the beveridge curve and unemployment fluctuations in canada

The Beveridge curve and unemploymentfluctuations in Canada

Richard Archambault Human Resources Development CanadaMario Fortin Département d’économique, Université de Sherbrooke

Abstract.We estimate the impact of cyclical, sectoral, and participation shocks and that ofthe trend on both the Canadian unemployment rate and the job vacancy rate over the 1969 to1998 period. We conclude that a rise in the Canadian unemployment rate of almost 5 per-centage units occurred between 1972 and 1982 because of participation rate shocks and atrend movement. Because the trend also explains the leftward shift of the Beveridge curveobserved in the 1990s, this shift cannot be interpreted as a decline in the natural unemploy-ment rate.

La courbe de Beveridge et les fluctuations du chômage au Canada. Les auteurs évaluent lacontribution des chocs sectoriels, des changements dans le taux d’activité, des fluctuationscycliques et de la tendance sur les mouvements du taux de chômage et du taux de postesvacants pendant la période 1969–98. Ils concluent que le taux de chômage non conjoncturels’est accru de près de 5 points de pourcentage entre 1972 et 1982 en raison de chocs dans letaux d’activité et d’un mouvement tendanciel. Le déplacement vers la gauche de la courbe deBeveridge observé dans les années 1990 est pour sa part le résultat d’une baisse tendancielledes postes vacants et n’indique nullement que le taux de chômage naturel ait diminué pen-dant cette période.

1. Introduction

There is an important debate surrounding the explanation of the high unemploy-ment rate observed during the 1990s in Canada. A cyclical interpretation, bestrepresented by Fortin ~1996!, maintains that because Bank of Canada’s inflationtargets were set too low, monetary conditions were over restrictive during the first

The authors wish to thank Paul Storer, Alain Guay, Louis Grignon, David Johnson, and two anon-ymous referees for their invaluable comments and suggestions, as well as those persons attendingthe UQAM and ARB lunch meetings. We are also grateful to Louise Casault for editing theEnglish version of this paper. Email: [email protected]; [email protected]

Canadian Journal of Economics 0 Revue canadienne d’Economique, Vol. 34, No. 1February 0 février 2001. Printed in Canada 0 Imprimé au Canada

0008-4085 0 01 0 58–81 0 r Canadian Economics Association

half of the decade. These tight monetary conditions would explain why the employ-ment ratio plummeted in Canada between 1990 and 1995, in absolute terms as wellas in comparison with the United States. Moreover, building on a model developedby Akerlof, Dickens, and Perry ~1996!, he argues that with an inflation rate below 2per cent, nominal downward wage rigidity has raised the non-accelerating inflationrate of unemployment ~NAIRU!.

Freedman and Macklem ~1998! dispute this explanation. They judge that mon-etary conditions were not too tight during the 1992–96 period and that there is noneed to rely on downward wage rigidity to explain a rising NAIRU. Rather, theystate that the restructuring of the Canadian economy weakened the output and employ-ment responses to monetary conditions during this period. In a reply, Fortin ~1999!sees no evidence whereby restructuring was an important cause for the sluggisheconomy. He points out, in particular, the behaviour of the Beveridge curve, that is,the inverse relation between the unemployment rate and the vacancy rate. As Abra-ham and Katz ~1986! showed, sectoral or regional reallocation of jobs implies thatvacant jobs are less easily filled by unemployed workers. If restructuring explainedthe decline in employment in the 1990s, the vacancy rate would have been higherthan in the previous decade. In effect, however, the help-wanted index ~HWI! sharplydeclined in Canada during the 1990–92 recession and never returned to its previoushigh of 1988. Moreover, the Beveridge curve shifted to the left between 1993 and1997. This is an indication that the NAIRU has declined since 1990, not risen, asFreedman and Macklem claim.1

Our goal in this paper is to proceed to a formal statistical analysis of the move-ments in the Beveridge curve to identify their sources. This task is important becausethe Beveridge curve has now been used many times to distinguish between cyclicaland non-cyclical fluctuations to the Canadian unemployment rate.2 In all cases,however, the analysis was carried out by the mean of a simple visual decompositionin the unemployment and job vacancy space. Such an approach presents two impor-tant weaknesses. First, because the slopes of a pure cyclical and a pure sectoralshock remain arbitrarily determined, their respective contribution cannot be pre-cisely quantified. Secondly, since a decomposition carried out in a two-dimensionalspace cannot identify more than two types of shocks, no distinction can be drawnbetween a shift in the Beveridge curve due to a sectoral shock and a shift producedby an exogenous change in the participation rate.

This last weakness is a source of confusion in interpreting the movements of theBeveridge curve. Indeed, recent research has shown that the participation rate fellafter 1989 in response to demographic and social policy changes ~Fortin and Fortin1999; Dugan and Robidoux 1999; Archambault and Grignon 1999; Beaudry and

1 Note that after Samson ~1985! initially found empirical support for the sectoral shock hypothesis,many studies have subsequently concluded that these shocks are not an important cause of thefluctuations in the Canadian unemployment rate. See, for example Burns ~1990!, Fortin and Araar~1997!, or Lu ~1997!.

2 See Riddell ~1986!, Blanchard ~1991!, or Samson ~1994!.

The Beveridge curve 59

Lemieux 1999!. As Card and Riddell ~1992! have pointed out, until 1989 an increasein the participation rate was called for to explain the high Canadian unemploymentrate. Such an increase, they argued, was a reaction to the high subsidization rate tounstable jobs after the 1971 unemployment insurance ~UI! reform, an outcomeconsistent with the prediction of the theoretical models of Fortin ~1984! or Mil-bourne, Purvis, and Scoones ~1991!. A series of modifications introduced between1990 and 1996, however, have drastically reduced the UI subsidy. If, as in Riddell~1986!, the rightward shift to the Beveridge curve between 1972 and 1978 is seen asa consequence of the 1971 UI reform, it becomes tempting to interpret the leftwardshift between 1992 and 1996 as a reversal due to the tightening in the program. Asa logical consequence, it becomes possible that both Fortin ~1999! and Freedmanand Macklem ~1998! are simultaneously right: sectoral shocks could have increasedthe NAIRU since 1990, while the participation rate could have declined sufficientlyto more than offset its impact on the unemployment rate.

Our analysis relies on a methodology developed by Blanchard and Diamond~1989; hereinafter BD!. Starting with a reduced-form VAR on the labour force,employment and a vacancy proxy based on the help wanted index ~HWI!, we utilizetheoretical restrictions to the short-term responses in order to identify, in additionto a time trend, three types of structural shocks: that is, structural, cyclical, andparticipation shocks.

Our approach differs from that of BD in one important regard. Despite the factthat the variables were not stationary, BD still estimated a VAR in level form. Thishas serious consequences on the estimation, since, as shown by Phillips ~1995!, theimpulse responses and historical decomposition then become random and do notconverge asymptotically. When the Johansen ~1988! method is used, the Canadiandata support the hypothesis of a cointegration relation between employment, labourforce, and vacancies. We have therefore estimated a vector error correction model~VECM!, which, in addition to ensuring convergence of the impulse response, presentsthe additional advantages that the long-run equilibrium and the short-term dynam-ics pushing the system back to its balance growth path can be described and ana-lysed. As a by-product, our work also shows that job vacancies is the only variablethat reacts to the labour market disequilibrium and, at the same time, is the soleimpulse that pushes the unemployment rate back to its stable long-run equilibriumlevel.

The paper is organized as follows. In order to outline the main elements involvedin the identification of the structural shocks, in the next section we show in a non-technical way how these shocks influence the determination of employment, labourforce, and vacancy, with an emphasis given to the presence of a long-run relationbetween the variables. In the third section we present the data, and the reduced-form model is set out in Section 4. The most important result of section 4, whichgives support to the matching model, is the crucial role vacancies play to correct thelabour market disequilibrium. In the fifth section we explain the identification ofthe shocks and analyse the impulse-response of the structural VAR. In section 6 wesee how the model decomposes the unemployment fluctuations into four components:

60 R. Archambault and M. Fortin

cyclical, sectoral, participation, and trend. We conclude with comments and sug-gestions for future research.

2. Job vacancy and unemployment in the matching model

In their influential paper, BD studied unemployment fluctuations together with jobvacancies by means of an aggregate matching function that describes how unemployedworkers fill vacant jobs. This approach has the interesting ability of analysing thestock of unemployed and employed workers together with the flow of separationsand matches.3 To understand how various types of shocks influence the unemploy-ment rate, we outline the matching model’s mechanisms. The interested reader isreferred to BD for a complete description of the model.

Let us define the labour force as L [ E 1 U, where E and L are respectively thenumber of jobs held and the number of unemployed, and the number of productivejobs as J [ E 1V, where V is the number of job vacancies. The aggregate matchingfunction m~U,V ! describes how job search by unemployed workers and recruitingactivities by firms translate into matches by which a vacant job is filled. It is usuallyassumed that this function displays constant returns to scale and that an increasingnumber of matches results from higher unemployment ~]m0]U . 0! or vacancies~]m0]V . 0!.4

This simple framework distinguishes sectoral from cyclical shocks in the fol-lowing way. A change in aggregate activity, which includes aggregate demand aswell as productivity shocks, occurs when the number of productive jobs changes.Because new productive jobs are initially vacant, the instantaneous impact of, let ussay, an increase in J, is to raise V. Because ]m0]V . 0, the number of matchesincreases and E progressively rises with a parallel reduction in V. Moreover, if oneassumes a pro-cyclical participation rate, L expands with E while U falls, given thatdL0dE , 1. In the end, the steady state is displaced towards higher levels of vacancy,employment, and labour force, and a lower unemployment level. When aggregateactivity declines, the system progressively returns to its initial state. If the unemploy-ment rate is measured on the horizontal axis and the vacancy rate is on the verticalaxis, a complete economic cycle is visually characterized by a counter-clockwisemovement. The rotation comes from the fact that since job vacancy leads unemploy-ment as well in expansion as in contraction, the path followed during the contrac-tion is below that followed during the recovery.

A pure sectoral shock is defined as a higher rate of job creation and destructionwithout any change in J. These larger labour market flows displace the long-run

3 Jones and Riddell ~1993! estimated with Canadian data the extent of these flows and some of theirproperties. Storer ~1992! estimated a matching function using provincial data for Alberta andOntario. A calibrated matching model was used by Hornstein and Yuan ~1999! to analyse thecontribution of UI changes to the unemployment rate.

4 This formulation deals only with the unemployed job search. See Broersma and Van Ours ~1999!for an analysis of how an employed job search affects the properties of the matching function andthe estimated return to scale.


equilibrium towards higher numbers of job vacancy and unemployment and a loweremployment level. Finally, an increase in the participation rate initially raisesunemployment. With the higher number of matches that ensues, the employmentlevel progressively increases with a concomitant reduction in unemployment andvacancy. In the end, the new equilibrium is characterized by a higher level of bothemployment and unemployment and a lower job vacancy.

In addition to describing the impact of these shocks on the model, we note thatemployment, unemployment, and job vacancies do not drift apart but rather movetogether in the long run. Indeed, let us assume that the fraction of filled jobs becom-ing vacant each period is p. Thus, the number of transitions from employment tounemployment is simply pE. The long-run relation between employment, unemploy-ment, and vacancy is obtained when these transitions just counter-balance the num-ber of matches, that is, when the following equation holds:

am~U,V ! 5 pE. ~1!

As discussed in the next section, this long-run relation has an important impli-cation on the empirical model.

3. The data

We estimate the model with seasonally adjusted quarterly data on E, L, and V. TheE and L series currently published in the Labour Force Survey have been revised in1994 and begin in 1976. Because of the possible impact of the 1971 UI reform onthe labour market, we sought to use continuous series that covered this period. Tothis end, we consolidated the actual E and L series with those published in CAN-SIM prior to the 1994 adjustment, these series having begun in 1966.5 With regardto job vacancies, no Canadian estimate is currently available, but a job vacancysurvey was carried out by Statistics Canada between 1971 and 1978.6 Like Zagor-sky ~1993! and Samson ~1994!, we developed a job vacancy proxy based on theHWI.

In order to get a Canadian vacancy estimate that also covers the 1970s, we con-solidated as a basis two different HWI series. The old series, which began in the firstmonth of 1962 and ended in the last month of 1988, was proportional to the spaceoccupied by job offers in the country’s leading daily newspapers. The current series,available from the first month of 1981, indicates instead the number of job offers inthose newspapers – the same concept as that used in the United States. We consol-idated these two series by effecting a reverse flow forecast of the number of jobs

5 If LEN is the logarithm of the new employment series and LEO is the former series, values priorto 1976 have been adjusted by the equation LEN 5 20.54861508 1 1.0632822*LEO, which hasbeen estimated on the monthly data from January 1976 to April 1994. The labour force logarithmhas been adjusted by the equation LLN 5 20.53398816 1 1.0611226*LLO.

6 Statistics Canada estimated the number of vacancies at approximately 31,000 in a 1995 surveyconducted in Quebec to determine employers’ hiring intentions. This suggests that the vacancyrate is less than 1 per cent.


advertised prior to 1981, the details of which are described in appendix A. Afterobtaining a continuous HWI series for 1966–98, we adjusted its level so that it wasidentical to the mean number of job vacancies estimated by Statistics Canada between1971 and 1978, namely, 127,600.

Our job vacancy proxy differs from that of Zagorsky ~1993! and that of Samson~1994! in the following way. Zagorsky used a consolidated HWI series calibratedon the level of the 1971–78 job vacancy survey. The absolute level of Zagorsky’sseries, then, is similar to our proxy. To consolidate both HWI series from 1962 to1990, however, he made two annual forecasts of the old HWI series on the newHWI. A reverse flow forecast is probably more suitable to the time period coveredin our analysis. Indeed, this method allows the addition of new observations as theybecome available. This is why we selected this type of forecast. Moreover, ourestimate is based on an HWI that is conceptually identical to the HWI used in theUnited States. Since Abraham ~1987! has shown that the American HWI seriesclosely follows the short-term movements in the number of job vacancies in theUnited States, it is legitimate to believe that this conclusion also applies in Canada.7

Samson ~1994! on the other hand, used only the old HWI to obtain a vacancy proxyfrom 1962 to 1988. The main difference, however, is that the level was adjusted sothat the number of job vacancies was equal to the number of unemployed in the firstquarter of 1966. This calibration is based on the assumption that the unemploymentrate was then entirely frictional. This method produces a vacancy rate almost threetimes higher than our proxy and therefore overestimates by a factor of three thenumber of vacancies as estimated by the job vacancy survey of 1971–78.

Based on the data on L, E, and V, it is easy to calculate the unemployment rateUR 5 100*~L 2 E !0L and the vacancy rate VR 5 100*V0~E 1V !. From these rateswe trace the Beveridge curve, presented in figure 1, enabling us to make three majorobservations. First, the vacancy rate is much smaller than the unemployment rate.Although the highest vacancy rate was 1.6 per cent in both 1974 and 1988, it hasrarely been higher than to 0.8 per cent in the 1990s. As noted by Zagorsky, thesevalues are, on average, lower than the vacancy rate estimated in the United States byAbraham ~1987!. Second, the biggest unemployment rate changes are negativelyrelated to vacancy rate changes and look very much like the counter-clockwiseloops theoretically associated with aggregate activity shocks with peaks in 1974and 1988 and troughs in 1983 and 1993. Third, the Beveridge curve slides to theright between 1970 and 1976, a shift already identified by Riddell ~1986!, Blan-chard ~1989!, and Samson ~1994!, but this movement is reversed by a displacementto the left beginning in 1994. As indicated previously, such lateral shifts suggestthat either reallocation shocks or changes in the participation rate – or even bothtypes of shocks – caused unemployment to increase in the early 1970s and decreaseafter 1993. Their net impact, however, is that in 1998 the natural rate of unemploy-

7 The shortcoming of the proxy is that no time series on job vacancies in Canada or in the UnitedStates is long enough to validate or invalidate the similarity of the trend growth rate of HWI withthat of job vacancies in the near future.


ment did not seem to be much different from its 1970–71 level. These observationsneed to be refined by the empirical investigation presented in the following sections.

4. The reduced-form model and the labour market disequilibrium

The model was estimated on a sample of 140 quarterly observations starting in1966:1 and ending in 1998:4 on the logarithm of L, E and V, designed, respectively,as l, e, and v.8 Although all series were available on a monthly basis, we neverthelesspreferred to work with quarterly data because this lower frequency allows a moreparsimonious representation without significantly reducing the power of statisticaltests. To select the appropriate model, we conducted many specification tests, thedetails of which are presented in appendix B. Our investigation led us to estimate aVECM model with eleven lags and a single, cointegrating relation whose onlydeterministic elements are the constants in the variables’ growth rate. As indicatedpreviously, a VECM of cointegrated9 variables is preferable, since, as shown by

8 Since U 5 L 2 E, estimating the model with the level of variables L, E, and V would be equivalentto estimating it with U, E, and V. However, a logarithmic transformation is required to make theseries variance stationary. The consequence of this non-linear transformation is that the two repre-sentations are no longer equivalent. When estimating a model in u, e, and v, the u residualspresent a leptokurtic distribution, while the model estimated with l, e, and v easily passes thenormality test.

9 Random variables are said to be cointegrated of order k if it is possible to find k different combi-nations of random integrated variables with an order of integration less than that of each variabletaken separately. If variables U, V, and E are I~1!, the combination am~U,V ! 2pE 5 0 has thisproperty, since 0 is integrated of order zero.

FIGURE 1 The Beveridge curveSource: Statistics Canada and author’s calculations


Phillips ~1995!, the impulse response and the historical decomposition do not asymp-totically converge when a VAR is estimated on non-stationary variables, while thisproblem does not arise if the variables are cointegrated. The representation of themodel is

Dxt 5 m0 1 (i51

p

Gi Dxt2i 1 Pxt21 1 ut . ~2!

In this equation, x 5 ~l e v!' is a 3 3 1 vector of observations at time t, m0 is a3 31 vector of constant terms, Gi is a 3 3 3 matrix capturing the short-run dynamicat lag i, P is a 3 3 3 matrix capturing the long-run effect, and ut is the vector oferror terms ~ul ue uv!. The P matrix itself can be written as a product of twovectors ab ', where b is the cointegrating relation and a is called the loading factor.Both a and b play an important role in the model. The product b 'xt21 defines thecointegration error, that is, the deviation from the long-run balance growth pathb 'xt21 5 0. The cointegration error provides a gross measure of the system’s devi-ation from its long-run balance growth path.10 As for ai , it shows the impact ofb 'xt21 in the ith equation. This provides a measure of the speed at which short-rundisturbances are absorbed and the ith variable returns to its long-run equilibrium.

Table 1 shows the estimated covariance and correlation, the latter in italics,between d~e!, d~v!, d~l ! in the upper part of the table, and between the series’innovations in the lower part. As BD found with U.S. data, the series d~e! and d~l !are highly correlated ~0.74! and the correlation remains as high between the inno-

10 The term ‘gross’ is used because it recovers the impact of the deterministic variables as well asthe transitory effects related to the model’s short-run dynamics. One obtains the net disequilib-rium after these transitory and deterministic effects are removed. If R defines the vector formedby the residuals of the regressions by ordinary least squares of e, v, and l on the deterministicvariables and the variables lagged in difference, the net disequilibrium is equal to b 'R. It is thelatter series, containing much less persistence than b 'x, that is tested for stationarity in theJohansen procedure.

TABLE 1Covariances and correlations of the reduced form

Variables d~e! D~v! d~l !

d~e! 0.0000398 0.6359 0.7357d~v! 0.0002370 0.0003484 0.1933d~l ! 0.0000219 0.0000539 0.0000223

ue uv ul

ue 0.0000130 0.3593 0.7720uv 0.0000345 0.0007090 0.1585ul 0.0000092 0.0000140 0.0000109


vations in both series. The close link between these two variables is explained bythe pro-cyclical participation rate and also by the fact that employment innovationsare in many cases determined by labour force innovations.11 We also note a rela-tively high correlation of 0.64 between d~e! and d~v!, but the correlation betweenthe series’ innovations drops to 0.36. Finally, the correlation between d~l ! and d~v!is low ~0.19!, and we observe that v is approximately ten times more volatile than eand fifteen times more volatile than l.

The cointegration error b 'xt21 5 be et21 1 bvvt21 1 bl lt21 is the empiricalcounterpart of equation ~1! which, after an arbitrary normalization with respect toe, becomes et21 2 0.0646 3 vt21 2 0.9007 3 lt21. Although we do not present itstime plot, we have verified that this normalized cointegrating error is pro-cyclicaland closely follows the peaks and troughs identified by Statistics Canada accordingto per capita GDP.12

The first three lines in table 2 report global statistics for each equation while therest of the table contains elements related to the analysis of the system’s self-correcting dynamic. The fourth line indicates the estimated values of ai , with itst-statistics in parentheses. The coefficient indicates the variable’s responsiveness tothe previous period’s deviation from long-run disequilibrium. The dominant featureis that this speed of adjustment is extremely small for both e ~ae 5 0.032! and l~al 5 0.017! and is not statistically different from zero in each case. A joint test thatai 5 0 simultaneously in d~e! and d~l ! equations, not reported in the table, has ax2~2! 5 2.02, which is well below the critical 5 per cent value of 5.99. This means

11 This is best illustrated by the hundreds of professorial jobs that have been eliminated in Canadianuniversities in recent years as those who held these jobs retired.

12 See the February 1996 issue of the Canadian Economic Observer, in which different criteria fordating the recessions in Canada are compared.

TABLE 2Results on the reduced-form model

Equations

d~e! d~v! d~a!

R2 0.678 0.808 0.492Sum of Squared Residuals 0.0016 0.0852 0.0013Standard Error of Regression 0.0043 0.0316 0.0039

ai0.032 20.452a 0.017

~t-statistic of H0: ai50! ~1.23! ~22.34! ~0.73!x~11!

2 of H0: Gd~e1! to Gd~e11! 5 0 6.33 24.24a 2.96x~11!

2 of H0 : Gd~v1! to Gd~v11! 5 0 58.48b 113.14b 20.58a

x~11!2 of H0 : Gd~a1! to Gd~a11! 5 0 13.82 12.14 19.98a

a Significant at the 5 per cent levelb Significant at the 1 per cent level


that neither employment nor labour force directly reacts to labour market disequi-librium. However, av5 20.452 and is statistically different from zero at the 5 percent level. Thus, v is the only variable whose speed of adjustment towards long-runequilibrium is significant. Its negative sign implies a counter-cyclical reaction of vto the labour market disequilibrium.

The three following lines in table 2 help to identify the contribution of eachvariable to the model’s short-run dynamic. To carry out this task, we tested whetherthe eleven lags of d~e!, d~v! and d~l ! are simultaneously zero in each equation.These tests show that the lags of d~e! are jointly significant only in the job vacancyequation, while the lags of d~l ! are significant only in the d~l ! equation. The moststriking feature of these tests, however, is the importance of d~v! lags in all equa-tions, which are significant at the 5 per cent level in d~l ! equation and at the 1 percent level in d~e! and d~v! equations. Although we do not report all coefficients ofthe system, it is worth while to underline that, individually, the first lag of d~v! ispositive and highly significant in all equations.

Taken together, these tests enable us to describe how the employment and labourforce adjustments towards equilibrium take place. Consider, for instance, a negativecointegration error, that is, an unusually slack labour market. The only direct impactof this slack is transmitted to the system by av, and this impact raises v one periodlater. Then, because of the short-run dynamic, the increase in vt11 pushes up allvariables in the second period, and this positive response has a permanent effect. Asshown in figure 2, however, because the response of l to a shock on v is smaller thanthat of e, there is a permanent reduction in the unemployment rate. Since this adjust-ment takes place only because d~v! reacts to b 'xt21, the unemployment rate wouldnot return to a stable long-run value if job vacancies were not reacting to the labourmarket disequilibrium. The essential contribution of job vacancies in the correctionof labour market disequilibrium constitutes a strong support of the matching model.

FIGURE 2 Response to a shock on v


In the next section, we will see how the analysis can be extended to measure thecontribution of cyclical and non-cyclical factors to unemployment rate changes.

5. The structural form model

We now introduce the three types of structural shocks discussed previously: sec-toral shocks ~s!, changes to the participation rate ~ f ! and aggregate activity shocks~c!, which we call cyclical. We use the term cyclical to describe any fluctuation inaggregate activity without any distinction between transitory shocks, usually asso-ciated with aggregate demand fluctuations, and productivity shocks, which mayhave long-lasting or even permanent impacts.13 We define a vector of structuralshocks « 5 ~s c f !' that are not serially correlated.14 The contemporaneous impactof structural shocks on reduced-form shocks u 5 ~ue uv ul ! comprises the elementsof matrix A such that u 5 A«. In principle, the model can be identified by a set ofrestrictions on either A or «, and these restrictions are given by the theoreticalreactions of variables to s, c, f . These are defined in A in the form:

3me

mv

m l

4 5 321 a l

b 1 0

2u ua 14 3

s

c

f4 . ~3!

The elements on the main diagonal are a normalization and indicate that, at im-pact, the effect on ue of a shock to s is unitary, that a shock to c increases uv by 1, andthat a shock to f increases ul by 1.The coefficient u captures the systematic labour forcereaction to employment changes, and l is the instantaneous impact of participationrate shocks on ue. The crucial distinction here should be made between cyclical andsectoral shocks.This is dictated by the values given to a, the effect of a cyclical shockon ue, and to b which is the increase in uv following a shock to s. Finally, we assumethat the shocks to f have no contemporaneous impact on v. The whole identificationprocess takes place by combining different values of the parameters a, b, l, and u.Following the approach used by BD, we identify the parameters of A on two grounds,which are calibration from other research and theoretical impulse responses.To achievethe task of selecting appropriate values for these coefficients, we proceed sequen-tially with solutions to be given first to u and l and then to a and b.

13 King et al. ~1991! used the premise that in a three-variable model with one cointegrating relation,only one shock has temporary impacts while two have permanent impacts. Because temporaryimpacts are associated with aggregate demand fluctuations, they can be identified with a restric-tion on the long-run variance matrix. Since our ‘cyclical’ shocks cover both aggregate demandfluctuations and productivity shocks, however, this identification strategy cannot be used in ourmodel.

14 A formal distinction should be made between structural disturbances and the shocks to thesedisturbances purged of the effects of their own dynamic. As BD pointed out, however, they cannotbe empirically distinguished. The only dynamic that can be observed is the one captured by thereduced form, which is a convolution of the structural disturbances dynamic and the systemdynamic. The short-cut we take here is a valid representation based on the hypothesis that thestructural disturbances are white noise processes.


If f had no instantaneous impact on ue, we could estimate u by a regression of ul

on ue, which would give an estimate of 0.707 of the pro-cyclical reaction of theparticipation rate. This is implausibly high. Indeed, because employment reactspositively to the labour force shocks, OLS provide an upward biased estimate of u.Fortin and Fortin ~1997! have estimated u by instrumental variables to correct thebias. Their finding that the participation rate’s sensitivity to change in the employ-ment ratio varies between 0.20 and 0.70, depending on the demographic group,implies that the average value is significantly lesser than 0.7. Variable l can beindirectly calibrated on the analysis of labour market flows by Jones and Riddell~1993!. They estimate that the number of transitions between non-participation andemployment is about half of those between non-participation and labour force.15 Ifall these transitions were due to changes in the participation rate, this would implythat l is approximately 0.5. Since these flow rates are also the consequence of thepro-cyclical reaction of the participation rate, however, this value is likely to over-estimate l. For purposes of identification, we have set l 5 0.4 and u 5 0.46.

The coefficients a and b were chosen so that the system’s responses to sectoraland cyclical shocks are consistent with theoretical expectations over as many quar-ters as possible. Pairs of values are sought that produce a response to the s shocksthat is positive for v and negative for e, although the response of these two variablesto a c shock must remain positive. The values adopted for identification are a 50.05 and b 5 2. These were adopted through a grid exploration of the differentparameter combinations. This exploration revealed that to obtain impulse responsesto an s shock compatible with theoretical expectations, b can be given a wide rangeof value, while a must lie within a very narrow band. Indeed, the response of theunemployment rate to a sectoral shock decreases when a rises, so that if we set a inthe range of approximately 0.20, the system’s responses to a sectoral shock lookslike a cyclical shock, with U and V moving in opposite directions. On the otherhand, if we set a too low, for example, at 0.02, both U and V responses to cyclicalshocks are positive, so that a cyclical shock mimics a sectoral shock. The combi-nation chosen is thus the most plausible scenario for the simultaneous presence ofsectoral and cyclical shocks.16 The exploration also reveals that the system’s impulseresponses are not very different when l varies between 0.3 and 0.5 and u is setbetween 0.4 and 0.5.

Our identification strategy differs from BD on one important point. Unlike them,we do not assume that the structural shocks are orthogonal because we do not think

15 We have used the March 1984 data from their sample, which provides details on the size of thedifferent groups. By weighting the flow rates of marginally attached and unattached persons, onefinds that the probability of flow during one month from non-participation to employment is0.0323, while the probability to participation is 0.0657, for a ratio of 49.1 per cent.

16 BD have also observed that it is difficult to find a pair of values capable of producing impulseresponses consistent with theoretical expectations of sectoral shocks. The values they haveselected are very different from those we have used. Nevertheless, the responses to shocks in theBeveridge space are very similar on impact. The difference in coefficients between the two studiesprobably arises from the data transformations, since BD normalize their observations and reducethe variance of the three series so that it is unitary.


FIGURE 3 Impulse-responses to structural shocks

this hypothesis can be strongly justified. For example, Davis and Haltiwanger ~1991!showed that job displacements are negatively correlated with cyclical fluctuations.In order, nevertheless, to obtain the impulse responses for the structural shocks, weorthogonalized the structural shocks using a Choleski decomposition, imposing theorder s, c, f. Because it was difficult to find parameter values compatible with thetheoretical effects of sectoral shocks, those were given the first position in theCholeski decomposition in order to allow them the greatest possible contribution.Shocks to f have been placed last, so as to measure the minimum contribution thatcan be assigned to labour force shocks in unemployment fluctuations. Unexpect-edly, despite the fact that the shocks were not constrained to be independent, thecorrelation between pairs of structural shocks remains weak. Indeed, as shown initalics in the upper part of table 3, the absolute value of the correlation between theshocks never exceeds 0.154. Because of these low correlations, the order in whichone performs the Choleski decomposition will have but a limited impact on theestimated contribution of each shock.

Figure 3 shows the system’s impulse responses to the structural shocks. Only theresponse of vacancies to the labour force shock is inconsistent with theoreticalexpectations. Indeed, although a larger number of job seekers should theoreticallyreduce the number of job vacancies, we found, on the contrary, that job vacanciesincrease following an f shock. We investigated the model at length, but were unsuc-cessful in identifying a combination of parameters able to produce a negative reac-tion of v to an f shock.17 One possible explanation may be that the data simply donot support the kind of labour force shock that we theoretically defined. Indeed, inthese shocks, new entrants adopt the same job search behaviour as those already inthe labour force. If the increase in the size of the labour force is a reaction to achange in UI benefits, however, it will be accompanied by a simultaneous reductionin the intensity of job search. In such a case, it is not impossible to observe apositive response of v to an f shock.18

17 One could easily constrain A to have a negative instantaneous reaction of v to an f shock. Thiswould not provide a satisfactorily solution, however, because the response of v is growing withtime, not falling, as predicted by the matching function.

18 See Riddell ~1999! for a thorough discussion on the sources of the change in the participation ratein Canada and the role UI may have played.

TABLE 3Covariances and correlations of the structural shocks

Variables us uc uf

us 0.0000084 20.154 20.153uc 20.0000116 0.0006731 20.082uf 20.0000011 20.0000051 0.0000056


6. Decomposing unemployment fluctuations

The historical decomposition serves to disentangle the variation of e, v, and l vari-ables over the period 1969–98 into four components, namely, those deriving fromthe three structural shocks and the impact of the deterministic components. Toexplain the observed level of unemployment, the contribution of each componentmust then be cumulated, and, in so doing, the level given to each series is arbitrarilychosen. Figure 4 shows how the unemployment rate would have evolved since 1969when the level is adjusted to the actual unemployment rate observed in 1968. Thus,each series shows how the Canadian unemployment rate would have changed since1968 if all other shocks had been nil throughout the entire period.

Cyclical shocks have been the most important source of short-run unemploy-ment fluctuations. They contributed to a fall in the unemployment rate of 1.5 per-centage points between 1970 and 1980. This was followed by an increase of 3.6points between 1981 and 1983, but the unemployment rate entirely recovered fromthis rise during the 1983–89 expansion, falling by 3.7 points. It then rose againduring the cyclical downturn of 1989–93, this time by 3.1 percentage points, but didnot significantly recover until 1996. Finally, aggregate activity improved enough in1997 and 1998 to pull down the cyclical unemployment rate by 1.1 percentagepoints in each year. It is interesting to note that although the slump of the 1990s wasnot as severe as that of 1981–82, the absence of a significant recovery for almostthree years following the trough made its length exceptional. Clearly, then, deficientaggregate demand was an important factor behind the persistent high unemploy-ment rate observed between 1990 and 1996.

The model indicates that the most important sectoral shock was observed in1980–82, with a 1.8 point increase in the unemployment rate. It also shows three

FIGURE 4 Historical decomposition of the unemployment rate


periods over which sectoral shocks raised the unemployment rate by 1 percentagepoint, that is, in 1974–76, 1990–92, and 1996–98. However, none of these fourepisodes was long lasting. Indeed, all were quickly followed by an equivalent fallover a period of two to three years, so that the periods with the lowest frictionalunemployment were in 1979–80, 1986–88, and, most recently, in 1996. Thus, sec-toral shocks may have contributed to the rising unemployment rate at the begin-ning of the 1990s, but it cannot explain why it remained high for so long thereafter.It is also of interest to note that we find that sectoral shocks have significantlycontributed to a rise in the unemployment rate during the course of the last tworecessions.

The participation shocks’ estimated impact on labour force and employment isthe most distinctive result compared with the results obtained by BD. They did notfind any significant impact of these shocks in the United States, but we estimate thatparticipation shocks have had a substantial effect on the Canadian unemploymentrate. The historical decomposition shows that the labour force began to grow at afaster pace in 1972, culminating in 1980 with an increase of nearly 800,000 per-sons. Subsequently this movement partly subsided so that, progressively throughoutthe 1980s and the 1990s, the labour force declined, with the decrease slightly accel-erating as of 1993. Employment shows a similar movement, with a less pronouncedamplitude, so that it rose by approximately 550,000 between 1970 and 1980. Insummary, participation shocks contributed to a 1.8 percentage point increase in theunemployment rate between 1970 and 1980, with a particularly rapid rise of 1.2points over the 1976–78 period. By 1994, however, two-thirds of the rise had evap-orated in a cumulative fall, since 1981, of 1.2 percentage points. Thus, if one com-pares the period of the mid-1990s with that at the end of the 1960s, participationshocks explain a rise in the unemployment rate of no more than 0.6 points. More-over, these shocks were helping to push down the unemployment rate in the 1990s,with a decline of 0.5 percentage points between 1990 and 1998.19

The last part of the decomposition is the contribution of the deterministic com-ponents. In the United States, BD found a trend increase of about 2 per cent betweenthe 1950s and 1980s. We find that the trend explains a rise to the unemployment rateof 3.9 percentage points between 1969 and 1989. As can be seen in figure 4, theincrease was particularly rapid between 1974 and 1980, and the trend’s impactreached a plateau in 1990. The deterministic components almost entirely capturethe secular rise in the unemployment rate that has been observed over the last thirtyyears. Unfortunately, our approach uses high-frequency correlations to identify shocksand consequently does not allow us to shed light on the cause of this upward trendin the unemployment rate. We suggest that there might have been a slow but significant

19 Since the amplitude of the employment change following an f shock is dictated by the value givento l, the change in employment would have been lower and the rise in unemployment more pro-nounced if l had been smaller. For example, when l 5 0.3, the rise in the unemployment ratewould have been 2.2 points between 1970 and 1980 rather than 1.8 points, and 1.4 points ifl 5 0.5.


deterioration of the degree of agreement between jobs created and jobs destroyed,or a progressive change in the composition of the Canadian labour force.

Two additional elements, derived from the historical decomposition, enable us toassess the extent of the recovery during the 1990s. First, we can trace the cyclicalBeveridge curve, that is, the movements of the vacancy rate and of the unemploy-ment rate over the last thirty years, in response to cyclical shocks. Once again, thelevel of the cyclical unemployment rate has been adjusted to the actual unemploy-ment rate in 1968. The counter-clockwise loops, clearly displayed in figure 5 arounda negatively sloped straight line, implies that at a given vacancy rate, the unemploy-ment rate is 1 per cent higher during a recovery than during a contraction. Thestraight line, which is a simple OLS estimate of the impact of the cyclical unemploy-ment rate on the cyclical vacancy rate, shows that a 1 per cent increase in theunemployment rate is accompanied by a reduction of 0.16 per cent in the vacancyrate. Figure 5 reveals that the cyclical position in 1998 is highly similar to that of1989. This aggregate activity pressure on the labour market is not overtly displayedin the Beveridge curve presented in figure 1. Indeed, because of the deterministiccomponent, the actual vacancy rate experienced an important downward trend duringthe 1990s.20 This trend is responsible for the leftward drift in the Beveridge curve,which gives the visual impression that the natural rate of unemployment was declin-ing in the 1990s. When this trend effect is abstracted from, the vacancy rate rose

20 The downward trend in the vacancy rate results from a negative impact of the deterministic com-ponent on job vacancies since 1985, although it remained positive on employment.

FIGURE 5 The cyclical Beveridge curve


extremely rapidly in both 1997 and 1998. There is then a call for caution for thosewho would be tempted to interpret the leftward shift in the Beveridge curve in the1990s as a sign that the natural rate of unemployment has been falling.

So, how did the natural rate of unemployment evolve? The second interestingelement is that we can provide an estimate of the changes in the non-cyclicalunemployment rate. To this end, we presume that in 1989, when the cyclicalunemployment rate was at its lowest value, all cyclical unemployment had beenwiped out of the labour market, so that the non-cyclical unemployment rate wasequal to the actual unemployment rate. This crude assumption does not allow aprecise definition of the absolute level of the non-cyclical unemployment rate, butit does provide a means to identify how it changed over time. Figure 6 shows thatthe non-cyclical unemployment rate rose quickly between 1973 and 1981, from 3.1to 8.3 per cent. Thereafter, however, it showed no tendency to move upward ordownward, fluctuating between 7 and 8 per cent. Because this definition makes nouse of a link with the inflationary process, it cannot be interpreted as a measure ofthe NAIRU. Nevertheless, it would suggest that, as of 1998, the Canadian economywas close to full employment, with aggregate activity as high as it was in 1981 and1989. It also shows that, taken together, non-cyclical factors did not play an impor-tant role in the rise of the unemployment rate during the first half of the 1990s.

7. Conclusion

The main question dealt with in this paper is how the Beveridge curve can best beused to identify the contribution of cyclical fluctuations and of two non-cyclicalshocks to variations in the Canadian unemployment rate. We were particularly inter-ested in explaining the unemployment of the 1990s. To achieve this goal we needed

FIGURE 6 The non-cyclical unemployment rate


to estimate a reduced-form VECM model between labour force, employment, and aproxy of the number of vacancies. We showed that job vacancies alone directlyreact to the labour market disequilibrium and are also the main source of the mech-anism needed to push the labour market back to its equilibrium.

After identification of three structural shocks, we concluded that the impact ofnon-cyclical factors was substantial in the late 1970s and early 1980s. Sectoralshocks may cause the unemployment rate to fluctuate by as much as 2 percentagepoints over the course of two years, but these shocks are short lived. Shocks to theparticipation rate increased the unemployment rate by almost 2 percentage pointsbetween 1970 and 1981, and their impact was long-lasting. Nevertheless, two-thirds of the impact progressively disappeared during the 1980s and the 1990s. Wesuggest that the 1971 UI reform and the massive influx of baby boomers into thelabour market are likely explanations for the general shape of labour force shocks.

Our analysis shows that, although sectoral shocks might have contributed to aslight increase in the unemployment rate from 1990 to 1992, a decline in aggregateactivity was the main element in the sustained high level of the unemployment rateobserved until 1996. As for shocks to the participation rate, they contributed to adecline of 0.5 percentage point during this interval. The explanation for this highlevel of unemployment in comparison with that at the end of the 1960s remainsobscure, however, since it is attributed by the model to some unexplained determin-istic factors. We can say only that, whatever these factors might be, their impact hasrapidly grown during the 1970s to cause the unemployment rate to rise by almost 4percentage points as of 1981 and has stabilized since then.

Our approach makes intensive use of the help-wanted index as a proxy for agenuine vacancy index. Our analysis pointed out that the vacancy rate proxy basedon the help-wanted index contains a surprisingly strong and stable informative con-tent over the future course of the labour market. Since the last direct estimates ofvacancies date back more than twenty years, it is possible that our estimate of thevacancy rate is currently biased. We believe that this affects our conclusions onlyslightly, since our model permits independent trend movements in each series. If theHWI poorly captures the inherent vacancy trend, as Abraham’s ~1987! study sug-gests, this should be translated chiefly by a different constant in the vacancy equa-tion so that the non-deterministic components would have to remain similar.Nonetheless, in view of the role we assign them in the correction of disequilibrium,it is still important for a survey on job vacancies to be carried out.

Appendix A: Consolidation of the help-wanted index

Although the two HWI measures are related, they are also conceptually distinct.One result is that the new series behaves more smoothly than the old one at highfrequencies. We preferred to consolidate the series by making a reverse forecastfrom the new series rather than a forecast from the old one since 1981 for tworeasons. First, the relevance of the HWI as a vacancy rate proxy has been demon-strated by Abraham ~1987! using the number of help-wanted ads. Hence, it seemed


to us preferable to use the same concept. Furthermore, by doing the forecast back-wards, new observations can be automatically added to the consolidated series insteadof continually adding forecasts in place of new observations.

The method adopted for effecting the forecast is to estimate a link capable ofproducing independent prediction errors during the period common to the two series.To do this, we estimated on monthly data the logarithm of the new index LHWIN onthe logarithm of the old index LHWIO. Since this simple regression produced auto-correlated errors, we progressively added lead values, and at each stage the equa-tion’s residuals were subjected to the LM test to detect any autocorrelation. Withthree leading values, no serial correlation was detectable. Analysis by recursiveleast squares was then used to test temporal stability, and only late 1982 seemssubject to some stability problems. Since this problem could not be remedied byextending the lead structure, we adopted the equation with three leading values.This equation is LHWIN 5 0.108 1 1.135*LHWIN~11! 2 0.154*LHWIN~12! 20.062*LHWIN~13! 1 0.197*LHWIO 2 0.0117*LHWIO~11! 2 0.047*LHWIO~12! 2 0.073*LHWIO~13!. We then did a reverse-flow forecast, and theseries obtained was returned to level form by an inverse logarithmic transformation.After the consolidation, variability at high frequencies becomes almost identicalbefore and after 1981, since the standard deviation of the change in the seriesvariation rate is 3.69 per cent before 1981 and 4.05 per cent from 1981 to 1996.

Appendix B: The specification of the reduced form model

The results of the unit root tests on the level of the series, given in table B1, do notallow to reject the presence of a unit root in each series. Although not reported inthe table, the ADF and PP unit root tests were also applied to the series in firstdifference. Since the calculated statistics were superior to Mackinnon’s critical valueat the 1 per cent level, we rejected the presence of a unit root in the first difference,thus ruling out the possibility that the series are I~2!.

Before testing for model selection, we determined p, the number of lags neededto capture the short-run dynamic. If eleven lags are included, both the LM~4! test

TABLE B1

Variable pa Deterministicb ADF PPCriticalvalue ~5%!c

Criticalvalue ~1%!

log ~L! 1 C 21.43 21.99 22.88 23.48log ~E ! 2 C, t 23.00 22.17 23.44 24.03log ~V ! 1 C, t 20.83 20.57 23.44 24.03

a p is the number of lags used in the ADF test.b C is the constant and t is a time trend. For e and v, we reject the null hypothesis of no deterministic and

no stochastic trend.c Mackinnon’s critical value to reject the null hypothesis of a unit root


and the autocorrelograms confirm that the residuals are not serially correlated in allequations, while the Bera and Jarque tests do not allow rejection of the normality.

Having proceeded with a model that includes eleven short-run matrices, we sequen-tially tested the non-rejection of a general model versus a more restrictive modelwith the help of the lmax test and the trace test. Different formulations of the model’sdeterministic components are possible. Models with no deterministic component orthat allow for a quadratic trend were excluded a priori because they are not plausi-ble. We therefore compared a general model with a constant in the first difference ofthe series and a trend in the cointegrating relation, a model with no deterministiccomponents in the cointegrating relation but with a constant in the difference of thevariables, and a model with a constant in the cointegrating relation but no deter-ministic elements in the first difference of the series.

These comparisons led us to select a model with one cointegrating relation whoseonly deterministic elements are the constants in the variables’ growth rate equa-tions. The uniqueness of the cointegrating relation was confirmed by looking atadditional information. The two largest systems’ characteristic roots have a normthat lies close to the unit circle, which suggest the presence of two stochastic trendsand a single cointegrating relation. Second, when we allow for additional cointe-grating relations, a visual inspection shows that the cointegration error associatedwith those relations is not mean stationary. Third, the recursively estimated tracetest shows only one cointegrating relation increasing with time, as is consistentwith the existence of a single vector of cointegration. The recursive estimates alsoconfirm the system’s ability to forecast one period ahead the endogenous variables.Indeed, only six vacancy predictions are outside the 95 per cent confidence interval,among them three in 1981. Given the strong recession of that time and the fact thatthe frequency of rejection is not superior to the confidence level of the test, we donot consider this as a serious indication of instability problems. On the whole, then,no major specification problems can be identified.

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the beveridge curve and unemployment fluctuations in canada

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