Symmetric and asymmetric stationarity tests of productivity.

Studies of productivity in the operations and engineering management literature have typically focused on identifying the drivers of productivity and how best to manage resources. To date, the issues of the time-series behavior and the stochastic structure of productivity have largely been overlooked. This article examines the times-series properties of productivity utilizing several unit root and stationarity tests including one that allows for asymmetric adjustments to equilibrium. The findings suggest that productivity is a nonstationary process and first-differencing is necessary to render a stationary series. Moreover, we find some evidence of an asymmetric adjustment process in the productivity growth rates of manufacturing.

INTRODUCTION

While studies on managing productivity and its drivers abound in the literature, little attention has been placed on how productivity evolves and changes over time. The focus of this article is on how productivity behaves over time and, more specifically, how it responds to unexpected changes or disturbances. Snyder (2006) pointed out how firms operate in a changing environment and that many of the factors traditionally used in strategic decision-making are highly uncertain. Knowledge of what leads to movements in productivity, how long shocks may last, as well as differences between sectors should be useful to firms interested in pursuing sustainable competitive advantage. The results from this article may shed light on some of the uncertainty associated with using productivity in strategic decision-making.

Clearly, productivity is not constant over time (see Figures 1 and 2) and thus managers of firms would benefit from knowing how productivity likely responds following a disturbance or unexpected change. Oftentimes, economic time series are found to be integrated of order one, denoted I(1). However, the story does not necessarily end there as the process of reverting to the mean (or attractor) may be either symmetric or asymmetric. Consider the case in which the series returns to its long-run value following a disturbance. If this variable is important to firms in their decision-making processes, then it may be informative to know how long this reversion process takes and does it matter if the disturbance was negative or positive. If the direction of the disturbance matters (e.g., higher or lower than expected), then the adjustment process is asymmetric; otherwise, it is symmetric. Thus, this article seeks to determine the time-series properties of productivity by utilizing several standard symmetric tests in addition to a recently proposed asymmetric test. The results will provide insight as to the stochastic structure of productivity and may have important implications for the operations of businesses.

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Generally speaking, productivity measures the output produced to inputs used and labor productivity is defined as output per hour of work. The link between productivity growth and the growth in living standards is well documented (e.g., see Russell and Taylor 2006, Romer 2006). Increases in the value added in production through improved technology and production efficiency lead to more resources available for other uses (e.g., income distribution, etc.). At the national level, increases in productivity growth raise living standards as more available income allows consumers to purchase more goods and services. However, productivity growth is also important at the industry and firm level. At the firm level, a productivity increase can be redistributed back to the workforce through higher wages and increased profits to shareholders. For this reason, firms are continually looking for ways to improve efficiency through improvements in supply chain management, operations, technology, and organizational structure. These factors have an overall effect on the industry and aggregate economy. Consequently, an accurate understanding of the time-series dynamics of productivity is called for. In particular, whether shocks to productivity are temporary or permanent has implications for forecasting, long-term planning, and performance. As such, we specifically test for unit roots (i.e., stationarity) in four different categories/measures of productivity using a variety of tests, including one that allows for asymmetry in the process. In what follows, we describe the data used in the study and the tests used to examine stationarity. We then discuss the results and offer some concluding remarks.

DATA

Productivity is defined as output per hour of work and shows the amount of goods and services produced per hour. This study uses quarterly measures of labor productivity over the period 1987:1-2006:1. The four measures of productivity are for the following sectors of the economy: business (BUS), nonfarm business (NONFARM), manufacturing (MAN), and nonfinancial corporations (NONFIN). In the analyses that follow, we utilize the natural logarithm of each seasonally adjusted productivity series. (1)

Descriptive statistics on productivity in levels and growth rates (i.e., first differences) are reported in Table 1. The average (quarterly) rate of productivity growth over the sample period is the lowest in the NONFARM category and highest in the MAN category. However, the variability of productivity growth as measured by the standard deviation indicates that the NONFARM category is the most stable of all the categories and MAN is the least stable category. Figures 1 and 2 provide plots of productivity and changes in productivity, respectively.

TESTS FOR STATIONARITY

To check the stationarity of the productivity series, we employ the Dickey-Fuller (DF; 1979), the DF-GLS, the Phillips-Perron (PP; 1988), and the Kwiatkowski et al. (KPSS; 1992) unit root tests, as well as, the threshold autoregressive (TAR) and momentum-TAR (M-TAR) unit root testing procedures that allow for asymmetric adjustments. The augmented DF test is based on the ordinary least squares regression of Equation (1).

[DELTA] [P.sup.j.sub.t] = [[rho].sub.0] + ([[rho].sub.1] - 1) [P.sup.j.sub.t-1] + [[rho].sub.2]t + [[summation].sup.m.sub.k=1] [[delta].sub.k] [DELTA] [P.sup.j.sub.t-k] + [e.sub.t] (1)

where [P.sup.j.sub.t] is the measure of productivity under investigation (i.e., NON-FARM, BUS, MAN, NONFIN), [DELTA] is the first-difference operator, t is a linear time trend, [e.sub.t] is a covariance stationary random error, and m is determined by Akaike's information criterion to ensure serially uncorrelated residuals. The null hypothesis is that [P.sup.j.sub.t] a nonstationary time series and is rejected if ([[rho].sub.1] - 1) < 0 and statistically significant. The finite sample critical values for the ADF test developed by MacKinnon (1996) are used to determine statistical significance.

Elliott et al. (1996) proposed a modification of the standard DF test that estimates Equation (1) with detrended series. The DF-GLS unit root test estimates the standard DF Equation (1) but substitutes [P.sup.j.sub.t] with the GLS detrended series, [[??].sup.j.sub.t]. (2) Since the asymptotic distribution of the DF-GLS t-ratio differs from the DF distribution, the critical values provided by Elliot et al. (1996) are used.

An alternative unit root test developed by Phillips and Perron (1988) allows for weak dependence and heterogeneity in the error term and is robust to a wide range of serial correlation and time-dependent heteroskedasticity. The test is based on the following regression:

[P.sub.j.sub.t] = [[eta].sub.0] + [[eta].sub.1] (t - T/2) + [lambda][P.sup.j.sub.t-1] + [v.sub.t] (2)

where (t - T/2) is the time trend with T representing the sample size and [v.sub.t] is the error term. The null hypothesis of a unit root, [H.sub.o] : [lambda] = 1, is tested against the alternative hypothesis that [P.sup.j.sub.t] is stationary around a deterministic trend ([H.sub.a] : [lambda] < 1). As in the DF test, MacKinnon critical values may be used to determine statistical significance for the PP test.

Unlike the DE DF-GLS, and PP unit root tests, the KPSS test differs in that the productivity series is assumed to be (trend) stationary under the null hypothesis. The KPSS unit root test statistic is obtained from the residuals by regressing [P.sup.j.sub.t] on a constant and a trend. The KPSS statistic is defined as the Lagrange multiplier (LM) statistic:

KPSS = ([T.sup.-2] [T.summation over (t = 1)] [[??].sup.2.sub.t]) / [[??].sup.2] (3)

where [[??].sub.t] is the sum of the residuals on the regression, [[??].sup.2] is the consistent estimate of the long-run variance, and T is the sample size. Critical values from the asymptotic distributions for the KPSS test statistic are provided in Kwiatkowski et al. (1992). The null hypothesis of stationarity is rejected if the KPSS test statistic exceeds the respective critical value.

Recently, the literature on time-series econometrics and business cycles has focused on how variables may respond asymmetrically to positive and negative shocks. Standard unit root tests (e.g., DF, DF-GLS, PR KPSS, etc.) assume that positive and negative shocks have the same effect on a series. However, the existence of transactions costs, capital stock adjustment costs, and market and institutional frictions may make this assumption unrealistic.

Enders and Granger (1998) generalized the DF unit root test methodology to allow for asymmetric adjustment. The data series are regressed on the deterministic components and the resulting residuals, [[??].sub.t], are used in the estimation as follows:

[DELTA] [[??].sub.t] = [I.sub.t] [[rho].sub.1] [[??].sub.t+1] + (1 - [I.sub.t]) [[rho].sub.2] [[??].sub.t-1] + [l.summation over (i=1)] [[gamma].sub.i] [DELTA] [[??].sub.t-i] + [[??].sub.t] (4)

where [[??].sub.t] ~ I.I.D (O, [[sigma].sup.2]) and the lagged values of [DELTA] [[??].sub.t] are meant to yield uncorrelated residuals. The Heaviside indicator function for the TAR model is denoted as follows:

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