Labor market inefficiency and economic restructuring: evidence from cross-sectoral data.

1. Introduction

It has been widely noted that the U.S. labor market was relatively weak during the recovery from the 2001 recession. Although the recession itself was comparatively mild and short-lived (March-November 2001, according to the National Bureau of Economic Research), and although four million additional workers entered the labor force during the recovery, even by the most generous measure employment did not return to prerecession levels for over two and a half years. (1) One might imagine that the "jobless recovery" reflected a weak macroeconomy, but real gross domestic product growth averaged a healthy 3.2% in the three years after the recession. Nevertheless, job creation was weak, and unemployment rates recovered slowly even after employers began to post vacant jobs.

Because there was also a considerable reallocation of jobs across industries during this period, some economists have recently speculated that the two trends might be related (Groshen and Potter 2003; Minehan 2004; Groshen, Potter, and Sela 2005). According to the "Sectoral Shift Hypothesis" (generally attributed to Lilien [1982]), extensive restructuring can create inefficiencies in the labor market, either by causing a skills mismatch between employers and employees or by forcing them to search in less familiar territory. If so, changes in relative labor demand across industries would raise unemployment, even if there were no change in aggregate labor demand. Consequently, aggregate demand stimulus would likely not be effective in combating unemployment, but there may be more useful roles for retraining programs or industrial policy that attempted to balance growth (or even contraction) more evenly across sectors of the economy.

That said, such restructuring is only one of the possible explanations for the jobless recovery. For example, Minehan (2004) also discusses potential roles for trends in international trade, labor costs, and an increase in uncertainty caused by geopolitical events. If such factors proved to be more important, policy discussions might focus more fruitfully on topics like trade agreements, exchange rate policy, the financing of health care and pensions, payroll taxes, diplomacy, the treatment of investments in the tax code, or interest rates.

This paper contributes to that debate by presenting new evidence on the relevance of the Sectoral Shift Hypothesis during this business cycle. Previous empirical work on the hypothesis has produced mixed results--many papers find support for the hypothesis, but many others find evidence against it. As Schwerin (2003) has argued in his extensive literature review, one explanation for this inconsistency could be that the hypothesis has greater merit in some periods than others. For example, the nature of structural change likely varies across business cycles, and its effects might well depend on the similarity between the skills demanded by expanding and contracting industries, the extent of job-specific human capital or on-the-job training, or the geographical composition of the structural change.

Accordingly, it would seem desirable to use an empirical methodology that allows the effects of restructuring to vary over time. Unfortunately, until recently, that has been difficult because the best data available consisted of aggregate time series. That data permitted researchers to address the Sectoral Shift Hypothesis by looking for a relationship between the timing of changes in some measure of labor market inefficiency (often just aggregate unemployment rates) and some economy-wide index of structural change (e.g., a dissimilarity index of industries' growth rates). However, given sample size limitations, the only reasonable identification strategy rested on an implicit assumption that a given degree of reallocation has similar effects on unemployment across different business cycles. If that assumption were violated, researchers could reject the Sectoral Shift Hypothesis as a general proposition even if it had substantial explanatory power in some subset of business cycles.

To avoid this concern, this paper takes a different approach, one that is more nearly cross-sectional than intertemporal. Thanks to new data, we are able to disaggregate across supersectors of the U.S. economy. The additional sample size this provides allows us to examine a single business cycle in isolation, and we can then test the hypothesis by quantifying the changes in the inefficiency of sectoral labor markets and comparing them to the extent of structural change in those sectors. In essence, we ask whether reallocation and increased inefficiency happen not only at the same time, but in the same sectors. We know of only one previous paper that has attempted to address the Sectoral Shift Hypothesis with disaggregated data; part of a paper by Abraham (1987) analyzes unemployment and vacancy rates across U.S. states. We argue that the approach in this paper provides a more direct test of the Sectoral Shift Hypothesis.

The outcome of that test will cast doubt on the relevance of this hypothesis in the business cycle that started with the 2001 recession. The analysis will reveal that the sectors that experienced the largest structural change did not have unusually large changes in inefficiency, and that most of the sectors that suffered the largest increases in labor market inefficiency had relatively mild structural change. Admittedly, this conclusion emerges from a sample of just 12 sectors, so it would be reasonable to regard it with some caution. That said, previous studies based their conclusions on just one (aggregate) data series, albeit over a few business cycles.

In addition to the main finding, this paper also makes two ancillary contributions. First, it develops a new method to measure changes in labor market inefficiency: a variation on principal components analysis. This method allows us to identify the relationship between unemployment and vacancies (what is often called the "Beveridge curve" or "u-v curve") with relatively sparse data and without strong identifying assumptions. Changes in unemployment and vacancy rates can then be decomposed into two parts, one reflecting changes in the position of the Beveridge curve relative to the origin (which will serve as our measure of labor market inefficiency) and the other reflecting movements along the Beveridge curve (and thus "pure" business cycle effects, holding labor market inefficiency constant). Because this is a nonstandard approach, we also attempt to gauge the potential for bias and conclude that it is unlikely to affect our results materially.

The other ancillary contribution is to present the first evidence on sectoral Beveridge curves, something that is only possible because of the new data set we examine. Although we will find that Beveridge curves have similar slopes across sectors, we will document substantial differences in the efficiency of sectoral labor markets and the extent to which that efficiency fluctuated over this business cycle. At the least, these findings should provide a useful point of comparison for future studies of sectoral labor markets during business cycles yet to come.

The presentation proceeds as follows. Section 2 introduces the underlying theory and outlines the paper's empirical strategy. The data used in the empirical analysis are described in section 3, and section 4 explains the method we use to quantify labor market inefficiency and investigates the potential for bias. Section 5 presents results of that methodology, with an emphasis on cross-sectoral differences in the extent to which labor market inefficiency rose over this business cycle. We then use those measured increases in sectoral labor market inefficiency to test the Sectoral Shift Hypothesis in section 6. Specifically, the first subsection of section 6 shows that the cross-sectoral variation in those fluctuations cannot be explained by an important alternative explanation, but a second subsection shows that it also does not correlate well with the extent of sectoral reallocation--which is why we conclude that there is little support for the Sectoral Shift Hypothesis. The argument is summarized in section 7, and the paper ends with a brief discussion of the implications and issues for future research.

2. Background

The Sectoral Shift Hypothesis (Bowden 1980; Lilien 1982) posits that labor markets clear more slowly in the presence of economic restructuring. The idea is that there is an important difference between workers who are unemployed because of purely temporary fluctuations and those who are unemployed because of a structural change that reallocates jobs between sectors (or for that matter, between industries within the same sector). In contrast to the former group, structurally unemployed workers cannot reasonably hope to be rehired when the economy rebounds, so they are forced to cast a wider net. During the time needed to realize that the shock is persistent, skills are mismatched and there is a general lack of familiarity with the expanding employers, so such workers are unlikely to find new jobs as quickly as other unemployed workers.

The hypothesis thus proposes a specific reason for an increase in what is often called "matching inefficiency." That term refers to any factors that slow the rate at which new employment matches form from given stocks of unemployed workers and vacant jobs. In other words, the hypothesis proposes a reason why search frictions are more severe in some situations than in others.

Matching inefficiency is difficult to observe directly, so we shall infer it by observing the behavior of sectoral Beveridge curves and ruling out some other major explanations for what we find. Once it is established that the observed patterns could plausibly represent matching inefficiency, we can test the Sectoral Shift Hypothesis by asking whether the increases in matching inefficiency are larger in sectors that have experienced more extensive restructuring.

For readers who might not be familiar with the Beveridge curve, it is the downward-sloping convex locus along which unemployment and vacancy rates tend to move over the business cycle. Named after the first economist to identify it as an empirical regularity (Beveridge 1945), it does not represent a causal relationship between unemployment and vacancies, but rather that both are affected by the same underlying forces, usually in opposite directions. According to theoretical models, such factors include productivity shocks, possibly combined with wage rigidities, and thus the curve's slope represents the relative speeds at which employers and workers respond to those shocks (Solow 1998; Hall 2003; Shimer 2005). (2)

The relevance of the Beveridge curve for the Sectoral Shift Hypothesis lies in the fact that the curve's position relative to the origin is often considered a useful summary statistic describing the level of inefficiency in the labor market (Solow 1998). Intuitively, when unemployment and vacancies are both elevated, there are workers who want a job and employers who want to hire workers, yet they do not pair with one another, so the market is thus inefficient in the sense that there are apparently unrealized gains from trade. Consequently, we shall call the distance from sectoral Beveridge curves to the origin the "labor market inefficiency," and we shall denote it n.

Labor market inefficiency is a broader concept than matching inefficiency. Whereas matching inefficiency pertains only to the severity of search frictions, an increase in labor market inefficiency can also reflect increased exposure to search frictions. For example, even if the length of the average unemployment spell did not change, an increase in the job destruction rate (3) would raise the number of workers who must pass through that period of unemployment, and thus the measured unemployment rate would rise.

To see this formally, it is helpful to reference the "matching function": the relationship between the rate at which new employment matches occur and existing rates of unemployment and vacancies. In practice, it is often specified with a Cobb-Douglas form,

h(U, V) = A [U.sup.[alpha]][V.sup.1-[alpha]], (1)

where h is the hiring rate (hires/labor force), U is the unemployment rate, V is the vacancy rate, and A is a constant. (4) Equation 1 suggests two reasons why the Beveridge curve could shift, at least temporarily. First, steady-state levels of U and V would rise if A decreased. Because this would mean that fewer new matches would result from given levels of U and V, such a change would represent an increase in matching inefficiency. Second, if U, V, or both temporarily changed because of job creation or destruction shocks, they would return to their initial levels only gradually (not immediately), except in the unlikely case in which those increases are exactly offset by the new hiring. This would represent an increase in labor market inefficiency, but not an increase in matching inefficiency.

The dynamics of the second mechanism warrant a bit more attention because they create cycles in labor market inefficiency, which appear as counterclockwise "loops" around the central axis of the Beveridge curve. Such loops are depicted in Figure 1 and have been frequently noted in empirical studies. Although a complete description of the theory is quite complex and varies somewhat across models (Hansen 1970; Bowden 1980; Pissarides 1985, 2000; Blanchard and Diamond 1989; Shimer 2005), the core of the story does not seem to be terribly controversial. In short, the idea is that employment shocks cause an initial increase in U, V, or both, and the hiring rate responds to this change via the matching function, but generally not so much that U and V return to their original levels immediately. In more concrete terms, firms facing a recession often eliminate vacant positions before laying off workers, so there is initially a larger decrease in vacancies and a smaller increase in unemployment. Likewise, when the recovery begins, employers must create vacant jobs before they can fill them, and in that phase, vacancies rise before unemployment falls. Thus, over the course of a typical business cycle, the level of labor market inefficiency oscillates around its average level, first improving, then weakening, and finally returning to its initial position. Although it is temporary, this outward shift of the Beveridge curve typically persists for a few years at a time, so it could represent a potentially important cost of business cycles--as illustrated by the concern over the jobless recovery.

[FIGURE 1 OMITTED]

Although theory clearly indicates that employment shocks can create such loops in the Beveridge curve, it is less clear how matching inefficiency would affect the Beveridge curve. Most discussions suggest that restructuring causes a permanent increase in inefficiency, manifesting itself as a persistent outward shift of the Beveridge curve. Previous research has shown that the Beveridge curve has indeed shifted slowly over time, moving further from the origin in the 1970s and closer in the 1980s and 1990s (Abraham 1987; Blanchard and Diamond 1989; Bleakley and Fuhrer 1997). The existence of such trends does not necessarily demonstrate a role for sectoral reallocation because the trends could alternatively be due to permanently higher rates of job creation and destruction or to the growth of the labor force (Bleakley and Fuhrer 1997; Pissarides 2000; Hall 2003; Shimer 2005). However, such trends could provide evidence for the Sectoral Shift Hypothesis if they correlated with patterns of restructuring.

On the other hand, it is also possible that the effects of restructuring might appear as temporary increases in labor market inefficiency (i.e., as wider loops in the Beveridge curve than one might otherwise expect). This could happen if restructuring affected matching inefficiency more strongly during some phases of the business cycle than others. For instance, it is plausible that unemployed workers might be more adept at job searches in industries in which they have more experience, so that they find jobs at a slower rate (conditional on the unemployment/ vacancy ratio) if they are forced to search elsewhere. That reasoning suggests that structural change, by encouraging workers to search in new markets, pushes the entire Beveridge loop further from the origin. However, if it took some time for those unemployed workers to recognize that a shock was structural rather than cyclical, or if it took some time to adapt their search, then the effect would grow with the average duration of unemployment. Such a dynamic would presumably cause the upper portion of the Beveridge loop to shift further than the lower portion, creating a wider loop.

At any rate, because we cannot be sure of the expected duration of the response to reallocation, the empirical work herein will consider both possibilities. The analysis will establish that, although many features of the data are reassuringly consistent with the "employment shocks" mechanism, we cannot rule out the possibility that matching inefficiency increased during the last recovery. We then ask (i) whether there are trends in sectoral labor market inefficiency, conditional on employment flows, and (ii) whether restructuring correlates with the thickness of sectoral Beveridge loops. Because all of the relevant correlations are weak, we will conclude that there is little evidence of an important role for restructuring.

3. Data

The main reason that no previous work has examined sectoral differences in labor market inefficiency has been a lack of data. Although the Bureau of Labor Statistics (BLS) has for many years computed monthly unemployment rates at several levels of aggregation from the Consumer Population Survey (CPS), until recently, the best available measure of vacancies has been the Conference Board's help-wanted index, which can only be disaggregated geographically. However, since December 2000, the BLS has conducted the Job Openings and Labor Turnover Survey (JOLTS), which includes monthly counts of vacancies that are disaggregated at the supersector level of the North American Industrial Classification System. (5) This is the data we will use to examine the sector-specific Beveridge curves.

In addition to its disaggregation, the JOLTS data on vacancies has some other advantages over the help-wanted index. JOLTS' measure is based on an actual count of job openings from a nationally representative sample of employers, whereas the help-wanted index is an indirect measure and not geographically representative. Moreover, although an adjusted version of the help-wanted index has proved useful in earlier work (Abraham 1987), some economists have begun to question whether the relationship between the help-wanted index and the actual number of vacancies could have changed in recent years because of, for example, the growing role of the internet in the job matching process (Bleakley and Fuhrer 1997; Shimer 2005).

Another useful aspect of the JOLTS data is that it contains data on employment flows, including monthly hiring and separation rates by sector, as well as a decomposition of separations into quits and layoffs/discharges. The results section will use this information to investigate whether increases in labor market inefficiency can be attributed to those employment flows or, alternatively, whether matching inefficiency might have increased.

The JOLTS also has some disadvantages. For one thing, it would have been desirable to disaggregate the data by occupation and state as well, but JOLTS only distinguishes between supersectors. Another disadvantage is that the data is still relatively new. We have 65 monthly observations on each sector, from December 2000 to April 2006. This prevents us from investigating trends in labor market inefficiency across several business cycles, as previous papers have endeavored. That said, this paper is premised on the idea that different inefficiencies can arise during different business cycles, and it has the more modest goal of understanding only the cycle that began in 2001. The data series is long enough to achieve that objective because it begins a few months before the recession and continues for four and a half years of the subsequent recovery and expansion. Also, as noted below, Figures 2 and 3 suggest that the cycle was nearing completion by the end of our sample period, with the last unemployment and vacancy observations beginning to resemble the first observations in most sectors.

[FIGURES 2-3 OMITTED]

Some readers might also consider it a disadvantage that JOLTS attributes unemployed workers to their previous sector until they find a new job. Thus, the sectoral Beveridge curves we measure capture employment flows that are not entirely intrasectoral. (6) That said, we do not view this as especially problematic. Although theoretical models of unemployment dynamics frequently ignore the possibility that workers can enter or exit a labor market, this has always been more of a modeling convenience than a strong stance on actual behavior. After all, even the aggregate labor market experiences inflows and outflows as workers enter and exit the labor force, but this has not stopped economists from studying aggregate Beveridge curves. Moreover, we argue that this accounting attributes the labor market inefficiency to the proper sector. For example, if manufacturing firms were looking for employees, and unemployed workers who previously worked in manufacturing were searching for jobs in another sector, it strikes us as much more reasonable to consider those facts as evidence of inefficiency in manufacturing's labor market.

Table 1 provides some summary statistics for each sector: average seasonally adjusted unemployment and vacancy rates, average seasonally adjusted employment, and employment growth over the sample period. (7) There is considerable variability across sectors. Unemployment rates average less than 3.4% in government, education and health, and finance, but more than 8% in leisure and construction. Vacancy rates are typically smaller in magnitude, but their range is proportionally similar, with the highest rate (3.5% in education and health) roughly triple the lowest (1.3% in natural resources). Consistent with substantial restructuring, rates of employment growth also vary markedly across sectors. Although aggregate employment grew only 1.9% over our sample period, employment grew by 15.3% in education and health, but it fell by just over 17% in both information and manufacturing.

To provide a clearer sense of the data, Figure 2 plots the log unemployment and vacancy rates for the manufacturing sector. Note that the data follow a path similar to the stylized Beveridge curve "loop" shown in Figure 1 and that the loop appears to be nearing completion. This suggests that our data cover nearly a full business cycle. As expected, the major axis of the loop is downward sloping. That said, the width of the loop is also considerable, suggesting that labor market inefficiency increased meaningfully over this business cycle.

Figure 3 presents the same series for all 12 sectors. The curves look qualitatively similar for all of the industries, although the cycles for two sectors (financial services and information) appear to be less complete than those of the other sectors. Most of these curves have similar slopes, although the curve for natural resources appears relatively flatter and the curve for the leisure sector is a bit steeper. More significantly, some curves are much closer to the origin than others, indicating large differences in labor market inefficiency. In particular, the curves for the leisure and professional and business service sectors are about twice as far from the origin as that for the government sector. Finally, it appears that some sectors experienced larger changes in labor market inefficiency over the business cycle than others. For example, the loops for the trade, information, and financial services sectors appear rather wide, whereas the data for the government, leisure, and education and health services sectors nearly lie along a straight line.

4. Methodology

With only five and a half years of data and a limited selection of variables available at the sectoral level, it is not practical to use an elaborate structural model such as that of Blanchard and Diamond (1989). As an alternative, this section proposes a much simpler, reduced form method to identify the location and movement of the Beveridge curve. Although it is surely subject to at least some of the criticisms Yellen (1989) has levied against previous reduced form methods, our approach has several virtues. Besides its feasibility with limited data, it is also easy to understand, and its results are readily interpreted.

The empirical strategy developed herein is motivated by the goal of decomposing movements in unemployment and vacancy rates into two parts: one associated with movements along the (central tendency of the) Beveridge curve and one associated with shifts toward or away from the origin. As explained below, we use a variation on principal components analysis to identify those components. Although that technique is most often used to reduce the number of regressors in the presence of multicollinearity (see Theil [1971, pp. 46-55] for an introduction), we use both of the principal components from our data. The Beveridge curve is identified as the downward-sloping principal component, which is also the dominant principal component for all sectors. Because the two components are necessarily orthogonal, one can think of this approach as finding the correct angle to rotate the axes so that the Beveridge curve is parallel to one of the new axes and perpendicular to the other.

One might wonder why we do not simply identify the Beveridge curve by regressing log vacancy rates on log unemployment rates. The main reason is that a regression approach would require making an assumption about the very topic this paper endeavors to test. That is, because the underlying theory posits that the Beveridge curve shifts out and back in over the course of the business cycle, the curve could only be properly identified if some regressor were available to use as a "shifter" variable. Although theory does suggest some factors that could cause the curve to shift, the use of those variables would effectively assume the hypothesis that this paper is trying to examine.

For the sake of comparison, our method estimates that the elasticity of the aggregate Beveridge curve over this period is d(log V)/d(log U) = -1.18 (0.04). Estimates in the previous literature, although reported infrequently and often without great precision, have generally implied an elasticity around -1.00 (Bleakley and Fuhrer 1997; Shimer 2005). These estimates are fairly similar and, at any rate, it is not surprising that there are some differences in that we use a different measure of vacancies, our data covers a different period, and Bleakley and Fuhrer report that the slope of the Beveridge curve varies substantially over time (e.g., it fell by 50% between the 1960s and early 1990s).

Specification

The Beveridge curve is often taken to be log-linear, which is consistent with both a Cobb-Douglas matching function and the empirical evidence (Petrongolo and Pissarides 2001). If we define u and v to be (respectively) the log unemployment and log vacancy rates, a Beveridge curve can thus be written as v = [v.sub.o] zu, and its elasticity is d(log V)/d(log U) = -z < 0 (where U and V are the [nonlogged] rates). A movement along the curve can then be represented as a multiple of some downward-sloping unit vector (a, -b) in (u, v) space, with b/a = z. That vector forms an orthonormal basis with (b, a), so any combination of log unemployment and vacancy rates (u, v) can be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2)

Here m indicates a position along the Beveridge curve (i.e., the u/v ratio), and n represents the distance between the Beveridge curve and the origin, which serves as our measure of labor market inefficiency. See Figure 4 for a diagram.

[FIGURE 4 OMITTED]

Equation 2 implies that the covariance matrix is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3)

Under the assumption that m and n are uncorrelated (not necessarily independent), Equation 3 has the form of an eigenvector decomposition. (8) It is thus reasonable to identify the Beveridge curve vector (a, -b) and its perpendicular (b, a) as the eigenvectors of the covariance matrix; that is, they can be interpreted as the principal components of the data. The associated eigenvalues (call them [[lambda].sub.m] and [[lambda].sub.n]) represent the variances of movements along and against the Beveridge curve ([[sigma].sup.2.sub.m] and [[sigma].sup.2.sub.n] respectively). Standard errors for all of these estimates can also be computed under the assumption that the data is joint-normally distributed. (9) Finally, once (a, -b) is estimated, m and n can be computed by inverting Equation 2,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (4)

Note that this transformation just rotates the axes to align with the Beveridge curve.

Bias

The validity of this technique depends on the assumption that m and n are uncorrelated. Over a long period, this is not controversial--movements along a properly defined Beveridge curve should be uncorrelated with long-run shifts of that curve. However, our data apparently cover a bit less than a full business cycle, and the hypothesized dynamics in which (u, v) follows a counterclockwise loop suggests that m and n might be correlated in the short run.

The Appendix attempts to place some bounds on the possible correlation p between m and n, and it derives formulas for the biases that would result from any given p. It concludes that [rho] is unlikely to exceed 0.1 (and could be much smaller yet), and at correlations of that magnitude, the estimates are quite accurate. Specifically, if [rho] = 0.1, the angle between the true and estimated Beveridge curves would be less than seven degrees for all sectors, and usually much smaller. Every individual estimate of n, across all sectors and months, would be within 2.6% of its true value, and over 80% of the individual values would be accurate to within 1%. Most importantly, the widths of the sectoral Beveridge curve loops (call them [DELTA]n) would be estimated to within 5.5% of the actual width for all sectors and to within 3% for all but two sectors. This precision is critical because [DELTA]n is our measure of changes in labor market inefficiency over the business cycle.

Thus, although we cannot entirely rule out the presence of some bias, whatever biases remain are likely small and unlikely to skew the analysis.

5. Inefficiency in Sectoral Labor Markets

This section presents the most immediate results from the procedure developed above. The main reason for discussing these results in some depth (especially the various measures of An reported in the second half of this section) is that they will form the basis of the tests conducted in section 6. Beyond that, because this is the first study to examine Beveridge curves at the sectoral level, the discussion provides new information about differences between sectors and could provide a valuable baseline for future studies. These results also strike us as fairly reasonable, so they could lend some credence to the method developed above and to the idea of analyzing sectoral Beveridge curves more generally.

Sectoral Beveridge Curves

Table 2 presents the results of our eigenvector decompositions of the sectoral unemployment and vacancy data. The first two columns list the estimated elasticities of the sectoral Beveridge curves and their standard errors. Although there is some variation, most are quite similar. The largest estimated elasticities are -2.37 in the leisure services sector and -1.66 in the financial services sector, and the smallest are -0.82 in other services and -0.68 in natural resources. The estimates for the other eight sectors lie within a fairly small range from about -1.0 to -1.5. In all, the 95% confidence intervals for 10 of the 12 estimates include points in that range, consistent with the casual observation that most of the curves plotted in Figure 3 had fairly similar slopes.

The third column of Table 2 reports average distance (call it [bar.n]) between the sectoral Beveridge curves and the origin. This distance varies considerably across sectors, with the differences paralleling those depicted in Figure 3. Specifically, the results indicate that the labor market is typically more than twice as efficient (in this sense) in natural resources and government than in leisure and professional and business services.

Finally, Table 2 presents the eigenvalues [[lambda].sub.m] and [[lambda].sub.n]. As Equation 7 shows, for small [rho], these eigenvalues are good estimates of the variation along ([[sigma].sup.2.sub.m]) and around ([[sigma].sup.2.sub.n]) the Beveridge curve, so we call those values the "business cycle variation" and the "efficiency variation," respectively. For all industries, the motion along the Beveridge curve explains at least 90% of the total variation in unemployment and vacancy rates. There is a positive correlation (0.56) between [[lambda].sub.m] and [[lambda].sub.n] across industries, and both sources of variation are greater in some industries (natural resources, information) than others (government, education and health services, wholesale and retail trade, leisure). That said, the correlation is hardly perfect, in that some other sectors exhibit relatively high business cycle variation alongside relatively low efficiency variation (manufacturing, professional and business services).

Sectoral Differences in Labor Market Inefficiency

The theory in question is not so much about intersectoral differences in normal levels of labor market inefficiency ([bar.n]) as it is about the changes in labor market inefficiency ([DELTA]n) that arise as the market adjusts to new circumstances. According to the theory discussed earlier, those changes in labor market inefficiency can be measured by the size of the "loop" in the Beveridge curve. Because there is not a standard definition of this concept, Table 3 presents several measures of the size of that loop. Our assessment of the proposed hypotheses in section 6 will use all of those measures to verify that the findings are robust.

The first measure reported in Table 3 is the width of the loop. For some sectors (e.g., construction), the maximum n's preceded the minimum, possibly reflecting some carryover from earlier business cycles (which are not necessarily perfectly aligned across sectors). Thus, the widths have been computed as the difference between the first local minimum n and the maximum subsequent n. The second column normalizes this width by [bar.n], expressing it relative to its baseline level. As one might imagine, the loop widths are highly correlated (0.75) with the "efficiency variation" [[lambda].sub.n] from Table 2.

One possible objection to these measures is that the lower portion of the loop actually corresponds to an improvement in labor market inefficiency. In essence, over that range, unemployment is growing relatively slower than vacancies are contracting, so on the net, fewer agents are looking to match. Including this portion of the curve thus tends to overstate the true increase in inefficiency that occurs as the business cycle resolves itself. Thus, it might be more useful to consider an alternative measure that focuses attention on the top half of the loop, where labor market inefficiency is worse than its usual level. Thus, the third column of Table 3 presents this measure of (half-)loop width (max n - [bar.n]), and the fourth column normalizes it by [bar.n]. This half-loop width is highly correlated (0.84) with the full loop width, and for 11 of the 12 sectors it is 33-71% of the full loop width.

Regardless of which measure one prefers, the change in labor market inefficiency is substantially larger in some sectors than others. For example, the largest change was in the financial services and information sectors, and inefficiency increased substantially more in transportation and utilities and natural resources than it did in construction or government.

Another way to assess the importance of increased labor market inefficiency is to compare the change in unemployment predicted by the "loop" motion to the actual range in unemployment observed over the business cycle. The fifth column of Table 3 thus translates the half-loop width reported in the third column into a predicted increase in unemployment rates, (10) and the next two columns compare this to the actual maximum amount by which the sectoral unemployment rates exceeded their average level. For all but one sector, the increase in n explains at least a 0.14 percentage point increase in unemployment rates above the average level (often more than 0.30 percentage points). In many sectors these increases account for more than a quarter of the actual increase in unemployment, and they explain more than three-quarters of the actual change in leisure and other services. At the other extreme, increased labor market inefficiency explains less than 20% of the range in unemployment rates in natural resources, construction, manufacturing, and government.

Still another dimension of inefficiency is the length of time that labor market inefficiency remains elevated. The penultimate column in Table 3 reports the number of months that n remained above its average level--in essence, the duration of the top half of the loop. Because Figure 3 showed that some of these loops are not yet closed, these numbers likely underestimate the true duration for some sectors. Nevertheless, some clear differences emerge, with the inefficiency being relatively short-lived in construction and leisure, but much longer in financial services and information.

The final measure of inefficiency in Table 3 is the area of the top half of the Beveridge loop, computed as [[summation].sub.n>[bar.n](n - [bar.n]). This measure varies widely across sectors and corresponds closely to what one might expect from the graphical evidence in Figure 3. It is also highly correlated (0.85) with the first measure of loop width, confirming the concepts' similarity.

6. Evaluation of the Sectoral Shift Hypothesis

Having found that sectors experienced different fluctuations in labor market inefficiency, we now investigate whether those fluctuations could be due to reallocation. As discussed in section 2, labor market inefficiency is a broader concept than matching inefficiency because it also includes changes that can be attributed to changes in employment flows. Thus, the first task is to determine whether the various measures of [DELTA]n presented in Table 3 can be explained merely by changes in employment flows. Although the results conform in many ways with the view that such flow changes create the Beveridge curve dynamics, we will find that employment flows do not explain the intersectoral variation in [DELTA]n very well. Thus, we conclude that it is plausible that [DELTA]n at least partially reflects matching inefficiency. The second half of this section then asks the main question of this paper: Is the variation in [DELTA]n due to restructuring?

Tests of Employment Flows

As noted in section 2, the theory of the Beveridge curve assigns an important role to employment flows, which we will denote here by X. Standard models indicate that such flows have an important effect on labor market inefficiency (i.e., X is a major determinant of [bar.n]) and that fluctuations in those flows generate the loops in the Beveridge curve. In light of these considerations, it is reasonable to imagine that fluctuations in those flows (call them [DELTA]X) might be a strong predictor of the Beveridge loop thickness ([DELTA]n) reported in Table 3. If that were true, it would seem more appropriate to describe those changes in labor market inefficiency as the product of those fluctuations in employment flows, rather than an increase in matching inefficiency (as postulated by the Sectoral Shift Hypothesis).

We begin by reassuring the reader that our results are indeed consistent with the standard predictions. First of all, sectoral average level of labor market inefficiency [bar.n] is highly correlated with the average level of every available measure of employment flows: hiring rates (correlation = 0.76), separation rates (0.77), quit rates (0.80), layoffs and discharges (0.60), gross worker reallocation rates (0.77), and the excess worker reallocation rates (0.77). (11) Thus, it seems clear that such flows have an important effect on the average position of sectoral Beveridge curves relative to the origin. (12)

Likewise, our results do not contradict the standard theory that job creation shocks create the loops in the Beveridge curve. As a simple test, we have run sector-specific regressions of the monthly n's against monthly net job creation rates (computed as the first difference of log[employment + vacancies]), with an autoregressive moving average (2, 2) error process to accommodate persistence dynamics flexibly. (13) In all 12 sectors, the estimated coefficient had the expected positive sign, and it was statistically significant in nine cases (and nearly so in a tenth). In 11 of 12 sectors, a one standard deviation (within-sector) change in job creation is associated with at least a 0.14 standard deviation change in n, and a majority of the coefficients lie between 0.31 and 0.56, suggesting reasonable predictive power.

It is not surprising that the data are consistent with these standard predictions, but it does not necessarily follow that the magnitude of sectoral variation in labor market inefficiency over the business cycle ([DELTA]n) simply reflects the magnitude of their fluctuations in employment flows ([DELTA]X). That possibility is explored in Table 4. The upper panel of that table presents the within-sector standard deviation of several different measures of employment flows, and the lower panel displays the cross-sectoral correlations between those standard deviations and several measures of inefficiency fluctuations ([DELTA]n) from Table 3. Even a quick scan of those correlations should convince the reader that employment flows cannot explain more than a modest share of the cross-sectoral variation in [DELTA]n. (14)

We are thus left to conclude that employment shocks could raise labor market inefficiency by a larger magnitude in some sectors than others. This conclusion is consistent with the considerable variation in parameter estimates observed in the sector-specific regressions discussed above, with the statistically significant coefficients ranging from 3.9 to 17.5 (although the second largest was 8.8).

As noted in section 2, one possible source of that variation is matching inefficiency arising from reallocation. That hypothesis is the principal focus of this paper, and we test it in the following section. However, it should be noted in advance that this is only one possible explanation for the different rates at which sectors translate employment shocks into labor market inefficiency. That difference could also be the result of any number of sectoral characteristics (e.g., geographic dispersion, unionization, specialization of human and physical capital, or any economic rents that discourage workers from changing jobs), or it could simply mean that something is different in the nature of the forces that are affecting the various sectors. Because we lack data on those factors and because we can only use the variation across 12 sectors anyway, consideration of these hypotheses must be left to future research.

Restructuring and Labor Market Inefficiency

As explained in section 2, the effects of restructuring could be manifest either as an ongoing trend in labor market inefficiency or as wider Beveridge loops. We now present evidence on both of those possibilities.

Trends in Labor Market Inefficiency

If reallocation were pushing Beveridge curves away from the origin, we would expect to find stronger positive trends in n in sectors that experienced greater reallocation. Because such a trend might be obscured by the dynamic responses of u and v to employment shocks, we must control for those dynamics when we measure trends in n. To do this, we can again run sector-specific regressions of n's against monthly net job creation rates (as discussed above), but now we add a linear time trend to the specification.

None of those regressions support this version of the Sectoral Shift Hypothesis. All of the estimated trends are small, only three are statistically significant, and of those, only the trend for financial services is positive. (The parameters from the original regressions are largely unchanged as well.) Because there is little indication of extensive structural change in financial services (see Table 5), we suspect that the estimated trend actually reflects the fact that the Beveridge loop for that sector appears less complete than the others (see Figure 3). It should also be noted that one of the statistically significant negative trends was for education and health, a sector that did experience relatively large structural change.

Thus, although we hesitate to make a definitive pronouncement about long-run trends from five years of data, we find little indication of ongoing trends in labor market inefficiency and no indication whatsoever of trends that are positively correlated with restructuring.

Fluctuations in Labor Market Inefficiency

The last remaining possibility is that structural change causes larger fluctuations in labor market inefficiency over the business cycle (i.e., wider loops in the Beveridge curve). If this variant of the Sectoral Shift Hypothesis has merit, there should be a significant relationship between the measures of [DELETA]n from Table 3 and some measures of restructuring.

To investigate, the upper panel of Table 5 presents seven measures of reallocation for each sector. The first statistic reported is the absolute value of each sector's net percent change in employment over the sample period. However, this metric does not capture reallocation within sectors, and it measures only net changes in employment, whereas gross flows would capture the full extent to which workers enter or exit sectors (including, for better or worse, some flows that are not related to differences in sectoral growth rates). To provide some information on those aspects, the next two columns report the percentage of workers who left the sector during a typical year and the average flow of workers entering a new sector. (The difference is the percentage of workers who left the labor force.) The fourth column then reports the average annual percentage of workers who switched industries within each supersector, a measure of intrasectoral reallocation. (15)

A shortcoming of these measures is that they do not distinguish between structural and cyclical changes, whereas earlier we noted that structural change might reasonably be expected to cause larger inefficiencies. The final three columns of Table 5 address this concern with a simple measure of intersectoral structural change proposed by Groshen and Potter (2003). In short, they deem an industry's employment fluctuations to be primarily cyclical if the industry's employment falls relative to other industries during the recession and rises by a similar percentage during the recovery, but they call the change primarily structural if the trend in relative employment is in the same direction during both phases of the cycle. Specifically, define [r.sub.i] to be industry i's percent change in employment during the recession and [s.sub.i] to be the same statistic during the recovery, and let [bar.r] and [bar.s] be the corresponding changes in employment for the aggregate economy. Then if we define [[??].sub.i] [equivalent to] [r.sub.i] - [bar.r] and [[??].sub.i] - [[bar.s], the structural percent change in employment is computed as 0.5([[??].sub.i] + [[??].sub.i]) and the cyclical change is 0.5([[??].sub.i] - [[??].sub.i]). The absolute value of the structural component is reported in the fifth column of Table 4, and the remaining two columns translate this into (i) an absolute change in the level of employment because of structural change and (ii) the percentage of employment flows that are structural. Like the measures discussed previously, these statistics are imperfect, and they are not the only reasonable ways to distinguish between structural and cyclical changes. (16) Even so, we think there is some value in choosing a simple measure that has recent precedent, especially considering that it comes from a paper that encourages consideration of the Sectoral Shift Hypothesis.

Because each of the metrics in Table 5 is intended to represent a different aspect of restructuring, it is not terribly surprising that they are not always highly correlated. Nevertheless, taken together, they seem to indicate a fairly consistent pattern of restructuring--in particular, several measures indicate considerable structural growth in the education and health sector and strong structural contraction in manufacturing and information.

The lower panel of Table 5 presents the correlations between these measures of reallocation and the measures of [DELTA]n from Table 3. Some of these correlations are at least mildly supportive of the Sectoral Shift Hypothesis, but most are not. The most favorable results are that four of the five measures of [DELTA]n are moderately correlated (0.25-0.50) with both the rate at which workers leave for other sectors and the share of employment flows that are structural. On the other hand, four of the five measures of An are negatively correlated with the absolute number of jobs added or removed by structural change, and three of those four negative correlations are reasonably large. Of the remaining 15 correlations in Table 5, seven are negative--including the only four that are larger than 0.2 in magnitude. Indeed, only six of the 35 reported correlations exceed 0.3, none exceed 0.5, and the average of all the correlations in the table is just 0.08.

In light of these findings, it seems difficult to argue that structural change is a major determinant of the increase in labor market inefficiency over this business cycle. This is not to say that the Sectoral Shift Hypothesis should be dismissed as a general proposition. After all, at least a few of the correlations in Table 5 were moderately large, and anyway, our methodology was predicated on the idea that the hypothesis might have greater merit in some periods than in others. Nevertheless, if structural change were an important cause of the jobless recovery, we would certainly expect it to be more strongly associated with measures of increased labor market inefficiency than it appears to be.

7. Conclusion

To summarize, this paper has explored the hypothesis that labor markets became considerably less efficient because of extensive economic restructuring in the years immediately after the U.S. recession of 2001. The evidence was not favorable--although this was apparently a period with extensive structural change, we were unable to identify strong ongoing trends in labor market inefficiency, and we found at most a mild relationship between structural change and temporary increases in labor market inefficiency. We thus conclude that such reallocation likely had little to do with the jobless recovery that followed the 2001 recession.

Of course, this finding only begs the question of what did cause the jobless recovery. As noted in the introduction, three other proposed explanations center on international trade, employment costs, and increased uncertainty arising from geopolitical events. It is not possible to comment on any of those hypotheses here because the relevant data do not line up well with our sectoral data, but that should not preclude future research. Sectoral data was necessary in this analysis because the hypothesis under investigation was fundamentally about reallocation across sectors, but it does not seem essential for these alternative hypotheses. For example, Cutler and Madrian (1998) have used microdata to measure employers' responses to higher health care costs, and it is easy to imagine that similar follow-up studies could support or reject the hypothesis that rising employment costs are discouraging hiring.

On the other hand, if one wanted to reconsider the Sectoral Shift Hypothesis during this period, it might be promising to investigate restructuring along other dimensions that we have not been able to consider, like occupational or spatial reallocation. Another potential innovation is to expand the model to account for transitions into and out of the labor force. As Bradbury (2005) has noted, reduced labor force participation was an important aspect of this business cycle, and it seems plausible that structural change could provide a stronger push out of the labor force than cyclical changes do. If so, such forces might have obscured a relationship between structural change and labor market inefficiency.

At any rate, we hope that future research will identify the causes of jobless recoveries. Although business cycles might be unavoidable, it seems plausible that public policy could help to reduce the sluggishness in the labor market once employers start to post vacant jobs. However, until we know why the labor market recovers so slowly, it is difficult to know which policies could be most effective. About all we can say at this point--at least if we accept the conclusion that sectoral reallocation has not had an important effect--is that it does not appear promising to subsidize worker retraining or to promote economic growth that is more balanced across sectors. The viability of the other policy instruments in the introduction will depend on what future research reveals.

Finally, we are intrigued by our finding that net job creation shocks apparently increase labor market inefficiency by a larger magnitude in some sectors than others, especially because that difference does not appear related to the factor we identified. Presumably this means that some other characteristics allow some sectoral labor markets to adjust to shocks more easily than others. If those characteristics can be identified, they might suggest additional ways to reduce the cost of those adjustments.

Appendix: Bias

As noted in the text, our estimators (e.g., Eqn. 4) might be biased if the correlation p between m and n is not zero. This Appendix derives formulas for the bias that would result if [rho] [not equal to] 0.

Although there is no way to measure [rho], we can gain insight into its magnitude by considering the somewhat extreme case in which (m, n) follows a stylized perfect ellipse: (re(t), n(t)] = ([k.sup.2.sub.1] cost, [k.sup.2.sub.2] sin t). Even then, [rho] = 0 over any half-cycle. After 3/4 of a cycle, [absolute value of [rho]] cannot exceed 0.33, and it remains possible that [rho] = 0, although for most parameters, [absolute value of [rho]] is closer to 0.1. (17) The bound shrinks rapidly as the path grows longer: [absolute value of [rho]] cannot exceed 0.11 after nine-tenths of a cycle (even as new cycles begin), and it is typically closer to zero. Because our data seem to cover nearly a complete cycle, it seems unlikely that the correlation between m and n is very large. More to the point, although one might squint at the data and detect a vaguely elliptical pattern, the progression is much less orderly than the stylized model would suggest. In most sectors, there is a significant period of "meandering," during which the two rates bounce around almost randomly. (See Figure 2 for an example.)

In light of these factors, we would be quite surprised if P exceeded 0.1 in our data, and we suspect that it is considerably smaller. Even so, we should ask whether such a correlation would seriously bias our estimates.

To investigate the potential bias in the empirical techniques developed in section 4, let (x, - y) be our estimate of the true Beveridge curve vector (a, - b). The estimate is found from an eigenvector decomposition of the observed covariance matrix of (u, v)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (A1)

Because the left-hand side of Equations 3 and AI are identical, equating the right-hand sides leads to a formula for the bias q in our slope estimator [??] [equivalent to] y/x,

[eta] [equivalent to] [??] - z = - [rho] + z (square root of [k.sup.2]+[[rho].sup.2]] - k)/2[a.sup.2](k + z[rho]) + ([square root of [k.sup.2]+[[rho].sup.2]] - k), (A2)

where k [equivalent to] ([[sigma].sup.2.sub.m], - [[sigma].sup.2.sub.m])/2 [[sigma].sub.m][[sigma].sub.n]. For some purposes, it might be more instructive to consider the angle [theta] between the estimated and true Beveridge curves: [theta] = [tan.sup.-1]([??]) - [tan.sup.-1]([??] + [eta]).

Furthermore, note that the eigenvalues are also a function of k:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A3)

This leads to a formula by which k can be measured for any hypothesized correlation [rho]:

k = [square root of [([[lambda].sub.m] - [[lambda].sub.n]).sup.2]/4[[lambda].sub.m][[lambda].sub.n] (1 - [[rho].sup.2]) - [[rho].sup.2]]. (A4)

Thus, because [a.sup.2] [equivalent to] [[1 + [z.sup.2]].sup.-1] and the method itself provides an estimate of z, Equation A2 shows how we can estimate the bias q under any assumption about [rho].

Before computing this bias, it is worth noting that Equation A4 imposes an observable upper bound on the magnitude of [rho]: [[absolute value of [rho]] [less than or equal to] ([[lambda].sub.m] - [[lambda].sub.m])/([[lambda].sub.m] + [[lambda].sub.n]). For most sectors examined in Table 2, the implied upper bound is much higher than we would ever expect anyway, on the order of 0.3-0.7. However, the bound implies that [absolute value of [rho]] [less than or equal to] 0.17 for government, corroborating the suggestion above that [absolute value of [rho]] is unlikely to exceed 0.1.

For the remaining sectors, Equations A2 through A4 imply the estimated slopes are fairly accurate, even with a large correlation of p = 0.25. Under that assumption, the largest angle between the true and estimated slopes would still be less than 13[degrees] (for construction), the smallest would be less than 4[degrees] (for the leisure sector), and all of the true slopes would still lie within the reported 95% confidence intervals. At a more reasonable value of p = 0.1, the largest angle between true and estimated slopes would be 6.7[degrees] (again, construction), and the gap would be less than 3.0[degrees] for half of the other industries. For smaller p yet, all of these biases approach zero as p does.

The bias in n is [??] - n = (y - b)u + (x - a)v. Because a = [(1 + [z.sup.2]).sup.-1/2] and b = z[(1 + [z.sup.2]).sup.- 1/2], Taylor approximations yield x - a [approximately equal to] - [??][(1 + [[??].sup.2]).sup.-3/2] and y - b [approximately equal to] [(1 + [[??].sup.2]).sup.3/2], so

[[??].sub.it] - [n.sub.it] [approximately equal to] [u.sub.it] - [??][v.sub.it]/[(1 + [[??].sup.2]).sup.3/2]. (A5)

For [rho] = 0.1, Equation A5 implies that the average bias in n across our sample would be 0.008, which is less than 0.5% of the average estimate. Nearly 80% of the individual [[??].sub.it] would be within 1% of their true values, and the largest percent bias for any individual observation would be 2.6%. This bias is generally in the same direction for all observations in a sector, so differences ([[??].sub.i,t+k] - [[??].sub.it]) have smaller biases than individual observations. Thus, with [rho] = 0. 1, the widths of the sectoral Beveridge curve loops would be estimated to within 5.5% of the actual width for all sectors and to within 3% for all but two sectors.

We are especially grateful to Hagen Schwerin for stimulating our interest in the topic and for many thoughtful comments. We also thank Swarnjit Arora, Keith Bender, John Berdell, Hoyt Bleakley, Robert Dixon, Erica Groshen, Julie Hotchkiss (the editor), Matthew McGinty, Mike Miller, John Rust, staff at the Bureau of Labor Statistics and the Bureau of the Census, three anonymous referees, and seminar participants at DePaul University and the University of Wisconsin-Milwaukee for helpful comments and information. Any remaining deficiencies are solely our responsibility.

Received January 2006; accepted January 2007.

References

Abraham, Katharine G. 1987. Help-wanted advertising, job vacancies, and unemployment. Brookings Papers on Economic Activity 1987:207-43.

Beveridge, William Henry. 1945. Full employment in a free society. New York: W. W. Norton.

Blanchard, Olivier Jean, and Peter Diamond. 1989. The Beveridge curve. Brookings Papers on Economic Activity 1989:1-60.

Bleakley, Hoyt, and Jeffrey C. Fuhrer. 1997. Shifts in the Beveridge curve, job matching, and labor market dynamics. New England Economic Review September:3-19.

Bowden, Roger J. 1980. On the existence and secular stability of u-v loci. Economica 47:35-50.

Bradbury, Katharine. 2005. Additional slack in the economy: The poor recovery of labor force participation during this business cycle. Federal Reserve Bank of Boston Public Policy Brief 05-2.

Cutler, David M., and Brigitte C. Madrian. 1998. Labor market responses to rising health insurance costs: Evidence on hours worked. RAND Journal of Economics 29:509-30.

Davis, Steven J., John C. Haltiwanger, and Scott Schuh. 1996. Job creation and destruction. Cambridge, MA: MIT Press.

Groshen, Erica L., and Simon Potter. 2003. Has structural change contributed to a jobless recovery? Current Issues in Economics and Finance 9:1-7.

Groshen, Erica L., Simon Potter, and Rebecca Paul Sela. 2005. Was the 2001 recession different in the labor market? Comparing measures. Paper presented at the 2005 Annual Meetings of the American Economic Association, Philadelphia, PA.

Hall, Robert E. 2003. Modern theory of unemployment fluctuations: Empirics and policy applications. American Economic Review 93:145-50.

Hansen, Bent. 1970. Excess demand, unemployment, vacancies, and wages. Quarterly Journal of Economics 84: 1-23.

King, Miriam, Steven Ruggles, Trent Alexander, Donna Leicach, and Matthew Sobek. 2004. Integrated public use microdata series, current population survey: Version 2.0. [Machine-readable database]. Minneapolis, MN: Minnesota Population Center. Accessed 27 June 2006. Available http://www.ipums.org/cps.

Lilien, David M. 1982. Sectoral shifts and cyclical unemployment. Journal of Political Economy 90:777-93.

Minehan, Cathy E. 2004. Labor markets: What we know and what we don't. Speech given to Money Marketeers, New York, March 24, 2004. Accessed 14 January 2006. Available at http://www.bos.frb.org/news/speeches/cem/2004/032404.htm.

Petrongolo, Barbara, and Christopher A. Pissarides. 2001. Looking into the black box: A survey of the matching function. Journal of Economic Literature 39:390-431.

Pissarides, Christopher A. 1985. Short-run equilibrium dynamics of unemployment, vacancies, and real wages. American Economic Review 75:676-90.

Pissarides, Christopher A. 2000. Equilibrium unemployment theory. 2nd edition. Cambridge, MA: MIT Press.

Schwerin, Hagen. 2003. Empirical tests of the sectoral shift hypothesis. M.A. thesis, University of Wisconsin Milwaukee, Milwaukee, WI.

Shimer, Robert. 2005. The cyclical behavior of equilibrium unemployment and vacancies. American Economic Review 95:2549.

Solow, Robert M. 1998. What is labour-market flexibility? What is it good for? Proceedings of the British Academy 4:189-211.

Theil, Henri. 1971. Principles of econometrics. New York: John Wiley & Sons.

Yellen, Janet. 1989. Comment [on Blanchard and Diamond (1989)]. Brookings Papers on Economic Activity 1989:65-71.

(1) As many have noted, the household (Current Population Survey [CPS]) and payroll (Current Employment Survey [CES]) data diverge over part of this period, but both indicate that employment recovered slowly after the recession. According to the CPS, it took 32 months (until October 2003) for seasonally adjusted employment to permanently surpass pre-recession levels; it took 47 months (until January 2005) in the CES data.

(2) Hall (2003) thus computes the slope of the Beveridge curve as the relative responses of vacancies and unemployment to a productivity shock z: dV/dU = [[partial derivative]V/[partial derivative](log z)]/[[partial derivative]U/[partial derivative](log z)].

(3) Of course, job creation and destruction are endogenous, caused by more fundamental factors that are not observed directly (e.g., demand or productivity shocks affecting either individual firms or entire industries).

(4) Petrongolo and Pissarides (2001) thoroughly review the literature on the matching function. As they note, in principle, the matching function could have either decreasing or increasing returns to scale, although many (not all) of the studies they survey cannot reject the hypothesis of constant returns to scale.

(5) The JOLTS vacancy data is disaggregated a bit more finely than the CPS unemployment data, but the supersector level is the finest level at which the two series can be matched. JOLTS separates wholesale and retail trade, but we have recombined them to match the unemployment data. Also, JOLTS' natural resources sector includes both mining and logging, whereas the CPS reports data on mining alone, but we have matched those series because logging accounts for only 1/10 of the natural resource sector. JOLTS also disaggregates geographically, but only into four large regions.

(6) This is another way that reallocation can influence the measured efficiency of sectoral labor markets, albeit one having more to do with accounting than behavior.

(7) The BLS publishes seasonally adjusted vacancy data for only a few sectors, so we have seasonally adjusted all data by computing 12-month moving averages of unadjusted data. Where a comparison is possible, there is very little difference between the published data and our seasonally adjusted series.

(8) The eigenvectors must be orthogonal, justifying (b, a) as the second basis vector.

(9) The relevant command in Stata 9.0 is "'pca, vce(normal)."

(10) Denote the relevant loop width by W, the average rate of unemployment by [bar.U], and the estimated Beveridge curve elasticity by z. Because the curve shifts along an axis with slope (-1/z), the predicted increase in log U is [DELTA]u = W cos[arctan(-1/z)]. Thus, the predicted increase in unemployment rates because of the decrease in labor market efficiency is calculated as [bar.U][exp([DELTA]u) - 1].

(11) The last two measures are defined by Davis, Haltiwanger, and Schuh (1996) as, respectively, the sum of hiring and separation rates and the gross worker reallocation rate minus the absolute value of net employment growth. Those authors emphasize reallocation of jobs rather than workers, so their excess reallocation measure captures variation in manufacturing plants' employment growth around the industry trend. In principle, such a measure might be taken to reflect the sort of reallocation considered in the next subsection. However, our data permit analysis of only excess worker reallocation, which is more appropriately considered here because it measures employment flows divorced from intersectoral reallocation of jobs.

(12) Although this evidence suggests that turnover raises labor market inefficiency, as we have defined the term, it should be noted that turnover (especially job-to-job quits) could also raise the quality of matches between workers and jobs, which is a different and possibly more important type of efficiency.

(13) Ideally, one would like to check this by comparing within-sector changes in n against gross rates of both job creation and job destruction. However, net rates are easier to obtain, and they suffice for our simple purpose of verifying the consistency between the theory and the data.

(14) It is perhaps also worth noting that, for example, net job creation is actually much more strongly correlated with the "business cycle variation" [[lambda].sub.m] (0.73) than with the "efficiency variation" [[lambda].sub.n] (0.23), suggesting that the correlations in Table 4 are probably not seriously attenuated by some measurement error.

(15) The data in the second through fourth columns of Table 5 were computed from the CPS (through the Integrated Public Use Microdata Series [King et al. 2004]) by comparing agents' responses about their current industry and their industry one year earlier. Industries were reclassified in 2003, so the figures use only data from the 2003-2005 surveys (covering transitions from 2002-2005). However, the rates are fairly constant across years, and where a direct comparison was possible the data for 2001-2002 were quite comparable. Intrasectoral reallocation is not well defined for the construction sector because it consists of only one industry.

(16) Groshen, Potter, and Sela (2005) discuss alternative measures of structural change. Following their method, the structural share of employment flows is computed as [([[??].sub.i] + [[??.sub.i]).sup.2]/[]([[??].sub.i] + [[??].sub.i]).sup.2] + ([[??].sub.i] - [[??.].sub.i]).sup.2]].

(17) Note that 9 depends on both the length of the cycle and the angle to between the starting point and the m-axis. Thus, after 3/4 of a cycle (3[pi]/2 radians), [[sigma].sub.mn] is proportional to ([sin.sup.2] to [cos.sup.2] to). [rho] = 0 if [t.sub.0] is an odd-integer multiple of [pi]/4.

Chad D. Cotti, University of South Carolina, Moore School of Business, 1705 College Street, Columbia, SC 29201, USA; E- mail [e-mail].

Scott Drewianka, University of Wisconsin-Milwaukee, Department of Economics, Bolton Hall 860, 3210 N. Maryland Avenue. Milwaukee, WI 53211, USA; E-mail [e-mail]; corresponding author.

Table 1. Summary Statistics

                                         Average
                                       Unemployment       Average
Industrial Sector                          Rate        Vacancy Rate

Construction                               8.23            1.64
Education and health services              3.30            3.52
Financial services                         3.23            2.42
Government                                 2.52            1.85
Information                                5.72            2.32
Leisure                                    8.11            3.30
Manufacturing                              5.69            1.59
Natural resources                          4.74            1.34
Other services                             4.94            2.45
Professional and business services         6.97            3.42
Transportation and utilities               4.59            1.86
Wholesale and retail trade                 5.56            2.11
Aggregate economy                          5.35            2.47
Source                                     CPS             JOLTS

                                         Average        % Change in
                                        Employment    Employment over
Industrial Sector                      (thousands)     Sample Period

Construction                                 6931          10.6
Education and health services              16,592          15.3
Financial services                           7977           7.3
Government                                 21,539           5.3
Information                                  3272         -17.2
Leisure                                    12,331           8.5
Manufacturing                              14,951         -17.1
Natural resources                             600          13.0
Other services                               5369           3.9
Professional and business services         16,410           2.3
Transportation and utilities                 4870          -2.4
Wholesale and retail trade                 20,795          -0.8
Aggregate economy                         133,470           0.0
Source                                       CES            CES

The data are seasonally adjusted and cover the period December
2000-April 2006.

Table 2. Beveridge Curves by Sector

                                                            Average
                            Estimated                    Distance from
                           Elasticity,                      Origin
Industrial Sector        d(ln V)/d(ln U)        SE         (mean n)

Construction                  -1.45            0.10          2.01
Education and health
  services                    -1.25            0.07          1.71
Financial services            -1.66            0.17          1.45
Government                    -1.53            0.08          1.10
Information                   -1.22            0.07          1.85
Leisure                       -2.37            0.09          2.39
Manufacturing                 -1.18            0.04          1.61
Natural resources             -0.68            0.03          1.07
Other services                -0.82            0.08          1.63
Professional and
  business services           -1.02            0.05          2.24
Transportation and
  utilities                   -1.49            0.11          1.60
Wholesale and retail
  trade                       -1.53            0.13          1.84

                                                          Efficiency
                             Business                      Variation
                              Cycle                       ([[lambda].
                            Variation                       sub.n])
Industrial Sector       ([[lambda].sub.m])      SE          (x100)

Construction                  0.034            0.006         0.186
Education and health
  services                    0.020            0.004         0.080
Financial services            0.037            0.006         0.375
Government                    0.023            0.004         0.075
Information                   0.082            0.014         0.368
Leisure                       0.025            0.004         0.031
Manufacturing                 0.045            0.008         0.080
Natural resources             0.144            0.025         0.323
Other services                0.022            0.004         0.244
Professional and
  business services           0.034            0.006         0.123
Transportation and
  utilities                   0.028            0.005         0.198
Wholesale and retail
  trade                       0.021            0.004         0.184

                                              Average
                                SE           Education
Industrial Sector             (x100)          (years)

Construction                  0.032            12.4
Education and health
  services                    0.014            14.9
Financial services            0.065            14.3
Government                    0.013            14.6
Information                   0.064            14.3
Leisure                       0.005            12.4
Manufacturing                 0.014            13.1
Natural resources             0.056            12.3
Other services                0.042            13.1
Professional and
  business services           0.021            14.4
Transportation and
  utilities                   0.034            13.2
Wholesale and retail
  trade                       0.032            13.1

Table 3. Measures of the Change in Labor Market Efficiency over the
Business Cycle

                                               Ratio of        Loop
                                                 Loop         Width
                                   Loop        Width to       above
Industrial Sector                  Width      Average n      Average

Construction                       0.04          0.02          0.02
Education and health services      0.13          0.08          0.05
Financial services                 0.26          0.18          0.08
Government                         0.10          0.09          0.02
Information                        0.23          0.13          0.08
Leisure                            0.10          0.04          0.07
Manufacturing                      0.10          0.06          0.04
Natural resources                  0.18          0.17          0.07
Other services                     0.10          0.06          0.04
Professional and business
  services                         0.13          0.06          0.05
Transportation and utilities       0.18          0.11          0.08
Wholesale and retail trade         0.14          0.08          0.07

                                                              Actual
                                              Change in      Range in
                                   Ratio     Unemployment  Unemployment
                                  of Loop        Rate      Rates above
                                Width above  Predicted by    Average,
                                Average to    Half-Loop       Max U
Industrial Sector                Average n      Width      minus Avg U

Construction                       0.01          0.14          1.3
Education and health services      0.03          0.14          0.4
Financial services                 0.06          0.24          0.5
Government                         0.02          0.04          0.3
Information                        0.04          0.36          1.3
Leisure                            0.03          0.54          0.7
Manufacturing                      0.03          0.18          1.0
Natural resources                  0.07          0.20          2.6
Other services                     0.03          0.14          0.2
Professional and business
  services                         0.02          0.28          1.4
Transportation and utilities       0.05          0.30          0.7
Wholesale and retail trade         0.04          0.33          0.6

                                Percent of
                                  Actual      Number of
                                  Change     Postminimum   Integral of
                                 Explained   Months with   Postminimum
                                by Changes     n above      n's above
Industrial Sector                  in n        Average       Average

Construction                       10.2           10           0.13
Education and health services      35.4           34           0.72
Financial services                 53.8           45           1.46
Government                         15.4           23           0.20
Information                        27.3           40           1.60
Leisure                            77.7            7           0.15
Manufacturing                      18.1           36           0.79
Natural resources                   7.7           23           0.59
Other services                     81.5           23           0.45
Professional and business
  services                         20.3           36           0.95
Transportation and utilities       44.9           32           1.18
Wholesale and retail trade         53.0           29           1.15

Table 4. Variability of Employment Flows and Labor Market Efficiency
Across Sectors

                                 Net Job                    Excess
                                Creation   Gross Worker     Worker
                                  Rate     Reallocation  Reallocation
Industrial Sector                (x 10)        Rate          Rate

A. Standard deviation of monthly employment flows

Construction                      2.23         0.34          0.38
Education and health services     0.53         0.28          0.24
Financial services                0.75         0.21          0.21
Government                        0.60         0.15          0.13
Information                       2.62         0.27          0.35
Leisure                           1.53         0.73          0.75
Manufacturing                     2.76         0.17          0.21
Natural resources                 3.55         0.28          0.37
Other services                    0.89         0.45          0.47
Professional and business
  services                        2.66         0.85          0.85
Transportation and utilities      2.08         0.63          0.67
Wholesale and retail trade        1.22         0.31          0.29

B. Cross-industry correlation of standard deviations
   with measures of labor market efficiency

Efficiency variation
  ([[lambda].sub.n])              0.23        -0.29         -0.19
Loop width                        0.07        -0.15         -0.11
Loop width above average          0.16         0.21          0.25
Number of postminimum months
  with n above average           -0.01        -0.25         -0.25
Integral of postminimum n's
  above average                   0.11        -0.09         -0.06

                                 Hiring     Separation
Industrial Sector                 Rate         Rate       Quit Rate

A. Standard deviation of monthly employment flows

Construction                      0.24         0.18          0.16
Education and health services     0.15         0.14          0.12
Financial services                0.12         0.11          0.10
Government                        0.10         0.06          0.04
Information                       0.20         0.16          0.18
Leisure                           0.48         0.31          0.32
Manufacturing                     0.14         0.24          0.09
Natural resources                 0.27         0.21          0.14
Other services                    0.25         0.22          0.20
Professional and business
  services                        0.47         0.39          0.20
Transportation and utilities      0.47         0.18          0.15
Wholesale and retail trade        0.19         0.15          0.17

B. Cross-industry correlation of standard deviations
   with measures of labor market efficiency

Efficiency variation
  ([[lambda].sub.n])             -0.21        -0.27         -0.13
Loop width                       -0.10        -0.25         -0.16
Loop width above average          0.29         0.10          0.29
Number of postminimum months
  with n above average           -0.32        -0.18         -0.49
Integral of postminimum n's
  above average                  -0.10        -0.13         -0.16

                                             Absolute
                                 Layoff/    Change in
                                Discharge   Employment
Industrial Sector                 Rate        (x 10)

A. Standard deviation of monthly employment flows

Construction                      0.29         0.64
Education and health services     0.04         0.55
Financial services                0.07         0.35
Government                        0.04         0.51
Information                       0.16         1.73
Leisure                           0.12         0.30
Manufacturing                     0.22         2.40
Natural resources                 0.24         1.45
Other services                    0.10         0.56
Professional and business
  services                        0.27         0.62
Transportation and utilities      0.16         0.84
Wholesale and retail trade        0.06         0.37

B. Cross-industry correlation of standard deviations
   with measures of labor market efficiency

Efficiency variation
  ([[lambda].sub.n])              0.10         0.16
Loop width                       -0.22         0.11
Loop width above average         -0.16         0.01
Number of postminimum months
  with n above average           -0.16         0.31
Integral of postminimum n's
  above average                  -0.11         0.24

The net job creation rate is the monthly difference in log(employment
+ vacancies). The separation rate is the rate at which
employer-employee matches end, including both quits and layoffs/
discharges. The gross worker reallocation rate is the sum of the
separation and hiring rates (as in Davis, Haltiwanger, and Schuh
[1996]), and the excess worker reallocation rate is the gross worker
reallocation rate minus the absolute value of the change in employment
rates (i.e., the flow of worker reallocation that is not directly
caused by changes in employment levels).

Table 5. Sectoral Reallocation and Labor Market Efficiency

                                                     Average Annual
                                   Absolute Value     % of Workers
                                  of % Net Change    who Left Sector
                                   in Employment,    for Any Reason
Industrial Sector                  12/2000-4/2006         (CPS)

A. Measures of reallocation
  Construction                          10.6              12.8
  Education and health
    services                            15.3              11.3
  Financial services                     7.3              14.1
  Government                             5.3              23.1
  Information                           17.2              19.4
  Leisure                                8.5              23.7
  Manufacturing                         17.1              12.7
  Natural resources                     13.0              21.3
  Other services                         3.9              17.3
  Professional and business
    services                             2.3              18.6
  Transportation and utilities           2.4              14.4
  Wholesale and retail trade             0.8              18.5
B. Cross-industry correlation with measures of labor market efficiency
  Efficiency variation
    ([[lambda].sub.n])                   0.10             -0.05
  Loop width                             0.08              0.00
  Loop width above average              -0.02              0.10
  Number of postminimum
    months with n above
    average                              0.12             -0.38
  Integral of postminimum n's
    above average                        0.03             -0.25

                                                     Average Annual
                                                      % of Workers
                                  Average Annual %     Who Changed
                                   of Workers who      Subsectors
                                  Changed Sectors    within Sectors
Industrial Sector                      (CPS)              (CPS)

A. Measures of reallocation
  Construction                           7.8              n.a.
  Education and health
    services                             6.0               4.4
  Financial services                    10.5               2.4
  Government                             9.8               1.1
  Information                           14.8               1.2
  Leisure                               12.6               2.3
  Manufacturing                          9.1               5.2
  Natural resources                     14.3               1.3
  Other services                        12.9               1.3
  Professional and business
    services                            12.6               2.1
  Transportation and utilities          10.5               1.8
  Wholesale and retail trade            11.6               6.1
B. Cross-industry correlation with measures of labor market efficiency
  Efficiency variation
    ([[lambda].sub.n])                   0.48             -0.37
  Loop width                             0.41             -0.23
  Loop width above average               0.50             -0.04
  Number of postminimum
    months with n above
    average                              0.02              0.17
  Integral of postminimum n's
    above average                        0.27              0.14

                                     Absolute %         Absolute
                                     Change in         Structural
                                   Employment Due        Change
                                   to Structural      in Employment
Industrial Sector                      Change          (thousands)

A. Measures of reallocation
  Construction                          3.8                258
  Education and health
    services                            6.0                913
  Financial services                    2.3                179
  Government                            1.3                270
  Information                          10.0                372
  Leisure                               3.2                380
  Manufacturing                         9.2               1588
  Natural resources                     4.2                 25
  Other services                        0.9                 48
  Professional and business
    services                            0.5                 86
  Transportation and utilities          1.8                 92
  Wholesale and retail trade            1.1                238
B. Cross-industry correlation with measures of labor market efficiency
  Efficiency variation
    ([[lambda].sub.n])                  0.12              -0.45
  Loop width                            0.16              -0.25
  Loop width above average              0.08              -0.24
  Number of postminimum
    months with n above
    average                             0.27               0.21
  Integral of postminimum n's
    above average                       0.25              -0.03

                                   Share of Total
                                  Change that was
Industrial Sector                    Structural

A. Measures of reallocation
  Construction                           51
  Education and health
    services                             88
  Financial services                     96
  Government                             36
  Information                            72
  Leisure                                70
  Manufacturing                          85
  Natural resources                      47
  Other services                         12
  Professional and business
    services                              2
  Transportation and utilities           83
  Wholesale and retail trade             60
B. Cross-industry correlation with measures of labor market efficiency
  Efficiency variation
    ([[lambda].sub.n])                  0.10
  Loop width                            0.41
  Loop width above average              0.43
  Number of postminimum
    months with n above
    average                             0.30
  Integral of postminimum n's
    above average                       0.39

Data are seasonally adjusted and cover the period December 2000-April
2006. See text for definitions of measures of structural change. The
CPS definitions of sectors and subsectors changed in 2003, so those
columns use only data from the 2003-2005 surveys (reflecting
employment transitions in 2002-2005). The data are relatively constant
across those years and are quite similar for 2001-2002 for sectors in
which a clear comparison was possible. n.a., not applicable.