Okun’s law revisited in the time–frequency domain: introducing unemployment into a wavelet-based control model

Abstract

This paper integrates the Okun’s law (OL) relationship into a wavelet-based control (WBC) model to compare simulated optimal fiscal and monetary policy for the US when the policymakers place varying emphasis on the primary macroeconomic targets of unemployment, output growth, and inflation. The simulation results show that the unemployment rate is impacted differently across frequency ranges. We find that fiscal policy is the most aggressive when economic growth is emphasized as a policy objective, whereas monetary policy is relatively more aggressive when the inflation rate is emphasized. Given that the US inflation rate was below target for the start of the simulation exercises, when it is emphasized, that leads to lower interest rates, a depreciated exchange rate, and larger aggregate investment. We also find that introducing OL into a WBC model leads to less expansionary fiscal and monetary policies when unemployment is initially low.

Appendix

Tables 8, 9, and 10 show the estimation results for Eq. (7), using different proxies for the natural rate of unemployment, where the potential GDP was estimated as a semi-logarithmic trend using data from the post-Bretton-Woods sample period. In Table 8, the natural rate of unemployment was taken to be 4.5%. In Table 9, the FRED data series for the Natural Rate of Unemployment was used as a proxy. In Table 10, we use the smooth trend series from the wavelet decomposition for the natural rate.

Table 8 Okun’s Law Unemployment coefficient estimates from Eq. (7), with (p values) $un_{j,k} {-} un_{j,k}^{*} = \beta_{un,0} + \beta_{un,1} (\% \Delta Y_{j,k} {-} \% \Delta Y_{j,k}^{*} ) + \omega_{un,j,k}$ un* = 4.5%

Table 9 Okun’s Law Unemployment coefficient estimates from Eq. (7), with (p values) $un_{j,k} {-} un_{j,k}^{*} = \beta_{un,0} + \beta_{un,1} (\% \Delta Y_{j,k} {-} \% \Delta Y_{j,k}^{*} ) + \omega_{un,j,k}$ un* is taken from the FRED database

Table 10 Okun’s Law Unemployment coefficient estimates from Eq. (7), with (p values) $un_{j,k} {-} un_{j,k}^{*} = \beta_{un,0} + \beta_{un,1} (\% \Delta Y_{j,k} {-} \% \Delta Y_{j,k}^{*} ) + \omega_{un,j,k}$ un* taken from the wavelet smooth trend

The empirical results presented in Tables 8 to 10 show that the slope values $\beta_{un,j,1}$ are similar for a given frequency range across the different proxies for the natural rate of unemployment. The R2 values are also similar across the three tables, with the largest values generally occurring in Table 9.

Since Eq. (7) is estimated as a modified version of OL where the output gap measured in percentage change form, rather than in logarithms, the results are not directly comparable to previous studies. After converting from percentages, the coefficients in Table 8 are closer to the 0.3 value in Okun (1962) than the values of 0.5 in Ball et al. (2017), and the values at lower-frequency ranges (but only in the levels version of OL) of 0.8 in Aguiar-Conraria et al. (2020).

Table 11 shows the estimation results for Eq. (8). Our empirical results, especially for Eq. (8), do align with the general findings of Aguiar-Conraria et al. (2020), where the statistical significance and magnitude of the OL coefficients are larger for the medium cycle frequencies from 2 to 8 years have the largest impact on unemployment. Our reduced form state-space model contains a 1-quarter lag for most wavelet-decomposed variables, with other variables also including a 2-quarter lag. Thus, the manner in which the results are integrated within our WBC model is also consistent with Aguiar-Conraria et al. (2020), which finds that although the lead or lag relationship of output to unemployment does vary over time and across frequencies, a lag in the unemployment response of 4 months is generally sufficient. Moreover, our state-space model actually computes these output lags by recombining the GDP components in the first-order matrix difference equation system, which is consistent with Anderton et al. (2014), which finds that using GDP component data enhances the OL relationship.

Table 11 Okun’s Law Unemployment coefficient estimates from Eq. (8), with (p values) $d_{un,j,k} = \beta_{un,j,0} + \beta_{un,j,1} d_{Y,j,k} + \omega_{un,j,k}$

Whereas the purpose of Aguiar-Conraria et al. (2020) is to improve the estimates of OL by utilizing wavelets to capture the time–frequency structure, the thrust of this analysis shows that a WBC model can be built which also captures the time–frequency aspects of OL. Thus, our model could employ the estimates obtained by any of these studies as the parameter values in the simulations. As a further extension, these coefficients could also be made time-varying within the model framework. If the model were employed to aid in real-time forecasting, then these coefficients would be updated during each quarter as the wavelet decomposition incorporated the new data as it became available.

We also conducted robustness checks when the OL coefficients were relatively larger at the longest cycles and relatively smaller at the shorter cycles. Aguiar-Conraria et al. (2020) found evidence that the magnitude of the OL response coefficients is larger for the frequency ranges 2 to 8 years, but also found evidence that the coefficients for the longer cycle of 8 to 16 years have also been large. Thus, we also ran simulations when the OL coefficients are adjusted as follows: coefficient on frequency range 1 (6 months to 1 year) is reduced by 10%, coefficient 2 (1–2 years) is reduced by 5%, coefficient 3 (2 to 4 years) is increased by 5%, coefficient 4 (4–8 years) is increased by 10%, and coefficient 5 (8–16 years) is increased by 25%.

Table 12 shows the cumulative simulation comparisons for the robustness checks on the OL coefficients. The first column compares the percentage changes in the cumulative values of the variables across the entire horizon when the new coefficients are substituted for the original coefficients in the base case. Although the cumulative changes are very small, this shows that fiscal policy is slightly more expansionary, and monetary policy is slightly more contractionary when unemployment is driven by relatively longer cycles.

Table 12 Cumulative simulations in the base case with greater OL responsiveness at the longer cycles

The other columns in Table 12 use the new coefficients to compare the cumulative changes when economic growth and then inflation are more heavily emphasized. All of these relative changes are similar to those in the original base case reported in Table 3. When economic growth is emphasized, the percentage changes in government spending, the interest rate, and the money supply are smaller compared to the base case with the new coefficients than for the original values. Thus, changing the emphasis from unemployment to economic growth has less of an impact on fiscal and monetary policies when the longer cycles are more prevalent in unemployment.

When inflation is emphasized, however, the fiscal and monetary policy changes are greater in Table 12 than in Table 3. Thus, when unemployment is driven by the longer cycles, shifting toward an inflation emphasis leads to slightly more aggressive fiscal and monetary policies. Comparing Table 3 to Table 12 also shows that when policymakers shift the emphasis toward either economic growth or inflation, cumulative investment increases more, and consumption increases less, when the OL coefficients place more emphasis in the longer cycles.