Mean Reversion of Unemployment Rates: Evidence for Colombia Using Regional Data

2009 Oxford Business & Economics Conference ProgramISBN : 978-0-9742114-1-1

Mean reversion of unemployment rates:

Evidence for Colombia using regional data

February 2009

Abstract

In this paper, we test for the stationarity of unemployment rates in Colombia over the period 1984Q1 to 2004Q4, using a panel of seven cities. Our testing strategy addresses key concerns with regard to unit root panel data testing, namelythe presence of serial correlation and cross-sectional dependency across the unemployment rates in the panel. To address these concerns, we employ an AR-based bootstrap approach that allows us to test the null hypothesis of joint stationarity allowing for cross sectional dependencies. In contrast to the existing literature, we find that unemployment rates in Colombia can be characterised as stationary processes, providing support to the view that they exhibit mean reversion, and therefore appear consistent with the natural rate hypothesis.

JEL classification: C12; C15; C22; C23; E31

1. INTRODUCTION

The question of whether unemployment rates should be treated as stationary or non-stationary processes has received a great deal of attention ever since the publication of the paper by Nelson and Plosser (1982). Indeed, classifying unemployment rates as one or the other has a number of important statistical and economic implications. From a statistical point of view, stationary series have a finite (non-zero) variance and exhibit temporary memory in the sense that the effect of a perturbation disappears as time passes. In contrast, non-stationary serieshave an unbounded variance and exhibit a permanent memory. From an economic perspective, a stationary unemployment rate has been commonly associated with the natural rate of unemployment hypothesis (NAIRU), for it may be viewed as a mean reverting process that fluctuates around its long-run equilibrium value, while non-stationarity has been associated with the hysteresis hypothesis, whereby long lasting unemployment effects arise from cyclical fluctuations (see Blanchard and Summers, 1986).

For these reasons, empirical examinations of unemployment behaviour have typically fallen into one of two categories. The first category of studies has examined the possibility of non-stationarity in unemployment by conducting univariate tests of unit roots.Studies such as Blanchard and Summers (1986), Brunello(1990), Jaeger and Parkinson (1994) and Røed (1996) consistently reject the null hypothesis of a unit root for the US (favouring the natural rate hypothesis)but are unable to reject the unit root hypothesis for other OECD countries(supporting the presence of unemployment hysteresis effects).Within a regional context, and allowing for the presence of structural breaks, Clemente et al. (2005) find favourable evidence to the rejection of the unit root hypothesis in US states, while Reis and Gomes (2008) find hysteresis effects in five out of six Brazilian regional unemployment rates.

The second group of studies has applied panel unit root techniques to test for stationarity since univariate tests suffer from low power. In recent years a number of alternative procedures have been proposed to test for the presence of unit roots in panels that combine information from the time-series dimension with that from the cross-section dimension, such that fewer time observations are required for these tests to have power. The most commonly used unit root test applied to panels include Maddala and Wu (MW) (1999) and Im, Pesaran and Shin (IPS) (2003), which test the joint null hypothesis of a unit root against the alternative of at least one stationary series, by using the augmented Dickey–Fuller (ADF) (1979) statistic across the cross-sectional units of the panel. A recent study that employs panel data methods is León-Ledesma (2002),who confirms that hysteresis for 11 EU countries and the natural rate for 51 US states are the most plausible hypothesis using the IPS test. It should, however, be noted that IPS (2003, p.73) warn that due to the heterogeneous nature of the alternative hypothesis in their test, one needs to be careful when interpreting the results, because the null hypothesis that there is a unit root in each cross section may be rejected when only a fraction of the series in the panel are stationary. A further issue of concern is that the presence of cross-sectional dependencies can undermine the asymptotic normality of the IPS test and lead to over-rejection of the null hypothesis of joint non-stationarity. To some extent, these concerns are addressed by Camarero and Tamarit (2004) and Chang et al. (2005) who conduct ADF unit root tests within a seemingly unrelated regression framework. Camarero and Tamarit (2004) reject hysteresis effects in 12 out of 19 OECD countries, while Chang et al. (2005) confirm the hysteresis hypothesis for 8 out of 12 European countries.

This paper aims to study regional unemployment in Colombia. In particular, we test for stationarity of unemployment rates using data for a panel of seven metropolitan areas. Since unit root tests applied to single series suffer from low power, panel unit root techniques offer a way forward in terms of enhanced test power. A distinctive feature of our analysis is that we apply the Hadri (2000) tests of the null hypothesis that all individual series are stationary, against the alternative of at least a single unit root in the panel. The Hadri tests offer the key advantage that if the null hypothesis is not rejected, we may conclude that all the city relative prices in the panel are stationary. In addition to this, an important novel feature of our analysis is that we allow for the presence of potential cross-sectional dependencies. More specifically, we consider a procedure based on a bootstrap of the Hadri tests because failure to account for cross-sectional dependencies leads to size distortion and over-rejection by the Hadri test.

We believe that the study of the Colombian case is interesting because Colombian regions are often characterised by a “centre–periphery” dichotomy, where the central region (which consists of the three main cities of Bogotá, Medellín and Cali) comprises the largest concentration of population, economic activity and infrastructure (see e.g. Galvis, 2008). In earlier papers,Arango and Posada (2002) examined the univariate time series properties of the unemployment rate in Colombia and were unable to reject the unit root hypothesis. Gamarra (2006) also finds support for the unit root hypothesis when using unemployment rates for the seven main metropolitan areas of the country. Our paper differs in that we study the stationarity issue within a panel context and allowing for the potential effects of cross-sectional dependencies.

The paper is organised as follows. Section 2 presents an overview of the Hadri (2000) panel stationarity tests. Section 3 describes the dataset and presents the results of applying the panel stationarity tests to the city unemployment rates in major Colombian cities. Section 4 concludes.

2. TESTING FOR PANEL STATIONARITY

Hadri (2000) proposes residual-based Lagrange Multiplier tests for the null hypothesis that all the time series in the panel are stationary (either around a level or a deterministic time trend), against the alternative that some of the series are nonstationary. The Hadri tests are panel versions of the Kwiatkowski, Phillips, Schmidt and Shin (KPSS) (1992) stationarity tests. Following Hadri (2000), consider the models:

(1)

and

(2)

where is a random walk, , and and are mutually independent normal distributions. Also, and are across and over , with , , , , and . The null hypothesis that all the series are stationary is given by , , while the alternative that some of the series are nonstationary is , and , .

Let be the residuals from the regression of on an intercept, for model (1)(or on an intercept and a linear trend term, for model (2)). Then, the individual univariate KPSS stationarity test is given by:

wheredenotes the partial sum process of the residuals given by and is a consistent estimator of the long-run variance of from the appropriate regression. In their original paper, KPSS propose a nonparametric estimator of based on a Bartlett window having a truncation lag parameter of , with . However, Caner and Kilian (2001) have pointed out that stationarity tests, like the KPSS tests, exhibit very low power after correcting for size distortions. Thus, in our paper we follow recent work by Sul, Phillips and Choi (2005), who propose a new boundary condition rule that improves the size and power properties of the KPSS stationarity tests. In particular, Sul et al. suggest the following procedure. First, an AR model for the residuals is estimated, that is:

(3)

where the lag length of the autoregression can be determined for example using the general-to-specific algorithm proposed by Campbell and Perron (1991). Second, the long-run variance estimate of is obtained with the boundary condition rule:

,

where denotes the autoregressive polynomial evaluated at . In turn, is the long-run variance estimate of the residuals in equation (3) that is obtained using a quadratic spectral window Heteroscedastic and Autocorrelation Consistent (HAC) estimator.[1]

The Hadri (2000) panel stationarity test statistic is given by the simple average of individual univariate KPSS stationarity tests:

which after a suitable standardisation, using appropriate moments, follows a standard normal limiting distribution.[2] That is:

where and .

The Monte Carlo experiments of Hadri (2000) illustrate that these tests have good size properties for and sufficiently large. However, Giulietti et al. (2008) show that even for relatively large and the Hadri (2000) tests suffer from severe size distortions in the presence of cross-sectional dependence, the magnitude of which increases as the strength of the cross-sectional dependence increases. This finding is in line with the results obtained by Strauss and Yigit (2003) and Pesaran (2007) on both the Im, Pesaran and Shin and the Maddala and Wu panel unit root tests. In order to correct for the size distortion caused by cross-sectional dependence, Giulietti et al. (2008) apply the bootstrap method and find that the bootstrap Hadri tests are approximately correctly sized.

To implement the bootstrap method in the context of the Hadri tests, we start off by correcting for serial correlation using equation (2) and obtain , which are centred around zero. Next, as suggested in Maddala and Wu (1999), the residuals are resampled with replacement with the cross-section index fixed, so that their cross-correlation structure is preserved; the resulting bootstrap innovation is denoted . Then, is generated recursively as:

,

where, in order to ensure that initialisation of , i.e. the bootstrap samples of , becomes unimportant, we follow Chan (2004) who advocates generating a largenumber of , say values and discard the first values of (for our purposes we choose ). Lastly, the bootstrap samples of are calculated by adding to the deterministic component of the corresponding model, and the Hadri LM statistic is calculated for each . The results shown in Table 1 are based on 1,000 bootstrap replications used to derive the empirical distribution of the LM statistic.

3. DATA AND EMPIRICAL ANALYSIS

The data set, obtained from Departamento Administrativo Nacional de Estadística–DANE, consists of seasonally unadjusted monthly observations on unemployment rates for the seven major metropolitan areas in Colombia: Bogotá, Medellín, Cali, Barranquilla, Bucaramanga, Manizales and Pasto. The sample period runs from 1984Q1 to 2004Q4 (for a total of 84 observations), and the series are considered in percentage terms.

Table 1 presents the results of applying the KPSS stationarity test to the unemployment rates of the metropolitan areas listed above (based on the model with intercept only). As can be seen from the table, the null hypothesis of stationarity is rejected for three out of the seven metropolitan areas under consideration, namelyBogotá, Bucaramanga and Cali. The evidence here is mixed and does not provide a clear indication of stationarity.

Next, we apply the Hadri test to the unemployment rates of Bogotá, Bucaramanga and Cali, which are found to be non-stationary on the basis of the individual KPSS tests discussed before. The results indicate that the null hypothesis that all of series in the panel are stationary is clearly rejected (the test statistic is 2.053; p-value 0.000). However, as indicated above, an important assumption underlying the Hadri test is that of cross section independence among the individual time series in the panel. To allow for potential cross section dependence and subsequent size distortion, we apply the bootstrap method to the Hadri test as described in the previous section. In particular, based on the results of the bootstrap Hadri test reported in the second line of Table 2, we are now unable to reject the null hypothesis of panel stationarity for a panel consisting of the metropolitan areas of Bogotá, Bucaramanga and Cali. This finding provides support to the view that the unemployment rates exhibit mean reversion, and therefore appear consistent with the natural rate hypothesis.

4. CONCLUDING REMARKS

This paper applies the Hadri (2000) tests for panel stationarity to examine evidence on stationarity for unemployment rates in seven Colombian cities. The Hadri tests offer the key advantage insofar as we may conclude that all the city relative prices in the panel are stationary, if the joint null hypothesis is not rejected. In addition to this, another important feature of our analysis is that we allow for the presence of serial correlation and cross-sectional dependency across the city relative prices in the panel, by means of the implementation of an AR-based bootstrap.In contrast to the existing literature for Colombia, we find support for the view that unemployment rates appear stationary in the long run after allowing for cross sectional dependencies. An implication of this finding is that it provides support to the view that the unemployment rates exhibit mean reversion, and therefore appear consistent with the natural rate hypothesis.

Table 1. Individual KPSS tests for mean stationarity

* and ** indicate 10 and 5 per cent levels of significance, respectively, based on the response surfaces in Sephton (1995).

Table 2.Hadri test for unemployment rates

The p-value of the Hadri test is based on the standard normal distribution. The p-value of the bootstrap Hadri test was calculated as described in Section 2, using 2,000 replications.

REFERENCES

Blanchard, O., & Summers, L. (1986) Hysteresis and the European unemployment problem.In S. Fischer (Ed.), NBER Macroeconomics Annual 1986, CambridgeMA: MIT Press, 15-78.

Brunello, G. (1990) Hysteresis and the Japanese unemployment problem: A preliminary investigation, Oxford Economic Papers, 42, 483–500.

Camarero, M., & Tamarit, C. (2004) Hysteresis vs. natural rate of unemployment: New evidence for OECD countries. Economics Letters, 84, 413-417.

Campbell, J., & Perron, P. (1991), Pitfalls and opportunities: what macroeconomists should know about unit roots. In O. Blanchard and S. Fischer (Eds.), NBER Macroeconomics Annual 1991. CambridgeMA: MIT Press, 141-201

Caner, M., & Kilian L. (2001). Size distortions of tests of the null hypothesis of stationarity: Evidence and implications for the PPP debate. Journal of International Money and Finance, 20, 639–657.

Carrion-i-Silvestre, J.L., &Sansó, A. (2006). A guide to the computation of stationarity tests. Empirical Economics, 31, 433-448.

Chang, Y. (2004). Bootstrap unit root tests in panels with cross-sectional dependency. Journal of Econometrics, 120, 263-293.

Chang, T., Lee, K-C., Nieh, Ch-Ch., &Wei, Ch-Ch. (2005). An empirical note on testing hysteresis in unemployment for ten European countries: Panel SURADF approach. Applied Economics Letters, 12, 881-886.

Clemente, J., Lanaspa, L., & Montañés, A. (2005). The unemployment structure of the US states. The Quarterly Review of Economics and Finance, 45, 848-868.

Dickey, D.A., & Fuller, W.A. (1979). Distribution of the estimators for autoregressive time series with a unit root. Journal of the American Statistical Association, 74, 427-431.

Giulietti, M., Otero, J., &Smith, J. (2008). Testing for stationarity in heterogeneous panel data in the presence of cross section dependence. Journal of Statistical Computation and Simulation, 79, 195-203.

Hadri, K. (2000). Testing for stationarity in heterogeneous panels. The Econometrics Journal, 3, 148-161.

Hadri, K., &Larsson, R. (2005). Testing for stationarity in heterogeneous panel data where the time dimension is finite. The Econometrics Journal, 8, 55-69.

Im, K., Pesaran, M.H., &Shin, Y. (2003). Testing for unit roots in heterogeneous panels. Journal of Econometrics, 115, 53-74.

Jaeger, A., Parkinson, M. (1994) Some evidence on hysteresis in unemployment rates, European Economic Review, 38, 329–42.

Kwiatkowski, D., Phillips, P.C.B., Schmidt, P., &Shin, Y. (1992). Testing the null hypothesis of stationarity against the alternative of a unit root. Journal of Econometrics, 54, 159-178.

León-Ledesma, M. (2002) Unemployment hysteresisin the US states and the EU: a panel approach,Bulletin of Economic Research, 54, 95–103.

Maddala, G.S., &Wu, S. (1999). A comparative study of unit root tests with panel data and a new simple test. Oxford Bulletin of Economics and Statistics, 61, 631-652.

Mitchell, W. F. (1993) Testing for unit roots and persistence in OECD unemployment rates, Applied Economics, 25, 1489–501.

Nelson, C.R., &Plosser, C.I. (1982). Trends and random walks in macroeconomic time series: Some evidence and implications. Journal of Monetary Economics, 10, 139-162.

Reis, F.A., & Gomes, C. (2008). Hysteresis versus NAIRU and convergence versus divergence: The behaviour of regional unemployment rates in Brazil. The Quarterly Review of Economics and Finance, forthcoming.

Røed, K. (1996) Unemployment hysteresis – macro evidence from 16 OECD countries, Empirical Economics, 21, 589–600.

Sul, D., Phillips, P.C.B., &Choi, C.Y. (2005). Prewhitening bias in HAC estimation. Oxford Bulleting of Economics and Statistics, 67, 517-546.

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June 24-26, 2009
St. Hugh’s College, OxfordUniversity, Oxford, UK