Following the recent financial crisis and its subsequent Great Recession, the issue of a sluggish US employment was raised by economic observers. In a previous post on Econbrowser, Menzie Chinn pointed out the usefulness of the Okun’s law in assessing the potential level of employment after the recession. Especially, Menzie shows that:
- If one does not account for the long-term relationship between GDP and employment (i.e.; if one focuses only on the relationship in differences), then the bounce-back in employment after the 2008-09 recession cannot be captured.
- A standard error-correction model (ECM) is able to reproduce the general evolutions, but misses a large part of the recovery after the end of the recession.
- Accounting for the US business cycle by incorporating a dummy variable that takes the value 1 during recessions and 0 otherwise, according to the NBER Dating Committee dating, enables a better reproduction of stylized facts.
In this work, we reconsider this ECM approach but we do not impose any dummy variable and propose to let the data speak through a non-linear ECM. More specifically, we retain the following non-linear specification:
where LEMP denotes private nonfarm payroll employment in logarithm, LGDP the logarithm of GDP, εt is iid, G denotes the transition function, γ is the slope parameter that determines the smoothness of the transition from one regime to the other, c is the threshold parameter, and Z denotes the transition variable.
This non-linear ECM takes the long-term relationship into account, as well as two regimes for the short-term relationship. More specifically, the economy can evolve in two states, but the transition between those two regimes is smooth: the two regimes are associated with large and small values of the transition variable relative to the threshold value, the switches from one regime to the other being governed by the transition variable Z. The smooth transition function, bounded between 0 and 1, takes a logistic form given by:
To account for the business cycle, we select as transition variable the average GDP growth over 2 quarters, lagged by 2 quarters, i.e.
We estimate this model over the period 1960q1 – 2007q4. The implementation of Johansen’s test leads to conclude to the existence of a cointegrating relationship between LEMP and LGDP. Once the estimation of this long-term relationship realized, we integrate the lagged residuals in our main non-linear equation.
The estimation of the non-linear specification is made following the methodology proposed by Teräsvirta (1994). We start by testing for the null hypothesis of linearity using the test introduced by Luukkonen et al. (1988). Once the null of linearity has been rejected, we select the specification of the transition function using the test sequence presented in Teräsvirta (1994). We refer for example to van Dijk, Teräsvirta and Franses (2002) for details on smooth transition models. We then estimate the non-linear ECM, leading to the results presented in the following table:
The estimated value cˆ of the threshold c is close to zero, delimiting thus periods of expansion and periods of recession, as shown by the transition function displayed in figure 1. The transition function is slightly lagged over the business cycle, underlining the point that the switch in regime intervenes at the end of the recession, or just after.
Figure 1: Estimated transition function and US recessions. Sources: authors’ calculations, and NBER for the dating of recessions. The gray bands represent US recessions.
This result confirms that the business cycle plays a non-negligible role in the short-term relationship between employment and output.
Second, we compute conditional dynamic forecasts of the variable LEMP over the period 2008q1 – 2012q3. This means that we forecast employment based on the knowledge of the ex post path for LGDP. The figure 2 below presents the observed employment (green line), as well conditional forecasts from a standard (linear) ECM (blue line) and from the non-linear ECM (red line).
Figure 2: Conditional forecasts of employment (in logs) and observed employment (in logs). Source: Log private non-farm employment (green), conditional forecasts from standard error correction model (blue) and from non-linear error correction model (red), seasonally adjusted. Quarterly employment figures are average of monthly figures. Source BLS via FRED and authors’ calculations. Quarterly GDP data used to estimate the models are seasonally adjusted, expressed in billions of chained 2005 dollars. Source BEA via FRED.
We clearly see that both ECMs are able to reproduce the main movements in employment during and after the recession. However, there is a persistent gap between the observed employment (green line) and the conditional forecasts from both models (red and blue lines), meaning that the employment is currently well below what it should be according to the models. When comparing both ECM models, taking the non-linear business cycle into account through the non-linear ECM (red line) leads to reduction in the gap. But the contribution of the non-linear cycle to the employment is low (the difference between the blue and the red lines is around 1.2 % in 2010) and tends to diminish (the red line tends to the blue line).
This leads us to conclude that there has been indeed an effect of the Great Recession on the long-term employment. Specifically, on average, since the exit of the recession (2009q2), we get that the employment is 2.7 % below its potential level (according to the non-linear ECM), meaning that, from a structural point of view, around 3 millions of jobs have been lost after the recession. This interpretation is in line with the recent literature on this topic, as pointed out for example by Chen, Kannan, Loungani and Trehan (2011) or Stock and Watson (2012) that put forward various explanations.
As in the linear framework, it is noteworthy that a non-linear model estimated in differences, that is without integrating a long-term relationship, also leads to unrealistic results. This underlines the usefulness of having the long-term relationship into the model.
Luukkonen, R., Saikkonen, P. and Teräsvirta, T. (1988), Testing linearity against smooth transition autoregressive models, Biometrika 75, 491-499.
Teräsvirta, T. (1994), Specification, estimation, and evaluation of smooth transition autoregressive models, Journal of the American Statistical Association 89, 208-218.
This post written by Laurent Ferrara and
and Valérie Mignon.