Tests for segmented cointegration: an application to US governments budgets

Abstract

There is a growing literature documenting that the persistence of time series may change over time, and as a consequence, shifts in the long-run equilibrium of macroeconomic variables are expected. An important example is the significant increase in public debt in certain periods of time due to increases in government expenditures which are not matched by revenue counterparts. In this paper, new residual-based Wald-type tests are proposed which are designed to detect segmented cointegration, i.e., subsamples during which equilibrium relations exist. We derive the asymptotic properties of the tests, tabulate critical values for models with different deterministic components, and show by simulations that the tests display good finite sample performance in many relevant setups. Our empirical application provides a thorough examination of the main components of US governments’ budgets at two administrative levels (Federal, and State and Local) and concludes that until Bill Clinton’s presidency government budgets components never moved together.

Technical appendix

1.1 Preliminary results

Considering $\xi _{t}$ as defined in Assumption 1 and $\varepsilon _{0}=0$, the following standard multivariate FCLT can be stated,

$$\begin{aligned} T^{-1/2}\sum _{t=1}^{\left[ rT\right] }\mathbf {\xi }_{t}\Rightarrow \mathbf { \Omega }^{1/2}\mathbf {W}\left( r\right) =:\mathbf {B}\left( r\right) , \end{aligned}$$

where $\mathbf {W}\left( r\right) :=\left( W_{1}\left( r\right) ,\mathbf {W}_{ \mathbf {2}}\left( r\right) ^{\prime }\right) ^{\prime },$ $W_{1}\left( r\right)$ is a standard Wiener process, $\mathbf {W}_{\mathbf {2}}\left( r\right)$ is a K vector of standard Wiener processes and $\mathbf {\Omega } :=\left( \begin{array}{cc} \omega _{yy} &{} \omega _{xy}^{\prime } \\ \omega _{xy} &{} \mathbf {\Omega }_{\mathbf {x}\mathbf {x}} \end{array} \right) .$ Thus, $T^{-1/2}\varepsilon _{\left[ rT\right] }\Rightarrow \omega _{11\cdot 2}{}^{1/2}W_{11\cdot 2}\left( r\right) ,$ where $\omega _{11\cdot 2}:=\omega _{yy}-\omega _{xy}^{\prime }\mathbf {\Omega }_{\mathbf {x}\mathbf {x} }^{-1}\omega _{xy}$ and $W_{11\cdot 2}\left( r\right) :=W_{1}\left( r\right) +\left( \frac{R^{2}}{1-R^{2}}\right) ^{1/2}\mathbf {W}_{2}\left( r\right) ,$ is a scalar Wiener process, $\mathbf {W}_{2 }\left( r\right) :=K^{-1/2}\sum _{i=1}^{K}W_{i}\left( r\right) ,$ K is the number of exogenous regressors considered in (2.1), and $R^{2}=\Lambda '\Lambda$ with $\Lambda =\Omega _{\mathbf {x}\mathbf {x}}^{-1/2}\omega _{xy}\omega _{yy}^{-1/2}$ and $0 \le R^{2} \le 1$; see e.g., Perron and Rodriguez (2016).

Under $H_{0}:c_{j}=0$ and $\gamma _{j}=0,$ for all j, considering $\widehat{\mathbf {b}}^{\prime }:=\left( 1,-\widehat{\mathbf {\ }\varvec{ \beta }}^{\prime }\right) ,$ where $\widehat{\varvec{\beta }}$ is the OLS estimate of $\varvec{\beta }$ in (2.1) and $\widehat{ \varvec{\beta }}^{\prime }\mathbf {x}_{t}=e_{t},$ it follows that $\widehat{\mathbf {b}}\rightarrow \mathbf {b},$ where $\mathbf {b}:=\left( 1,- \varvec{\beta }^{\prime }\right)$. Furthermore,

$$\begin{aligned} T^{-2}\sum _{t=1}^{T}e_{t}^{2}\Rightarrow \mathbf {b}^{\prime }\left[ \int _{0}^{1}\mathbf {B}(r)\mathbf {B}(r)^{\prime }dr\right] \mathbf {b}, \end{aligned}$$

where $\mathbf {B}\left( r\right) :=\left[ B_{1}\left( r\right) _{1\times 1}, \text { } \mathbf {B}_{2}\left( r\right) _{K \times 1}\right] ^{\prime }$ is a $(K+1)\times 1$ vector Brownian motion with covariance matrix $\mathbf {\Omega }=\Omega _{0}+\Omega _{1}+\Omega _{1}^{\prime }.$

To remove the nuisance parameters present in the distributions of the test statistics, consider $\mathbf {\Omega }:=\mathbf {LL}^{\prime },$ where $\mathbf {L}:=\left( \begin{array}{cc} l_{11} &{} 0 \\ l_{21} &{} \mathbf {L}_{22} \end{array} \right)$ with $l_{11}:=\left( \omega _{yy}-\omega _{xy}^{\prime }\mathbf { \Omega }_{\mathbf {xx}}^{-1}\omega _{xy}\right) ^{1/2};$ $l_{21}:=\mathbf { \Omega }_{\mathbf {xx}}^{-1/2}\omega _{xy};$ and $\mathbf {L}_{22}:=\mathbf {\Omega }_{\mathbf {xx}}^{1/2}$ (see Phillips and Ouliaris 1990). Moreover, define

$$\begin{aligned} \mathbf {A}:= & {} \int _{0}^{1}\mathbf {B}(r)\mathbf {B}(r)^{\prime }dr=\left( \begin{array}{cc} a_{yy} &{} a_{xy}^{\prime } \\ a_{xy} &{} \mathbf {A}_{\mathbf {xx}} \end{array} \right) ; \\ \mathbf {F}:= & {} \int _{0}^{1}\mathbf {W}(r)\mathbf {W}(r)^{\prime }dr=\left( \begin{array}{cc} f_{yy} &{} f_{xy}^{\prime } \\ f_{xy} &{} \mathbf {F}_{\mathbf {xx}} \end{array} \right) ,\text { } \end{aligned}$$

where $\mathbf {W}(r)$ is a vector of standard Brownian motions. Then,

$$\begin{aligned}&\kappa :=\left( 1,-f_{xy}^{\prime }\mathbf {F}_{\mathbf {xx}}^{-1}\right) =\left( 1,- \int _{0}^{1}W_{1}(r)\mathbf {W}_{2}(r)^{\prime }dr \left( \int _{0}^{1}\mathbf {W}_{2}(r)\mathbf {W}_{2}(r)^{\prime }dr\right) ^{-1}\right) , \\&Q(r):=W_{1}(r)- \int _{0}^{1}W_{1}(r)\mathbf {W}_{2}(r)^{\prime }dr \left( \int _{0}^{1}\mathbf {W}_{2}(r)\mathbf {W}_{2}(r)^{\prime }dr\right) ^{-1} \mathbf {W}_{2}(r), \\&\omega _{11\cdot 2} :=\omega _{yy}-\omega _{xy}^{\prime }\mathbf {\Omega }_{ \mathbf {xx}}^{-1}\omega _{xy}=l_{11}^{2}; a_{11\cdot 2}:=a_{yy}-a_{xy}^{\prime }\mathbf {A}_{\mathbf {xx}}a_{xy}. \end{aligned}$$

Furthermore, $\mathbf {B}\left( r\right) =\mathbf {L}^{\prime }\mathbf {W} \left( r\right) ;$ $\mathbf {Lb}=l_{11}\kappa ;$ $\mathbf {b}^{\prime }\mathbf { \Omega b}=\omega _{11\cdot 2}\kappa ^{\prime }\kappa ;$ $\mathbf {b}^{\prime } \mathbf {B}\left( r\right) =l_{11}Q\left( r\right) ;$ $\mathbf {b}^{\prime }\left( \int _{0}^{1}\mathbf {B}d\mathbf {B}^{\prime }\right) \mathbf {b}=\omega _{11\cdot 2}\left( \int _{0}^{1}Q(r)dQ(r)^{\prime }\right) ,$ and $\mathbf {b} ^{\prime }\mathbf {Ab}=a_{11\cdot 2}=\omega _{11\cdot 2}\int _{0}^{1}Q(r)^{2}$ so that $\left( 1,-a_{\mathbf {x}y}^{\prime }\mathbf {A}_{\mathbf {xx} }^{-1}\right) =\mathbf {b}^{\prime }.$

1.2 Proof of Theorem 1

Considering Assumption 1 under the null hypothesis of no cointegration, $\varepsilon _{t}=\varepsilon _{t-1}+\nu _{t}$ with $\nu _{t}$ white noise, the test regression we consider is:

$$\begin{aligned} \Delta e_{t}=\gamma _{j}e_{t-1}+\eta _{t}^{*}, \end{aligned}$$

(A.1)

where $e_{t}$ are the full sample LS residuals from (2.1). For the sake of simplicity and with no loss of generality, we consider the case of a test regression with no deterministics; however, this will be generalized below.

Recall that the residual-based Wald test for $H_{1A}$ (which corresponds to the case where the first regime is I(1)) is,

$$\begin{aligned} F_{A}\left( \tau ,m^{*}\right) :=\left\{ \begin{array}{ll} \left( T-m^{*}-p_{T}\right) \\ \qquad \quad \times \left( SSR_{0}-SSR_{A,m^{*}}\right) /\left( m^{*}SSR_{A,m^{*}}\right) , \qquad \text { if }m^{*}\text { is even} \\ \left( T-m^{*}-1-p_{T}\right) \\ \qquad \times \left( SSR_{0}-SSR_{A,m^{*}}\right) /\left( \left( m^{*}+1\right) SSR_{A,m^{*}}\right) , \qquad \text { if } m^{*}\text { is odd} \end{array} \right. , \end{aligned}$$

where $\tau :=\left( \tau _{1},\ldots ,\tau _{m^{*}}\right)$ with $\tau _{j}:=T_{j}/T$ and the number of changes is fixed, $m=m^{*}.$ Thus, under $H_{0}:\gamma _{j}=0,$ $j=1,\ldots ,m+1,$ it follows from (A.1) that

$$\begin{aligned} SSR_{0}=\sum _{t=1}^{T}\left( \Delta e_{t}\right) ^{2}=\widehat{\mathbf {b}} ^{\prime }\left( \sum _{t=1}^{T}\mathbf {\xi }_{t}\mathbf {\xi }_{t}^{\prime }\right) \widehat{\mathbf {b}} \end{aligned}$$

(A.2)

where $\xi _{t}:=(\varepsilon _{t},\mathbf {u}_{t}^{\prime })^{\prime }.$ Moreover, under the alternative hypothesis and for $m^{*}$ fixed and even,

$$\begin{aligned} SSR_{A,m^{*}}=\sum _{j=1}^{m^{*}/2}\sum _{t=T_{2j-1}+1}^{T_{2j}}\bigg [ \Delta e_{t}-\overline{\Delta e}_{2j}-\widehat{\gamma }_{2j}\left( e_{t-1}- \overline{e}_{2j,-1}\right) \bigg ]^{2}+\sum _{j=0}^{m^{*}/2}\sum _{t=T_{2j}+1}^{T_{2j+1}}\big ( \Delta e_{t}\big ) ^{2}. \end{aligned}$$

Noting that $\sum _{j=0}^{m^{*}/2}\sum _{t=T_{2j}+1}^{T_{2j+1}}\left( \Delta e_{t}\right) ^{2}=\sum _{t=1}^{T}\left( \Delta e_{t}\right) ^{2}-\sum _{j=1}^{m^{*}/2}\sum _{t=T_{2j-1}+1}^{T_{2j}}\left( \Delta e_{t}\right) ^{2},$ it follows that

$$\begin{aligned} SSR_{A,m^{*}}= & {} \sum _{j=1}^{m^{*}/2}\sum _{t=T_{2j-1}+1}^{T_{2j}}\bigg [ \Delta e_{t}-\overline{\Delta e}_{2j}-\widehat{\gamma }_{2j}\left( e_{t-1}- \overline{e}_{2j,-1}\right) \bigg ]^{2}\\&+\sum _{t=1}^{T}\left( \Delta e_{t}\right) ^{2}-\sum _{j=1}^{m^{*}/2}\sum _{t=T_{2j-1}+1}^{T_{2j}}\left( \Delta e_{t}\right) ^{2}. \end{aligned}$$

Since

$$\begin{aligned}&\sum _{t=T_{2j-1}+1}^{T_{2j}}\bigg [ \left( \Delta e_{t}-\overline{\Delta e} _{2j}\right) -\widehat{\gamma }_{2j}\left( e_{t-1}-\overline{e} _{2j,-1}\right) \bigg ]^{2} \\&\quad =\sum _{t=T_{2j-1}+1}^{T_{2j}}\left( \Delta e_{t}-\overline{\Delta e} _{2j}\right) ^{2}-2\widehat{\gamma }_{2j}\sum _{t=T_{2j-1}+1}^{T_{2j}}\left( \Delta e_{t}-\overline{\Delta e}_{2j}\right) \left( e_{t-1}-\overline{e} _{2j,-1}\right) \\&\qquad +\widehat{\gamma }_{2j}^{2}\sum _{t=T_{2j-1}+1}^{T_{2j}} \left( e_{t-1}-\overline{e}_{2j,-1}\right) ^{2} \end{aligned}$$

and given that $\widehat{\gamma }_{2j}=\frac{\sum _{t=T_{2j-1}+1}^{T_{2j}} \left( e_{t-1}-\overline{e}_{2j,-1}\right) \left( \Delta e_{t}-\overline{ \Delta e}_{2j}\right) }{\sum _{t=T_{2j-1}+1}^{T_{2j}}\left( e_{t-1}-\overline{ e}_{2j,-1}\right) ^{2}},$ and

$$\begin{aligned} \widehat{\gamma }_{2j}^{2}\sum _{t=T_{2j-1}+1}^{T_{2j}}\left( e_{t-1}- \overline{e}_{2j,-1}\right) ^{2}= & {} \widehat{\gamma }_{2j} \sum _{t=T_{2j-1}+1}^{T_{2j}}\left( \Delta e_{t}-\overline{\Delta e} _{2j}\right) \left( e_{t-1}-\overline{e}_{2j,-1}\right) \\= & {} \frac{\bigg [ \sum _{t=T_{2j-1}+1}^{T_{2j}}\left( e_{t-1}-\overline{e} _{2j,-1}\right) \left( \Delta e_{t}-\overline{\Delta e}_{2j}\right) \bigg ] ^{2}}{\sum _{t=T_{2j-1}+1}^{T_{2j}}\left( e_{t-1}-\overline{e}_{2j,-1}\right) ^{2}}\\=: & {} \Xi \end{aligned}$$

we establish that

$$\begin{aligned} \sum _{t=T_{2j-1}+1}^{T_{2j}}\left[ \left( \Delta e_{t}-\overline{\Delta e} _{2j}\right) -\widehat{\gamma }_{2j}\left( e_{t-1}-\overline{e} _{2j,-1}\right) \right] ^{2}=\sum _{t=T_{2j-1}+1}^{T_{2j}}\left( \Delta e_{t}- \overline{\Delta e}_{2j}\right) ^{2}-\Xi . \end{aligned}$$

Therefore,

$$\begin{aligned} SSR_{A,m^{*}}= & {} \sum _{j=1}^{m^{*}/2}\sum _{t=T_{2j-1}+1}^{T_{2j}} \left[ \Delta e_{t}-\overline{\Delta e}_{2j}-\widehat{\gamma }_{2j}\left( e_{t-1}-\overline{e}_{2j,-1}\right) \right] ^{2}\\&+\sum _{t=1}^{T}\left( \Delta e_{t}\right) ^{2}-\sum _{j=1}^{m^{*}/2}\sum _{t=T_{2j-1}+1}^{T_{2j}}\left( \Delta e_{t}\right) ^{2} \\= & {} \sum _{j=1}^{m^{*}/2}\left( \sum _{t=T_{2j-1}+1}^{T_{2j}}\left( \Delta e_{t}-\overline{\Delta e}_{2j}\right) ^{2}-\Xi \right) +\sum _{t=1}^{T}\left( \Delta e_{t}\right) ^{2}-\sum _{j=1}^{m^{*}/2}\sum _{t=T_{2j-1}+1}^{T_{2j}}\left( \Delta e_{t}\right) ^{2} \\= & {} \sum _{j=1}^{m^{*}/2}\left( \sum _{t=T_{2j-1}+1}^{T_{2j}}\left[ \left( \Delta e_{t}-\overline{\Delta e}_{2j}\right) ^{2}-\left( \Delta e_{t}\right) ^{2}\right] -\Xi \right) +\sum _{t=1}^{T}\left( \Delta e_{t}\right) ^{2} \\= & {} \sum _{j=1}^{m^{*}/2}\left( -\frac{1}{T_{2j}-T_{2j-1}}\left( \sum _{t=T_{2j-1}+1}^{T_{2j}}\Delta e_{t}\right) ^{2}-\Xi \right) +SSR_{0}, \end{aligned}$$

where

$$\begin{aligned} \overline{e}_{2j}:=\frac{1}{T_{2j}-T_{2j-1}} \sum _{t=T_{2j-1}+1}^{T_{2j}}e_{t}=\widehat{\mathbf {b}}^{\prime }\overline{ \mathbf {X}}_{2j}, \end{aligned}$$

and

$$\begin{aligned} \overline{e}_{2j,-1}:=\frac{1}{T_{2j}-T_{2j-1}} \sum _{t=T_{2j-1}+1}^{T_{2j}}e_{t-1}=\widehat{\mathbf {b}}^{\prime }\overline{ \mathbf {X}}_{2j,-1}, \end{aligned}$$

for $j=1,\ldots ,m^{*}/2\$and $\overline{\Delta \mathbf {X}}_{2j}:=\frac{1}{ T_{2j}-T_{2j-1}}\sum _{t=T_{2j-1}+1}^{T_{2j}}\Delta \overline{\mathbf {X}}_{t}= \overline{\mathbf {\xi }}_{2j},$ with $\mathbf {X}_{t}:=(y_{t},$ $\mathbf {x} _{t}^{\prime })^{\prime }$.

Hence, under Assumptions 1 and the null hypothesis of no cointegration it follows that

$$\begin{aligned}&SSR_{0}-SSR_{A,m^{*}} \\= & {} \sum _{j=1}^{m^{*}/2} \left\{ \Xi +\frac{1}{T_{2j}-T_{2j-1}}\left( \sum _{t=T_{2j-1}+1}^{T_{2j}}\Delta e_{t}\right) ^{2}\right\} \\= & {} \sum _{j=1}^{m^{*}/2}\Bigg \{ \frac{\left( \sum _{t=T_{2j-1}+1}^{T_{2j}}\left( e_{t-1}-\overline{e}_{2j,-1}\right) \left( \Delta e_{t}-\overline{\Delta e}_{2j}\right) \right) ^{2}}{ \sum _{t=T_{2j-1}+1}^{T_{2j}}\left( e_{t-1}-\overline{e}_{2j,-1}\right) ^{2}}\\&\quad + \frac{T}{T_{2j}-T_{2j-1}}\left( \mathbf {b}^{\prime }\frac{1}{\sqrt{T}} \sum _{t=T_{2j-1}+1}^{T_{2j}}\mathbf {\xi }_{t}\right) ^{2}\Bigg \} \\\Rightarrow & {} \sum _{j=1}^{m^{*}/2}\left\{ \frac{\left[ \mathbf {b} ^{\prime }\left( \int _{\tau _{2j-1}}^{\tau _{2j}}\mathbf {B}^{\left( 2j\right) }(r)d\mathbf {B(r)}^{\prime }+\left( \tau _{2j}-\tau _{2j-1}\right) \mathbf {\Omega }_{1}\right) \mathbf {b}\right] ^{2}}{\mathbf {b}^{\prime }\left( \int _{\tau _{2j-1}}^{\tau _{2j}}\mathbf {B}^{\left( 2j\right) }(r) \mathbf {B}^{\left( 2j\right) }(r)^{\prime }\right) \mathbf {b}}\right. \\&\left. +\mathbf {b}^{\prime }\frac{1}{\tau _{2j}-\tau _{2j-1}}\left( \mathbf {B}\left( \tau _{2j}\right) -\mathbf {B}\left( \tau _{2j-1}\right) \right) \left( \mathbf {B}\left( \tau _{2j}\right) -\mathbf {B}\left( \tau _{2j-1}\right) \right) ^{\prime }\mathbf {b}\right\} . \end{aligned}$$

Note from (A.2) that

$$\begin{aligned} \frac{1}{T}SSR_{A,m^{*}}=\widehat{\mathbf {b}}^{\prime }\left( \frac{1}{T} \sum _{t=1}^{T}\mathbf {\xi }_{t}\mathbf {\xi }_{t}^{\prime }\right) \widehat{ \mathbf {b}}+o_{p}\left( 1\right) \Rightarrow \mathbf {b}^{\prime }\mathbf { \Omega }_{0}\mathbf {b.} \end{aligned}$$

Hence, for $m^{*}$ fixed and even, and assuming $p_{T}=0,$ it follows that

$$\begin{aligned} F_{A}\left( \tau ,m^{*}\right)&=\frac{1}{m^{*}}\frac{\left( SSR_{0}-SSR_{A,m^{*}}\right) }{SSR_{A,m^{*}}/\left( T-m^{*}\right) } \nonumber \\&\Rightarrow \frac{1}{m^{*}}\frac{1}{\mathbf {b}^{\prime }\mathbf {\Omega }_{0}\mathbf {b}}\sum _{j=1}^{m^{*}/2}\Bigg \{ \frac{\left( \mathbf {b} ^{\prime }\left( \int _{\tau _{2j-1}}^{\tau _{2j}}\mathbf {B}^{\left( 2j\right) }(r)d\mathbf {B}(r)^{\prime }+\left( \tau _{2j}-\tau _{2j-1}\right) \mathbf {\Omega }_{1}\right) \mathbf {b}\right) ^{2}}{\mathbf {b}^{\prime }\left( \int _{\tau _{2j-1}}^{\tau _{2j}}\mathbf {B}^{\left( 2j\right) }(r) \mathbf {B}^{\left( 2j\right) }(r)^{\prime }dr\right) \mathbf {b}}\Bigg . \nonumber \\&\quad \Bigg . +\mathbf {b}^{\prime }\frac{1}{\tau _{2j}-\tau _{2j-1}}\left[ \mathbf {B}\left( \tau _{2j}\right) -\mathbf {B}\left( \tau _{2j-1}\right) \right] \left[ \mathbf {B}\left( \tau _{2j}\right) -\mathbf {B}\left( \tau _{2j-1}\right) \right] ^{\prime }\mathbf {b}\Bigg \} . \end{aligned}$$

(A.3)

We can straightforwardly establish from the results in Sect. 1 that

$$\begin{aligned} \mathbf {b}^{\prime }\mathbf {B}^{\left( 2j\right) }(r)&=\mathbf {b}^{\prime } \left[ \mathbf {B}\left( r\right) -\frac{1}{\tau _{2j}-\tau _{2j-1}} \int _{\tau _{2j-1}}^{\tau _{2j}}\mathbf {B}\left( r\right) dr\right] \nonumber \\&=:l_{11}Q^{\left( 2j\right) }; \end{aligned}$$

(A.4)

$$\begin{aligned} \mathbf {b}^{\prime }\left( \int _{\tau _{2j-1}}^{\tau _{2j}}\mathbf {B} ^{\left( 2j\right) }(r)\mathbf {B}^{\left( 2j\right) }(r)^{\prime }\right) \mathbf {b}&=\omega _{11\cdot 2} \int _{\tau _{2j-1}}^{\tau _{2j}}Q^{\left( 2j\right) }(r)Q^{\left( 2j\right) }(r)^{\prime }dr ; \end{aligned}$$

(A.5)

$$\begin{aligned}&\quad \times \mathbf {b}^{\prime }\left( \int _{\tau _{2j-1}}^{\tau _{2j}}\mathbf {B} ^{\left( 2j\right) }(r)d\mathbf {B}(r)^{\prime }+\left( \tau _{2j}-\tau _{2j-1}\right) \mathbf {\Omega }_{1}\right) \mathbf {b} \nonumber \\&=\omega _{11\cdot 2} \int _{\tau _{2j-1}}^{\tau _{2j}}Q^{\left( 2j\right) }(r)dQ(r)^{\prime } +\left( \tau _{2j}-\tau _{2j-1}\right) \mathbf {b} ^{\prime }\mathbf {\Omega }_{1}\mathbf {b} \end{aligned}$$

(A.6)

and

$$\begin{aligned}&\mathbf {b}^{\prime }\frac{1}{\tau _{2j}-\tau _{2j-1}}\left( \mathbf {B} \left( \tau _{2j}\right) -\mathbf {B}\left( \tau _{2j-1}\right) \right) \left( \mathbf {B}\left( \tau _{2j}\right) -\mathbf {B}\left( \tau _{2j-1}\right) \right) ^{\prime }\mathbf {b} \nonumber \\= & {} \frac{\omega _{11\cdot 2}}{\tau _{2j}-\tau _{2j-1}}\left( Q\left( \tau _{2j}\right) -Q\left( \tau _{2j-1}\right) \right) \left( Q\left( \tau _{2j}\right) -Q\left( \tau _{2j-1}\right) \right) ^{\prime }. \end{aligned}$$

(A.7)

Hence, from (A.5) to (A.7), we establish that under joint convergence, for $m^{*}$ fixed and even,

$$\begin{aligned} F_{A}\left( \tau ,m^{*}\right)\Rightarrow & {} \frac{1}{m^{*}}\frac{1}{ \mathbf {b}^{\prime }\mathbf {\Omega }_{0}\mathbf {b}}\sum _{j=1}^{m^{*}/2} \Bigg \{ \frac{\omega _{11\cdot 2}\left( \int _{\tau _{2j-1}}^{\tau _{2j}}Q^{\left( 2j\right) }(r)dQ(r)^{\prime } +\left( \tau _{2j}-\tau _{2j-1}\right) b^{\prime }\Omega _{1}b\right) ^{2}}{\omega _{11\cdot 2} \int _{\tau _{2j-1}}^{\tau _{2j}}Q^{\left( 2j\right) }(r)Q^{\left( 2j\right) }(r)^{\prime }dr }\Bigg . \nonumber \\&\Bigg . +\frac{\omega _{11\cdot 2}}{\tau _{2j}-\tau _{2j-1}}\left( Q\left( \tau _{2j}\right) -Q\left( \tau _{2j-1}\right) \right) \left( Q\left( \tau _{2j}\right) -Q\left( \tau _{2j-1}\right) \right) ^{\prime }\Bigg \} \nonumber \\= & {} \frac{1}{m^{*}}\frac{1}{\mathbf {b}^{\prime }\mathbf {\Omega }_{0} \mathbf {b}}\sum _{j=1}^{m^{*}/2}\Bigg \{\frac{\omega _{11\cdot 2}\left( \int _{\tau _{2j-1}}^{\tau _{2j}}Q^{\left( 2j\right) }(r)dQ(r)^{\prime } +\left( \tau _{2j}-\tau _{2j-1}\right) b^{\prime }\Omega _{1}b\right) ^{2}}{ \omega _{11\cdot 2} \int _{\tau _{2j-1}}^{\tau _{2j}}Q^{\left( 2j\right) }(r)^{2} }\Bigg . \nonumber \\&\Bigg . +\frac{\omega _{11\cdot 2}}{\tau _{2j}-\tau _{2j-1}}\left( Q\left( \tau _{2j}\right) -Q\left( \tau _{2j-1}\right) \right) ^{2}\Bigg \} . \end{aligned}$$

(A.8)

When $m^{*}$ is odd, $F_{A}\left( \tau ,m^{*}\right) :=\frac{\left( T-m^{*}-1\right) \left( SSR_{0}-SSR_{k,m^{*}}\right) }{\left( m^{*}+1\right) SSR_{k,m^{*}}},$ so that there is the correction of $m^{*}+1$ replacing $m^{*}$ in the previous expression. By the same token,

$$\begin{aligned} F_{B}\left( \tau ,m^{*}\right):= & {} \left\{ \begin{array}{ll} \frac{\left( T-m^{*}-2\right) \left( SSR_{0}-SSR_{k,m^{*}}\right) }{ \left( m^{*}+2\right) SSR_{k,m^{*}}}, &{}\text { if }m^{*}\text { is even} \\ \\ \frac{\left( T-m^{*}-1\right) \left( SSR_{0}-SSR_{k,m^{*}}\right) }{ \left( m^{*}+1\right) SSR_{k,m^{*}}}, &{}\text { if }m^{*}\text { is odd } \end{array} \right. , \end{aligned}$$

(A.9)

and all results in Theorem 1 hold true knowing that $\mathbf {b}^{\prime } \mathbf {\Omega b}=\omega _{11\cdot 2}\kappa ^{\prime }\kappa$ from Sect. 1 and with $\xi _{t}$ not autocorrelated, $\mathbf {\Omega } _{1}=0,$ so that $\mathbf {\Omega }=\mathbf {\Omega }_{0}$. The distributions become free of nuisance parameters even if we have endogeneity, $\omega _{xy}\ne 0$. For a single variable, these correspond to the distributions in Kejriwal et al. (2013), as $Q=W_{1}\equiv W$ and $\kappa =1.$ The case of $m^{*}=0$ (standard cointegration) cannot be obtained from our distributions.

Remark A.1. Note that $Q\$is a scalar and that in general the distribution of these distributions is non-standard and depends on the nuisance parameters $\Omega _{0},$ $\Omega _{1}.$ This is still the case even if $\Omega _{0}$ and $\Omega _{1}$ are block diagonal. This a is typical feature in limit theory for (non)cointegrating regressions. In this context, we can propose transformations of the statistic which involve consistent estimates of the nuisance parameters (Phillips and Park 1988), or we can try FM-OLS optimal estimation of Phillips and Hansen (1990) which introduces nonparametric corrections in the OLS estimator. $\diamondsuit$

For the case of $\xi _{t}$ autocorrelated, $\mathbf {\Omega }_{1}\ne 0,$ we present now a correction that eliminates the nuisance parameters and thus keeps the validity of the limit distributions presented above. To correct for the nuisance parameters, consider instead of $SSR_{A,m^{*}}$ the following corrected sum of squared residuals:

$$\begin{aligned}&SSR_{A,m^{*}}^{\#}=\sum _{j=1}^{m^{*}/2}\Bigg \{ \sum _{t=T_{2j-1}+1}^{T_{2j}}\left[ \Delta e_{t}-\overline{\Delta e} _{2j}-\left( \widehat{\phi }_{2j}-\widehat{\Upsilon }_{1}\right) \left( e_{t-1}-\overline{e}_{2j,-1}\right) \right] ^{2}\Bigg \} \\&\quad +\sum _{j=0}^{m^{*}/2}\sum _{t=T_{2j}+1}^{T_{2j+1}}\left( \Delta e_{t}\right) ^{2}, \end{aligned}$$

where the scalar correction term $\widehat{\Upsilon }_{1}$ is defined as,

$$\begin{aligned} \widehat{\Upsilon }_{1}:= & {} \frac{\left( \frac{T_{2j}-T_{2j-1}}{T}\right) \widehat{\mathbf {b}}^{\prime }\widehat{\mathbf {\Omega }}_{1}\widehat{\mathbf { b}}}{\sum _{t=T_{2j-1}+1}^{T_{2j}}\left( e_{t-1}-\overline{e}_{2j,-1}\right) ^{2}}\\= & {} \frac{\left( \frac{T_{2j}-T_{2j-1}}{T}\right) \widehat{\mathbf {b}} ^{\prime }\widehat{\mathbf {\Omega }}_{1}\widehat{\mathbf {b}}}{\widehat{ \mathbf {b}}^{\prime }\left( \sum _{t=T_{2j-1}+1}^{T_{2j}}\left( x_{t-1}- \overline{x}_{2j,-1}\right) \left( x_{t-1}-\overline{x}_{2j,-1}\right) ^{\prime }\right) \widehat{\mathbf {b}}}. \end{aligned}$$

The statistic considering $p_{T}=0,$ and $m^{*}$ even is,

$$\begin{aligned} F_{A}^{\#}\left( \tau ,m^{*}\right) =\left( T-m^{*}\right) \frac{ \left( SSR_{0}-SSR_{A,m^{*}}^{\#}\right) }{m^{*}\left( SSR_{A,m^{*}}^{\#}+\widehat{\Upsilon }_{2}\right) }, \end{aligned}$$

(A.10)

where the second (scalar) correction term, $\widehat{\Upsilon }_{2}$, in the denominator of (A.10) is,

$$\begin{aligned} \widehat{\Upsilon }_{2}:=T\widehat{\mathbf {b}}^{\prime }\left( \widehat{ \mathbf {\Omega }}_{1}+\widehat{\mathbf {\Omega }}_{1}^{\prime }\right) \widehat{\mathbf {b}}=T\widehat{\mathbf {b}}^{\prime }\left( \widehat{\mathbf { \Omega }}-\widehat{\mathbf {\Omega }}_{0}\right) \widehat{\mathbf {b}}. \end{aligned}$$

Hence, for the numerator of (A.10), we observe that

$$\begin{aligned}&SSR_{0}-SSR_{A,m^{*}}^{\#} \nonumber \\= & {} \sum _{j=1}^{m^{*}/2}\left\{ \frac{ \left( \sum _{t=T_{2j-1}+1}^{T_{2j}}\left( e_{t-1}-\overline{e} _{2j,-1}\right) \left( \Delta e_{t}-\overline{\Delta e}_{2j}\right) -\left( \frac{T_{2j}-T_{2j-1}}{T}\right) \widehat{\mathbf {b}}^{\prime }\widehat{ \mathbf {\Omega }}_{1}\widehat{\mathbf {b}}\right) ^{2}}{ \sum _{t=T_{2j-1}+1}^{T_{2j}}\left( e_{t-1}-\overline{e}_{2j,-1}\right) ^{2}} \right. \nonumber \\&\quad \left. +\frac{T}{T_{2j}-T_{2j-1}}\left( \mathbf {b}^{\prime }T^{-1/2}\sum _{t=T_{2j-1}+1}^{T_{2j}}\xi _{t}\right) ^{2}\right\} \nonumber \\\Rightarrow & {} \sum _{j=1}^{m^{*}/2}\left\{ \frac{\left( \mathbf {b} ^{\prime } \int _{\tau _{2j-1}}^{\tau _{2j}}\mathbf {B}^{\left( 2j\right) }(r)d \mathbf {B}(r)^{\prime } \mathbf {b}\right) ^{2}}{\mathbf {b}^{\prime } \int _{\tau _{2j-1}}^{\tau _{2j}}\mathbf {B}^{\left( 2j\right) }(r)\mathbf {B} ^{\left( 2j\right) }(r)^{\prime } \mathbf {b}}\right. \nonumber \\&\left. +\mathbf {b}^{\prime }\frac{1}{\tau _{2j}-\tau _{2j-1}}\left( \mathbf {B}\left( \tau _{2j}\right) -\mathbf {B}\left( \tau _{2j-1}\right) \right) \left( \mathbf {B}\left( \tau _{2j}\right) -\mathbf {B}\left( \tau _{2j-1}\right) \right) ^{\prime }\mathbf {b}\right\} \end{aligned}$$

(A.11)

and

$$\begin{aligned} \frac{SSR_{A,m^{*}}^{\#}+\widehat{\Upsilon }_{2}}{T}=\widehat{\mathbf {b}} ^{\prime }\left( \frac{1}{T}\sum _{t=1}^{T}\xi _{t}\xi _{t}^{\prime }\right) \widehat{\mathbf {b}}+o_{p}\left( 1\right) +\widehat{\mathbf {b}}^{\prime }\left( \widehat{\Omega }-\widehat{\Omega }_{0}\right) \widehat{\mathbf {b}} \Rightarrow \mathbf {b}^{\prime }\mathbf {\Omega b}. \end{aligned}$$

(A.12)

Thus, under joint convergence of (A.11) and (A.12), we establish that

$$\begin{aligned} F_{A}^{\#}\left( \tau ,m^{*}\right)\Rightarrow & {} \frac{1}{m^{*}} \frac{1}{\mathbf {b}^{\prime }\mathbf {\Omega b}}\sum _{j=1}^{m^{*}/2} \Bigg \{ \frac{\left( \omega _{11\cdot 2} \int _{\tau _{2j-1}}^{\tau _{2j}}Q^{\left( 2j\right) }(r)dQ(r)^{\prime } \right) ^{2}}{\omega _{11\cdot 2} \int _{\tau _{2j-1}}^{\tau _{2j}}Q^{\left( 2j\right) }(r)Q^{\left( 2j\right) }(r)^{\prime }dr }\Bigg . \\&\Bigg . +\frac{\omega _{11\cdot 2}}{\tau _{2j}-\tau _{2j-1}}\left( Q\left( \tau _{2j}\right) -Q\left( \tau _{2j-1}\right) \right) \left( Q\left( \tau _{2j}\right) -Q\left( \tau _{2j-1}\right) \right) ^{\prime }\Bigg \} \\= & {} \frac{1}{m^{*}}\frac{1}{\kappa ^{\prime }\kappa }\sum _{j=1}^{m^{*}/2}\left[ \frac{\left( \int _{\tau _{2j-1}}^{\tau _{2j}}Q^{\left( 2j\right) }(r)dQ(r)^{\prime }\right) ^{2}}{\int _{\tau _{2j-1}}^{\tau _{2j}}Q^{\left( 2j\right) }(r)^{2}dr}+\frac{\left( Q\left( \tau _{2j}\right) -Q\left( \tau _{2j-1}\right) \right) ^{2}}{\tau _{2j}-\tau _{2j-1}}\right] , \end{aligned}$$

as in the case of $F_{A}\left( \tau ,m^{*}\right)$ with $\xi _{t}$ not autocorrelated.