Background
If
is the dependent (autoregressive) variable,
are
distributed-lag explanatory variables, and
are
exogenous, potentially deterministic variables, the
Intertemporal Dynamics (ITD) representation of an ARDL(
) model is given by:
| (29.1) |
where
are the innovations, and
,
, and
are the coefficients associated with the exogenous variables,
lags of
, and
lags of the
distributed lag regressors
, respectively.
Let
be the usual lag operator and define the lag polynomials:
| (29.2) |
Substituting into
Equation (29.1) yields:
| (29.3) |
Noting that any series
may be written as
and performing a Beveridge-Nelson decomposition on both
and the
in
Equation (29.3) produces the
Conditional Error Correction (CEC) representation of the ARDL,
| (29.4) |
which, with a bit of manipulation, may be rewritten as
| (29.5) |
where
| (29.6) |
and
| (29.7) |
Since CEC
Equation (29.5) and
Equation (29.8) are derived from ITD
Equation (29.1), there is an obvious one-to-one correspondence between the two. As with the vector error correction (VEC) form of a VAR, the CEC form offers easy identification of a cointegrating relationship between the dependent variable and the explanatory variables in the ARDL. We discuss this parallel in greater depth in
“Relationship to Vector Error Correction (VEC) Models”.
Rearrange terms, we may re-write
Equation (29.8) as
| (29.8) |
If we define the equilibrium error correction term,
| (29.9) |
then
Equation (29.8) may be written in Error Correction (EC) form:
| (29.10) |
where
is the error correction parameter, and the long-term equilibrium parameters for the explanatory variables are given by
, for
.
Conveniently, the coefficients in both the ITD and the CEC representations of the ARDL model may be estimated via least squares.
Relationship to Vector Error Correction (VEC) Models
Assuming the same lag across the distributed-lag regressors
and that the deterministics
consist of a simple constant and linear trend, Pesaran (2001) demonstrates that the ARDL CEC representation in
Equation (29.8) is in fact the CEC of the VAR(
) model:
| (29.11) |
where
is a
vector of endogenous variables,
and
are the
vectors of intercept and trend coefficients, respectively, and
| (29.12) |
is the
matrix lag polynomial.
Invoking the BN decomposition on
and with following some rearrangement, the CEC representation of this VEC may be written as
| (29.13) |
where
| (29.14) |
which is equivalent to
Equation (29.5).
Nonlinear (asymmetric) ARDL
The classical ARDL framework assumes that the long-run relationship
is a symmetric linear combination of regressors. While this is a natural starting assumption, it does not match the behavioral finance and economics literature approach to modeling nonlinearity and asymmetry (Kahneman, Tversky, and Shiller, 1979). In response, Shin (2014) proposes a nonlinear ARDL (NARDL) framework in which short-run and long-run nonlinearities are modeled as positive and negative partial sum decompositions of the explanatory variables.
Consider the partial sum decomposition of a variable
around a threshold of
as
where
and
are the partial sum processes of positive and negative
changes in
, respectively:
| (29.15) |
The ITD representation of a NARDL(
) model is given by:
| (29.16) |
where
are coefficients for the initial conditions, and where
and
are coefficients associated with the asymmetric distributed-lag variables.
We may an obtain a CEC representation of the ITD NARDL model,
| (29.17) |
where the
are asymmetric analogues of the coefficients in
Equation (29.6).
We may rearrange terms so that
Equation (29.17) becomes
| (29.18) |
Then, define the asymmetric equilibrium error correction term,
| (29.19) |
so that the CEC
Equation (29.18) may be written in EC form:
| (29.20) |
where
is the error correction parameter, the long-term equilibrium parameters for the explanatory variables are given by
and
for
. The short-run parameters for the explanatory variables are given by the
.
Notice that because the CEC representation decomposes the effect of the distribution lag variables into short and long-run components, it allows for asymmetries in various combinations of short and long-run dynamics. This flexibility does not exist in the ITD representation.
As with their symmetric counterparts, NARDL models may be estimated via least squares. This result is appealing since nonlinear models often require iterative estimation routines. Furthermore, bounds testing procedures (
“Bounds Test View”) remain valid and require no meaningful adjustments.