Has U.S. Labor Productivity Undergone High-Frequency Changes?

Transcription

1 Has U.S. Labor Productivity Undergone High-Frequency Changes? Luiggi Donayre Department of Economics University of Minnesota-Duluth Neil A. Wilmot Department of Economics University of Minnesota-Duluth May 12th, 2014 Abstract While many empirical studies have documented the so-called productivity slowdown in the U.S., an interesting common feature of all break date estimates is that they different from the widely accepted date of Because productivity and its determinants are likely described by a long-run relationship, we argue that the different estimates reflect the inability of current structural break tests to detect changes that occur at a higher frequency and, therefore, more gradually. In this paper, we propose a two-step approach to examine the nature and timing of breaks in U.S. labor productivity growth. Monte Carlo simulations show that the proposed approach exhibits low coverage at the beginning and end of the interval over which parameters change gradually, but a much higher middle coverage. When applied to U.S. labor productivity, we find a conservative confidence interval which spans from March 1957 through January Notably, it includes the widely accepted breakdate of 1973, unlike those found in previous studies. These results, taken together, suggest that U.S. labor productivity slowed down gradually. JEL Classification Code: C15, C22, E24 Keywords: Labor Productivity, Structural Breaks, Confidence Intervals, Likelihood ratio. We thank Yamin Ahmad, Yunjong Eo, Claude Lopez, Irina Panovska and the participants at the 2013 Symposium of the Society for Nonlinear Dynamic Economics and the 2013 Meetings of the Midwest Econometrics Group for helpful comments. All remaining errors are our own. Corresponding author: Department of Economics, Labovitz School of Business and Economics, University of Minnesota-Duluth, 1318 Kirby Drive, Duluth, Minnesota, 55812, U.S.A; Tel. +1 (218) ; adonayre@d.umn.edu. Department of Economics, Labovitz School of Business and Economics, University of Minnesota-Duluth, 1318 Kirby Drive, Duluth, Minnesota, 55812, U.S.A; Tel. +1 (218) ; nwilmot@d.umn.edu 1

2 1 Introduction The notion that U.S. labor productivity experienced a slowdown in the early 1970s has received considerable attention in the economic time series literature. For example, the empirical issue of timing the so-called productivity slowdown in the U.S. has motivated a large number of studies (Perron, 1989; Zivot and Andrews, 1992; Bai, Lumsdaine, and Stock, 1998; Hansen, 2001; Eo and Morley, 2014). 1 However, an interesting common feature of all breakdate estimates for productivity growth is that they are different from the widely accepted break date of The empirical motivation for our study stems from this common feature. In particular, we investigate whether the parameters governing the dynamics of U.S. labor productivity could have undergone a change occurring slowly over a long time horizon, thus making its detection and accurate timing more difficult. If we accept the premise that this productivity slowdown was caused by the adoption of new technologies in the 1950s and 1960s that implied the acquisition of new skills, it is likely that the slowdown occurred gradually over time as workers slowly started to acquire such new skills. 2 To the extent that all existing structural break tests assume that the change in parameters occurs abruptly, the different results found could be a consequence of the current literature s inability to correctly identify and time change that occurs more gradually in economic time series. 3 While there have been substantial theoretical innovations on the problem of identifying unknown break dates and making inference in the context of multiple breaks (Andrews, 1993; Andrews and Ploberger, 1994; Bai, 1997; Bai and Perron, 1998; Bai et al., 1998; Elliott and Müller, 2007; Qu and Perron, 2007), the central assumption of an instantaneous change that underlies all parameter constancy tests can often be violated in applied settings. It is likely that a structural break might occur immediately and abruptly in the context of exogenous, unan- 1 Some of these studies focus on dating the slowdown in post-war U.S. output growth and consumption growth. To the extent that labor productivity growth affects both otuput and consumption, we deem the studies comparable. 2 In this sense, we can think of the productivity slowdown as occurring gradually in terms of the parameters governing the behavior of U.S. labor productivity undergoing a relatively large number of high-frequency changes over a given period. 3 One exception that finds a confidence interval that includes the widely accepted date of 1973 is Eo and Morley (2014). However, the multivariate environment they consider leads them to find four break dates in their output-consumption vector error correction model whose confidence intervals nearly cover the entire admissible set of possible breaks, given a 15% trimming. This can be interpreted as a large number of high-frequency breaks occurring over the sample period. 2

3 ticipated shocks that could affect economic conditions in the short-run. However, parameter instability of the abrupt type is less reasonable in environments where long-run determinants of economic relationships change over time. Many macroeconomic variables in the postwar U.S. economy appear to display patterns of parameter instability that occur more gradually. In this paper, we propose a two-step approach to address this issue. The approach we propose combines a sequence of tests for multiple structural breaks, in the first step, and the subsequent construction of a conservative likelihood-ratio-based confidence interval, in the second. Specifically, our procedure follows Bai and Perron (1998), who discuss a test of l versus l + 1 breaks which can be used as the basis of a sequential testing procedure in the first step. Because gradual change can be interpreted as a sequence of high-frequency, smaller, multiple breaks over a time interval, the sequential testing procedure is relevant as important improvements are obtained by iterative refinements. In the second step, we then obtain confidence intervals for each estimated break date and construct a conservative confidence interval defined as the union of all confidence intervals for each individual break date. We assess the performance of the proposed approach by means of a Monte Carlo experiment. We then apply our approach to examine the nature and timing of structural breaks in the growth rate of U.S. labor productivity and find evidence of three breaks in April 1958, December 1966 and May By considering a conservative confidence interval over multiple break dates, our proposed approach includes a wider set of potential break dates (from March 1957 through January 1982). Notably, our conservative confidence interval does include the widely accepted break date of 1973, contrary to previous studies. The interpretation of these results suggests that the so-called productivity slowdown may have occurred less abruptly than previously thought. The rest of the paper is organized as follows. Section 2 describes the two-step approach we propose. In section 3, we report the results of the Monte Carlo experiment to assess the performance of our proposed approach. We then examine the nature and timing of strutural breaks in U.S. labor productivity in section 5. Section 6 concludes. 3

4 2 Econometric Approach In section 2.1, we motivate the distinction between an abrupt break and breaks that occur at a higher frequency and, therefore, more gradually. Section 2.2 describes the econometric specification and the two-step approach we propose to making inferences about the nature and timing of structural breaks in U.S. labor productivity growth. 2.1 Motivation Many macroeconomic relationships can be described by parameters that change over time, both abruptly and gradually. This distinction becomes relevant in the context of a vast literature that has focused, primarily, on the identification, timing and inference of structural breaks that occur abruptly (Perron, 1989; Andrews, 1993; Bai, 1997; Bai and Perron, 1998; Qu and Perron, 2007; Eo and Morley, 2014). To the extent that such literature is unable to identify and time changes in parameter values that occur at a higher frequency, and therefore more gradually, inferences and policy implications in those environments can be misleading. To gain some insight regarding the behavior of economic time series that exhibit parameter changes of both types, and to motivate their distinction, we simulate artificial series that undergo changes in their parameter values which occur abruptly and more gradually. The series are displayed in figure 1. The left panel of figure 1 shows an artificial series that exhibits an abrupt break at period 150. The right panel of figure 1 plots the same series, except that the change starts at period 150 and occurs gradually until period 400. With the exception of this disctintion, both series were created under the same data-generating process (DGP) for 500 observations. As it is clear from figure 1, the type of structural break can be, sometimes, easily distinguishable. In this case, it is likely that existing structural break tests are able to identify and time parameter instability, at least partially. Often, on the contrary, economic relationships undergo changes that are more difficult to distinguish and, as a consequence, to detect and time. For example, figure 2 plots two artificial series that were constructed using the same DGP, except that one was allowed to experience a break. In this way, the left panel of figure 2 shows an artificial series with parameters that are stable over time, while the right panel of 4

5 Figure 1: Two series exhibiting easily-distinguishable parameter instability (a) Abrupt change (b) Gradual change Both series are generated under the same data-generating process (DGP) except that one of the parameters changes abruptly in the left panel and gradually, in the right one. Both series were generated for 500 observations. figure 2 exhibits a series that undergoes a gradual change. However, distinguishing between the two series in this case is not as straightforward. With this discussion in mind, we therefore deem that our objective of determining whether U.S. labor productivity has experienced structural breaks at higher frequencies (and, hence, more gradually) is pertinent. Figure 2: Series exhibiting stability and instability (a) No change (b) Gradual change Both series are generated under the same data-generating process (DGP) with the difference that relevant model parameter in the left panel is stable over time, while the one in the right panel changes gradually. Both series were generated for 500 observations. 5

6 2.2 Econometric Specification We consider a modified version of the fairly general linear econometric model from Bai and Perron (1998). In particular, our linear econometric model allows coefficient parameters to change gradually, according to: y t = x tβ j + z tδ + ɛ t (1) where ɛ t i.i.d. N (0, σ 2 ), T denotes the sample size, t = T j 1 + 1,..., T j for j = {1, M + 1} and t = T j for j = 2,..., M. In this model, y t is the observed dependent variable at time t; both x t (p 1) and z t (q 1) are vectors of covariates and β j (j = 1,..., M + 1) and δ are the corresponding vectors of coefficients. The indices (T 1,..., T M ) are explicitly treated as unknown and we follow the convention that T 0 = 0 and T M+1 = T. Here, β j undergoes a change in value between times T 1 and T M. 4 Gradual change in the model parameters can, then, be interpreted as a sequence of N multiple, discrete changes that occur at a high frequency at dates T j, for j = 1,..., M between T 1 and T M. Hence, β 1 if 1 t T 1 β j = β 2, if T 1 < t T 2. (2) β M, if T M 1 < t T M β M+1, if T M < t T M+1 so that the vector of coefficients β j can have M breaks and the m th regime s changing coefficient is β m. Specifically, the vector of coefficients β j for j = 1,..., M + 1 is given by β k if T 1,k < k T 1,k+1 for k = 0, 1,..., M, where T 1,0 = T 1, T 1,M+1 = T M and M is the number of breaks for β j. 4 For expositional purposes, we only allow β j to undergo a break. However, it is straightforward to generalize equation (1) to the case where both { β j, σ 2 j} undergo changes at high frequencies. 6

7 From model (1), z t and δ represent regressors and their respective coefficients that are stable across the sample period. In practice, the potential break dates are restricted between the middle (1 2λ) portion of the sample period, Π = [λt, (1 λ)t ], to avoid end-of-sample distortions. However, β j can have two different values in two consecutive break dates that are less than a distance of λ j T observations apart, for j = {1, M + 1}. 5 In this econometric framework, the approach we propose to examine the nature and timing of U.S. labor productivity growth series involves, first, a test of multiple structural breaks and, then, constructing a conservative likelihood-ratio-based confidence interval. In the first step, we follow Bai (1997), and Bai and Perron (1998, 2003), and consider a multiple linear regression model, as in equation (1), with M breaks and M + 1 regimes. There are T observations and M is assumed to be known. Breaks occur at dates {T 1,..., T M }. The purpose is to estimate the unknown regression coefficients together with the break points when T observations on (y t, x t, z t ) are available. The multiple linear regression model in (1) can be expressed in matrix form as y t = Xβ + Zδ + Ξ (3) where Y = (y 1,..., y T ), X = (x 1,..., x T ), Ξ = (ɛ 1,..., ɛ T ), and β = ( β 1,..., β M+1). The method of estimation is based on the least-squares principle. For each m-partition {T 1,..., T M }, the associated least-squares estimates of β j and δ are obtained by minimizing the sum of squared residuals according to (Y Xβ Zδ) (Y Xβ Zδ) = M+1 i=1 T i t=t i 1 +1 [ yt x tβ i ζ tδ ] 2 Let ˆβ ({T j }) and ˆδ ({T j }) denote the estimates based on the given m-partition {T 1,..., T M } denoted ({T j }). Substituting these in the objective function and denoting the resulting sum of 5 An important assumption behind the asymptotic theory to develop strutural break tests is that two break dates occur at least λ jt observations apart, with λ j < λ j+1 and 0 < λ 1, λ M+1 < 1. To generate high-frequency breaks and, therefore, a more gradual change in parameter values, we allow breaks to occur less than a distance of λ jt observations apart. 7

8 { } squared residuals as S T {T 1,..., T M }, the estimated break points ˆT 1,..., ˆT M are such that { ˆT 1,..., ˆT M } = argmin S T {T 1,..., T M } (4) {T 1,...,T M } where the minimization is taken over some set of admissible partitions. Thus the breakpoint estimators are global minimizers of the objective function. The regression parameter ({ }) ({ }) estimates are the estimates associated with the m-partition ˆT j, i.e. ˆβ = ˆβ ˆT j and ({ }) ˆδ = ˆδ ˆT j. The procedure we adopt to test for multiple breaks follows Bai and Perron (1998), who discuss a test of l versus l+1 breaks which can be used as the basis of a sequential testing procedure. For details, we refer the reader to appendix A. Once all M break dates are identified, we proceed to construct a conservative confidence interval in the second step. For break dates {T 1,..., T M }, the confidence interval for T i is given by [τ 1,i, τ 2,i ] for i = 1,..., M. Then, we define the conservative confidence interval for {T 1, T M } as CI(T 1, T M ) = [τ 1, τ 2 ] (5) where τ 1 = inf τ i,1 [ τ i,1 : τ i,1 ] M τ i,1 i=1 and τ 2 = sup τ i,2 [ τ i,2 : τ i,2 ] M τ i,2 i=1 (6) There are a number of ways to construct [τ 1,i, τ 2,i ] for each break date. 6 In doing so, we follow Eo and Morley (2014) who propose the use of likelihood-ratio-based confidence intervals constructed by inverting the likelihood ratio test. They consider a range of Monte Carlo simulations in order to evaluate the finite-sample performance of the asymptotic inverted likelihood ratio (ILR) and other competing methods for constructing confidence intervals for 6 Many different bootstrap and asymptotical methods have been proposed to constructing confidence intervals for the timing of structural breaks in time series. See Eo and Morley (2014) for further details. 8

9 the timing of structural break dates. The Monte Carlo analysis supports the asymptotic results in the sense that the ILR confidence intervals always exhibit the shortest expected length while, at the same time, maintenaing accurate, if somewhat conservative, coverage rates. Although bootstrap methods, in principle, are available for complicated models of structural breaks, a Monte Carlo analysis of bootstrap methods is computationally infeasible. This is exacerbated in the case of multiple structural breaks as the number of bootstrap data sets would increase considerably. 7 Meanwhile, other asympototic methods have been shown to perform poorly in the case of structural breaks that are too small or too large, or in finite samples. For example, Bai (1997) employs the asymptotic thought-experiment of a slowly shrinking magnitude of a break to derive the distribution of a break date estimator in a linear time series regression model and uses a related statistic to construct confidence intervals for the timing of breaks. These confidence intervals, however, tend to undercover in finite samples, even for moderately-sized breaks. Meanwhile, Elliott and Müller (2007) propose a different approach based on the inversion of a test for an additional break under the null hypothesis of a given break date, still employing a similar asymptotic thought-experiment of a quickly shrinking magnitude of a break. While their approach produces a confidence set that has accurate coverage rates in finite samples, it is only applicable for a single break and tends to produce prohibitively wide confidence sets, even when breaks are large. In their analysis, Eo and Morley (2014) derive the asymptotic distribution of and provide critical values for a likelihood ratio test of a hypothesized break date. Likewise, they also derive an anlaytical expression for the expected length of a likelihood-based confidence set based on inverting the likelihood ratio test. For brevity, we omit the specifics of their derivation and encourage interested readers to refer to appendix B for details. Finally, it is important to note that some of the assumptions for which the likelihoodbased confidence interval is asymptotically valid do not hold in the case of structural change of the gradual type. In particular, Eo and Morley (2014), following Qu and Perron (2007), 7 Consider, for example, a simple linear regression model with only 100 possible break dates. For each bootstrap data set, we would need to consider 10,000 (= ) combinations of two break dates and simultaneously estimate break dates, as well as model parameters. For 199 bootstrap samples only, 1,990,000 (=199 10,000) ML estimations are necessary. Even if it takes just one second per estimation, 1,990,000 cases would take more than 3 weeks. A Monte Carlo analysis of this simply multiplies this computational burden by the number of simulations. 9

10 restrict the break dates to be asymptotically distinct, which implies that we should allow for a reasonable number of observations between the break dates (specifically, λt ). This is clearly violated in the case of gradual change. Further, they also assume that, while the magnitude of the structural change shrinks as the sample size increases, it should be large enough so that limiting distributions for the estimates of break dates can be derived. Again, this assumption is violated when the break occurs gradually. See Qu and Perron (2007) and Eo and Morley (2014) for further details. 3 Monte Carlo Analysis In this section, we evaluate the performance of our two-step approach using artificial data by means of a Monte Carlo experiment. In each experiment, we report the number of estimated breaks and examine the coverage rate accuracy of the conservative confidence interval and its average length. In settings of abrupt structural breaks, the model parameters change in value at one break date. In the case of high frequency change, however, the break occurs gradually over a time interval, T 1 and T M, and not just at one break date. To cope with this issue, we report three relevant coverage rates: (i) the coverage rate, cr 1, for the beginning of the interval over which model parameters start to gradually change, T 1 ; (ii) the coverage rate, cr 2 for the end point of said interval, T M ; and (iii) the coverage rate, cr 3, for the entire time interval, [T 1, T M ]. In each case, the coverage rate is defined as the number of instances that the confidence interval includes the true break date (or true time interval over which the gradual change takes place) across the number of Monte Carlo repetitions. Its accuracy is, then, compared a specified nominal confidence level. The average length is defined as the length of the conservative confidence interval averaged across Monte Carlo repetitions. The nominal confidence level is set at 95% and we evaluate the performance of our proposed approach for different sample sizes: 40, 60, 120, 240 and 480. The trimming value for possible break dates, λ, is set to We consider 1,000 replications for each experiment. In the Monte Carlo experiments, the model in (1) is re-written as follows: 10

11 ( ) α(t ψt + 1) y t = µ + 1 [ψt t < (ψ + ζ)t ] + α1 [(ψ + ζ)t t] + e t (7) ζt where e t i.i.d. N (0, σ 2 ), 1[. ] is an indicator function, ψ determines the fraction of the sample, T, at which the gradual change starts, ζ denotes the fraction of the sample over which the gradual change extends (i.e., the given parameter changes gradually over ζt periods) and α is the magnitude of the total change in µ. Hence, the model in equation (7) has a mean given by µ up until period ψt. Then, the mean gradually changes from µ, at period ψt, to µ + α, by period (ψ + ζ)t. The mean then remains at µ + α after period (ψ + ζ)t. In terms of the parameterization of the data generating process described in (7), we set consider ψ = {0.2, 0.4, 0.6}, ζ = {0.1, 0.2, 0.3}, α = { 0.3, 0.6, 1.2}, µ = 1 and σ 2 = 1. The results are presented in tables (2) through (4). From tables (2) through (4), we can identify six main observations. The first one refers to the fact that the number of estimated breaks, m, increases with the magnitude of the break, α, and the sample size, T. When the magnitude of the break is large (i.e., α = 1.2), m also increases with the length of the break, ζ. The relationship between the number of breaks, m, and the length over which the gradual change occurs, ζ, is less clear for smaller magnitudes of the break (low α). Intuitively, tests will be able to identify structural breaks more accurately when the magnitude of the break or the sample size is larger. Indeed, if the period over which the gradual change occurs is long, then the change in parameter values will be smaller at each given date, t. Therefore, tests will be able to identify structural breaks over longer periods only when the overall change in parameter values is large enough (high α). The second observation refers to the fact that the beginning of the period over which parameters change gradually, ψ, determines the accuracy with which tests will be able to identify the beginning or the end of such period. In all cases, the coverage rate of the entire interval over which parameters change gradually, cr 3, is substantially higher than cr 1 and cr 2, although always below the 95% nominal level. When the gradual change starts early in the period under consideration (ψ = 0.2), structural break tests are less likely to cover the initial break date, T 1. Conversely, when the gradual change starts late in the sample (ψ = 0.6), 11

12 structural break tests are less likely to cover the end break date, T M. That is, the coverage rates at the extremes of the gradual change period are poor, as there is less information at the ends of the sample period. This situation could also arise because there are instances where the sequential procedure to identify multiple breaks does not work (Bai and Perron, 2003). 8 Additionally, because the nature of the structural change is gradual, the limiting distributions derived in Bai (1997) and Bai and Perron (1998) no longer hold. Specifically, because the distance between two break dates is less than λt, the sequential procedure is unable to identify those breaks. However, given the large number of structural breaks, the sequential procedure is more likely to identify those that occurred in the middle of the sample, as the parameter values would be, then, considerably different relative to the values prior to the beginning of the gradual change. To some extent, this problem is lessened by our procedure introducing iterative refinements where potential break dates are re-estimated based on refined samples. The Monte Carlo experiments show that the approach we propose is able to cover a large fraction of the entire period over which the gradual change occurs (i.e., cr 3 is relatively high). This coverage rate increases with the sample size, T, which is the third observation we infer from our Monte Carlo analysis. In part, this is explained by the fact that the average number of estimated breaks across Monte Carlo repetitions increases with T. Indeed, for ψ = 0.4, α = 1.2 and ζ = 0.3 for example, the average number of estimated breaks increases from 1.54, when T = 60, to 5.12, when T = 480. Our fourth observation states that coverage rates decrease with ζ when α is low. When α is high, the relationship weakens, although the coverage rates tend to increase with ζ. A higher value of ζ implies that the period over which parameters change gradually is longer, which makes it more difficult to identify multiple, smaller breaks that occur at a higher frequency. Hence, when α is small, the change in parameter values becomes even smaller, lowering coverage rates. Meanwhile, a higher value of α implies a larger magnitude of the break, which tests are able to more easily identify. However, the reason why this relationship is 8 The problem is that, in the presence of multiple breaks, certain configurations of changes are such that it is difficult to reject the null of l versus l + 1 breaks, but it is not difficult to reject the null hypothesis of l versus a number of breaks higher than l

13 weaker is because tests more easily identify larger, abrupt breaks. When the change occurs gradually, the break in parameter values, even if large, would be spread out over longer periods. It is important to note that inferences in small samples can be misleading. Specifically, the fifth observation refers to the fact that the identification and timing of structural breaks in small samples is distorted by the small number of observations. This small sample result also drives the sixth observation: a smaller value of α, especially in small samples, could lead to higher coverage rates because our proposed approach is unable to distinguish a small gradual change over a short period of time from an abrupt change. Overall, the results from the Monte Carlo simulations suggest that our proposed approach could partially capture parameter instability that occurs at higher frequencies and, therefore, more gradually. Even though the coverage rates are far from target nominal levels, especially at the beginning and end of the interval over which parameters change, the longer conservative confidence interval is able to capture a large fraction of the sample over which parameters change gradually. 4 Structural Breaks in U.S. Labor Productivity Using our proposed two-step approach, we examine the nature and timing of structural breaks in U.S. labor productivity. Specifically, we estimate break dates and model parameters by maximum likelihood estimation and apply the approach proposed in section 2 in order to construct a conservative confidence interval for structural breaks, possibly of the gradual type, in post-war U.S. labor productivity growth. Under the assumption that the natural logarithm of U.S. labor productivity exhibits a stochastic trend with a drift and a finite-order AR representation, the model can be specified as follows: P y t = µ + φ j y t j + e t, e t i.i.d. N (0, σ 2 ) (8) j=1 where, for simplicity, E(e 2 t ) = σ 2. From the model in (8), the number of autoregressive 13

14 coefficients, P, is determined by the Akaike Information Criterio (AIC). We then proceed to estimate break dates and construct the conservative confidence interval as explained in section 2. We measure U.S. labor productivity as the growth rate of the ratio of the Industrial Production index for consumer durable goods to average weekly labor hours. 9 The growth rates are expressed as natural logarithm differences and the sample goes from January 1947 through December 2012, corresponding to 792 observations. Based on a symmetric 15% trimming value for λ, the candidate break dates lie in the interval from November 1956 through September All data are seasonally adjusted from their sources, the Board of Governors and the Bureau of Labor Statistics. Table 1: AR(3) model of U.S. labor productivity growth: Regime Break date Growth rate Test statistic April December January Growth rate means for an AR(3) model of U.S. labor productivity, estimated according to equation (8). To avoid end-sample-distortions, λ was set to 0.15, corresponding to an effective sample that ranges from November 1956 through September Based on the AIC, three lagged differences (P = 3) are necessary to capture serial correlation. Table 1 reports the estimates of the U.S. labor productivity growth rate and test statistics for the model in equation (8) with three structural breaks. The procedure identifies, initially, a break date in April 1958 with a corresponding test statistc of Thus, the null hypothesis of no breaks is rejected against the alternative of one break. We then proceed to test for an additional break, with the previous alternative as the new null hypothesis. The null of one break is rejected in favor of the alternative of an additional break, estimated in December 1966, with a corresponding test statistc of The sequential procedure detailed in section 2 finds an additional break date in May 1980, with a corresponding test statistc 9 Average weekly hours is determined as the number of employees in the durable good sector multiplied by average weekly hours of production in the manufacturing sector, as measured by the Bureau of Labor Statistics. 14

15 of 15.38, and is unable to reject the null of three breaks against the alternative of four break dates. The years before the first break in April 1958 correspond to a period of relative volatility in U.S. labor productivity, reflected in a low long-run growth rate in labor productivity of 1.62 percentage points. The second break in December 1966 is associated with the peak of U.S. labor productivity before the so-called slowdown, with an average long-run growth rate of 4.38 percentage points. After December 1966, the growth rate of U.S. labor productivity declined by 3 percentage points until the third break found by our procedure in January After this productivity slowdown period, the growth rate of U.S. labor productivity increased 1.96 percentage points, but the increase was not large enough to reach the levels prevailing prior to the slowdown period. We then construct the proposed conservative confidence interval, which spans the periods between March 1957 through January 1982, corresponding to 299 periods. Hence, the proposed approach finds evidence of three breaks, which could be understood as a gradual change in model parameters. This evidence is further supported by a relatively wide confidence interval, as depicted in figure 3. Figure 3: U.S. labor productivity (logs), The figure shows U.S. labor productivity (in logs) and the conservative confidence interval estimated as in equation (5), spanning from March 1957 through January In this way, by considering a conservative confidence interval over multiple break dates, 15

16 the approach proposed includes a wider set of break dates and a longer period of parameter instability, which can be understood as a process of gradual change. In general, the results are in line with those found in the literature. For example, Bai et al. (1998) estimate a break in U.S. labor productivity in 1969:Q1 and find a 90% confidence interval that ranges between [1966:Q2, 1971:Q4]. Similarly, Hansen (2001) finds evidence of a structural break in labor productivity in December 1963, with a 90% confidence interval that ranges between [1959, 1971]. 10 If we accept the premise that this productivity slowdown could have been caused by the adoption of new technologies in the 1950s and 1960s that implied the acquisition of new skills, it is likely that the slowdown occurred gradually as workers slowly started to acquire such new skills. As a consequence, the different results found in the literature could stem from the current structural break literature s inability to correctly identify and time gradual change in economic time series. Notably, our conservative confidence interval does include the widely accepted break date of 1973, unlike those found by Bai et al. (1998) and Hansen (2001). Meanwhile, it is also important to mention that Eo and Morley (2014) find a break in the fourth quarter of 1972, with a 95% confidence interval that ranges between [1971:Q2, 1973:Q1]. By taking cointegration into account, they consider a two-variable vector error-correction model that includes the growth rates of U.S. real GDP and consumption and find evidence of four breaks: 1961:Q3, 1972:Q4, 1982:Q4 and 1996:Q1. While they argue that their evidence suggests four abrupt breaks, our Monte Carlo simulations cast some doubt on whether they are in fact abrupt breaks or, rather, a large number of small breaks occuring at a high frequency that current structural break tests are not able to correctly identify or time. This is more evident if we note that, taken together, the confidence intervals for all break dates nearly cover the entire admissible set of possible breaks, given a 15% trimming. 11 These results are also in line with the findings from our Monte Carlo experiments. When 10 He also finds evidence of two other structural breaks. The second one in January 1982, with a 90% confidence interval of [1977, 1988], and the third one in April 1994, with associated 90% confidence interval of [1992, 1996]. Notice that none of his intervals are very precise. 11 Their confidence intervals span from the second quarter in 1957 through the fourth quarter in 1997, if we were to think of them in terms of our conservative approach. This covers 87.6% (= 162/185) of their admissible set of possible breaks. 16

17 a series experiences a change in parameters that occurs gradually, we found that our approach is less likely to correctly identify the beginning and the end of the gradual change period. Accordingly, if we assume that the productivity slowdown began around 1973, the fact that this date lies well in the middle of our conservative confidence interval could be interpreted as evidence that U.S. labor productivity did decline, gradually, during the late 1960s. Further, considering the large sample size, T = 792, and the fact that the fraction covered by the conservative confidence interval (303/792 = 0.376) implies a high value of ζ, our Monte Carlo results would suggest a coverage rate of around 80% for the entire period over which parameters change gradually, although the coverage of the beginning and end of such period would be closer to 30%. Therefore, the results from our Monte Carlo analysis, together with the results from this section, seem to support a slow decline in U.S. labor productivity. 5 Concluding Remarks The so-called productivity slowdown in the U.S. has been documented by a large number of documents in the literature. Even though there seems to be a consensus of a break date that occurred in 1973, none of the break dates or confidence intervals estimated in those studies include this date. Because the long-run determinants of productivity are likely to evolve slowly over time, we argue that the different results found could arise because current structural break tests are unable to correctly identify and time changes that occur at a higher frequency and, therefore, more gradually. In this paper, we have proposed a two-step approach to examine the nature and timing of structural breaks in U.S. labor productivity that takes into account the possibility that they may have occurred gradually. The Monte Carlo simulations suggest that our approach can partially identify higher-frequency changes. Indeed, our proposed approach is able to cover a large fraction of the entire period over which gradual change occurs. However, we find evidence of low coverage at the beginning and at the end of the period over which gradual change in model parameters occurs. This is exacerbated in the case of large breaks, longer periods and large sample sizes. In the end, we believe that this result could shed some light on the reasons behind the difficulty to time the slowdown in U.S. labor productivity. Specifically, 17

18 our conservative confidence interval spans from March 1957 through January Notably, it includes the widely accepted break date of 1973, unlike those found in previous studies. We believe that these results, taken together, would suggest that U.S. labor productivity slowed down gradually. Finally, it is important to notice that identifying and timing gradual change in economic time series is difficult because existing asymptotic theories become invalid when break dates are less than a distance λt apart. However, addressing high frequency change that may occur more gradually is paramount to make correct inferences and appropriate policy recommendations. We anticipate that future research will help us understand and time gradual change better, especially with respect to the beginning and end of such periods. 18

19 References Andrews, D. W. (1993), Tests for Parameter Instability and Structural Change with Unknown Change Point, Econometrica, 61(4), Andrews, D. W. and Ploberger, W. (1994), Optimal Tests When a Nuisance Parameter is Present Only under the Alternative, Econometrica, 62(6), Bai, J. (1997), Estimating Multiple Breaks One at a Time, Econometric Theory, 13(3), Bai, J., Lumsdaine, R. L., and Stock, J. H. (1998), Testing for and Dating Common Breaks in Multivariate Time Series, Review of Economic Studies, 64(3), Bai, J. and Perron, P. (1998), Estimating and Testing Linear Models with Multiple Structural Changes, Econometrica, 66(1), (2003), Computation and Analysis of Multiple Structural Change Models, Journal of Applied Econometrics, 18(1), Elliott, G. and Müller, U. (2007), Confidence Sets for the Date of a Single Break in Linear Time Series Regressions, Journal of Econometrics, 141, Eo, Y. and Morley, J. (2014), Likelihood-Ratio-Based Confidence Sets for the Timing of Structural Breaks, Working Paper, University of New South Wales. Hansen, B. E. (2000), Sample Splitting and Threshold Estimation, Econometrica, 68, (2001), The New Econometrics of Structural Change: Dating Breaks in U.S. Labor Productivity, Journal of Economic Perspectives, 15(4), Perron, P. (1989), The Great Crash, the Oil Price Shock, and the Unit Root Hypothesis, Econometrica, 57(6), Qu, Z. and Perron, P. (2007), Estimating and Testing Structural Changes in Multivariate Regressions, Econometrica, 75(2),

20 Zivot, E. and Andrews, D. W. K. (1992), Further Evidence on the Great Crash, the Oil Price Shock and the Unit Root Hypothesis, Journal of Business and Economic Statistics, 10(3),

21 Appendix A: Structural break sequential tests In this appendix, we provide details of the sequential test of l versus l + 1 breaks, following Bai and Perron (1998). The econometric environment is given by the linear model in equation 1. The parameter coefficient of interest is β j, which undergoes a potentially multiple breaks and, hence, can be desribed by equation 2. For the model with l breaks, the estimated break points denoted by { ˆT 1,..., ˆT l } are obtained by a global minimization of the sum of squared residuals. The strategy proceeds by testing for the presence of an additional break in each of the l + 1 segments, obtained using { } the estimated partition ˆT 1,..., ˆT l. More precisely, the test is defined by: { } ( ) ( ) W T (l + 1 l) = S T ˆT 1,..., ˆT l min inf S T ˆT 1,..., ˆT i 1, t, ˆT i+1..., ˆT l /ˆσ 2 1 i l+1t Λ i,η (A-1) where S T (.) denotes the sum of squared residuals, and Λ i,ɛ = {t : ˆT i 1 + ( ˆT i ˆT i 1 )η t ˆT i ( ˆT i ˆT i 1 )η} (A-2) where η > 0 and ˆσ 2 is a consistent estimate of σ 2 under the null hypothesis. Note that for i = 1, S T ( ˆT 1,..., ˆT i 1, t, ˆT i+1,..., ˆT l ) is understood as S T (t, ˆT 1,..., ˆT l ) and, for i = l + 1, as S T ( ˆT 1,..., ˆT l, t). It is important to note that one can allow different distributions across segments for the regressors and the errors. The limit distribution of the test is related to the limit distribution of a test for a single change. The test amounts to the application of l + 1 tests of the null hypothesis of no structural change versus the alternative hypothesis of a single change. It is applied to each segment containing the observations ˆT i to ˆT i for i = 1,..., l + 1. We conclude for a rejection in favor of a model with l + 1 breaks if the overall minimal value of the sum of squared residuals (over all segments where an additional break is included) is sufficiently smaller than the sum of 21

22 squared residuals from the l breaks model. The break date thus selected is the one associated with this overall minimum. Intuitively, one simply needs to apply the tests successively starting from l = 0, until a non-rejection occurs. That is, one first starts by testing for a single structural break. If the test rejects the null hypothessis that there is no structural break, the sample is split in two (based on the break date estimate) and the test is reapplied to each subsample. This sequence continues until each subsample test fails to find evidence of a break. The key insight is that when there are multiple structural breaks, the sum of squared residuals (as a function of the break date) can have a local minimum near each break date. Thus, the global minimum can be used as a break date estimator, and the other local minima can be viewed as candidate break date estimators. Bai (1997) shows that important improvements are obtained by iterative refinements: reestimation of break dates based on refined samples. Once all M break dates are identified, we proceed to construct a conservative confidence interval as specified in section 2. 22

23 Appendix B: Likelihood-ratio-based confidence intervals Under fairly general assumptions about regressors and error terms, Eo and Morley (2014) derive the asymptotic distribution of the likelihood ratio (LR) test for a structural break date. Additionally, they also derive the confidence set for the break date, and its expected length, based on inverting the LR test. More generally, Eo and Morley (2014) show that the likelihoodratio-based confidence intervals are valid for the broad setting of a system of multivariate linear regression equations and allowing for multiple breaks in mean and variance parameters. The likelihood-ratio-based confidence intervals are constructed under the assumption that the magnitude of the break shrinks as the sample size increases, so that the actual coverage may exceed the desired level (1 α) when the magnitude of the break is large (Hansen, 2000). In order to construct confidence intervals, Eo and Morley (2014) consider the following inverted LR statistic with a non-standard distribution: lr j ( ˆT j T 0 j ) d max υ Z(υ) (B-1) where the superscript 0 denotes true parameters, T j is the j th break date, j = 1,..., M, and lr j (r) = r 2 ( ln Σ 0 j ln Σ 0 j+1 ) τ 0 j +r 1 (y t x 2 tβj 0 )(Σ 0 j) 1 (y t x tβj 0 ) (y t x tβj+1)(σ 0 0 j+1) 1 (y t x tβj+1) 0 t=τj 0+1 for r = T j T 0 j > 0, where y t is n 1 vector of data, x t is matrix of regressors, β j is a p 1 vector of regression coefficients, Σ j is the covariance matrix of the error term of the regression equation, and 23

24 Z(υ) = ω 1,j [ 1 2 υ + W j(υ) ] for υ (, 0] ω 2,j [ 1 2 υ + W j(υ) ] for υ (0, ) where W j is a standard Wiener process defined over R and ω 1,j, ω 2,j are parameters. Hence, Z(υ) establishes the asymptotic distribution of the LR test. Eo and Morley (2014) show how to construct the likelihood-ratio-based confidence interval for the j th break date from the LR test in equation (B-1). In particular, it is given by C j (y) = { t max Tj ln L(τ j y) ln L(t y) κ α,j } (B-2) where the asymptotic critical value used to construct the (1 α) likelihood-ratio-based confidence interval is κ α,j such that 1 exp κ α,j [ ( )] ( )] ω 1,j [1 exp κ α,j ω 2,j = 1 α. Meanwhile, as α 0, the expected length of a (1 α) likelihood-ratio-based confidence interval is shown to be 2 +2 ( Γ 2 1,j Ψ 2 1,j ( Γ 2 2,j Ψ 2 2,j ) [ 1 exp ( κ α,j ω 1,j ) [ 1 exp ( κ α,j ω 2,j )] { κα,j 1 ω 1,j 2 )] { κα,j 1 ω 2,j 2 [ ( 1 exp κ α,j [ 1 exp ω 1,j ( κ α,j ω 2,j )]} )]} (B-3) where Γ 2 1,j, Γ2 2,j, Ψ2 1,j and Ψ2 2,j are parameters. The lenght is calculated by measuring the expected size of the break dates T j such that lr j ( ˆT j T j ) κ α,j. 24

25 Table 2: Coverage rates and average length for a gradual change in mean, Ψ = 0.2 T Length of change, ζ ζ = 0.1 ζ = 0.2 ζ = 0.3 Coefficient change, α cr cr cr [τ 1, τ 2 ] [ ] [11.6, 19.6] [12.9, 17.0] [5.8, 10.5] [13.4, 22.2] [15.1, 20.7] [7.3, 13.4] [15.6, 26.0] [17.3, 24.9] Expected length Av. number breaks cr cr cr [τ 1, τ 2 ] [16.6, 31.5] [25.8, 36.7] [27.1, 33.1] [19.5, 36.4] [30.6, 43.6] [30.6, 42.2] [22.6, 39.4] [35.2, 51.8] [33.0, 52.3] Expected length Av. number breaks cr cr cr [τ 1, τ 2 ] [46.3, 78.3] [53.8, 67.6] [54.2, 69.3] [54.5, 92.3] [63.9, 83.1] [56.5, 94.1] [63.2, 106.6] [72.3, 100.6] [59.3, 115.0] Expected length Av. number breaks cr cr cr [τ 1, τ 2 ] [105.0, 148.3] [111.0, 132.8] [103.0, 157.9] [123.2, 174.2] [126.7, 170.6] [103.6, 199.7] [143.3, 205.1] [134.0, 212.3] [109.6, 238.7] Expected length Av. number breaks Coverage rates and average lengths of proposed conservative confidence interval (CCI) estimated as in equation (5). The DGP is constructed according to equation (7) for µ = σ 2 = 1 and different values for the size of the break, α, the length of the change, ζ, and the sample size, T. Results are shown for high frequency changes starting in period ΨT, for Ψ = 0.2. cr1 corresponds to the coverage rate of the CCI associated with the beginning of the interval over which the gradual change occurs. cr2, for the end point of said interval. And, cr3, for the entire time interval. 25

26 Table 3: Coverage rates and average length for a gradual change in mean, Ψ = 0.4 T Length of change, ζ ζ = 0.1 ζ = 0.2 ζ = 0.3 Coefficient change, α cr cr cr [τ 1, τ 2 ] [ ] [22.0, 30.4] [24.7, 28.2] [11.0, 17.5] [23.9, 33.1] [26.9, 32.3] [11.8, 18.1] [25.8, 36.3] [28.6, 36.1] Expected length Av. number breaks cr cr cr [τ 1, τ 2 ] [32.7, 51.2] [48.8, 59.3] [50.8, 56.8] [35.9, 55.9] [53.5, 66.1] [53.8, 66.0] [38.7, 58.1] [57.0, 72.6] [55.1, 75.6] Expected length Av. number breaks cr cr cr [τ 1, τ 2 ] [90.3, 123.6] [101.3, 114.4] [101.2, 114.9] [97.2, 135.4] [110.6, 128.9] [100.2, 137.1] [105.5, 147.0] [118.2, 143.1] [102.6, 159.1] Expected length Av. number breaks cr cr cr [τ 1, τ 2 ] [195.0, 236.5] [204.8, 226.6] [198.6, 237.8] [214.8, 265.6] [218.7, 263.1] [193.6, 286.8] [232.4, 291.7] [222.0, 300.4] [197.5, 325.4] Expected length Av. number breaks Coverage rates and average lengths of proposed conservative confidence interval (CCI) estimated as in equation (5). The DGP is constructed according to equation (7) for µ = σ 2 = 1 and different values for the size of the break, α, the length of the change, ζ, and the sample size, T. Results are shown for high frequency changes starting in period ΨT, for Ψ = 0.4. cr1 corresponds to the coverage rate of the CCI associated with the beginning of the interval over which the gradual change occurs. cr2, for the end point of said interval. And, cr3, for the entire time interval. 26