Environmental Engineering Reference
In-Depth Information
uence of the
AR(1) and AR(2) processes on the carbon price, but also of the macroeconomic
activity proxy lagged one and two periods (all at the 1 % level). When the industrial
production is on the uptake, we detect positive in
In the higher-regime, we are able to detect not only the statistical in
uences on the carbon price,
according to the underlying economic mechanisms at stake (as described in
Sect.
3.1
). Therefore, it seems that the carbon-macroeconomy relationship varies
nonlinearly with respect to the threshold identi
ed.
In the lower regime, the carbon price could be related mainly to institutional
events (Conrad et al. 2012), while the higher-regime
ndings are conform to pre-
vious literature (see, among others, [
25
,
21
,
26
]). Alberola et al. [
2
] noted previ-
ously that the relationship between the carbon price and its main drivers changes
before and after the occurrence of structural breaks. We are able to con
rm their
intuition based on the TVAR nonlinear model, which speci
es the presence of
several regimes in the data.
Hence, our interpretation in terms of macroeconomic drivers for the carbon
market hold both during the lower- and higher regimes, which has been docu-
mented recently in the literature [
5
,
9
11
,
24
]. Taken together, these results yield to
new insights into the relationship between the CO
2
price and the macroeconomy
compared to the linear regression framework [
2
,
3
].
-
5.1 Diagnostic Test
Here, we discuss some formal statistical approaches to model diagnostics via
residual analysis [
15
]. Namely, we consider the generalization of the portmanteau
test based on some overall measure of the magnitude of the residual autocorrelations.
The dependence of the residuals necessitates the employment of a quadratic form of
the residual autocorrelations:
B
m
¼
T
eff
X
X
m
m
q
i
;
j
q
i
q
j
ð
3
Þ
i
¼1
j
¼1
where
T
eff
=
T
−
max
(p
1
,
p
2
,d)
is the effective sample size, (
p
1
,
p
2
) are the lag
orders
,d
is the delay parameter,
q
i
is the
i
th lag sample autocorrelation of the
standardized residuals, and
q
i,j
some model-dependent constants given in [
15
]. If
the true model is a TVAR model, the
q
i
are likely close to zero and so is
B
m
, but
B
m
tends to be large if the model speci
cation is incorrect. The quadratic form is
designed so that
B
m
is approximately distributed as
2
with
m
degrees of freedom. In
practice, the
p
-value of
B
m
may be plotted against
m
over a range of
m
values to
provide a more comprehensive assessment of the independence assumption on the
standardized errors.
Model diagnostics are shown in Fig.
4
. The top panel represents the time series
plot of
χ
the standardized residuals of
the TVAR model
for EUAs and
Search WWH ::
Custom Search