with the same length as SARLags.
SMA	Seasonal, moving average coefficients for the error model, specified as the comma-separated pair consisting of 'SMA' and a cell vector. The coefficient must yield an invertible polynomial. If you specify SMALags, then SMA is an equivalent-length cell vector of coefficients associated with the lags in SMALags. For example, if SMALags = [1, 4], SMA = {0.2, 0.1}, and Seasonality = 4, then, ignoring all other specifications, the error model is $$\left(1-L^4\right)u_t=\left(1+0.2L+0.1L^4\right){\epsilon }_t.$$ If you do not specify SMALags, then SMA is a cell vector of coefficients at lags 1,2,...,qs, the seasonal, moving average polynomial degree. For example, if SMA = {0.2, 0.1} and Seasonality = 4, and you do not specify SMALags, then, ignoring all other specifications, the error model is $$\left(1-L^4\right)u_t=\left(1+0.2L+0.1L^2\right){\epsilon }_t.$$ The coefficients in SMA correspond to coefficients in an underlying LagOp lag operator polynomial, and are subject to a near-zero tolerance exclusion test. If you set a coefficient to 1e–12 or below, regARIMA excludes that coefficient and its corresponding lag in SMALags from the model. Default: Cell vector of NaNs with the same length as SMALags.
SARLags	Lags associated with the SAR coefficients in the error model, specified as the comma-separated pair consisting of 'SARLags' and a vector of positive integers. Default: Vector of integers 1,2,...,ps, the seasonal, autoregressive polynomial degree.
SMALags	Lags associated with the SMA coefficients in the error model, specified as the comma-separated pair consisting of 'SMALags' and a vector of positive integers. Default: Vector of integers 1,2,...,qs, the seasonal moving average polynomial degree.
D	Nonseasonal differencing polynomial degree (i.e., nonseasonal integration degree) for the error model, specified as the comma-separated pair consisting of 'D' and a nonnegative integer. Default: 0 (no nonseasonal integration)
Seasonality	Seasonal differencing polynomial degree for the error model, specified as the comma-separated pair consisting of 'Seasonality' and a nonnegative integer. Default: 0 (no seasonal integration)
Variance	Variance of the model innovations εt, specified as the comma-separated pair consisting of 'Variance' and a positive scalar. Default: NaN
Distribution	Conditional probability distribution of the innovation process, specified as the comma-separated pair consisting of 'Distribution' and the distribution name or structure array describing the distribution. Distribution Distribution name Structure array Gaussian 'Gaussian' struct('Name','Gaussian') Student’s t 't' By default, DoF is NaN. struct('Name','t','DoF',DoF) DoF > 2 or DoF = NaN Default: 'Gaussian'
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) Error Model (Gaussian Distribution)".

Note

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 of the error model. Associated lags are 1,2,...,p, which is the nonseasonal, autoregressive polynomial degree, or as specified in ARLags.
Beta	Real vector of regression coefficients corresponding to the columns of the predictor data matrix.
D	Nonnegative integer indicating the nonseasonal integration degree of the error model.
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.
Intercept	Scalar intercept in the error model.
MA	Cell vector of nonseasonal moving average coefficients corresponding to an invertible polynomial of the error model. Associated lags are 1,2,...,q to the degree of the nonseasonal moving average polynomial, or as specified in MALags.
P	Scalar, compound autoregressive polynomial degree of the error model. P is the total number of lagged observations necessary to initialize the autoregressive component of the error 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. P does not necessarily conform to standard Box and Jenkins notation [1]. If D = 0, Seasonality = 0, and SAR = {}, then P conforms to the standard notation.
Q	Scalar, compound moving average polynomial degree of the error model. 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. Q does not necessarily conform to standard Box and Jenkins notation [1]. If SMA = {}, then Q conforms to the standard notation.
SAR	Cell vector of seasonal autoregressive coefficients corresponding to a stable polynomial of the error model. Associated lags are 1,2,...,ps, which is the seasonal autoregressive polynomial degree, or as specified in SARLags.
SMA	Cell vector of seasonal moving average coefficients corresponding to an invertible polynomial of the error model. Associated lags are 1,2,...,qs, which is the seasonal moving average polynomial degree, or as specified in SMALags.
Seasonality	Nonnegative integer indicating the seasonal differencing polynomial degree for the error model.
Variance	Positive scalar variance of the model innovations.

Object Functions

estimate	Estimate parameters of regression models with ARIMA errors
infer	Infer innovations of regression models with ARIMA errors
summarize	Display estimation results of regression model with ARIMA errors
simulate	Monte Carlo simulation of regression model with ARIMA 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 regression model with ARIMA errors
arima	Convert regression model with ARIMA errors to ARIMAX model

Copy Semantics

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

Examples

Specify a 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)"
    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. Note that P = p + D = 3, indicating that you need three presample observations to initialize the model for estimation.

Modify a 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)"
    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)"
    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 a 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,'SAR',{0.5,0.2},'SARLags',[4, 8],...
    'SMA',{0.05,0.01},'SMALags',[4 8],'D',1,...
    '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)"
    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,'SAR',{0.5,0.2},'SMA',{0.05,0.01},...
    'D',1,'Seasonality',4,'Variance',1)

Mdl = 
  regARIMA with properties:

     Description: "Regression with ARIMA(1,1,1) Error Model Seasonally Integrated with Seasonal AR(2) and MA(2) (Gaussian Distribution)"
    Distribution: Name = "Gaussian"
       Intercept: 1
            Beta: [6]
               P: 8
               D: 1
               Q: 3
              AR: {0.2} at lag [1]
             SAR: {0.5 0.2} at lags [1 2]
              MA: {0.1} at lag [1]
             SMA: {0.05 0.01} at lags [1 2]
     Seasonality: 4
        Variance: 1

More About

Regression Model with ARIMA Time Series Errors

A model that explains the behavior of a response using a linear regression model with predictor data, though the errors have autocorrelation indicative of an ARIMA process.

The model has the following 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.