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Vector autoregression (VAR) to vector error-correction model (VEC)
[VEC,C]
= vartovec(VAR)
Given a vector autoregression (VAR) model, [VEC,C] = vartovec(VAR) converts VAR to an equivalent vector error-correction (VEC) model. A VAR(p) model of a time series y(t) has the form:
The equivalent VEC(q) model, with q = p − 1, has the form:
where z(t) = y(t) − y(t − 1) and C is the error-correction coefficient.
VAR |
The VAR(p) model to be converted to an equivalent VEC(q) model, with q = p − 1. VAR is specified by a (p + 1)-element cell vector of square matrices {A0 A1 ... Ap} associated with coefficients at lags 0, 1, ..., p. To represent a univariate model, VAR may be specified as a double-precision vector. Alternatively, VAR may be specified as a LagOp object or a vgxset object. |
VEC |
The VEC representation of the input VAR model. The data type and orientation of VEC is consistent with that of VAR |
C |
The error-correction coefficient. C is a square matrix the same size as the coefficients of the associated VEC. |
Specify a VAR(2) model of time series y_{t}:
The coefficients are:
Enter the coefficients from the difference equation directly into a cell array:
VAR = {eye(2) [-0.1 0.3 ; 0.2 -0.1] ...
[-0.2 0.8 ; -0.7 -0.4]};
Use vartovec to convert the VAR(2) model to an equivalent VEC(1) model:
[VEC, C] = vartovec(VAR);
Since the original VAR model was specified as a cell array, the VEC model is also a cell array. The error correction coefficient argument is a matrix.
You can express the same VAR(2) model as a lag operator polynomial:
Specify the model with the LagOp constructor:
VAR_LAG = LagOp({eye(2) [0.1 -0.3 ; -0.2 0.1] ...
[0.2 -0.8 ; 0.7 0.4]});
Use vartovec to convert the VAR(2) model to an equivalent VEC(1) model:
[VEC_LAG, C_LAG] = vartovec(VAR_LAG)
VEC_LAG = 2-D Lag Operator Polynomial: ----------------------------- Coefficients: [Lag-Indexed Cell Array with 2 Non-Zero Coefficients] Lags: [0 1] Degree: 1 Dimension: 2 C_LAG = -1.3000 1.1000 -0.5000 -1.5000
Since the input model is a lag operator polynomial the output model is also is a lag operator polynomial. See Specify Lag Operator Polynomials for more information on lag operator polynomials.
[1] Hamilton, J. D. "Time Series Analysis." Princeton, NJ: Princeton University Press, 1994.
[2] Lutkepohl, H. "New Introduction to Multiple Time Series Analysis." Springer-Verlag, 2007.