The bootCT package
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
A(L)(zt − μ − ηt) = εt, εt ∼ NK + 1(0, Σ), t = 1, 2, …, T
$$\mathbf{A}(L)=\left(\mathbf{I}_{K+1}- \sum_{j=1}^{p}\mathbf{A}_j\mathbf{L}^j\right) $$ Here, Aj are square (K + 1) matrices, zt a vector of (K + 1) variables, μ and η are (K + 1) vectors representing the drift and the trend respectively, and det(A(z)) = 0 for |z| ≥ 1. If the matrix $\mathbf{A}(1)=\mathbf{I}_{K+1}-\sum_{j=1}^{p}\mathbf{A}_{j}$ is singular, the components of zt 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 A (suppressing the bracket term for simplicity’s sake) contains the long-run cointegrating relationships among the variables, while the Γj’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
α0 = Aμ + (Γ(1) − A)η, α1 = Aη It is important to highlight now the five possible cases in the intercept and trend specifications:
- Case I: no intercept, no trend, for which μ = η = 0
- Case II: restricted intercept, no trend, for which α0 = Aμ, η = 0
- Case III: unrestricted intercept, no trend, for which α0 ≠ Aμ, η = 0
- Case IV: unrestricted intercept, restricted trend, for which α0 ≠ Aμ, α1 = Aη
- Case V: unrestricted intercept, unrestricted trend, for which α0 ≠ Aμ, α1 ≠ Aη.
We now introduce the key concept of conditioning in the VECM system. To study the adjustment to the equilibrium of a single variable yt, given the other xt variables, the vectors zt and εt 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 α0, α1, the matrix A and the polynomial matrix Γ(L) are partitioned conformably to zt
$$\boldsymbol{\alpha}_0=\begin{bmatrix} \underset{(1,1)}{\alpha_{0y}} \\ \underset{(K,1)}{\boldsymbol{\alpha}_{0x}} \end{bmatrix}, \enspace \enspace \enspace \boldsymbol{\alpha}_1=\begin{bmatrix} \underset{(1,1)}{\alpha_{1y}} \\ \underset{(K,1)}{\boldsymbol{\alpha}_{1x} } \end{bmatrix}$$
$$\mathbf{A}=\begin{bmatrix} \underset{(1,K+1)}{\mathbf{a}^{'}_{(y)}} \\ \underset{(K,K+1)}{\mathbf{A}_{(x)}} \end{bmatrix} =\begin{bmatrix} \underset{(1,1)}{a_{yy}} & \underset{(1,K)}{\mathbf{a}^{'}_{yx}} \\ \underset{(K,1)}{\mathbf{a}_{xy}} & \underset{(K,K)}{\mathbf{A}_{xx} } \end{bmatrix}, \enspace \enspace \enspace \boldsymbol{\Gamma}(L)=\begin{bmatrix} \underset{(1,K+1)}{\boldsymbol{\gamma}^{'}_{y}(L)} \\ \underset{(K,K+1)}{\boldsymbol{\Gamma}_{(x)}(L)} \end{bmatrix} =\begin{bmatrix} \underset{(1,1)}{\gamma_{yy}(L)} & \underset{(1,K)}{\boldsymbol{\gamma}^{'}_{yx}(L)} \\ \underset{(K,1)}{\boldsymbol{\gamma}_{xy}(L)} & \underset{(K,K)}{\boldsymbol{\Gamma}_{xx}(L) } \end{bmatrix} $$ 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 yt on xt, we define the conditional variance σy.x = σyy − σyx′Σxx−1σxy = σyy − ω′σxy And thus εyt = ω′εxt + νyt ∼ N(0, σy.x) where νyt is independent of εxt. Accordingly, conditioning can be applied to the entire system, obtaining
a(y).x′ = a(y)′ − ω′A(x), γy.x′(L) = γy′(L) − ω′Γ(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 yt and xt 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 xt variables Δxt = α0x + α1xt + A(x)zt − 1 + Γ(x)(L)Δzt + εxt while the equation in the conditional ARDL specification for yt 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, ECt − 1, expressing the long-run equilibrium relationship between yt and xt, is given by ECt − 1 = yt − 1 − θ0 − θ1t − θ′xt − 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 θ0 = θ1 = α0.y = α1.y = 0
- Case II: restricted intercept, no trend. Since α0 = Aμ, θ0 ≠ 0 and the intercept term appears in ECt − 1. Therefore, α0.y = 0. In addition, θ1 = α1.y = 0
- Case III: unrestricted intercept, no trend, Since α0 ≠ Aμ, θ0 = 0 and α0.y = α0y − ω′α0x ≠ 0. In addition, θ1 = α1.y = 0
- Case IV: unrestricted intercept, restricted trend, for which α0.y = α0y − ω′α0x ≠ 0.
Since
α1 = Aη, θ1 ≠ 0 and the trend term appear in ECt − 1. - Case V: unrestricted intercept, unrestricted trend. Since α0 ≠ Aμ, θ0 = 0 and α0.y = α0y − ω′α0x ≠ 0. In addition, α1 ≠ Aη, θ1 = 0 and α1.y = α1y − ω′α1x ≠ 0
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 Σ
- The short-run parameters, Γ1, Γ2
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, ayy, ayx, Axx that will eventually form the matrix $\widetilde{\mathbf A}$
- The intercept and trend terms (μ, η, α0 and α1), which depend on the chosen case. For instance, in Case II:
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., α0.y, α1.y, θ0, θ1). 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 6Notice 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:
Bootstrapping the ARDL test for cointegration
The usual test based on the conditional ARDL model examines the following null hypotheses:
$$ H_0^{F_{ov}}: a_{yy}=0\enspace\cap \enspace\widetilde{\mathbf a}_{y.x}=\mathbf 0 $$ $$ H_0^{F_{ind}}: \widetilde{\mathbf a}_{y.x}=\mathbf 0\qquad \text{(Degeneracy of first type)} $$
H0t : ayy = 0 (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 H0Find is rejected, it might be that ay.x = 0, while ω′Axx ≠ 0. In that case, no cointegration is really in place.
For this reason, the Find 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 (Δxt), along with their coefficient ω′. Its null hypothesis is
H0, UCFind : ayx = 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 Fov and t, while Goh et al. (2017) derived the bound test for Find. However:
- The t and Find 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: Fov, t, Find and Find, 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 cα, 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 ayy, $\widetilde{\mathbf a}_{y.x}$, or both). Thus, the residuals of the ARDL and VECM models are calculated. Naturally, for the Find, UC model, the restriction applies to ayx.
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 Fov(b), t(b) and Find(b) is calculated and stored, for b = 1, …, B. Find, UC(b) 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 Find, 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 Fov, t, Find and Find, 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 0The 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 ω′) 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 -0.003256883We 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 4.030This 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.