Goethe University Frankfurt
Advanced Econometrics 2, Part 2
Sommersemester 2016
Prof. Michael Binder, Ph.D.

I. Vector Autoregressions and Vector Error Correction Models
3. Estimation and Inference with and without Parameter Restrictions
Cointegrated VAR – Case of a Single Cointegrating Relationship

Special Case of One Cointegrating Relationship:
Weak Exogeneity and ARDL Models
When the cointegration rank of a cointegrated VAR is one, then under certain conditions it is feasible to work with a notably more parsimonious model, namely the so-called Autoregressive
Distributed Lag (ARDL) model of the form: p q


1 − ∑ φ j Lj  yt =∑ θ j ' xt − j + ε t ,

= 1= 0 j j



iid

(

(76)

)

with ε t ~ 0, σ 2 .
Note: For simplicity of notation only, in (76) model deterministic components are irgnored and it is assumed that all elements of x enter with the same lag order, namely q.
68

Goethe University Frankfurt
Advanced Econometrics 2, Part 2
Sommersemester 2016
Prof. Michael Binder, Ph.D.

I. Vector Autoregressions and Vector Error Correction Models
3. Estimation and Inference with and without Parameter Restrictions
Cointegrated VAR – Case of a Single Cointegrating Relationship

To understand when we can reduce a cointegrated VAR to an ARDL model, let us carefully derive the ARDL model in (76) from a system of equations in zt = ( yt xt )′ . Suppose zt is generated by

{ }

p

j εt ,
 I − ∑ Φ j L  zt =
 j =1


(77)

iid

t = 1, 2,..., T , with ε t ~ ( 0, Ω ) . We assume that the elements of zt are either I(0) or I(1).
Defining

p
 p

Π = and Γ j =
− ∑ Φk ,
 ∑ Φj − I  , k = j +1
 j= 1


(78)

j = 1, 2,..., p −1, we can re-write (77) in error correction form as
69

Goethe University Frankfurt
Advanced Econometrics 2, Part 2
Sommersemester 2016
Prof. Michael Binder, Ph.D.

I. Vector Autoregressions and Vector Error Correction Models
3. Estimation and Inference