2.1 NARDL and Error-Correction Models

We consider a NARDL( $\!\! p,q$ ) process augmented by a time trend:

(1) $$ \begin{align} y_{t}= \alpha_{0\ast} + \xi_{\ast} t + \sum_{j=1}^{p}\phi _{j\ast}y_{t-j}+\sum_{j=0}^{q}(\boldsymbol{\theta}_{j\ast}^{+\prime }\boldsymbol{x}_{t-j}^{+}+\boldsymbol{\theta}_{j\ast}^{-\prime }\boldsymbol{x}_{t-j}^{-}) + e_{t}, \end{align} $$

where $\boldsymbol {x}_{t}\in \mathbb {R}^{k}\, (k\in \mathbb {N})$ , $\boldsymbol {x}_{t}^{+}:=\sum _{j=1}^{t}\Delta \boldsymbol {x}_{j}^{+}$ , $\boldsymbol {x}_{t}^{-}:=\sum _{j=1}^{t}\Delta \boldsymbol {x}_{j}^{-}$ , $\Delta \boldsymbol {x}_{t}^{+}:=\max [\mathbf {0},\Delta \boldsymbol {x}_{t}]$ , $\Delta \boldsymbol {x}_{t}^{-}:=\min [\mathbf {0},\Delta \boldsymbol {x}_{t}]$ , $\{e_{t},\mathcal {F}_{t}\}$ is a martingale difference array (MDA), and $\mathcal {F}_{t}$ is the smallest $\sigma $ -algebra driven by $\{y_{t-1},\boldsymbol {x}_{t}^{+},\boldsymbol {x}_{t}^{-},\, y_{t-2}, \boldsymbol {x}_{t-1}^{+},\boldsymbol {x}_{t-1}^{-},\ldots \}$ such that $\Delta \boldsymbol {x}_{t}$ is a stationary process, and $\Delta $ denotes the differencing operator, so that $\Delta \boldsymbol {x}_{t} := \boldsymbol {x}_{t} - \boldsymbol {x}_{t-1}$ . Here, the $\max $ and $\min $ operators applied to a vector operate element-wise. The NARDL process in (1) is defined by introducing the asymmetric effect to the autoregressive distributed lag (ARDL) process proposed by Pesaran and Shin (Reference Pesaran, Shin and Strom1998). By supposing that the response of dependent variable $y_{t}$ to the positive and negative parts of $\boldsymbol {x}_{t}$ is possibly different, the NARDL process generalizes the ARDL process. If $\boldsymbol {\theta }_{j*}^{+} = \boldsymbol {\theta }_{j*}^{-}$ for each $j$ , the NARDL process reduces to the ARDL process. In addition, the NARDL process in (1) is more general than those defined by Shin et al. (Reference Shin, Yu, Greenwood-Nimmo, Horrace and Sickles2014) and Cho, Greenwood-Nimmo, and Shin (Reference Cho, Greenwood-Nimmo and Shin2023b), because the latter do not allow for a time trend on the right side.

The NARDL process is closely related to an error-correction representation. We first rewrite (1) as follows:

$$ \begin{align*} \Delta y_{t} &= \rho_{\ast}y_{t-1} + \boldsymbol{\theta}_{\ast}^{+\prime }\boldsymbol{x}_{t-1}^{+}+\boldsymbol{\theta}_{\ast}^{-\prime }\boldsymbol{x}_{t-1}^{-}+ \xi_{\ast} (t-1) + \alpha_{\ast} + \sum_{j=1}^{p-1}\varphi_{j\ast}\Delta y_{t-j}\\&\quad +\sum_{j=0}^{q-1}\left( \boldsymbol{\pi}_{j\ast}^{+\prime }\Delta \boldsymbol{x}_{t-j}^{+}+\boldsymbol{\pi}_{j\ast}^{-\prime }\Delta \boldsymbol{x}_{t-j}^{-}\right) + e_{t}, \end{align*} $$

where $\rho _{\ast }$ , $\boldsymbol {\theta }_{\ast }^{+}$ , $\boldsymbol {\theta }_{\ast }^{-}$ , $\varphi _{j\ast }$ ( $\!\! j=1,2,\ldots ,p-1$ ), $\boldsymbol {\pi }_{j\ast }^{+}$ , and $\boldsymbol {\pi }_{j\ast }^{-}$ ( $\!\! j=0,1,\ldots ,q-1$ ) are defined by the parameters in (1), as given in Cho et al. (Reference Cho, Greenwood-Nimmo and Shin2023b). Specifically, $\rho _{*} := \sum _{j=1}^{p} \phi _{j*} - 1$ , $\boldsymbol {\theta }_{*}^{+} := \sum _{j=0}^{q} \boldsymbol {\theta }_{j*}^{+}$ , $\boldsymbol {\theta }_{*}^{-} := \sum _{j=0}^{q} \boldsymbol {\theta }_{j*}^{-}$ , $\boldsymbol {\pi }_{0*}^{+} := \boldsymbol {\theta }_{0*}^{+},\;\;\;\;\boldsymbol {\pi }_{0*}^{-} := \boldsymbol {\theta }_{0*}^{-}$ , and for $\ell =1, 2, \ldots , p-1$ and $j = 1, 2, \ldots q-1$ , $\varphi _{\ell *} := - \sum _{i=\ell +1}^{p} \phi _{i*}$ , $\boldsymbol {\pi }_{j*}^{+} := -\sum _{i=j+1}^{q}\boldsymbol {\theta }_{i*}^{+}$ , $\boldsymbol {\pi }_{j*}^{-} := -\sum _{i=j+1}^{q}\boldsymbol {\theta }_{i*}^{-}$ , and $\alpha _{*} = \alpha _{0*} + \xi _{*}$ . By plugging these parameters into (2), the NARDL( $\!\! p$ , $q$ ) process in (1) is obtained. For notational simplicity, we further rewrite this as

(2) $$ \begin{align} \Delta y_{t}=\rho_{\ast} y_{t-1} + \boldsymbol{\theta}_{\ast}^{\prime} \ddot{\boldsymbol{x}}_{t-1} + \xi_{\ast}(t-1) + \alpha_{\ast} + \boldsymbol{\kappa}_{\ast}^{\prime} \boldsymbol{z}_{2t} + e_{t} \end{align} $$

by letting $\boldsymbol {\theta }_{\ast } := [\boldsymbol {\theta }_{*}^{+\prime }, \boldsymbol {\theta }_{*}^{-\prime }]^{\prime }$ , $\ddot {\boldsymbol {x}}_{t-1} := [\boldsymbol {x}_{t-1}^{+\prime },\boldsymbol {x}_{t-1}^{-\prime }]^{\prime }$ , $\boldsymbol {\kappa }_{*} :=[\boldsymbol {\varphi }_{*}^{\prime }, \boldsymbol {\pi }_{*}^{\prime }]^{\prime }$ , and $\boldsymbol {z}_{2t}:= [ \Delta \boldsymbol {y}_{t-1}^{\prime }, \Delta \widetilde {\boldsymbol {x}}_{t}^{\prime }]^{\prime }$ , such that

$$ \begin{align*} \boldsymbol{\varphi}_{\ast} &:= [\varphi_{1\ast}, \ldots, \varphi_{p-1\ast}]^{\prime },\;\;\; \Delta \boldsymbol{y}_{t-1} := [\Delta y_{t-1}, \ldots, \Delta y_{t-p+1}]^{\prime },\\ \boldsymbol{\pi}_{*} &:= [\boldsymbol{\pi}_{*}^{+\prime}, \;\;\; \boldsymbol{\pi}_{*}^{-\prime}]^{\prime}, \;\;\; \boldsymbol{\pi}_{\ast}^{+} := [\boldsymbol{\pi}_{0\ast}^{+\prime}, \ldots, \boldsymbol{\pi}_{q-1\ast}^{+\prime}]^{\prime},\;\;\; \boldsymbol{\pi}_{\ast}^{-} := [\boldsymbol{\pi}_{0\ast}^{-\prime}, \ldots, \boldsymbol{\pi}_{q-1\ast}^{-\prime}]^{\prime },\\ \Delta \widetilde{\boldsymbol{x}}_{t} &:= [\Delta \widetilde{\boldsymbol{x}}_{t}^{+ \prime}, \Delta \widetilde{\boldsymbol{x}}_{t}^{- \prime}]^{\prime},\;\;\; \Delta \widetilde{\boldsymbol{x}}_{t}^{+} := [\Delta \boldsymbol{x}_{t}^{+ \prime}, \ldots, \Delta \boldsymbol{x}_{t-q+1}^{+ \prime}]^{\prime},\;\;\;\text{and}\\ \Delta \widetilde{\boldsymbol{x}}_{t}^{-} &:= [\Delta \boldsymbol{x}_{t}^{- \prime}, \ldots, \Delta \boldsymbol{x}_{t-q+1}^{- \prime}]^{\prime}. \end{align*} $$

Here, if $y_{t-1}$ is cointegrated with $\ddot {\boldsymbol {x}}_{t-1}$ , (2) can be rewritten into the following error-correction form:

(3) $$ \begin{align} \Delta y_{t} = \rho_{\ast}u_{t-1} + \gamma_{\ast} + \boldsymbol{\kappa}_{\ast}^{\prime} \boldsymbol{z}_{2t} + e_{t}, \end{align} $$

such that the cointegration error is defined as $u_{t-1} := y_{t-1} - \boldsymbol {\beta }_{\ast }^{\prime } \ddot {\boldsymbol {x}}_{t-1} - \zeta _{\ast } (t-1) - \nu _{\ast }$ with $ \boldsymbol {\beta }_{*}:= [\boldsymbol {\beta }_{*}^{+\prime }, \boldsymbol {\beta }_{*}^{-\prime }]^{\prime }$ , $\boldsymbol {\beta }_{\ast }^{+}:=-(\boldsymbol {\theta }_{\ast }^{+}/\rho _{\ast })$ , $\boldsymbol {\beta }_{\ast }^{-}:=-(\boldsymbol {\theta }_{\ast }^{-}/\rho _{\ast })$ , and $\zeta _{\ast } := -(\xi _{\ast }/\rho _{\ast })$ . Here, the intercept $\nu _{\ast }$ is introduced so that $\mathbb {E}[u_{t}] = 0$ , and $u_{t}$ is assumed to be stationary and possibly correlated with $\Delta \boldsymbol {y}_{t-1}$ and $\Delta \widetilde {\boldsymbol {x}}_{t}$ . From this, it follows that $\gamma _{\ast } = \alpha _{\ast } + \rho _{\ast } \nu _{\ast }$ .

The NARDL process captures an asymmetric cointegrating relationship between nonstationary processes. If we let $\boldsymbol {\mu }_{*}:=[\boldsymbol {\mu }_{\ast }^{+\prime }, \boldsymbol {\mu }_{\ast }^{-\prime }]^{\prime }$ , $\boldsymbol {\mu }_{\ast }^{+} := \mathbb {E}[\Delta \boldsymbol {x}_{t}^{+}],$ and $\boldsymbol {\mu }_{\ast }^{-} := \mathbb {E}[\Delta \boldsymbol {x}_{t}^{-}]$ , then $\boldsymbol {\mu }_{\ast }^{+} + \boldsymbol {\mu }_{\ast }^{-} \equiv \mathbb {E}[\Delta \boldsymbol {x}_{t}]$ by construction. Therefore, if we further let $\boldsymbol {s}_{t}^{+} := \Delta \boldsymbol {x}_{t}^{+} - \boldsymbol {\mu }_{\ast }^{+}$ and $\boldsymbol {s}_{t}^{-} := \Delta \boldsymbol {x}_{t}^{-} - \boldsymbol {\mu }_{\ast }^{-}$ , it follows that

(4) $$ \begin{align} \ddot{\boldsymbol{x}}_{t} &= \boldsymbol{\mu}_{\ast} t + \mathbf{m}_{t} \;\;\;\text{by letting}\;\;\; \mathbf{m}_{t}:= [\mathbf{m}_{t}^{+\prime}, \mathbf{m}_{t}^{-\prime}]^{\prime},\nonumber\\ \mathbf{m}_{t}^{+} &:= \sum_{j=1}^{t} \boldsymbol{s}_{j}^{+}\;\;\;\text{and}\;\;\; \mathbf{m}_{t}^{-} := \sum_{j=1}^{t} \boldsymbol{s}_{j}^{-}. \end{align} $$

From (4), $\ddot {\boldsymbol {x}}_{t}$ is clearly a unit-root process with nonzero time trends. Moreover, $\Delta y_{t}$ is not necessarily distributed around zero, even when $\boldsymbol {x}_{t}$ is a unit-root process without a time trend. From (3), we have

$$ \begin{align*} \delta_{\ast}& := \mathbb{E}[\Delta y_{t}] = \frac{1}{\varrho_{\ast}} \left[\gamma_{\ast} + \sum_{j=0}^{q-1} \boldsymbol{\pi}_{j*} \boldsymbol{\mu}_{\ast} \right],\;\;\;\text{where}\;\;\; \boldsymbol{\pi}_{j*}:= [ \boldsymbol{\pi}_{j\ast}^{+ \prime}, \boldsymbol{\pi}_{j\ast}^{- \prime } ]^{\prime}\;\;\;\text{and}\\\varrho_{\ast}& := 1 - \sum_{j=1}^{p-1} \varphi_{j\ast}, \end{align*} $$

so that if $d_{t} := \Delta y_{t} - \delta _{\ast }$ , then

(5) $$ \begin{align} y_{t} = \delta_{\ast} t + \sum_{j=1}^{t} d_{j}. \end{align} $$

This fact implies that $y_{t}$ is a unit-root process with a deterministic time trend, which has further implications. First, even when $\mathbb {E}[\boldsymbol {x}_{t}] = \mathbf {0}$ , the NARDL model captures a cointegrating relationship between the variables without and with time trend, viz., $\boldsymbol {x}_{t}$ and $y_{t}$ . Second, the NARDL model includes the time trend in (2) as an explanatory variable by noting that the trend in $y_{t}$ could also be influenced by exogenous factors other than the NARDL effect. If the estimated coefficient of the time trend is insignificant, it signals that the trend in $y_{t}$ is sorely sourced from the NARDL effect.

2.2 OLS for NARDL and Asymptotic Singular Matrix

Shin et al. (Reference Shin, Yu, Greenwood-Nimmo, Horrace and Sickles2014) estimate (2) by OLS without distinguishing the long- and short-run equations, which is now very popular in the empirical literature. For this examination, we first let

$$ \begin{align*} \boldsymbol{z}_{t} &:= {[ \begin{array}{c:c} \boldsymbol{z}_{1t}^{\prime} & \boldsymbol{z}_{2t}^{\prime} \end{array} ]}^{\prime } := {[\begin{array}{cc:c} y_{t-1} & \boldsymbol{r}_{t-1}^{\prime} & \boldsymbol{z}_{2t}^{\prime} \end{array}]}^{\prime} := {[\begin{array}{cccc:c} y_{t-1} & \ddot{\boldsymbol{x}}_{t-1}^{\prime} & (t-1) & 1 & \boldsymbol{z}_{2t}^{\prime} \end{array}]}^{\prime}, \end{align*} $$

so that $\boldsymbol {r}_{t-1} = [\ddot {\boldsymbol {x}}_{t-1}^{\prime }, (t-1), 1]^{\prime }$ . Here, $\boldsymbol {z}_{t} \in \mathbb {R}^{2 + p + 2k(1+q)}$ is partitioned into two variables such that $\boldsymbol {z}_{1t} \in \mathbb {R}^{3+2k}$ and $\boldsymbol {z}_{2t} \in \mathbb {R}^{p+2kq-1}$ . Next, we let $\boldsymbol {\alpha }_{\ast } := [ \begin {array}{cccc:c} \rho _{\ast } & \boldsymbol {\theta }_{\ast }^{\prime } & \xi _{\ast } & \alpha _{\ast } & \boldsymbol {\kappa }_{\ast }' \end {array} ]^{\prime }$ . From this, the OLS estimator is written as

$$ \begin{align*} \widehat{\boldsymbol{\alpha}}_{T} := [ \begin{array}{cccc:c} \widehat{\rho}_{T} & \widehat{\boldsymbol{\theta}}_{T}^{\prime} & \widehat{\xi}_{T} & \widehat{\alpha}_{T} & \widehat{\boldsymbol{\kappa}}_{T}' \end{array} ]^{\prime } := \left( \sum_{t=1}^{T} \boldsymbol{z}_{t} \boldsymbol{z}_{t}^{\prime }\right)^{-1} \sum_{t=1}^{T} \boldsymbol{z}_{t} \Delta y_{t}. \end{align*} $$

For later purposes, we partition $\widehat {\boldsymbol {\theta }}_{T}$ and $\widehat {\boldsymbol {\kappa }}_{T}$ into $[\widehat {\boldsymbol {\theta }}_{T}^{+\prime }, \widehat {\boldsymbol {\theta }}_{T}^{-\prime }]^{\prime }$ and $[\widehat {\boldsymbol {\varphi }}_{T}^{\prime }, \widehat {\boldsymbol {\pi }}_{T}^{\prime }]^{\prime } \equiv [\widehat {\boldsymbol {\varphi }}_{T}^{\prime }, \widehat {\boldsymbol {\pi }}_{T}^{+\prime }, \widehat {\boldsymbol {\pi }}_{T}^{-\prime }]^{\prime }$ , respectively.

Despite its popularity, there is no theoretical foundation for the OLS. This is mainly attributed to the asymptotically singular matrix problem. Even under mild regularity conditions, the OLS estimator suffers from an asymptotically singular matrix problem. For this discussion, we impose the following conditions, which are standard in the literature.

We now show how the asymptotically singular matrix problem is associated with the OLS. For this, we provide the following lemma.

Here, $\mathbf {M}$ is singular because its first $(1+2k)$ columns are proportional to the $(2+2k)$ -th column.

3.1 Alternative Representation

We achieve the alternative representation in two steps. First, we estimate the long- and short-run parameters using OLS and represent the OLS estimator $\widehat {\boldsymbol {\alpha }}_{T}$ as a bilinear transformation of the long- and short-run OLS estimators. Next, we express the long- and short-run OLS estimators as other bilinear transformations of further primitive estimators that do not suffer from an asymptotically singular matrix problem.

3.1.1 First Step: Representing OLS Estimator by Long- and Short-Run Estimators

We estimate the long-run parameter using OLS. That is, after specifying the long-run equation

(6) $$ \begin{align} y_{t-1} = \boldsymbol{r}_{t-1}^{\prime} \boldsymbol{\upsilon}_{*} + u_{t-1} = \boldsymbol{\beta}_{\ast}^{\prime} \ddot{\boldsymbol{x}}_{t-1} + \zeta_{\ast} (t-1) + \nu_{\ast} + u_{t-1} \end{align} $$

by letting $\boldsymbol {\upsilon }_{\ast } := [\boldsymbol {\beta }_{\ast }^{\prime }, \zeta _{\ast }, \nu _{\ast } ]'$ , we let the long-run OLS estimator be

$$ \begin{align*} \widetilde{\boldsymbol{\upsilon}}_{T} := [ \widetilde{\boldsymbol{\beta}}_{T}^{\prime},\, \widetilde{\zeta}_{T},\, \widetilde{\nu}_{T} ]^{\prime } := \left( \sum_{t=1}^{T} \boldsymbol{r}_{t-1} \boldsymbol{r}_{t-1}^{\prime }\right)^{-1} \sum_{t=1}^{T} \boldsymbol{r}_{t-1} y_{t-1}. \end{align*} $$

For later purposes, we let $\widetilde {u}_{t-1} := y_{t-1} - \boldsymbol {r}_{t-1}^{\prime } \widetilde {\boldsymbol {\upsilon }}_{T}$ and partition $\widetilde {\boldsymbol {\beta }}_{T}$ such that $\widetilde {\boldsymbol {\beta }}_{T} \equiv [\widetilde {\boldsymbol {\beta }}_{T}^{+\prime }, \widetilde {\boldsymbol {\beta }}_{T}^{-\prime }]^{\prime }$ .

We next estimate the short-run parameter by letting $\boldsymbol {\omega }_{*} := [\rho _{*}, \boldsymbol {\eta }_{*}^{\prime }, \psi _{*}, \gamma _{*}, \boldsymbol {\kappa }_{*}^{\prime } ]^{\prime }$ , such that $\boldsymbol {\eta }_{\ast } := \boldsymbol {\theta }_{\ast } + \rho _{\ast }\boldsymbol {\beta }_{\ast }$ and $\psi _{\ast } := \xi _{\ast } + \rho _{\ast } \zeta _{\ast }$ , and $\dot {\boldsymbol {z}}_{t} := [ u_{t-1}, \boldsymbol {r}_{t-1}^{\prime }, \boldsymbol {z}_{2t}^{\prime } ]^{\prime }$ to specify the error-correction model

(7) $$ \begin{align} \Delta y_{t} = \dot{\boldsymbol{z}}_{t}^{\prime} \boldsymbol{\omega}_{*}+ e_{t} = \rho_{\ast}u_{t-1} + \boldsymbol{\eta}_{\ast}^{\prime} \ddot{\boldsymbol{x}}_{t-1} + \psi_{\ast}(t-1) + \gamma_{\ast} + \boldsymbol{\kappa}_{\ast}^{\prime} \boldsymbol{z}_{2t} + e_{t}. \end{align} $$

We obtain this equation by combining (2) and (6) and estimate the parameters in (7) by

$$ \begin{align*} \widetilde{\boldsymbol{\omega}}_{T} := [\widetilde{\rho}_{T}, \widetilde{\boldsymbol{\eta}}_{T}^{\prime}, \widetilde{\psi}_{T}, \widetilde{\gamma}_{T}, \widetilde{\boldsymbol{\kappa}}_{T}^{\prime} ]^{\prime} := \left( \sum_{t=1}^{T} \widetilde{\boldsymbol{z}}_{t} \widetilde{\boldsymbol{z}}_{t}^{\prime }\right)^{-1} \sum_{t=1}^{T} \widetilde{\boldsymbol{z}}_{t} \Delta y_{t}, \;\;\text{where}\;\; \widetilde{\boldsymbol{z}}_{t} := [ \widetilde{u}_{t-1}, \boldsymbol{r}_{t-1}^{\prime}, \boldsymbol{z}_{2t}^{\prime} ]^{\prime}, \end{align*} $$

estimating the short-run parameter $\boldsymbol {\omega }_{*}$ . Note that $\widetilde {\boldsymbol {\omega }}_{T}$ is obtained by replacing $u_{t-1}$ with $\widetilde {u}_{t-1}$ . For later purposes, we partition $\boldsymbol {\eta }_{*}$ into $[\boldsymbol {\eta }_{*}^{+\prime }, \boldsymbol {\eta }_{*}^{-\prime }]^{\prime }$ .

The long- and short-run OLS estimators have a regular relationship in terms of the OLS estimator $\widehat {\boldsymbol {\alpha }}_{T}$ . The following proposition reveals the relationship.

Proposition 1. Under Assumption 1, $\widehat {\boldsymbol {\alpha }}_{T} = \mathbf {R}_{T} \widetilde {\boldsymbol {\omega }}_{T}$ , where

$$ \begin{align*} \mathbf{R}_{T} := \left[ \begin{array}{@{}cc@{}} \mathbf{R}_{T}^{11} & \mathbf{0} \\ \mathbf{0} & \mathbf{I}_{p+2kq-1} \end{array} \right],\;\;\;\;\;\text{and}\;\;\;\;\; \mathbf{R}_{T}^{11} := \left[ \begin{array}{@{}cc@{}} 1 & \mathbf{0}_{1\times (2+2k)} \\ -\widetilde{\boldsymbol{\upsilon}}_{T} & \mathbf{I}_{2+2k} \end{array} \right]. \end{align*} $$

Remarks.

1. (a) By Proposition 1, the OLS estimator $\widehat {\boldsymbol {\alpha }}_{T}$ is now represented as a bilinear transformation between the long- and short-run OLS estimators $\widetilde {\boldsymbol {\upsilon }}_{T}$ and $\widetilde {\boldsymbol {\omega }}_{T}$ . That is,

   $$ \begin{align*} \widehat{\boldsymbol{\alpha}}_{T} = [\widetilde{\rho}_{T}, \, -\widetilde{\rho}_{T} \widetilde{\boldsymbol{\beta}}_{T}^{\prime} + \widetilde{\boldsymbol{\eta}}_{T}^{\prime}, \, -\widetilde{\rho}_{T} \widetilde{\zeta}_{T} + \widetilde{\psi}_{T}, \, -\widetilde{\rho}_{T} \widetilde{\nu}_{T} + \widetilde{\gamma}_{T}, \, \widetilde{\boldsymbol{\kappa}}_{T}^{\prime} ]^{\prime}. \end{align*} $$

   If both $\widetilde {\boldsymbol {\upsilon }}_{T}$ and $\widetilde {\boldsymbol {\omega }}_{T}$ are consistent for $\boldsymbol {\upsilon }_{*}$ and $\boldsymbol {\omega }_{*}$ , respectively, it follows that

   $$ \begin{align*} \widehat{\boldsymbol{\alpha}}_{T} \stackrel{\mathbb{P}}{\to} [\rho_{*}, \, -\rho_{*} \boldsymbol{\beta}_{*} + \boldsymbol{\eta}_{*}^{\prime}, \, -\rho_{*} \zeta_{*} + \psi_{*}, \, -\rho_{*} \nu_{*} + \gamma_{*}, \, \boldsymbol{\kappa}_{*}^{\prime} ]^{\prime} = [\rho_{*}, \, \boldsymbol{\theta}_{*}, \, \xi_{*}, \, \alpha_{*}, \, \boldsymbol{\kappa}_{*}^{\prime} ]^{\prime} \end{align*} $$

   by noting that $\boldsymbol {\eta }_{*} = \mathbf {0}$ and $\psi _{*} = 0$ .

2. (b) Proposition 1 follows by applying Lemma A.2 in the Supplementary Material. By letting $y_{t}$ , $\boldsymbol {x}_{t}$ , $\boldsymbol {z}_{t}$ , and $\boldsymbol {w}_{t}$ of Lemma A.2 in the Supplementary Material be $\Delta y_{t}$ , $y_{t-1}$ , $\boldsymbol {r}_{t-1}$ , and $\boldsymbol {z}_{2t}$ of (7), respectively, Proposition 1 follows. Here, $\widehat {\boldsymbol {v}}_{t}$ in Lemma A.2 in the Supplementary Material corresponds to $\widetilde {u}_{t-1}$ . Lemma A.2 in the Supplementary Material is established by elaborating on the FWL projection method in Park and Phillips (Reference Park and Phillips1988, Reference Park and Phillips1989). It complements the FWL method by augmenting a representation for the estimator ignored by FWL, so that the OLS estimator $\widehat {\boldsymbol {\alpha }}_{T}$ could be written without omitting any of its element as given in Proposition 1. By this feature, we call Lemma A.2 in the Supplementary Material the augmented FWL (AFWL). For the goal to derive the joint limit distribution of $\widehat {\boldsymbol {\alpha }}_{T}$ , it is important to derive the joint limit distributions of the long- and short-run OLS estimators without omitting any of its element, but FWL omits the representation of the coefficient, if its variable is concentrated out. For this reason, we complement FWL using AFWL and provide the alternative representation for $\widehat {\boldsymbol {\alpha }}_{T}$ .

3.1.2 Second Step: Representing Long- and Short-Run Estimators by Primitive Estimators

Although Proposition 1 represents $\widehat {\boldsymbol {\alpha }}_{T}$ as a bilinear transformation between the long- and short-run OLS estimators, they both suffer from asymptotically singular matrix problems. It is mainly because $\widetilde {\boldsymbol {\upsilon }}_{T}$ and $\widetilde {\boldsymbol {\omega }}_{T}$ are defined by regressors including $\ddot {\boldsymbol {x}}_{t-1}$ and $(t-1)$ that are driven by the time trend (see Lemma A.3 in the Supplementary Material). Therefore, we again represent both estimators using other primitive estimators that do not suffer from an asymptotically singular matrix problem.

Long-Run Estimator: We first represent the long-run OLS estimator $\widetilde {\boldsymbol {\upsilon }}_{T}$ using other estimators. Note that both (4) and (6) imply that

(8) $$ \begin{align} y_{t-1} = \boldsymbol{r}_{t-1}^{\prime}\boldsymbol{\upsilon}_{*} + u_{t-1} = \boldsymbol{\beta}_{\ast}^{\prime} \mathbf{m}_{t-1} + \vartheta_{\ast} (t-1) + \nu_{\ast} + u_{t-1}, \;\;\;\text{where}\;\;\; \vartheta_{\ast} := \boldsymbol{\beta}_{\ast}^{\prime} \boldsymbol{\mu}_{\ast} + \zeta_{\ast}. \end{align} $$

This representation is obtained by collecting the deterministic time trend $(t-1)$ and the remainder $\mathbf {m}_{t-1}$ in (4) as regressors. However, $\mathbf {m}_{t-1}$ is unobservable. So, we first predict it by estimating the unknown parameters $\boldsymbol {\mu }_{*}$ in (4):

$$ \begin{align*} \ddot{\boldsymbol{\mu}}_{T} := \left( \sum_{t=1}^{T} (t-1)^{2} \right)^{-1} \sum_{t=1}^{T} (t-1) \ddot{\boldsymbol{x}}_{t-1},\;\;\;\text{obtaining}\;\;\; \ddot{\mathbf{m}}_{t-1} := \ddot{\boldsymbol{x}}_{t-1} - (t-1) \ddot{\boldsymbol{\mu}}_{T}, \end{align*} $$

and regressing $y_{t-1}$ against $\ddot {\boldsymbol {r}}_{t-1}:= [ \ddot {\mathbf {m}}_{t-1}^{\prime }, (t-1), 1 ]^{\prime }$ , to obtain

$$ \begin{align*} \ddot{\boldsymbol{\upsilon}}_{T} := [ \ddot{\boldsymbol{\beta}}_{T}^{\prime}, \,\ddot{\vartheta}_{T},\, \ddot{\nu}_{T} ]^{\prime} := \left( \sum_{t=1}^{T} \ddot{\boldsymbol{r}}_{t-1}\ddot{\boldsymbol{r}}_{t-1}^{\prime} \right)^{-1} \sum_{t=1}^{T} \ddot{\boldsymbol{r}}_{t-1} y_{t-1}, \;\;\;\text{estimating}\;\;\; \bar{\boldsymbol{\upsilon}}_{\ast} := [\boldsymbol{\beta}_{\ast}^{\prime}, \vartheta_{\ast}, \nu_{\ast}]'. \end{align*} $$

For later purposes, we partition $\ddot {\boldsymbol {\beta }}_{T}$ such that $\ddot {\boldsymbol {\beta }}_{T} \equiv [\ddot {\boldsymbol {\beta }}_{T}^{+\prime }, \ddot {\boldsymbol {\beta }}_{T}^{-\prime }]^{\prime }$ and let $\ddot {u}_{t-1} := y_{t-1} - \ddot {\boldsymbol {r}}_{t-1}^{\prime } \ddot {\boldsymbol {\upsilon }}_{T}$ .

The interrelationship between the long-run OLS and primitive estimators is stated in the following proposition.

Proposition 2. Given Assumption 1, $\widetilde {\boldsymbol {\upsilon }}_{T} := [ \widetilde {\boldsymbol {\beta }}_{T}^{\prime },\, \widetilde {\zeta }_{T},\, \widetilde {\nu }_{T} ]^{\prime } = \mathbf {P}_{T} \ddot {\boldsymbol {\upsilon }}_{T} = [ \ddot {\boldsymbol {\beta }}_{T}^{\prime }, \, \ddot {\vartheta }_{T} -\ddot {\boldsymbol {\mu }}_{T}^{\prime } \ddot {\boldsymbol {\beta }}_{T}, \, \ddot {\nu }_{T} ]^{\prime }$ , where

$$ \begin{align*} \mathbf{P}_{T} := \left[ \begin{array}{@{}cc@{}} \mathbf{I}_{2k} & \mathbf{0} \\ \mathbf{P}_{T}^{21} & \mathbf{I}_{2} \\ \end{array} \right] \;\;\;\;\text{and}\;\;\;\;\; \mathbf{P}_{T}^{21} := \left[ \begin{array}{@{}c@{}} -\ddot{\boldsymbol{\mu}}_{T}^{\prime} \\ \mathbf{0}_{1 \times 2k} \end{array} \right]. \end{align*} $$

Remarks.

1. (a) By Proposition 2, the long-run OLS estimator $\widetilde {\boldsymbol {\upsilon }}_{T}$ is represented as a bilinear transformation between the primitive estimators, $\ddot {\boldsymbol {\mu }}_{T}$ and $\ddot {\boldsymbol {\upsilon }}_{T}$ . This representation is particularly useful, because neither $\ddot {\boldsymbol {\mu }}_{T}$ nor $\ddot {\boldsymbol {\upsilon }}_{T}$ suffers from an asymptotically singular matrix problem (see Lemma 2 below).

2. (b) If both $\ddot {\boldsymbol {\mu }}_{T}$ and $\ddot {\boldsymbol {\upsilon }}_{T}$ are consistent for $\boldsymbol {\mu }_{*}$ and $\bar {\boldsymbol {\upsilon }}_{*}$ , respectively,

   $$ \begin{align*} \widetilde{\boldsymbol{\upsilon}}_{T} = [ \ddot{\boldsymbol{\beta}}_{T}^{\prime}, \, \ddot{\vartheta}_{T} -\ddot{\boldsymbol{\mu}}_{T}^{\prime} \ddot{\boldsymbol{\beta}}_{T}, \, \ddot{\nu}_{T} ]^{\prime} \stackrel{\mathbb{P}}{\to} [ \boldsymbol{\beta}_{*}^{\prime}, \, \vartheta_{*} - \boldsymbol{\mu}_{*}^{\prime} \boldsymbol{\beta}_{*}, \, \nu_{*} ]^{\prime} = [ \boldsymbol{\beta}_{*}^{\prime}, \, \zeta_{*}, \, \nu_{*} ]^{\prime} =: \boldsymbol{\upsilon}_{*} \end{align*} $$

   by noting that $\vartheta _{\ast } := \boldsymbol {\beta }_{\ast }^{\prime } \boldsymbol {\mu }_{\ast } + \zeta _{\ast }$ , leading to the consistency of $\widetilde {\boldsymbol {\upsilon }}_{T}$ for $\boldsymbol {\upsilon }_{*}$ . Furthermore, $\widetilde {u}_{t-1} = \ddot {u}_{t-1}$ using the definition of $\ddot {\mathbf {m}}_{t-1} := \ddot {\boldsymbol {x}}_{t-1} - (t-1) \ddot {\boldsymbol {\mu }}_{T}$ .

3. (c) Proposition 2 is also established by AFWL. If we let $y_{t-1}$ , $\ddot {\boldsymbol {x}}_{t-1}^{\prime }$ , $(t-1)$ , and $1$ be $y_{t}$ , $\boldsymbol {x}_{t}$ , $\boldsymbol {z}_{t}$ , and $\boldsymbol {w}_{t}$ , respectively, in Lemma A.2 in the Supplementary Material, then Proposition 2 follows.

Short-Run Estimator: Next, we represent the short-run OLS estimator $\widetilde {\boldsymbol {\omega }}_{T}$ as a bilinear transformation between two other estimators that do not suffer from an asymptotically singular matrix problem. This representation is parallel to the one for the long-run OLS estimator. As both $\ddot {\boldsymbol {x}}_{t-1}$ and $(t-1)$ on the right side of (7) are driven by the deterministic trend, we reparameterize (7) using (4):

(9) $$ \begin{align} \Delta y_{t} = \rho_{\ast} u_{t-1}+ \boldsymbol{\eta}_{\ast}^{\prime}\mathbf{m}_{t-1} + \varsigma_{\ast} (t-1) +\gamma_{\ast} + \boldsymbol{\kappa}_{\ast}^{\prime}\boldsymbol{z}_{2t} + e_{t}, \;\;\;\text{where}\;\;\; \varsigma_{\ast} := \boldsymbol{\mu}_{\ast}^{\prime}\boldsymbol{\eta}_{\ast} +\psi_{\ast}. \end{align} $$

Here, the coefficient of $(t-1)$ is obtained as $\varsigma _{*}$ by collecting the coefficients of the time trends in $\ddot {\boldsymbol {x}}_{t-1}$ and $(t-1)$ , but the coefficient of $\mathbf {m}_{t-1}$ keeps the same as that of $\ddot {\boldsymbol {x}}_{t-1}$ on the right side of (7).

We estimate the unknown parameters in (9) by OLS. For this, we replace $u_{t-1}$ and $\mathbf {m}_{t-1}$ in (9) with $\ddot {u}_{t-1}$ and $\ddot {\mathbf {m}}_{t-1}$ , respectively, as they are unobservable. By letting $\ddot {\boldsymbol {z}}_{t} := [ \begin {array}{c:c:cc} \ddot {u}_{t-1} & \ddot {\boldsymbol {r}}_{t-1}^{\prime } & \boldsymbol {z}_{2t}^{\prime } \end {array} ]^{\prime }$ , the parameter estimator is obtained as follows:

$$ \begin{align*} \ddot{\boldsymbol{\tau}}_{T}& := [ \begin{array}{c:c:c} \ddot{\rho}_{T} & \ddot{\boldsymbol{\tau}}_{1T}^{\prime} & \ddot{\boldsymbol{\kappa}}_{T}^{\prime} \end{array} ]^{\prime} := [ \begin{array}{c:ccc:c} \ddot{\rho}_{T} & \ddot{\boldsymbol{\eta}}_{T}^{\prime} & \ddot{\varsigma}_{T} & \ddot{\gamma}_{T} & \ddot{\boldsymbol{\kappa}}_{T}^{\prime} \end{array} ]^{\prime}\\ & := \left( \sum_{t=1}^{T} \ddot{\boldsymbol{z}}_{t}\ddot{\boldsymbol{z}}_{t}^{\prime} \right)^{-1} \sum_{t=1}^{T} \ddot{\boldsymbol{z}}_{t} \Delta y_{t},\end{align*} $$

estimating

$$ \begin{align*} \boldsymbol{\tau}_{\ast} := [ \begin{array}{c:c:c} \rho_{\ast} & \boldsymbol{\tau}_{1\ast}^{\prime} & \boldsymbol{\kappa}_{\ast}^{\prime} \end{array} ]^{\prime} := [ \begin{array}{c:ccc:c} \rho_{\ast} & \boldsymbol{\eta}_{\ast}^{\prime} & \varsigma_{\ast} & \gamma_{\ast} & \boldsymbol{\kappa}_{\ast}^{\prime} \end{array} ]^{\prime}. \end{align*} $$

Here, for later purposes, we let $\ddot {\boldsymbol {\eta }}_{T} \equiv [\ddot {\boldsymbol {\eta }}_{T}^{+\prime }, \ddot {\boldsymbol {\eta }}_{T}^{-\prime } ]^{\prime }$ and $\ddot {\boldsymbol {\kappa }}_{T} \equiv [\ddot {\boldsymbol {\varphi }}_{T}^{\prime }, \ddot {\boldsymbol {\pi }}_{T}^{\prime }]^{\prime } \equiv [\ddot {\boldsymbol {\varphi }}_{T}^{\prime }, \ddot {\boldsymbol {\pi }}_{T}^{+\prime }, \ddot {\boldsymbol {\pi }}_{T}^{-\prime }]^{\prime }$ . In addition, we separately estimate $\psi _{\ast }$ by $\ddot {\psi }_{T} := -\ddot {\boldsymbol {\mu }}_{T}^{\prime } \ddot {\boldsymbol {\eta }}_{T} + \ddot {\varsigma }_{T}$ by applying the plug-in principle to $\varsigma _{*} := \boldsymbol {\mu }_{*}^{\prime } \boldsymbol {\eta }_{*} + \psi _{*}$ .

The interrelationship between the short-run OLS and primitive estimators, $\widetilde {\boldsymbol {\omega }}_{T}$ and $\ddot {\boldsymbol {\tau }}_{T}$ , is stated in the following proposition.

Proposition 3. Given Assumption 1, $\widetilde {\boldsymbol {\omega }}_{T} = \mathbf {Q}_{T} \ddot {\boldsymbol {\tau }}_{T}$ , where

$$ \begin{align*} \mathbf{Q}_{T} := \left[ \begin{array}{@{}cc@{}} \mathbf{I}_{1+2k} & \mathbf{0} \\ \mathbf{Q}_{T}^{21} & \mathbf{I}_{p+2kq +1}\end{array} \right]\;\;\;\;\;\text{and}\;\;\;\;\;\; \mathbf{Q}_{T}^{21} := \left[ \begin{array}{@{}cc@{}} 0 & -\ddot{\boldsymbol{\mu}}_{T}^{\prime} \\ \mathbf{0}_{(p+2kq) \times 1} & \mathbf{0}_{(p+2kq) \times 2k} \end{array} \right]. \end{align*} $$

Remarks.

1. (a) By Proposition 3, the short-run OLS estimator $\widetilde {\boldsymbol {\omega }}_{T}$ is represented as a bilinear transformation between $\ddot {\boldsymbol {\mu }}_{T}$ and $\ddot {\boldsymbol {\tau }}_{T}$ , which do not suffer from asymptotically singular matrix problems (see Lemma 2 below).

2. (b) If both $\ddot {\boldsymbol {\mu }}_{T}$ and $\ddot {\boldsymbol {\tau }}_{T}$ are consistent for $\boldsymbol {\mu }_{*}$ and $\boldsymbol {\tau }_{*}$ , respectively, it follows that

   $$ \begin{align*} \widetilde{\boldsymbol{\omega}}_{T} &= [\ddot{\rho}_{T}, \, \ddot{\boldsymbol{\eta}}_{T}^{\prime}, \, -\ddot{\boldsymbol{\mu}}_{T}^{\prime} \ddot{\boldsymbol{\eta}}_{T} + \ddot{\varsigma}_{T}, \, \ddot{\gamma}_{T}, \, \ddot{\boldsymbol{\kappa}}_{T}^{\prime} ]^{\prime} \\ &\stackrel{\mathbb{P}}{\to} [\rho_{*}, \, \boldsymbol{\eta}_{*}^{\prime}, \, -\boldsymbol{\mu}_{*}^{\prime} \boldsymbol{\eta}_{*} + \varsigma_{*}, \, \gamma_{*}, \, \boldsymbol{\kappa}_{*}^{\prime} ]^{\prime} = [\rho_{*}, \, \boldsymbol{\eta}_{*}^{\prime}, \, \psi_{*}, \, \gamma_{*}, \, \boldsymbol{\kappa}_{*}^{\prime} ]^{\prime} =: \boldsymbol{\omega}_{*} \end{align*} $$

   by noting that $\varsigma _{*} := \boldsymbol {\mu }_{*}^{\prime } \boldsymbol {\eta }_{*} + \psi _{*}$ .

3. (c) As before, Proposition 3 is established by AFWL. By letting $\Delta y_{t}$ , $\ddot {\boldsymbol {x}}_{t-1}$ , $(t-1)$ , and $[\ddot {u}_{t-1}, 1, \boldsymbol {z}_{2t}^{\prime }]'$ be $y_{t}$ , $\boldsymbol {x}_{t}$ , $\boldsymbol {z}_{t}$ , and $\boldsymbol {w}_{t}$ , respectively, in Lemma A.2 in the Supplementary Material, Proposition 3 follows.

OLS Estimator: Using the alternative representations for the long- and short-run OLS estimators, we now represent the OLS estimator $\widehat {\boldsymbol {\alpha }}_{T}$ as a transform of the primitive estimators: $\ddot {\boldsymbol {\mu }}_{T}$ , $\ddot {\boldsymbol {\upsilon }}_{T}$ , and $\ddot {\boldsymbol {\tau }}_{T}$ . The following proposition reveals it.

Proposition 4. Given Assumption 1, $\widehat {\boldsymbol {\alpha }}_{T} = \mathbf {T}_{T} \ddot {\boldsymbol {\tau }}_{T}$ , where $\mathbf {T}_{T} := \mathbf {R}_{T} \mathbf {Q}_{T}$ such that

$$ \begin{align*} \mathbf{R}_{T} \mathbf{Q}_{T} = \mathbf{T}_{T} := \left[ \begin{array}{@{}cc@{}} \mathbf{T}_{T}^{11} & \mathbf{0} \\ \mathbf{T}_{T}^{21} & \mathbf{I}_{p+2kq+1} \end{array} \right], \;\;\;\; \mathbf{T}_{T}^{11} := \left[ \begin{array}{@{}cc@{}} 1 & \mathbf{0}_{1\times 2k} \\ -\widetilde{\boldsymbol{\beta}}_{T} & \mathbf{I}_{2k} \end{array} \right] \;\;\;\;\text{and} \end{align*} $$

$$ \begin{align*} \mathbf{T}_{T}^{21} := \left[ \begin{array}{@{}cc@{}} -\widetilde{\zeta}_{T} & -\ddot{\boldsymbol{\mu}}_{T}^{\prime} \\ -\widetilde{\nu}_{T} & \mathbf{0}_{1\times 2k} \\ \mathbf{0}_{(p+2kq-1) \times 1} & \mathbf{0}_{(p+2kq-1) \times 2k}\end{array} \right]. \end{align*} $$

Remarks.

1. (a) Proposition 4 follows by combining Propositions 1 and 3, in which we demonstrated that $\widehat {\boldsymbol {\alpha }}_{T} = \mathbf {R}_{T} \widetilde {\boldsymbol {\omega }}_{T}$ and $\widetilde {\boldsymbol {\omega }}_{T} = \mathbf {Q}_{T} \ddot {\boldsymbol {\tau }}_{T}$ , respectively, so that

   (10) $$ \begin{align} \widehat{\boldsymbol{\alpha}}_{T} = \left[ \ddot{\rho}_{T}, -\ddot{\rho}_{T} \ddot{\boldsymbol{\beta}}_{T}^{\prime} + \ddot{\boldsymbol{\eta}}_{T}^{\prime}, -\ddot{\rho}_{T} (\ddot{\vartheta}_{T} - \ddot{\boldsymbol{\mu}}_{T}^{\prime} \ddot{\boldsymbol{\beta}}_{T}) -\ddot{\boldsymbol{\mu}}_{T}^{\prime} \ddot{\boldsymbol{\eta}}_{T} + \ddot{\varsigma}_{T}, -\ddot{\rho}_{T} \ddot{\nu}_{T} + \ddot{\gamma}_{T}, \ddot{\boldsymbol{\kappa}}_{T}^{\prime} \right]^{\prime} \end{align} $$

   by further noting Proposition 2: $[ \widetilde {\boldsymbol {\beta }}_{T}^{\prime },\, \widetilde {\zeta }_{T},\, \widetilde {\nu }_{T} ]^{\prime } = [ \ddot {\boldsymbol {\beta }}_{T}^{\prime }, \, \ddot {\vartheta }_{T} -\ddot {\boldsymbol {\mu }}_{T}^{\prime } \ddot {\boldsymbol {\beta }}_{T}, \, \ddot {\nu }_{T} ]^{\prime }$ . An alternative representation of $\widehat {\boldsymbol {\alpha }}_{T}$ is now provided without omitting any of its elements.

2. (b) If $\ddot {\boldsymbol {\mu }}_{T}$ , $\ddot {\boldsymbol {\upsilon }}_{T}$ , and $\ddot {\boldsymbol {\tau }}_{T}$ are consistent for $\boldsymbol {\mu }_{*}$ , $\boldsymbol {\upsilon }_{\ast }$ , and $\boldsymbol {\tau }_{\ast }$ , respectively, then

   (11) $$ \begin{align} \widehat{\boldsymbol{\alpha}}_{T} &\stackrel{\mathbb{P}}{\to} \left[\rho_{\ast}, -\rho_{\ast} \boldsymbol{\beta}_{\ast}^{\prime} + \boldsymbol{\eta}_{\ast}^{\prime}, -\rho_{\ast} \zeta_{\ast} - \boldsymbol{\mu}_{\ast}^{\prime} \boldsymbol{\eta}_{\ast} + \varsigma_{\ast}, -\rho_{\ast} \nu_{\ast} + \gamma_{\ast}, \boldsymbol{\kappa}_{\ast}^{\prime} \right]^{\prime} \notag \\ &= \left[ \rho_{\ast}, -\rho_{\ast} \boldsymbol{\beta}_{\ast}^{\prime}, -\rho_{\ast} \zeta_{\ast}, -\rho_{\ast} \nu_{\ast} + \gamma_{\ast}, \boldsymbol{\kappa}_{\ast}^{\prime} \right]^{\prime} = [\rho_{\ast}, \boldsymbol{\theta}_{\ast}^{\prime}, \xi_{\ast}, \alpha_{\ast}, \boldsymbol{\kappa}_{\ast}^{\prime} ]^{\prime} =: \boldsymbol{\alpha}_{*}, \end{align} $$

   where the first equality holds because $\zeta _{*} = \vartheta _{*} - \boldsymbol {\mu }_{*}^{\prime }\boldsymbol {\beta }_{*}$ , $\boldsymbol {\eta }_{\ast } = \mathbf {0,}$ and $\varsigma _{*} = 0$ . In addition, the second equality holds by the definitions of $\boldsymbol {\beta }_{\ast }$ , $\zeta _{\ast }$ , and $\gamma _{\ast }$ . From this, the consistence of $\widehat {\boldsymbol {\alpha }}_{T}$ can be established.

3. (c) By Proposition 4, we can obtain the weak limit of $\widehat {\boldsymbol {\alpha }}_{T}$ using the weak limits of the primitive estimators: $\ddot {\boldsymbol {\mu }}_{T}$ , $\ddot {\boldsymbol {\upsilon }}_{T}$ , and $\ddot {\boldsymbol {\tau }}_{T}$ (see Theorem 1).

4. (d) The convergence rate of the OLS estimator $\widehat {\boldsymbol {\alpha }}_{T}$ is slower than $\mathbf {D}$ in Lemma 1. For example, for the long-run OLS estimator $\widehat {\boldsymbol {\theta }}_{T}$ , we have $\widehat {\boldsymbol {\theta }}_{T} = -\ddot {\rho }_{T} \ddot {\boldsymbol {\beta }}_{T} + \ddot {\boldsymbol {\eta }}_{T}$ from (10), and the convergence rates of $\ddot {\rho }_{T}$ and $(\ddot {\boldsymbol {\beta }}_{T}, \ddot {\boldsymbol {\eta }}_{T})$ are $T^{1/2}$ and $T$ , respectively, by Lemmas 3 and 4 given below. From this, if for $j=1, 2, \ldots , k$ , $\theta _{j*}^{\pm } \neq 0$ , then the convergence rate of $\widehat {\boldsymbol {\theta }}_{T}$ is determined as $T^{1/2}$ , which is slower than $T^{3/2}$ given in $\mathbf {D}$ . Similar arguments apply to the other elements in $\widehat {\boldsymbol {\alpha }}_{T}$ . This slower convergence rate is the same effect as expected when the asymptotic distribution of an estimator is obtained by a higher-order expansion (e.g., Teräsvirta, Reference Teräsvirta1994; Cho and White, Reference Cho and White2007, Reference Cho and White2010; Cho and Phillips, Reference Cho and Phillips2018). Theorem 1 below elaborates on the convergence rates under various parameter conditions.

3.2 Nonsingular Primitive Estimators

We now show that the primitive estimators $\ddot {\boldsymbol {\tau }}_{T}$ and $\ddot {\boldsymbol {\upsilon }}_{T}$ do not suffer from an asymptotically singular matrix problem by deriving their limit distributions. By the definition of $\ddot {\boldsymbol {\mu }}_{T}$ , it does not suffer from an asymptotic singularity. We therefore focus on the limit distributions of $\ddot {\boldsymbol {\tau }}_{T}$ and $\ddot {\boldsymbol {\upsilon }}_{T}$ . For this, we let

$$ \begin{align*} \boldsymbol{\mathcal{B}}(\cdot) := [ \boldsymbol{\mathcal{B}}_{s}(\cdot)^{\prime}, \mathcal{B}_{u}(\cdot), \mathcal{B}_{e}(\cdot), \mathcal{B}_{ue}(\cdot), \boldsymbol{\mathcal{B}}_{\boldsymbol{z} e}(\cdot)^{\prime} ]^{\prime} := \boldsymbol{\Sigma}_{\ast}^{1/2} \boldsymbol{\mathcal{W}}(\cdot), \end{align*} $$

where $\boldsymbol {\mathcal {W}}(\cdot )$ is a vector of $(2 + p + 2k(1+q))$ independent standard Wiener processes, and $\boldsymbol {\Sigma }_{\ast }$ is the global covariance matrix given in Assumption 1. Here, $\boldsymbol {\mathcal {B}}(\cdot )$ is the Brownian motion obtained by applying FCLT to $\mathbf {B}_{T}(\cdot ) := T^{-1/2} \sum _{t=1}^{\lfloor (\cdot ) T \rfloor } \boldsymbol {w}_{t}$ (see Lemma B.1 in the Supplementary Material). For later purposes, we also partition $\mathbf {B}_{T}(\cdot )$ in parallel to $\boldsymbol {\mathcal {B}}(\cdot )$ :

$$ \begin{align*} \mathbf{B}_{T}(\cdot)& := [ \mathbf{B}_{sT}(\cdot)^{\prime}, B_{uT}(\cdot), B_{eT}(\cdot), B_{ueT}(\cdot), \mathbf{B}_{\boldsymbol{z} eT}(\cdot)^{\prime} ]^{\prime}\\& := \frac{1}{\sqrt{T}} \sum_{t=1}^{\lfloor (\cdot) T \rfloor} [ \boldsymbol{s}_{t-1}^{\prime}, u_{t-1}, e_{t}, u_{t-1} e_{t}, \boldsymbol{z}_{2t}' e_{t} ]^{\prime } \end{align*} $$

by noting that $\boldsymbol {w}_{t} := [ \boldsymbol {s}_{t-1}^{\prime }, u_{t-1}, e_{t}, u_{t-1} e_{t}, \boldsymbol {z}_{2t}' e_{t} ]^{\prime }$ .

The following lemma shows that neither $\ddot {\boldsymbol {\tau }}_{T}$ nor $\ddot {\boldsymbol {\upsilon }}_{T}$ suffers from an asymptotically singular matrix problem.

Both weak limits $\boldsymbol {\mathcal {M}}_{11}$ and $\boldsymbol {\mathcal {M}}$ given in Lemmas 2(ii and vi) are nonsingular almost surely, meaning that the weak limits of $(\ddot {\mathbf {D}}_{1}^{-1}\sum _{t=1}^{T} \ddot {\boldsymbol {r}}_{t-1} \ddot {\boldsymbol {r}}_{t-1}^{\prime }\ddot {\mathbf {D}}_{1}^{-1})^{-1}$ and $(\ddot {\mathbf {D}}^{-1}\sum _{t=1}^{T} \ddot {\boldsymbol {z}}_{t} \ddot {\boldsymbol {z}}_{t}^{\prime }\ddot {\mathbf {D}}^{-1})^{-1}$ are obtained as $\boldsymbol {\mathcal {M}}_{11}^{-1}$ and $\boldsymbol {\mathcal {M}}^{-1}$ , respectively, where the inverse matrices are those within the long- and short-run OLS estimators, $\ddot {\boldsymbol {\upsilon }}_{T} :=(\sum _{t=1}^{T} \ddot {\boldsymbol {r}}_{t-1} \ddot {\boldsymbol {r}}_{t-1}^{\prime })^{-1}\sum _{t=1}^{T} \ddot {\boldsymbol {r}}_{t-1} y_{t-1}$ and $\ddot {\boldsymbol {\tau }}_{T} := (\sum _{t=1}^{T} \ddot {\boldsymbol {z}}_{t} \ddot {\boldsymbol {z}}_{t}^{\prime })^{-1} \sum _{t=1}^{T} \ddot {\boldsymbol {z}}_{t} \Delta y_{t}$ . From this, it follows that neither $\ddot {\boldsymbol {\upsilon }}_{T}$ nor $\ddot {\boldsymbol {\tau }}_{T}$ suffers from an asymptotically singular matrix problem.

4.1 Weak Limit of the Long-Run OLS Estimator

We derive the weak limits of $\ddot {\boldsymbol {\upsilon }}_{T}$ and $\widetilde {\boldsymbol {\upsilon }}_{T}$ using Proposition 2: $\widetilde {\boldsymbol {\upsilon }}_{T} = \mathbf {P}_{T} \ddot {\boldsymbol {\upsilon }}_{T}$ . Here,

$$ \begin{align*} \mathbf{P}_{T}\stackrel{\mathbb{P}}{\to} \mathbf{P} := \left[ \begin{array}{@{}cc@{}} \mathbf{I}_{2k} & \mathbf{0} \\ \mathbf{P}^{21} & \mathbf{I}_{2} \\ \end{array} \right]\;\;\;\;\text{with}\;\;\;\;\; \mathbf{P}^{21} := \left[ \begin{array}{@{}cc@{}} -\boldsymbol{\mu}_{*}^{\prime} \\ \mathbf{0}_{1 \times 2k} \end{array} \right], \end{align*} $$

as $\ddot {\boldsymbol {\mu }}_{T}$ is consistent for $\boldsymbol {\mu }_{\ast }$ . Next, we obtain the limit distribution of the primitive estimator $\ddot {\boldsymbol {\upsilon }}_{T}$ . For this, we note that $\mathbf {m}_{t-1} = \ddot {\mathbf {m}}_{t-1} + (\ddot {\boldsymbol {\mu }}_{T} - \boldsymbol {\mu }_{\ast }) (t-1)$ and rewrite (8) as

(12) $$ \begin{align} y_{t-1} &= \ddot{\boldsymbol{r}}_{t-1}^{\prime} \bar{\boldsymbol{\upsilon}}_{T*} + u_{t-1} = \boldsymbol{\beta}_{\ast}^{\prime}\ddot{\mathbf{m}}_{t-1} + \vartheta_{T\ast} (t-1) + \nu_{\ast} + u_{t-1}, \;\;\text{where}\nonumber\\ \bar{\boldsymbol{\upsilon}}_{T\ast} &:= [ \boldsymbol{\beta}_{\ast}^{\prime}, \vartheta_{T\ast}, \nu_{\ast} ]^{\prime} \end{align} $$

with $\vartheta _{T\ast } := \boldsymbol {\beta }_{\ast }^{\prime } (\ddot {\boldsymbol {\mu }}_{T} - \boldsymbol {\mu }_{\ast }) + \vartheta _{\ast }$ . This rearranges (8) so that the right-hand side variables in the long-run equation are the same as the regressors employed for $\ddot {\boldsymbol {\upsilon }}_{T}$ . From this, it follows that

(13) $$ \begin{align} \ddot{\boldsymbol{\upsilon}}_{T} := \left( \sum_{t=1}^{T} \ddot{\boldsymbol{r}}_{t-1}\ddot{\boldsymbol{r}}_{t-1}^{\prime} \right)^{-1} \sum_{t=1}^{T} \ddot{\boldsymbol{r}}_{t-1} y_{t-1} = \bar{\boldsymbol{\upsilon}}_{T\ast} + \left( \sum_{t=1}^{T} \ddot{\boldsymbol{r}}_{t-1}\ddot{\boldsymbol{r}}_{t-1}^{\prime} \right)^{-1} \sum_{t=1}^{T} \ddot{\boldsymbol{r}}_{t-1} u_{t-1}. \end{align} $$

Using this, we derive the weak limits of $\widetilde {\boldsymbol {\upsilon }}_{T}$ and $\ddot {\boldsymbol {\upsilon }}_{T}$ as follows.

4.2 Weak Limit of the Short-Run OLS Estimator

We derive the weak limit of the primitive estimator $\ddot {\boldsymbol {\tau }}_{T}$ . For this purpose, we first rewrite (9) as

$$ \begin{align*} \Delta y_{t} = \boldsymbol{\tau}_{T\ast}^{\prime}\ddot{\boldsymbol{z}}_{t} + e_{t} = \rho_{\ast} \ddot{u}_{t-1} &+ (\boldsymbol{\eta}_{\ast}+\rho_{\ast}(\ddot{\boldsymbol{\beta}}_{T} - \boldsymbol{\beta}_{\ast}))^{\prime} \ddot{\mathbf{m}}_{t-1} \\&+ (\psi_{\ast} + \boldsymbol{\eta}_{\ast}^{\prime}\ddot{\boldsymbol{\mu}}_{T} + \rho_{\ast}(\ddot{\vartheta}_{T} - \vartheta_{T\ast})) (t-1)\\ & + (\gamma_{\ast} + \rho_{\ast}(\ddot{\nu}_{T} - \nu_{\ast})) + \boldsymbol{\kappa}_{\ast}^{\prime} \boldsymbol{z}_{2t} + e_{t}, \end{align*} $$

where

$$ \begin{align*}\boldsymbol{\tau}_{T\ast} &:= [ \rho_{\ast}, \boldsymbol{\tau}_{1T}^{\prime}, \boldsymbol{\kappa}_{\ast}^{\prime} ]^{\prime}\;\;\;\text{and} \\ \boldsymbol{\tau}_{1T} &:= [ (\boldsymbol{\eta}_{\ast}+\rho_{\ast}(\ddot{\boldsymbol{\beta}}_{T} - \boldsymbol{\beta}_{\ast}))^{\prime}, \psi_{\ast} + \boldsymbol{\eta}_{\ast}^{\prime}\ddot{\boldsymbol{\mu}}_{T} + \rho_{\ast}(\ddot{\vartheta}_{T} - \vartheta_{T\ast}), \gamma_{\ast} + \rho_{\ast}(\ddot{\nu}_{T} - \nu_{\ast}) ]^{\prime}. \end{align*} $$

This equality is established by noting that

$$ \begin{align*} u_{t-1} = \ddot{u}_{t-1} + (\ddot{\boldsymbol{\beta}}_{T} - \boldsymbol{\beta}_{\ast})^{\prime} \ddot{\mathbf{m}}_{t-1} + (\ddot{\vartheta}_{T} - \vartheta_{T*} ) (t-1) + (\ddot{\nu}_{T} - \nu_{\ast}), \end{align*} $$

which follows from (12) and the definition of $\ddot {u}_{t-1} := y_{t-1} - \ddot {\boldsymbol {r}}_{t-1}^{\prime } \ddot {\boldsymbol {\upsilon }}_{T}$ . This representation is obtained so that the right-hand side variables in the short-run equation are the same as the regressors employed for $\ddot {\boldsymbol {\tau }}_{T}$ . That is, we can decompose the primitive estimator $\ddot {\boldsymbol {\tau }}_{T}$ into $\boldsymbol {\tau }_{T*}$ and a remainder using this representation:

(14) $$ \begin{align} \ddot{\boldsymbol{\tau}}_{T} := \left( \sum_{t=1}^{T} \ddot{\boldsymbol{z}}_{t}\ddot{\boldsymbol{z}}_{t}^{\prime} \right)^{-1} \sum_{t=1}^{T} \ddot{\boldsymbol{z}}_{t} \Delta y_{t} = \boldsymbol{\tau}_{T\ast} + \left( \sum_{t=1}^{T} \ddot{\boldsymbol{z}}_{t}\ddot{\boldsymbol{z}}_{t}^{\prime} \right)^{-1} \sum_{t=1}^{T} \ddot{\boldsymbol{z}}_{t} e_{t}. \end{align} $$

We now provide the weak limit of the primitive estimator $\ddot {\boldsymbol {\tau }}_{T}$ in the following lemma.

4.3 Weak Limit of the OLS Estimator

We finally derive the weak limit of the OLS estimator $\widehat {\boldsymbol {\alpha }}_{T}$ . For this derivation, we first note that

(15) $$ \begin{align} (\widehat{\boldsymbol{\alpha}}_{T} - \boldsymbol{\alpha}_{\ast}) = (\mathbf{T}_{T} - \mathbf{T}) \boldsymbol{\tau}_{\ast} + \mathbf{T} (\ddot{\boldsymbol{\tau}}_{T} - \boldsymbol{\tau}_{\ast}) + (\mathbf{T}_{T} - \mathbf{T}) (\ddot{\boldsymbol{\tau}}_{T} - \boldsymbol{\tau}_{\ast}), \end{align} $$

by (10) and (11), where $\mathbf {T}$ is the probability limit of $\mathbf {T}_{T}$ . That is,

$$ \begin{align*} \mathbf{T} &:= \left[ \begin{array}{@{}cc@{}} \mathbf{T}^{11} & \mathbf{0} \\ \mathbf{T}^{21} & \mathbf{I}_{p+2kq+1} \end{array} \right],\;\; \mathbf{T}^{11} := \left[ \begin{array}{@{}cc@{}} 1 & \mathbf{0}_{1\times 2k} \\ -\boldsymbol{\beta}_{\ast} & \mathbf{I}_{2k} \end{array} \right],\\\mathbf{T}^{21}& := \left[ \begin{array}{@{}cc@{}} -\zeta_{\ast} & -\boldsymbol{\mu}_{\ast}^{\prime} \\ -\nu_{\ast} & \mathbf{0}_{1\times 2k} \\ \mathbf{0}_{(p+2kq-1) \times 1} & \mathbf{0}_{(p+2kq-1) \times 2k} \end{array} \right]. \end{align*} $$

Here, the weak limits of $\mathbf {T}_{T}$ and $\ddot {\boldsymbol {\tau }}_{T}$ are already provided in Lemmas 3 and 4, respectively. Using these weak limits, we derive the weak limit of the OLS estimator $\widehat {\boldsymbol {\alpha }}_{T}$ .

Before stating the weak limit, we note that (15) identifies the channel through which the unknown parameters and primitive estimators affect the weak limit. It is difficult to identify a priori how they affect the limit distribution of $\widehat {\boldsymbol {\alpha }}_{T}$ , but (15) represents $\widehat {\boldsymbol {\alpha }}_{T}$ as a transformation of the unknown parameters and the primitive estimators for the long- and short-run OLS estimators, enabling us to identify how they affect the weak limit. For example, if for each $j= 1, 2, \ldots , k$ , $\beta _{j\ast }^{\pm } \neq 0$ and $\zeta _{\ast } \neq 0$ , we can rewrite (15) as

(16) $$ \begin{align} \sqrt{T} (\widehat{\boldsymbol{\alpha}}_{T} - \boldsymbol{\alpha}_{\ast}) = \boldsymbol{c}_{\ast} \sqrt{T}(\ddot{\rho}_{T} - \rho_{\ast}) + \sqrt{T} \boldsymbol{d}_{T} + o_{\mathbb{P}}(1), \end{align} $$

where $\boldsymbol {c}_{\ast }:= [1, -\boldsymbol {\beta }_{\ast }^{\prime }, -\zeta _{\ast }, -\nu _{\ast }, \mathbf {0}^{\prime }]^{\prime }$ and $\boldsymbol {d}_{T} := [0, \mathbf {0}_{2k \times 1}^{\prime }, 0, \{ (\ddot {\gamma }_{T} - \gamma _{\ast }) - \rho _{\ast }(\ddot {\nu }_{T} - \nu _{\ast }) \}, (\ddot {\boldsymbol {\kappa }}_{T} - \boldsymbol {\kappa }_{\ast })^{\prime }]^{\prime }$ . This representation implies that the weak limit is determined by the weak limits of $\ddot {\rho }_{T}$ and $\boldsymbol {d}_{T}$ . However, if $\boldsymbol {\beta }_{*}$ or $\zeta _{*}$ is zero, we cannot use (16) to find the weak limit, because the estimator corresponding to $\boldsymbol {\beta }_{*}$ or $\zeta _{*}$ is asymptotically negligible. That is, $\sqrt {T}(\widehat {\boldsymbol {\theta }}_{T} - \boldsymbol {\theta }_{\ast }) = o_{\mathbb {P}}(1)$ or $\sqrt {T}(\widehat {\xi }_{T} - \xi _{*}) = o_{\mathbb {P}}(1)$ , respectively. For such a case, we have to exploit the next-order terms to derive the weak limit of $(\widehat {\boldsymbol {\theta }}_{T} - \boldsymbol {\theta }_{*})$ , demonstrating that the weak limit depends on the parameter value condition on $\boldsymbol {\beta }_{*}$ and $\zeta _{*}$ .

In the following theorem, we collect the weak limits obtained under various parameter conditions. For notational simplicity, we let $\boldsymbol {\mathcal {D}}_{2} \equiv [\boldsymbol {\mathcal {D}}_{2}^{+\prime }, \boldsymbol {\mathcal {D}}_{2}^{-\prime }]^{\prime }$ . Recall that $\boldsymbol {\mathcal {D}}_{2}$ is the weak limit of $\ddot {\boldsymbol {\eta }}_{T} - (\boldsymbol {\eta }_{*}+ \rho _{*}(\ddot {\boldsymbol {\beta }}_{T} - \boldsymbol {\beta }_{*}))$ .

Before closing this section, we note that the NARDL model can also be estimated by instrumental variable (IV) estimation by overcoming singular matrix problems similarly to OLS (e.g., Phillips and Hansen, Reference Phillips and Hansen1990). We leave this as a future research topic.

5.1 Testing $\mathcal {H}_{0}^{\prime }$ versus $\mathcal {H}_{1}^{\prime }$

We first apply the Wald test principle to test $\mathcal {H}_{0}^{\prime }$ . The standard Wald test applied to OLS is defined as follows:

$$ \begin{align*} W_{T}^{(1)} := \widehat{\boldsymbol{\alpha}}_{T}^{\prime}\widehat{\mathbf{R}}_{1}^{\prime} \left( \widehat{\mathbf{W}}_{T}^{(1)} \right)^{-1} \widehat{\mathbf{R}}_{1} \widehat{\boldsymbol{\alpha}}_{T},\;\;\;\text{where}\;\;\; \widehat{\mathbf{W}}_{T}^{(1)} := \widehat{\sigma}_{e,T}^{2} \widehat{\mathbf{R}}_{1} \left( \sum_{t=1}^{T} \boldsymbol{z}_{t}\boldsymbol{z}_{t}^{\prime} \right)^{-1} \widehat{\mathbf{R}}_{1}^{\prime}, \end{align*} $$

$\widehat {\sigma }_{e,T}^{2} := T^{-1} \sum _{t=1}^{T} (\Delta y_{t} - \boldsymbol {z}_{t}^{\prime } \widehat {\boldsymbol {\alpha }}_{T} )^{2}$ , and $\widehat {\mathbf {R}}_{1} := [\mathbf {0}_{k \times 1}, \mathbf {I}_{k}, -\mathbf {I}_{k}, \mathbf {0}_{k \times (1+p+2kq)}]$ . As we assume that $\boldsymbol {\theta }_{\ast }^{+} \neq \mathbf {0}$ and/or $\boldsymbol {\theta }_{\ast }^{-} \neq \mathbf {0}$ , we cannot suppose the environment assumed in Theorems 1(ii or iii) under the null. Furthermore, we cannot directly apply the limit distribution of the OLS estimator in Theorems 1(i and iv) to obtain the null limit distribution of the test. Note that $\widehat {\mathbf {R}}_{1} \widehat {\boldsymbol {\alpha }}_{T} = (\widehat {\boldsymbol {\theta }}_{T}^{+} - \widehat {\boldsymbol {\theta }}_{T}^{-})$ , so that $\sqrt {T}(\widehat {\boldsymbol {\theta }}_{T}^{+} - \widehat {\boldsymbol {\theta }}_{T}^{-}) \Rightarrow (\boldsymbol {\beta }_{\ast }^{-} - \boldsymbol {\beta }_{\ast }^{+}) \mathcal {D}_{1}$ . Thus, $\sqrt {T}(\widehat {\boldsymbol {\theta }}_{T}^{+} - \widehat {\boldsymbol {\theta }}_{T}^{-})$ is asymptotically negligible under the null. The null limit behavior has to be obtained by the next-order term of $(\widehat {\boldsymbol {\theta }}_{T}^{+} - \widehat {\boldsymbol {\theta }}_{T}^{-})$ . Specifically, (10) implies that

(18) $$ \begin{align} (\widehat{\boldsymbol{\theta}}_{T}^{+} - \widehat{\boldsymbol{\theta}}_{T}^{-}) &= (\ddot{\boldsymbol{\eta}}_{T}^{+} - \ddot{\boldsymbol{\eta}}_{T}^{-}) - \ddot{\rho}_{T}(\widetilde{\boldsymbol{\beta}}_{T}^{+} - \widetilde{\boldsymbol{\beta}}_{T}^{-}) \\ &= (\ddot{\boldsymbol{\eta}}_{T}^{+} - \ddot{\boldsymbol{\eta}}_{T}^{-}) - (\boldsymbol{\eta}_{\ast}^{+} - \boldsymbol{\eta}_{\ast}^{-}) - \rho_{\ast}(\widetilde{\boldsymbol{\beta}}_{T}^{+} - \widetilde{\boldsymbol{\beta}}_{T}^{-}) - (\ddot{\rho}_{T} - \rho_{\ast})(\widetilde{\boldsymbol{\beta}}_{T}^{+} - \widetilde{\boldsymbol{\beta}}_{T}^{-}) \notag \\ &= (\ddot{\boldsymbol{\eta}}_{T}^{+} - \ddot{\boldsymbol{\eta}}_{T}^{-}) - (\boldsymbol{\eta}_{T\ast}^{+} - \boldsymbol{\eta}_{T\ast}^{-}) + o_{\mathbb{P}}(T^{-1}) \notag \end{align} $$

under $\mathcal {H}_{0}^{\prime }$ , where $\boldsymbol {\eta }_{T\ast } := \boldsymbol {\eta }_{*}+ \rho _{*}(\ddot {\boldsymbol {\beta }}_{T} - \boldsymbol {\beta }_{*})$ , and (18) is established by letting $\boldsymbol {\eta }_{T\ast } := [\boldsymbol {\eta }_{T\ast }^{+\prime }, \boldsymbol {\eta }_{T\ast }^{-\prime }]^{\prime }$ and noting that $\widetilde {\boldsymbol {\beta }}_{T} = \ddot {\boldsymbol {\beta }}_{T}$ . Therefore, the null limit distribution of $W_{T}^{(1)}$ is determined not by Theorem 1(i or iv), but by Lemma 4. All these imply $(\widehat {\boldsymbol {\theta }}_{T}^{+} - \widehat {\boldsymbol {\theta }}_{T}^{-}) = O_{\mathbb {P}}(T^{-1})$ , so that if we let $\mathbf {R}_{1}:= [\mathbf {0}_{k\times 1}, \mathbf {I}_{k}, -\mathbf {I}_{k}, \mathbf {0}_{k\times (1+p+2kq)}]$ , $T(\widehat {\boldsymbol {\theta }}_{T}^{+} - \widehat {\boldsymbol {\theta }}_{T}^{-}) \Rightarrow \mathbf {R}_{1} \boldsymbol {\mathcal {D}}$ under $\mathcal {H}_{0}^{\prime }$ by Lemma 4. However, its alternative behavior is differently determined. From $\widehat {\mathbf {R}}_{1} \widehat {\boldsymbol {\alpha }}_{T} = (\widehat {\boldsymbol {\theta }}_{T}^{+} - \widehat {\boldsymbol {\theta }}_{T}^{-})$ , its divergence rate is determined by Theorem 1(i or iv), meaning that $\widehat {\mathbf {R}}_{1} \widehat {\boldsymbol {\alpha }}_{T} = O_{\mathbb {P}}(T^{-1/2})$ . Therefore, the convergence and divergence rates of the test basis are unbalanced between $\mathcal {H}_{0}^{\prime }$ and $\mathcal {H}_{1}^{\prime }$ , respectively.

We supplement the unbalanced rates by applying the Wald test principle to the primitive estimator $\ddot {\boldsymbol {\upsilon }}_{T}$ . This is mainly because the primitive estimator $\ddot {\boldsymbol {\upsilon }}_{T}$ enables us to define another Wald test using the next-order terms. Because testing $\mathcal {H}_{0}^{\prime }$ is equivalent to testing

$$ \begin{align*} \mathbb{H}_{0}^{\prime}: \boldsymbol{\beta}_{\ast}^{+} - \boldsymbol{\beta}_{\ast}^{-} = \mathbf{0}, \end{align*} $$

we apply the Wald test principle to the primitive estimator $\ddot {\boldsymbol {\beta }}_{T}$ . If we let $\ddot {\mathbf {R}}_{1} := [ \mathbf {I}_{k}, -\mathbf {I}_{k}, \mathbf {0}_{k\times 2} ]$ , $\ddot {\mathbf {R}}_{1} \bar {\boldsymbol {\upsilon }}_{T\ast } = \boldsymbol {\beta }_{\ast }^{+} - \boldsymbol {\beta }_{\ast }^{-}$ , which we can estimate by $\ddot {\boldsymbol {\beta }}_{T}^{+} - \ddot {\boldsymbol {\beta }}_{T}^{-} = \ddot {\mathbf {R}}_{1} \ddot {\boldsymbol {\upsilon }}_{T}$ . Using this, we define another Wald test:

$$ \begin{align*} {\mathcal{W}}_{T}^{(1)} &:= \ddot{\boldsymbol{\upsilon}}_{T}^{\prime} \ddot{\mathbf{D}}_{1}\ddot{\mathbf{R}}_{1}^{\prime} (\ddot{\mathbf{W}}_{T}^{(1)})^{-1} \ddot{\mathbf{R}}_{1}\ddot{\mathbf{D}}_{1} \ddot{\boldsymbol{\upsilon}}_{T},\;\;\;\text{where}\\ \ddot{\mathbf{W}}_{T}^{(1)} &:= \ddot{\sigma}_{u, T}^{2} \ddot{\mathbf{R}}_{1} \ddot{\mathbf{D}}_{1} \left(\sum_{t=1}^{T} \ddot{\boldsymbol{r}}_{t-1}\ddot{\boldsymbol{r}}_{t-1}^{\prime} \right)^{-1} \ddot{\mathbf{D}}_{1} \ddot{\mathbf{R}}_{1}^{\prime} \end{align*} $$

and $\ddot {\sigma }_{u, T}^{2} := T^{-1} \sum _{t=1}^{T} \ddot {u}_{t}^{2}$ . Here, $(\ddot {\boldsymbol {\beta }}_{T}^{+} - \ddot {\boldsymbol {\beta }}_{T}^{-})$ and $(\widehat {\boldsymbol {\theta }}_{T}^{+} - \widehat {\boldsymbol {\theta }}_{T}^{-})$ are different by the presence of $(\ddot {\boldsymbol {\eta }}_{T}^{+} - \ddot {\boldsymbol {\eta }}_{T}^{-})$ and $\ddot {\rho }_{T}$ as given in (18) such that $(\widetilde {\boldsymbol {\beta }}_{T}^{+} - \widetilde {\boldsymbol {\beta }}_{T}^{-}) = (\ddot {\boldsymbol {\beta }}_{T}^{+} - \ddot {\boldsymbol {\beta }}_{T}^{-})$ . Furthermore, $(\ddot {\boldsymbol {\beta }}_{T}^{+} - \ddot {\boldsymbol {\beta }}_{T}^{-}) = O_{\mathbb {P}}(T^{-1})$ under both hypotheses, letting the convergence and divergence rates of ${\mathcal {W}}_{T}^{(1)}$ be identical under both $\mathcal {H}_{0}^{\prime }$ and $\mathcal {H}_{1}^{\prime }$ .

5.2 Testing $\mathcal {H}_{0}^{\prime \prime }$ versus $\mathcal {H}_{1}^{\prime \prime }$

We next apply the Wald test principle to test $\mathcal {H}_{0}^{\prime \prime }$ . The Wald test principle applied to $\widehat {\boldsymbol {\alpha }}_{T}$ delivers the following test:

$$ \begin{align*} W_{T}^{(2)} := \widehat{\boldsymbol{\alpha}}_{T}^{\prime} \widehat{\mathbf{R}}_{2}^{\prime} (\widehat{\mathbf{W}}_{T}^{(2)})^{-1} \widehat{\mathbf{R}}_{2} \widehat{\boldsymbol{\alpha}}_{T}, \;\;\;\text{where}\;\;\; \widehat{\mathbf{W}}_{T}^{(2)} := \widehat{\sigma}_{e,T}^{2} \widehat{\mathbf{R}}_{2} \left(\sum_{t=1}^{T} \boldsymbol{z}_{t}\boldsymbol{z}_{t}^{\prime} \right)^{-1} \widehat{\mathbf{R}}_{2}^{\prime} \end{align*} $$

and $\widehat {\mathbf {R}}_{2} := [\mathbf {0}_{kq \times (2+p+2k)}, \mathbf {I}_{kq}, -\mathbf {I}_{kq} ]$ , so that it follows that $\widehat {\mathbf {R}}_{2} \widehat {\boldsymbol {\alpha }}_{T} = \widehat {\boldsymbol {\pi }}_{T}^{+} - \widehat {\boldsymbol {\pi }}_{T}^{-}$ and $\sqrt {T} \widehat {\mathbf {R}}_{2} \widehat {\boldsymbol {\alpha }}_{T} \Rightarrow \widehat {\mathbf {R}}_{2} \boldsymbol {\mathcal {D}}$ using the limit distribution in Theorem 1.

We define another supplementary Wald test using the primitive estimator. From Proposition 4, it follows that $\widehat {\boldsymbol {\kappa }}_{T} = \ddot {\boldsymbol {\kappa }}_{T}$ . That is, the short-run OLS and its primitive estimator are identical, so that we can test $\mathcal {H}_{0}^{\prime \prime }$ by using $\ddot {\boldsymbol {\tau }}_{T}$ . By Lemma 2(vi), $\ddot {\boldsymbol {\tau }}_{T}$ is not associated with an asymptotically singular matrix problem. We define the following Wald test:

$$ \begin{align*} {\mathcal{W}}_{T}^{(2)} := \ddot{\boldsymbol{\tau}}_{T}^{\prime} \ddot{\mathbf{D}}\widehat{\mathbf{R}}_{2}^{\prime} (\ddot{\mathbf{W}}_{T}^{(2)})^{-1} \widehat{\mathbf{R}}_{2}\ddot{\mathbf{D}} \ddot{\boldsymbol{\tau}}_{T}, \;\;\;\text{where}\;\;\; \ddot{\mathbf{W}}_{T}^{(2)} := \widehat{\sigma}_{e,T}^{2} \widehat{\mathbf{R}}_{2} \ddot{\mathbf{D}} \left(\sum_{t=1}^{T} \ddot{\boldsymbol{z}}_{t}\ddot{\boldsymbol{z}}_{t}^{\prime} \right)^{-1} \ddot{\mathbf{D}} \widehat{\mathbf{R}}_{2}^{\prime}. \end{align*} $$

Although the test basis of the Wald test ${\mathcal {W}}_{T}^{(2)}$ is the same as that of $W_{T}^{(2)}$ , from the fact that $\ddot {\boldsymbol {\pi }}_{T}^{+} - \ddot {\boldsymbol {\pi }}_{T}^{-} = \widehat {\boldsymbol {\pi }}_{T}^{+} - \widehat {\boldsymbol {\pi }}_{T}^{-}$ , the weight matrix of ${\mathcal {W}}_{T}^{(2)}$ is different from $W_{T}^{(2)}$ . The weight matrix $\ddot {\mathbf {W}}_{T}^{(2)}$ is more relevant to the test basis, because $\ddot {\boldsymbol {\tau }}_{T}$ is obtained by regressing $\Delta y_{t}$ on $\ddot {\boldsymbol {z}}_{t}$ .

5.3 Testing $\mathcal {H}_{0}^{\prime \prime \prime }$ versus $\mathcal {H}_{1}^{\prime \prime \prime }$

Finally, we apply the Wald test principle to $\widehat {\boldsymbol {\alpha }}_{T}$ as follows:

$$ \begin{align*} W_{T}^{(3)} := \widehat{\boldsymbol{\alpha}}_{T}^{\prime} \widehat{\mathbf{R}}_{3}^{\prime} (\widehat{\mathbf{W}}_{T}^{(3)})^{-1} \widehat{\mathbf{R}}_{3} \widehat{\boldsymbol{\alpha}}_{T}, \;\;\;\text{where}\;\;\; \widehat{\mathbf{W}}_{T}^{(3)} := \widehat{\sigma}_{e,T}^{2} \widehat{\mathbf{R}}_{3} \left(\sum_{t=1}^{T} \boldsymbol{z}_{t}\boldsymbol{z}_{t}^{\prime} \right)^{-1} \widehat{\mathbf{R}}_{3}^{\prime} \end{align*} $$

and $\widehat {\mathbf {R}}_{3}:= \textrm {diag}[\widehat {\mathbf {R}}_{1}, \widehat {\mathbf {R}}_{2}]$ , so that $\widehat {\mathbf {R}}_{3} \widehat {\boldsymbol {\alpha }}_{T} = (\widehat {\boldsymbol {\theta }}_{T}^{+\prime } - \widehat {\boldsymbol {\theta }}_{T}^{-\prime }, \widehat {\boldsymbol {\pi }}_{T}^{+\prime } - \widehat {\boldsymbol {\pi }}_{T}^{-\prime })^{\prime }$ . Here, the null weak limit of $\widehat {\mathbf {R}}_{3} \widehat {\boldsymbol {\alpha }}_{T}$ is obtained by combining the fact that $T(\widehat {\boldsymbol {\theta }}_{T}^{+} - \widehat {\boldsymbol {\theta }}_{T}^{-}) \Rightarrow \mathbf {R}_{1}\boldsymbol {\mathcal {D}}$ and $\sqrt {T}(\widehat {\boldsymbol {\pi }}_{T}^{+} - \widehat {\boldsymbol {\pi }}_{T}^{-}) \Rightarrow \widehat {\mathbf {R}}_{2} \boldsymbol {\mathcal {D}}$ under $\mathcal {H}_{0}^{\prime \prime \prime }$ . Therefore, if we let $\mathbf {R}_{3}:= \textrm {diag}[\mathbf {R}_{1}, \widehat {\mathbf {R}}_{2}]$ , $(T(\widehat {\boldsymbol {\theta }}_{T}^{+\prime } - \widehat {\boldsymbol {\theta }}_{T}^{-\prime }), \sqrt {T}(\widehat {\boldsymbol {\pi }}_{T}^{+\prime } - \widehat {\boldsymbol {\pi }}_{T}^{-\prime }))^{\prime } \Rightarrow \mathbf {R}_{3} \boldsymbol {\mathcal {D}}$ under $\mathcal {H}_{0}^{\prime \prime \prime }$ . However, $\widehat {\mathbf {R}}_{3} \widehat {\boldsymbol {\alpha }}_{T} = O_{\mathbb {P}}(T^{-1/2})$ under $\mathcal {H}_{1}^{\prime \prime \prime }$ because $(\widehat {\boldsymbol {\theta }}_{T}^{+ } - \widehat {\boldsymbol {\theta }}_{T}^{- }) = O_{\mathbb {P}}(T^{-1/2})$ as discussed above.

We define another supplementary Wald test and overcome the unbalanced convergence and divergence rates of $W_{T}^{(3)}$ between $\mathcal {H}_{0}^{\prime \prime \prime }$ and $\mathcal {H}_{1}^{\prime \prime \prime }$ . For this, we first reformulate $\mathcal {H}_{0}^{\prime \prime \prime }$ against $\mathcal {H}_{1}^{\prime \prime \prime }$ into

$$ \begin{align*} \mathbb{H}_{0}^{\prime\prime\prime}: \boldsymbol{\beta}_{\ast}^{+} = \boldsymbol{\beta}_{\ast}^{-}\;\;\;\text{and}\;\;\;\; \boldsymbol{\pi}_{\ast}^{+} = \boldsymbol{\pi}_{\ast}^{-}\;\;\;\;\text{vs.}\;\;\;\; \mathbb{H}_{1}^{\prime\prime\prime}: \boldsymbol{\beta}_{\ast}^{+} \neq \boldsymbol{\beta}_{\ast}^{-}\;\;\;\text{or}\;\;\;\; \boldsymbol{\pi}_{\ast}^{+} \neq \boldsymbol{\pi}_{\ast}^{-}, \end{align*} $$

and define the following supplementary Wald test:

$$ \begin{align*} {\mathcal{W}}_{T}^{(3)} := \ddot{\boldsymbol{\delta}}_{T}^{\prime} \bar{\mathbf{D}} \ddot{\mathbf{R}}_{3}^{\prime} (\ddot{\mathbf{W}}_{T}^{(3)})^{-1} \ddot{\mathbf{R}}_{3} \bar{\mathbf{D}} \ddot{\boldsymbol{\delta}}_{T}, \end{align*} $$

where $\ddot {\boldsymbol {\delta }}_{T} := (\ddot {\boldsymbol {\upsilon }}_{T}^{\prime }, \ddot {\boldsymbol {\tau }}_{T}^{\prime } )^{\prime }$ , $\bar {\mathbf {D}} := \textrm {diag}[ \ddot {\mathbf {D}}_{1}, \ddot {\mathbf {D}} ]$ , $\ddot {\mathbf {R}}_{3}:= \textrm {diag}[\ddot {\mathbf {R}}_{1}, \widehat {\mathbf {R}}_{2}]$ , and $\ddot {\mathbf {W}}_{T}^{(3)} := \textrm {diag}[\ddot {\mathbf {W}}_{T}^{(1)}, \ddot {\mathbf {W}}_{T}^{(2)} ]$ . The Wald test ${\mathcal {W}}_{T}^{(3)}$ is defined by applying the Wald test principle to both primitive estimators $\ddot {\boldsymbol {\upsilon }}_{T}$ and $\ddot {\boldsymbol {\tau }}_{T}$ and by noting that $\ddot {\mathbf {R}}_{3}\bar {\mathbf {D}} \ddot {\boldsymbol {\delta }}_{T} = (T (\ddot {\boldsymbol {\beta }}_{T}^{+}- \ddot {\boldsymbol {\beta }}_{T}^{-})^{\prime }, \sqrt {T}(\ddot {\boldsymbol {\pi }}_{T}^{+} - \ddot {\boldsymbol {\pi }}_{T}^{-})^{\prime })^{\prime }$ . It straightforwardly follows that ${\mathcal {W}}_{T}^{(3)} = {\mathcal {W}}_{T}^{(1)} + {\mathcal {W}}_{T}^{(2)}$ from the fact that $\ddot {\mathbf {W}}_{T}^{(3)}$ is block-diagonal. From this, its null and alternative limit behaviors are determined by those of ${\mathcal {W}}_{T}^{(1)}$ and ${\mathcal {W}}_{T}^{(2)}$ .

We now provide the limit behaviors of the Wald tests. For notational simplicity, we partition $\mathbb {H}_{0}^{\prime \prime \prime }$ into $\mathbb {H}_{01}^{\prime \prime \prime }: \boldsymbol {\beta }_{\ast }^{+} = \boldsymbol {\beta }_{\ast }^{-}$ and $\mathbb {H}_{02}^{\prime \prime \prime }: \boldsymbol {\pi }_{\ast }^{+} = \boldsymbol {\pi }_{\ast }^{-}$ . In parallel, we also partition $\mathbb {H}_{1}^{\prime \prime \prime }$ into $\mathbb {H}_{11}^{\prime \prime \prime }: \boldsymbol {\beta }_{\ast }^{+} \neq \boldsymbol {\beta }_{\ast }^{-}$ and $\mathbb {H}_{12}^{\prime \prime \prime }: \boldsymbol {\pi }_{\ast }^{+} \neq \boldsymbol {\pi }_{\ast }^{-}$ . These partitions are provided because the power of $\boldsymbol{\mathcal {W}}_{T}^{(3)}$ depends on the sub-hypotheses from the fact that the divergence rate of $(\ddot {\boldsymbol {\beta }}_{T}^{+}- \ddot {\boldsymbol {\beta }}_{T}^{-})$ differs from that of $(\ddot {\boldsymbol {\pi }}_{T}^{+} - \ddot {\boldsymbol {\pi }}_{T}^{-})$ . We also let $\sigma _{e}^{2}:= \mathbb {E}[e_{t}^{2}]$ . The following theorem summarizes their limit behaviors.

Before moving to the next section, we summarize the use of the Wald tests introduced in this section. Table 1 provides their summary. We contain the formulas of the Wald tests along with their null hypotheses and null limit distributions. In case the null limit distribution is mixed chi-square, the researcher can obtain the asymptotic critical values from the chi-square distribution table. Otherwise, the residual bootstrap is applicable. Section 6 details the residual bootstrap procedure.

Table 1

Summary of the Wald tests

Table 1 Long description

The table consists of four columns: Wald test, Null hypothesis, Symmetry, and Null limit distribution.

* Row 1: Wald test W sub T super (1) defined as alpha hat T prime R hat 1 prime (W hat T super (1)) inverse R hat 1 alpha hat T. Null hypothesis H sub 0 prime: theta star super plus equals theta star super minus. Symmetry: Long-run. Null limit distribution: Mixed chi-square.

* Row 2: Wald test W sub T super (2) defined as alpha hat T prime R hat 2 prime (W hat T super (2)) inverse R hat 2 alpha hat T. Null hypothesis H sub 0 prime prime: pi star super plus equals pi star super minus. Symmetry: Short-run. Null limit distribution: Mixed chi-square.

* Row 3: Wald test W sub T super (3) defined as alpha hat T prime R hat 3 prime (W hat T super (3)) inverse R hat 3 alpha hat T. Null hypothesis H sub 0 prime prime prime: beta star super plus equals beta star super minus and pi star super plus equals pi star super minus. Symmetry: Long- and short-run. Null limit distribution: Mixed chi-square.

* Row 4: Wald test script W sub T super (1) defined as upsilon double-dot T prime D double-dot 1 R hat 1 prime (W double-dot T super (1)) inverse R hat 1 D double-dot 1 upsilon double-dot T. Null hypothesis H sub 0 prime: beta star super plus equals beta star super minus. Symmetry: Long-run. Null limit distribution: Nonstandard.

* Row 5: Wald test script W sub T super (2) defined as tau double-dot T prime D double-dot R hat 2 prime (W double-dot T super (2)) inverse R hat 2 D double-dot tau double-dot T. Null hypothesis H sub 0 prime prime: pi star super plus equals pi star super minus. Symmetry: Short-run. Null limit distribution: Mixed chi-square.

* Row 6: Wald test script W sub T super (3) defined as delta double-dot T prime D bar R hat 3 prime (W double-dot T super (3)) inverse R hat 3 D bar delta double-dot T. Null hypothesis H sub 0 prime prime prime: beta star super plus equals beta star super minus and pi star super plus equals pi star super minus. Symmetry: Long- and short-run. Null limit distribution: Nonstandard.

Note: This table summarizes the hypotheses and the null limit distributions of the Wald tests defined in Section 5. $W_{T}^{(1)}$ , $W_{T}^{(2)}$ , and $W_{T}^{(3)}$ are defined by the OLS estimator $\widehat {\boldsymbol {\alpha }}_{T}$ , whereas ${\mathcal {W}}_{T}^{(1)}$ , ${\mathcal {W}}_{T}^{(2)}$ , and ${\mathcal {W}}_{T}^{(3)}$ are defined by the primitive estimators, viz., $\ddot {\boldsymbol {\upsilon }}_{T}$ , $\ddot {\boldsymbol {\tau }}_{T}$ , and $\ddot {\boldsymbol {\delta }}_{T}$ , respectively. When the null limit distribution of the Wald test is mixed chi-square, the asymptotic critical values of the test can be selected from the chi-square distribution. However, for the Wald test whose null limit distribution is nonstandard, the residual bootstrap can be applied to the test. Section 6 illustrates the application of the residual bootstrap to the Wald test.

7.1 Literature Review and Empirical Motivation

Estimating the effects of fiscal policy on output is challenging, because many fiscal factors that lead to tax changes are correlated with output, and this lets the OLS estimator suffer from endogenous bias. Blanchard and Perotti (Reference Blanchard and Perotti2002) address this by using structural vector autoregression (SVAR). They attempt to solve the bias problem by assuming that policymakers do not respond to shocks contemporaneously, and by using information on the elasticity of revenue to create cyclically adjusted revenues. From this, they estimate that the effect of a tax cut on U.S. GDP is around 1%. However, Romer and Romer (Reference Romer and Romer2010) and Cloyne (Reference Cloyne2013) argue that the structural assumptions used in the SVAR model may be unrealistic in estimating the effect of fiscal policy on output.

Romer and Romer (Reference Romer and Romer2010) analyze the effects of tax changes correlated with GDP differently by performing a narrative analysis. They examine the motivations behind each tax change from 1945 to 2007 using sources, such as the Economic Report of the President and the Congressional Record. They categorize legislated tax changes that altered tax liabilities from one quarter to the next into four categories: (i) tax changes to counteract changes in government spending, (ii) tax changes to offset other factors affecting near-term output, (iii) tax changes to address inherited budget deficits, and (iv) tax changes to promote long-term growth. The first two categories, considered countercyclical, are motivated by restoring output growth reduced by other factors, making it difficult to classify them as purely exogenous. In contrast, the last two categories, based on policymakers’ perceptions of prudent fiscal policies or focus on increasing long-term growth, may be classified as exogenous. Consequently, they identified 54 exogenous tax changes through narrative analysis during the same period. Using these exogenous fiscal shocks, they develop a time-series model and estimate that GDP would increase by approximately 3% over three years following a tax cut of 1% of GDP. This estimate differs significantly from that of Blanchard and Perotti (Reference Blanchard and Perotti2002).

Narrative analysis is widely used to examine the effects of fiscal shocks on GDP. For example, Cloyne (Reference Cloyne2013) applies this approach to U.K. legislation and estimates that a 1% tax cut, as a percentage of GDP, increases output by nearly 2.5% over the following three years, similar to findings in the United States of America. Mertens and Olea (Reference Mertens and Olea2018) use narrative analysis to determine that the short-run tax elasticity of income is approximately 1.2% in the United States of America by analyzing exogenous variations in the marginal tax rate. Additionally, Gunter et al. (Reference Gunter, Riera-Crichton, Vegh and Vuletin2021) expand the use of narrative analysis to estimate the value-added-tax multipliers for 51 countries, showing that the effect of tax changes on output is highly nonlinear.

Narrative analysis is also used to specialize the time-series model in Romer and Romer (Reference Romer and Romer2010) for specific economic models under various environments. For example, Mertens and Ravn (Reference Mertens and Ravn2012) differentiate between surprise and anticipated tax changes to analyze the dynamic effects of tax changes on GDP, finding that anticipated tax cuts lead to a contraction in GDP. Demirel (Reference Demirel2021) and Ghassibe and Zanetti (Reference Ghassibe and Zanetti2021) examine the state-dependent effects of exogenous tax changes on GDP, allowing for varying tax multiplier estimations during recessions and expansions. Researchers widely apply narrative analysis to examine the effect of tax changes on GDP in other fields, enabling comparisons with conventional analysis outcomes.

The current study extends its focus beyond short-run relationships to estimate the long-run relationship between GDP and fiscal shocks. Previous studies have primarily concentrated on the short-term effects of fiscal policy. For example, one benchmark model specified by Romer and Romer (Reference Romer and Romer2010) is given as

(19) $$ \begin{align} \Delta y_{t} = \gamma_{\ast} + \sum_{j=0}^{q-1} \pi_{j\ast} \Delta \tau_{t-i} + \sum_{j=1}^{p-1} \varphi_{j\ast} \Delta y_{t-j} + e_{t} \end{align} $$

to examine how GDP responds to exogenous tax changes, where $y_{t}$ is the logarithm of real GDP and $\Delta \tau _{t}$ is the logarithm of an exogenous tax change. Note that all variables in (19) are differenced. Thus, it characterizes the short-run relationship between $\Delta y_{t}$ and $\Delta \tau _{t}$ .

Because $y_{t}$ and $\tau _{t}$ are both observable, we can estimate their long-run relationship by applying cointegration analysis. To this end, we augment the cointegration error on the right side of (19) as follows:

(20) $$ \begin{align} \Delta y_{t} = \gamma_{\ast} + \rho_{\ast} u_{t-1} + \sum_{j=0}^{q-1} \pi_{j\ast} \Delta \tau_{t-i} + \sum_{j=1}^{p-1} \varphi_{j\ast} \Delta y_{t-j} + e_{t}, \end{align} $$

where $u_{t} := y_{t} - \beta _{\ast } \tau _{t} - \zeta _{\ast } t - \nu _{\ast }$ . Note that the long-run relationship between $y_{t}$ and $\tau _{t}$ can be found by estimating the long-run coefficient $\beta _{\ast }$ , and the short-run relationship can be found by estimating the coefficients of $\pi _{j\ast }$ and $\varphi _{j\ast }$ . To estimate the unknown parameters, we also convert (20) as follows:

(21) $$ \begin{align} \Delta y_{t} = \alpha_{\ast} + \rho_{\ast} y_{t-1} + \theta_{\ast} \tau_{t-1} + \xi_{\ast}t + \sum_{j=0}^{q-1} \pi_{j\ast} \Delta \tau_{t-i} + \sum_{j=1}^{p-1} \varphi_{j\ast} \Delta y_{t-j} + e_{t}. \end{align} $$

For this estimation, we first examine the partial sum processes of the exogenous tax changes used in our empirical analysis. Figure 2 illustrates the partial sum processes due to the exogenous tax changes.Footnote ¹ The solid and dashed lines represent the partial sum processes of tax changes for deficit reduction ( $\!\! \tau _{1t}$ ) and long-run growth ( $\!\! \tau _{2t}$ ), respectively, and the dotted line represents the partial sum process of their sum ( $\!\! \tau _{t}$ ).Footnote ²

Figure 2

Partial sum process formed by exogenous fiscal shocks. The solid, dashed, and dotted lines represent deficit reduction, long-run growth, and exogenous fiscal shocks, respectively.

Figure 2 Long description

The line graph plots values on a Y-axis ranging from -100 to 120 against an X-axis representing years from 1947 to 2007. A legend in the top-left identifies three series.

* Deficit Reduction is represented by a solid line with circular markers. It remains relatively flat near zero until 1959, then begins a significant, step-like upward climb, reaching a peak of approximately 105 by 2007.

* Long-run Growth is represented by a dashed line with square markers. It follows a downward trajectory, beginning a sharp decline around 1962 and reaching its lowest point of approximately -100 around 2003 before a slight recovery.

* Exogenous Fiscal Shocks is represented by a dotted line with diamond markers. It fluctuates significantly below the zero line, dropping to nearly -50 in the mid-1980s before trending upward to settle around -10 by 2007.

The three lines diverge significantly after 1960, with Deficit Reduction moving positively while the other two variables remain in negative territory.

The exogenous tax changes exhibit characteristics suitable for NARDL analysis. First, tax changes for budget deficits always result in tax increases, so $\Delta \tau _{1t}$ is always positive. Second, most tax changes aimed at long-run economic growth involve tax decreases; out of 31 legislated tax changes for long-run growth, only six of them result in tax increases. Consequently, overall, the partial sum processes for deficit reduction and long-run growth remain in positive and negative regions, respectively. Although the NARDL model assumptions do not perfectly align with the characteristics of the exogenous fiscal shocks, we use the approximation of $\Delta \tau _{t}^{+}:= \max [0, \Delta \tau _{t}]$ for tax changes due to budget deficit and $\Delta \tau _{t}^{-}:= \min [0, \Delta \tau _{t}]$ for tax changes due to long-run growth.

Next, we specify the following NARDL model and estimate the long- and short-run parameters:

(22) $$ \begin{align} \Delta y_{t} = \gamma_{\ast} + \rho_{\ast} u_{t-1} + \sum_{j=0}^{q-1} \left( \pi_{j\ast}^{+} \Delta \tau_{t-i}^{+} + \pi_{j\ast}^{-} \Delta \tau_{t-i}^{-} \right) + \sum_{j=1}^{p-1} \varphi_{j\ast} \Delta y_{t-j} + e_{t}, \end{align} $$

where $u_{t} := y_{t} - \beta _{\ast }^{+} \tau _{t}^{+} - \beta _{\ast }^{-} \tau _{t}^{-} - \zeta _{\ast } t - \nu _{\ast }$ . This can be rewritten as

(23) $$ \begin{align} \Delta y_{t} &= \alpha_{\ast} + \rho_{\ast} y_{t-1} + \theta_{\ast}^{+} \tau_{t-1}^{+} + \theta_{\ast}^{-} \tau_{t-1}^{-} + \xi_{\ast} t + \sum_{j=0}^{q-1} \left( \pi_{j\ast}^{+} \Delta \tau_{t-i}^{+} + \pi_{j\ast}^{-} \Delta \tau_{t-i}^{-} \right)\nonumber\\&\quad + \sum_{j=1}^{p-1} \varphi_{j\ast} \Delta y_{t-j} + e_{t}, \end{align} $$

which we estimate by OLS. If $\beta _{\ast }^{+} = \beta _{\ast }^{-}$ (or $\theta _{\ast }^{+} = \theta _{\ast }^{-}$ ) and $\pi _{j\ast }^{+} = \pi _{j\ast }^{-}$ , (22) reduces to (20). Economically, this implies that the relationship between tax changes aimed at deficit reduction and long-run growth in real GDP is roughly symmetric in both the long and short run.

7.2 Empirical Results

This section presents the estimation and inference results using the tax change data outlined in the previous section. We have limited the sample period to 1947Q1–2007Q4 by excluding periods with missing observations.

Before presenting the estimation and inference results, we provide the basic statistical characteristics of the data. The logarithm of the GDP quantity index, multiplied by 100, is represented by $y_t$ , and $\tau _{t}$ , $\tau _{1,t}$ , and $\tau _{2,t}$ are defined as in footnote 2. The descriptive statistics of $\Delta y_{t}$ , $\Delta \tau _{1t}$ , $\Delta \tau _{2t}$ , and $\Delta \tau _{t}$ can be found in Table A.9 in the Supplementary Material. Furthermore, our unit-root test on $y_{t}$ , $\tau _{1t}$ , $\tau _{2t}$ , and $\tau _{t}$ follows the method of Phillips and Perron (Reference Phillips and Perron1988), including or excluding the time trend. The test results indicate that we cannot reject the unit-root hypothesis for the series.

We report the estimation results in Table 4. The columns marked “Exo” give the parameter estimates obtained by OLS for the NARDL and ARDL models. That is, the unknown parameters in (23) and (21) are estimated by OLS. Orders for the NARDL model are based on the Akaike information criterion (AIC), with $p=3$ and $q=1$ for both the NARDL and the ARDL models. Standard errors are listed in parentheses below the parameter estimates. Except for the coefficient of $y_{t-1}$ , we use the asymptotic critical values from the mixed normal distribution. For the $t$ -test on the coefficient of $y_{t-1}$ , we use the asymptotic critical values provided by Banerjee et al. (Reference Banerjee, Dolado and Mestre1998). Furthermore, we test whether all coefficients of $y_{t-1}$ , $\tau _{t-1}^{+}$ , and $\tau _{t-1}^{-}$ are equal to zero by applying the $F$ -test of Pesaran et al. (Reference Pesaran, Shin and Smith2001). Finally, we test the hypotheses of symmetry between long-run parameters, short-run parameters, or both by using the Wald tests given in Section 5. The results are presented in the two bottom panels. We summarize the results in Table 4 as follows:

1. (a) The coefficient of $y_{t-1}$ is significant at the 10% and 5% levels for the NARDL and ARDL models, respectively, by the $t$ -test. Although we cannot reject the hypothesis of no cointegration by the $F$ -test for the NARDL model, it is significant at the 10% level for the ARDL model. Our analysis using the 2SNARDL model also suggests a cointegrating relationship between the log real GDP and the log of an exogenous log tax shock.

2. (b) The NARDL model estimation indicates that an increase in an exogenous tax shock measured by $\tau _{t-1}^{+}$ reduces the log real GDP. In contrast, a decrease in an exogenous tax shock measured by $\tau _{t-1}^{-}$ increases the log of real GDP. The ARDL model shows the same relationship between the log of an exogenous tax shock and the log real GDP, aligning with standard economic theory.

3. (c) The estimated coefficients of $\tau _{t-1}^{+}$ and $\tau _{t-1}^{-}$ are almost equal in magnitude, suggesting no long-run asymmetry between the log real GDP and the log of an exogenous tax shock. We confirm this using the Wald tests. Both Wald tests ${\mathcal {W}}_{T}^{(1)}$ and $W_{T}^{(1)}$ provide $p$ -values that make it difficult to reject the symmetry hypothesis.

4. (d) Short-run symmetry is confirmed by the Wald tests ${\mathcal {W}}_{T}^{(2)}$ and $W_{T}^{(2)}$ , because they do not reject the symmetry hypothesis. Moreover, neither the long-run nor the short-run symmetry hypothesis is challenged by the Wald test ${\mathcal {W}}_{T}^{(3)}$ or $W_{T}^{(3)}$ . As such, we conclude that the ARDL model is appropriate for studying the relationship between the log real GDP and the log of an exogenous tax shock.

5. (e) We present the estimation results for the log of an endogenous tax shock, calculated in the same way as for the log of an exogenous tax shock. Columns labeled “Endo” give the estimation and inference results obtained using the log of the endogenous tax shock data. Similarly, the columns labeled “Sum” show the estimation and inference results obtained using the logs of both exogenous and endogenous tax shocks. The estimated signs of $\tau _{t-1}^{+}$ , $\tau _{t-1}^{-}$ , and $\tau _{t-1}$ are inconsistent with the parameter values posited by standard economic theory. When using the logs of both exogenous and endogenous tax shocks jointly, there is little evidence of cointegration. These estimation results indicate that only the log of an exogenous tax shock can be used to estimate the coefficients having signs consistent with standard economic theory.

Table 4

OLS estimation of the NARDL and ARDL models

Table 4 Long description

The table is divided into two main sections: N A R D L model on the left and A R D L model on the right. Each section contains columns for Exo. (Exogenous), Endo. (Endogenous), and Sum. (Summation).

Key variable rows include:

* y sub t minus 1: N A R D L values range from minus 0.0497 to minus 0.0817; A R D L values range from minus 0.0404 to minus 0.0755.

* tau sub t minus 1 super plus: N A R D L values are minus 0.0123, 0.0158, and 0.0016.

* tau sub t minus 1: A R D L values are minus 0.0139, 0.0142, and minus 0.0053.

* Trend: Coefficients range from 0.0324 to 0.0631 across both models.

* Constant: Values range from 0.6974 to 0.8543.

* Delta y sub t minus 1 and Delta y sub t minus 2: Show positive coefficients around 0.11 to 0.31.

* Delta tau sub t super plus and Delta tau sub t super minus: Specific to the N A R D L model, showing asymmetric short-run effects.

Statistical diagnostics at the bottom include:

* A I C: Approximately minus 6.54 for both models.

* B I C: Approximately minus 6.42 to minus 6.53.

* t-test and F-test: Reported for both models to indicate significance.

* Wald tests (W sub T and w sub T) with p-values in parentheses, ranging from 0.0000 to 32.3642.

Significance levels are indicated by symbols: dagger (25%), asterisk (10%), double asterisk (5%), and triple asterisk (1%). Standard errors are provided in parentheses below the coefficients.

Note: This table presents the OLS estimation using the quarterly data in Romer and Romer (Reference Romer and Romer2010). The left and right panels display estimated parameters for (23) and (21), respectively. Figures in parentheses indicate standard errors of the OLS estimates. At the bottom of the top panels, AIC, BIC, $t$ -test, and Pesaran et al.’s (Reference Pesaran, Shin and Smith2001) $F$ -test are reported. $^{\dagger }$ , $^{\ast }$ , $^{\ast \ast }$ , and $^{\ast \ast \ast }$ indicate significance at 25%, 10%, 5%, and 1% levels, respectively. Wald tests in the last two bottom panels show the Wald tests in Section 5 and the standard Wald tests. Figures in parentheses below the Wald tests show $p$ -values. They are obtained from 100,000 bootstrap iterations.

Table 5

2SNARDL estimation of the NARDL and ARDL models

Table 5 Long description

The table is divided into two main sections: the N A R D L model on the left and the A R D L model on the right. Each section contains three columns: Exo., Endo., and Sum.

Long-run variables:

* Constant: N A R D L values range from 3.7413 to 4.9730; A R D L values range from 4.8528 to 6.0785.

* Tau sub t minus 1 super plus (N A R D L) / Tau sub t minus 1 (A R D L): N A R D L shows values like minus 0.4329 and 0.2759; A R D L shows minus 0.2328 and 0.1564.

* Tau sub t minus 1 super minus: Only present in N A R D L with values from minus 0.3202 to 0.2598.

* Trend: Highly consistent across both models, ranging from 0.8261 to 0.8461.

Short-run variables:

* u sub t minus 1: Negative coefficients across all columns, ranging from minus 0.0404 to minus 0.0817.

* Constant: Values range from 0.5683 to 0.6918.

* Delta y sub t minus 1: Consistent values around 0.30 to 0.31.

* Delta y sub t minus 2: Values range from 0.1091 to 0.1446.

* Delta tau sub t super plus (N A R D L) / Delta tau sub t (A R D L): N A R D L values range from minus 0.0029 to 0.0890; A R D L values range from minus 0.0444 to minus 0.0122.

* Delta tau sub t super minus: Only in N A R D L with values from minus 0.0780 to minus 0.0302.

Standard errors are provided in parentheses below each coefficient. Significance levels are indicated by symbols: dagger (25%), single asterisk (10%), double asterisk (5%), and triple asterisk (1%).

Note: This table presents the 2SNARDL estimation using the quarterly data in Romer and Romer (Reference Romer and Romer2010). The left and right panels display estimated parameters for (22) and (20), respectively. $^{\dagger }$ , $^{\ast }$ , $^{\ast \ast }$ , and $^{\ast \ast \ast }$ imply that the tests are significant at 25%, 10%, 5%, and 1%, respectively.

Next, we estimate the NARDL and ARDL models using the 2SNARDL method proposed by Cho et al. (Reference Cho, Greenwood-Nimmo and Shin2023a). For the ARDL model, we apply 2SNARDL estimation by imposing the short- and long-run parameter symmetry conditions. The results are presented in Table 5, which is structured similarly to Table 4. This separate investigation is conducted to corroborate the inference results presented in Table 4. We summarize the results in Table 5.

1. (a) The NARDL and ARDL models estimated by 2SNARDL show that the coefficient of $u_{t-1}$ for the exogenous tax shock is statistically significant. The significance levels are 10% and 5% for the NARDL and ARDL models, respectively. Moreover, we apply the unit-root test of Phillips and Perron (Reference Phillips and Perron1988) to the cointegration residuals obtained from both models. The results reject the unit-root hypothesis. The $p$ -values are 10.19% and 2.58% for the NARDL and ARDL models, respectively. This confirms the cointegrating relationship between the log real GDP and the log of an exogenous tax shock.

2. (b) For the log of an exogenous tax shock, the NARDL model indicates that the long-run coefficient of the log tax increase is $-$ 0.4329, whereas that of the log tax decrease is $-$ 0.3202. These signs are consistent with standard economic theory and are statistically significant at the 1% and 10% levels, respectively. For the ARDL model, the long-run coefficient of the log tax change is $-$ 0.2328. This sign is also consistent with standard economic theory, and the estimated coefficient is statistically significant at the 1% level.

3. (c) Our findings for the endogenous and aggregate log tax shocks align with the results presented in Table 4. The coefficients of $\tau _{t-1}^{+}$ , $\tau _{t-1}^{-}$ , and $\tau _{t-1}$ for the log of endogenous tax shocks are statistically significant, but their signs are inconsistent with standard economic theory. However, for the log of the aggregate tax shock, none of these coefficients are statistically significant. Moreover, the coefficient of $u_{t-1}$ is insignificant. We believe these inconsistent results are due to the correlation between an endogenous shock and a structural error.

The results in Tables 4 and 5 suggest that by using the log of exogenous tax shocks, we can properly identify the relationship between the log real GDP and a fiscal shock. The findings indicate limited statistical support for the asymmetry between tax shocks for deficit reduction and those for long-run growth. Moreover, the OLS and 2SNARDL estimations produce qualitatively similar results.

In the Supplementary Material, we provide additional empirical results. Rather than using the tax change logarithm $\tau _t$ , we use $\Delta r_t := (\Delta T_{1t} + \Delta T_{2t})/NY_{t}$ , which represents the tax change-to-nominal GDP ratio, to measure the effect of fiscal policy and to specify the models corresponding to (20)–(23). From this, we obtain empirical results similar to those using log tax data and further show that an increase in an exogenous tax shock measured by $r_{t-1}$ reduces the long-run log real GDP by about 3%. This is close to the result of Romer and Romer (Reference Romer and Romer2010), estimating that GDP will increase by approximately 3% over three years following a tax cut of 1% of GDP (see Section A.5 of the Supplementary Material).