Is the natural unemployment rate hypothesis valid for the United States? Evidence from recent unit root tests.

Murthy, Vasudeva N.R.

This research paper tests the natural unemployment rate hypothesis for the United States using a long span of monthly data on the unemployment rate for the period 1950-January through March, 2011 by applying a battery of unit root tests that includes the recent nonlinear and structural break unit root tests. The empirical findings reveal that the natural rate of unemployment hypothesis is valid in the United States. Policy implications of the findings are discussed.

Keywords: Unemployment hysteresis, structural breaks, natural unemployment rate, non-linear unit root tests

I. INTRODUCTION

Economists and applied econometricians have always been interested in understanding the stochastic nature of unemployment time series. Whether the natural unemployment rate itself is changing and whether the U.S. beverage curve has been shifting are some important questions that many economists are still examining. If the unemployment series of a country in its level is found to be non-stationary, meaning that its first three statistical moments such as the mean, variance and autocorrelation are dependent on time, then any shocks to the series are going to be permanent. This phenomenon would lead to permanent changes in the natural unemployment rate or NAIRU. In that case, the series is said to be integrated of the order one, I(1) and in the language of macroeconomics, the unemployment hysteresis hypothesis hold valid for the country. On the other hand, if the unemployment series is found to be stationary, any shocks to the series are temporary and the series will be mean-reverting. The evidence of a stationary unemployment series in an economy means that the natural rate hypothesis or the structuralist hypothesis is valid for the economy (See, Blanchard and Summers, 1986: Phelps, 1968 and Friedman, 1968, Caner and Hansen, 2001). In this case, the changes in the unemployment rate are transitory.

While there exists a plethora of econometric studies testing the natural rate unemployment hypothesis (see, Tamarit, 2004) for the Organization for Economic Co-operation and Development (OECD) countries, one finds a small number of studies dealing with the phenomenon for the United States (see Mitchell, 1993; Roed, 1996; Tsay(1997), Caner and Hsmsen (2001), Camerero et al. (2010). Most of these studies, with the exceptions of Tsay (1997) and Caner and Hansen (2001), besides using relatively small sample sizes, have used the ADF unit root test. It has been demonstrated that the ADF test, which is a linear unit root test, suffers from low power. Furthermore, with the exceptions of Tsay (1997) and Caner and Hansen (2001), many of the above mentioned studies have ignored the existence of frictions, regulations, transaction costs and broken deterministic trends as structural changes that might cause the data generating process of the unemployment rate to display nonlinear behavior in its course of path to the long run equilibrium. The present paper attempts to test the natural unemployment rate hypothesis by applying the recent nonlinear unit root tests developed by Kapetanios et al. (2003), the modified Kapetanios nonlinear unit root test of Kruse (2011) and the Lee-Strazicieh minimum LM unit root test with Structural Breaks (2003). Another innovative contribution of the paper is that in order to test the hypothesis under investigation, it uses a very long span of data on the 15.S. monthly unemployment rate for the period 1950- January, 2011: March.

II. METHODOLOGY AND DATA

In order to accommodate nonlinearities in the data generating process [x.sub.t] Kapetanios et al. (2003) specify a reparameterised, exponential smooth transition autoregressive (ESTAR) model of the following type data generating process model, [x.sub.t] as follows:

[DELTA][x.sub.t] = [gamma][x.sub.t-1] (1-exp{-[theta][(xt-1-c).sup.2]}) + [[epsilon].sub.t] (1)

Where in model (1), the speed of mean reversion parameter [theta] = 0 under the null hypothesis of the presence of a unit root and [theta] > 0 under the alternative hypothesis of a nonlinear but globally stationary data generating process (see, for mathematical details, Kapetanios et al., 2003 and Kruse, 2011). While Kapetanios et al. assume that the location parameter c in (1) is zero; Kruse (2011) considers c to be non-zero in most of the real world situations. Since testing the null hypothesis is not feasible because [gamma] is not identified in (1) under the null, by using the first-order Taylor approximation and assuming that c is zero, Kapetanios et al. suggest the following estimable auxiliary regression model with the cubic term, [delta][[x.sup.3].sub.t-1] approximating the ESTAR nonlinear function, in the presence of serially correlated errors, with j augmentations:

[DELTA][x.sub.t] = [delta][x.sup.3.sub.i-1] + [j.summation over (i=1)][[rho].sub.i] [DELTA][x.sub.t-i] + [error.sub.t] = (2)

In model (2), the maintained null hypothesis is that [delta] = 0 against the alternative hypothesis of [delta] < 0. This test is also called KSS test. We use the Ordinary Least Squares (OLS) estimator to estimate (2). The test statistic [t.sub.NL], the observed KSS test statistic, is derived by dividing the estimated a and by its standard error in (2) and is presented as [t.sub.NL] = [??] / S.e. ([??]). As the asymptotic standard normal distribution critical values are not defined for these tests, Kapetanios et al. (2003) report the 1%, 5% and 10% bootstrapped critical values for the raw, demeaned and de-trended data series, depending on the deterministic terms specified in the auxiliary regression model (2). Furthermore, they demonstrate that the KSS tests have better power and size properties. The data on seasonally unadjusted monthly unemployment rate used in this paper are gathered from the U.S. Bureau of Labor Statistics (also, see Economic Data, Federal Reserve Bank of St. Louis, Economic data).

III. RESULTS

The results of the ADF, ADF-GLS and the Ng and Perron's MZa and [MZ.sub.t] tests, reported in Table 1, clearly show that the null hypothesis of the presence of a unit root can be rejected at the 5% and 10% levels of significance. The ADF, ADF-GLS, MZ[alpha] and [MZ.sub.t] tests maintain the null hypothesis of the presence of a unit root in the data generating process. The ADF-GLS test is a modified ADF test that transforms the time series by a generalized regression before conducting the unit root test (See, Elliot et al. 1996). The ADF-GLS is a more powerful test than the ADF test, especially when the alternative hypothesis is stationarity and a time trend is included in the regression as a deterministic term. Ng and Perron (2001) have developed two modified tests, MZ[alpha] and [MZ.sub.t] unit root tests that are more powerful and exhibit fewer distortions. The results from the KPSS tests indicate that the null of stationarity cannot be rejected at the 1%. The KPSS unit root test states the null hypothesis as stationarity or the absence of a unit root in the data generating process. Thus, the overall conclusion from these unit root tests is that the time series on unemployment rate is stationary in its level. Thus, we can infer that the natural unemployment hypothesis is valid for the United States. Furthermore, we employ the nonlinear unit root tests, KSS, recently developed by Kapetaneos et al., (2003) and the "tau' test formulated by Kruse (2011) and the results for the demeaned unemployment series are presented in Table 2. The result from the KSS test shows that since the observed [t.sub.NLT] value is greater than the 10% critical value, we can reject the null of non-stationarity in favor of the alternative of nonlinear but globally stationary. The results from the Kruse's "tau' test results indicate that the null hypothesis of nonstationarity can be rejected both at the 5% and 10% significance levels. Thus, the results from these nonlinear unit root tests also confirm that the unemployment series in the United States during the period of observation is stationary and therefore it is integrated of the order zero, I (0).

As the above discussed and reported unit root tests in Table 2 do not consider the presence of structural breaks in the data generating process, we conduct the Lee and Strazicich's minimum Lagrange multiplier unit root test (2003) with two breaks. This test has many desirable econometric features. The main advantage of this test is that it allows explicitly structural breaks under both the null and alternative hypotheses. Under this test, the rejection of the null hypothesis unambiguously means that the data generating process is stationary with broken trends. In running this test, the optimal number of lags is determined by following the general to specific approach of Ng and Perron (1995), starting with a maximum number of 8 lagged terms. We use the 10% asymptotic value in deciding the significance level. The results of the Lee and Strazicich's test are shown in Table 3. The test statistic is significant at the 10% level of significance revealing that we can reject the null of the presence of a unit root process. The results also indicate that the unemployment series is stationary with two significant structural breaks associated with the recession of 1975 and the onset of a bubble in the housing market in 2005, preceding the financial crisis of 2007-2009.

IV. CONCLUSIONS

By conducting a battery of unit root tests that includes the nonlinear unit root tests recently developed by Kapetanios et al., and Kruse and the Lee and Strazicich's minimum Lagrange multiplier unit root test with two breaks, this paper extends the literature on stochastic properties of the United States monthly unemployment series during a long span of period, 1950: 1-2011: 3.

A noteworthy feature of the period examined in the present paper is that it is devoid of any exogenous outliers such as the World Wars and the great depression. The results overwhelmingly show that in the United States, the natural unemployment rate hypothesis is valid. The findings of the paper are meaningful as the United States is a predominantly market economy, where one finds less frictions in the labor market, in the presence of a well developed information and communication infrastructure and relatively speaking, the decline of unionism.

The results also indicate that monetary and fiscal policy actions are less mandatory and furthermore, they will not lead to any permanent changes in the unemployment rate. The evidence presented in the paper is also consistence with those of other studies on unemployment rate in the United States, using different time periods and various methodologies.

References

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VASUDEVA N. R. MURTHY *

* Professor of Economics, Creighton University, Omaha, Nebraska 68178, USA,

E-mail: vmurthy@creighton.edu

Table 1
Linear Unit Root Tests for the Level Series

Series         ADF           ADF-GLS        KPSS

UR        -3.020 (a)(4)   -3.440 (a)(4)   0.562 (b)

Series     [MZ.sub.d]      [MZ.sub.t]

UR        -8.867 (a)(4)   -2.073 (a)(4)

Notes: (a,b) denote significance respectively at the 5% and 10%
levels in rejecting the null hypothesis. The figures in parentheses
are the optimal lags. The Schwartz Information Criterion (SIC)
(2001) was used to determine the lags for the ADF tests.
Deterministic terms include the intercept. The KPSS test critical
value at the 1%, level is 0.739.

Table 2
Nonlinear Unit Root Tests

Series        [KSS.sub.DM]   [KRUSE.sub.DM]

[t.sub.NLT]      -2.720 **         11.980 *
1% CV               -3.480           13.750
5%CV                -2.930           10.170
10%                 -2.660            8.600

Notes: The asterisks * and ** denote the significance at the
5% and 10%, levels respectively.

Table 3

Lee-Strazicich Minimum LM Unit Root Test of Two Structural Breaks

Series   [S.sub.t-1]   [B.sub.lt]   [DT.sub.lt]   [B.sub.2t]

UR            -0.076        0.177         0.183       -0.544
          (-5.358) *       (0.381)    (3.538) *     (-1.164)

Series   [DT.sub.2t]   [TB.sub.1]   [TB.sub.2]

UR             0.121       1975.1       2005.3
          (1.990) **

Notes: Break locations, [[lambda].sub.1] = [TB.sub.1]/T) and
[[lambda].sub.2] = TB 2/T. [DT.sub.i] demote breaks Critical values
for unit root tests on the co-efficient of [S.sub.t-1] for
[[lambda].sub.1=0.4] and [[lambda].sub.2=0.8] at the 1%, 5% and
10% levels are respectively, are -6.42, -5.65 and -5.32 [Model CC]
(Lee and Strazicich, 2003, Table 2). Significance at the 1% and 5%
levels (Observed t-value in parentheses)