Instability in spatial error models: an application to the hypothesis of convergence in the European case

Abstract

This paper focuses on the hypothesis of stability in the mechanisms of spatial dependence that are usually employed in spatial econometric models. We propose a specification strategy for which the first step is to solve a local estimation algorithm, called the Zoom estimation. The aim of this stage is to detect problems of heterogeneity in the parameters and to identify the regimes. Then we resort to a battery of formal Lagrange Multipliers to test the assumption of stability in the processes of spatial dependence. The alternative hypothesis consists of the existence of several regimes in these parameters. A small Monte Carlo serves to confirm the behaviour of this strategy in a context of finite size samples. As an illustration, we solve an application to the case of the hypothesis of convergence for the per capita income in the European regions. Our results reveal the existence of a strong Centre-Periphery dichotomy in which instability extends to all the elements (coefficients of regression as well as parameters of spatial dependence) that intervene in a classical conditional β-convergence model.

Appendix: Tests for the existence of a structural break in the mechanisms of spatial interaction

In this Appendix, we are going to obtain the expressions of the Lagrange Multipliers of Eqs. 4 and 6 with which we test for the existence of a break in the parameter of spatial dependence. The first part of the appendix is dedicated to the SLM and the second to a SEM.

1.1 The case of the SLM⁴

The equation for this case appears in expression given by Eq. 4 in the text:

$$ \left. \begin{array}{l} y = \mathop \gamma \nolimits_{0} {\mathbf{W}}y + \mathop \gamma \nolimits_{1} \mathop {\mathbf{W}}\nolimits^{*} y + x\beta + u \hfill \\ u\sim N(0,\mathop \sigma \nolimits^{2} I) \hfill \\ \end{array} \right\} $$

(A1)

whose log-likelihood function is:

$$ l(y;\varphi ) = -\, {\frac{R}{2}}{ \log }(2\pi ) - {\frac{R}{2}}{ \log }(\mathop \sigma \nolimits^{2} ) - {\frac{{\left( {{\mathbf{B}}y - x\beta } \right)^{\prime } \left( {{\mathbf{B}}y - x\beta } \right)}}{{2\mathop \sigma \nolimits^{2} }}} + { \log }\left| {\mathbf{B}} \right| $$

(A2)

where φ is the vector of parameters of the model, φ = [β, γ₀, γ₁, σ²]′, and B is a square matrix of order (R, R), ${\mathbf{B}} = I - \mathop \gamma \nolimits_{0} {\mathbf{W}} - \mathop \gamma \nolimits_{1} \mathop {\mathbf{W}}\nolimits^{*}$. The score vector is the following:

$$ g(y;\varphi ) = {\frac{\partial l(y;\varphi )}{\partial \varphi }} = {\frac{1}{{\mathop \sigma \nolimits^{2} }}}\left[ {\begin{array}{*{20}c} {x^{\prime}\left( {{\mathbf{B}}y - x\beta } \right)} \\ {y^{\prime}{\mathbf{W^{\prime}}}\left( {{\mathbf{B}}y - x\beta } \right) + \mathop \sigma \nolimits^{2} {\frac{{\partial { \log }\left| {\mathbf{B}} \right|}}{{\partial \mathop \gamma \nolimits_{0} }}}} \\ {y^{\prime}{\mathbf{W}}^{*\prime } \left( {{\mathbf{B}}y - x\beta } \right) + \mathop \sigma \nolimits^{2} {\frac{{\partial { \log }\left| {\mathbf{B}} \right|}}{{\partial \mathop \gamma \nolimits_{1} }}}} \\ { - {\frac{R}{2}} + {\frac{{\left( {{\mathbf{B}}y - x\beta } \right)^{\prime } \left( {{\mathbf{B}}y - x\beta } \right)}}{{2\mathop \sigma \nolimits^{2} }}}} \\ \end{array} } \right] $$

(A3)

Under the assumption that there is no break in the coefficient of spatial dependence, this vector simplifies to:

$$ \left. {\begin{array}{*{20}c} {\mathop H\nolimits_{0} :\mathop \gamma \nolimits_{1} = 0} \\ {\mathop H\nolimits_{A} :\mathop \gamma \nolimits_{1} \ne 0} \\ \end{array} } \right\} \Rightarrow \mathop {g(y;\varphi )}\nolimits_{{\left| {\mathop H\nolimits_{0} } \right.}} = \left[ {\begin{array}{*{20}c} 0 \\ 0 \\ {{\frac{{y^{\prime}{\mathbf{W}}^{*\prime } \tilde{u}}}{{\mathop {\tilde{\sigma }}\nolimits^{2} }}} - {\text{tr}}\left[ {\mathop {\left( {I - \mathop {\tilde{\gamma }}\nolimits_{0} {\mathbf{W}}} \right)}\nolimits^{ - 1} \mathop {\mathbf{W}}\nolimits^{*} } \right]} \\ 0 \\ \end{array} } \right] $$

(A4)

where $\mathop {\tilde{\gamma }}\nolimits_{0}$ and $\mathop {\tilde{\sigma }}\nolimits^{2}$ are the ML estimations of γ₀ and σ² and $\tilde{u}$ is the corresponding series of residuals of the restricted model:

$$ \left. \begin{array}{l} y = \mathop \gamma \nolimits_{0} {\mathbf{W}}y + x\beta + u \hfill \\ u\sim N(0,\mathop \sigma \nolimits^{2} I) \hfill \\ \end{array} \right\} $$

(A5)

The structure of the Hessian matrix is a bit complex:

$$ \begin{gathered} {\frac{{\mathop \partial \nolimits^{2} l(y;\varphi )}}{{\partial \varphi \partial \varphi^{\prime}}}} = - {\frac{1}{{\mathop \sigma \nolimits^{2} }}}\left[ {\begin{array}{*{20}c} {x^{\prime}x} & {x^{\prime}{\mathbf{W^{\prime}}}y} & {x^{\prime}{\mathbf{W}}^{*\prime } y} & {{\frac{1}{{\mathop \sigma \nolimits^{2} }}}x^{\prime}u} \\ {y^{\prime}{\mathbf{W}}x} & {y^{\prime}{\mathbf{W^{\prime}W}}y - \mathop \sigma \nolimits^{2} {\frac{{\mathop \partial \nolimits^{2} { \log }\left| {\mathbf{B}} \right|}}{{\partial \mathop \gamma \nolimits_{0}^{2} }}}} & {y^{\prime}{\mathbf{W^{\prime}}}\mathop {\mathbf{W}}\nolimits^{*} y - \mathop \sigma \nolimits^{2} {\frac{{\mathop \partial \nolimits^{2} { \log }\left| {\mathbf{B}} \right|}}{{\partial \mathop \gamma \nolimits_{0} \partial \mathop \gamma \nolimits_{1} }}}} & {{\frac{1}{{\mathop \sigma \nolimits^{2} }}}y^{\prime}{\mathbf{W^{\prime}}}u} \\ {y^{\prime}\mathop {\mathbf{W}}\nolimits^{*} x} & {y^{\prime}{\mathbf{W}}^{*\prime } {\mathbf{W}}y - \mathop \sigma \nolimits^{2} {\frac{{\mathop \partial \nolimits^{2} { \log }\left| {\mathbf{B}} \right|}}{{\partial \mathop \gamma \nolimits_{0} \partial \mathop \gamma \nolimits_{1} }}}} & {y^{\prime}{\mathbf{W}}^{*\prime } \mathop {\mathbf{W}}\nolimits^{*} y - \mathop \sigma \nolimits^{2} {\frac{{\mathop \partial \nolimits^{2} { \log }\left| {\mathbf{B}} \right|}}{{\partial \mathop \gamma \nolimits_{1}^{2} }}}} & {{\frac{1}{{\mathop \sigma \nolimits^{2} }}}y^{\prime}{\mathbf{W}}^{*\prime } u} \\ {{\frac{1}{{\mathop \sigma \nolimits^{2} }}}u^{\prime}x} & {{\frac{1}{{\mathop \sigma \nolimits^{2} }}}u^{\prime}{\mathbf{W}}y} & {{\frac{1}{{\mathop \sigma \nolimits^{2} }}}u^{\prime}\mathop {\mathbf{W}}\nolimits^{*} y} & { - {\frac{R}{{\mathop {2\sigma }\nolimits^{2} }}} + {\frac{{u^{\prime}u}}{{\mathop \sigma \nolimits^{4} }}}} \\ \end{array} } \right] \hfill \\ \end{gathered} $$

(A6)

The information matrix evaluated under the null hypothesis is the following:

$$ \begin{gathered} {\mathbf{I}}(\varphi ) = - \mathop {E\left[ {{\frac{{\mathop \partial \nolimits^{2} l(y;\varphi )}}{{\partial \varphi \partial \varphi^{\prime}}}}} \right]}\nolimits_{{\left| {\mathop H\nolimits_{0} } \right.}} = {\frac{1}{{\mathop \sigma \nolimits^{2} }}}\left[ {\begin{array}{*{20}c} {x^{\prime}x} & {x^{\prime}{\mathbf{W^{\prime}}}\mathop {\mathbf{B}}\nolimits^{ - 1} x\beta } & {x^{\prime}{\mathbf{W}}^{*\prime } \mathop {\mathbf{B}}\nolimits^{ - 1} x\beta } & 0 \\ {\beta^{\prime}x^{\prime}\mathop {{\mathbf{B^{\prime}}}}\nolimits^{ - 1} {\mathbf{W}}x} & \begin{gathered} \left\{ {\beta^{\prime}x^{\prime}\mathop {{\mathbf{B^{\prime}}}}\nolimits^{ - 1} \mathop {\mathbf{W}}\nolimits^{'} } \right.{\mathbf{W}}\mathop {\mathbf{B}}\nolimits^{ - 1} x\beta \hfill \\ \left. { + 2\mathop \sigma \nolimits^{2} {\text{tr}}\mathop {\mathbf{B}}\nolimits^{ - 1} {\mathbf{W}}\mathop {\mathbf{B}}\nolimits^{ - 1} {\mathbf{W}}} \right\} \hfill \\ \end{gathered} & \begin{gathered} \left\{ {\beta^{\prime}x^{\prime}\mathop {{\mathbf{B^{\prime}}}}\nolimits^{ - 1} {\mathbf{W^{\prime}}}} \right.\mathop {\mathbf{W}}\nolimits^{*} \mathop {\mathbf{B}}\nolimits^{ - 1} x\beta \hfill \\ \left. { + 2\mathop \sigma \nolimits^{2} {\text{tr}}\mathop {\mathbf{B}}\nolimits^{ - 1} {\mathbf{W}}\mathop {\mathbf{B}}\nolimits^{ - 1} \mathop {\mathbf{W}}\nolimits^{*} } \right\} \hfill \\ \end{gathered} & {{\text{tr}}\mathop {{\mathbf{B^{\prime}}}}\nolimits^{ - 1} {\mathbf{W}}} \\ {\beta^{\prime}x^{\prime}\mathop {{\mathbf{B^{\prime}}}}\nolimits^{ - 1} \mathop {\mathbf{W}}\nolimits^{*} x} & \begin{gathered} \left\{ {\beta^{\prime}x^{\prime}\mathop {{\mathbf{B^{\prime}}}}\nolimits^{ - 1} {\mathbf{W^{\prime}}}} \right.\mathop {\mathbf{W}}\nolimits^{*} \mathop {\mathbf{B}}\nolimits^{ - 1} x\beta \hfill \\ \left. { + 2\mathop \sigma \nolimits^{2} {\text{tr}}\mathop {\mathbf{B}}\nolimits^{ - 1} {\mathbf{W}}\mathop {\mathbf{B}}\nolimits^{ - 1} \mathop {\mathbf{W}}\nolimits^{*} } \right\} \hfill \\ \end{gathered} & \begin{gathered} \left\{ {\beta^{\prime}x^{\prime}\mathop {\mathbf{B}}\nolimits^{ - 1} {\mathbf{W}}^{*\prime } } \right.\mathop {\mathbf{W}}\nolimits^{*} \mathop {\mathbf{B}}\nolimits^{ - 1} x\beta \hfill \\ \left. { + 2\mathop \sigma \nolimits^{2} {\text{tr}}\mathop {\mathbf{B}}\nolimits^{ - 1} \mathop {\mathbf{W}}\nolimits^{*} \mathop {\mathbf{B}}\nolimits^{ - 1} \mathop {\mathbf{W}}\nolimits^{*} } \right\} \hfill \\ \end{gathered} & {{\text{tr}}\mathop {{\mathbf{B^{\prime}}}}\nolimits^{ - 1} \mathop {\mathbf{W}}\nolimits^{*} } \\ 0 & {{\text{tr}}{\mathbf{W^{\prime}}}\mathop {\mathbf{B}}\nolimits^{ - 1} } & {{\text{tr}}{\mathbf{W}}^{*\prime } \mathop {\mathbf{B}}\nolimits^{ - 1} } & {{\frac{R}{{\mathop {2\sigma }\nolimits^{2} }}}} \\ \end{array} } \right] \hfill \\ \end{gathered} $$

(A7)

The Lagrange Multiplier for the test of Eq. A4 is obtained in the usual way:

$$ \begin{aligned} \left. {\begin{array}{*{20}l} {\mathop H\nolimits_{0} :\mathop \gamma \nolimits_{1} = 0} \\ {\mathop H\nolimits_{A} :\mathop \gamma \nolimits_{1} \ne 0} \\ \end{array} } \right\} &\Rightarrow \mathop {{\rm LM}}\nolimits_{{\rm break}}^{{\rm LAG}} = \left[ {\mathop {g\left( \varphi \right)}\nolimits_{{\left| {\mathop H\nolimits_{0} } \right.}} } \right]^{\prime } \mathop {\left[ {\mathop {I\left( \varphi \right)}\nolimits_{{\left| {\mathop H\nolimits_{0} } \right.}} } \right]}\nolimits^{ - 1} \left[ {\mathop {g\left( \varphi \right)}\nolimits_{{\left| {\mathop H\nolimits_{0} } \right.}} } \right]\mathop \sim \limits_{{\rm as}} \mathop \chi \nolimits^{2} (1) \\ \mathop {{\rm LM}}\nolimits_{{\rm break}}^{{\rm LAG}} &= {\frac{{\mathop {\left[ {{\frac{{y^{\prime } \mathop {\mathbf{W}}\nolimits^{*} \tilde{u}}}{{\mathop {\tilde{\sigma }}\nolimits^{2} }}} - {\rm tr}\mathop {{\tilde{\mathbf{B}}}}\nolimits^{ - 1} \mathop {\mathbf{W}}\nolimits^{*} } \right]}\nolimits^{2} }}{{\mathop {\tilde{\sigma }}\nolimits_{{\rm lag}}^{2} }}}\mathop \sim \limits_{{\rm as}} \mathop \chi \nolimits^{2} (1) \\ \end{aligned} $$

(A8)

The term $\mathop {\tilde{\sigma }}\nolimits_{\text{lag}}^{2}$ in the denominator is the asymptotic variance of the restriction corresponding to the null hypothesis, obtained as:

$$ \begin{gathered} {\tilde{\sigma }}_{\text{lag}}^{2} = \tilde{a} - \tilde{b}^{\prime} {{\tilde{\mathbf{V}}}\left[ {\tilde{\varphi }} \right]}_{{\left| { H_{0} } \right.}} \tilde{b} \hfill \\ \begin{array}{*{20}c} \to & {\tilde{a} = {\frac{{\tilde{\beta }^{\prime}x^{\prime}{{\tilde{\mathbf{B}}}}^{ - 1} {\mathbf{W}}^{\prime *} {\mathbf{W}}^{*} {{\tilde{\mathbf{B}}}}^{ - 1} x\tilde{\beta }}} {{{\tilde{\sigma }}^{2} }}} + 2{\text{tr}}{{\tilde{\mathbf{B}}}}^{ - 1} {\mathbf{W}}^{*} {{\tilde{\mathbf{B}}}}^{ - 1} {\mathbf{W}}^{*} } \\ \end{array} \hfill \\ \begin{array}{*{20}c} \to & {\tilde{b} = {\frac{1}{{{\tilde{\sigma }}^{2} }}} \left[ {\begin{array}{*{20}c} {x^{\prime}{\mathbf{W}}^{\prime *} {{\tilde{\mathbf{B}}}}^{ - 1} x\tilde{\beta }} \\ {\tilde{\beta }^{\prime}x^{\prime} {{\tilde{\mathbf{B}^{\prime}}}} ^{ - 1} {\mathbf{W^{\prime}}} {\mathbf{W}}^{*} {{\tilde{\mathbf{B}}}}^{ - 1} x\tilde{\beta } + 2 {\tilde{\sigma }}^{2} {\text{tr}} {{\tilde{\mathbf{B}}}}^{ - 1} {\mathbf{W}} {{\tilde{\mathbf{B}}}}^{ - 1} {\mathbf{W}}^{*} } \\ {{\text{tr}} {{\tilde{\mathbf{B}^{\prime}}}}^{ - 1} {\mathbf{W}}^{*} } \\ \end{array} } \right]} \\ \end{array} \hfill \\ \end{gathered} $$

(A9)

Matrix $\mathop {{\tilde{\mathbf{V}}}\left[ {\tilde{\varphi }} \right]}\nolimits_{{\left| {\mathop H\nolimits_{0} } \right.}}$ is the ML estimation of the corresponding variance matrix of the vector of parameters under the null hypothesis of Eq. A8. The Lagrange Multiplier of Eq. A8 can easily be generalized to the more general case in which there are p different regimes in the coefficient of autocorrelation. The extended model corresponding to this case is:

$$ \left. \begin{array}{l} y = \mathop \gamma \nolimits_{0} {\mathbf{W}}y + \sum\limits_{s = 1}^{p} {\mathop \gamma \nolimits_{s} \mathop {\mathbf{W}}\nolimits_{s}^{*} y} + x\beta + u \hfill \\ u\sim N(0,\mathop \sigma \nolimits^{2} I) \hfill \\ \end{array} \right\} $$

(A10)

The null hypothesis will be of the joint type:

$$ \begin{aligned} & \left. {\begin{array}{*{20}l} {\mathop H\nolimits_{0} :\mathop \gamma \nolimits_{1} = \mathop \gamma \nolimits_{2} = \ldots = \mathop \gamma \nolimits_{p} = 0} \hfill \\ {\mathop H\nolimits_{A} :\exists \mathop \gamma \nolimits_{j} \ne 0} \hfill \\ \end{array} } \right\} \hfill \\ & \quad \Rightarrow \mathop {{\rm LM}}\nolimits_{{\rm break}}^{{\rm LAG}} = \left[ {\mathop {g\left( \varphi \right)}\nolimits_{{\left| {\mathop H\nolimits_{0} } \right.}} } \right]^{\prime } \mathop {\left[ {\mathop {{\mathbf{I}}\left( \varphi \right)}\nolimits_{{\left| {\mathop H\nolimits_{0} } \right.}} } \right]}\nolimits^{ - 1} \left[ {\mathop {g\left( \varphi \right)}\nolimits_{{\left| {\mathop H\nolimits_{0} } \right.}} } \right]\mathop \sim \limits_{{\rm as}} \mathop \chi \nolimits^{2} (p) \hfill \\ \end{aligned} $$

(A11)

The score will reproduce the structure indicated in Eq. A4, with an equation such as ${\frac{{y^{\prime}\mathop {\mathbf{W}}\nolimits_{s}^{*} \tilde{u}}}{{\mathop {\tilde{\sigma }}\nolimits^{2} }}} - {\text{tr}}\mathop {{\tilde{\mathbf{B}}}}\nolimits^{ - 1} \mathop {\mathbf{W}}\nolimits_{s}^{*}$ for each regime. The Hessian and information matrices must be similarly extended.

1.2 The case of the SEM

The specification that we must use in this case is that of Eq. 5:

$$ \left. \begin{array} {l}y = x\beta + u \hfill \\ u = \mathop \rho \nolimits_{0} {\mathbf{W}}u + \mathop \rho \nolimits_{1} \mathop {\mathbf{W}}\nolimits^{*} u + \varepsilon \hfill \\ \varepsilon \sim N(0,\mathop \sigma \nolimits^{2} I) \hfill \\ \end{array} \right\} $$

(A12)

The log-likelihood function is the following:

$$ l(y;\varphi ) = - \,{\frac{R}{2}}{ \log }(2\pi ) - {\frac{R}{2}}{ \log }(\mathop \sigma \nolimits^{2} ) - {\frac{{\left( {y - x\beta } \right)^{\prime } {\mathbf{B^{\prime}B}}\left( {y - x\beta } \right)}}{{2\mathop \sigma \nolimits^{2} }}} + { \log }\left| {\mathbf{B}} \right| $$

(A13)

where φ is the vector of parameters φ = [β, ρ₀, ρ₁,σ²]′ and B is the diffusion matrix, of order (R, R), ${\mathbf{B}} = I - \mathop \rho \nolimits_{0} {\mathbf{W}} - \mathop \rho \nolimits_{1} \mathop {\mathbf{W}}\nolimits^{*}$. The score vector is:

$$ g(y;\varphi ) = {\frac{\partial l(y;\varphi )}{\partial \varphi }} = {\frac{1}{{\mathop \sigma \nolimits^{2} }}}\left[ {\begin{array}{*{20}c} {x^{\prime}{\mathbf{B^{\prime}}}\varepsilon } \\ {u^{\prime}{\mathbf{W^{\prime}}}\varepsilon + \mathop \sigma \nolimits^{2} {\frac{{\partial { \log }\left| {\mathbf{B}} \right|}}{{\partial \mathop \rho \nolimits_{0} }}}} \\ {u'{\mathbf{W}}^{*\prime } \varepsilon + \mathop \sigma \nolimits^{2} {\frac{{\partial { \log }\left| {\mathbf{B}} \right|}}{{\partial \mathop \rho \nolimits_{1} }}}} \\ { - {\frac{R}{{2\mathop \sigma \nolimits^{2} }}} + {\frac{{\varepsilon^{\prime } \varepsilon }}{{2\mathop \sigma \nolimits^{2} }}}} \\ \end{array} } \right] $$

(A14)

with $u = y - x\beta$ and $\varepsilon = {\mathbf{B}}u = {\mathbf{B}}(y - x\beta )$. Under the assumption that there is no break in the coefficient of spatial dependence, the model of Eq. A12 simplifies to:

$$ \left. \begin{array} {l}y = x\beta + u \hfill \\ u = \mathop \rho \nolimits_{0} {\mathbf{W}}u + \varepsilon \hfill \\ \varepsilon \sim N(0,\mathop \sigma \nolimits^{2} I) \hfill \\ \end{array} \right\} $$

(A15)

so the score becomes:

$$ \left. {\begin{array}{*{20}c} {\mathop H\nolimits_{0} :\mathop \rho \nolimits_{1} = 0} \\ {\mathop H\nolimits_{A} :\mathop \rho \nolimits_{1} \ne 0} \\ \end{array} } \right\} \Rightarrow \mathop {g(y;\varphi )}\nolimits_{{\left| {\mathop H\nolimits_{0} } \right.}} = \left[ {\begin{array}{*{20}c} 0 \\ 0 \\ {{\frac{{\tilde{u}^{\prime}\mathop {\mathbf{W}}\nolimits^{*} {\mathbf{B}}\tilde{u}}}{{\mathop {\tilde{\sigma }}\nolimits^{2} }}} - {\text{tr}}\left[ {\mathop {\left( {I - \mathop {\tilde{\rho }}\nolimits_{0} {\mathbf{W}}} \right)}\nolimits^{ - 1} \mathop {\mathbf{W}}\nolimits^{*} } \right]} \\ 0 \\ \end{array} } \right] $$

(A16)

where $\mathop {\tilde{\rho }}\nolimits_{0}$ and $\mathop {\tilde{\sigma }}\nolimits^{2}$ are the ML estimations of ρ₀ and σ² in the model of Eq. A15 and $\tilde{u}$ is the associated series of residuals. From the score of Eq. A14, it is straightforward to obtain the Hessian matrix:

$$ \begin{gathered} {\frac{{\mathop \partial \nolimits^{2} l(y;\varphi )}}{{\partial \varphi \partial \varphi^{\prime}}}} = - \frac{1}{{\mathop \sigma \nolimits^{2}}} \hfill \\ \left[ {\begin{array}{*{20}c} {x^{\prime}{\mathbf{B^{\prime}B}}x} & {x^{\prime}{\mathbf{WB}}u} & {x^{\prime}\mathop {\mathbf{W}}\nolimits^{*} {\mathbf{B}}u} & {{\frac{1}{{\mathop \sigma \nolimits^{2} }}}x^{\prime}{\mathbf{B^{\prime}B}}u} \\ {u^{\prime}{\mathbf{B^{\prime}W^{\prime}}}x} & {u^{\prime}{\mathbf{W^{\prime}W}}u - \mathop \sigma \nolimits^{2} {\frac{{\mathop \partial \nolimits^{2} { \log }\left| {\mathbf{B}} \right|}}{{\partial \mathop \rho \nolimits_{0}^{2} }}}} & {u^{\prime}{\mathbf{W^{\prime}}}\mathop {\mathbf{W}}\nolimits^{*} u - \mathop \sigma \nolimits^{2} {\frac{{\mathop \partial \nolimits^{2} { \log }\left| {\mathbf{B}} \right|}}{{\partial \mathop \rho \nolimits_{0} \partial \mathop \rho \nolimits_{1} }}}} & {{\frac{1}{{\mathop \sigma \nolimits^{2} }}}u^{\prime}{\mathbf{W^{\prime}B}}u} \\ {u^{\prime}{\mathbf{B^{\prime}W}}^{{*^{\prime}}} x} & {u^{\prime } \mathop {\mathbf{W}}\nolimits^{*\prime } {\mathbf{W}}u - \mathop \sigma \nolimits^{2} {\frac{{\mathop \partial \nolimits^{2} { \log }\left| {\mathbf{B}} \right|}}{{\partial \mathop \rho \nolimits_{0} \partial \mathop \rho \nolimits_{1} }}}} & {u^{\prime}\mathop {\mathbf{W}}\nolimits^{*\prime } \mathop {\mathbf{W}}\nolimits^{*} u - \mathop \sigma \nolimits^{2} {\frac{{\mathop \partial \nolimits^{2} { \log }\left| {\mathbf{B}} \right|}}{{\partial \mathop \rho \nolimits_{1}^{2} }}}} & {{\frac{1}{{\mathop \sigma \nolimits^{2} }}}u^{\prime}\mathop {{\mathbf{W^{\prime}}}}\nolimits^{*} {\mathbf{B}}u} \\ {{\frac{1}{{\mathop \sigma \nolimits^{2} }}}u^{\prime}{\mathbf{B^{\prime}B}}x} & {{\frac{1}{{\mathop \sigma \nolimits^{2} }}}u^{\prime}{\mathbf{B^{\prime}W}}u} & {{\frac{1}{{\mathop \sigma \nolimits^{2} }}}u^{\prime}\mathop {{\mathbf{B^{\prime}W}}}\nolimits^{*} u} & { - {\frac{R}{{\mathop {2\sigma }\nolimits^{2} }}} + {\frac{{u^{\prime}{\mathbf{B^{\prime}B}}u}}{{\mathop \sigma \nolimits^{4} }}}} \\ \end{array} } \right] \hfill \\ \end{gathered} $$

(A17)

as well as the information matrix corresponding to the null hypothesis:

$$ \begin{gathered} {\mathbf{I}}(\varphi ) = - \mathop {E\left[ {{\frac{{\mathop \partial \nolimits^{2} l(y;\varphi )}}{{\partial \varphi \partial \varphi^{\prime}}}}} \right]}\nolimits_{{\left| {\mathop H\nolimits_{0} } \right.}} = {\frac{1}{{\mathop \sigma \nolimits^{2} }}}\left[ {\begin{array}{*{20}c} {x^{\prime}{\mathbf{B^{\prime}B}}x} & 0 & 0 & 0 \\ {0^{\prime}} & {2\mathop \sigma \nolimits^{2} {\text{tr}}\left( {\mathop {{\mathbf{B^{\prime}}}}\nolimits^{ - 1} {\mathbf{W^{\prime}}}\mathop {\mathbf{B}}\nolimits^{ - 1} {\mathbf{W}}} \right)} & {2\mathop \sigma \nolimits^{2} {\text{tr}}\left( {\mathop {{\mathbf{B^{\prime}}}}\nolimits^{ - 1} \mathop {\mathbf{W}}\nolimits^{*'} \mathop {\mathbf{B}}\nolimits^{ - 1} {\mathbf{W}}} \right)} & {{\text{tr}}\mathop {{\mathbf{B^{\prime}}}}\nolimits^{ - 1} {\mathbf{W}}} \\ {0^{\prime}} & {2\mathop \sigma \nolimits^{2} {\text{tr}}\left( {\mathop {{\mathbf{W^{\prime}B^{\prime}}}}\nolimits^{ - 1} \mathop {\mathbf{W}}\nolimits^{*} \mathop {\mathbf{B}}\nolimits^{ - 1} } \right)} & {2\mathop \sigma \nolimits^{2} {\text{tr}}\left( {\mathop {{\mathbf{B^{\prime}}}}\nolimits^{ - 1} \mathop {\mathbf{W}}\nolimits^{*\prime } \mathop {\mathbf{B}}\nolimits^{ - 1} \mathop {\mathbf{W}}\nolimits^{*} } \right)} & {{\text{tr}}\mathop {{\mathbf{B^{\prime}}}}\nolimits^{ - 1} \mathop {\mathbf{W}}\nolimits^{*} } \\ 0 & {{\text{tr}}{\mathbf{W^{\prime}}}\mathop {\mathbf{B}}\nolimits^{ - 1} } & {{\text{tr}}\mathop {\mathbf{W}}\nolimits^{\prime *} \mathop B\nolimits^{ - 1} } & {{\frac{R}{{\mathop {2\sigma }\nolimits^{2} }}}} \\ \end{array} } \right] \hfill \\ \end{gathered} $$

(A18)

Finally, the Lagrange Multiplier for the hypothesis of Eq. A16 takes the following expression:

$$ \begin{aligned} \left. {\begin{array}{*{20}c} {\mathop H\nolimits_{0} :\mathop \rho \nolimits_{1} = 0} \\ {\mathop H\nolimits_{A} :\mathop \rho \nolimits_{1} \ne 0} \\ \end{array} } \right\} &\Rightarrow \mathop {{\rm LM}}\nolimits_{{\rm break}}^{{\rm ERR}} = \left[ {\mathop {g\left( \theta \right)}\nolimits_{{\left| {\mathop H\nolimits_{0} } \right.}} } \right]^{\prime } \mathop {\left[ {\mathop {\mathbf{I}\left( \theta \right)}\nolimits_{{\left| {\mathop H\nolimits_{0} } \right.}} } \right]}\nolimits^{ - 1} \left[ {\mathop {g\left( \theta \right)}\nolimits_{{\left| {\mathop H\nolimits_{0} } \right.}} } \right]\mathop \sim \limits_{{\rm as}} \mathop \chi \nolimits^{2} (1) \\& \mathop {{\rm LM}}\nolimits_{{\rm break}}^{{\rm ERR}} = {\frac{{\mathop {\left[ {{\frac{{\tilde{u}^{\prime}\mathop {\mathbf{W}}\nolimits^{*} {\tilde{\mathbf{B}}}\tilde{u}}}{{\mathop {\tilde{\sigma }}\nolimits^{2} }}} - {\rm tr}\mathop {{\tilde{\mathbf{B}}}}\nolimits^{ - 1} \mathop {\mathbf{W}}\nolimits^{*} } \right]}\nolimits^{2} }}{{\mathop {\tilde{\sigma }}\nolimits_{{\rm err}}^{2} }}}\mathop \sim \limits_{{\rm as}} \mathop \chi \nolimits^{2} (1) \\ \end{aligned} $$

(A19)

$\mathop {\tilde{\sigma }}\nolimits_{err}^{2}$ is the asymptotic variance of the restriction of the null hypothesis, which is equal to:

$$ \begin{array}{l} \mathop {\tilde{\sigma }}\nolimits_{err}^{2} = \mathop {\left[ {{\frac{{\mathop {\tilde{\sigma }}\nolimits^{2} }}{{2\mathop {\tilde{\sigma }}\nolimits^{2} {\text{tr}}\mathop {{\tilde{\mathbf{B}^{\prime}}}}\nolimits^{ - 1} \mathop {\mathbf{W}}\nolimits^{*} \mathop {{\tilde{\mathbf{B}}}}\nolimits^{ - 1} \mathop {{\mathbf{W^{\prime}}}}\nolimits^{*} - \mathop {\tilde{b}{\tilde{\mathbf{D}}}}\nolimits^{ - 1} \mathop {\tilde{b}}\nolimits^{{}} }}}} \right]}\nolimits^{ - 1} \hfill \\ \to \tilde{b} = \left[ {\begin{array}{*{20}l} {2\mathop \sigma \nolimits^{2} {\text{tr}}\mathop {{\tilde{\mathbf{B}^{\prime}}}}\nolimits^{ - 1} {\mathbf{W^{\prime}}}\mathop {{\tilde{\mathbf{B}}}}\nolimits^{ - 1} \mathop {\mathbf{W}}\nolimits^{*} } \\ {\mathop {{\text{tr}}{\tilde{\mathbf{B}^{\prime}}}}\nolimits^{ - 1} \mathop {\mathbf{W}}\nolimits^{*} } \\ \end{array} } \right] \hfill \\ \to {\tilde{\mathbf{D}}} = \left[ {\begin{array}{*{20}c} {2\mathop \sigma \nolimits^{2} {\text{tr}}\mathop {{\tilde{\mathbf{B}^{\prime}}}}\nolimits^{ - 1} {\mathbf{W}}\mathop {{\tilde{\mathbf{B}}}}\nolimits^{ - 1} {\mathbf{W^{\prime}}}} & {\mathop {{\text{tr}}{\tilde{\mathbf{B}^{\prime}}}}\nolimits^{ - 1} {\mathbf{W}}} \\ {\mathop {{\text{tr}}{\tilde{\mathbf{B}^{\prime}}}}\nolimits^{ - 1} {\mathbf{W}}} & {{\frac{R}{{2\mathop {\tilde{\sigma }}\nolimits^{2} }}}} \\ \end{array} } \right] \hfill \\ \end{array} $$

(A20)

As in the previous case, the Multiplier of Eq. A19 can be generalized to treat with p different regimes. The extended model will be:

$$ \left. \begin{array} {l}y = x\beta + u \hfill \\ u = \mathop \rho \nolimits_{0} {\mathbf{W}}u + \sum\limits_{s = 1}^{p} {\mathop \rho \nolimits_{s} \mathop {\mathbf{W}}\nolimits_{s}^{*} u} + \varepsilon \hfill \\ \varepsilon \sim N(0,\mathop \sigma \nolimits^{2} I) \hfill \\ \end{array} \right\} $$

(A21)

The null hypothesis is again of the joint type:

$$ \begin{gathered} \left. {\begin{array}{*{20}l} {\mathop H\nolimits_{0} :\mathop \rho \nolimits_{1} = \mathop \rho \nolimits_{2} = \cdots = \mathop \rho \nolimits_{p} = 0} \hfill \\ {\mathop H\nolimits_{A} :\exists \mathop \rho \nolimits_{j} \ne 0} \hfill \\ \end{array} } \right\} \hfill \\\quad \Rightarrow \mathop {{\rm LM}}\nolimits_{{\rm break}}^{{\rm ERR}} = \left[ {\mathop {g\left( \varphi \right)}\nolimits_{{\left| {\mathop H\nolimits_{0} } \right.}} } \right]^{\prime } \mathop {\left[ {\mathop {{\mathbf{I}}\left( \varphi \right)}\nolimits_{{\left| {\mathop H\nolimits_{0} } \right.}} } \right]}\nolimits^{ - 1} \left[ {\mathop {g\left( \varphi \right)}\nolimits_{{\left| {\mathop H\nolimits_{0} } \right.}} } \right]\mathop \sim \limits_{{\rm as}} \mathop \chi \nolimits^{2} (p) \hfill \\ \end{gathered} $$

(A22)

The score will be obtained as in A16, with an equation of the type ${\frac{{\tilde{u}^{\prime}\mathop {\mathbf{W}}\nolimits_{s}^{*} {\tilde{\mathbf{B}}}\tilde{u}}}{{\mathop {\tilde{\sigma }}\nolimits^{2} }}} - {\text{tr}}\mathop {{\tilde{\mathbf{B}}}}\nolimits^{ - 1} \mathop {\mathbf{W}}\nolimits_{s}^{*}$ for each of the p regimes considered. The Hessian and information matrices must be similarly extended.