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A Nonparametrsc Analysis of Regional UnemploymentDynamics in BritainMarco Bianchi a & Gylfi Zoega ba Monetary Analysis, Bank of England , London , EC2R 8AH , United Kingdom E-mail:b Department of Economics , Birkbeck College , London , W1P 2LL , United Kingdom E-mail:Published online: 02 Jul 2012.
To cite this article: Marco Bianchi & Gylfi Zoega (1999) A Nonparametrsc Analysis of Regional Unemployment Dynamics inBritain, Journal of Business & Economic Statistics, 17:2, 205-216, DOI: 10.1080/07350015.1999.10524811
To link to this article: http://dx.doi.org/10.1080/07350015.1999.10524811
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rvjarco BaaNgxi &qonsisiy Wiiaiysls, 3;nB.c. 0.f Ecgland, Bandon, EC2R 8AH, United Mii?gdorn jbi;~nchi.@globzii?ef~ca ukj
Cy;f: Z o ~ ~ t 2 ~ Dspartmenl of Econofi~ics, Birkksck College, Lordon, W1 P 218, United Kivigdarn ([email protected])
- 1 his article esiiiliates the probability distribritio!~ of relative county u:ler!?ploqInent in aritain for the years :981-1995. We liiid that the distribution is iiilimodal in all )ear,. with a falling varlancc jetvee, 1989 and 1994. Mie use bootstrap methods to deternine critical values for rlic tn-o tails of :he distributioil and ana1)zs iiltradistribuiiol1 dynamics. Tlv'e calculate transition probabilities and 5-iil that t!?e 1;1.obabilit)- of leaving any given state is Xery low. Vi'e aisc f i ~ d that high (low) ~!ne:xplo) ment regions have a higher probability of enrering a irate of lower (hipher) unemplo~~men! ti?an a state of higher (lower) unemplo~rneni.
KEY ViOWDS: Intra-distribution dynamics: Qrnei densit! eitimation: Pailel data: iJnemplo>- mext dynamics.
111 this ei-tick l~vs a s a:ocpararnetric methods to describe + LA. 3- n county-level dy~.anaics of unemployment in Britain. We
want te analyze Erst the distributior? of relative uriem- ployme?: to find ou; whether it is multi:?aodal. Tlze high- ~er~empioyrnent areas 01 Sco1:land; PTaies. and laor~laerr! Eng- land are o f~en coiatrasled wkh tire Eower ul-neanpioynnent ar- eas in the Sor:th; cori~ties in :hose regions might forE a gi-oap separzte from the yes:. Second. we would like to k:aow 7+vherher the -variance of ~cernplojrmerat across coun- ties has changed In Iecen! years, A Ia-llli~rg variance could either imply :laat hfgh-~~!ilrs~p~oymenb counties are recov- ering due to migrcticn and capital xover~ents or that ihe geographic2 disiribneion of shocks has changed over time. T'his is the case of CJ coiwergence (see Barro and Salai- Martin 1995). Fiidly. >,:;e ,?.re interested iaz assessirag the relative fortunes of different coimties by identifying those e3joyilag ~ e r s i s e e ~ t prosperity and those sagering persistent unempioyaent. Yy7e vm:t io know^ for exz-~ple* whether it is nore cificuit to recover from a (relative) depression than i t is to .fail kmm (relak've) p.;,asperlty and whether there is a Icndcncy lo :iro.vc toward she m a n . Thc iatter is recerred to zs 3 convergence. -
I here exists a s~bstantiel literature studying regional tan- enngioymenl persistence for diEeren: countries, Blanchard and Kafz ( 1 992) S;rnsdied the IJniied Stares, Sirrueno and Ben- telila I t 995) spadied Spain, a id Decressin and Fa:& (1995) studied Enaropean regiol-us. The I-esu:ts suggest that migrs- tlon plays a key Tole in h e Vnired States so that regiena; iebor-demand shocks nave onaiy a small transitory efkct oil regional s~nerq!o>!rnent. In Europe, however, it is through changes in Iabo--farce gartic;petlon-these include early re- tirement al.ld dlsekllity peiasions-that elxploy~rnaent is af- fected in the short run anc t2iorrgh migration in the long rrn. There is no !ong-?:ti9 effect on ~~nemployment in either case, S p i n is an e:ncepliol. according to JirYeno and Sento- 3 . rrla; ;!nernpIoyrnect ~espcnds mcre to labor-ciernand shocks.
Bianchi and Zoega (1996) used sinrilar com~entionai ~~ethudology to look at regional ~ir~employment data for Britain to measure tile persistence of relative t~ne~xploy- meni rates in the 10 regions anci the response to changes in regional labor- demand. We ~xrmeasured steady-state unem- ployment rates for each of the regior,~, and the speed of adjustzen: toward these steady states following regional shocks. Regional unernploynent appears either to be non- statlorary or. if there is any convergence over rime. St is ex- tremely slow. The point esiirne'ies imply that, if e~ncmploy- aenr in any one regloia is 5% higher than its steady-state value (i.e., 500 basis poifits higher), the unernployme~t rete \rlilE fall by o"~J' 65 basis poiilt~ t11e first year (65%) and that it a70iald lalie ixcre :ban 12 years for unc;mll;3ioq,rncn: to return to steady state. If the rate started out 10% higher tncn the steady-s:ate value, it njol.:li take more than 17 years to return to tile iteady-state ;.ajue. Thj-~s, 8:-itisla I-egiona! -a- bor ~i~asltets appear to be much less integrated thaii labor markets in continental Europe and in the United States. lil the latter. rnigrzcion eliminates une~xployrnent diEereniials within 4-6 years.
These resulfs lend support to earlier studies of re- gional labor maxJ<eis ic Britain (see Biacliaby and Manning 1990: Hughes and 5,4,cCorn1air,k i987. 1994; Evans and b/Tc- Cormicli 1994: Pissrrides aa~d McIM!asher 1998; Pissarides and VTadsworth 1989; Jackman and Sa~~our i 1992: ZPecca.,rel. 1994;. Pissarides and l~ch4akaster. usi~llg in~erregional xi- gration data. forind rhx miratio;? xespo:?ds very slou.ly re diEerences in regional u~aax;~ioyr~eni. Their rescZts i m ~ l y that it can take more than 2C years fol- an smemployment differential in a depressed region to disrgpeai-. Evans and McCormick loand an in:egrated :abor- nlarhel lor nonmarn- iaals in which migration equaJYzes r-egiorlai ur:e~nr;i~?;~mcnt
F> 1999 American Siz':Estica! Wsscc;s,ian - Journal of Business % Ecoi~omic BletislEes
arad its sc!~anges iasT longer. &pri3 "99, i!o!" 17, NO. 2
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rates? but they found the market for manuals to be local- ized with persistent unemployment differentials across re- gions and hardly any interregional migration in response to unemployment differences.
The objective of this article is to look more closely at re- gional labor-market dynamics in Britain using nonparamet- ric methods. In doing so, we attempt to pro~iide an alterna- tive methodology for analyzing regional developments. This involves an analysis of changes in the entire cross-section distribution of relative unemployment rather than looking at simple summary statistics of the distribution. This type of an approach was advocated by Quah (1996a.b) in the context of the literature on economic growth and has the advantage of accounting for mobility patterns and intradis- eribution dynamics.
In this article, we follow a similar approach to that of Bianchi (2997) of modeling the entire cross-section distri- bution by nonparametric density estimation. We use Lter- nel density estimation to estimate the probability distribu- tion of relative unemployment rates across 64 counties for the years 1981-1995. This enables us to test for the exis- tence of modes in the distribution and to observe changes in the variance of relative unemployment during this period. Moreover, we Book at movements of counties within the distribution and calculate transition probabilities of leaving states of different relative unemployment.
The organization of this article is as follows. Section 1 de- scribes the statistical framework (our definitions of shoclcs of nnernpioyment transitions. nonparametric density esti- mation, the construction of bootstrap confidence intervals and classification analysis, the analysis of intradistribution dynamics. etc.) Section 2 has the empirical results. Section 3 concludes.
1. THE STATiSTICAI FRAMEWORK
1.1 Intradistribution Dynamics
We have longitudinal data on unemployment rates for n = 64 counties from 1981 to i995 (Source: Employmerzt Cazzetle, various issues.) We are interested in establishing empirical facts about shocks to their relative unemploy- ment. TO this end, we construct a formai statistical def- inition of shocks based on ~Iassification analysis and the concept of intradistribution dynamics.
c1 c2 5
Figure 1. Classification Analysis.
W'e denote by xi the ratio of nnemployment in county i to the aggregate British unemployment rate and by f (x) its probability distribution at time t. In Figure P it is assumed chat we know the probability distribution of the data, and we fix a value of o: to define the area in each tail of the distribution. This allows us to identify two crifcai values, cl and c2, on the real line such tkat
Then. we define county i to be in the state of
Low Unemployment = So gh -x < .c2 I c,
Average Unemployment = SI + cl < x, 5 c2 High Unemployment = S2 @ c2 < x, < 3~
We construct an indicator variable for county z at time t . I,(t). which takes the values 0, I. and 2 (respectively, the states of low. average. and high unemployment); that is.
By doing the preceding classification analysis for each county (i = 1,. . . . n) and every year ( t = 1951,. . . ,1995), we are able to focus on intradistribution dynamics. We now have the following definition:
Dejnition. A county 2 is hit by a positive shock at rirne t if I,(t + 1) < I , j t); it is affected by a negative shock if I% ( t + 1 ) > 1% ( t ) .
7.2 Nsnpararneiric Density Estimation and Classification Mnaiysis
Given the data and a positive value cr; inference on in- tradistribution dynamics requires us to know the true prob- ability f(x) As a matter of fact, the true probability is never known, but we can sepiace f(x) by a nonparametric estimate
where h > 0 is the bandwidth (governing the degree of smoothness of the estimate, with larger values of h pro- ducing a smoother density estimate) and K ( u ) = l/d%. ex-p(-1/271,~) being the Gaussian kernel (see Silverman 1986; Hkdle 1990).
In this way. we obtain esti~nntes El and E2 of the criti- cal values, whereas our classification analysis depends on the "true" critical values el and C> MJe therefore construct confidence intervals for el and c2; that is.
where c is the coverage probability (e.g.. r = 30). The situation is summarized in Frgure 2. which shows
two regions of indetersninancy; if x, is In the interval ig,. E I ]
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Bianchi and Zoega: Regional Unemployment Dynamics in Britain
Indeterminancv
Figure 2. Classificaiion Analysis With lndeterminancy Regions.
or [c,, cz] . it is unltnown whether county i should be allo- cated to state So or .S1. Counties with unemployment ratios smaller than c,, howeverl are allocated to the state of low unemp!oyment with probability c; and counties with unem- ployment ratios bigger than ca are allocated to (he state of high unemployment. In other words; with probability c, a county i .with El < i, < c, at time t but x, > C:, at time t+ 1 can be defined to have been affected by a negative shock at time t. For a county s in the same situation at time t (i.e., with r - ~ < dS < g2) and with z, > g2 but z, < Cz at time t + 1, however, the switch from S1 to S2 could be generated by noise rather than by a genuine negative shock. A more complete treatment of the problem would require a joint. rather than pointwise. confidence interval for the quantiles across time.
Given the preceding considerations, we define the five areas in the distribution according to
where 5 represents the code of the indicator in the regions of indeterminancy (regardless of whether we are in between So an;d 4, or SI and S2).
As the density of the data is estimated nonparametrically, we ;Lase the bootstrap apprsach to construct the confidence intervals. This means fhat, given an optimal bandwidth h calculated using the plug-in method of Sheather and Jones (1991), we resample with replacement from the original data. By 'bpgtimal," we Eean the minimization of the trade- oiT betwee2 the bias and the variance of the estimator in an AlMISE (asymptotic mean integrated squared error) sense, as, for example, in the work of Park and Turlach (19921, Sheather and Jones (1991), and Bianchi (1995). Due to an efiec~ of the Gaussian lternel, bootstrap samples drawn from f h have a variance larger than the sample variance of the data; so the following transformation is required (see Efron and Tibshirani 1993, pp. 231 and 234, for details):
where y* = ( y f ; . . . : y:)' are sampled with replacement from x = (zl,. . . . z,,,jl; y* = mean(y"), 6' is the sample variance of x; and e, are standard normal variables gener- ated by the computer.
The construction of the confidence intervals and the im- plementation of the classification analysis can be summa- rized as follows:
0. Select bandwidth h from the sample data using a data- driven bandwidth selector (e.g., Sheather and Bones 1991).
1. Draw B bootstrap samples y* of size n from x by sampling with replacement.
2. Define the rescaled bootstrap samples x* as in (5). 3. For each bootstrap sample x", estimate the den-
sity fh(xr) and, given a, obtain the critical value pairs {e;(b). EZ ( b ) ) , B _ , ~
4. Derive the confidence Piminits c,; 4 by raking the {(B - Bc)}th and the ( 3 ~ 1 t h largest of the B replicates E ; , k = 1 .2 .
5. Construct the indicator I , ( t ) as in (4) for i = 1,. . . . n and t = 2, . . . , T [this gives an rL x (T - 1) design matrix with elements 49, 1, 2: or 51.
6. At t = 1: fix tir, = median{E~(b))f=,, for k = 1 ;2 . and classify the ith county in So if x, 5 E l , Sz if x, > t2, or S1 otherwise.
7. For t = 2 , . . . T, allocate counties falling in the in- determinancy regions to the same state they were at % - 1; this gives an n x T design matrix, D, with elements 0, 1, or 2. A move from one indeterminancy regiufi to another indeterminancy region is recorded, however: as a change in state.
The design matrix D summarizes most information on intradistribution dynamics that is relevant for making in- ference about transition probabilities (these describe the probability of a county leaving one state of relative unehn- pioyrnent for another). For these, we analyze the col~tmizs of the matrix; at each point in time, from t = 2 , . . . , T, we count the proportion of counties that moved from SI to S,, for 1 : rn = 1,2,3, between t - 1 and t . This will en- able us to compare the transition probabilities for the two tails. We will also do the analysis for five states. This will allow us to address the issue of 8 convergence: This im- plies that counties with high (low) relative unemployment should have higher transition probabilities to states of lower
Figure 3. Example of Tvvo Extreme Cases of Intradistribuiion Gy- namics.
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208 Journal of Business & Economic Statistics, April 1999
(higher) relative unemployment tha.n counties with higher (lower) unemployment rates.
To address the issue of a convergence, nonparametric density estimation cam be used first to test for the number of modes in the true probability distribution f (x) . It is of interest whether the counties can be grouped into high- and low-unemployment areas in that way. Thus, it is possible that certaic regions of the country :~ave high mean rrnem- ployanent while others have a much lower mean rate. A for- mal iese for unimodaliey can easily be implemented by the bootstrap method. as discussed by Silverman ( 4 98 1, 1983, 9986) and Efron and Tibshirani (1993). A semmmary of the proce~ure is reported in Appendix A. kiiavlng detected the fiuanber of modes, we then measure changes in the variance across the co~nnties over time. If the variance is falling over time, we have a case of a convergence. This might either suggest the operation of adjustment mechanisms. such as intercounty labor migration, or changes in the geographical distribution of shoclts.
Finally, we can also record the a7ating of the different shoclts and. by looking ah the kernel density estimates for these daies. check whether positive and negative shoclts oc- curred at dlfferent times.
1.3 Generalization and Model Selection
Before turning to the empirical analysis of unemploy- ment data, we briefly discuss the advantages of our sta- tistical methodology. Vie also generalize the model to an arbitrary number of states and discuss the choice of the model's parameters.
The advantage of nonparametric methods in the context of our analysis is clearly flexibility, insofar as virtually any shape of the density can be accounted for by the method. Regardless of whether the empirical hiistribution is well ap- proximated by a Gaussian distribution or has fat tails, skew- ness. and/or kurtosis, nonparametric density estimation will automatically estimate the different shapes. &'sing paramet- ric methods, on the other hand, one would have to try a
Figure 4. Top: Uneml~loyment Rates in Percentage Points in 64 U.K. Counties (left) ano' in the United Kingdom (right) From 1981 -1995; Bottom: Unemployment Ratios (left) I/l/iih the Corresponding Boxplots (right).
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Bianchi and Zoega: Regional Unemployment Dynamics in Britain 209
variety of parametric families and choose the family that best fits ;he data in cach year. Moreover. It is very un- likely that the way conpararnetrics are implemented may Pnfiuence the results: it is avell known that the choice of the kernel %action is 3ot of great significance and that fhe choice of tire bandwidth need no5 be subjective because data-driven methods svi'rh optimal statistical properties are readily avaiiabie-scch as, for example, the plug-in method of Slaeatel- end Jones (I 99?). This method is statistically op- timal in the S G ~ S C of miniinizing the mean squ2.red er:ror of the estinetor. Park and Turlach (1992) reported the results of a comprehenskve simnlation stady showing the excellent perforinance of the SJ bandwidth selector in s~nall samples.
The choice of the nursiber of ennemployrnent states and the areas in the tails of the distribution, hov~ever, deserves a mcre detailed explanation. The analysis presented in pre- vious sections (the algorithm in 1.2) is based. in fact, on a threefo9d classification of unempioyment (high, low, aver- age) with a gizm value of a. This is only for expositional purposes because the underlying mode? can be generalized to include a higher :lumber of states with diEereint proba- bility areas in each state. The generalization requires us lo carefully consider ~ w o exzerine situations depicted in Figure 3 :p. 207;. The first situation is :;ha; of a c o ~ i ~ t y moving from a state of Iow (high) to a stale of average unemployment in a. glven time period-c tnovement from u to b in Figure 3- to revert back to the low (high) xnernpIoyment state next period-movement from h to C. In both cases, we have very snral movements jn tl7e neighborl~ood of the cat point cl. which, inrui;ian suggests. should nor be recorded as gen- uine siate transitions. Such spurious transitions will not be detected 13 our fraa~euvork thanks to the construction of the indetemk~ancy regions obtained fl-om bootstrap confidence intervals. Whcn a county juaps within the state of alerage ~~nemploysnent fa-om a level close to el to a Ievel close to c3? however, this may be a c a s e for concern. The move-
ment from d to e in Figure 3, in fact, should bc detected as a truly genuine transition in our analysis of intradistri- bution dynamics, but it may not be detected in practice. 111 the context of a three-state model. the risk of failing to detect a large movement of this kind is higher xhe lower the value of a. For this reason, the largest possible value of a minimizing the distance between the two cut points cl and cz. a = li3. should be selected to minimize :;his risk. An alternative may he to allow for a higher number of ~nemployment states, such as, for exzmple, 5 (lower, low, average, high. higher) instead of 3. This will enable us to analyze richer patterns of ir~tradistribution dynamics.
To summarize. we can represent our general statistical model by the parameter set -11 = ( h . S. al.. . . .as. c). where h is the bandwidth for the nonparametric density estimate of (relatiire) unen-ployanent races; S is the lluill- 'ser of unemploymeni states; a, is the probsbility area for
S the sth state. with C,=, a, = I; and c is the coverage probability for the confidence intervals. He is clear from the preceding discussion that there is a natural choice for most of our parameters. which is as follo\.vs: ii = hsJ. where hsJ is the bandwidth selected by the method of $heater and Jones (199!), ai-rd nl = n 2 = a3 = 113 or a1 = = ~ $ 3 = a4 = as = 115. The choice of the cover- age probability c for the construction of confidence intervals remains more subjective because any number between .80 and .95 could be selected. Nevertheless, the closer the value of c is to unity, the wider the confidence interval for the cut points. A value of r equal to .85 or .90 may lead. therefore, to an overlap of the indeterminancy regions. In our applaca- tion, we have selected a value of c = ,230 to avoid this, but we also check the robustless of our results by recakcniating them with larger values of c such as 3 5 and .90.
Foi'lowing tine statistical methodology &scribed in Sec- tion 1, we take a look at the levels of unemployment in
- Table I. List of Counties
- S o ~ ~ l h East South West Yorkshire and Humberside Wales
1 Bedfordshire 16 .Avon 32 South York. Met. 45 Clwyd 2 Berkshire 17 Cornwail 33 West York. Met. 46 Dyfed 3 Buckingharnshire 48 Devon 34 Humberside 47 Gwent @> East Sussex I9 Dorset 35 North Yorkshire 48 Gwynedd 5 Essex 20 Gloucestershire 49 Mid G!amorgan 6 Gretiter London 24 Somerset North West 50 Powys 7 Hampshire 22 Wiltshire 36 Greater Manchester 51 South Glamorgar? 8 Mertfordshiie 37 ivlerseyside Met. 52 West Glamorgan 9 Isle of Wight West Midiand 38 Cheshire
10 Kent 23 lli. Midlands Met. 39 Lancashire Scotland 1 1 Oxfordsnire 24 fiereford and Worcester 53 Borders 12 West Sussex 25 Shropshire North 54 Central
26 Siafiordshire 40 Cleveland 55 Dumfries and Galloway East Angiia 41 Cumbria 56 Fife
4 3 Cambridgeshire East Wlidlard 4-2 Durham 57 Grampian 14 Norfolk 27 Derbyshire 43 Norlhumberland 58 Highiands 15 SuRolk 28 ieicestersh~re 44 Tyne and Wear Met. 59 Lothians
29 Lincoinshire 60 Qrkneys 30 Northamptonshire 61 Shetlands 34 INottinghamshire 62 Strathclyde
63 Tayside 64 Western isle
NOTE- Surre), (South East) and VVsrwlckshlre (West Mldiands) are not ,ncluded due to wiss'ng observations for some years
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Figure 5. Kernel Density Estimates for the Unemployment Ratios of 64 Counties in Different Years Using the Bandwidths Reported in Table I.
64 counties from 1981-1995. The data are shown in Figure 4, page 208 (top panels)--the county unemployinent rates in the left panel. the aggregate unemployment rate in the right panel. Because we are interested in looking at the un- employment problem in different counties relative to the national aggregate, the counties' unemployment rates are divided by the British aggregate; this leads to the series plot- ted in the bottom left panel of Figure 4. Of course, we could have used absolute unemployment rates rather than relative unemplojiment rates and this would not have changed the results of our analysis. In fact, densities of absolute and rel- ative unemployment rates are identical. apart from a scaling factor. En any given year the absolute unemployment rate can be derived from the relative rate by multiplying every observation by the average British unemployment rate. But this would just shift the mode of our density.
The bottom right panel shows the boxplot representation of these series. which presents the distribution of the data in different years. A drop in the median of the distributions (represented by the horizontal line in the box) can be no-
ticed after 1982 to a value closer to unity; there is also a significant reduction in the dispersion of the distributions (represented by the size of the box) over the last 4-5 years. For most years. the densities appear somewhat skewed to- ward large values but with very few outliers.
For the unemployment ratios, the bandwidths selected by the method of Shearher and Bones (see Table 1) give the density estimates shown in Figure 5 . The probability distribution is unimodal in all years. Formal multimodality tests using nonparametric kernel density estimation and the bootstrap reject ~xultimodality in all years-see the results reported in Appendix A. There is a clearly visible fail in the mean (relative) unemployment in the early 1980s. The fall in the variance around 1990 and the subsequent increase in the first half of 1990s is a reflection of the regional distri- bution of shocks. Both the recovery in the late 1980s and the recession in the 1990s were concentrated in the South.
The densities, together with a fixed value for the area in the tails of the distribution, a = 1/S with S = 3 and S = 5, give the intradistribution dynamics in Tables 2 and 3.
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Bianchi and Zoega: Regional Unemployment Dynainics in Britain 21 I
Table 2. Results of our Classification Analysis I\lonparametric Density Estimation With Bandwidths h Reported in Table A. 7, Appendix A, S = 3, a = 1/3, c = 30, and B = 1,000.
County 87 82 83 84 85 86 87 88 89 90 91 92 93 94 95
1 I 5 5 5 5 5 5 0 0 5 1 1 4 1 1 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 5 0 0 0 4 1 1 1 I 1 1 5 5 5 1 5 2 2 2 2 5 1 1 1 I 1 1 1 5 5 1 4 2 2 2 5 6 0 0 5 - c 5 5 5 1 1 1 1 2 2 2 2 7 0 5 0 5 5 0 5 0 0 0 5 1 I 5 5 8 0 0 0 0 0 0 0 0 0 0 0 1 5 5 0 9 0 4 4 ? 1 1 1 4 1 f 5 2 2 2 2 10 1 1 1 1 1 1 1 5 5 5 1 5 5 5 5 11 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 12 o o o a o o o o o o o o o o o
NOTE: Flve represents the outcome for the ndetermtnancy regton. Horizontal lines In the table distinguish the 10 Britlsh regions
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21 2 Journal of Business & Economic Statistics. April 1999
Table 3. Results of our Classification Analysis by IVonparametric Density Estimation With Bandwidths h Reported in Table A. I , Appendix A, S = 5, a = I i 5 , c = .80, and B = 1.000.
County 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95
NOTE Horizontal lines In the tabe distlngush the 10 British repions lnoetermlnancy reglons have been allocated to unernoloyrnent stares
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Bianchi and Zoega: Regional Unemployment Dynamics in Britain
Figure 6. Plot of Ur7em,oloj/,neni Ratios for 10 Counties Us~ng S = 3. Note: The x symbols mark the indeierminancy region for c l ; ine + s!/rnbols represent the indeterrninancy region for c2.
Figure 4 plots the intra,disrribution dynamics for 11) coun- tles in the case of three states, one for each of the 10 regions-we ncce Chat the seven metropolitan districts are, strict"; speaking, not counties because each has more than one local government. The first panel shows the iatradistri- bution dynamics for Greater London. London starts out in the area of low nne:np!symeni but in 1983 enters the zone of indeaerminancy 'sezween "'how" and "normal" unemploy- naent. In 4988 it enters the area of normal unemployment and finally in 1993 the area of high unemployment.
Table 4 sssmvnariaes the dating of shocks. Figure 7 then shcws the density estimate sf the dales of the shoclts. I[ appears ;hat positive shoclts occurred most Gequently around 1989. whereas most negative shocks occurred in 1981 and 1991. Thus posi',ive shocks to relative unemploy- ment tended to bc most freqaent during the boom^' years
of the late 1980s but negative sl~osks tended to 36- cur in h e recessions of the early 1980s and the early 1990s.
It is interesting to exaxine whether it is more or less likely that a given county finding itself in the left tail of the distribution moves to the center than fm a similar county in the high-unemployment sight tail of the distribution. The 3 x 3 matrix in the top part of Table 5 has the transition probabilities between three states, which is caiculated as the average probability over the 14 years. The first row refers to So, the second row to SI, and the third to S2. The first nuxber ir, row 1 shows the probability that a county in region So in year t - 1 will remain in that same state in year- t . The second number is the probability of moving from state ,9b to S1, and the last number is the probability of moving to state S2.
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214. Journal of Business & Economic Statistics, Apri! 1999
Table 4. Summary Results From the Ciassif~cation Analysis Concerning the Dating
Positive shocks Negative shocks
Index County Date Index County Date
1 Bedfordshire (SE) 1987 1 Bedfordshire (SE) 1990 8 Hertfordshire (SE) 1994 4 East Sussex (SE) 1991
4 9 Dorset (SW) 1988 5 Essex (SE) 1994 25 Shropshire (WM) 1986 6 Gr. London (SE) 1987 30 Northamptonshire 1987 6 Gr. London (Sf ) 1991
(EW 36 Greater Manch (NW) 1994 7 Hampshire (SE) 1991 38 Cheshire (NW) 1983 8 Hertfordshire (SE) 1991 41 Cumbria (N) 1990 9 Isle of Wight (SE) 7981 43 Northumbershire (N) 1987 9 lsle of Wight (SE) 1991 45 Clwyd (Wales) 4989 19 Dorset (SW) 1990 50 Powys (Wales) 1990 20 Gloucerstershire 1992
(SW) 52 West Glam (Wales) 1991 21 Somerset (SW) 1990 53 Borders (SC) 1990 31 Nottingham (EM) 1993 54 Central (SC) 1991 41 Cumbria (N) 1993 56 Fife (SC) 1982 53 Borders (SC) 4 988 56 Fife (SC) 1985 57 Granpian (SC) 1987 57 Grampian (SC) 1989 58 Highlands (SC) 1983 60 Orkneys (SC) 1990 58 Highlands (SC) 1990 63 Tayside (SC) 1991 60 Orkneys (SC) 1981
NOTE: SE = South East. SW = South West. WM = West Midlands, EM = Easi Midlands. N = North NW = North West. SC = Scotland
We see that changes in the relative county unemployment rates are very persistent. The probability that a county stays in the current state between ally two years is 96.8% for So, 94.9% for S1, and 96.7% for state S2. The probability of getting out of a bad state, Sz, is only 3.3% and about the same as the probability of getting out of the good one, 3.2%.
The 5 x 5 matrix in the middle part of Table 5 has the analogous transition probabilities for the case of five states with a = 1/5. Again we find that relative unemployment is very persistent. We also find, however, that a county with a high (Pow) level of relative unemployment has a higher probability of moving to a state of lower (higher) unem- ployment. This implies 3 convergence and is shown in the last three lines of the table. For example, a county finding
r seo 1981 1990 1905
Dmtlng ot =hose
Figure 7. Density Estimate of the Dating of Positive and Negat~ve Shocks. Note: h = 6 for the density estimate of both positive andnegative shocks: --, positive; - - - -, negative.
Table 5. Transition Probabilities With S = 3 and S = 5 States: a = 1 / S and c = .80
so $1 s2
so ,947 ,053 ST .027 ,881 s 2 .009 ,076 5-3 ,005 000 s 4 ,000 ,000
Cum prob + - 2.7% Cum prob - 5.3% 9.2%
Cum prob +. - 11.9%
itself in the second lowest unemployment state has a proba- bility of 2.7% of moving to the lowest state and a probabil- ity of 9.2% of moving to one of the three states with higher unemployment. Similarly, a county in the second highest state of unempioymeni has a probability of 5.5% of mov- ing to the state of higher unemployment and a probability of 9.2% of moving to a state of lower unemployment. Note that the probability of moving io a different state-higher or lower-is greater for the high unemployment states.
%Ve conclude that the key results of our analysis of intradistribution dynamics-the persistence of relative unemployment-is not sensitive to our choice of the num- ber of states. Using five states, however, allows us to take a closer loolc at the dynamics. We also derived the transition probabilities using coverage probabilities (c) for both three and five states of .&5 and 90, with similar results.
3. CONCLUSIONS
This article has used nonpararnetric methods to analyze unemployment persistence. We used data on unemployment at the county level in Britain from 1981-1995 to test for the number of modes in the probability distribution across the counties. We found the distribution to be unimodal in every year. Although in any given year there are counties with very high and very low unemployment rates, we do not detect significant subgroups in the data with either high or low rates.
By constructing confidence intervals for quantiles (criti- cal values) in the density that leave 33% of observations in each tail of the distribution, we analyzed the persistence of movements of counties between the three states of unem- ployment (low, average, high), corresponding to the mid- dle and the two tails of the dist-ribution. We found that the transition probabilities were the same for the two tails. This implies that the probability of falling from a state cf rela- tive prosperity is equal to the probability of recovering in a state of relative depression.
The transition probabilities confirmed a high degree of persistence of changes in relative unemployment. Thus the probability that a county finding itself in a state of high unemployment will stay in that state in the following year is around 97%. This implies that the regional adjustment mechanisms of intercounty migration and capital move-
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Bianchi and Zoega: Regional Unemployment Dynamics in Britain 21 5
malts are very weak. When using five states of unemploy- ment instead of three, we again found unemployment per- sistence but also a marked tendency for high-unemployment regions to recover and Zw+v-unemployment regions to ex- perience rising unemployment. This presents evidence in support of weak 3 convergence.
We conclude that, although (here is some evidence for the existence of 3 convergence, this is not sufficiently strong for individual counties to recover quickly following neg- a.iiwe shocks. Nor is it strong enough for the variance of unemployment across counties to fa11 over time-the ef- fect of contemporaneous shocks on changes in the variance dominate any effect of the adjustment mechanisms of labor migration and capital movement.
ACKNOWLEDGMENTS
We thank Sigbert Klinke, Danny Quah. Won Smith, and Howard Ifill for comments and suggestions. The data and the programs (written in the GAUSS programming language) are available on request from the authors. The first author acknov;ledges the l-~ospitality of the Institiit fur Statistili und 6kononqetrie at Humboldt University, Berlin. The usual disclaimer applies. The views expressed in this article are those of the authors and do not necessarily reflect those of the Bank of England.
APPENDIX A: BOOTSTRAP MbLTBBllsBDALITY TESTS
A formal mimodality test is constructed based on the concept of ci-iticai band~vidth introduced by Silverman (1981: 1983, 1986). A critical bandwidth h, is defined as the smallest possible h producing a density with at most 1.1 rmodes, which means that for a41 h < h , the estimated density fh has at Beast rn + 1 modes. This idea of critical smoothing is nalurally related to hypothesis testing and, in particular, to multimodality tests. Indeed, if the true under- lying density has two modes, a large value of hl is expected because a considerable amount of smoothing is required to obtain a unimodal dei2si:y estimate from a bimodal density. This suggests that h, can be used as a statistic to test
i-9, f ( x ) has rrL modes versuc
HI f(s) Raq more than m modes. (A.1)
Table A. I. Bootstrap Mult;modali~'y Tests With B = 1.000 Replications
Year EL (m = I ) EL (m = 2)
1981 .50 .53 1982 .20 .63 1983 .77 .57 1984 .52 .24 1985 .79 .85 1986 .76 .31 1987 .49 .27 1988 .79 .47 1989 .76 .74 1990 .60 .98 1991 .98 .84 1992 .92 .63 1993 .59 .79 1994 .88 .49 1995 .36 .36
Here, a '"large" value of h,, indicates more than m modes, thus rejecting the null. How large is large in this context is assessed by the bootstrap, as discussed by Silverman and, among others, by Hzenman and S o m e r (1988) and Efron and Tibshirani (1 993).
The steps to test for mulfnnodality can then be summa- rized as follows:
1. Draw B bootstrap samples x* of size n using (5). 2. For each bootstrap sample x", compute the critical
bandwidth consistent with rn modality, h ~ . Denote the val- ues of h:, by h,;(l), h k ( 2 ) : . . . . h & ( ~ ) .
3. Obtain an estimate of - the achieved significance level (or p value) of the test as ASL, = #{,$&(b) 2 h , , ) / ~ .
4. Fail to reject the null hypothesis of 'rn modes in the density whenever ASL,, is larger than standard levels of significance.
By implementing the preceding test la each year, we have obtained the results shown in Table A.1. In all years, we far1 to reject unimodality.
APPENDIX B: BANDWIDTH SELECTION FOR DENSITY ESTIMATION
Using the method of Sheater and Jones (1991). we have calculated the bandwidths reported in Table B. 1.
Table B. 1. Bandwjdth Selected by Sheather and Jones (1991) Plug-in Method
81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 . ..
h .22 .20 . I6 .17 .20 .19 .20 .24 .26 .22 .17 .I2 .11 .14 . I5
(Received Jzily 1996. Re1,ised June 1998.1
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