AutoRegressive Distributed Lag (ARDL) Estimation. Part 1 - Theory

One of our favorite bloggers, Dave Giles often writes about current trends in econometric theory and practice. One of his most popular topics is ARDL modeling, and he has a number of fantastic posts about it.

Since we have recently updated ARDL estimation in EViews 9.5, and are in the midst of adding some enhanced features to ARDL for the next version of EViews, EViews 10, we thought we would jot down our own thoughts on the theory and practice of ARDL models, particularly in regard to their use as a cointegration test.

This blog post will be in three parts. The first will discuss the theory behind ARDL models, the second will present the theory behind correct inference of the Bounds test, while the third will bring everything together with an example in EViews.

Overview

ARDL models are linear time series models in which both the dependent and independent variables are related not only contemporaneously, but across historical (lagged) values as well. In particular, if $y_t$ is the dependent variable and $x_1, \ldots, x_k$ are $k$ explanatory variables, a general ARDL$(p,q_1,\ldots,q_k)$ model is given by: \begin{align} y_t = a_0 + a_1t + \sum_{i=1}^p{\psi_i y_{t-i}} + \sum_{j=1}^{k}\sum_{l_j=0}^{q_j}{\beta_{j,l_j}x_{j,t-l_j}} + \epsilon_t \label{eq.ardl.1} \end{align} where $\epsilon_t$ are the usual innovations, $a_0$ is a constant term, and $a_1, \psi_i,$ and $\beta_{j,l_j}$ are respectively the coefficients associated with a linear trend, lags of $y_t$, and lags of the $k$ regressors $x_{j,t}$ for $j=1,\ldots k$. Alternatively, let $L$ denote the usual lag operator and define $\psi(L)$ and $\beta_j(L)$ as the lag polynomials: $$\psi(L) = 1 - \sum_{i=1}^{p}{\psi_iL^i} \quad \text{and} \quad \beta_j(L) = \sum_{l_j=0}^{q_j}{\beta_{j,l_j}L^{l_j}}$$ Then, equation (\ref{eq.ardl.1}) above can also be written as: \begin{align} \psi(L)y_t = a_0 + a_1t + \sum_{j=1}^{k}{\beta_{j}(L)x_{j,t}} + \epsilon_t \label{eq.ardl.1.lag} \end{align} Although ARDL models have been used in econometrics for decades, they have gained popularity in recent years as a method of examining cointegrating relationships. Two seminal contributions in this regard are Pesaran and Shin (1998, PS(1998)) and Pesaran, Shin and Smith (2001, PSS(2001)). In particular, they argue that ARDL models are especially advantageous in their ability to handle cointegration with inherent robustness to misspecification of integration orders of relevant variables. In this regard, we have three cases of interest:

• 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.

It is precisely in this last case where traditional cointegration methodologies of Engle-Granger (1987), Phillips and Ouliaris (1990) or Johansen (1995), typically fail since all variables need to have identical orders of integration, usually I$(1)$. This requires pre-testing for the presence of a unit root in each of the variables under consideration, which is clearly subject to misclassification, particularly since unit root tests are known to suffer size and power problems in many cases of interest; see Perron and Ng (1996).

Alternatively, the PSS(2001) bounds test for cointegration is not subject to such limitations and readily accommodates the nuances of the third case. The test is in fact a parameter significance test on the long-run variables in the ECM of the underlying vector autoregression (VAR) model, and works when all or some variables are I$(0)$, I$(1)$, or even mutually cointegrated. Since there exists a one-to-one correspondence between an ECM of a VAR model and an ARDL model (see Banerjee et. al., 1993), and since ARDL models are estimated and interpreted using familiar least squares techniques, ARDL models are de facto the standard of estimation when one chooses to remain agnostic about the orders of integration of the underlying variables. It is precisely in this regard where the ARDL methodology shines.

Specification

Although the general ARDL model is specified in (\ref{eq.ardl.1}), there exist three alternative representations. While all three can be used for parameter estimation, the first is typically used for intertemporal dynamic estimation, the second for post-estimation derivation of the long-run (equilibrium) relationship, whereas the third is a reduction of equation (\ref{eq.ardl.1}) to the conditional error correction (CEC) representation in the PSS(2001) bounds test; see Banerjee et. al. (1993).

All three representations require some preliminary results. Using principles underlying the famous Beveridge-Nelson decomposition, recall that $\psi(L)$ and $\beta_j(L)$ can always be decomposed as: \begin{align*} \psi(L) = \psi(1) + (1-L)\widetilde{\psi}(L) \quad \text{and} \quad \beta_j(L) = \beta_j(1) + (1-L)\widetilde{\beta}_j(L) \end{align*} where \begin{align*} \widetilde{\psi}(L) = \sum_{i=0}^{p-1}{\widetilde{\psi}_{i}L^{i}} &\quad \text{and} \quad \widetilde{\psi}_i = -\sum_{r=i+1}^{p}\psi_r\\ \widetilde{\beta}_j(L) = \sum_{l_j=0}^{q_j-1}{\widetilde{\beta}_{j,l_j}L^{l_j}} &\quad \text{and} \quad \widetilde{\beta}_{j,l_j} = -\sum_{s=l_j+1}^{q_j}\beta_{j,s} \end{align*} and $$\psi(1) = 1 - \sum_{i=1}^{p}{\psi_i} \quad \text{and} \quad \beta_j(1) = \sum_{l_j=0}^{q_j}{\beta_{j,l_j}}$$ Next, note that $\psi(L) = 1 - \psi^\star(L)$ where $\psi^\star(L) = \sum_{i=1}^{p}{\psi_iL^i}$. Furthermore, observe that: $$ \psi^\star(L) = \sum_{i=1}^{p}{\psi_iL^i} = \left(\sum_{i=1}^{p}{\psi_iL^{i-1}}\right)L = \left(\psi^\star(1) + (1-L)\widetilde{\psi^\star}(L)\right)L $$ where $\widetilde{\psi^\star}(L) = \sum_{i=1}^{p-1}{\widetilde{\psi^\star}_{i}L^{i-1}}$, $\widetilde{\psi^\star}_i = -\sum_{r=i+1}^{p}\psi_r$, and $\psi^\star(1) = \sum_{i=1}^{p}{\psi_i}$. Finally, note that for any series $z_t$ one can always write: $$z_t = z_{t-1} + \Delta z_t$$

First Representation: (Intertemporal Dynamics Regression)

The typical starting point for most ARDL applications is the estimation of intertemporal dynamics. In this form, one is interested in estimating the relationship between $y_t$ on both its own lags as well as the contemporaneous and lagged values of the $k$ regressors $x_{j,t}$. This in fact the basis of the ARDL model studied in PS(1998). In particular, we cast equation (\ref{eq.ardl.1}) into the following representation: \begin{align} y_t &= a_0 + a_1t + \sum_{i=1}^p{\psi_i y_{t-i}} + \sum_{j=1}^{k}{\beta_j(L)x_{j,t}} + \epsilon_t \notag\\ &= a_0 + a_1t + \sum_{i=1}^p{\psi_i y_{t-i}} + \sum_{j=1}^{k}{\left(\beta_j(1) + (1-L)\widetilde{\beta}_j(L)\right)x_{j,t}} + \epsilon_t \notag\\ &= a_0 + a_1t + \sum_{i=1}^p{\psi_i y_{t-i}} + \sum_{j=1}^{k}{\beta_j(1)x_{j,t}} + \sum_{j=1}^{k}{\widetilde{\beta}_j(L)\Delta x_{j,t}} + \epsilon_t \label{eq.ardl.2} \end{align} where we use the first difference notation $\Delta = (1-L)$. Since equation (\ref{eq.ardl.2}) does not explicitly solve for $y_t$, it is typically interpreted as a regression for intertemporal dynamics. Of course, the model above uses theoretical coefficients whereas in a practical regression setting, it would be represented as: \begin{align} y_t &= a_0 + a_1t + \sum_{i=1}^p{b_{0,i} y_{t-i}} + \sum_{j=1}^{k}{b_{j}x_{j,t}} + \sum_{j=1}^{k}{\sum_{l_j=1}^{q_j-1}c_{j,l_j}\Delta x_{j,t-l_j}} + \epsilon_t \label{eq.ardl.2.reg} \end{align}

Second Representation: (Post-Regression Derivation of Long-Run Dynamics)

The second representation is in essence an attempt to derive the long-run relationship between $y_t$ and the $k$ regressors. As such, the representation solves for $y_t$ in terms of $x_{j,t}$. In particular, starting from equation (\ref{eq.ardl.1}), note that: \begin{align*} \psi(L)\Delta y_t &= (1-L)\psi(L)y_t\\ &= (1-L) \left(y_t - \sum_{i=1}^p{\psi_i y_{t-i}}\right)\\ &= (1-L)\left(a_0 + a_1t + \sum_{j=1}^{k}{\beta_j(L)x_{j,t}} + \epsilon_t\right)\\ &= a_1 + \sum_{j=1}^{k}{\beta_j(L)\Delta x_{j,t}} + \Delta\epsilon_t \end{align*} Next, assuming $\psi(L)$ is in fact invertible, that is, the roots of the characteristic polynomial $1 - \sum_{i=1}^{p}{\psi_iz^i} = 0$ all fall outside the unit circle and a stable relationship between $y_t$ and $x_{1,t}, \ldots, x_{k,t}$ does indeed exist, it holds that: $$\Delta y_t = \psi^{-1}(L) \left(a_1 + \sum_{j=1}^{k}{\beta_j(L)\Delta x_{j,t}} + \Delta\epsilon_t\right)$$ Furthermore, noting that $\psi(L)y_t = \psi(1)y_t + \widetilde{\psi}(L)\Delta y_t$, rewrite equation (\ref{eq.ardl.1}) as follows: \begin{align*} \psi(1)y_t &= \psi(L)y_t - \widetilde{\psi}(L)\Delta y_t\\ \psi(1)y_t &= a_0 + a_1t + \sum_{j=1}^{k}{\beta_j(1)x_{j,t}} + \sum_{j=1}^{k}{\widetilde{\beta}_j(L)\Delta x_{j,t}} - \widetilde{\psi}(L)\Delta y_t + \epsilon_t\\ &= a_0 + a_1t + \sum_{j=1}^{k}{\beta_j(1)x_{j,t}} + \sum_{j=1}^{k}{\widetilde{\beta}_j(L)\Delta x_{j,t}} - \widetilde{\psi}(L)\psi^{-1}(L)\left( a_1 + \sum_{j=1}^{k}{\beta_j(L)\Delta x_{j,t}} + \Delta\epsilon_t \right) + \epsilon_t\\ &= a_0^\star + a_1t + \sum_{j=1}^{k}{\beta_j(1)x_{j,t}} + \sum_{j=1}^{k}{\widetilde{\beta}^\star_j(L)\Delta x_{j,t}} + \epsilon_t^\star \end{align*} where \begin{align*} a_0^\star &= a_0 - \widetilde{\psi}(L)\psi^{-1}(L)a_1\\ &= a_0 - \widetilde{\psi}(1)\psi^{-1}(1)a_1\\ \widetilde{\beta}^\star_j(L) &= \widetilde{\beta}_j(L) - \widetilde{\psi}(L)\psi(L)^{-1}\beta_j(L)\\ \epsilon_t^\star &= \epsilon_t - \widetilde{\psi}(L)\psi(L)^{-1}\Delta\epsilon_t \end{align*} At last, the second representation is formulated as: \begin{align} y_t &= \psi^{-1}(1)\left( a_0^\star + a_1t + \sum_{j=1}^{k}{\beta_j(1)x_{j,t}} + \sum_{j=1}^{k}{\widetilde{\beta}^\star_j(L)\Delta x_{j,t}} + \epsilon_t^\star \right) \notag\\ &= \alpha_0 + \alpha_1 t + \sum_{j=1}^{k}{\theta_j(1)x_{j,t}} + \sum_{j=1}^{k}{\widetilde{\theta}_j(L)\Delta x_{j,t}} + \xi_t \label{eq.ardl.3} \end{align} where \begin{align*} \alpha_0 &= \psi^{-1}(1) \left(a_0 - \widetilde{\psi}(1)\alpha_1 \right)\\ \alpha_1 &= \psi^{-1}(1) a_1\\ \theta_j(1) &= \psi^{-1}(1)\beta_j(1)\\ \widetilde{\theta}_j(L) &= \psi^{-1}(1) \left(\widetilde{\beta}_j(L) - \widetilde{\psi}(L)\psi(L)^{-1}\beta_j(L)\right)\\ \xi_t &= \psi^{-1}(1)\left(\epsilon_t - \widetilde{\psi}(L)\psi(L)^{-1}\Delta\epsilon_t\right) \end{align*} From equation (\ref{eq.ardl.2}), we are typically interested in the long-run (trend) parameters captured by $\alpha_1$ and $\theta_j(1)$, for $j=1,\ldots,k$. In fact, given the one-to-one correspondence between the parameter estimates obtained in (\ref{eq.ardl.2}) and equation (\ref{eq.ardl.3}), having estimated the regression model (\ref{eq.ardl.2.reg}), one can use the parameter formulas above to derive estimates of the long-run parameters post-estimation. In particular, if $\widehat{a}_1,\widehat{b}_{0,1},\ldots,\widehat{b}_{0,p},\widehat{b}_1\ldots,\widehat{b}_k$ denote the relevant subset of estimated coefficients from the regression model (\ref{eq.ardl.2.reg}), then, a post-regression estimate of the long-run parameters is derived as follows: $$\widehat{\alpha}_1 = \frac{\widehat{a}_1}{1 - \displaystyle\sum_{i=1}^{p}{\widehat{b}_{0,i}}} \quad \text{and} \quad \widehat{\theta}_j(1) = \frac{\widehat{b}_j}{1 - \displaystyle\sum_{i=1}^{p}{\widehat{b}_{0,i}}}$$

Third Representation: (Conditional Error Correction Form and the Bounds Test)

The final representation is arguably the most interesting and one that typically receives the most attention in applied work. The objective here is to test for cointegration by reducing a typical VAR framework to its corresponding conditional error correction (CEC) form. As it happens to be, the CEC model of interest is in fact an ARDL model with a one-to-one correspondence with the model in (\ref{eq.ardl.2}). To see this, substitute the right hand side of equation (\ref{eq.ardl.1}) for $y_t$ in line 2 below. In particular: \begin{align} \Delta y_t &= y_t - y_{t-1} \notag\\ &= a_0 + a_1t + \psi^\star(L)y_t + \sum_{j=1}^{k}{\beta_j(L)x_{j,t}} + \epsilon_t - y_{t-1}\notag\\ &= a_0 + a_1t - y_{t-1} + \psi^\star(1)y_{t-1} + \widetilde{\psi^\star}(L)\Delta y_{t-1} + \sum_{j=1}^{k}{\beta_j(L)x_{j,t}} + \epsilon_t \notag\\ &= a_0 + a_1t - \left(1 - \psi^\star(1)\right)y_{t-1} + \widetilde{\psi^\star}(L)\Delta y_{t-1} + \sum_{j=1}^{k}{\beta_j(L)\left(x_{j,t-1} + \Delta x_{j,t}\right)} + \epsilon_t \notag\\ &= a_0 + a_1t - \psi(1)y_{t-1} + \widetilde{\psi^\star}(L)\Delta y_{t-1} + \sum_{j=1}^{k}{\left(\beta_j(1) + (1-L)\widetilde{\beta}_j(L)\right)x_{j,t-1}} + \sum_{j=1}^{k}{\beta_j(L)\Delta x_{j,t}} + \epsilon_t \notag\\ &= a_0 + a_1t - \psi(1)y_{t-1} + \sum_{j=1}^{k}{\beta_j(1)x_{j,t-1}} \notag\\ &+ \left(\widetilde{\psi^\star}(L)\Delta y_{t-1} + \sum_{j=1}^{k}{\widetilde{\beta}_j(L)\Delta x_{j,t-1}}\right) + \sum_{j=1}^{k}{\beta_j(L)\Delta x_{j,t}} + \epsilon_t \label{eq.ardl.4} \end{align} Equation (\ref{eq.ardl.4}) above is the CEC form derived from the ARDL model in equation (\ref{eq.ardl.1}). Rewriting this equation as: \begin{align} \Delta y_t &= a_0 + a_1t - \psi(1)\left( y_{t-1} - \sum_{j=1}^{k}{\frac{\beta_j(1)}{\psi(1)}x_{j,t-1}}\right) \notag\\ &+ \left(\widetilde{\psi^\star}(L)\Delta y_{t-1} + \sum_{j=1}^{k}{\widetilde{\beta}_j(L)\Delta x_{j,t-1}}\right) + \sum_{j=1}^{k}{\beta_j(L)\Delta x_{j,t}} + \epsilon_t \notag\\ &= a_0 + a_1t - \psi(1)EC_{t-1} \notag\\ &+ \left(\widetilde{\psi^\star}(L)\Delta y_{t-1} + \sum_{j=1}^{k}{\widetilde{\beta}_j(L)\Delta x_{j,t-1}}\right) + \sum_{j=1}^{k}{\beta_j(L)\Delta x_{j,t}} + \epsilon_t \label{eq.ardl.5} \end{align} it is readily verified that the error correction term, typically denoted as $EC_t$, is also the cointegrating relationship when $y_t$ and $x_{1,t},\ldots,x_{k,t}$ are cointegrated. In fact, PSS(2001) demonstrate that equation (\ref{eq.ardl.4}) is in fact (abstracting from differing lag values) the CEC of the VAR$(p)$ model: $$\pmb{\Phi}(L)(\pmb{z}_t - \pmb{\mu} - \pmb{\gamma}t) = \pmb{\epsilon}_t$$ where $\pmb{z}_t$ is a $(k+1)$-vector $(y_t,x_{1,t},\ldots, x_{k,t})^\top$ and $\pmb{\mu}$ and $\pmb{\gamma}$ are respectively the $(k+1)$-vectors of intercept and trend coefficients, and $\pmb{\Phi}(L) = \pmb{I}_{k+1} - \sum_{i=1}^{p}\pmb{\Phi}_iL^i$ is the $(k+1)$ square matrix lag polynomial. This is particularly important as the CEC is typically used as a platform for testing for the presence of cointegration.

Traditionally, the cointegration tests of Engle-Granger (1987), Phillips and Ouliaris (1990) or Johansen (1995), typically require all variables in the VAR to be I$(1)$. This clearly requires a battery of pre-testing for the presence of a unit root in each of the variables under consideration, and is clearly subject to misclassification, particularly since unit root tests are known to suffer size and power problems in many cases of interest. In contrast, PSS(2001) propose a test for cointegration that is not only robust to whether variables of interest are I$(0)$, I$(1)$, or mutually cointegrated, but is significantly easier to implement as it only requires estimation and inferential procedures used in the familiar least squares regressions. In this regard, PSS(2001) discuss the famous bounds test for cointegration as a test on parameter significance in the cointegrating relationship of the CEC model (\ref{eq.ardl.4}). In other words, the test is a standard $F-$ or Wald test for the following null and alternative hypotheses: \begin{align*} H_0 &: \quad \psi(1) \cap \left \{ \beta_j(1) \right \}_{j=1}^{k} = 0 \quad \text{(variables are not cointegrated)}\\ H_A &: \quad \psi(1) \cup \left \{ \beta_j(1) \right \}_{j=1}^{k} \neq 0 \quad \text{(variables are cointegrated)} \end{align*} Once the test statistic is computed, it is compared to two asymptotic critical values corresponding to polar cases of all variables being purely I$(0)$ or purely I$(1)$. As such, these critical values lie in the lower and upper tails, respectively, of a non-standard mixture distribution involving integral functions of Brownian motions. When the test statistic is below the lower critical value, one fails to reject the null and concludes that cointegration is not possible. In contrast, when the test statistic is above the upper critical value, one rejects the null and concludes that cointegration is indeed possible. In either of these two cases, knowledge of the cointegrating rank is not necessary. Alternatively, should the test statistic fall between the lower and upper critical values, testing is inconclusive, and knowledge of the cointegrating rank is required to proceed further.

We also remark here that it has been argued in Narayan (2005) that asymptotic critical values presented in PSS (2001) are usually unrealistic for most practical implementations since they are derived for sample sizes of $T=1000$. Accordingly, Narayan (2005) presents critical values for sample sizes ranging from $T=30$ to $T=80$ in increments of 5, which they argue ought to improve inferential reliability under most finite sample settings. EViews 10 will offer the user a choice of whether they'd like to use the PSS (2001) or the Narayan (2005) critical values.

Here it is also important to highlight that PSS(2001) offer five alternative interpretations of the CEC model (\ref{eq.ardl.4}), distinguished by whether deterministic terms integrate into the error correction term. When deterministic terms contribute to the error correction term, they are implicitly projected onto the span of the cointegrating vector. In other words, $\pmb{\mu}$ and $\pmb{\gamma}$ of the VAR$(p)$ model are restricted to a linear combination of the elements in the cointegrating vector. This clearly implies that $a_0$ and $a_1$ in equation (\ref{eq.ardl.4}) must too be similarly restricted. Below are summaries of the theoretical (DGP) and practical regression (REG) models, respectively, for each of the five interpretations along with the appropriate cointegrating relationship ($EC_t$) and the bounds test null-hypothesis ($H_0$).

Case 1: (No Constant and No Trend): $a_0 = a_1 = 0$, that is, ($\pmb{\mu} = \pmb{\gamma} = \pmb{0}$)

DGP: \begin{align*} \Delta y_t &=-\psi(1)y_{t-1} + \sum_{j=1}^{k}{\beta_j(1)x_{j,t-1}}\\ &+ \left(\widetilde{\psi^\star}(L)\Delta y_{t-1} + \sum_{j=1}^{k}{\widetilde{\beta}_j(L)\Delta x_{j,t-1}}\right) + \sum_{j=1}^{k}{\beta_j(L)\Delta x_{j,t}} + \epsilon_t \\ EC_t &= y_{t} - \sum_{j=1}^{k}{\frac{\beta_j(1)}{\psi(1)}x_{j,t}} \\ H_0 &:\quad \psi(1) = \beta_j(1) = 0, \quad \forall j \end{align*} Regression: \begin{align} \Delta y_t &= b_0y_{t-1} + \sum_{j=1}^{k}{b_{j}x_{j,t-1}} \notag\\ &+ \sum_{i=1}^{p-1}{c_{0,i}\Delta y_{t-i}} + \sum_{j=1}^{k}\sum_{l_j=1}^{q_j-1}{c_{j,l_j}\Delta x_{j,t-l_j}} + \sum_{j=1}^{k}{d_{j}\Delta x_{j,t}} + \epsilon_t \label{eq.ardl.6}\\ EC_t &= y_{t} - \sum_{j=1}^{k}{\frac{b_j}{b_0}x_{j,t}} \notag\\ H_0 &: \quad b_0 = b_j = 0, \quad \forall j \notag \end{align}
Case 2: (Restricted Constant and No Trend): $a_0 = \psi(1)\mu_y + \sum_{j=1}^{k}\beta_j(1)\mu_{x_j}$ and $a_1 = 0$ so that $\pmb{\gamma} = \pmb{0}$

DGP: \begin{align*} \Delta y_t &=-\psi(1)\left(y_{t-1} - \mu_y\right) + \sum_{j=1}^{k}{\beta_j(1)\left(x_{j,t-1} - \mu_{x_j}\right)} \\ &+ \left(\widetilde{\psi^\star}(L)\Delta y_{t-1} + \sum_{j=1}^{k}{\widetilde{\beta}_j(L)\Delta x_{j,t-1}}\right) + \sum_{j=1}^{k}{\beta_j(L)\Delta x_{j,t}} + \epsilon_t \\ EC_t &= y_{t} - \mu_y - \sum_{j=1}^{k}{\frac{\beta_j(1)}{\psi(1)}\left(x_{j,t} - \mu_{x_j}\right)} \\ H_0 &:\quad \psi(1) = \beta_j(1) = 0, \quad \forall j \end{align*} Regression: \begin{align} \Delta y_t &= a_0 + b_0y_{t-1} + \sum_{j=1}^{k}{b_{j}x_{j,t-1}} \notag\\ &+ \sum_{i=1}^{p-1}{c_{0,i}\Delta y_{t-i}} + \sum_{j=1}^{k}\sum_{l_j=1}^{q_j-1}{c_{j,l_j}\Delta x_{j,t-l_j}} + \sum_{j=1}^{k}{d_{j}\Delta x_{j,t}} + \epsilon_t \label{eq.ardl.7}\\ EC_t &= y_{t} - \sum_{j=1}^{k}{\frac{b_j}{b_0}x_{j,t}} - \frac{a_0}{b_0} \notag\\ H_0 &: \quad a_0 = b_0 = b_j = 0, \quad \forall j \notag \end{align}
Case 3: (Unrestricted Constant and No Trend): $a_0 \neq 0$ and $a_1 = 0$ so that $\pmb{\mu} \neq \pmb{0}$ and $\pmb{\gamma} = \pmb{0}$

DGP: \begin{align*} \Delta y_t &=a_0 -\psi(1)y_{t-1} + \sum_{j=1}^{k}{\beta_j(1)x_{j,t-1}} \\ &+ \left(\widetilde{\psi^\star}(L)\Delta y_{t-1} + \sum_{j=1}^{k}{\widetilde{\beta}_j(L)\Delta x_{j,t-1}}\right) + \sum_{j=1}^{k}{\beta_j(L)\Delta x_{j,t}} + \epsilon_t \\ EC_t &= y_{t} - \sum_{j=1}^{k}{\frac{\beta_j(1)}{\psi(1)}x_{j,t}} \\ H_0 &:\quad \psi(1) = \beta_j(1) = 0, \quad \forall j \end{align*} Regression: \begin{align} \Delta y_t &= a_0 + b_0y_{t-1} + \sum_{j=1}^{k}{b_{j}x_{j,t-1}} \notag\\ &+ \sum_{i=1}^{p-1}{c_{0,i}\Delta y_{t-i}} + \sum_{j=1}^{k}\sum_{l_j=1}^{q_j-1}{c_{j,l_j}\Delta x_{j,t-l_j}} + \sum_{j=1}^{k}{d_{j}\Delta x_{j,t}} + \epsilon_t \label{eq.ardl.8}\\ EC_t &= y_{t} - \sum_{j=1}^{k}{\frac{b_j}{b_0}x_{j,t}} \notag\\ H_0 &: \quad b_0 = b_j = 0, \quad \forall j \notag \end{align}
Case 4: (Unrestricted Constant and Restricted Trend): $a_0 \neq 0$ so that $\pmb{\mu} \neq \pmb{0}$ and $a_1 = \psi(1)\gamma_y + \sum_{j=1}^{k}\beta_j(1)\gamma_{x_j}$

DGP: \begin{align*} \Delta y_t &=a_0 - \psi(1)\left(y_{t-1} - \gamma_y t\right) + \sum_{j=1}^{k}{\beta_j(1)\left(x_{j,t-1} - \gamma_{x_j}t\right)} \\ &+ \left(\widetilde{\psi^\star}(L)\Delta y_{t-1} + \sum_{j=1}^{k}{\widetilde{\beta}_j(L)\Delta x_{j,t-1}}\right) + \sum_{j=1}^{k}{\beta_j(L)\Delta x_{j,t}} + \epsilon_t \\ EC_t &= y_{t} - \gamma_yt - \sum_{j=1}^{k}{\frac{\beta_j(1)}{\psi(1)}\left(x_{j,t} - \gamma_{x_j}t\right)} \\ H_0 &:\quad \psi(1) = \beta_j(1) = 0, \quad \forall j \end{align*} Regression: \begin{align} \Delta y_t &= a_0 + a_1t + b_0y_{t-1} + \sum_{j=1}^{k}{b_{j}x_{j,t-1}} \notag\\ &+ \sum_{i=1}^{p-1}{c_{0,i}\Delta y_{t-i}} + \sum_{j=1}^{k}\sum_{l_j=1}^{q_j-1}{c_{j,l_j}\Delta x_{j,t-l_j}} + \sum_{j=1}^{k}{d_{j}\Delta x_{j,t}} + \epsilon_t \label{eq.ardl.9}\\ EC_t &= y_{t} - \sum_{j=1}^{k}{\frac{b_j}{b_0}x_{j,t}} - \frac{a_1}{b_0}t \notag\\ H_0 &:\quad a_1 = b_0 = b_j = 0, \quad \forall j \notag \end{align}
Case 5: (Unrestricted Constant and Unrestricted Trend): $a_0 \neq 0$ and $a_1 \neq 0$ so that $\pmb{\mu} \neq \pmb{0}$ and $\pmb{\gamma} \neq \pmb{0}$

DGP: \begin{align*} \Delta y_t &=a_0 + a_1t - \psi(1)y_{t-1} + \sum_{j=1}^{k}{\beta_j(1)x_{j,t-1}} \notag\\ &+ \left(\widetilde{\psi^\star}(L)\Delta y_{t-1} + \sum_{j=1}^{k}{\widetilde{\beta}_j(L)\Delta x_{j,t-1}}\right) + \sum_{j=1}^{k}{\beta_j(L)\Delta x_{j,t}} + \epsilon_t\\ EC_t &= y_{t} - \sum_{j=1}^{k}{\frac{\beta_j(1)}{\psi(1)}x_{j,t}}\\ H_0 &:\quad \psi(1) = \beta_j(1) = 0, \quad \forall j \end{align*} Regression: \begin{align} \Delta y_t &= a_0 + a_1t + b_0y_{t-1} + \sum_{j=1}^{k}{b_{j}x_{j,t-1}} \notag\\ &+ \sum_{i=1}^{p-1}{c_{0,i}\Delta y_{t-i}} + \sum_{j=1}^{k}\sum_{l_j=1}^{q_j-1}{c_{j,l_j}\Delta x_{j,t-l_j}} + \sum_{j=1}^{k}{d_{j}\Delta x_{j,t}} + \epsilon_t \label{eq.ardl.10}\\ EC_t &= y_{t} - \sum_{j=1}^{k}{\frac{b_j}{b_0}x_{j,t}} \notag\\ H_0 &:\quad b_0 = b_j = 0, \quad \forall j \notag \end{align}
Whilst EViews 9.5 only supports the first four cases, EViews 10 will support all 5.

References:

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