Understanding the Link Between Aggregated Industrial Production and the Carbon Price - Green Energy and Efficiency: An Economic Perspective - page 123

4.2 TVAR Model

Next, we present formally the TVAR model used. We build here on [ 4 ] ' s notations:

X t ¼ A 0 ðLÞX t 1 þ ½ A 1

ðLÞX t 1 Iðc td [ rÞþe t ð 1 Þ

where X t denotes a vector a time series, A 0 (L) and A 1 ( L) are lag polynomials, and

ε t
is the error term. c t - d is the threshold variable that determines which regime the
system is in, r is the threshold critical value, I( c t - d > r) is an indicator function that
equals 1 when c t - d > r , and zero otherwise. The threshold value r is not known a
priori, and must be estimated (see [ 18 ]).
Before estimating the TVAR model, we need to implement a nonlinearity test in
order to test formally whether the threshold-type behavior is rejected, or not. The
test is the multivariate extension by Hansen [ 19 , 23 ] of linearity test against various
thresholds. As in the univariate case, the rst threshold parameter is estimated by

Conditional Least Squares (CLS) upon a grid of potential values for the threshold
and the delays. Then, for the second threshold, a conditional search with one
iteration is performed. Instead of a F -test comparing the Sum of Squared Residuals
(SSR) for the univariate case, a Likelihood Ratio (LR) test comparing the covari-
ance matrix of each model is computed:

LR ij ¼ Tð ln ð det R i Þð ln ð det R j ÞÞ ð 2 Þ

with R i the estimated covariance matrix of the model with i -regimes (and i -
thresholds), det the notation for the determinant of the matrix, and T the number of
observations. Three tests are presented:

1. Test 1 versus 2: Linear VAR versus 1-threshold TVAR;
2. Test 1 versus 3: Linear VAR versus 2-threshold TVAR;
3. Test 2 versus 3: 1-threshold TVAR versus 2-threshold TVAR.

rst whether a purely linear model is rejected (in favor
of one or two thresholds). In the second step, once the presence of the threshold(s)
has been con

The goal is to determine

rmed, we aim at identifying whether a model with one or two
thresholds is preferable (see [ 29 ] for more details).
The model hyper-parameters (i.e. the possible thresholds and delays value) are
determined by running an automatic search upon a grid of potential values 11

(for

more details, see [ 23 ]). For a xed threshold variable, the model is linear, so that the

estimation of the two higher- and lower-regimes can be done directly by CLS. The
standard errors coef cients provided for this model are taken from the linear

regression theory, and are to be considered asymptotical [ 17 , 30 ].

11 An exhaustive search is conducted over all the possible combinations of values of the speci ed
hyper-parameters. These results are not shown here to conserve space, and may be obtained upon
request to the author.