Demonstrating That the Autoregressive Distributed Lag Bounds Test Can Detect a Long-Run Levels Relationship When the Dependent Variable Is I(0)

Abstract

The autoregressive distributed lag bounds t-test and F-test for a long-run relationship that allows level variables to be either $I\left(1\right)$ or $I\left(0\right)$ is widely used in the literature. However, a long-run levels relationship cannot be detected when the dependent variable is $I\left(0\right)$, because both tests will always reject their null hypotheses. It has subsequently been argued that a third test determines whether the dependent variable is $I\left(1\right)$, such that when all three tests reject their null hypotheses, a cointegrating equation with an $I\left(1\right)$ dependent variable is identified. It is argued that all three tests rejecting their null hypotheses rules out the possibility that the dependent variable is $I\left(0\right)$, implying that the three tests cannot detect an equilibrium when the dependent variable is $I\left(0\right)$. Our first contribution is to demonstrate and explain that rejection of all three tests’ null hypotheses can also indicate an equilibrium when the dependent variable is $I\left(0\right)$ and not only when it is $I\left(1\right)$. Our second contribution is to produce previously unavailable critical values for the third test in the cases where an intercept or trend is restricted into the equilibrium.

1. Introduction

Pesaran et al. (2001), PSS hereafter, introduced the autoregressive distributed lag (ARDL) bounds testing procedure for a long-run levels relationship (also called an equilibrium, hereafter) when the dependent variable and regressors can all be $I\left(1\right)$, all be $I\left(0\right)$, or be a mixture of $I\left(1\right)$ and $I\left(0\right)$ variables, possibly involving cointegration.1 (A variable is integrated of the order $d$, denoted $I\left(d\right)$, if it needs differencing $d$ times to become stationary). The justification for allowing the variables to be some combination of $I\left(1\right)$ and $I\left(0\right)$ variables is that there is uncertainty over many economic variables’ orders of integration. For example, the low power problem of integration order tests means variables that such tests indicate to be $I\left(1\right)$ may instead be $I\left(0\right)$ with relatively high probability.

The PSS procedure involves two tests. The first is a t-test for the significance of the coefficient on the lagged level of the dependent variable. The second is an F-test for the joint significance of the coefficients on all lagged level variables. PSS simulate lower bound critical values when the dependent variable is $I\left(1\right)$ and all regressors are $I\left(0\right)$ and upper bound critical values when both the dependent variable and all regressors are $I\left(1\right)$ for both the t-test and F-test. The dependent variable is $I\left(1\right)$ under the null hypothesis. Under the alternative hypothesis, either the dependent variable is $I\left(0\right)$ and there is no equilibrium levels relation, or the linear combination of all level variables is $I\left(0\right)$ and a long-run levels relationship exists.

The two PSS tests can indicate a long-run levels relationship when the dependent variable is $I\left(1\right)$, which necessarily requires cointegration. However, if the dependent variable is $I\left(0\right),$ the t-test and F-test suggested by PSS should always reject their respective null hypotheses because the null hypotheses of these tests are that the dependent variable or linear combination of all level variables are $I\left(1\right)$. Hence, rejecting the null hypotheses of both of PSS’s tests cannot detect the existence of an equilibrium when the dependent variable is $I\left(0\right)$.

McNown et al. (2018) and Sam et al. (2019), which are papers by the same authors, suggest an additional F-test for the joint significance of all the lagged level regressors’ coefficients. It is argued that if this third (F-)test’s null hypothesis is rejected, it rules out the possibility that the dependent variable is $I\left(0\right)$. They therefore suggest that the rejection of all three tests’ null hypotheses indicates the existence of a long-run levels relationship through cointegration, with the dependent variable being $I\left(1\right)$.

However, McNown et al. (2018) and Sam et al. (2019) do not recognise that (or explain how) the three tests can be used to detect an equilibrium without cointegration when the dependent variable is $I\left(0\right)$. We argue that the additional third test (in conjunction with the two PSS tests) does not exclusively test whether an equilibrium exists through cointegration involving an $I\left(1\right)$ dependent variable. Rather, this additional third test generalises the PSS procedure such that it can detect an equilibrium when any of the level variables (regressors and the dependent variable) are $I\left(0\right)$ or $I\left(1\right)$, including when the dependent variable is $I\left(0\right)$.2

Demonstrating that the use of all three tests gives an ARDL bounds testing procedure that can detect an equilibrium when the dependent variable is $I\left(0\right)$ is the first contribution of this paper. The significance is that the ARDL model is demonstrated as facilitating the detection of an equilibrium to a broader set of models (including those where the dependent variable is $I\left(0\right)$) than previously.

A second contribution of this paper is that we produce critical values for the third (F-)test when there is either an intercept or trend restricted into the equilibrium relation. Sam et al. (2019) do not report critical values for these two cases, and we therefore fill this gap in the literature.

The rest of this paper is organised as follows. Section 2 discusses how the ARDL method detects an equilibrium when the dependent variable is $I\left(0\right)$. Section 3 presents simulation results that, firstly, give new critical values and, secondly, demonstrate that the tests can identify a long-run levels relationship when the dependent variable is $I\left(0\right)$. In Section 4 we apply the ARDL bounds method to test the law of one price (LOP) hypothesis. Section 5 concludes the paper.

2. The ARDL Bounds Test Method When the Dependent Variable Is I(0)

PSS make the following assumptions about their ARDL bounds testing approach.3 First, the single dependent variable $\left(y_t\right)$ and regressors $\left(x_{k,t}\right)$ can be $I\left(0\right)$ or $I\left(1\right),$ while $x_{k,t}$ can also be mutually cointegrated; however, no variable can have an order of integration above one or seasonal unit roots. Second, the error term of the test Equation (2), $u_t$, is normally distributed and satisfies the standard white noise assumptions, $E\left(u_t\right)=0$; $E\left(u_t^2\right)={\sigma }^2$; and $E(u_t,u_{t-s})=0$, $s\ne 0$. Third, all $K$ regressors are long-run forcing variables of $y_t$, such that $y_{t-1}$ does not determine $\Delta x_{k,t}$ for all $k=1, 2, \dots , K$ (where $\Delta {\mathit{z}}_t={\mathit{z}}_t-{\mathit{z}}_{t-1}$ and ${\mathit{z}}_t=\left(y_t, x_{1,t}, \dots , x_{K,t}\right)$). Hence, there is, at most, one long-run levels relationship between $y_t$ and $x_{k,t}$. PSS consider five different deterministic specifications that are nested within the following most general (case 5) single equation ARDL models:

$$y_t=c_0+c_{1}t+{\sum }_{i=1}^p{\alpha }_{yi}{y}_{t-i}+{\sum }_{i=0}^{q_1}{\beta }_{1i}{x}_{1,t-i}+\dots +{\sum }_{i=0}^{q_K}{\beta }_{Ki}{x}_{K,t-i}+u_t,$$

(1)

$$\begin{array}{c}\Delta y_t=c_0+c_{1}t+{\pi }_y{y}_{t-1}+{\sum }_{k=1}^K{\pi }_{xk}{x}_{k,t-1}+{\sum }_{i=1}^{p-1}{\psi }_{yi}\Delta y_{t-i}+{\sum }_{k=1}^K{\omega }_k\Delta x_{k,t}+\\ {\sum }_{i=1}^{q_1-1}{\psi }_{1i}\Delta x_{1,t-i}+\dots +{\sum }_{i=1}^{q_K-1}{\psi }_{Ki}\Delta x_{K,t-i}+u_t,\end{array}$$

(2)

where $t=1, 2, \dots , T$, ${\pi }_y={\sum }_{i=1}^p{\alpha }_{yi}-1$, ${\pi }_{xk}={\sum }_{i=0}^{q_k}{\beta }_{ki}$, ${\omega }_k={\beta }_{k0}$, ${\psi }_{yi}=-{\sum }_{j=i+1}^p{\alpha }_{yj}$, and ${\psi }_{xi}=-{\sum }_{j=i+1}^{q_k}{\beta }_{kj}$.

If an equilibrium exists, the long-run relation is nested within

$$y={\theta }_0+{\theta }_{1}t+{\sum }_{k=1}^K{\theta }_{xk}{x}_k, {\theta }_0=-{\displaystyle \frac{c_0}{{\pi }_y}}, {\theta }_1=-{\displaystyle \frac{c_1}{{\pi }_y}}, {\theta }_{xk}=-{\displaystyle \frac{{\pi }_{xk}}{{\pi }_y}}.$$

(3)

The test equation for the five different deterministic term specifications are as follows: case 1 (no intercept, no trend), $c_0=c_1=0$; case 2 (restricted intercept, no trend), $c_0=-{\pi }_y{\theta }_0$ and $c_1=0$; case 3 (unrestricted intercept, no trend), $c_0\ne 0$ and $c_1=0$; case 4 (unrestricted intercept, restricted trend), $c_0\ne 0$ and $c_1=-{\pi }_y{\theta }_1$; and case 5 (unrestricted intercept, unrestricted trend), $c_0\ne 0$ and $c_1\ne 0$. PSS suggest two tests applied to (2) for the five deterministic cases. The first is an F-test (denoted $F_{xy}$) of the null hypothesis $H_0^{xy}= H_0^y \cap H_0^x$ ($\cap$ is the intersection symbol) and alternative hypothesis $H_1^{xy}= H_1^y \cup H_1^x$ ($\cup$ is the union symbol), where

$$\begin{array}{cc}{H}_0^y: {\pi }_y=0& case 1, 2, 3, 4 \& 5\end{array},$$

(4)

$$H_0^x: \left\{\begin{array}{cc}{\pi }_{x1}={\pi }_{x2}=\dots {\pi }_{xK}=0& case 1, 3 \& 5\\ c_0={\pi }_{x1}={\pi }_{x2}=\dots {\pi }_{xK}=0& case 2\\ c_1={\pi }_{x1}={\pi }_{x2}=\dots {\pi }_{xK}=0& case 4\end{array}\right.,$$

(5)

$$\begin{array}{cc}{H}_1^y: {\pi }_y\ne 0& case 1, 2, 3, 4 \& 5\end{array},$$

(6)

$$H_1^x: \left\{\begin{array}{cc}{\pi }_{x1}\ne 0 \cup {\pi }_{x2}\ne 0 \cup \dots \cup {\pi }_{xK}\ne 0& case 1, 3 \& 5\\ c_0\ne 0 \cup {\pi }_{x1}\ne 0 \cup {\pi }_{x2}\ne 0 \cup \dots \cup {\pi }_{xK}\ne 0& case 2\\ c_1\ne 0 \cup {\pi }_{x1}\ne 0 \cup {\pi }_{x2}\ne 0 \cup \dots \cup {\pi }_{xK}\ne 0& case 4\end{array}\right..$$

(7)

The second is a t-test (denoted $t_y$) of the null hypothesis $H_0^y$ against the alternative hypothesis $H_1^y$.4 If both $F_{xy}$ and $t_y$ reject their respective null hypotheses, there is evidence of cointegration assuming $y_t~I\left(1\right)$. Rejection of $H_0^{xy}$ is insufficient on its own to find cointegration because $F_{xy}$’s alternative hypothesis, $H_1^{xy}$, is that at least one of the lagged level variables’ coefficients is non-zero and not that all these coefficients are jointly non-zero. Hence, it is possible that $H_0^{xy}$ is rejected because some (or all) of the ${\pi }_{xk}$ are non-zero and not because ${\pi }_y$ is non-zero. In this case, ${\pi }_y$ is not significant, which means that $y_t$ does not form an equilibrium with the regressors (the degenerate lagged dependent variable case). Therefore, the rejection of ${\pi }_y$ being zero (through the application of the $t_y$ test) is additionally required to indicate an equilibrium and rule out the degenerate lagged dependent variable case.

Similarly, rejection of $H_0^{xy}$ could be due to only ${\pi }_y$ being non-zero ($H_0^y$ is false), with all the ${\pi }_{xk}$s being jointly zero ($H_0^x$ is true). This is the degenerate lagged independent variable case. In this case $t_y$ should reject $H_0^y$, which suggests $y_t~I\left(0\right)$ because (2) becomes a generalised (augmented) Dickey–Fuller (ADF) test equation. Solely using the two PSS tests cannot rule out this possibility. That is, rejection of both $H_0^{xy}$ and $H_0^y$ suggests the following two possibilities when $y_t~I\left(0\right)$.5 First, $y_t~I\left(0\right)$ and is significantly correlated with at least one regressor such that there is a long-run levels relation.6 Second, $y_t~I\left(0\right)$ and is not significantly correlated with any of the regressors, so there is no equilibrium relationship (the degenerate lagged independent variable case). Hence, a drawback of solely using the two PSS tests is that rejection of both $H_0^{xy}$ and $H_0^y$ does not necessarily indicate the existence of an equilibrium when $y_t~I\left(0\right)$ due to the possibility of the degenerate lagged independent variable case.

McNown et al. (2018) propose a third (F) test (denoted $F_x$) of the null hypothesis, $H_0^x$, that all the ${\pi }_{xk}$ coefficients are jointly zero against the alternative hypothesis, $H_1^x$, that at least one of the ${\pi }_{xk}$ coefficients is significant.7 It is argued that rejection of $H_0^x$, in addition to rejecting $H_0^{xy}$ and $H_0^y$, rules out the degenerate lagged independent variable case by ensuring that $y_t$ is not $I\left(0\right)$. McNown et al. (2018) and Sam et al. (2019), which are papers by the same authors, suggest that, in contrast, PSS rule out the degenerate lagged independent variable case by assuming $y_t~I\left(1\right)$. Hence, they contend that PSS’s tests augmented with the $F_x$ test indicates evident cointegration if $H_0^x$, $H_0^{xy}$, and $H_0^y$ are all rejected without requiring PSS’s assumption that $y_t$ is known to be $I\left(1\right)$. For example, Sam et al. (2019, p. 131) contend that augmenting the two PSS tests with $F_x$ removes “… the reliance on the assumption of an $I\left(1\right)$ dependent variable to rule out the degenerate case. … By combining this new test with the two tests presented by PSS, we gain a complete picture of the cointegration status of the system. If all the three tests (overall F-test on lagged level variables, t-test on the lagged level of the dependent variable, and F-test on the lagged levels of the independent variable(s)) are found to be significant, we can conclude that there is cointegration.” Sam et al. (2019, p. 135) further state the following (where MSG denotes McNown et al., 2018), “If any variable is $I\left(0\right)$, the ARDL equation using this $I\left(0\right)$ variable as the dependent variable violates PSS’ assumption. However, as pointed out by MSG, one should proceed to test for cointegration using the full three-test framework in the ARDL, even if a variable is $I\left(0\right)$. The tests would suggest a degenerate lagged independent variable case if the dependent variable is indeed $I\left(0\right)$.”8

This is also evident in the following statement by McNown et al. (2018, p. 1512), with our text in square brackets, “To rule out degenerate case #1, one also needs to make sure that the dependent variable is integrated of order 1 or $I\left(1\right)$. If the F-test is significant and the dependent variable is $I\left(1\right)$, then the coefficient[s on ${\pi }_{x1}$, ${\pi }_{x2}$, …, ${\pi }_{xK}$] must be [jointly] significant, ruling out degenerate case #1.” Similarly, McNown et al. (2018, p. 1519) state that by using all three tests “… we can have a better insight into whether the relationship between the dependent variable and independent variables is one of cointegration, non-cointegration or a degenerate case.”9 Thus, in both McNown et al. (2018) and Sam et al. (2019), an equilibrium is only seen as occurring when there is cointegration with $y_t~I\left(1\right)$.10

In both papers, rejection of the $F_x$ test’s null hypothesis rules out $y_t~I\left(0\right)$ and indicates cointegration given that both PSS tests reject $H_0^{xy}$ and $H_0^y$. Hence, according to both McNown et al. (2018) and Sam et al. (2019), the three bounds tests cannot detect an equilibrium existing when $y_t~I\left(0\right)$, because if $y_t~I\left(0\right)$, they believe that this indicates the degenerate lagged independent variable case. This implies that when all three tests reject their null hypotheses, an equilibrium can only occur by cointegration with it being established that $y_t~I\left(1\right)$. Our paper challenges this contention by arguing that rejection of all three null hypotheses is also consistent with the possibility that an equilibrium exists when $y_t~I\left(0\right)$.

To understand why an equilibrium can exist when $y_t~I\left(0\right)$, first assume that there is a long-run levels relationship and $y_t~I\left(0\right)$. In this case, the $F_{xy}$ and $t_y$ tests should reject $H_0^{xy}$ and $H_0^y$, respectively, because this indicates that $y_t$ is continuously forced to an equilibrium (there is valid error correction behaviour when ${\pi }_y<0$). Further, rejection of $H_0^x$ with the $F_x$ test indicates that at least one of the regressors is correlated with the $I\left(0\right)$ dependent variable. Hence, the interpretation of rejecting all three null hypotheses ($H_0^{xy}$, $H_0^y$, and $H_0^x$) is that the stationary dependent variable is significantly correlated with at least one of the regressors in a dynamic relationship that has a long-run solution in the form of (3)—this outcome is summarised in row 1 of

Table 1. Bounds test decisions.

Row		${\mathit{F}}_{\mathit{x}}$	${\mathit{F}}_{\mathit{x}\mathit{y}}$	${\mathit{t}}_{\mathit{y}}$
1	Equilibrium: $y_t~I\left(0\right)$	Reject $H_0^x$	Reject $H_0^{xy}$	Reject $H_0^y$
2	No equilibrium: $y_t~I\left(0\right)$	Accept $H_0^x$	Reject $H_0^{xy}$	Reject $H_0^y$
3	Equilibrium: $y_t~I\left(1\right)$	Reject $H_0^x$	Reject $H_0^{xy}$	Reject $H_0^y$
4	No equilibrium: $y_t~I\left(1\right)$	Accept $H_0^x$	Accept $H_0^{xy}$	Accept $H_0^y$

$F_x$, $F_{xy}$, and $t_y$ indicate which of the three bounds tests is being referred to in each column. The second column indicates whether an equilibrium exists (labelled “Equilibrium”) or not (labelled “No equilibrium”) and whether $y_t~I\left(0\right)$ or $y_t~I\left(1\right)$. The decision of whether to “accept” or reject each test is indicated in the body of the table.

.11 If $y_t~I\left(0\right)$ and is significantly correlated with at least one regressor, this suggests that this/these regressors are also $I\left(0\right)$ (or form a stationary linear combination),12 given that variables with different orders of integration should not be significantly correlated (unless spuriously). Thus, rejection of all three null hypotheses can occur when a stationary dependent variable is significantly correlated with at least one of the regressors in a dynamic relationship that has an equilibrium without cointegration. Hence, rejection of all three null hypotheses in the ARDL bounds testing procedure need not exclusively imply cointegration involving an $I\left(1\right)$ dependent variable, as previously indicated by McNown et al. (2018) and Sam et al. (2019). By demonstrating and explaining why an equilibrium with an $I\left(0\right)$ dependent variable can be detected by the three tests, we provide a contribution beyond that of McNown et al. (2018) and Sam et al. (2019).

Now assume no long-run levels relation exists so that the ARDL test Equation (2) becomes the generalised ADF test equation and $y_t~I\left(0\right)$. Since $y_t~I\left(0\right)$, the $F_{xy}$ and $t_y$ tests for the respective null hypotheses $H_0^{xy}$ and $H_0^y$ should be rejected by construction, because both these null hypotheses specify ${\pi }_y=0$ (indicating that $y_t~I\left(1\right)$).13 In contrast, the $F_x$ test for the null hypothesis $H_0^x$ should not be rejected, because if no equilibrium exists, all the $x_{k,t-1}$’s coefficients in (2) should be (jointly) insignificant, indicating they are not correlated with $y_t$. Therefore, when no equilibrium exists and $y_t~I\left(0\right),$ the two hypotheses $H_0^{xy}$ and $H_0^y$ will be rejected, while $H_0^x$ will not be rejected. This outcome is summarised in row 2 of Table 1.14

Hence, the $F_x$ test is crucial for determining whether a long-run levels relation exists when $y_t~I\left(0\right)$. This is because using only the two tests for $F_{xy}$ and $t_y$ (as with the PSS method) does not facilitate the determination of whether an equilibrium exists when $y_t~I\left(0\right)$, since these tests’ null hypotheses are both rejected when there is an equilibrium and when there is not. Our demonstration that the addition of the $F_x$ test to the bounds testing procedure enables the detection of an equilibrium when $y_t~I\left(0\right)$ therefore represents a contribution of our paper beyond that of PSS.

Thus, the first contribution of our paper is the demonstration and explanation of the use of the $F_x$ test, in addition to $F_{xy}$ and $t_y$, to generalise the ARDL bounds method such that it can detect an equilibrium when $y_t~I\left(0\right)$. The two PSS tests cannot do this, and McNown et al. (2018) and Sam et al.’s (2019) discussion of the addition of the third $F_x$ test does not recognise this possibility.

Kripfganz and Schneider (2023) suggest that, given $H_0^{xy}$ and $H_0^y$ are rejected, a Wald (F or t) test directly on the long-run coefficients in (3) can be used as an alternative to $F_x$—we denote the F version of this test with $F_{x_{\theta }}$.15 The null and alternative hypotheses of $F_{x_{\theta }}$ are specified as follows:

$$H_0^{x_{\theta }}: \left\{\begin{array}{cc}{\theta }_{x1}={\theta }_{x2}=\dots ={\theta }_{xK}=0& case 1, 3 \& 5\\ {\theta }_0={\theta }_{x1}={\theta }_{x2}=\dots ={\theta }_{xK}=0& case 2\\ {\theta }_1={\theta }_{x1}={\theta }_{x2}=\dots ={\theta }_{xK}=0& case 4\end{array}\right.,$$

(8)

$$H_1^{x_{\theta }}: \left\{\begin{array}{cc}{\theta }_{x1}\ne 0\cup {\theta }_{x2}\ne 0\cup \dots \cup {\theta }_{xK}\ne 0& case 1, 3 \& 5\\ {\theta }_0\ne 0 \cup {\theta }_{x1}\ne 0\cup {\theta }_{x2}\ne 0\cup \dots \cup {\theta }_{xK}\ne 0& case 2\\ {\theta }_1\ne 0 \cup {\theta }_{x1}\ne 0\cup {\theta }_{x2}\ne 0\cup \dots \cup {\theta }_{xK}\ne 0& case 4\end{array}\right..$$

(9)

That is, if $H_0^{x_{\theta }}$ can be rejected in favour of $H_1^{x_{\theta }}$, the degenerate lagged independent variable case is ruled out, and there is an evident equilibrium. This test uses the estimated short-run coefficients from (2) to obtain the long-run coefficients given by (3): ${\theta }_0=-{\displaystyle \frac{c_0}{{\pi }_y}}$, ${\theta }_1=-{\displaystyle \frac{c_1}{{\pi }_y}}$, ${\theta }_{xk}=-{\displaystyle \frac{{\pi }_{xk}}{{\pi }_y}}$. The variance–covariance matrix for $\mathit{\theta }={\left({\theta }_0, {\theta }_1, {\theta }_{xk}\right)}^{\prime }$ can be calculated using the delta method and standard t; F or chi-squared distributions can be used because Pesaran and Shin (1998) demonstrate that the ordinary least squares (OLS) estimator of $\mathit{\theta }$ has a normal distribution asymptotically, regardless of whether the regressors are $I\left(0\right)$ or $I\left(1\right)$.16 In Kripfganz and Schneider’s (2023) applied example, they only use t-tests to determine whether at least one of the elements of $\mathit{\theta }$ are significant to support an equilibrium given that $H_0^{xy}$ and $H_0^y$ are rejected. Kripfganz and Schneider (2023, pp. 989–990) suggest that, provided none of the regressors are cointegrated with each other, the integration properties of the regressors determine the order of integration of the dependent variable. However, they do not explicitly state that a long-run relationship can exist when the dependent variable is $I\left(0\right)$, and their focus is on cointegration with an $I\left(1\right)$ dependent variable.

We explicitly explain and demonstrate that the ARDL method can detect a long-run levels relationship with $y_t~I\left(0\right)$.

3. Simulation Methods and Results

This section conducts two sets of simulations. The first set assumes no equilibrium and generates critical values for cases 2 and 4 of the $F_x$ test, given Sam et al. (2019) only produce critical values for cases 1, 3, and 5. While PSS and Kripfganz and Schneider (2020) provide critical values for the $F_{xy}$ and $t_y$ tests, they do not produce critical values for $F_x$. We also calculate rejection rates of the $F_x$ test using our 5% critical values, assuming no equilibrium and all variables are $I\left(0\right)$. The second set assumes a long-run levels relationship exists with $x_{k,t}~I\left(0\right)$. All our simulations use $N=\mathrm{100,000}$ replications (after discarding the first 100 replications to reduce the impact of initial values on the simulated series) and are produced for various sample sizes.

3.1. Simulations Without an Equilibrium

We simulate 5% non-standard upper and lower bound critical values for cases 2 and 4 of the $F_x$ test with $K=1, 2, \dots , 12$ regressors.17 Producing critical values for cases 2 and 4 of the $F_x$ test represents the second contribution of our paper. We assume $y_t~I\left(1\right)$ and no equilibrium exists, which implies that the three ARDL bounds tests’ null hypotheses are true. All level variables in the model are independently generated processes (with zero initial values when they are $I\left(1\right)$), and error terms (${\epsilon }_{y,t}$ and ${\epsilon }_{k,t}$) are drawn from independent standard normal distributions using a pseudo random number generator. That is

$$y_t={\rho }_y{y}_{t-1}+{\epsilon }_{y,t}, y_0=0 (\mathrm{iff} {\rho }_y=1), {\epsilon }_{y,t}~N(0, 1),$$

(10)

$$x_{k,t}={\rho }_k{x}_{k,t-1}+{\epsilon }_{k,t}, x_{k,0}=0 (\mathrm{iff} {\rho }_k=1), {\epsilon }_{k,t}~N(0, 1).$$

(11)

The upper bound of the test is considered when ${\rho }_y={\rho }_k=1$ and all level variables are $I\left(1\right)$ processes, whereas the lower bound is given when ${\rho }_y=1$ and ${\rho }_k=0$, such that all $x_{k,t}$ are pure $I\left(0\right)$ processes while $y_t~I\left(1\right)$. The critical values (reported in

Table 2. $F_x$-test 5% critical values: case 2 (restricted intercept, no trend).

	Sample Size (T)
K	30	40	50	60	70	80	90	100	200	500	1000	10,000
	Lower bound
1	5.332	5.187	5.093	5.049	5.001	4.973	4.984	4.963	4.883	4.824	4.836	4.810
2	4.195	4.049	3.993	3.942	3.914	3.890	3.875	3.845	3.792	3.741	3.749	3.722
3	3.655	3.488	3.409	3.376	3.357	3.332	3.307	3.288	3.243	3.185	3.197	3.161
4	3.317	3.165	3.083	3.031	3.003	2.981	2.961	2.936	2.897	2.853	2.846	2.828
5	3.098	2.923	2.844	2.800	2.773	2.739	2.720	2.698	2.653	2.614	2.603	2.593
6	2.949	2.761	2.691	2.638	2.603	2.571	2.561	2.530	2.477	2.446	2.436	2.416
7	2.844	2.648	2.565	2.504	2.470	2.437	2.424	2.400	2.349	2.313	2.301	2.287
8	2.787	2.557	2.453	2.403	2.367	2.345	2.321	2.299	2.249	2.212	2.202	2.190
9	2.740	2.488	2.376	2.323	2.293	2.259	2.239	2.213	2.161	2.128	2.113	2.103
10	2.699	2.427	2.318	2.258	2.223	2.198	2.175	2.144	2.086	2.055	2.049	2.033
11	2.677	2.389	2.266	2.205	2.169	2.140	2.119	2.095	2.030	1.996	1.986	1.973
12	2.676	2.355	2.223	2.162	2.128	2.092	2.071	2.046	1.981	1.944	1.936	1.920
	Upper bound
1	6.329	6.098	6.015	5.948	5.884	5.863	5.836	5.811	5.726	5.667	5.641	5.619
2	5.457	5.230	5.130	5.072	4.998	4.951	4.946	4.926	4.836	4.784	4.775	4.727
3	5.003	4.767	4.649	4.582	4.528	4.498	4.452	4.436	4.374	4.306	4.303	4.247
4	4.738	4.490	4.367	4.276	4.221	4.195	4.163	4.156	4.050	4.008	4.015	3.959
5	4.583	4.313	4.165	4.077	4.024	3.983	3.956	3.948	3.832	3.788	3.801	3.750
6	4.470	4.174	4.025	3.951	3.879	3.850	3.814	3.797	3.683	3.624	3.625	3.594
7	4.416	4.091	3.926	3.833	3.776	3.733	3.703	3.684	3.555	3.499	3.493	3.468
8	4.369	4.011	3.842	3.748	3.685	3.639	3.602	3.589	3.459	3.399	3.384	3.374
9	4.356	3.963	3.779	3.679	3.607	3.559	3.526	3.506	3.377	3.309	3.299	3.285
10	4.379	3.925	3.729	3.612	3.547	3.501	3.458	3.441	3.315	3.237	3.226	3.209
11	4.397	3.900	3.682	3.572	3.495	3.443	3.404	3.381	3.255	3.176	3.168	3.146
12	4.432	3.901	3.670	3.536	3.457	3.411	3.371	3.347	3.202	3.125	3.113	3.092

Number of replications, $N=\mathrm{100,000}$. $T$ is the number of observations in the sample (after 100 discarded initial observations). DGPs: $y_t=y_{t-1}+{\epsilon }_{y,t}$, ${\epsilon }_{y,t}~N\left(\mathrm{0,1}\right)$; lower bound, $x_{k,t}={\epsilon }_{k,t}$, ${\epsilon }_{k,t}~N\left(\mathrm{0,1}\right)$; and upper bound, $x_{k,t}=x_{k,t-1}+{\epsilon }_{k,t}$. The regression is $\Delta y_t=c_0+{\pi }_y{y}_{t-1}+{\sum }_{k=1}^K{\pi }_{xk}{x}_{k,t-1}+u_t$. Critical values are given for an $F_x$ test of the null hypothesis of no equilibrium, $H_0^x:c_0={\pi }_{x1}=\dots ={\pi }_{xK}=0$.

and

Table 3. $F_x$-test 5% critical values: case 4 (unrestricted intercept, restricted trend).

	Sample Size (T)
K	30	40	50	60	70	80	90	100	200	500	1000	10,000
	Lower bound
1	6.108	5.979	5.904	5.834	5.768	5.771	5.721	5.709	5.646	5.596	5.570	5.579
2	4.732	4.585	4.502	4.423	4.379	4.367	4.331	4.326	4.265	4.217	4.182	4.212
3	4.051	3.851	3.787	3.723	3.681	3.674	3.644	3.641	3.567	3.512	3.504	3.510
4	3.621	3.433	3.364	3.294	3.261	3.242	3.208	3.204	3.137	3.097	3.086	3.087
5	3.366	3.157	3.073	3.026	2.983	2.947	2.918	2.914	2.846	2.810	2.795	2.795
6	3.177	2.965	2.882	2.825	2.782	2.755	2.721	2.718	2.646	2.611	2.596	2.592
7	3.042	2.820	2.732	2.665	2.628	2.595	2.565	2.565	2.487	2.460	2.441	2.441
8	2.949	2.709	2.608	2.548	2.504	2.477	2.440	2.437	2.370	2.328	2.317	2.311
9	2.873	2.628	2.519	2.451	2.406	2.381	2.345	2.335	2.271	2.230	2.216	2.213
10	2.841	2.563	2.443	2.366	2.327	2.300	2.263	2.251	2.193	2.145	2.130	2.130
11	2.815	2.515	2.378	2.303	2.258	2.232	2.201	2.186	2.128	2.080	2.065	2.062
12	2.811	2.470	2.324	2.249	2.205	2.176	2.146	2.129	2.066	2.022	2.006	2.002
	Upper bound
1	7.030	6.848	6.745	6.671	6.608	6.547	6.559	6.535	6.428	6.376	6.332	6.373
2	5.889	5.667	5.561	5.497	5.442	5.414	5.368	5.369	5.276	5.223	5.187	5.220
3	5.292	5.113	4.963	4.890	4.836	4.801	4.786	4.772	4.669	4.600	4.577	4.593
4	4.971	4.717	4.600	4.532	4.479	4.425	4.401	4.381	4.305	4.220	4.187	4.216
5	4.738	4.463	4.342	4.265	4.207	4.166	4.149	4.136	4.040	3.954	3.931	3.941
6	4.616	4.306	4.169	4.076	4.024	3.984	3.954	3.930	3.840	3.762	3.750	3.747
7	4.518	4.192	4.040	3.951	3.877	3.841	3.808	3.795	3.694	3.619	3.588	3.595
8	4.457	4.102	3.939	3.838	3.761	3.719	3.697	3.679	3.568	3.497	3.476	3.475
9	4.446	4.051	3.867	3.753	3.669	3.634	3.604	3.586	3.475	3.408	3.379	3.371
10	4.437	4.003	3.809	3.684	3.610	3.551	3.528	3.507	3.392	3.324	3.299	3.290
11	4.486	3.971	3.761	3.629	3.552	3.493	3.459	3.434	3.324	3.255	3.228	3.226
12	4.547	3.942	3.727	3.594	3.502	3.445	3.405	3.378	3.264	3.203	3.170	3.166

Number of replications, $N=\mathrm{100,000}$. $T$ is the number of observations in the sample (after 100 discarded initial observations). DGPs: $y_t=y_{t-1}+{\epsilon }_{y,t}$, ${\epsilon }_{y,t}~N\left(\mathrm{0,1}\right)$; lower bound, $x_{k,t}={\epsilon }_{k,t}$, ${\epsilon }_{k,t}~N\left(\mathrm{0,1}\right)$; and upper bound, $x_{k,t}=x_{k,t-1}+{\epsilon }_{k,t}$. The regression is $\Delta y_t=c_0+c_{1}t+{\pi }_y{y}_{t-1}+{\sum }_{k=1}^K{\pi }_{xk}{x}_{k,t-1}+u_t$. Critical values are given for an $F_x$ test of the null hypothesis of no equilibrium, $H_0^x:c_1={\pi }_{x1}=\dots ={\pi }_{xK}=0$.

) are calculated from the OLS estimation of test equations nested within the following:

$$\Delta y_t=c_0+c_{1}t+{\pi }_y{y}_{t-1}+{\sum }_{k=1}^K{\pi }_{xk}{x}_{k,t-1}+u_t.$$

(12)

We also simulate rejection rates of the $F_x$ test for case 2 and 4 when both $x_t$ and $y_t$ are independently generated (no equilibrium) pure $I\left(0\right)$ processes. That is, we set ${\rho }_y={\rho }_k=0$. We specify the test equation used for these calculations to be consistent with the simulations that estimate rejection rates assuming an equilibrium (see Section 3.2 below). That is, the first difference in each regressor is added to (12), thus18

$$\Delta y_t=c_0+c_{1}t+{\pi }_y{y}_{t-1}+{\sum }_{k=1}^K{\pi }_{xk}{x}_{k,t-1}+{\sum }_{k=1}^K{\omega }_k\Delta x_{k,t}+u_t.$$

(13)

Table 4. Rejection rates at a 5% nominal significance level of the $F_x$ test with $I\left(0\right)$ independently generated variables for cases 2 and 4.

	Sample Size (T)
K	30	50	80	100	200	500	30	50	80	100	200	500
	Case 2 using lower bound critical values						Case 2 using upper bound critical values
1	0.01064	0.00977	0.00937	0.00848	0.00828	0.00843	0.00544	0.00444	0.00417	0.00412	0.00383	0.00372
2	0.01489	0.01240	0.01193	0.01197	0.01103	0.01103	0.00429	0.00364	0.00316	0.00311	0.00265	0.00270
3	0.01738	0.01512	0.01377	0.01404	0.01378	0.01329	0.00411	0.00284	0.00249	0.00217	0.00190	0.00197
4	0.02103	0.01748	0.01517	0.01585	0.01506	0.01493	0.00398	0.00260	0.00188	0.00161	0.00140	0.00147
5	0.02376	0.01876	0.01776	0.01788	0.01605	0.01625	0.00401	0.00229	0.00173	0.00144	0.00127	0.00098
6	0.02808	0.01958	0.01846	0.01847	0.01808	0.01680	0.00435	0.00204	0.00138	0.00123	0.00101	0.00086
7	0.03399	0.02151	0.02007	0.01984	0.01860	0.01856	0.00555	0.00145	0.00087	0.00082	0.00064	0.00060
8	0.04021	0.02437	0.02116	0.02121	0.01952	0.02007	0.00779	0.00182	0.00091	0.00069	0.00058	0.00053
9	0.05105	0.02689	0.02206	0.02218	0.02036	0.02034	0.01162	0.00168	0.00065	0.00051	0.00045	0.00032
10	0.06826	0.02795	0.02122	0.02155	0.02048	0.02110	0.01735	0.00183	0.00070	0.00048	0.00023	0.00030
11	0.09862	0.03124	0.02374	0.02367	0.02216	0.02223	0.03295	0.00153	0.00057	0.00030	0.00028	0.00019
12	0.15387	0.03366	0.02369	0.02301	0.02160	0.02346	0.06967	0.00184	0.00057	0.00032	0.00019	0.00011
	Case 4 using lower bound critical values						Case 4 using upper bound critical values
1	0.00761	0.00585	0.00489	0.00521	0.00413	0.00384	0.00430	0.00280	0.00231	0.00230	0.00226	0.00177
2	0.01084	0.00806	0.00739	0.00707	0.00618	0.00575	0.00421	0.00280	0.00201	0.00188	0.00178	0.00158
3	0.01399	0.01049	0.00923	0.00853	0.00749	0.00741	0.00400	0.00231	0.00207	0.00167	0.00150	0.00107
4	0.01731	0.01268	0.01113	0.01080	0.00987	0.00876	0.00406	0.00217	0.00145	0.00116	0.00113	0.00104
5	0.02128	0.01530	0.01286	0.01211	0.01092	0.01076	0.00504	0.00200	0.00133	0.00105	0.00061	0.00061
6	0.02635	0.01686	0.01419	0.01317	0.01220	0.01099	0.00568	0.00164	0.00108	0.00078	0.00069	0.00045
7	0.03360	0.01886	0.01556	0.01427	0.01421	0.01288	0.00754	0.00168	0.00077	0.00061	0.00041	0.00046
8	0.04269	0.02056	0.01589	0.01520	0.01416	0.01435	0.00989	0.00164	0.00077	0.00057	0.00032	0.00046
9	0.05727	0.02262	0.01684	0.01687	0.01569	0.01555	0.01588	0.00175	0.00061	0.00043	0.00026	0.00025
10	0.08010	0.02423	0.01839	0.01799	0.01594	0.01572	0.02631	0.00176	0.00081	0.00043	0.00022	0.00020
11	0.11959	0.02746	0.01962	0.01948	0.01554	0.01669	0.05017	0.00211	0.00050	0.00039	0.00019	0.00014
12	0.20218	0.03194	0.02054	0.01957	0.01763	0.01680	0.10938	0.00222	0.00050	0.00043	0.00009	0.00016

Number of replications, $N=\mathrm{100,000}$. $T$ is the number of observations in the sample (after 100 discarded initial observations). DGPs: $x_{k,t}={\epsilon }_{k,t}$, ${\epsilon }_{k,t}~N\left(\mathrm{0,1}\right)$; $y_t={\epsilon }_{y,t}$, ${\epsilon }_{y,t}~N\left(\mathrm{0,1}\right)$. The regression is $\Delta y_t=c_0+c_{1}t+{\sum }_{k=1}^K{\pi }_{xk}{x}_{k,t-1}+{\pi }_y{y}_{t-1}+{\sum }_{k=1}^K{\omega }_k\Delta x_{k,t}+u_t$ ($c_1=0$ for case 2). Rejection rates are given for an $F_x$ test of the null hypothesis of no equilibrium, $H_0^x:c_0={\pi }_{x1}=\dots ={\pi }_{xK}=0$ (case 2), $H_0^x:c_1={\pi }_{x1}=\dots ={\pi }_{xK}=0$ (case 4). Critical values for the $F_x$ test are taken from Table 2 (case 2) and Table 3 (case 4).

reports the empirical rejection rates for the $F_x$ test using a nominal 5% level of significance (based on critical values from Table 2 and Table 3). The empirical rejection rates are all well below 0.050; hence, $H_0^x$ is very rarely (incorrectly) rejected when there is no equilibrium.19 These results are consistent with row 2 of Table 1 and support the fundamental contention of our paper that the $F_x$ test can identify whether or not a long-run relation exists when $y_t~I\left(0\right)$.

3.2. Simulations with an Equilibrium

The second set of simulations assume an equilibrium exists. We simulate the rejection rate of the $F_x$ test for case 2 and 4 (using the 5% lower and upper bound critical values reported in Table 2 and Table 3). The data generation processes (DGPs) for $y_t$ are nested within

$$y_t=c_0+c_{1}t+{\beta }_{10}{x}_{1,t}+{\alpha }_{y1}{y}_{t-1}+{\beta }_{11}{x}_{1,t-1}+{\epsilon }_{y,t}, {\epsilon }_{y,t}~N(0, 1).$$

(14)

We can reparametrize (14) as

$$\Delta y_t=c_0+c_{1}t+({\beta }_{10}+{\beta }_{11})x_{1,t-1}-(1-{\alpha }_{y1})y_{t-1}+{\beta }_{10}\Delta x_{1,t}+{\epsilon }_{y,t}.$$

(15)

This is nested within (2) with $K=1$, $p=q_k=1$, ${\pi }_y=-(1-{\alpha }_{y1})$, ${\pi }_{xk}={\beta }_{k0}+{\beta }_{k1}$, ${\omega }_k={\beta }_{k0}$, and $u_t={\epsilon }_{y,t}$. The static equilibrium is nested within

$$y={\theta }_0+{\theta }_{1}t+{\theta }_{x1}{x}_1, {\theta }_0={\displaystyle \frac{c_0}{1-{\alpha }_{y1}}}, {\theta }_1={\displaystyle \frac{c_1}{1-{\alpha }_{y1}}}, {\theta }_{x1}={\displaystyle \frac{{\beta }_{10}+{\beta }_{11}}{1-{\alpha }_{y1}}}.$$

(16)

We consider six sets of different parameter values for (14), which are specified in

Table 5. Parameter values for generating $y_t$ when $K=1$.

	${\mathit{\beta }}_{10}$	${\mathit{\alpha }}_{\mathit{y}1}$	${\mathit{\beta }}_{11}$	${\mathit{\pi }}_{\mathit{x}\mathit{k}}$	${\mathit{\pi }}_{\mathit{y}}$	${\mathit{\theta }}_0$	${\mathit{\theta }}_1$	${\mathit{\theta }}_{\mathit{x}1}$
DGP 1 and 7	0.07	0.90	0.03	0.10	−0.10	10.00	0.10	1.00
DGP 2 and 8	0.20	0.75	0.05	0.25	−0.25	4.00	0.04	1.00
DGP 3 and 9	0.30	0.50	0.20	0.50	−0.50	2.00	0.02	1.00
DGP 4 and 10	0.15	0.90	−0.05	0.10	−0.10	10.00	0.10	1.00
DGP 5 and 11	0.30	0.75	−0.05	0.25	−0.25	4.00	0.04	1.00
DGP 6 and 12	1.00	0.50	−0.50	0.50	−0.50	2.00	0.02	1.00

The table gives the parameter values of ${\beta }_{10}$, ${\alpha }_{y1}$, and ${\beta }_{11}$ used for generating $y_t$ from (14), $y_t=c_0+c_{1}t+{\beta }_{10}{x}_{1,t}+{\alpha }_{y1}{y}_{t-1}+{\beta }_{11}{x}_{1,t-1}+{\epsilon }_{y,t}$, for DGPs 1 to 12. The implied population values for ${\pi }_y$ and ${\pi }_{xk}$ in (15) are also given, along with the corresponding equilibrium coefficients on the intercept (for case 2 only), trend (for case 4 only), and $x_1$ in (16) in the columns headed ${\theta }_0$, ${\theta }_1$, and ${\theta }_{x1}$, respectively. The coefficients of deterministic terms are $c_0=1.00$ (case 2 and case 4), $c_1=0.00$ (case 2), and $c_1=0.01$ (case 4).

, and two different ways of generating $x_{1,t}$, giving twelve DGPs. For DGPs 1 to 6, $x_{1,t}={\epsilon }_{1,t}$ with ${\epsilon }_{1,t}~N(0, 1)$, and, following Cho et al. (2015, p. 288), DGPs 7 to 12 specify ${\epsilon }_{1,t}=b_{\epsilon 1}{W}_{t-1}+(1-b_{\epsilon 1}^2)W_t$, with $W_t~N(0, 1)$ and $b_{\epsilon 1}=0.5$ in the generation of $x_{1,t}$. Because $x_{1,t}~I\left(0\right)$ for all DGPs, this implies that $y_t~I\left(0\right)$.

Empirical rejection rates of the $F_x$ test for case 2 and 4’s deterministic specifications are reported in

Table 6. Rejection rates at a 5% nominal significance level of the $F_x$ test with $I\left(0\right)$ variables where an equilibrium exists for cases 2 and 4.

	Sample Size (T)
DGP	30	50	80	100	200	500	30	50	80	100	200	500
	Case 2 using lower bound critical values						Case 2 using upper bound critical values
1	0.08267	0.12618	0.22688	0.31123	0.81175	1.00000	0.04457	0.07077	0.13709	0.20055	0.66720	1.00000
2	0.34798	0.65055	0.92905	0.98479	1.00000	1.00000	0.19524	0.44571	0.81156	0.93482	1.00000	1.00000
3	0.81880	0.98924	0.99997	1.00000	1.00000	1.00000	0.62618	0.95167	0.99955	0.99998	1.00000	1.00000
4	0.26552	0.40174	0.60217	0.72783	0.98695	1.00000	0.11525	0.19189	0.34202	0.45354	0.91319	1.00000
5	0.53890	0.84428	0.98672	0.99852	1.00000	1.00000	0.27413	0.59343	0.91108	0.97908	1.00000	1.00000
6	0.80913	0.98982	0.99995	1.00000	1.00000	1.00000	0.52375	0.92550	0.99932	0.99999	1.00000	1.00000
7	0.36685	0.53756	0.75025	0.85151	0.99712	1.00000	0.14992	0.26348	0.44621	0.57413	0.96159	1.00000
8	0.63652	0.91384	0.99574	0.99970	1.00000	1.00000	0.33854	0.69123	0.95197	0.99135	1.00000	1.00000
9	0.92448	0.99875	1.00000	1.00000	1.00000	1.00000	0.72723	0.98320	0.99991	1.00000	1.00000	1.00000
10	0.39359	0.59700	0.80098	0.89239	0.99874	1.00000	0.15143	0.29297	0.49083	0.62171	0.97289	1.00000
11	0.64779	0.92979	0.99705	0.99981	1.00000	1.00000	0.31747	0.69949	0.95753	0.99300	1.00000	1.00000
12	0.86382	0.99688	1.00000	1.00000	1.00000	1.00000	0.55233	0.95311	0.99988	1.00000	1.00000	1.00000
	Case 4 using lower bound critical values						Case 4 using upper bound critical values
1	0.04717	0.05978	0.11763	0.18147	0.69006	0.99998	0.02945	0.03678	0.07635	0.11833	0.56051	0.99997
2	0.09941	0.16688	0.39248	0.60868	0.99983	1.00000	0.05571	0.09871	0.26313	0.45907	0.99893	1.00000
3	0.29196	0.53306	0.84703	0.95615	1.00000	1.00000	0.18011	0.38219	0.73822	0.90310	1.00000	1.00000
4	0.18250	0.25176	0.42410	0.56510	0.97356	1.00000	0.08764	0.12720	0.23841	0.34539	0.87972	1.00000
5	0.20723	0.33136	0.61661	0.80837	0.99999	1.00000	0.10060	0.18191	0.41374	0.62929	0.99981	1.00000
6	0.35509	0.58214	0.85614	0.95493	1.00000	1.00000	0.19277	0.38245	0.70533	0.87622	1.00000	1.00000
7	0.26433	0.36151	0.56685	0.70888	0.99268	1.00000	0.12344	0.18490	0.33415	0.46082	0.94441	1.00000
8	0.27615	0.43460	0.72576	0.88647	1.00000	1.00000	0.13349	0.24667	0.52331	0.73580	0.99999	1.00000
9	0.49778	0.76589	0.95654	0.99173	1.00000	1.00000	0.29465	0.57778	0.88242	0.96847	1.00000	1.00000
10	0.29258	0.41499	0.62800	0.76942	0.99628	1.00000	0.13015	0.20873	0.37876	0.51199	0.96300	1.00000
11	0.30009	0.47762	0.76393	0.90787	1.00000	1.00000	0.13400	0.26801	0.55033	0.75565	0.99996	1.00000
12	0.45528	0.72499	0.93450	0.98370	1.00000	1.00000	0.22459	0.49470	0.81873	0.93793	1.00000	1.00000

Number of replications, $N=\mathrm{100,000}$. $T$ is the number of observations in the sample (after 100 discarded initial observations). $I\left(0\right)$ regressor DGP: $x_{1,t}={\epsilon }_{1,t}$, DGPs 1 to 6, and ${\epsilon }_{1,t}~N\left(\mathrm{0,1}\right)$; DGPs 7 to 12, ${\epsilon }_{1,t}=b_{\epsilon 1}{W}_{t-1}+(1-b_{\epsilon 1}^2)W_t$, $W_t~N\left(\mathrm{0,1}\right)$, and $b_{\epsilon 1}=0.5$; and $y_t=c_0+c_{1}t+{\beta }_{10}{x}_{1,t}+{\alpha }_{y1}{y}_{t-1}+{\beta }_{11}{x}_{1,t-1}+{\epsilon }_{y,t}$, ${\epsilon }_{y,t}~N\left(\mathrm{0,1}\right)$. DGP 1 and 7: ${\beta }_{10}=0.07$, ${\alpha }_{y1}=0.90$, and ${\beta }_{11}=0.03$; DGP 2 and 8: ${\beta }_{10}=0.20$, ${\alpha }_{y1}=0.75$, and ${\beta }_{11}=0.05$; DGP 3 and 9: ${\beta }_{10}=0.30$, ${\alpha }_{y1}=0.50$, and ${\beta }_{11}=0.20$; DGP 4 and 10: ${\beta }_{10}=0.15$, ${\alpha }_{y1}=0.90$, and ${\beta }_{11}=-0.05$; DGP 5 and 11: ${\beta }_{10}=0.30$, ${\alpha }_{y1}=0.75$, and ${\beta }_{11}=-0.05$; and DGP 6 and 12: ${\beta }_{10}=1.00$, ${\alpha }_{y1}=0.50$, and ${\beta }_{11}=-0.50$. The coefficients of deterministic terms for all DGPs are $c_0=1.00$ (case 2 and case 4), $c_1=0.00$ (case 2), and $c_1=0.01$ (case 4). The regression is $\Delta y_t=c_0+c_{1}t+{\pi }_{x1}{x}_{1,t-1}+{\pi }_y{y}_{t-1}+{\omega }_1\Delta x_{1,t}+u_t$ ($c_1=0$ for case 2). Rejection rates for the $F_x$ test using a 5% nominal significance level with case 2 and case 4 critical values taken from Table 2 and Table 3, respectively. The null hypothesis of no equilibrium for $F_x$ is $H_0^x:c_0={\pi }_{x1}=0$ (case 2) and $H_0^x:c_1={\pi }_{x1}=0$ (case 4).

using 5% critical values from Table 2 (case 2) and Table 3 (case 4). For sample sizes of 500 observations, virtually all empirical rejection rates in Table 6 equal 1.00000, suggesting that $H_0^x$ is typically rejected 100,000 times out of the 100,000 replications for both deterministic specifications and all DGPs considered when $y_t~I\left(0\right)$. Given that $H_0^{xy}$ and $H_0^y$ are necessarily rejected when $y_t~I\left(0\right)$, the generally 100% power of the $F_x$ test indicates that all three tests reject their respective null hypotheses ($H_0^x$, $H_0^{xy}$, and $H_0^y$) and infer an equilibrium for $T=500$ when a long-run levels relationship exists and $y_t~I\left(0\right)$. This is consistent with row 1 of Table 1.20 Meanwhile, the $F_x$ test does not reject $H_0^x$ when an equilibrium does not exist and $y_t~I\left(0\right)$, as previously discussed in Section 3.1 with reference to Table 4.

Hence, our simulation results corroborate the fundamental contribution of this paper, because they show that an equilibrium can be inferred when $y_t~I\left(0\right)$ using the three bounds tests. This contrasts with PSS, where an equilibrium can only be identified by cointegration when $y_t~I\left(1\right)$. It also contrasts with McNown et al. (2018) and Sam et al. (2019), who suggest that the rejection of the null hypothesis for all three tests means that the level variables cointegrate and the levels’ dependent variable is $I\left(1\right)$. We show that rejection of all three null hypotheses can occur when $y_t~I\left(0\right)$ and does not necessarily require the equilibrium to involve cointegration.

However, the following smaller sample (200 observations or less) implications of the simulation results reported in Table 6 are noteworthy. First, the rejection rates for the $F_x$ test obtained using the lower bound critical values are never below those using the upper bound critical values. This is expected, because the former critical values are smaller than the latter. Second, rejection rates decline as the sample size falls and can be below 5% for the smallest sample size considered $(T=30)$. (Very near) 100% power is only assured when $T\ge 500$. Therefore, the power of the $F_x$ test varies substantially with the sample size. Third, simulations with faster speeds of adjustment $({\pi }_y=-0.50)$, see DGPs 3, 6, 9, and 12, have higher power than simulations with slower adjustment speeds $({\pi }_y=-0.10)$, see DGPs 1, 4, 7, and 10. For example, the power of $F_x$ based on the upper bound critical value for case 4 when $T=100$ is 11.8% for simulation 1 $({\pi }_y=-0.10)$, 45.9% for simulation 2 $\left({\pi }_y=-0.25\right),$ and 90.3% for simulation 3 $({\pi }_y=-0.50)$. Hence, the power of the test can vary substantially with the speed of adjustment in small samples. Fourth, simulations where ${\beta }_{10}$ and ${\beta }_{11}$ are both positive (DGPs 1, 2, 3, 7, 8, and 9) generally have lower power than simulations where ${\beta }_{10}>0$ and ${\beta }_{11}<0$ (DGPs 4, 5, 6, 10, 11, and 12). These differences are due to short-run dynamics in the regressors and not long-run effects, given that ${\theta }_{x1}=1$ for all DGPs considered. Fifth, simulations where the error term follows a moving average (MA) process (DGPs 7 to 12) generally have greater power than those where the error terms are not MA processes (DGPs 1 to 6).21 Sixth, the power of the $F_x$ test is typically greater for the case 2 deterministic specification than for case 4.

Overall, when all level variables are $I\left(0\right)$, the main variations in the power of the $F_x$ test are due to the sample size and the magnitude of the adjustment coefficient. When $\left|{\pi }_y\right|=0.25$, good power (at least 90%) of the $F_x$ test is obtained for case 2 when $T\ge 100$ and for case 4 when $T\ge 200$. Larger sample sizes will likely be required to obtain good power for slower speeds of adjustment. Therefore, to allow for the possibility that $y_t~I\left(0\right)$, reasonably large sample sizes may be required to uncover an equilibrium when it exists.

4. Empirical Application with Results and Discussion

Absolute purchasing power parity (PPP) modified to permit constant trade barriers and constant transport costs ($\delta \ne 1$) without proportionality (${\theta }_{x1}=1$) imposed is $S_t=\delta {\left({\displaystyle \frac{P_{1,t}}{P_{2,t}}}\right)}^{{\theta }_{x1}}$. Where $S_t$ is the nominal exchange rate in period $t$ (units of country 1’s currency per unit of country 2’s currency), $P_{1,t}$ denotes country 1’s price level, and $P_{2,t}$ is country 2’s price level. Our empirical application is based on the following logarithmic form of PPP with stochastic error term, $u_t$, added.

$$ln{S}_t={\theta }_0+{\theta }_{1}t+{\theta }_{x1}ln\left({\displaystyle \frac{P_{1,t}}{P_{2,t}}}\right)+u_t, {\theta }_0=ln\left(\delta \right).$$

(17)

We consider specifications with a time trend added to allow for different weights used in different countries’ price indices and/or the Balassa–Samuelson effect that incorporate some non-traded goods (see A. M. Taylor, 2002, p. 143). However, because our use of disaggregated price data may render the trend redundant, we also consider specifications without a trend.

4.1. PPP Literature Review

PPP holding in the short run is generally rejected because exchange rates are more volatile than aggregate relative prices (A. M. Taylor & Taylor, 2004), and that price stickiness can cause $S_t$ to deviate from its PPP equilibrium in the short run while returning to this value in the long run (Dornbusch, 1976). Testing PPP since the late 1970s has focused on whether it holds in the long run, using a variety of unit root, stationarity, and cointegration tests.

Examples in the literature where there is general evidence supporting long-run PPP include Bahmani-Oskooee et al. (2015a), Bahmani-Oskooee et al. (2017), Bahramian and Saliminezhad (2021), De Villiers and Phiri (2022), A. M. Taylor and Taylor (2004), M. P. Taylor (2009), and Yilanci et al. (2024). There are also many examples where the evidence for long-run PPP holding is mixed. These include Bahmani-Oskooee and Hegerty (2009), Huang and Yang (2015), Bahmani-Oskooee et al. (2013), Bahmani-Oskooee et al. (2014a, 2014b), Bahmani-Oskooee et al. (2015b), Boundi-Chraki and Mateo Tomé (2022), She et al. (2021), Wu et al. (2018), and Zhang (2024). Further, Rogoff’s (1996) ‘PPP puzzle’ is that even when the real exchange rate $\left({RER}_t={\displaystyle \frac{S_t{P}_{2,t}}{P_{1,t}}}\right)$ is mean reverting the speed of adjustment is too slow (as measured by half-lives).

This evidence on whether long-run PPP holds does not generally refer to the strictest (core) form of absolute PPP, being $S_t={\displaystyle \frac{P_{1,t}}{P_{2,t}}}$. For example, Crownover et al. (1996) suggest that $ln\left({RER}_t\right)$ is usually constructed using price indices, which means that relative PPP is being tested. However, the specific form of relative PPP being tested when using price indices is equivalent to testing whether $ln\left({RER}_t\right)$ converges to a constant. We argue that this is a test of absolute PPP modified for constant transportation costs and barriers to trade (strong absolute PPP), that is, (17) with $\delta \ne 1$ and ${\theta }_{x1}=1$.

Testing PPP using price indices is also criticised by Imbs et al. (2005) because of aggregation bias, which can explain the large half-lives given in the PPP literature. A subsequent strand of research on the law of one price (LOP) and PPP used disaggregated price data (Crucini & Shintani, 2008; Bergin et al., 2013; and Robertson et al., 2014) with general conclusions that include the following. There is general support for long-run PPP/LOP with traded goods and little support with non-traded goods. Support for strong absolute PPP (with proportionality imposed, ${\theta }_{x1}=1$) is mixed, although weak absolute PPP (strong PPP without proportionality imposed, ${\theta }_{x1}\ne 1$) is generally supported.22

Pelagatti and Colombo (2015) demonstrate that even when LOP holds for the individual products that comprise a price index (such as the consumer price index, CPI), PPP will only hold if RER is a time invariant function of the price vectors in each nation’s price index.23 “Probably the main implication of our results is that the use of individual prices should be preferred to aggregate prices” (Pelagatti & Colombo, 2015, p. 914). In our empirical application, we use highly disaggregated data from Eurostat (the same source employed by Pelagatti and Colombo (2015)) to test weak absolute LOP (as specified by (17)) using the ARDL method.

4.2. Empirical Results and Discussion

Our empirical application involves two variables (the natural logarithms of the nominal exchange rate and relative prices). We consider specifications with trend ($c_1\ne 0$) and without trend ($c_1=0$), thereby applying tests based on cases 2, 3, and 4 utilising the following version of (2) with $K=1$:

$$\begin{array}{c}\Delta y_t=c_0+c_{1}t+{\pi }_y{y}_{t-1}+{\pi }_{x1}{x}_{1,t-1}+{\sum }_{i=1}^{p-1}{\psi }_{yi}\Delta y_{t-i}+{\omega }_1\Delta x_{1,t}+\\ {\sum }_{i=1}^{q_1-1}{\psi }_{1i}\Delta x_{1,t-i}+u_t.\end{array}$$

(18)

Cho et al. (2015, p. 293) suggest that any endogeneity of the contemporaneous differenced regressor, $\Delta x_{1,t}$, may be depicted as $u_t=d_1\Delta x_{1,t}+e_t$, which, after substitution into (18), gives

$$\begin{array}{c}\Delta y_t=c_0+c_{1}t+{\pi }_y{y}_{t-1}+{\pi }_{x1}{x}_{1,t-1}+{\sum }_{i=1}^{p-1}{\psi }_{yi}\Delta y_{t-i}+f_1\Delta x_{1,t}+\\ {\sum }_{i=1}^{q_1-1}{\psi }_{1i}\Delta x_{1,t-i}+e_t, f_1=\left({\omega }_1+d_1\right)\end{array}.$$

(19)

Estimation of (19) has two effects. First, it removes any contemporaneous correlation between $\Delta x_{1,t}$ and $e_t$ by construction, thereby avoiding any endogeneity problems from the inclusion of $\Delta x_{1,t}$. Second, because the coefficient on $\Delta x_{1,t}$ is $f_1$, the coefficients ${\omega }_1$ and $d_1$ are not separately identified. In our application, knowing the estimated values of ${\omega }_1$ and $d_1$ is not of concern because our interest is in testing whether an LOP equilibrium exists and identifying the parameters of any evident long-run levels relation (that are derived from the $c_0$, ${\pi }_y$, and ${\pi }_{x1}$ coefficients). Nevertheless, (19) features $y_{t-1}$ as an independent variable, which is non-contemporaneously correlated with the error term. This makes OLS applied to (19) intrinsically biased if it is consistent because $\Delta x_{1,t}$ is not contemporaneously correlated with $e_t$ in (19).

Assumption 3 in PSS on p. 293 specifies unidirectional causality among the level variables with the regressors determining the dependent variable and no (long-run) reverse causality, as well as restricting the maximum number of levels relationships between $y_t$ and the regressors to be one. If this assumption does not hold and $y_{t-1}$ determines $\Delta x_{1,t}$, the OLS estimation of (19) violates weak exogeneity. However, McNown et al.’s (2018) simulation evidence suggests little difference in terms of the size and power of the ARDL bounds tests when weak exogeneity is violated and when it is not (the bounds test is not adversely affected by the violation of weak exogeneity). Based on this, we estimate the single Equation (19) with $y_t=ln{S}_t$ and $x_{1,t}=ln\left({\displaystyle \frac{P_{1,t}}{P_{2,t}}}\right)=ln{RP}_t$ when testing LOP.

When we report estimated long-run models, we employ the delta method to calculate long-run coefficient standard errors. Pesaran and Shin (1997, pp. 10, 11, 23) suggest that the latter are “asymptotically valid” irrespective of whether $x_{k,t}$ are $I\left(1\right)$ or $I\left(0\right)$ and present simulation evidence demonstrating that the delta method can be “reliably” employed for testing hypotheses on long-run coefficients in small samples.24

We use monthly bilateral exchange rates and individual consumption item price index data from the Eurostat database (https://ec.europa.eu/eurostat/web/main/data/database, accessed on 14 September 2023). The natural logarithms of the number of Danish krone per euro (LS_DE) and number of Bulgarian lev per Danish krone (LS_BD) are our dependent variables (generically denoted $ln{S}_t$).25 Our regressors (generically denoted $ln{RP}_t$) are the natural logarithms of the ratio of Danish to Euro area prices for household textiles (LRP_HT_DE) and recreational and sporting services (LRP_RR_DE), as well as the logarithm of the ratio of Bulgarian to Danish pharmaceutical product prices (LRP_PH_BD). Our Denmark-euro (Bulgaria–Denmark) data has a maximum matched sample period of January 1999 to June 2023 (December 1999 to June 2023).26 This data was selected to illustrate the application of the ARDL bounds method when the dependent variable is likely to be $I\left(0\right)$.

We apply the ADF, Phillips–Perron (PP), GLS detrended Dickey–Fuller (DF-GLS) and Kwiatkowski et al. (1992), KPSS order of integration tests. Although such tests are not required to determine whether variables are $I\left(0\right)$ or $I\left(1\right)$ with the ARDL procedure, they are useful to ensure they are not $I\left(2\right)$, and because we are interested in the situation when $y_t=ln{S}_t~I\left(0\right)$ in a potential equilibrium.

The following procedure for testing for an equilibrium LOP relationship is employed. First, we estimate 156 ARDL test equation combinations assuming case 3 deterministics nested within a maximum lag order of $p=q_1=12$.

Second, we apply six sets of misspecification tests for autocorrelation, nonlinearity, non-normality, heteroscedasticity, autoregressive conditional heteroscedasticity (ARCH), and structural instability. Unreported misspecification test results (which are available online in the Supplementary Materials, along with the short-run equations to which they relate) indicate that no equations are free from all six forms of misspecification. For the two Denmark-euro models, there are numerous ARDL regressions where the only form of evident misspecification is non-normality.27 Given our relatively large sample size (exceeding 280 observations), we appeal to the central limit theorem and assume that coefficient distributions are asymptotically as expected, despite non-normal residuals. From the models where non-normality is the only form of evident misspecification, we select the ARDL specification for bounds testing with the minimum Akaike information criterion (AIC). For the Bulgaria–Denmark equations, we select the model for bounds testing with the minimum AIC from the models with the fewest (three) forms of misspecification (non-normality, heteroscedasticity, and ARCH). Evident heteroscedasticity is addressed with White’s heteroscedasticity consistent standard errors, and evident ARCH effects are explicitly modelled. For the model with ARCH effects, we utilise Bollerslev and Wooldridge (1992) covariances that are robust to non-normality.28

Third, the selected ARDL model for case 3 (which is the same as for case 2) is re-estimated over the maximum available sample, as is the same ARDL model, except with a time trend added to accommodate case 4 testing. Fourth, we apply the bounds testing procedure for case 2, 3, and 4, because the disaggregated price data are indices that make a non-zero intercept and/or trend possible.29 If $H_0^y$, $H_0^{xy}$, and $H_0^x$ are all rejected, this is consistent with the existence of a weak absolute LOP (WALOP) without the proportionality restriction imposed equilibrium. We also apply $F_{x_{\theta }}$ to test $H_0^{x_{\theta }}: {\theta }_0={\theta }_{x1}=0$ (case 2), $H_0^{x_{\theta }}: {\theta }_{x1}=0$ (case 3), and $H_0^{x_{\theta }}: {\theta }_1={\theta }_{x1}=0$ (case 4).30 Our implementation of such joint coefficient tests extends the procedure suggested by Kripfganz and Schneider (2023), who applied single coefficient t-tests in the equilibrium equation. Fifth, if an equilibrium is supported, we test the proportionality restriction (${\theta }_{x1}=1$ in (17)) to determine whether a long-run strong absolute LOP (SALOP) holds.

Table 7. Order of integration tests.

Currency	Bulgaria–Denmark
Variable	LS_BD		LRP_PH_BD
Hypotheses	$\mathit{I}\left(\mathbf{1}\right) \mathbf{v} \mathit{I}\mathbf{\left(}\mathbf{0}\mathbf{\right)}$	$\mathit{I}\left(\mathbf{2}\right) \mathbf{v} \mathit{I}\mathbf{\left(}\mathbf{1}\mathbf{\right)}$	$\mathit{I}\left(\mathbf{1}\right) \mathbf{v} \mathit{I}\mathbf{\left(}\mathbf{0}\mathbf{\right)}$	$\mathit{I}\left(\mathbf{2}\right) \mathbf{v} \mathit{I}\mathbf{\left(}\mathbf{1}\mathbf{\right)}$
ADF	−3.427 *	−18.602 *	−3.858 *	−2.762
PP	−4.656 *	−20.285 *	−5.579 *	−15.163 *
DF-GLS	−1.897	−3.418 *	0.282	−2.791 *
KPSS	0.231	0.051	0.900 *	0.685 *
Currency	Denmark-Euro
Variable	LS_DE		LRP_HT_DE		LRP_RR_DE
Hypotheses	$\mathit{I}\left(\mathbf{1}\right) \mathbf{v} \mathit{I}\mathbf{\left(}\mathbf{0}\mathbf{\right)}$	$\mathit{I}\left(\mathbf{2}\right) \mathbf{v} \mathit{I}\mathbf{\left(}\mathbf{1}\mathbf{\right)}$	$\mathit{I}\left(\mathbf{1}\right) \mathbf{v} \mathit{I}\mathbf{\left(}\mathbf{0}\mathbf{\right)}$	$\mathit{I}\left(\mathbf{2}\right) \mathbf{v} \mathit{I}\mathbf{\left(}\mathbf{1}\mathbf{\right)}$	$\mathit{I}\left(\mathbf{1}\right) \mathbf{v} \mathit{I}\mathbf{\left(}\mathbf{0}\mathbf{\right)}$	$\mathit{I}\left(\mathbf{2}\right) \mathbf{v} \mathit{I}\mathbf{\left(}\mathbf{1}\mathbf{\right)}$
ADF	−3.737 *	−17.828 *	−1.477	−24.412 *	−0.136	−12.564 *
PP	−3.733 *	−17.917 *	−6.864 *	−48.650 *	−1.326	−23.863 *
DF-GLS	−2.617 *	−3.487 *	−1.457	−0.648	3.135	−12.241 *
KPSS	0.213	0.028	0.442	0.171	2.048 *	0.126

The currencies involved in the exchange rate are specified in the row labelled “Currency”. The variable to which the tests refer is given in the row labelled “Variable”. The row labelled “Hypotheses” specify the orders of integration that constitute the null and alternative hypotheses. The higher order of integration of the two specified in the “Hypotheses” row represents the null hypothesis for the ADF, PP, and DF-GLS tests. The lower order of integration of the two specified in the “Hypotheses” row represents the null hypothesis for the KPSS test. Test statistics are reported in the body of the table. The 5% critical values are −2.871 or −2.872 for the ADF and PP tests (respectively), −1.942 for the DF-GLS test, and 0.463 for the KPSS test. The modified AIC is used to choose the number of lagged dependent variables for the ADF and DF-GLS tests. An asterisk (*) denotes rejection of the null hypothesis at the 5% level of significance. All results were produced using EViews 11.

reports integration order test results. The dependent variable for the Bulgaria–Denmark models (LS_BD) is $I\left(0\right)$ according to the ADF, PP, and KPSS tests, whereas the DF-GLS test suggests it is $I\left(1\right)$. Hence, LS_BD is very likely $I\left(0\right)$ and almost certainly not $I\left(2\right)$. There is also uncertainty over the order of integration of LRP_PH_BD. It is likely $I\left(0\right)$ according to the ADF and PP tests, $I\left(1\right)$ according to DF-GLS, and $I\left(2\right)$ according to KPSS. Low power may cause some tests to suggest a higher order of integration than should be assigned, and we cannot discount the possibility that LRP_PH_BD is $I\left(0\right)$ and can therefore have an equilibrium relationship with LS_BD. The dependent variable for the Denmark-euro equations (LS_DE) is $I\left(0\right)$ according to all tests, while LRP_RR_DE is unambiguously $I\left(1\right),$ suggesting that there should be no equilibrium between LS_DE and LRP_RR_DE. LRP_HT_DE is $I\left(0\right)$ according to the PP and KPSS tests, $I\left(1\right)$ according to ADF, and $I\left(2\right)$ according to DF-GLS. Once again, the low power of tests means we cannot discount the possibility that LRP_HT_DE is $I\left(0\right)$ and can therefore have an equilibrium relationship with LS_DE.

Table 8. Bounds tests.

Currency	Bulgaria–Denmark		Denmark-Euro
Regressor	LRP_PH_BD	LRP_PH_BD	LRP_HT_DE	LRP_RR_DE
Model	ARDL(1,1)	ARDL(1,1) ARCH(5)	ARDL(1,3)	ARDL(1,1)
	Case 2 and Case 3
$t_y$	−5.635 {−2.868*, −3.224 *}	−6.008 {−2.868 *, −3.224 *}	−4.077 {−2.865 *, −3.226 *}	−3.662 {−2.869 *, −3.224 *}
	Case 2
$F_{xy}$	10.830 {3.636 *, 4.158 *}	12.346 {3.636 *, 4.158 *}	5.651 {3.629 *, 4.163 *}	4.490 {3.638 *, 4.155 *}
$F_x$	15.964 {4.866 *, 5.683 *}	18.316 {4.866 *, 5.683 *}	8.442 {4.847 *, 5.685 *}	6.720 {4.859 *, 5.674 *}
$F_{x_{\theta }}$	16,373,960.207 [0.000] *	19,065,236.642 [0.000] *	18,588,586.026 [0.000] *	11,858,472.960 [0.000] *
	Case 3
$F_{xy}$	15.960 {4.948 *, 5.758 *}	18.312 {4.948 *, 5.758 *}	8.441 {4.933 *, 5.761 *}	6.721 {4.945 *, 5.749 *}
$F_x$	5.615 {3.924 *, 7.179}	8.498 {3.924 *, 7.179 *}	2.072 {4.013, 7.194}	0.185 {3.914, 7.164}
$F_{x_{\theta }}$	6.384 [0.012] *	12.699 [0.000] *	2.599 [0.108]	0.190 [0.663]
	Case 4
$t_y$	−5.712 {−3.424 *, −3.704 *}	−6.653 {−3.424 *, −3.704 *}	−4.069 {−3.416 *, −3.703 *}	−3.642 {−3.422 *, −3.702}
$F_{xy}$	11.020 {4.733 *, 5.226 *}	17.011 {4.733 *, 5.226 *}	5.615 {4.712 *, 5.223 *}	4.467 {4.724 *, 5.214}
$F_x$	3.004 {5.611 *, 6.418}	7.138 {5.611 *, 6.418}	1.042 {5.607, 6.423}	0.096 {5.608, 6.410}
$F_{x_{\theta }}$	3.383 [0.035] *	9.892 [0.000] *	1.312 [0.271]	0.098 [0.907]
Observations	282	282	291	293

See notes for Table 7 except for the following. The bounds t-test for $H_0^y$ is denoted as $t_y$. $F_{xy}$ and $F_x$ denote the bounds F-tests for $H_0^{xy}$ and $H_0^x$, respectively, while $F_{x_{\theta }}$ represents the F-test of $H_0^{x_{\theta }}$ (where zero restrictions are applied to both intercept [trend] and slope coefficients for case 2 [case 4]). The delta method is used to calculate $F_{x_{\theta }}$. For the Bulgaria–Denmark equations, White’s heteroscedasticity consistent covariance is used for test statistics in the model without an ARCH process and the Bollerslev–Wooldridge covariance in the model with ARCH effects. The standard OLS covariance matrix is used to calculate test statistics for Denmark-euro equations. Probability values based on the conventional F-distribution are reported in square brackets, while lower (first number) and upper (second number) 5% critical values for the bound tests are reported in braces. Lower and upper bound critical values are simulated for the specific sample size (given in the row labelled “Observations”) and values of $p$ and $q_1$ of the estimated ARDL($p$,$q_1$) specification indicated in the row labelled “Model”. The simulated critical values do not account for any ARCH process or (robust) adjustment to the covariance matrix. An asterisk (*) denotes rejection of the null hypothesis at the 5% level of significance.

reports the bounds tests for an equilibrium and

Table 9. Equilibrium models.

Currency	Bulgaria–Denmark		Denmark-Euro
Regressor	LRP_PH_BD	LRP_PH_BD	LRP_HT_DE	LRP_RR_DE
Model	ARDL(1,1)	ARDL(1,1) ARCH(5)	ARDL(1,3)	ARDL(1,1)
	Case 2 and Case 3
Adjustment $\left({\pi }_y\right)$	−0.191	−0.171	−0.109	−0.089
Intercept $\left({\theta }_0\right)$	−1.338 (−5439.383) *	−1.338 (−5040.970) *	2.009 (3227.496) *	2.008 (4556.664) *
$lnRP$ (${\theta }_{x1}$)	0.005 (2.527) *	0.010 (3.564) *	0.021 (1.612)	0.003 (0.436)
Proportionality	−540.152 [0.000] *	−358.013 [0.000] *	−73.850 [0.000] *	−171.328 [0.000] *
	Case 4
Adjustment $\left({\pi }_y\right)$	−0.193	−0.185	−0.110	−0.089
Trend $\left({\theta }_1\right)$	$2.13 \times$ 10−6 (0.542)	$6.34 \times$ 10−6 (2.163) *	$5.66 \times$ 10−7 (0.141)	$-2.78 \times$ 10−6 (−0.082)
$lnRP$ (${\theta }_{x1}$)	0.004 (2.066) *	0.008 (3.179) *	0.021 (1.548)	0.006 (0.143)
Proportionality	−492.832 [0.000] *	−390.455 [0.000] *	−72.276 [0.000] *	−24.267 [0.000] *

See notes for Table 8 except for the following. The equilibrium coefficients with t-ratios in parentheses are reported for (3)—the equilibrium should exclude the intercept for case 3. The row labelled “Adjustment $\left({\pi }_y\right)$” is the adjustment coefficient in the short-run ARDL model (2). The delta method is used to calculate test statistics based on the forms of covariance matrices specified in the notes for Table 8. An asterisk (*) denotes rejection of the null hypothesis at the 5% level of significance.

gives the corresponding long-run ARDL models. For virtually all models for case 2, 3, and 4, the $t_y$ and $F_{xy}$ bounds tests reject their respective null hypotheses $H_0^y$ and $H_0^{xy}$. The exception is the Denmark-euro case 4 specification with the LRP_RR_DE regressor, where both tests are indecisive.

For case 2, the $F_x$ and $F_{x_{\theta }}$ tests reject $H_0^x$ and $H_0^{x_{\theta }}$ for all models which, given the rejection of $H_0^y$ and $H_0^{xy}$, suggests that a restricted intercept equilibrium exists for all specifications. However, in the equilibrium models reported in Table 9, we observe that, while the intercept is significant in all long-run models, the coefficient on $lnRP$ is only significant in the two Bulgaria–Denmark specifications. Hence, the evidence from case 2 supports an equilibrium LOP equation for the Bulgaria–Denmark models and rejects an LOP equilibrium for both Denmark-euro specifications (where the tests suggest $ln{S}_t$ is $I\left(0\right)$ around a non-zero mean). For both Bulgaria–Denmark models, the proportionality restriction (${\theta }_{x1}=1$) is rejected (see Table 9), suggesting WALOP is supported.

For case 3 and 4, $F_x$ and $F_{x_{\theta }}$ cannot reject $H_0^x$ and $H_0^{x_{\theta }}$ for both Denmark-euro models, which therefore rejects the existence of an equilibrium for this currency (confirming the case 2 results). In contrast, for both Bulgaria–Denmark case 3 models, $H_0^x$ and $H_0^{x_{\theta }}$ are rejected, except that $F_x$ lies between the bounds for the ARDL(1,1) specification without ARCH effects. However, since $F_{x_{\theta }}$ unambiguously rejects $H_0^{x_{\theta }}$ for the ARDL(1,1) specification without ARCH effects, an unrestricted intercept equilibrium is supported for this Bulgaria–Denmark specification. Further, the equilibrium coefficient on $lnRP$ is significantly different from both zero and one (rejecting proportionality) for both Bulgaria–Denmark equations (see Table 9). Overall, the case 3 results support WALOP for the Bulgaria–Denmark specifications.

For case 4, for both Bulgaria–Denmark models, the $F_x$ statistics are indecisive as to whether to reject $H_0^x$; however, the $F_{x_{\theta }}$ tests unambiguously reject $H_0^{x_{\theta }}$ (see Table 8). While the equilibrium coefficient on $lnRP$ is significantly different from zero (and one) for both Bulgaria–Denmark equations, the long-run trend coefficient is only significant in the specification with ARCH effects (see Table 9). This suggests that a restricted trend equilibrium is only supported for the Bulgaria–Denmark model with ARCH effects and is thereby consistent with WALOP.

In summary, an LOP equilibrium is rejected for both Denmark-euro specifications, whereas a weak LOP (without the proportionality restriction imposed) is supported for the Bulgaria–Denmark models for cases 2 and 3; this support only extends to case 4 for the equation with ARCH effects. Notably, our unit root tests suggested that both ${lnS}_t$ and ${lnRP}_t$ are likely $I\left(0\right)$ for the Bulgaria–Denmark models, indicating that our ARDL models have uncovered an equilibrium when all level variables are probably $I\left(0\right)$. This supports the contention of our paper that the three ARDL tests can detect an equilibrium when the dependent variable is $I\left(0\right)$. Our unit root tests also indicated that ${lnS}_t$ is likely $I\left(0\right)$ for the Denmark-euro equations, while ${lnRP}_t$ may be $I\left(0\right)$ for LRP_HT_DE and is unambiguously $I\left(1\right)$ for LRP_RR_DE. This is consistent with our rejection of an LOP equilibrium for both Denmark-euro models and with the ability of the three ARDL tests to reject an equilibrium when the dependent variable is $I\left(0\right)$—both when the regressor is $I\left(0\right)$ and when it is $I\left(1\right)$.

The speeds of adjustment (the magnitude of ${\pi }_y$) reported in Table 9 are in the range 0.171 to 0.193 per month for the Bulgaria–Denmark models. Our simulation results reported in Table 6 suggested that the sample size and speed of adjustment are the main determinants of power for the $F_x$ test when the dependent variable is $I\left(0\right)$, which we believe is likely in our empirical application. Good power is indicated by our simulations for the adjustment speeds and sample size of our Bulgaria–Denmark models (282 observations), which is consistent with our general finding of equilibria for these specifications. While the Denmark-euro models have similar sample sizes (291 to 293 observations) and lower adjustment speeds (0.089 to 0.110 per month), our simulations indicate that the $F_x$ test should still have good power, suggesting that our finding of no equilibrium LOP for the Denmark-euro models is unlikely due to low power.31

5. Conclusions

The first contribution of this paper is to explain and demonstrate that using all three ARDL bounds tests allows an equilibrium to be detected when the dependent variable is $I\left(0\right)$. This crucially requires a third test for the joint significance of all level regressors in either the short-run model (McNown et al., 2018; Sam et al., 2019) or long-run equation (Kripfganz & Schneider, 2020, 2023). However, none of the authors who suggest such a third test provide a justification for using all three tests to determine whether an equilibrium exists when the dependent variable is $I\left(0\right)$. We explicitly provide this justification and report simulation results to support our argument. Use of the three ARDL tests to determine whether an equilibrium exists when the dependent variable is $I\left(0\right)$ is illustrated with an application to LOP equations. In this application we are, as far as we are aware, the first to apply the $F_{x_{\theta }}$ test for joint restrictions for the restricted intercept and restricted trend cases.

The second contribution of our paper is the production of previously unavailable upper and lower bound critical values for the $F_x$ test for case 2 and case 4. These are provided for a broad range of sample sizes and number of regressors.

One possible direction for future research is to introduce a third test into other ARDL bounds testing methods, such as those discussed in Cho et al. (2023), so that an equilibrium can additionally be detected when the dependent variable is $I\left(0\right)$.

We are grateful to a reviewer for the following additional suggestions for further work: to develop ARDL testing procedures that can accommodate multiple structural breaks and regime switching mechanisms; to consider the sensitivity of size and power simulations to DGPs with near unit roots, conditional heteroscedasticity, and non-Gaussian errors, cointegrated regressors, fractional integration, other deterministic cases, and lag specifications. To produce surface response functions and corresponding probability values for the $F_x$ test and compare them with those produced by block and wild bootstrap methods. To produce a Monte Carlo experiment comparing the performance of the $F_x$ and $F_{x_{\theta }}$ tests to determine whether one should be preferred as the third bounds test.