# Did Okun's law die after the Great Recession?

Abstract This paper estimates Okun's law, focusing on piecewise non-linearity in the form of structural breaks and threshold dynamics, and obtains regime-dependent as well as threshold-dependent changes in the unemployment rate. We employ an autoregressive distributed lag version of Okun's law in first differences, which allows for delayed reactions of the unemployment rate to output growth. Applied to U.S. data over 1949Q1-2015Q4, the empirical analysis characterizes Okun's law as a three-regime relationship with the first structural break coinciding with the 1973 oil price shock, and the second structural break immediately following the end of the Great Recession. We find support for threshold dynamics in each regime, which suggests that Okun's law follows complex non-linear dynamics. Okun's law, as a linear and constant "rule of thumb," breaks down in each of the three regimes. In each regime, the unemployment rate responds asymmetrically to changes in output. In sum, Okun's law died during the Great Recession. Only time will tell whether resurrection is feasible.Keywords Okun's law * Structural breaks * Threshold effects

JEL classification C14 * E31 * C22

1 Introduction

Since Okun's (1962) seminal contribution, a large literature documents substantial evidence of the correlation between changes in real output and changes in the unemployment rate, an enduring relationship that has become known in the macroeconomic literature as Okun's law. (1) Okun's law, in its difference version, postulates that the growth rate in real gross domestic product (GDP) drives the change in the unemployment rate. Theoretically, the relationship links aggregate demand and the Phillips curve; empirically, the "Okun's coefficient" provides a useful "rule of thumb" in macroeconomics, economic forecasting, and policy modeling and evaluation. The value of the Okun's coefficient gives a benchmark for policy makers to measure the unemployment costs of higher growth fluctuations. Despite its popularity, however, the usefulness of Okun's law depends on the stability, linearity, and symmetry assumptions implicit in the 1962 version.

While conventional macroeconomics accepted Okun's law as an empirical regularity, recent research questions the robustness of the relationship and the validity of the underlying assumptions. (2) Moreover, strong evidence for structural breaks in the relationship exists, which Lee (2000) attributes to rising female labor force participation, productivity and wage slowdowns, and corporate restructuring. More recently, Owyang and Sekhposyan (2012) and Abdel-Raouf (2014), using a rolling regression approach, detect significant changes in the unemployment rate-output growth nexus during the Great Recession of 2008-2009.

Significant evidence also supports the idea that macroeconomic time series, in general, exhibit non-linear or asymmetric behavior over various phases of the business cycle. Silvapulle, Moosa, and Silvapulle (2004), Harris and Silverstone (2001), and more recently, Chinn, Ferrara, and Mignon (2014) and Valadkhani and Smyth (2015) note that ignoring asymmetry when it is present leads to a misspecified model, which produces not only erroneous inference in hypothesis testing but also poor policy evaluation results and inadequate economic policies.

Macroeconomic forecasting could benefit from a better understanding of structural breaks and non-linearities in Okun's law. Okun's law can provide policy makers with a benchmark to measure the relative cost of output in terms of the unemployment rate. Stabilization policies designed to mediate the effect of output on the unemployment rate during recessions could benefit from an understanding of how the responsiveness of the unemployment rate to output growth changes when a given level of the growth of output (or the unemployment rate) is exceeded. Such analysis can be done with the use of a relatively econometric tool that tests for threshold non-linearity. A process may behave differently when the value of a given variable surpasses a threshold value, because a different model specification may apply.

The remainder of the paper is organized as follows: Section 2 outlines the empirical methodology. Section 3 displays the findings of the empirical analysis with three sets of results. First, we present the results from the constant parameter linear model, assuming no breaks and no thresholds within the breaks. Second, we present the results from the structural break model without threshold. Finally, we present the results from the structural break model with threshold. In other words, we first investigate whether the model exhibits structural breaks. Upon finding two structural breaks and, thus, three regimes, we then examine each regime for evidence of threshold non-linearity. Section 4 concludes.

2 The empirical models

The standard difference version of Okun's Law, written as a linear regression model, is given by:

[DELTA][U.sub.t] = [[alpha].sub.0] + [[alpha].sub.1][DELTA][GDP.sub.t] + [[epsilon].sub.t], (1)

where [DELTA][U.sub.t] = [U.sub.t] - [U.sub.t-1] and [U.sub.t] is the unemployment rate, measured in percent, [DELTA][GDP.sub.t] = [GDP.sub.t] - [GDP.sub.t-1] and [GDP.sub.t] is real gross domestic product, measured in natural logarithms, and [[epsilon].sub.t] is the error term. (3) Okun (1962) used quarterly data from 1947Q2 to 1960Q4 on the unemployment rate and real gross national product (GNP), and obtained parameter estimates of [[alpha].sub.0] = 0.3 and [[alpha].sub.1] = -0.3. His results implied that a 1% increase in real GNP growth is associated with a 0.3 percentage point decrease in the unemployment rate. (Note that the ratio -([[alpha].sub.0]/[[alpha].sub.1]) gives the growth rate at which the unemployment rate is stable, except for random shocks.)

This "static version" of Okun's law (Knotek 2007) captures only the contemporaneous correlation, and ignores the rich dynamics between [DELTA][U.sub.t] and [DELTA][GDP.sub.t], such as the effect of past GDP growth on the current unemployment rate or the effect of the past unemployment rate on the current unemployment rate, as suggested by the literature on persistence of the unemployment rate (Barro 1988; Mortensen and Pissarides 1994) and the hysteresis hypothesis (Blanchard and Summers 1987; Leon-Ledesma 2002), where "history matters" (Lang and de Peretti 2009). Knotek (2007) shows that the business cycle and variation in the timing of the connection between GDP growth and the unemployment rate affects the version of Okun's law in Eq. (1). One argument favors including past GDP growth to capture the idea of "jobless recoveries." If true, then after a recession, the recovery of employment lags the recovery of output and, thus, both employment and the unemployment rates will not only depend on current output but also on past output values. Including past changes in the unemployment rate on the right-hand side of the dynamic version of Okun's law helps to eliminate serial correlation in the error terms that comes from regressing the differences (Weber 1995; Moosa 1997). For these reasons, we specify Okun's law within a simple autoregressive distributed lag framework (see Weber 1995; Sogner and Stiassny 2002) as follows:

[DELTA][U.sub.t] = [[alpha].sub.0] + [[alpha].sub.1][DELTA][GDP.sub.t] + [[alpha].sub.2][DELTA][GDP.sub.t-1] + [[alpha].sub.3][DELTA][GDP.sub.t-2] + [[alpha].sub.4][DELTA][GDP.sub.t-1] + [[epsilon].sub.t]. (2)

In this "dynamic version" of Okun's law (Knotek 2007; Owyang and Sekhposyan 2012), (4) the coefficient [[alpha].sub.1] measures the contemporaneous effect of output growth, whereas the sum [[alpha].sub.1] + [[alpha].sub.2] + [[alpha].sub.3] measures the total short-run effect. This specification allows us to calculate the long-run effect of output growth on the unemployment rate change as follows:

[theta] = [[alpha].sub.1] + [[alpha].sub.2] + [[alpha].sub.3]/1 - [[alpha].sub.4]. (3)

In addition to the short- and long-run dynamics of the link between the unemployment rate and output growth, the empirical literature has increasingly focused on the presence of discrete changes (structural breaks) and threshold non-linearity and asymmetry of the relationship.

Okun's law in either the static or dynamic versions assumes linearity and symmetry: expansions and contractions in output exert the same absolute effect on the unemployment rate. As mentioned by Lundbergh, Terasvirta, and van Dijk (2003), ample empirical evidence exists for both structural breaks and non-linearity in the dynamic properties of many macroeconomic series. Gordon (1984) and Evans (1989) suspect that the unemployment rate and output dynamics underwent structural change following the crude oil shocks in the 1970s, resulting in a structural break in Okun's relationship. As mentioned above, Lee (2000) attributes the developments to structural changes caused by corporate restructuring, productivity and wage slowdowns, and rising female labor force participation. Moreover, Harris and Siverston (2001) argue the importance of testing for asymmetry and non-linearity in the relation between the unemployment rate and output growth. How the unemployment rate reacts to changes in output has implications for the labor market and the appropriate monetary and fiscal policy responses. Viren (2001) argues that asymmetry and non-linearity in Okun's law can result in varying degrees of effectiveness of unemployment policies. Conceptually, the existence of asymmetries in the relationship between the unemployment rate and output growth does not necessarily invalidate Okun's law. Although the original specifications impose symmetric responses no theoretical reason exists to justify the presence of symmetric responses in the relationship between the unemployment rate and output growth.

To test the model for structural breaks and threshold effects, we proceed in two basic steps. First, following the Bai and Perron (1998, 2003) methodology, we test the "dynamic" version for multiple structural breaks or regime shifts. Second, within each regime identified by the Bai--Perron procedure, we test the "dynamic" version for threshold non-linearity. That is, we apply threshold estimation, which can capture complex non-linearities and complex dynamics described by observed variables crossing unknown thresholds. (5)

3 Data and empirical results

We employ quarterly data on real GDP growth rate (not annualized) and the quarterly average of the civilian unemployment rate from 1949Q1 to 2015Q4. (6) As a preliminary step, we analyze the time series properties of [DELTA][U.sub.t] and [DELTA][GDP.sub.t] to determine whether they contain unit roots. We implement the Augmented Dickey-Fuller (ADF) and the Phillips-Perron (PP) tests with a constant term, and the relevant test statistics easily reject the null hypothesis of a unit root in both [DELTA][U.sub.t] and [DELTA][GDP.sub.t], using both tests. This implies stationarity of both series, a necessary condition for valid Bai-Perron tests.

Figure 1 depicts the linear correlation between [DELTA][U.sub.t] and [DELTA][GDP.sub.t], using quarterly data for 1949Q1-1960Q4, which is, approximately, the original period used in Okun (1962). Each dot in the chart represents the observed change in unemployment rate and the growth rate of real GDP in a particular quarter. The correlation coefficient is -0.781. The solid line depicts the OLS regression, which captures the average slope (-0.332) of the relationship. This states that a 1% decrease (increase) in the real GDP growth rate translates into a 0.332% increase (decrease) in the unemployment rate. This estimate closely approximates the original Okun's estimate. Figure 2 extends the linear correlation analysis to cover 1949Q1-2015Q4. Now, the correlation coefficient is -0.684, and the slope of the fitted line implies that a 1% decrease (increase) in real GDP growth associates with a 0.282% increase (decrease) in the unemployment rate.

3.1 Results from the linear constant parameter model

For ease of exposition, we first present the empirical results for the linear constant parameter autoregressive distributed lag, as in Eq. (2), which is the model to which the structural break model collapses if the regression parameters are equal across the regimes.

Table 1 reports the OLS estimates of Eq. (2) with standard errors corrected for heteroskedasticity and autocorrelation using the Newey-West (1987) covariance matrix. The intercept is positive and significantly different from zero at the 1% level. The contemporaneous effect, Okun's coefficient on [DELTA][GDP.sub.t] is -0.204 and significantly different from zero at the 1% level. The total short-run effect, the sum of the estimates on the GDP growth terms, exceeds the contemporaneous effect in absolute value and significantly differs from zero at the 1% level. This value of -0.294 does not differ much from the findings of Okun (1962), Moosa (1997), and Crespo-Cuaresma (2003). The [[alpha].sub.4] estimate of 0.315 is significant at the 1% level and positive, suggesting a positive dependence on the previous quarter's unemployment rate and a monotonie adjustment toward equilibrium. The long-run estimates of Okun's relationship are significant at the 1% level. In the long-run, a stronger relationship exists between the unemployment rate and output growth, as a 1% increase (decrease) in real GDP growth associates with a -0.429% decrease (increase) in the rate of unemployment. Thus, overall, the OLS results of the dynamic version of Okun's law provide strong evidence in support of the inverse linear relationship between the unemployment rate and GDP growth. Some evidence of residual serial correlation at longer lags exists. The Ramsey Reset test indicates possible non-linearities, since we can reject the null hypothesis of linear specification at the 5% level.

3.2 Results from the structural break model without threshold

Testing for structural change has become an important issue in econometrics because a multitude of political and economic factors can cause the relationships among economic variables to change over time. We estimate Okun's law through a multiple endogenous break model, making use of the approach of Bai and Perron (1998, 2003). (7) The Bai-Perron procedure allows testing for multiple breaks at unknown dates. Table 2 contains the Bai-Perron tests of structural breaks applied to the dynamic version of Okun's law.

The sup [F.sub.T](l) tests uniformly reject at the 5% level the null hypothesis of no structural break against the alternative of l breaks (l = 1,..., 5). The double maximum test statistics, UD max and WD max, are significant at the 5% level, confirming the finding of at least one structural break. The sequential test statistics sup FT(l + 1|l), on the other hand, are not significant for l > 2 suggesting a model with only two breaks.

The overwhelming and consistent evidence suggests a model with three regimes, with the endogenous break dates estimated at 1973Q2 and 2009Q3. The first break date coincides with the first oil price shock, while the second break date coincides with the end of the Great Recession. Our findings differ from Weber (1995), who finds no indication of structural break in 1973. (8)

Figures 3, 4, and 5 depict the linear correlation between [DELTA][U.sub.t] and [DELTA][GDP.sub.t] using the three subsamples 1949Q1-1973Q2, 1973Q3-2009Q2, and 2009Q3-2015Q4, which emerge from the Bai-Perron tests for structural breaks. The linear correlation coefficients for the pre-first oil price shock and the period between the first oil price shock and the end of the Great Recession are -0.738 and -0.718, respectively. The associated slopes of the fitted lines are -0.294 and -0.312, respectively. The correlation coefficient for post-Great Recession is 0.031, which does not differ significantly from zero. The slope of the fitted line is 0.015, implying basically a complete dissociation between output growth and changes in the unemployment rate.

Given the strong evidence of structural breaks in 1973Q2 and 2009Q3, we now explore the characteristics of the three regimes--1949Q2-1973Q2 (with 96 observations), 1973Q3-2009Q2 (with 144 observations), and 2009Q3-2015Q4 (with 26 observations). Table 3 reports the OLS estimates of the dynamic version of Eq. (2) for the three regimes. We correct the standard errors for heteroskedasticity and autocorrelation, using the Newey-West (1987) covariance matrix. The adjusted [R.sup.2] indicates that the structural change model improves substantially over the constant parameter model, but evidence of serial correlation at longer lags still exists.

The estimates of the first and second regimes do not differ too much from the corresponding OLS constant parameter estimates in Table 1. In the first regime, 1949Q2-1973Q2, the analysis produces highly significant coefficients. In the second regime, 1973Q3-2009Q2, the contemporaneous effect of output growth on the unemployment rate nearly matches the contemporaneous effect in the first regime and the corresponding constant parameter estimate. The total effect in the second regime approximately matches that in the first regime, but the persistence of the unemployment rate, a measure of the speed of adjustment to equilibrium, exceeds that of the first regime.

The results for the third regime, 2009Q3-2015Q4, however, show strikingly that none of the explanatory variables are statistically significant. Furthermore, the long-run effect of [DELTA][GDP.sub.t] on [DELTA][U.sub.t] does not significantly differ from zero at the 5% level.

Thus, it appears that in the post-Great Recession period, Okun's law does not hold either in the short- or long-run. Although the fit of Okun's law in the first and second regimes remains good, the linear relationship appears to break down immediately after the Great Recession.

3.3 Results from the structural break model with threshold

This sub-section investigates whether asymmetries exist in the estimates of the Okun law within the identified regimes. We use threshold models to detect non-linearities and asymmetries in models that we otherwise treat as linear. See Tong (1983, 1990) for a broad treatment of the threshold autoregressive (TAR) and the self-exciting autoregressive threshold (SETAR) models.

We first consider the existence of a threshold effect. Table 4 presents the Bai and Perron global and sequential tests for the threshold models applied to each of the three regimes. The search procedure obtains [DELTA][GDP.sub.t-1] as the threshold parameter in the first regime, [DELTA][GDP.sub.t-4] in the second regime, and [DELTA][U.sub.t-3] in the third regime. These findings suggest that we use the TAR model in the first and second regimes, and the SETAR model in the third regime. Such a change in the threshold variable itself confirms that a structural change occurred.

The UD max and WD max statistics, as well as the sup [F.sub.T](k) statistics reject the null hypothesis of no threshold. That is, both tests indicate that at least one threshold exists, rejecting the linearity hypothesis in favor of the threshold model. The sup [F.sub.T](l + 1|) statistics, on the other hand, indicate that only one threshold state exists in each of the three regimes. Tables 5, 6, and 7 present the estimates of the threshold model applied to each regime. Importantly, the serial correlation problems present in the constant parameter model in Table 1, and in the structural break model in Table 3, no longer exist in the threshold estimates of the three regimes. (9)

Table 5 provides the threshold estimates of Okun's relationship in the first regime, 1949Q2-1973Q2. In both states, unemployment rate persistence does not significantly differ from zero. In both states, output growth exerts a significant negative effect on the unemployment rate. The threshold [DELTA][GDP.sub.t-1] value is 0.39. Okun's law is stronger when output growth falls below the threshold, and weaker when output growth exceeds, or equals, the threshold. Below the threshold, the total effect of output growth on the unemployment rate is -0.654, while above the threshold, the total effect is only -0.251. Thus, when the output growth in the previous period switches from below to above the threshold, the total short-run effect of output growth falls by more than half. The test of equality of the total effect of output growth across the first and second states rejects the null at the 1% level.

Table 6 provides the threshold estimates of Okun's relationship in the second regime, 1973Q3-2009Q2. Unemployment rate persistence does not significantly differ from zero in the first state, but does significantly differ at the 1% level in the second state. The difference between the two persistence estimates, however, does not significantly differ from zero. In both states, the contemporaneous effect of output growth on the unemployment rate is significant with the correct sign. The first lag of output growth is significant at the 1% level in the first state, while in the second state, the second lag of output growth is significant at the 1% level. The threshold [DELTA][GDP.sub.t-4] value is 0.26. Okun's law is stronger when output growth falls below the threshold, and weaker when output growth exceeds, or equals, the threshold. Below the threshold, the total effect of output on the unemployment rate is -0.647, while above the threshold the total effect is only -0.313. Thus, when output growth at lag four switches from below to above the threshold, the total short-run effect of output growth falls by more than one-half. The test of equality of the total effect of output growth across the first and second states, however, rejects the null only at the 10% level. Thus, the main difference between the first and second states in the second regime is limited to the contemporaneous effect of output growth on the unemployment rate.

Finally, Table 7 provides the threshold estimates of Okun's relationship in the third regime, 2009Q3-2015Q4. The threshold model shifts from a TAR to a SET AR. The main difference between the two states is the estimate of [DELTA][U.sub.t-1]. The threshold [DELTA][U.sub.t-3] value is -0.10. Below the threshold the estimate of the coefficient of [DELTA][U.sub.t-1] is negative and significant at the 5% level, while above the threshold, the estimate is positive and significant at the 5% level. Clear evidence supporting the hypothesis of threshold non-linearity is discovered, simultaneously indicating an uneven mean-reverting pattern and asymmetry around the threshold value. The convergence pattern oscillates below the threshold, while it follows a monotonie adjustment above the threshold. Below the threshold only the estimate of the coefficient of [DELTA][GDP.sub.t-2] differs significantly from zero at the 5% level and has the correct sign. Above the threshold only the estimate of the coefficient of [DELTA][GDP.sub.t-1] is significant and has the correct sign. Thus, in the post-Great Recession regime, no contemporaneous effect of output growth on the unemployment rate occurs, but only a short-run delayed effect exists. The cumulative effect is significant at the 1% level in both states and is approximately the same in both states. The long-run effects, however, differ significantly. (10)

4 Conclusion

In the original and more recent formulations of Okun's law, the link between growth and the unemployment rate is postulated to be linear and constant. Our findings show that the reaction of the unemployment rate to changes in real GDP growth differs substantially across regimes. In particular, the OLS estimates for the third regime indicate that Okun's estimates have become insignificant, casting doubts on the effective validity of the relationship.

The threshold analysis conducted inside each regime rejects the linearity hypothesis in favor of threshold asymmetry and segmented non-linearity. Taken together, the U.S. data confirm that the reaction of the unemployment rate to output growth depends on both the regime and the threshold. When we incorporate these dependencies, we find that Okun's law remains valid in the sense of statistical significance of the estimates of the total effect of output growth on the unemployment rate, despite the lack of statistical significance of the contemporaneous effect. We also find that "history matters," in that past shocks are not immediately erased from the economy's memory. An important implication of this "path-dependency" is that economic policies designed to reduce unemployment should be sufficiently sustained to erase the negative stocks that remain in the memory (Lang and de Peretti 2009).

Our finding that Okun's law experiences structural breaks and threshold switching challenges the linearity and stability of this widely believed empirical regularity. Future research may consider what factors caused the structural shift and the non-linearity and asymmetry observed in the recent data. What we know now is that several circumstances have converged to affect the labor markets. At the secular level, the leading edge of the baby boomers began retiring, and this significantly contributed to the decline in the rate of labor force participation. At the cyclical level, the financial crisis affected investment, and the lack of investment lowered the growth of total factor productivity. These distortions suggest that the connection between output and the unemployment rate may no longer conform to a stable, symmetric, and monotonie linear relationship.

In sum, enough evidence exists to claim that original form of the Okun's law died during the Great Recession and a more modified approach is in order.

DOI 10.1057/s11369-017-0045-1

Published online: 3 July 2017

Acknowledgements We thank two anonymous referees and the editor for their useful comments on prior versions.

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Giorgio Canarella is an economist at the Center for Business and Economic Research (CBER) and a Visiting Lecturer in the Economics Department in the Lee Business School at the University of Nevada, Las Vegas. Prior to that, he was a Professor of Economics at the California State University, Los Angeles. His research interests include macroeconomic theory and policy, time-series econometrics, applied forecasting, and international finance. Born in Italy, he received his doctorate in Economics (Laurea) at the Catholic University of Milan and his Ph.D. in Economics at the University of Virginia.

Stephen M. Miller is an initial founder of the Economic Club of Las Vegas and is currently a member and former Chair of the Club's Board of Directors. He also is the Director of the Center for Business and Economic Research, and a Professor of Economics and former Department Chair in the Lee Business School at the University of Nevada, Las Vegas. Prior to that, he was a Professor of Economics and Department Head at the University of Connecticut. He also held a visiting position at the Federal Reserve Bank of Boston and served as a visiting scholar at the Congressional Budget Office. His research interests span a wide range of economic topics, including macroeconomic theory and policy, bank scale and efficiency, central bank decision making, housing, forecasting, and regional economics. Born in Marion, Indiana, he received higher education training at Purdue University, receiving his bachelor's degree with distinction in Engineering Sciences Engineering (a part of the Aeronautical Engineering School), and at the State University of New York at Buffalo, receiving his M.A. and Ph.D. degrees in Economics.

Giorgio Canarella [1,2] * Stephen M. Miller [1,2]

([mail]) Stephen M. Miller

stephen.miller@unlv.edu

Giorgio Canarella

giorgio.canarella@unlv.edu

[1] Department of Economics, University of Nevada, Las Vegas, NV 89154-6005, USA

[2] Center for Business and Economic Research, University of Nevada, Las Vegas, NV 89154-6002, USA

(1) This empirical relationship forms a major part of every traditional macro-model, as the aggregate supply curve comes from combining Okun' s law with the Phillips curve. Moreover, this relationship also leads to important implications for macroeconomic policy. First, it documents what rate of growth of output leads to a reduction in the unemployment rate. Second, the effectiveness of disinflation policy depends on the responsiveness of the unemployment rate to output growth.

(2) For example, Ball, Leigh, and Loungani (2013) find evidence that Okun's law is remarkably stable, with no evidence of non-linearity. Additionally, several studies, including Gordon (1984), Prachowny (1993), Weber (1995), Moosa (1997), Lee (2000), Harris and Silverstone (2001), Sogner and Stiassny (2002), Osterholm (2016), support the empirical validity of the relationship, but the estimates of Okun's coefficient vary substantially across countries and time.

(3) We implement, in accordance with the majority of the literature (Lee 2000), a structured version of Okun's law, where output growth explains changes in unemployment.

(4) The distributed lag specification reduces the simultaneous equation bias for the total effect, as long as output growth is positively autocorrelated (Sogner and Stiassny 2002).

(5) For technical details of these steps, refer to a longer version of this paper posted online at the Social Science Research Network (see https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2826781).

(6) The data come from the FRED database at the Federal Reserve Bank of St. Louis and were downloaded in April 2016. See https://research.stlouisfed.org/fred2/.

(7) We apply, following the standard convention, a 10% trimming, so that each subsample contains at least approximately 26 observations (i.e., about seven years of data), and allow for a maximum of 5 breaks. We also incorporate heterogeneous error distributions across breaks. At the same time, since we include a lagged dependent variable as a regressor, we rule out serial correlation in the errors (see Assumption A4 in Bai and Perron 1998).

(8) To check further on the robustness of our results, we apply the Quandt-Andrew test to the residuals of the static and dynamic versions of Okun's law. This one-break test does not assume a given break date. In both the static and dynamic versions, we find a significant break at 2009Q2.

(9) Following the suggestion of one referee, we also estimate the threshold model over the entire sample. This ignores the structural breaks. The findings indicate that the appropriate model is a two-state TAR model, the value of the threshold variable [DELTA][GDP.sub.t-1] is 0.34. In the first state, the contemporaneous effect of output growth is -0.274, whereas in the second state is -0.172. In both states unemployment persistence is significant, but higher in the first (0.271) than in the second (0.187). The adjusted [R.sup.2] is 0.700, which is practically the same as the adjusted R-squared obtained in the structural break model in Table 3. As in the structural break model, however, problems of serial correlation also exist in the threshold model estimated over the entire sample, especially at 8 and 12 lags.

(10) We need to view the findings of the post-Great Recession regime with a degree of caution. The problem concerns the relative shortness of the sample. To address this problem, we re-estimated the threshold model deleting the variable that is not significant in both states (i.e., [DELTA][GDP.sub.t]). The estimates of the threshold as well as the estimates of the threshold regression are substantially unchanged, as one would expect, and are not reported for that reason. Still, a caveat is in order.

Caption: Fig. 1 Okun's Law: 1949Q1-1960Q4

Caption: Fig. 2 Okun's Law: 1949Q1-2015Q4

Caption: Fig. 3 Okun's Law: 1949Q1-1979Q2

Caption: Fig. 4 Okun's Law: 1979Q3-2009Q2

Caption: Fig. 5 Okun's Law: 2009Q3-2015Q4

Table 1 Constant parameter estimates of the dynamic linear model: 1949Q3-2015Q4 Variable Estimate Intercept 0.228 *** (0.038) [DELTA][GDP.sub.t] -0.204 *** (0.021) [DELTA][GDP.sub.t-1] -0.066 *** (0.023) [DELTA][GDP.sub.t-2] -0.024 (0.017) [DELTA][U.sub.t-1] 0.315 *** (0.059) Adj. R-squared 0.657 Q(4) 1.120 [0.891] Q(8) 13.633 [0.092] Q(12) 23.362 [0.025] Standard errors corrected for heteroskedasticity and autocorrelation (HAC) in parentheses, p-values in brackets. Q is the Ljung-Box statistic *** Significant at the 1% level Table 2 Bai-Perron specification test results for structural breaks: dynamic version Test statistic C.V. UD Max ** 142.388 [18.68] WD Max ** 142.388 [20.30] Sup F(l) ** 142.388 [18.68] Sup F(2) ** 74.794 [16.50] Sup F(3) ** 82.606 [15.07] Sup F(l/0) ** 142.388 [18.68] Sup F(2/l) ** 24.178 [20.57] Sup F(3/2) 18.598 [21.60] Max number of breaks 5 Number of breaks 2 Bai-Perron (2003) 5% critical values in brackets. The sup [F.sub.T](l) is the scaled F statistics from the Bai and Perron (1998) test of l globally optimized breaks against the null of no structural break. The sup [F.sub.T](t + 1|l) is the scaled F statistics from the Bai and Perron (1998) test of l breaks versus l + 1 breaks ** Significant at the 5% level Table 3 Bai-Perron estimates of the structural break regimes Variable 1949Q3-1973Q2 1973Q3-2009Q2 Intercept 0.386 *** 0.281 *** (0.071) (0.055) [DELTA][GDP.sub.t] -0.220 *** -0.225 *** (0.033) (0.041) [DELTA][GDP.sub.t-1] -0.127 *** -0.071 ** (0.029) (0.035) [DELTA][GDP.sub.t-2] -0.037 * -0.079 *** (0.020) (0.019) [DELTA][U.sub.t-1] 0.133 ** 0.232 *** (0.066) (0.085) Adj. R-squared 0.704 Q(4) 6.121 [0.190] Q(8) 13.637 [0.092] Q(12) 22.309 [0.034] Variable 2009Q3-2015Q4 Intercept -0.053 (0.052) [DELTA][GDP.sub.t] 0.004 (0.115) [DELTA][GDP.sub.t-1] -0.076 (0.059) [DELTA][GDP.sub.t-2] -0.150 (0.093) [DELTA][U.sub.t-1] 0.089 (0.159) Adj. R-squared Q(4) Q(8) Q(12) Standard errors corrected for heteroskedasticity and autocorrelation (HAC) in parentheses, p-values in brackets. Q is the Ljung-Box statistic * Significant at the 10% level ** Significant at the 5% level *** Significant at the 1% level Table 4 Bai Perron specification test results for thresholds 1949Q2-1973Q2 Statistic UD max 36.257 [17.76] WD max 43.588 [19.11] Sup F(l) * 33.461 [17.66] Sup F(2) * 36.257 [14.69] Sup F(l/0) * 33.461 [17.66] Sup F(2/l) 13.137 [19.50] Max number of thresholds 2 Number of thresholds 1 Threshold variable [DELTA][GDP.sub.t-1] Threshold value 0.39 1973Q3-2009Q2 Statistic UD max 39.716 [17.76] WD max 47.746 [19.11] Sup F(l) * 35.648 [17.66] Sup F(2) * 39.716 [14.69] Sup F(l/0) * 35.648 [17.66] Sup F(2/l) 10.128 [19.50] Max number of thresholds 2 Number of thresholds 1 Threshold variable [DELTA][GDP.sub.t-4] Threshold value 0.26 2009Q3-2015Q4 Statistic UD max 79.571 [17.14] WD max 99.799 [18.11] Sup F(l) * 39.609 [17.121 Sup F(2) * Sup F(l/0) * 39.609 [17.12] Sup F(2/l) Max number of thresholds 1 Number of thresholds 1 Threshold variable [DELTA][U.sub.t-3] Threshold value -0.1 Bai-Perron (2003) 5% critical values in brackets * Significant at the 10% level Table 5 Estimates of the TAR threshold model: 1949Q3-1973Q2 regime [DELTA][GDP.sub.t-1] < 0.39 Constant 0.559 *** (0.062) [DELTA][GDP.sub.t] -0.272 *** (0.021) [DELTA][GDP.sub.t-1] -0.251 ** (0.102) [DELTA][GDP.sub.t-2] -0.131 *** (0.046) [DELTA][U.sub.t-1] -0.005 (0.171) Adj. R-squared 0.814 Q(4) 2.532 [0.639] Q(8) 4.893 [0.769] Q(12) 11.718 [0.469] No. Obs 26 [DELTA][GDP.sub.t-1] [greater than or equal to] 0.39 Constant 0.159 ** (0.068) [DELTA][GDP.sub.t] -0.201 *** (0.028) [DELTA][GDP.sub.t-1] -0.026 (0.031) [DELTA][GDP.sub.t-2] -0.023 (0.025) [DELTA][U.sub.t-1] 0.106 (0.071) Adj. R-squared Q(4) Q(8) Q(12) No. Obs 70 Standard errors corrected for heteroskedasticity and autocorrelation (HAC) in parentheses, p-values in brackets. Q is the Ljung-Box statistic ** Significant at the 5% level *** Significant at the 1% level Table 6 Estimates of the TAR threshold model: 1973Q3-2009Q2 regime [DELTA][GDP.sub.t-4] < 0.26 Constant 0.325 *** (0.079) [DELTA][GDP.sub.t] -0.315 *** (0.041) [DELTA][GDP.sub.t-1] -0.173 *** (0.048) [DELTA][GDP.sub.t-2] 0.021 (0.029) [DELTA][U.sub.t-1] 0.301 (0.251) Adj. R-square 0.751 Q(4) 2.189 [0.701] Q(8) 6.803 [0.558] Q(12) 11.718 [0.469] No. Obs 28 [DELTA][GDP.sub.t-4] [greater than or equal to] 0.26 Constant 0.237 *** (0.044) [DELTA][GDP.sub.t] -0.165 *** (0.026) [DELTA][GDP.sub.t-1] -0.031 (0.026) [DELTA][GDP.sub.t-2] -0.115 *** (0.029) [DELTA][U.sub.t-1] 0.256 *** (0.055) Adj. R-square Q(4) Q(8) Q(12) No. Obs 116 Standard errors corrected for heteroskedasticity and autocorrelation (HAC) in parentheses. p-values in brackets. Q is the Ljung-Box statistic *** Significant at the 1% level Table 7 Estimates of the SETAR threshold model: 2009Q3-2015Q4 regime [DELTA][U.sub.t-3] < -0.10 Constant -0.219 ** (0.084) [DELTA][GDP.sub.t] -0.071 (0.067) [DELTA][GDP.sub.t-1] -0.051 (0.062) [DELTA][GDP.sub.t-2] -0.151 ** (0.063) [DELTA][U.sub.t-1] -0.562 ** (0.264) Adj. R-squared 0.548 Q(4) 1.208 [0.877] Q(8) 8.311 [0.404] Q(12) 12.477 [0.408] No. Obs 15 [DELTA][U.sub.t-3] [greater than or equal to] -0.10 Constant 0.127 (0.156) [DELTA][GDP.sub.t] -0.063 (0.278) [DELTA][GDP.sub.t-1] -0.275 *** (0.057) [DELTA][GDP.sub.t-2] 0.046 ** (0.096) [DELTA][U.sub.t-1] 0.282 *** (0.094) Adj. R-squared Q(4) Q(8) Q(12) No. Obs 11 Standard errors corrected for heteroskedasticity and autocorrelation (HAC) in parentheses, p-values in brackets. Q is the Ljung-Box statistic ** Significant at the 5% level *** Significant at the 1% level

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Comment: | Did Okun's law die after the Great Recession? |
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Author: | Canarella, Giorgio; Miller, Stephen M. |

Publication: | Business Economics |

Article Type: | Abstract |

Geographic Code: | 1USA |

Date: | Oct 1, 2017 |

Words: | 6770 |

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