The ins and outs of unemployment over different time horizons

Abstract

We devise a decomposition for the level of the unemployment rate that allows the assessment of the contributions of the various flow rates in the labor market over different time horizons. In particular, the decomposition allows one to recover the contributions of the flow rates in the long-run projection of the unemployment rate as in the steady-state decomposition widely used in the literature. We apply our methodology for data from the USA and Brazil. The results show that the relative contribution of the flows to the variance of the cyclical component of unemployment is sensitive to the time horizon of the projected unemployment rate. The changes in relative contributions of some flows are so prominent that their rankings change over time. These qualitative changes are more significant for Brazil than for the USA. Our results also evince that the steady-state approximation of the unemployment rate performs relatively worse in the former country. We take these findings as supporting evidence for considering the use of projected unemployment rate over shorter time horizons as an alternative to the steady-state approximation.

Appendices

Appendix A: deriving the convergence to the steady state

For each period $t$, let $\pi_{t}$ be the transition matrix, $p_{t}$ the state probability vector and $p_{t}^{*}$ its correspondent steady-state vector.

Let the canonical decomposition of the transition matrix be expressed by $\pi_{t} = \gamma_{t} \varLambda_{t} \gamma_{t}^{ - 1}$, where $\gamma_{t}$ represents the eigenvectors and $\varLambda_{t}$ the diagonal matrix of eigenvalues. As the sum of the columns of $\pi_{t}$ is equal to 1, it has incomplete rank. In the case of a model with three states, $\pi_{t}$ has rank equal to 2 and therefore $\varLambda_{t}$ has two eigenvalues whose values are less than 1 and one eigenvalue equal to 1, where the corresponding eigenvector has all components with equal values. This has been confirmed in our empirical analysis for both the USA and Brazil.

Let $\varLambda_{t} = \chi +\chi_{t}$, where $\chi = diag\left( {0,0,1} \right)$ and $\chi_{t} = diag\left( {\lambda_{t}^{1} ,\lambda_{t}^{2} ,0} \right)$, with $\lambda_{t}^{1} < 1$ and $\lambda_{t}^{2} < 1$. Denoting the state probability vector for the current period by $p_{t}$, the projection of $p$ for $h$ periods ahead is given by the following:

$$ p_{t + h} = \pi_{t}^{h} p_{t} = \gamma_{t} \varLambda_{t}^{h} \gamma_{t}^{ - 1} p_{t} = \gamma_{t} \left( { \chi^{h} +\chi_{t}^{h} } \right)\gamma_{t}^{ - 1} p_{t} = \left( {A_{t} + A_{t + h} } \right)p_{t} , $$

where

• $A_{t} = \gamma_{t} \chi^{h} \gamma_{t}^{ - 1} = \left( {a_{t} a_{t} a_{t} } \right)\;{\text{and}}\;A_{t} p_{t} = \left( {a_{t} a_{t} a_{t} } \right)p_{t} = a_{t} ,$ for any initial state $p_{t}$ since the sum of the components of $p_{t}$ is 1, and

• $A_{t + h} = (\gamma_{t} { \chi_{t}^{h}} \gamma_{t}^{ - 1} )$ is a matrix that converges to the zero matrix as $h$ grows to infinity. Thus, $p_{t + h} = a_{t} + A_{t + h} p_{t}$ and $\lim_{h \to \infty } p_{t + h} = a_{t}$, where $a_{t}$ represents the steady-state vector of the state probability, i.e., $a_{t} = p_{t}^{*}$.

Now, if $p_{t + h}$ converges to $p_{t}^{*}$, then any well-behaved function of $p_{t + h}$, say $g\left( {p_{t + h} } \right)$, should converge to $g\left( {p_{t}^{*} } \right)$. In particular, the unemployment rate function $u\left( {p_{t + h} } \right) = u\left( {\pi_{t}^{h} p_{t} } \right)$ should converge to $u\left( {p_{t}^{*} } \right) = u\left( {F(\pi_{t} } \right))$.

Following Shimer (2012), the steady-state unemployment rate $u\left( {F(\pi_{t} )} \right)$ can be decomposed into a sum of terms, one of which can be represented by $u(F\left( {\psi \left( {\pi_{t} \left( {i,j} \right);\bar{\pi }} \right)} \right) - u\left( {F\left( {\bar{\pi }} \right)} \right),$ where $\left( {i,j} \right)$ corresponds to a transition between the state $i$ to a state $j \ne i$ and the matrix $\psi \left( {\pi_{t} \left( {i,j} \right);\bar{\pi }} \right)$ is defined in Eq. (10) in the text. The contribution of the transition $\left( {i,j} \right)$ to steady-state unemployment is given by

$$ \beta_{ij}^{*} = \frac{{\text{cov} \langle u(F\left( {\psi \left( {\pi_{t} \left( {i,j} \right);\bar{\pi }} \right)} \right), u\left( {F\left( {\pi_{t} } \right)} \right)\rangle}}{{\text{var} (u\left( {F\left( {\pi_{t} } \right)} \right)}} . $$

(15)

The prediction of the unemployment rate for $h$ periods ahead can be written as Eq. (11) of the text:

$$ u_{t + h} \cong \kappa_{h} + \left[ {u\left( {\bar{\pi }^{h + 1} p_{t - 1} } \right) - \kappa_{h} } \right] + \mathop \sum \limits_{i \ne j} \{ u\left( {\psi^{h + 1} \left( {\pi_{t} \left( {i,j} \right);\bar{\pi }} \right)\bar{p}} \right) - \kappa_{h} \} = \kappa_{h} + u_{0t + h} + \mathop \sum \limits_{i \ne j} u_{ijt}^{h} ,\quad \forall h = 1,2, \ldots . $$

We have contributions from the following terms to the projected unemployment rate:

$$ {\text{Transition}}\;\left( {i,j} \right)\!:\beta_{ij}^{h} = \frac{{\text{cov} \langle u\left[ {\psi_{t}^{h + 1} \left( {\pi_{t} \left( {i,j} \right);\bar{\pi }} \right)\bar{p}} \right],u_{t + h} \rangle }}{{\text{var} \left( {u_{t + h} } \right)}}; $$

(16)

$$ {\text{Initial}}\;{\text{state}}\;{\text{of}}\;{\text{the}}\;{\text{labor}}\;{\text{market}}\!:\beta_{0}^{h} = \frac{{\text{cov} \langle u\left( {\bar{\pi }^{h + 1} p_{t - 1} } \right),u_{t + h} \rangle}}{{\text{var} \left( {u_{t + h} } \right)}}. $$

(17)

As $\psi \left( {\pi_{t} \left( {i,j} \right);\bar{\pi }} \right)$ is a transition matrix, the previous convergence results also apply here, and thus $u\left( {\psi^{h + 1} \left( {\pi_{t} \left( {i,j} \right);\bar{\pi }} \right)\bar{p}} \right)$ converges to $u\left( {F\left( {\psi \left( {\pi _{t} \left( {i,j} \right); \bar{\pi}} \right)} \right)} \right)$. Hence, all the elements in (16) converge to their counterparts in (15) (i.e., $\beta_{ij}^{h}$ converges to $\beta_{ij}^{*}$, $\forall i \ne j$) when $h$ grows to infinity.

As for $\beta_{0}^{h}$, note that $\bar{\pi }^{h + 1} p_{t - 1} = \left( {a + A^{h} } \right)p_{t - 1} = a + A^{h} p_{t - 1}$. As $A^{h} p_{t - 1}$ converges to zero with $h$, $\beta_{0}^{h} = \frac{{\text{cov} \langle u\left( {a + A^{h} p_{t - 1} } \right),u_{t + h} \rangle}}{{\text{var} \left( {u_{t + h} } \right)}}$ also converges to zero.

Appendix B: continuous time correction for time aggregation

In this Appendix, we apply the widely used continuous time correction method put forward by Shimer (2012). For the three-state context, the method converts a discrete time Markovian transition matrix into its continuous time counterpart. The main conditions for the existence and uniqueness of this conversion are that the eigenvalues of the discrete time transition matrix are distinct, real and nonnegative. Fortunately, this is the case for both the US and Brazilian datasets we use. We implement this correction method using the decomposition methodology set out in Sect. 2.2 and the same empirical filters presented in Sect. 3 (i.e., seasonal adjustment, quarterly averages and HP filter). Results for the USA and Brazil are presented for the long-run unemployment rate in Table 3 and for the short-run rate in Table 4.

Table 3 Decomposition of the long-run unemployment rate in the three-state context based on Shimer (2012)’s time aggregation correction method

Table 4 Decomposition of the short-run unemployment rate in the three-state context based on Shimer (2012)’s time aggregation correction method

The comparison of Table 3 and Table 1 in the main text shows that most estimates for the importance of the corresponding flows are quite similar for the steady-state decomposition for the USA. These results are qualitatively similar to those obtained by Shimer (2012, Table 2), but the magnitudes of the contributions are somewhat different. For instance, taking his results for the 1987–2010 period, the relative importance of the ue transition rate (0.51) to the eu rate (0.17) is much higher than the ones we find here (0.35 and 0.26, respectively). Our results show a more balanced role played by the outflow and the inflow rates into unemployment, a pattern also observed in Gomes (2015). Notwithstanding the difference in methodology, sample periods and the usage of a two-state context, this balancing between the outflow and the inflow rates is more in line with the results from other papers in the literature (e.g., Elsby et al. 2009; Fujita and Ramey 2009).

As for Brazil, the main changes between Table 3 and Table 1 are for the iu and ui transitions (0.28 to 0.35 and 0.08 to 0.03, respectively) but these differences do not change the relative importance of the flow rates for explaining the cyclicality of the unemployment rate in that country.

The comparison of Table 4 and Table 2 also shows quite small differences in the estimates for the short-run decomposition for both countries. We conclude that our discrete time results are robust to the continuous time correction proposed by Shimer (2012).²³

Appendix C: correlation between the current and the projected unemployment rates

In Shimer (2012), the long-run (steady-state) unemployment rate is taken as a good proxy for the current rate in the USA. In our method, one can choose any projection horizon—including the long-run one—as a proxy for the current rate. In this Appendix, we measure the correlation between the projected unemployment rate for different time horizons ($h$) and the current unemployment rate for both the USA and Brazil. The figures are displayed in Table 5 for $h = 1, \ldots ,24$ months ahead of the current period as well as for the long-run case. The results for the USA show a high and stable correlation throughout the projection horizons, while for Brazil the correlation is high in the short run, declines within the first semester and flattens thereafter. As in Figs. 1a and 1b in the main text, these results confirm that the steady-state unemployment rate is only a good proxy for the current rate in the case of the USA and that shorter time horizons are preferable for Brazil.

Table 5 Correlation between current and projected unemployment rates by time horizon