VAR / VECM models and the conditional ARDL specification
The starting point of the method which the bootCT package is
based on is the \(K+1\) dimensional
VAR\((p)\) model
\[
\mathbf{A}(L)(\mathbf{z}_t-\boldsymbol{\mu}-\boldsymbol{\eta}t)=\boldsymbol{\varepsilon}_t,
\enspace \enspace \enspace \boldsymbol{\varepsilon}_t\sim
N_{K+1}(\mathbf{0}, \boldsymbol{\Sigma}),\enspace \enspace \enspace
t=1,2,\dots,T\]
\[\mathbf{A}(L)=\left(\mathbf{I}_{K+1}- \sum_{j=1}^{p}\mathbf{A}_j\mathbf{L}^j\right)
\] Here, \(\mathbf{A}_j\) are
square \((K+1)\) matrices, \(\mathbf{z}_t\) a vector of \((K+1)\) variables, \(\boldsymbol{\mu}\) and \(\boldsymbol{\eta}\) are \((K+1)\) vectors representing the drift and
the trend respectively, and \(det(\mathbf{A}(z))=0\) for \(|z| \geq 1\). If the matrix \(\mathbf{A}(1)=\mathbf{I}_{K+1}-\sum_{j=1}^{p}\mathbf{A}_{j}\)
is singular, the components of \(\mathbf{z}_t\) turn out to be integrated
and possibly cointegrated.
The VAR model admits a VECM representation
\[
\Delta\mathbf{z}_t=\boldsymbol{\alpha}_{0}+\boldsymbol{\alpha}_{1}t-\mathbf{A}(1)\mathbf{z}_{t-1}+\sum_{j=1}^{p-1}\boldsymbol{\Gamma}_{j}\Delta
\mathbf{z}_{t-j}+\boldsymbol{\varepsilon}_t
\] where \(\boldsymbol{\Gamma}_{j}=-\sum_{i=j+1}^{p}\mathbf{A}_i\).
The matrix \(\mathbf{A}\) (suppressing
the bracket term for simplicity’s sake) contains the long-run
cointegrating relationships among the variables, while the \(\boldsymbol{\Gamma}_{j}\)’s contain the
short-run relationships. Defining now \(\boldsymbol{\Gamma}(1)=\mathbf{I}_{K+1}-\sum_{i=1}^{p-1}\boldsymbol{\Gamma}_{i}\),
the intercept and trend terms in the VECM specification are
\[\boldsymbol{\alpha}_0=\mathbf{A}\boldsymbol{\mu}+(\boldsymbol{\Gamma}(1)-\mathbf{A})\boldsymbol{\eta},
\enspace \enspace \enspace
\boldsymbol{\alpha}_1=\mathbf{A}\boldsymbol{\eta}
\] It is important to highlight now the five possible cases in
the intercept and trend specifications:
- Case I: no intercept, no trend, for which \(\boldsymbol\mu=\boldsymbol\eta=\mathbf
0\)
- Case II: restricted intercept, no trend, for which \(\boldsymbol{\alpha}_0=\mathbf{A}\boldsymbol{\mu}\),
\(\boldsymbol\eta=\mathbf 0\)
- Case III: unrestricted intercept, no trend, for which \(\boldsymbol{\alpha}_0\neq\mathbf{A}\boldsymbol{\mu}\),
\(\boldsymbol\eta=\mathbf 0\)
- Case IV: unrestricted intercept, restricted trend, for which \(\boldsymbol{\alpha}_0\neq\mathbf{A}\boldsymbol{\mu}\),
\(\boldsymbol{\alpha}_1=\mathbf{A}\boldsymbol{\eta}\)
- Case V: unrestricted intercept, unrestricted trend, for which \(\boldsymbol{\alpha}_0\neq\mathbf{A}\boldsymbol{\mu}\),
\(\boldsymbol{\alpha}_1\neq\mathbf{A}\boldsymbol{\eta}\).
We now introduce the key concept of conditioning in the VECM system.
To study the adjustment to the equilibrium of a single variable \(y_t\), given the other \(\mathbf{x}_t\) variables, the vectors \(\mathbf{z}_t\) and \(\boldsymbol{\varepsilon}_t\) are
partitioned:
\[
\mathbf{z}_t=\begin{bmatrix}
\underset{(1,1)}{y_{t}} \\ \underset{(K,1)}{\mathbf{x}_{t}}
\end{bmatrix}\enspace\boldsymbol{\varepsilon}_t=\begin{bmatrix}
\underset{(1,1)}{\varepsilon_{yt}}
\\ \underset{(K,1)}{\boldsymbol{\varepsilon}_{xt}} \end{bmatrix}
\] The vectors \(\boldsymbol{\alpha}_{0}\), \(\boldsymbol{\alpha}_{1}\), the matrix \(\mathbf{A}\) and the polynomial matrix
\(\boldsymbol{\Gamma}(L)\) are
partitioned conformably to \(\mathbf{z}_{t}\)
\[\boldsymbol{\alpha}_0=\begin{bmatrix}
\underset{(1,1)}{\alpha_{0y}} \\
\underset{(K,1)}{\boldsymbol{\alpha}_{0x}}
\end{bmatrix}, \enspace \enspace \enspace
\boldsymbol{\alpha}_1=\begin{bmatrix}
\underset{(1,1)}{\alpha_{1y}} \\
\underset{(K,1)}{\boldsymbol{\alpha}_{1x} }
\end{bmatrix}\]
\[\mathbf{A}=\begin{bmatrix}
\underset{(1,K+1)}{\mathbf{a}^{'}_{(y)}} \\
\underset{(K,K+1)}{\mathbf{A}_{(x)}}
\end{bmatrix}
=\begin{bmatrix}
\underset{(1,1)}{a_{yy}} &
\underset{(1,K)}{\mathbf{a}^{'}_{yx}} \\
\underset{(K,1)}{\mathbf{a}_{xy}} & \underset{(K,K)}{\mathbf{A}_{xx}
}
\end{bmatrix},
\enspace \enspace \enspace
\boldsymbol{\Gamma}(L)=\begin{bmatrix}
\underset{(1,K+1)}{\boldsymbol{\gamma}^{'}_{y}(L)} \\
\underset{(K,K+1)}{\boldsymbol{\Gamma}_{(x)}(L)}
\end{bmatrix}
=\begin{bmatrix}
\underset{(1,1)}{\gamma_{yy}(L)} &
\underset{(1,K)}{\boldsymbol{\gamma}^{'}_{yx}(L)} \\
\underset{(K,1)}{\boldsymbol{\gamma}_{xy}(L)} &
\underset{(K,K)}{\boldsymbol{\Gamma}_{xx}(L) }
\end{bmatrix}
\] while for the error term \[
\boldsymbol{\varepsilon}_t \sim N\Bigg(\mathbf{0}, \begin{bmatrix}
\underset{(1,1)}{\sigma_{yy}}&
\underset{(1,K)}{\boldsymbol{\sigma}_{yx}^{'}} \\
\underset{(K,1)}{\boldsymbol{\sigma}_{xy}} &
\underset{(K,K)}{\boldsymbol{\Sigma}_{xx}} \end{bmatrix}\Bigg)
\]
In order to condition \(y_t\) on
\(\mathbf x_t\), we define the
conditional variance \[\sigma_{y.x}=\sigma_{yy}-\boldsymbol{\sigma}^{'}_{yx}\boldsymbol{\Sigma}^{-1}_{xx}\boldsymbol{\sigma}_{xy}=
\sigma_{yy}-\boldsymbol{\omega}^{'}\boldsymbol{\sigma}_{xy}
\] And thus \[\varepsilon_{yt}=\boldsymbol{\omega}^{'}\boldsymbol{\varepsilon}_{xt}+\nu_{yt}
\sim N(0,\sigma_{y.x})\] where \(\nu_{yt}\) is independent of \(\boldsymbol{\varepsilon}_{xt}\).
Accordingly, conditioning can be applied to the entire system,
obtaining
\[
\mathbf{a}^{'}_{(y).x}=\mathbf{a}^{'}_{(y)}-\boldsymbol{\omega}^{'}\mathbf{A}_{(x)},
\enspace \enspace \enspace
\boldsymbol{\gamma}^{'}_{y.x}(L)=\boldsymbol{\gamma}_{y}'(L)-\boldsymbol{\omega}'\boldsymbol{\Gamma}_{(x)}(L)
\]
Therefore, the long-run relationships of the VECM turn out to be now
included in the matrix \[
{\mathbf{A}_c}=\begin{bmatrix}
\mathbf{a}^{'}_{(y).x} \\ \mathbf{A}_{(x)}
\end{bmatrix}=\begin{bmatrix}
a_{yy}-\boldsymbol{\omega}^{'}\mathbf{a}_{xy} &
\mathbf{a}_{yx}^{'}-\boldsymbol{\omega}^{'}\mathbf{A}_{xx} \\
\mathbf{a}_{xy}&\mathbf{A}_{xx}
\end{bmatrix}
\]
To rule out the presence of long-run relationships between \(y_{t}\) and \(\mathbf{x}_{t}\) in the marginal model, the
long-run matrix becomes \[
\widetilde{\mathbf{A}}=\begin{bmatrix}a_{yy} &
\mathbf{a}^{'}_{yx}-\boldsymbol{\omega}^{'}\mathbf{A}_{xx} \\
\mathbf{0} & \mathbf{A}_{xx}
\end{bmatrix}=\begin{bmatrix}
a_{yy} & \widetilde{\mathbf{a}}_{y.x}^{'} \\
\mathbf{0}&\mathbf{A}_{xx}\end{bmatrix}
\]
Putting everything together, the VECM system remains unaltered in the
\(\mathbf x_t\) variables \[
\Delta\mathbf{x}_{t}
= \boldsymbol{\alpha}_{0x} +\boldsymbol{\alpha}_{1x}t+
\mathbf{A}_{(x)}\mathbf{z}_{t-1}+
\boldsymbol{\Gamma}_{(x)}(L)\Delta\mathbf{z}_t+
\boldsymbol{\varepsilon}_{xt}
\] while the equation in the conditional ARDL specification for
\(y_t\) is
\[
\Delta y_{t}=\alpha_{0.y}+\alpha_{1.y}t -a_{yy}EC_{t-1}+
\sum_{j=1}^{p-1}\boldsymbol{\gamma}'_{y.x,j}\Delta\mathbf{z}_{t-j}+\boldsymbol{\omega}'\Delta\mathbf{x}_{t}+\nu_{yt}
\] The error correction term, \(EC_{t-1}\), expressing the long-run
equilibrium relationship between \(y_{t}\) and \(\mathbf{x}_{t}\), is given by \[
EC_{t-1}=y_{t-1}-\theta_{0}-\theta_{1}t-\boldsymbol{\theta}'\mathbf{x}_{t-1}
\] with \[\theta_{0}=\mu_{y}-\boldsymbol{\theta}'\boldsymbol{\mu}_{x},
\enspace
\theta_{1}=\eta_{y}-\boldsymbol{\theta}'\boldsymbol{\eta}_{x},
\enspace\boldsymbol{\theta}'=-\frac{\widetilde{\mathbf{a}'}_{y.x}}{a_{yy}}=-\frac{\mathbf{a}_{yx}'-\boldsymbol{\omega}'\mathbf{A}_{xx}}{a_{yy}}
\] Turning back to the cases specification, the ARDL equation can
be specialized on this basis:
- Case I: no intercept, no trend, therefore \(\theta_0=\theta_1=\alpha_{0.y}=\alpha_{1.y}=0\)
- Case II: restricted intercept, no trend. Since \(\boldsymbol{\alpha}_0=\mathbf{A}\boldsymbol{\mu}\),
\(\theta_0\neq 0\) and the intercept
term appears in \(EC_{t-1}\).
Therefore, \(\alpha_{0.y}=0\). In
addition, \(\theta_1=\alpha_{1.y}=0\)
- Case III: unrestricted intercept, no trend, Since \(\boldsymbol{\alpha}_0\neq\mathbf{A}\boldsymbol{\mu}\),
\(\theta_0 = 0\) and \(\alpha_{0.y}=\alpha_{0y}-\boldsymbol\omega'\boldsymbol\alpha_{0x}\neq0\).
In addition, \(\theta_1=\alpha_{1.y}=0\)
- Case IV: unrestricted intercept, restricted trend, for which \(\alpha_{0.y}=\alpha_{0y}-\boldsymbol\omega'\boldsymbol\alpha_{0x}\neq0\).
Since
\(\boldsymbol{\alpha}_1=\mathbf{A}\boldsymbol{\eta}\),
\(\theta_1\neq 0\) and the trend term
appear in \(EC_{t-1}\).
- Case V: unrestricted intercept, unrestricted trend. Since \(\boldsymbol{\alpha}_0\neq\mathbf{A}\boldsymbol{\mu}\),
\(\theta_0 = 0\) and \(\alpha_{0.y}=\alpha_{0y}-\boldsymbol\omega'\boldsymbol\alpha_{0x}\neq0\).
In addition, \(\boldsymbol{\alpha}_1\neq\mathbf{A}\boldsymbol{\eta}\),
\(\theta_1 = 0\) and \(\alpha_{1.y}=\alpha_{1y}-\boldsymbol\omega'\boldsymbol\alpha_{1x}\neq0\)
Generating a multivariate time series: the sim_vecm_ardl
function
As usual, we load the package
This first function allows to generate a multivariate time series
that follows a VECM/ARDL specification, and a particular case,
as detailed above. These are the parameters:
- The error covariance matrix \(\boldsymbol\Sigma\)
sigma.in = matrix(c(1.69, 0.39, 0.52,
0.39, 1.44, -0.3,
0.52, -0.3, 1.00),3,3)
- The short-run parameters, \(\boldsymbol\Gamma_1\), \(\boldsymbol\Gamma_2\)
gamma1 = matrix(c(0.60, 0.00, 0.20,
0.10, -0.3, 0.00,
0.00, -0.3, 0.20),3,3,T)
gamma2 = gamma1 * 0.3
gamma.in = list(gamma1,gamma2)
- The long-run parameters, \(a_{yy}\), \(\mathbf a_{yx}\), \(\mathbf A_{xx}\) that will eventually form
the matrix \(\widetilde{\mathbf
A}\)
ayy.in = 0.6
ayx.uc.in = c(0.4, 0.4)
axx.in = matrix(c(0.30, 0.50,
-0.4, 0.30),2,2,T)
- The intercept and trend terms (\(\boldsymbol\mu\), \(\boldsymbol\eta\), \(\boldsymbol\alpha_0\) and \(\boldsymbol\alpha_1\)), which depend on the
chosen case. For instance, in Case II:
case = 2
mu.in = c(2,2,2)
eta.in = c(0,0,0)
azero.in = c(0,0,0)
aone.in = c(0,0,0)
Calling the function to generate \(T=200\) observations:
data.vecm.ardl_2 =
sim_vecm_ardl(nobs=200,
case = 2,
sigma.in = sigma.in,
gamma.in = gamma.in,
axx.in = axx.in,
ayx.uc.in = ayx.uc.in,
ayy.in = ayy.in,
mu.in = mu.in,
eta.in = eta.in,
azero.in = azero.in,
aone.in = aone.in,
burn.in = 100,
seed.in = 999)
In the function output, there are several elements available. Some
represent the function arguments themselves, other are calculated as a
byproduct (e.g., \(\alpha_{0.y}\),
\(\alpha_{1.y}\), \(\theta_{0}\), \(\theta_{1}\)). The main points of interests
are naturally the data and its first difference
head(data.vecm.ardl_2$data)
#> dep_1_0 ind_1_0 ind_2_0 time
#> 104 -3.53399308 2.51204516 5.3842215 1
#> 105 -0.05750437 1.23912249 7.8679695 2
#> 106 -1.71303429 0.64048684 6.2980765 3
#> 107 -0.81595005 -0.03456059 5.4483791 4
#> 108 2.08297259 -0.78242332 4.1370603 5
#> 109 1.91584685 0.28597121 0.2817475 6
head(data.vecm.ardl_2$diffdata)
#> d_dep_1_0 d_ind_1_0 d_ind_2_0 time
#> 104 -0.7962310 -2.7284874 1.4603533 1
#> 105 3.4764887 -1.2729227 2.4837480 2
#> 106 -1.6555299 -0.5986356 -1.5698930 3
#> 107 0.8970842 -0.6750474 -0.8496974 4
#> 108 2.8989226 -0.7478627 -1.3113188 5
#> 109 -0.1671257 1.0683945 -3.8553128 6
Notice that the index of the data starts from burn.in +
length(gamma.in) + 1 .
The following code plots the data and the first difference:
df = data.vecm.ardl_2$data
meltdf <- melt(df,id="time")
ggplot(meltdf,
aes(x = time, y = value, colour = variable, group = variable)) +
geom_line() + ggtitle("CASE II - Level") + theme_bw()
df = data.vecm.ardl_2$diffdata
meltdf <- melt(df,id="time")
ggplot(meltdf,
aes(x = time, y = value, colour = variable, group = variable)) +
geom_line() + ggtitle("CASE II - Diff") + theme_bw()
To make the case in accordance with the intercept/trend specification
for Case III/IV/V, respectively:
# case III
case = 3
mu.in = c(0,0,0)
eta.in = c(0,0,0)
azero.in = c(2,2,2)
aone.in = c(0,0,0)
# case IV
case = 4
mu.in = c(0,0,0)
eta.in = c(0.6,0.6,0.6)
azero.in = c(2,2,2)
aone.in = c(0,0,0)
# case V
case = 5
mu.in = c(0,0,0)
eta.in = c(0,0,0)
azero.in = c(2,2,2)
aone.in = c(0.6,0.6,0.6)
Bootstrapping the ARDL test for cointegration
The usual test based on the conditional ARDL model examines the
following null hypotheses:
\[
H_0^{F_{ov}}: a_{yy}=0\enspace\cap \enspace\widetilde{\mathbf
a}_{y.x}=\mathbf 0
\] \[
H_0^{F_{ind}}: \widetilde{\mathbf a}_{y.x}=\mathbf 0\qquad
\text{(Degeneracy of first type)}
\]
\[
H_0^{t}: a_{yy}=0 \qquad \text{(Degeneracy of second type)}
\] Note that a form of spurious cointegration can occur, since
\(\widetilde{\mathbf a}_{y.x} = \mathbf
a_{y.x} - \boldsymbol\omega'\mathbf A_{xx}\).
If \(H_0^{F_{ind}}\) is rejected, it
might be that \(\mathbf a_{y.x} = \mathbf
0\), while \(\boldsymbol\omega'\mathbf A_{xx} \neq \mathbf
0\). In that case, no cointegration is really in place.
For this reason, the \(F_{ind}\)
test must be performed also in the unconditional ARDL model (UC), that
is the one omitting instantaneous differences of the independent
variables from the equation (\(\Delta\mathbf
x_t\)), along with their coefficient \(\boldsymbol\omega'\). Its null
hypothesis is
\[
H_{0,UC}^{F_{ind}}: {\mathbf a}_{yx}=\mathbf 0
\] On the model
\[
\Delta y_{t}=\alpha_{0.y}+\alpha_{1.y}t -a_{yy}EC_{t-1}+
\sum_{j=1}^{p-1}\boldsymbol{\gamma}'_{y.x,j}\Delta\mathbf{z}_{t-j}+\nu_{yt}
\] in which \(\boldsymbol{\theta}'=-\frac{{\mathbf{a}'}_{yx}}{a_{yy}}\).
The usual testing framework in Pesaran (2001) employs the ARDL bound
test on the conditional model for \(F_{ov}\) and \(t\), while Goh et al. (2017) derived the
bound test for \(F_{ind}\).
However:
- The \(t\) and \(F_{ind}\) tests are available only for Case
I-III-V
- Bound tests may lead to inconclusive results
For this reason, Bertelli et al. (2022) derived a comprehensive
bootstrap framework to analyze ARDL-based cointegration.
In order to perform this bootrap version, the usual “bootstrap
ingredients” are necessary
- The test statistics coming from the original data: \(F_{ov}\), \(t\), \(F_{ind}\) and \(F_{ind,UC}\).
- A sampling scheme on the original data.
- The simulated bootstrap distribution of each statistic under its
respective null hypothesis, of which the critical quantiles \(c^*_{\alpha,T}\) are calculated.
This bootstrap procedure does the following:
In calculating the test statistics, the most appropriate model in
the estimation phase for both the ARDL and VECM models has to be chosen.
This can either be imposed by the user or automatically detected via
information criteria
In deriving the bootstrap distribution of the statistics under
each null, the marginal VECM model remains unaltered, while the ARDL
model is re-estimated imposing the particular null hypothesis under
examination (i.e., discarding the variables related to \(a_{yy}\), \(\widetilde{\mathbf a}_{y.x}\), or both).
Thus, the residuals of the ARDL and VECM models are calculated.
Naturally, for the \(F_{ind,UC}\)
model, the restriction applies to \({\mathbf
a}_{yx}\).
In each bootstrap iteration, a random sample from the residuals
is extracted (of the same length of the original data), and synthetic
data is generated using the estimates obtained under each null. Finally
the bootstrap version \(F^{(b)}_{ov}\),
\(t^{(b)}\) and \(F^{(b)}_{ind}\) is calculated and stored,
for \(b = 1,\dots,B\). \(F^{(b)}_{ind,UC}\) is also calculated if
appropriate.
Bootstrap critical quantiles are calculated, and decision on each
test is based on the comparison between said critical value and the
statistic in the original data. Notably, the \(F_{ind,UC}\) and its bootstrap distribution
under the null are calculated only when it is mandatory to detect
spurious cointegration.
The boot_ardl function
We use the German macroeconomic dataset of Lutkepohl (2007). We start
by loading the data and visualizing it - also in first difference
data("ger_macro")
# Data preparation (log-transform)
LNDATA = apply(ger_macro[,-1], 2, log)
col_ln = paste0("LN", colnames(ger_macro)[-1])
LNDATA = as.data.frame(LNDATA)
colnames(LNDATA) = col_ln
LNDATA = LNDATA %>% select(LNCONS, LNINCOME, LNINVEST)
LNDATA$DATE = ger_macro$DATE
# First difference
lagdf1 = lag_mts(as.matrix(LNDATA[,-4]), k = c(1,1,1))
DIFF.LNDATA = na.omit(LNDATA[,-4] - lagdf1)
colnames(DIFF.LNDATA) = paste0("D_", colnames(LNDATA)[-4])
DIFF.LNDATA$DATE = ger_macro$DATE[-1]
# Graphs
dfmelt = melt(LNDATA, id = "DATE")
dfmelt = dfmelt%>%arrange(variable,DATE)
diff.dfmelt = melt(DIFF.LNDATA, id = "DATE")
diff.dfmelt = diff.dfmelt%>%arrange(variable,DATE)
ggplot(dfmelt,
aes(x = DATE, y = value, colour = variable, group = variable)) +
geom_line() + ggtitle("Level Variables (log-scale)") + theme_bw()
ggplot(diff.dfmelt,
aes(x = DATE, y = value, colour = variable, group = variable)) +
geom_line() + ggtitle("Diff. Variables (log-scale)") + theme_bw()
Calling the function using CONS as the dependent variable:
BCT_res_CONS = boot_ardl(data = LNDATA,
yvar = "LNCONS",
xvar = c("LNINCOME","LNINVEST"),
case = 3,
fix.ardl = NULL,
info.ardl = "AIC",
fix.vecm = NULL,
info.vecm = "AIC",
maxlag = 5,
a.ardl = 0.1,
a.vecm = 0.1,
nboot = 2000,
a.boot.H0 = c(0.01,0.05,0.1),
print = F)
Notably, the parameters fix.vecm and fix.ardl are
blank, meaning that the estimation phase is dealt with via an automatic
procedure that selects the best lag order for the short-run parameters
employing common information criteria such as the AIC, BIC, SC, FPE,
etc. In this case, the parameters identifying the criteria,
info.vecm and info.ardl are also null, which in turn
triggers the AIC as default value. The parameter maxlag sets
the maximum possible value in the lag search. Secondly, the parameters
a.vecm and a.ardl set the significance threshold for
each of the single parameters in the short-run part of the model
equation for the VECM and ARDL models, respectively. Coefficients for
which the \(p\)-value is greater than
this latter threshold are discarded. The argument a.boot.H0
sets the significance levels for which critical values of the bootstrap
distribution under the null of \(F_{ov}\), \(t\), \(F_{ind}\) and \(F_{ind,UC}\) (if appropriate) are
calculated, using nboot residual bootstrap replicates.
A summary method can be called to visualize subsets of the
output:
- The conditional ARDL model estimates
summary(BCT_res_CONS, out="ARDL")
#> CONDITIONAL ARDL MODEL
#>
#> Call:
#> lm(formula = formula.ardlx, data = na.omit(df.ardl.l[[cond]]))
#>
#> Residuals:
#> Min 1Q Median 3Q Max
#> -0.0152061 -0.0050872 0.0003464 0.0037709 0.0242904
#>
#> Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) 0.048190 0.012652 3.809 0.000267 ***
#> LNCONS.l1 -0.306508 0.054653 -5.608 2.62e-07 ***
#> LNINCOME.l1 0.296537 0.054579 5.433 5.42e-07 ***
#> LNINVEST.l1 -0.001366 0.011320 -0.121 0.904253
#> D_LNCONS.l1 -0.247543 0.078653 -3.147 0.002289 **
#> D_LNINCOME 0.470633 0.073736 6.383 9.45e-09 ***
#> D_LNINVEST 0.065370 0.019326 3.382 0.001098 **
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Residual standard error: 0.00773 on 83 degrees of freedom
#> Multiple R-squared: 0.5405, Adjusted R-squared: 0.5072
#> F-statistic: 16.27 on 6 and 83 DF, p-value: 2.724e-12
#>
#>
#> BEST LAG ORDER FOR LAGGED DIFFERENCES
#>
#> Series lag_dz
#> 1 LNCONS 1
#> 2 LNINCOME 0
#> 3 LNINVEST 0
The lagged levels of LNCONS and LNINCOME are
significant, hinting a possible cointegrating relationship. The
estimates of the instantaneous differences of LNINCOME and
LNINVEST (i.e., the parameters in \(\boldsymbol\omega'\)) are also
significant. Only the one-lagged difference of LNCONS is
significant.
- The VECM model estimates (displayed also for LNCONS for the
sake of completeness)
summary(BCT_res_CONS, out="VECM")
#> UNCONDITIONAL VECM MODEL
#>
#> VAR Estimation Results:
#> =========================
#> Endogenous variables: D_LNCONS, D_LNINCOME, D_LNINVEST
#> Deterministic variables: const
#> Sample size: 89
#> Log Likelihood: 749.335
#> Roots of the characteristic polynomial:
#> 0.5783 0.449 0.449 0.4405 0.3425 0.3425
#> Call:
#> vars::VAR(y = na.omit(dlag0), p = vecmsel, type = typevecm, exogen = na.omit(lagdata))
#>
#>
#> Estimation results for equation D_LNCONS:
#> =========================================
#> D_LNCONS = D_LNINCOME.l2 + D_LNINVEST.l2 + const + LNCONS.l1 + LNINCOME.l1 + LNINVEST.l1
#>
#> Estimate Std. Error t value Pr(>|t|)
#> D_LNINCOME.l2 0.16351 0.09243 1.769 0.080572 .
#> D_LNINVEST.l2 0.06112 0.02272 2.690 0.008633 **
#> const 0.05221 0.01577 3.311 0.001377 **
#> LNCONS.l1 -0.27421 0.07058 -3.885 0.000205 ***
#> LNINCOME.l1 0.27269 0.06950 3.923 0.000179 ***
#> LNINVEST.l1 -0.01094 0.01337 -0.819 0.415375
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#>
#> Residual standard error: 0.009436 on 83 degrees of freedom
#> Multiple R-Squared: 0.8216, Adjusted R-squared: 0.8087
#> F-statistic: 63.72 on 6 and 83 DF, p-value: < 2.2e-16
#>
#>
#> Estimation results for equation D_LNINCOME:
#> ===========================================
#> D_LNINCOME = D_LNCONS.l1 + D_LNINVEST.l1 + D_LNINVEST.l2 + const + LNINCOME.l1 + LNINVEST.l1
#>
#> Estimate Std. Error t value Pr(>|t|)
#> D_LNCONS.l1 0.21122 0.11311 1.867 0.0654 .
#> D_LNINVEST.l1 0.03549 0.02941 1.207 0.2310
#> D_LNINVEST.l2 0.04951 0.02749 1.801 0.0753 .
#> const 0.03345 0.01655 2.021 0.0465 *
#> LNINCOME.l1 -0.01668 0.01422 -1.173 0.2442
#> LNINVEST.l1 0.01622 0.01652 0.982 0.3291
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#>
#> Residual standard error: 0.01106 on 83 degrees of freedom
#> Multiple R-Squared: 0.7722, Adjusted R-squared: 0.7558
#> F-statistic: 46.9 on 6 and 83 DF, p-value: < 2.2e-16
#>
#>
#> Estimation results for equation D_LNINVEST:
#> ===========================================
#> D_LNINVEST = D_LNCONS.l1 + D_LNINVEST.l1 + D_LNCONS.l2 + const + LNINCOME.l1 + LNINVEST.l1
#>
#> Estimate Std. Error t value Pr(>|t|)
#> D_LNCONS.l1 0.89879 0.44170 2.035 0.0451 *
#> D_LNINVEST.l1 -0.18040 0.11117 -1.623 0.1084
#> D_LNCONS.l2 0.74424 0.43099 1.727 0.0879 .
#> const 0.03621 0.06580 0.550 0.5836
#> LNINCOME.l1 0.12380 0.05390 2.297 0.0241 *
#> LNINVEST.l1 -0.15237 0.06258 -2.435 0.0170 *
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#>
#> Residual standard error: 0.04289 on 83 degrees of freedom
#> Multiple R-Squared: 0.2557, Adjusted R-squared: 0.2019
#> F-statistic: 4.751 on 6 and 83 DF, p-value: 0.0003303
#>
#>
#>
#> Covariance matrix of residuals:
#> D_LNCONS D_LNINCOME D_LNINVEST
#> D_LNCONS 9.354e-05 5.932e-05 1.615e-04
#> D_LNINCOME 5.932e-05 1.286e-04 6.811e-05
#> D_LNINVEST 1.615e-04 6.811e-05 1.933e-03
#>
#> Correlation matrix of residuals:
#> D_LNCONS D_LNINCOME D_LNINVEST
#> D_LNCONS 1.0000 0.5409 0.3799
#> D_LNINCOME 0.5409 1.0000 0.1366
#> D_LNINVEST 0.3799 0.1366 1.0000
- The Johansen test for cointegration on the independent
variables
summary(BCT_res_CONS, out="cointVECM")
#> VECM MODEL JOHANSEN COINTEGRATION TESTS
#>
#> Test type: trace statistic , with linear trend
#>
#> Eigenvalues (lambda):
#> [1] 0.07694993 0.02379754
#>
#>
#> Values of test statistic and critical values of test:
#>
#> test 10pct 5pct 1pct
#> r <= 1 | 2.14 6.50 8.18 11.65
#> r = 0 | 9.27 15.66 17.95 23.52
#>
#>
#> Eigenvectors, normalised to first column:
#> (These are the cointegration relations)
#>
#> LNINCOME.l1 LNINVEST.l1
#> LNINCOME.l1 1.000000 1.000000
#> LNINVEST.l1 -1.155814 1.416763
#>
#>
#> Weights W:
#> (This is the loading matrix)
#>
#> LNINCOME.l1 LNINVEST.l1
#> LNINCOME.d -0.01610551 -0.001368126
#> LNINVEST.d 0.12129032 -0.003256883
#>
#>
#> =======================================
#>
#> Test type: maximal eigenvalue statistic (lambda max) , with linear trend
#>
#> Eigenvalues (lambda):
#> [1] 0.07694993 0.02379754
#>
#>
#> Values of test statistic and critical values of test:
#>
#> test 10pct 5pct 1pct
#> r <= 1 | 2.14 6.50 8.18 11.65
#> r = 0 | 7.13 12.91 14.90 19.19
#>
#>
#> Eigenvectors, normalised to first column:
#> (These are the cointegration relations)
#>
#> LNINCOME.l1 LNINVEST.l1
#> LNINCOME.l1 1.000000 1.000000
#> LNINVEST.l1 -1.155814 1.416763
#>
#>
#> Weights W:
#> (This is the loading matrix)
#>
#> LNINCOME.l1 LNINVEST.l1
#> LNINCOME.d -0.01610551 -0.001368126
#> LNINVEST.d 0.12129032 -0.003256883
We accept the null \(r=0\) for both
the trace and eigenvalue tests, therefore the independent variables are
not cointegrated.
- The ARDL bootstrap and bound tests
summary(BCT_res_CONS, out="cointARDL")
#>
#>
#>
#> Observations: 92
#> Number of Lagged Regressors (not including LDV) (k): 2
#> Case: 3
#> --------------------------------------------------------------------------
#> - PSS Fov-test -
#> --------------------------------------------------------------------------
#> <------- I(0) ----- I(1) ----->
#> 10% critical value 3.170 4.140
#> 5% critical value 3.790 4.850
#> 1% critical value 5.150 6.360
#>
#> F-statistic = 10.751
#>
#> Bootstrap critical values
#> 1 % 5 % 10 %
#> 5.610 3.920 3.160
#>
#>
#> Observations: 92
#> Number of Lagged Regressors (not including LDV) (k): 2
#> Case: 3
#> --------------------------------------------------------------------------
#> - PSS t-test -
#> --------------------------------------------------------------------------
#> <------- I(0) ----- I(1) ----->
#> 10% critical value -2.570 -3.210
#> 5% critical value -2.860 -3.530
#> 1% critical value -3.430 -4.100
#>
#> t-statistic = -5.608
#>
#> Bootstrap critical values
#> 1 % 5 % 10 %
#> -3.580 -2.930 -2.610
#>
#>
#> Observations: 92
#> Number of Lagged Regressors (not including LDV) (k): 2
#> Case: 3
#> -----------------------------------------------------------------------------
#> - SMG FInd-test -
#> -----------------------------------------------------------------------------
#> <------- I(0) ----- I(1) ----->
#> 10% critical value 2.31 4.33
#> 5% critical value 3.01 5.42
#> 2.5% critical value 3.74 6.42
#> 1% critical value 4.71 7.68
#>
#> F-statistic = 15.636
#>
#> Bootstrap critical values
#> 1 % 5 % 10 %
#> 7.760 4.970 4.030
This is the main output of the function. Since case=3 all
the bound tests can be performed. In this example, not only the
statistics exceed the I(1) threshold on the bound tests, but they also
exceed the bootstrap critical values reported at the bottom of each test
result. Cointegration is thus confirmed between LNCONS and the
other variables. No spurious (faux) cointegration is detected.