Documentation |
ARIMAX = arima(Mdl)
[ARIMAX,XNew]
= arima(Mdl,Name,Value)
ARIMAX = arima(Mdl) converts the univariate regression model with ARIMA time series errors Mdl to a model of type arima including a regression component (ARIMAX).
[ARIMAX,XNew] = arima(Mdl,Name,Value) returns an updated regression matrix of predictor data using additional options specified by one or more Name,Value pair arguments.
Mdl |
Regression model with ARIMA time series errors, as created by regARIMA or estimate. |
Specify optional comma-separated 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.
Let X denote the matrix of concatenated predictor data vectors (or design matrix) and β denote the regression component for the regression model with ARIMA errors, Mdl.
If you specify X, then arima returns XNew in a certain format. Suppose that the nonzero autoregressive lag term degrees of Mdl are 0 < a_{1} < a_{2} < ...< P, which is the largest lag term degree. The software obtains these lag term degrees by expanding and reducing the product of the seasonal and nonseasonal autoregressive lag polynomials, and the seasonal and nonseasonal integration lag polynomials
$$\varphi (L){(1-L)}^{D}\Phi (L)(1-{L}^{s}).$$
The first column of XNew is Xβ.
The second column of XNew is a sequence of a_{1} NaNs, and then the product $${X}_{{a}_{1}}\beta ,$$ where $${X}_{{a}_{1}}\beta ={L}^{{a}_{1}}X\beta .$$
The jth column of XNew is a sequence of a_{j} NaNs, and then the product $${X}_{{a}_{j}}\beta ,$$ where $${X}_{{a}_{j}}\beta ={L}^{{a}_{j}}X\beta .$$
The last column of XNew is a sequence of a_{p} NaNs, and then the product $${X}_{p}\beta ,$$ where $${X}_{p}\beta ={L}^{p}X\beta .$$
Suppose that Mdl is a regression model with ARIMA(3,1,0) errors, and ϕ_{1} = 0.2 and ϕ_{3} = 0.05. Then the product of the autoregressive and integration lag polynomials is
$$(1-0.2L-0.05{L}^{3})(1-L)=1-1.2L+0.02{L}^{2}-0.05{L}^{3}+0.05{L}^{4}.$$
This implies that ARIMAX.Beta is [1 -1.2 0.02 -0.05 0.05] and XNew is
$$\left[\begin{array}{ccccc}{x}_{1}\beta & NaN& NaN& NaN& NaN\\ {x}_{2}\beta & {x}_{1}\beta & NaN& NaN& NaN\\ {x}_{3}\beta & {x}_{2}\beta & {x}_{1}\beta & NaN& NaN\\ {x}_{4}\beta & {x}_{3}\beta & {x}_{2}\beta & {x}_{1}\beta & NaN\\ {x}_{5}\beta & {x}_{4}\beta & {x}_{3}\beta & {x}_{2}\beta & {x}_{1}\beta \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ {x}_{T}\beta & {x}_{T-1}\beta & {x}_{T-2}\beta & {x}_{T-3}\beta & {x}_{T-4}\beta \end{array}\right],$$
where x_{j} is the jth row of X.
If you do not specify X, then arima returns XNew as an empty matrix without rows and one plus the number of nonzero autoregressive coefficients in the difference equation of Mdl columns.