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Fit vector error-correction (VEC) model to data

uses additional
options specified by one or more name-value pair arguments. For example, `EstMdl`

= estimate(`Mdl`

,`Y`

,`Name,Value`

)`'Model',"H1*",'X',X`

specifies the H1* Johansen form of the deterministic terms and `X`

as exogenous predictor data for the regression component.

If 1 ≤

`Mdl.Rank`

≤`Mdl.NumSeries`

–`1`

, as with most VEC models, then`estimate`

performs parameter estimation in two steps.`estimate`

estimates the parameters of the cointegrating relations, including any restricted intercepts and time trends, by the Johansen method [2].The form of the cointegrating relations corresponds to one of the five parametric forms considered by Johansen in [2] (see

`'Model'`

). For more details, see`jcitest`

and`jcontest`

.The adjustment speed parameter (

*A*) and the cointegration matrix (*B*) in the VEC(*p*– 1) model cannot be uniquely identified. However, the product*Π*=*A***B**ʹ*is identifiable. In this estimation step,*B*=*V*_{1:r}, where*V*_{1:r}is the matrix composed of all rows and the first*r*columns of the eigenvector matrix*V*.*V*is normalized so that*V**ʹ***S*_{11}**V*=*I*. For more details, see [2].

`estimate`

constructs the error-correction terms from the estimated cointegrating relations. Then,`estimate`

estimates the remaining terms in the VEC model by constructing a vector autoregression (VAR) model in first differences and including the error-correction terms as predictors. For models without cointegrating relations (`Mdl.Rank`

= 0) or with a cointegrating matrix of full rank (`Mdl.Rank`

=`Mdl.Numseries`

),`estimate`

performs this VAR estimation step only.

You can remove stationary series, which are associated with standard unit vectors in the space of cointegrating relations, from cointegration analysis. To pretest individual series for stationarity, use

`adftest`

,`pptest`

,`kpsstest`

, and`lmctest`

. As an alternative, you can test for standard unit vectors in the context of the full model by using`jcontest`

.If

`1`

≤`Mdl.Rank`

≤`Mdl.NumSeries`

–`1`

, the asymptotic error covariances of the parameters in the cointegrating relations (which include*B*,*c*_{0}, and*d*_{0}corresponding to the`Cointegration`

,`CointegrationConstant`

, and`CointegrationTrend`

properties, respectively) are generally non-Gaussian. Therefore,`estimate`

does not estimate or return corresponding standard errors.In contrast, the error covariances of the composite impact matrix, which is defined as the product

*A***B**ʹ*, are asymptotically Gaussian. Therefore,`estimate`

estimates and returns its standard errors. Similar caveats hold for the standard errors of the overall constant and linear trend (*A***c*_{0}and*A***d*_{0}corresponding to the`Constant`

and`Trend`

properties, respectively) of the H1* and H* Johansen forms.

[1] Hamilton, James. D. *Time Series Analysis*. Princeton, NJ: Princeton University Press, 1994.

[2] Johansen, S. *Likelihood-Based Inference in Cointegrated Vector Autoregressive Models*. Oxford: Oxford University Press, 1995.

[3] Juselius, K. *The Cointegrated VAR Model*. Oxford: Oxford University Press, 2006.

[4] Lütkepohl, H. *New Introduction to Multiple Time Series Analysis*. Berlin: Springer, 2005.