Decoding unemployment persistence: an econometric framework for identifying and comparing the sources of persistence with an application to UK macrodata

Abstract

Most econometric analyses of persistence focus on the existence of non-stationary unemployment but not the origin of this. The present research contains a multivariate econometric framework for identifying and comparing different sources of unemployment persistence (e.g., slow adjustment versus a slowly moving equilibrium rate). The framework contains readily applicable formulas for testing different hypotheses of persistence and is used to identify the causes of this in the UK macroeconomy. The evidence suggests that persistence has been due to a slowly moving equilibrium, driven by the price of crude oil, and not to slow adjustment, for example as related to the process of wage formation, as has often been emphasized.

A Appendices

1.1 The partitioned structural ECM and its reduced ECM form

The point of departure is the SECM for the full process, $x_{t}\equiv (x_{1t}^{\prime },x_{2t}^{\prime })^{\prime },$ i.e.,

$$\begin{aligned} A\Delta x_{t}=m_{t}+Fx_{t-1}-C\Delta x_{t-1}+\varepsilon _{t}. \end{aligned}$$

(17)

This has the following block representation,

$$\begin{aligned} \left( \begin{array}{c@{\quad }l} A_{11} &{} A_{12} \\ 0 &{} A_{22} \end{array} \right) \left( \begin{array}{l} \Delta x_{1t} \\ \Delta x_{2t} \end{array} \right)= & {} \left( \begin{array}{l} m_{1t} \\ m_{2t} \end{array} \right) +\left( \begin{array}{c@{\quad }l} F_{11} &{} F_{12} \\ 0 &{} F_{22} \end{array} \right) \left( \begin{array}{l} x_{1t-1} \\ x_{2t-1} \end{array} \right) \nonumber \\&-\left( \begin{array}{c@{\quad }l} C_{11} &{} C_{12} \\ 0 &{} C_{22} \end{array} \right) \left( \begin{array}{l} \Delta x_{1t-1} \\ \Delta x_{2t-1} \end{array} \right) +\left( \begin{array}{l} \varepsilon _{1t} \\ \varepsilon _{2t} \end{array} \right) , \end{aligned}$$

(18)

corresponding to (2).

The corresponding reduced form ECM becomes,

$$\begin{aligned} \left( \begin{array}{l} \Delta x_{1t} \\ \Delta x_{2t} \end{array} \right)= & {} \left( \begin{array}{l} \mu _{1t} \\ \mu _{2t} \end{array} \right) +\left( \begin{array}{c@{\quad }l} \Pi _{11} &{} \Pi _{12} \\ 0 &{} \Pi _{22} \end{array}\right) \left( \begin{array}{l} x_{1t-1} \\ x_{2t-1} \end{array} \right) \nonumber \\&+\left( \begin{array}{c@{\quad }l} \Gamma _{11} &{} \Gamma _{12} \\ 0 &{} \Gamma _{22} \end{array} \right) \left( \begin{array}{l} \Delta x_{1t-1} \\ \Delta x_{2t-1} \end{array} \right) +\left( \begin{array}{l} \upsilon _{1t} \\ \upsilon _{2t} \end{array}\right) , \end{aligned}$$

(19)

with,

$$\begin{aligned} \mu _{t}\equiv & {} \left( \begin{array}{l} \mu _{1t} \\ \mu _{2t} \end{array} \right) =\left( \begin{array}{l} A_{11}^{-1}\left( m_{1}-A_{12}A_{22}^{-1}m_{2t}\right) \\ A_{22}^{-1}m_{2t} \end{array} \right) , \\ \Pi\equiv & {} \left( \begin{array}{c@{\quad }l} \Pi _{11} &{} \Pi _{12} \\ 0 &{} \Pi _{22} \end{array} \right) =\left( \begin{array}{c@{\quad }l} A_{11}^{-1}F_{11} &{} A_{11}^{-1}\left( F_{12}-A_{12}A_{22}^{-1}F_{22}\right) \\ 0 &{} A_{22}^{-1}F_{22} \end{array} \right) , \\ \Gamma _{1}\equiv & {} \left( \begin{array}{c@{\quad }l} \Gamma _{11} &{} \Gamma _{12} \\ 0 &{} \Gamma _{22} \end{array} \right) =\left( \begin{array}{c@{\quad }l} -A_{11}^{-1}C_{11} &{} -A_{11}^{-1}\left( C_{12}-A_{12}A_{22}^{-1}C_{22}\right) \\ 0 &{} -A_{22}^{-1}C_{22} \end{array}\right) , \end{aligned}$$

and $\Gamma$ which is defined as $I-\Gamma _{1}$ is thus,

$$\begin{aligned} \Gamma \equiv \left( \begin{array}{c@{\quad }l} I-\Gamma _{11} &{} -\Gamma _{12} \\ 0 &{} I-\Gamma _{22} \end{array} \right) =\left( \begin{array}{c@{\quad }c} I+A_{11}^{-1}C_{11} &{} A_{11}^{-1}\left( C_{12}-A_{12}A_{22}^{-1}C_{22}\right) \\ 0 &{} I+A_{22}^{-1}C_{22} \end{array} \right) . \end{aligned}$$

1.2 The UK application

1.2.1 The matrices

$$\begin{aligned} F= & {} \left( \begin{array}{l@{\quad }l@{\quad }l@{\quad }l@{\quad }l} \kappa _{1}+\kappa _{2}-1 &{} \rho _{1} &{} \rho _{21}+\rho _{22} &{} \rho _{5}+\rho _{6} &{} \rho _{3}+\rho _{4} \\ \phi _{2}+\phi _{3}+\phi _{4} &{} \kappa _{3}+\kappa _{4}-1 &{}\qquad 0 &{}\qquad 0 &{}\qquad 0 \\ \qquad 0 &{} \lambda _{1}+\lambda _{2}+\lambda _{3} &{} \kappa _{5}+\kappa _{6}-1 &{}\qquad 0 &{} \qquad 0 \\ \omega _{5}+\omega _{6} &{}\qquad 0 &{} \omega _{3}+\omega _{4} &{} \omega _{1}+\omega _{2}-1 &{}\qquad 0 \\ \qquad 0 &{}\qquad 0 &{}\qquad 0 &{}\qquad 0 &{} \pi _{2}+\pi _{3}-1 \end{array}\right) ,\\ A= & {} \left( \begin{array}{l@{\quad }l@{\quad }l@{\quad }l@{\quad }l} 1 &{} 0 &{} 0 &{} 0 &{} 0 \\ -\phi _{2} &{} 1 &{} 0 &{} 0 &{} 0 \\ 0 &{} -\lambda _{1} &{} 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} -\omega _{3} &{} 1 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 1 \end{array} \right) ,C=\left( \begin{array}{l@{\quad }l@{\quad }l@{\quad }l@{\quad }l} \kappa _{2} &{} 0 &{} \rho _{22} &{} \rho _{6} &{} \rho _{4} \\ \phi _{4} &{} \kappa _{4} &{} 0 &{} 0 &{} 0 \\ 0 &{} \lambda _{3} &{} \kappa _{6} &{} 0 &{} 0 \\ \omega _{6} &{} 0 &{} 0 &{} \omega _{2} &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} \pi _{3} \end{array} \right) ,\\ m_{t}= & {} \left( \begin{array}{l} \rho _{0}-g_{A}t+z_{t}^{p} \\ \phi _{1}+z_{t}^{d} \\ \lambda _{0}+z_{t}^{u} \\ \omega _{0}+z_{t}^{w} \\ \pi _{1}+z_{t}^{o} \end{array} \right) . \end{aligned}$$

1.2.2 Accumulated multiplier notation

See Table 5.

Table 5 Accumulated multipliers corresponding to the UK application

1.2.3 Description of the data

See Table 6.

Table 6 Description of the UK data

1.2.4 Misspecification tests

See Table 7.

Table 7 Misspecification vector tests