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Add-in Round Up for 2016 Q2/3

Overview

• All variables are I$(d)$ for some $0\leq d$ and are not cointegrated -- fractional orders of integration are in principle also possible. Here one can use familiar least squares techniques to estimate and interpret equation (\ref{eq.ardl.1}) in levels when $d=0$ and in appropriate differences when $d>0$.
• All variables are I$(1)$ and are cointegrated. Here one can:
  • use least squares to estimate the cointegrating (long-run) relationship by regressing $y_t$ on $x_{j,t}$ for $j=1,\ldots k$ in levels; and/or,
  • use least square to estimate speed of adjustment of short-run dynamics to the cointegrating relationship by regressing the appropriate error-correction model (ECM).
• Some variables are I$(0)$, others are I$(1)$, and amongst the latter, some are cointegrated.

Overview

Vector Auto-regression (VAR) and the Vector Error Correction Model (VECM)

Assumption 1:

Assumption 2:

• The Cointegrating Matrix is $\pmb{\Phi}(1)$: If the original VAR variables in (\ref{eq.ardl.11}), namely $\pmb{z}_t$, are I$(1)$, all variables in the VECM are I$(0)$, except possibly for $\pmb{z}_{t-1}$. Since orders of integration must balance, $\pmb{\Phi}(1)\pmb{z}_{t-1}$ must be I$(0)$. Since a set of I$(1)$ variables is said to be cointegrated if there exists a linear combination of said variables which is I$(0)$, it is clear that $\pmb{\Phi}(1)\pmb{z}_t$ is the matrix of cointegrating relationships and $\pmb{\Phi}(1)$ is the cointegrating matrix. In Economics, the concept is often referred to as a long-run relationship, motivating the example that while prices -- which are frequently I$(1)$ variables -- can drift apart in the short-run, economic forces will eventually force them to equilibrium.


• No Cointegration when $\pmb{\Phi}(1) = \pmb{0}$. Every variable in $\pmb{z}_t$ is I$(1)$: Recall that the rank of a matrix is the number of its linearly independent columns (or rows). The concept is frequently used in ordinary least squares (OLS) regression, and is typically exemplified using the dummy variable trap. In this regard, since $\pmb{\Phi}(1)$ is a $(k+1)$-square matrix, assume $\DeclareMathOperator{\rank}{\textbf{rk}}\rank\left(\pmb{\Phi}(1)\right) = r_z$, where $0 \leq r_z \leq (k+1)$, and $\rank(\cdot)$ denotes the rank operator. In other words, among the $(k+1)$ columns in $\pmb{\Phi}(1)$, only $r_z$ are linearly independent, and the ones which are not, are linear combinations of those $r_z$. Moreover, $r_z = 0$ if and only if $\pmb{\Phi}(1) = \pmb{0}_{(1+k)^2}$, where $\pmb{0}_{(1+k)^2}$ denotes the $(1+k)$-square matrix of zeros. When this is the case, the VECM reduces to: $$\Delta\pmb{z}_t = \pmb{a}_0 + \pmb{a}_1t + \widetilde{\pmb{\Phi}}^{\star}(L)\Delta\pmb{z}_t + \pmb{\epsilon}_t$$ Since all variables on the right-hand side (RHS) are I$(0)$, it follows that $\Delta\pmb{z}_t \sim \text{I}(0)$, and therefore $\pmb{z}_t \sim \text{I}(1)$. In other words, when $r_z = 0$, every variable in $\pmb{z}_t$ is I$(1)$, and since $\pmb{\Phi}(1) = \pmb{0}_{(1+k)^2}$, there are no cointegrating relationships.


• No Cointegration when $\pmb{\Phi}(1)$ has full rank. Every variable in $\pmb{z}_t$ is I$(0)$: When $r_z = (k+1)$, $\pmb{\Phi}(1)$ has full column rank (i.e. all columns (rows) are linearly independent). In this particular case, $\DeclareMathOperator{\spann}{\textbf{sp}} \pmb{\Phi}(1)\pmb{z}_{t-1} = \spann{\left(\pmb{z}_{t-1}\right)}$, where $\spann(\cdot)$ denotes the span -- the space of all unique linear combinations of $\pmb{z}_t$. This implies $\Delta \pmb{z}_t$ can be uniquely written as a linear combination of all variables in $\pmb{z}_t$, namely $\pmb{\Phi}(1)\pmb{z}_{t-1}$, plus the remaining deterministic and stationary ones. Since $\Delta \pmb{z}_t \sim \text{I}(0)$, this is only sensible when every variable in $\pmb{z}_t \sim \text{I}(0)$, and cointegration is not possible.


• VECM Estimates Speed of Convergence to Equilibrium: A classical result in linear algebra is that for any $m\times m$ matrix $\pmb{M}$ with rank $r$, there exist $m \times r$ matrices $\pmb{A}$ and $\pmb{B}$ such that $\pmb{M} = \pmb{AB}^\top$, where $\pmb{B}$ consists of the $r$ linearly independent columns of $\pmb{M}$. Thus, we can always write $\pmb{\Phi}(1) = \pmb{AB}^\top$, where $m=(1+k)$. More importantly, it implies that $\pmb{A}$ measures the rate of convergence to equilibrium. To see this, recall that if $\pmb{z}_t$ is cointegrated, then $\pmb{\Phi}(1)\pmb{z}_{t-1} \sim \text{I}(0)$. We can therefore factorize the cointegrated relationships as $\pmb{\Phi}(1)\pmb{z}_{t-1} = \pmb{A}\pmb{B}^\top \pmb{z}_{t-1} = \pmb{A}\pmb{\zeta}_{t-1}$ where $\pmb{\zeta}_{t-1}$ is a mean zero I$(0)$ process. This is because the cointegrating relationships are now captured by $\pmb{B}^\top \pmb{z}_{t-1}$. Observe further that when the system is in actual equilibrium, $\pmb{B}^\top \pmb{z}_{t-1} = \pmb{0}_{1+k}$, where $\pmb{0}_{1+k}$ is $(1+k)$-vector of zeros. This is because equilibrium requires not only stability, which follows from the stationarity of $\pmb{B}^\top \pmb{z}_{t-1}$, but also constancy, which manifests only when accumulated short-run dynamics $\widetilde{\pmb{\Phi}}^{\star}(L)\Delta\pmb{z}_t$, and the shocks to $\pmb{B}^\top \pmb{z}_{t-1}$, namely, $\pmb{\zeta}_{t-1}$, are zero as well. Accordingly, if the system was in equilibrium in the previous period, any current deviations from this state, namely $\Delta \pmb{z}_t$, must arise from systematic shocks $\pmb{\epsilon}_t$, where we assume $\pmb{a}_0 = \pmb{a}_1 = \pmb{0}_{1+k}$ for simplicity. Alternatively, when the system is in disequilibrium, $\pmb{B}^\top \pmb{z}_{t-1} = \pmb{\zeta}_{t-1} \neq \pmb{0}_{1+k}$. Thus, when $\pmb{B}^\top \pmb{z}_{t-1} < \pmb{0}_{1+k}$ $(\pmb{B}^\top \pmb{z}_{t-1} > \pmb{0}_{1+k})$, the impact on $\Delta\pmb{z}_t$ is of magnitude $\pmb{A}$ and positive (negative), since $\pmb{\Phi}(1)$ enters the VECM with a negative sign. In other words, $\Delta\pmb{z}_t$ adjusts toward equilibrium in the opposite direction to disequilibrium by a proportion equal to $\pmb{A}$.


• Cointegrating Relationships Include Constants and Trends: We have outlined this in Part 1 of this series. The restrictions in (\ref{eq.ardl.13}) indicate that $\pmb{a}_0$ and $\pmb{a}_1$ are linear functions of $\pmb{\Phi}(1)$. As such, they span the $r_z$ linearly independent columns of the cointegrating matrix $\pmb{\Phi}(1)$, and by extension, the cointegrating equation. This distinguishes the 5 data generating processes (DGPs) considered in Pesaran, Shin, and Smith (2001) and outlined in Part 1 of this series.


• Case I: $\pmb{\mu} = \pmb{\gamma} = \pmb{0}$ which implies $\pmb{a}_0 = \pmb{a}_1 = 0$. Accordingly, the VECM (\ref{eq.ardl.12}) reduces to: $$\Delta \pmb{z}_t = -\pmb{\Phi}(1)\pmb{z}_{t-1} + \widetilde{\pmb{\Phi}}^{\star}(L)\Delta\pmb{z}_t + \pmb{\epsilon}_t$$
• Case II: $\pmb{\mu} \neq \pmb{0}$, $\pmb{\gamma} = \pmb{0}$, and the restriction in (\ref{eq.ardl.13}) is imposed. This implies that $\pmb{a}_0 = \pmb{\Phi}(1)\pmb{\mu}$ and $\pmb{a}_1 = 0$. Accordingly, the VECM is just: $$\Delta \pmb{z}_t = -\pmb{\Phi}(1)\left(\pmb{z}_{t-1} - \pmb{\mu}\right) + \widetilde{\pmb{\Phi}}^{\star}(L)\Delta\pmb{z}_t + \pmb{\epsilon}_t$$
• Case III: $\pmb{\mu} \neq \pmb{0}$, $\pmb{\gamma} = \pmb{0}$, and the restrictions in (\ref{eq.ardl.13}) not imposed. This implies that $\pmb{a}_0 \neq 0$, $\pmb{a}_1 = 0$, while the VECM becomes: $$\Delta \pmb{z}_t = \pmb{a}_0 -\pmb{\Phi}(1)\pmb{z}_{t-1} + \widetilde{\pmb{\Phi}}^{\star}(L)\Delta\pmb{z}_t + \pmb{\epsilon}_t$$
• Case IV: $\pmb{\mu},\pmb{\gamma} \neq \pmb{0}$, and the restrictions in (\ref{eq.ardl.13}) are imposed only on $\pmb{a}_1$. This implies that $\pmb{a}_0 \neq 0$ and $\pmb{a}_1 = \pmb{\Phi}(1)\pmb{\gamma}$. The VECM is now: $$\Delta \pmb{z}_t = \pmb{a}_0 -\pmb{\Phi}(1)\left(\pmb{z}_{t-1} - \pmb{\gamma}t\right) + \widetilde{\pmb{\Phi}}^{\star}(L)\Delta\pmb{z}_t + \pmb{\epsilon}_t$$
• Case V: $\pmb{\mu},\pmb{\gamma} \neq \pmb{0}$, and the restrictions in (\ref{eq.ardl.13}) are not imposed. This implies that $\pmb{a}_0,\pmb{a}_1 \neq 0$ and the VECM is represented in (\ref{eq.ardl.12}).

• Derive an ECM for $y_t$, explicitly conditioning on all effects originating from $\pmb{x}_t$. Such a model must include not only explicit effects of $\pmb{x}_t$ on $y_t$ stemming from the VAR matrix polynomial $\pmb{\Phi}(L)$, but also any and all contemporaneous relationships between $y_t$ and $\pmb{x}_t$ implicit within the covariance matrix $\pmb{\Omega}$ of the error vector $\pmb{\epsilon}_t$.


• Ensure that $\pmb{x}_t$ are weakly exogenous.

Conditional ECM (CECM)

Assumption 3:

Relationship to ARDL

Inference

Assumption 4:

Analysis of the Null Hypotheses

• $\tau_F < \xi_{L,F} < \xi_{U,F}$: Here we fail to reject $H_{0,F}$ when $\pmb{x}_t$ is either I$(0)$ or I$(1)$. We are therefore assured that no cointegrating relationship between $y_t$ and $\pmb{x}_t$ exists.


• $\xi_{L,F} < \tau_F < \xi_{U,F}$: Here, $\xi_{L,F} < \tau_F$. Accordingly, we reject $H_{0,F}$ when $\pmb{x}_t \sim \text{I}(0)$. Nevertheless, since $\tau_F < \xi_{U,F}$, we fail to reject $H_{0,F}$ when $\pmb{x}_t \sim \text{I}(1)$. This indicates that cointegrating relationships between $y_t$ and $\pmb{x}_t$ may or may not exist for cases where $0 < r_x < k$. Accordingly, we cannot make any specific conclusions unless we know the rank of the system-wide cointegrating matrix (\ref{eq.ardl.19}).


• $\xi_{L,F} < \xi_{U,F} < \tau_F$: Here we reject $H_{0,F}$ when $\pmb{x}_t$ is either I$(0)$ or I$(1)$. Since $r_x = 0$ in this case, we know $\pmb{\Phi}_{xx}(1) = 0$. Moreover, since the maximal rank of the cointegrating matrix (\ref{eq.ardl.19}) is $r_z = 1 + r_x$, from the Abadir and Magnus (2005) result above, the remaining unity rank can arise from one of three possibilities:
  • $\phi_{yy} = 0$ and $\pmb{\phi}_{yx}(1) \neq \pmb{0}_k^\top$ in which case the equilibrating relationship between $y_t$ and $\pmb{x}_t$ is entirely nonsensical. In fact, looking at (\ref{eq.ardl.20}), it is undefined.
  
  
  • $\phi_{yy} \neq 0$ and $\pmb{\phi}_{yx}(1) = \pmb{0}_k^\top$, in which case the equilibrating relationship is defined but degenerate.
  
  
  • $\phi_{yy} \neq 0$ and $\pmb{\phi}_{yx}(1) \neq \pmb{0}_k^\top$ in which case the equilibrating relationship is well defined.
  
  
  This suggests an additional test for $\phi_{yy} = 0$ to exclude possibility (a) above. We discuss this in greater detail in the analysis of the alternative hypothesis below.

Analysis of the Alternative Hypotheses

• $H_{A_1,F}: \quad \phi_{yy}(1) = 0$ and $\pmb{\phi}_{yx}(1) - \pmb{\omega}_{yx}\pmb{\Omega}^{-1}_{xx}\pmb{\Phi}_{xx}(1) \neq \pmb{0}_k^\top$.
  
  Here, the result from Abadir and Magnus (2005) assures us that the cointegrating matrix (\ref{eq.ardl.19}) has rank $r_z = 1 + r_x$, and the CECM reduces to: $$ \Delta y_t = \alpha_{y0} + \alpha_{y1}t - \left(\pmb{\phi}_{yx}(1) - \pmb{\omega}_{yx}\pmb{\Omega}^{-1}_{xx}\pmb{\Phi}_{xx}(1)\right)\pmb{x}_{t-1} + \pmb{e}_1^\top\left((\pmb{I}_{k+1} - \pmb{\Psi})\widetilde{\pmb{\Phi}}^\star(L) + \pmb{\Psi}\right)\Delta\pmb{z}_t + u_{yt} $$ The cointegrating relationship (\ref{eq.ardl.20}) is here undefined since $\phi_{yy} = 0$. Moreover, since $\pmb{\Phi}_{xx}(1)$ is the only cointegrating matrix for $\pmb{x}_t$, it holds that $\pmb{\Phi}_{xx}(1)\pmb{x}_{t-1} \sim \text{I}(0)$, and therefore all RHS variables are I$(0)$ except possibly $\pmb{\phi}_{yx}(1)\pmb{x}_{t-1}$. However, since $\phi_{yy} = 0$, $\pmb{\phi}_{yx}(1)$ is not a cointegrating matrix for $\pmb{x}_t$ and therefore $\pmb{\phi}_{yx}(1)\pmb{x}_{t-1}$ may be I$(0)$ or I$(1)$. Either way, $y_t \sim \text{I}(1)$ regardless of the cointegrating rank $r_x$.


• $H_{A_2,F}: \quad \phi_{yy}(1) \neq 0$ and $\pmb{\phi}_{yx}(1) - \pmb{\omega}_{yx}\pmb{\Omega}^{-1}_{xx}\pmb{\Phi}_{xx}(1) = \pmb{0}_k^\top$
  
  In this case, the CECM assumes the form: $$ \Delta y_t = \alpha_{y0} + \alpha_{y1}t - \phi_{yy}(1)y_{t-1} + \left(\widetilde{\phi}^\star_{yy}(L) -\pmb{\omega}_{yx}\pmb{\Omega}^{-1}_{xx}\widetilde{\pmb{\phi}}^\star_{xy}(L)\right)\Delta y_t +\pmb{e}_1^\top\left((\pmb{I}_{k+1} - \pmb{\Psi})\widetilde{\pmb{\Phi}}^\star(L) + \pmb{\Psi}\right)\Delta\pmb{z}_t + u_{yt} $$ In fact, the equation is a special case of the Augmented Dickey-Fuller (ADF) regression. By Assumption 1, when $\pmb{\Phi}(1) = 0$, and therefore $\phi_{yy}(1) = 0$, the vector $\pmb{z}_t$, and therefore $y_t$, has a unit root. Under the alternative, however, $\phi_{yy}(1) \neq 0$ and we know that either $y_t \sim \text{I}(0)$ whenever $\alpha_{y1} = 0$, or $y_t$ is trend stationary should $\alpha_{y1} \neq 0$. Again, this holds regardless of the cointegrating rank $r_x$. Moreover, the result from Abadir and Magnus (2005) ensures that the cointegrating matrix (\ref{eq.ardl.19}) has rank $r_z = 1 + r_x$. It is important to note here that while the idea of a cointegrating relationship between $y_t$ and $\pmb{x}_t$ is not possible, there exists a relationship between $y_t$ and $\pmb{x}_t$ originating from the short-run dynamics manifesting through $\Delta \pmb{x}_t$. Since this is not an equilibrating relationship originating from $\pmb{x}_{t-1}$, the relationship is degenerate in equilibrium.


• $H_{A_3,F}: \quad \phi_{yy}(1) \neq 0$ and $\pmb{\phi}_{yx}(1) - \pmb{\omega}_{yx}\pmb{\Omega}^{-1}_{xx}\pmb{\Phi}_{xx}(1) \neq \pmb{0}_k^\top$
  
  Here, the result from Abadir and Magnus (2005) guarantees that $r_z = 1 + r_x$. Moreover, Abadir and Magnus (2005) ensures us there exist $(k+1\times r)$-matrices $\pmb{A}$ and $\pmb{B}$ such that one can write $\pmb{\phi}(1)$ in rank factorization as follows: \begin{align*} \begin{bmatrix} \phi_{yy}(1) & \pmb{\phi}_{yx}(1)\\ \pmb{0}_k & \pmb{\Phi}_{xx}(1) \end{bmatrix} &= \begin{bmatrix} A_{yy}\\ \pmb{0}_k \end{bmatrix} \begin{bmatrix} B_{yy} & \pmb{B}^\top_{yx} \end{bmatrix} + \begin{bmatrix} \pmb{A}_{yx}\\ \pmb{A}_{xx} \end{bmatrix} \begin{bmatrix} \pmb{0}_k & \pmb{B}^\top_{xx} \end{bmatrix}\\ &= \begin{bmatrix} A_{yy}B_{yy} & A_{yy}\pmb{B}^\top_{yx} + \pmb{A}_{yx}\pmb{B}^\top_{xx}\\ \pmb{0}_k & \pmb{A}_{xx}\pmb{B}^\top_{xx} \end{bmatrix} \end{align*} Thus, $\pmb{\phi}_{xy}(1) = A_{yy}\pmb{B}^\top_{yx} + \pmb{A}_{yx}\pmb{B}^\top_{xx}$, where $\pmb{B}_{xx}^\top$ comprises the cointegrating matrix underlying $\pmb{\Phi}_{xx}(1) = \pmb{A}_{xx}\pmb{B}^\top_{xx}$ of $\pmb{x}_t$, irrespective of $y_t$. Accordingly, any equilibrating link between $y_t$ and $\pmb{x}_t$ is due to the cointegrating matrix $\pmb{B}^\top_{yx}$. Accordingly, we have two possibilities.


• $\rank(\pmb{B}^\top_{yx},\pmb{B}^\top_{xx}) = r_x$. In this case, the cointegrating vector $\pmb{B}^\top_{yx}$ is subsumed by $\pmb{B}^\top_{xx}$ since $\rank(\pmb{\Phi}_{xx}(1)) = \rank(\pmb{B}_{xx}) = r_x$. Thus, the equilibrating relationship between $y_t$ and $\pmb{x}_t$ is not due to traditional cointegration, but is valid nonetheless. Here, $y_t \sim \text{I}(0)$ since $\phi_{yy}(1) \neq 0$.


• $\rank(\pmb{B}^\top_{yx},\pmb{B}^\top_{xx}) = 1 + r_x$. In this case, the cointegrating vector $\pmb{B}^\top_{yx}$ is not redundant, and drives the cointegrating link between $y_t$ and $\pmb{x}_t$. The equilibrating relationship is now of the traditional cointegration type, and therefore $y_t \sim \text{I}(1)$.


In either case, it is readily shown that the relationships which emerge are non-degenerate.

• $|\tau_t| < |\xi_{L,t}| < |\xi_{U,t}|$: As before, $\pmb{x}_t$ is either I$(0)$ or I$(1)$. Moreover, since $\tau_t < \xi_{L,t}$, we fail to reject $H_{0,t}$. Since we have already rejected $H_{0,F}$, this implies $\phi_{yy}(1) = 0$ and $\pmb{\phi}_{yx}(1) - \pmb{\omega}_{yx}\pmb{\Omega}^{-1}_{xx}\pmb{\Phi}_{xx}(1) \neq \pmb{0}_k^\top$. We conclude therefore that we $H_{A,F}$ manifests as $H_{A_1,F}$ and a nonsensical equilibrating relationship between $y_t$ and $\pmb{x}_t$ emerges.


• $|\xi_{L,t}| < |\tau_t| < |\xi_{U,t}|$: Here we reject $H_{0,t}$ when $\pmb{x}_t \sim \text{I}(0)$ but fail to do so for the case where $\pmb{x}_t \sim \text{I}(1)$ and $0 < r_x < k$. Thus, examples may emerge where the $\pmb{x}_t$ are mutually cointegrated for which we may or may not reject $H_{0,t}$. Unless we know the rank of the cointegrating matrix (\ref{eq.ardl.19}), little more can be inferred.


• $|\xi_{L,t}| < |\xi_{U,t}| < |\tau_t|$: In this case, we reject $H_{0,t}$ when $\pmb{x}_t$ is either I$(0)$ or I$(1)$, implying $\phi_{yy}(1) \neq 0$. Accordingly, unless we know that $\pmb{\phi}_{yx}(1) - \pmb{\omega}_{yx}\pmb{\Omega}^{-1}_{xx}\pmb{\Phi}_{xx}(1) = \pmb{0}_{k}^\top$, we must conclude that $H_{A,F}$ manifests either as $H_{A_2,F}$, or $H_{A_3,F}$. In either case, an equilibrating relationship emerges, albeit degenerate in case of $H_{A_2,F}$.