Area disparities in Britain: understanding the contribution of people versus place through variance decompositions 1

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1 Area disparities in Britain: understanding the contribution o people versus place through variance decompositions 1 Stephen Gibbons (LSE and SERC) Henry G. Overman (LSE and SERC) Panu Pelkonen (University o Sussex and SERC) Abstract: This paper considers methods or decomposing variation in wages into individual and group speciic components and applies these to British labour market areas. We discuss the relative merits o these methods, which are applicable to variance decomposition problems generally. We show how the relative magnitudes o the measures depend on the underlying variances and covariances, discuss how the measures should be interpreted and consider how they might relate to structural parameters o interest. We highlight that a clear-cut division o variation into components is strictly speaking impossible. The indings rom the empirical work show that, independent o the choice o decomposition, area eects contribute a very small percentage to the overall variation o wages in Britain. Keywords: wage, disparities, labour, variance decomposition JEL Classiications: R11, J31 1 An earlier version o this paper circulated as SERC DP0060 with the title Wage disparities: People or place This work was based on data rom the Annual Survey o Hours and Earnings, produced by the Oice or National Statistics (ONS) and supplied by the Secure Data Service at the UK Data Archive. The data are Crown Copyright and reproduced with the permission o the controller o HMSO and Queen's Printer or Scotland. The use o the data in this work does not imply the endorsement o ONS or the Secure Data Service at the UK Data Archive in relation to the interpretation or analysis o the data. This work uses research datasets which may not exactly reproduce National Statistics aggregates. Steve Gibbons, Department o Geography and Environment, London School o Economics, Houghton Street, London WCA AE, UK. s.gibbons@lse.ac.uk; Henry G. Overman, Department o Geography and Environment, London School o Economics, Houghton Street, London WCA AE, UK. h.g.overman@lse.ac.uk; Panu Pelkonen Department o Economics, University o Sussex Brighton BN1 9RF, UK. P.O.Pelkonen@sussex.ac.uk. 1

2 1 Introduction you are under no obligation to analyse variance into its parts i it does not come apart easily, and its unwillingness to do so naturally indicates that one s line o approach is not very ruitul. R. A. Fisher (1933) These warnings o R.A. Fisher notwithstanding, there are surely many situations where the decomposition o variance is potentially inormative, even i the variance does not come apart easily. This paper considers methods or decomposing the variance o an individual outcome into group components, where this individual outcome is determined by both individual characteristics and the characteristics o the group to which the individual belongs, and where individuals can sort across groups. In particular, we are interested in the importance o individual characteristics versus group membership in explaining variation in individual outcomes. Such decompositions are o interest in a wide variety o contexts. For example, labour economists have long been concerned with the role sorting on individual characteristics plays in explaining dierences in wages between groups o workers (particularly wage dierences across industries). See, or example, Krueger and Summers (1988), Gibbons and Katz (199) and Abowd, Kramarz and Margolis (1999). Another example occurs in urban economics when considering the extent to which sorting matters or individual and area disparities in wages. See, or example, Duranton and Monastiriotis (00), Taylor (006), Dickey (007) Mion and Naticchioni (009), Dalmazzo and Blasio (007) and Combes, Duranton and Gobillon (008). Education economics provides another example, where the interest is in inding out the contribution o schools to the variance o pupil test scores (e.g. Kramarz, Machin and Ouazad 009). Despite this widespread and growing interest in decompositions by geography, industries, irms, schools and other groups, the alternative methods and their interpretations have not been very clearly articulated in previous literature. The irst part o this paper discusses several methods or gauging the contribution o group eects to the dispersion o outcomes across individuals, where these group eects are treated as ixed eects. To make the discussion concrete, we ocus on wages as the individual outcome, and consider the role o individual characteristics and a group eect associated with the labour market By this we mean that we will allow the eects o area and individuals to be correlated. Researchers in some disciplines like to use multi-level/hierarchical models (Goldstein, 010) to provide variance decompositions, but these estimators assume that the various levels (e.g. areas, individuals) are uncorrelated random eects, which does not seem at all desirable in many contexts.

3 area o the individual. The methods we discuss are based on decomposing the variance o individual wages into between-group and within-group components in order to estimate the share o the individual variance attributable to group eects. These methods are easily extended to consider the share o group level variation attributable to group, as opposed to individual, eects. We consider the relationships between these dierent methods and how their results might be interpreted. The approaches that we consider are not novel, but the issues surrounding their application have not received careul attention in the literature. The second part o the paper applies these decompositions to data on employee wages in Britain. The results here illustrate that area eects make only a small contribution to overall wage inequality, and that observed disparities between mean wages in dierent areas are primarily due to sorting. Variance decompositions Assume we have data on wages w i and individual characteristics x i and that individuals are located in one o J areas. 3 Consider the (Mincerian) linear regression model or log wages: ln( w) x d (1) where are random errors, d is a J x 1 column vector with jth element equal to one i the worker is in area j and is a J x 1 vector o parameters. Variable d is the vector o 'area eects'. The vector o individual characteristics x could consist o observable variables (e.g. age), or be a vector o individual dummy variables, so that x is an 'individual eect. 4 Unless needed, we suppress individual i (and time t) subscripts and all constant terms (so variables are in deviation-rom-mean orm). It simpliies notation to deine d and x and rewrite equation (1) as: ln w () Our aim is to assess the contribution o area eects d to the variance o ln w. The area eect associated with an area j gives the expected gain in log wages that a randomly chosen individual could expect rom being assigned to area j (relative to some arbitrary baseline area). 3 We assume that individuals live and work in the same area. 4 A more general extension to panel data is simple and discussed in Appendix A. 3

4 .1 Decomposing the total variance Estimation o (1) by OLS (or some equivalent method) gives coeicient estimates ˆ, ˆ and residuals ˆ. We can decompose the variance o ln w as ollows: Var(ln w) Var( ˆ ˆ ˆ ) Var( ˆ ) Var( ˆ ) Cov( ˆ, ˆ ) Var( ˆ ) (3) where Var. is the sample variance,. ˆ y dˆ xˆ. Dividing through by Var(ln w ) gives: Cov the sample covariance, ˆ d ˆ and ˆ x ˆ and Var( ˆ ) Var( ˆ ) Cov( ˆ, ˆ ) Var( ˆ ) 1 (4) Var(ln w) Var(ln w) Var(ln w) Var(ln w) Using R (ln w; x, d ) to denote the R-squared rom a regression o ln w on x and d we get: Var( ˆ ) Var( ˆ ) Cov( ˆ, ˆ ) Var(ln w) Var(ln w) Var(ln w) R (ln w; x, d) (5) Note that R (ln w; x, d ) provides one estimate o area share o the variance o ln w i we assume that all variance in individual characteristics (including within-area variation) is caused by area. In most contexts this assumption is surely too extreme although, as is well known, the analysis o variance cannot be used to assess the validity o such causal assumptions.. The Raw Variance Share A more plausible upper estimate o area share is obtained i we assume area causes any dierences in average individual characteristics in each area. Then, the contribution o area is: Var( ) Var(ln w) R (ln w, d) (6) where d, is the coeicient vector and R (ln w, d ) the R-squared rom the regression o ln w on area dummies d, with no control or x (contrast ˆ d ˆ rom equation (1)). We reer to this as the raw variance share or RVS. RVS is simply the between-group variance in log wages divided by the total variance in log wages. I average individual characteristics vary by area and are 4

5 correlated with, but not caused by, area eects then is an upward biased estimate o and RVS provides an over-estimate o the area share o variance..3 The Correlated Variance Share A second possibility, based on the decomposition o total variance in equation (4) is: Var Var ˆ ln w (7) where ˆ d ˆ comes rom a regression o ln w on area dummies d and individual characteristics x (i.e. OLS estimation o equation (1)). That is, the area share is the ratio o the variance o area eects to the variance o log wage. This seems like an intuitive measure o the contribution o area eects. Note, however, that interpretation is complicated because even though equation (1) controls or individual x, area eects ˆ will still be correlated with x and ˆ i there is associative sorting across areas. This means that Var( ˆ ) / Var(ln w) excludes the direct contribution o sorting (which enters through the covariance term in equation (5)), but captures any indirect eect that sorting may have on the variance o the area eects themselves. That is, it captures the contribution o area eects including those induced by area composition but not the eects, i any, that area has in determining individual characteristics. We show this in more detail below in the section on Interpretations. For this reason, we reer to this measure as the Correlated Variance Share or CVS. The ratio Var( ˆ ) / Var(ln w) provides an analogous CVS estimate o the variance share attributable to individual characteristics, conditional on the area eects..4 Uncorrelated Variance Share We can also estimate the contribution o the components o area eects that are uncorrelated with individual characteristics. There are a number o equivalent methods or doing this that give a statistic that is oten reerred to as the semi-partial R-squared (and by Borcard and Legendre (00) as "Fraction a" partitioning). It is also related to the partial R-squared, as shown in Appendix B. We reer to this as the Uncorrelated Variance Share or UVS. Analogous methods can be used to estimate the corresponding UVS attributable to individual characteristics, but we describe only the process or area eects below. 5

6 .4.1 R-squared method (a) Regress ln w on x and d to get R (ln w; x, d ). This measures the share o overall variance explained by both individual characteristics and area eects; (b) Regress ln w on just x to get R (ln w; x ). This gives the share o overall variance explained by just the individual characteristics; (c) Calculate UVS R (ln w; x, d) R (ln w; x)..4. ANOVA (partial sum o squares) method ln ;, ln ; TSS ln w RSS ln w; x, d RSS ln w; x UVS R w x d R w x (8) where RSS is Regression Sum o Squares and TSS the Total Sum o Squares, so the numerator is the Partial Sum o Squares o d. These sums o squares are routinely computed by ANOVA sotware..4.3 Partitioned regression method (a) Regress ln w on x and d and predict ˆ d ˆ and ˆ x ˆ ; (b) Regress ˆ on ˆ and obtain a regression coeicient estimate ˆ and the uncorrelated residual area components ˆ ˆ ˆ ˆ ; (c) Regress ln w on the residual ˆ rom (b) and calculate UVS R ln w; ˆ. The UVS measures the amount o variation in log wage that is explained by the part o the area eect that is uncorrelated with individual characteristics that are included in x. Appendix C shows that these three methods o calculating the UVS are equivalent. Appendix D also provides a decomposition o total variance similar to equation (4) but based on the UVS rather than the CVS..5 Balanced Variance Share To consider the explanatory power o individual characteristics and group membership, Abowd, Kramarz and Margolis (1999) suggest reporting the standard deviation (across workers) o the eect o a variable and the correlation o the variable with wages as yet another variance analysis. The eect o a variable is calculated by multiplying the coeicient on that variable by the value o that variable or each observation. The variable is said to have large explanatory power when the eect has a large standard deviation and is highly correlated with wages. Combes, Duranton and Gobillon (008) perorm such an analysis to consider the importance o individual versus area in explaining area wage disparities. 6

7 These two components (the standard deviation o the eect and the correlation with wages) can be used to construct another variance share measure. Using ˆ d ˆ, ˆ x ˆ standard deviation and r. the Pearson correlation coeicient note that:,. s to denote sample ˆ ˆ s ˆ s Cov ln w, Cov ln w, ˆ r ln w, ˆ r ln w, s ln w s ln w Var ln w Var ln w ˆ ˆ Var ˆ Var Cov ˆ, ˆ R Var ln w Var ln w Var ln w ln w; x, d (9) Hence: rln w, ˆ ˆ s ˆ Cov ln w, ˆ Var ˆ Cov, ˆ (10) s ln w Var ln w Var ln w provides this other measure o the area share o the variance o ln w. An analogous equation to (10) would yield a measure o the individual share in the variance o ln w. The decomposition simply apportions the sorting component, Cov( ˆ, ˆ) in equation (4), equally to area eects and to individual characteristics, so we call this the Balanced Variance Share (BVS). Note that there is no obvious justiication or splitting the covariance in this way, apart rom to guarantee that the area and individual variance shares add up to the total R-squared R (ln w; x, d )..6 Contribution o area eects to area disparities So ar, we have ocused on deriving the contribution o area eects to overall wage disparities. We can, however, use the RVS, CVS, UVS or BVS to back out the contribution o area eects to area, rather than overall individual disparities. The ratio CVS/RVS or Var( ) / Var( ˆ ) rom equations (6) and (7) gives the contribution o area excluding the direct contribution o sorting. The ratio UVS/RVS or (ln ;, ) (ln ; ) / (ln ; ) R w x d R w x R w d gives the contribution o area excluding both the direct and indirect contribution o sorting. BVS/RVS provides a inal measure which adds in an additional contribution attributable to the covariance. The interpretations ollow directly rom the properties o CVS, UVS and BVS. 7

8 3 Comparisons and interpretation We have outlined several methods or calculating the individual and group shares o variance. In this section we consider the relative magnitudes o the measures and discuss how their interpretation depends on the underlying determinants o individual dierences in outcomes. 3.1 Magnitudes In Appendix C (equation A5), we show that (ln ; ) (1 ˆ ˆ ) ( ˆ) / (ln ) where ˆ R w x Var Var w is the coeicient rom the regression o ˆ d ˆ on ˆ x ˆ. An identical derivation gives: (ln, ) (1 ˆ ˆ ) ( ˆ) / (ln ) ˆ ˆ ˆ ˆ RVS R w d Var Var w ((1 ) Var( ) Cov(, )) / Var(ln w) (11) where ˆ is the coeicient rom a regression o immediate that ˆ x ˆ on ˆ d ˆ. Comparing to (7) it is RVS (1 ˆ ˆ ) CVS. Note that unless ˆ 0or Var( ˆ ) 0, RVS CVS. That is, when individual characteristics are positively correlated with area eects (i.e. ˆ 0 ), RVS is greater than CVS. In other words, the R-squared rom a regression o ln w on area dummies will attribute to area eects some o the contribution o individual characteristics. The same is true when sorting is negative, and strong relative to the variance o area eects (i.e. ˆ Cov( ˆ, ˆ ) / Var( ˆ ) ). In the range ˆ 0, negative sorting tends to cancel out the contribution o area eects when estimated using RVS. When ˆ 1, negative sorting exactly osets area eects, so regression o ln w on area dummies would yield an RVS o zero, whereas CVS would be positive. Appendix C (equation A9) shows that the uncorrelated variance share is: ˆ ˆ Var ˆ Var UVS (1) Var ln w where ˆ is the coeicient rom the sample regression o ˆ d ˆ on ˆ x ˆ. Comparing the numerators in the UVS in (1) and CVS in (7): ˆ ˆ ˆ ˆ CVS Var Var Var UVS (13) 8

9 with equality i ˆ is uncorrelated with ˆ. Thereore the CVS gives an area share that is bigger than that obtained by the UVS. As we said beore, the Balanced Variance Share, BVS adds ˆ, ˆ Cov to the CVS. Hence, with positive covariance between ˆ and ˆ, the BVS or area share is bigger than the CVS. From equation (11) it is also clear that the RVS or area share is bigger than BVS, when there is positive sorting (covariance). With negative sorting the BVS or area share is smaller than CVS, but it may be larger or smaller than UVS or RVS. Finally, all methods give the same result i ˆ and ˆ are uncorrelated. Naturally, these comparisons ollow through to the decomposition o the variance o ln w into the components attributed to area plus observable characteristics, versus unobservable actors. For example, with positive sorting, adding up the BVS or area ( BVS( ˆ )) and BVS or individual characteristics BVS( ˆ ) minimises the share o the variance attributed to unobservables (i.e. it is the standard R-squared) and attributes as much as possible to ˆ and ˆ. At the other extreme, adding UVS( ˆ ) to UVS( ˆ ) provides a more conservative estimate o the share attributable to area and individual characteristics based on the orthogonal components o ˆ and ˆ alone, and so assigns a greater share o ln w to unknown components. Given that all o the methods give a dierent answer when there is associative sorting o individuals across groups it is natural to ask how we should interpret the dierent calculations and whether we should preer any speciic method. This depends on the underlying actors determining individual and area outcomes as we now show. 3. Interpretations In order to gain some understanding o what contributes to each o these variance shares, it is helpul to ully speciy a system o linear equations which determine individual wages, area eects and individual characteristics. Continuing to use d and x : Individual wages: ln w (1) Area eects: (14) Individual characteristics: (15) 9

10 The equation or individual wages is identical to that used earlier in Section. As shown in equation (14) we allow area eects to be determined by average area composition (where the mean o is taken across individuals, within areas) and by exogenous variation (e.g. climate). Equation (15) allows individual characteristics to be determined by area and by exogenous individual speciic variation v (e.g. innate ability). Speciying equation (15) in terms o rather than x simpliies later notation. Averaging (15) at the area level gives area-mean individual characteristics, and it is correlation between these and area eects that biases estimations o the relative contributions o individual and area eects. Part o this correlation arises because o the interdependency between and explicit in (14) and (15) when 0 and/or 0. However, there is an additional correlation, present even when 0 and 0 i Cov, 0.That is, i exogenous area eects ( ) and area-mean exogenous individual characteristics ( v ) are correlated through sorting processes that are otherwise unrelated to wages (e.g. i higher skill people preer warmer climates). Note that all this assumes that the underlying sources o variance across areas are the exogenous additive components, and rules out multiple equilibria that arise in theoretical models. Even in this very restricted world, it turns out to be diicult to provide irm statements about the share o area eects or whether RVS, BVS, CVS or UVS provides the most appropriate empirical measure. In the Appendix E, we show that equations (1), (14) and (15) can be used to get expressions or the variance o area eects, mean individual characteristics and their covariance in terms o the underlying structural errors and parameters: Var( ) ( Var( v) Var( ) Cov( v, ) / (1 ) (16) Var( ) Var( v) Var( ) Cov( v, ) / (1 ) (17) Cov(, ) Var ( ) Var ( v) (1 ) Cov, v / (1 ) (18) I area aects individual characteristics ( 0 ) equations (17) and (18) show that the exogenous component o area eects ( ) enters into Var and, Cov. Similarly i area composition determines area eects ( 0 ), equations (16) and (18) show that the exogenous component o 10

11 individual characteristics ( ) enters into Cov, 0increases Var and, Var i area eects are increasing in ( 0 i individual characteristics are increasing in ( 0) Cov. (Positive) sorting, ) and increases Var. Positive sorting also increases, Cov, with the increase greater i both 0 and 0. These results regarding carry through to the distribution o individual characteristics because Cov, Cov, and Var Var Var, where represents all the within-area individual components o wages. In short, Var, Var and, Cov, which are observable, cannot be attributed to one or other o the potentially unobservable exogenous components, that are the underlying sources o variance across areas in the simple system represented in equations (1), (14) and (15). Turning to the our variance shares, it s useul to repeat the ormulas: 5 (1 ˆ ) Var( ˆ ) Cov( ˆ, ˆ ) RVS, Var w Var w ln ln Var( ˆ ) Cov( ˆ, ˆ ) BVS Var w Var w ln ln ˆ Var Var CVS, UVS Var ln w Var ˆ ˆ Var ˆ ln w It can be seen that equations (16) - (18) can, in principle, be used to express the our variance shares in terms o the underlying structural components. It should be clear, however, that even in this very basic linear world there is no simple relationship between these variance shares and the underlying structural components. Which variance share is most appropriate depends on a complex interplay between: (a) the underlying structural component in which we are interested; (b) the magnitudes and signs o the coeicients determining area eects and individual characteristics; and (c) sorting. One simple case occurs when composition does not cause area eects ( 0 ), in which case CVS gives the variance share o the area shocks. Correspondingly, when area does not cause changes 5 In these expressions, remember that ˆ is the coeicient rom the regression o ˆ on ˆ Cov 11 (i.e. ˆ ˆ, / Var( ˆ ) not the structural coeicient in equation (15)) and ˆ is the coeicient rom the regression o ˆ on ˆ (i.e. Cov ˆ, ˆ / Var( ˆ ) not the structural coeicient in equation (14)). We ignore this complication in the text, but note that the problems o linking variance shares to structural components persist even in the unlikely case that sample estimates equal their population values.

12 in individual characteristics ( 0 ), Var ˆ Var ln w characteristics in log wages. Note too that when 0 gives the share o individual,, Cov provides an unbiased estimate o the degree o sorting but otherwise combines the covariance caused by sorting and by the causal inluence o area on characteristics (and vice-versa). In all other contexts, pinning down the contribution o speciic sources o area-level variations is not possible, but as discussed above RVS, BVS, CVS and UVS can at least be used to provide a bounding exercise to guide us as to the likely contributions. 4 Application: wage disparities in Britain In this section we use the our variance shares to look at the contribution o individual and area eects to wage variation in Britain. We improve on existing UK work on regional inequalities (Duranton and Monastiriotis (00), Taylor (006) and Dickey (007)) by working with unctional labour market areas (rather than administrative boundaries) at a smaller spatial scale and by using panel data to control or unobserved individual characteristics. Outside o the UK, a number o existing studies have already used individual data to study spatial sorting and agglomeration economies. See, or example, Mion and Natticchioni (009), Dalmazzo and Blasio (007) and Combes, Duranton and Gobillon (008). This paper is most closely related to the last o these but pays more attention to the issue o the share o disparities attributable to area eects using the insights developed above. We use data or rom the Annual Survey o Hours and Earnings (ASHE) and its predecessor the New Earnings Survey (NES). NES/ASHE are constructed by the Oice o National Statistics (ONS) based on a 1% sample o employees on the Inland Revenue PAYE register or February and April. The sample is o employees whose National Insurance numbers end with two speciic digits and workers are observed or multiple years (up to 11 years given the time span o our sample). The sample is replenished as workers leave the PAYE system (e.g. to retirement) and new workers enter it (e.g. rom school). NES/ASHE include inormation on occupation, industry, whether the job is private or public sector, the workers age and gender and detailed inormation on earnings. We use basic hourly earnings as our measure o wages. NES/ASHE do not provide data on education but inormation on occupation works as a good proxy or our purposes. NES/ASHE provide national sample weights but as we are ocused on sub-national (TTWA) data we do not use them in the results we report below. 1

13 ASHE provides inormation on individuals including their home and work postcodes, while the NES only reports work postcodes. We use the National Statistics Postcode Directory (NSPD) to assign workers to Travel to Work Areas (TTWA) using their work postcode, allowing us to use the whole NES/ASHE sample over our study period. Given the way TTWA are constructed (so that 80% o the resident population also work within the area) the work TTWA will also be the home TTWA or the majority o workers. To ensure reasonable sample sizes, our analysis divides Britain into 157 labour market areas o which 79 are single urban TTWA and 78 are rural areas created by combining TTWAs. 6 We derive estimates o RVS, CVS, UVS and BVS, rom regression analysis o individual wages, based on equation (1) above, but adapted to include individual ixed eects and time dummies to take advantage o the panel dimension o our data: ' ' ln w ijt i t x it d jt ijt (19) where j indexes areas, i are individual ixed eects, the t are time dummies, x it is a vector o time varying individual characteristics, and everything else is as in equation (1). captures the impact o individual characteristics controlling or area. Likewise, captures the impact o area controlling or both time invariant unobserved individual characteristics ( ) and time varying observed individual characteristics ( x it ). These parameter estimates, allow us to obtain estimates o ˆ d and ˆ x ˆ rom which we can calculate RVS, CVS, UVS and BVS. i Including individual ixed eects means that identiication o the area ixed eects comes rom movers across areas. As usual, we cannot rule out the possibility that shocks to it are correlated with the decision to move so the area eect estimates need to be interpreted with some caution. In the absence o random allocation (or a policy change that as good as randomly assigns people) tracking individuals and observing the change in wages experienced when they move between areas is the best we can do to identiy causal area eects. 7 Estimating the area eects rom movers means 6 We reached this classiication in three steps: a) we identiied the primary urban TTWAs as TTWA centred around, or intersecting urban-ootprints with populations o 100,000 plus; b) we identiied TTWA with an annual average NES/ASHE sample size greater than 00 as stand-alone non-primary-urban TTWA (e.g. Inverness); and c) we grouped remaining TTWA (with sample sizes below 00) into contiguous units (e.g. North Scotland). 7 Note, that we track individuals in the panel or up to 10 years, so the wage changes we observe or individuals moving between areas are not just the one-o changes when the move occurs, but include longer-run changes (e.g. due to dierences in wage growth rates between areas). 13

14 we need to assume ixed-across-time area eects. 8 We shall show, however, that dropping individual ixed eects and controlling or observables only provides area eects estimates that are very stable over time. Given this, and as individual unobserved characteristics are important or explaining wages, our preerred speciication incorporates individual eects but assumes area eects are ixed across time. Some basic descriptive statistics are provided in Appendix F. We start by estimating equation (19) with an increasing set o control variables to show how allowing or sorting across areas aects the magnitude o estimated area eects. To summarise the distribution o eects we irst report the percentage change in wages when moving between dierent parts o the area eects distribution: the minimum to maximum and to mean, mean to maximum, the 10 th to the 90 th percentile and the 5 th to the 75 th percentile. Table 1 reports results based on several dierent speciications. The irst row reports results rom equation (19) when only including time dummies (which gives the upper bound estimates o area eects). The second row reports results when the observable variables are a set o age dummies, a gender dummy and a set o 1 digit occupation dummies. The third row uses a set o age dummies, a gender dummy, two digit occupation dummies, industrial dummies (three digit SIC) and dummies or public sector workers, part time workers and whether the worker is part o a collective agreement. Results using individual ixed eects are reported in rows our and ive. In row our, we simply include year dummies and individual eects. Row ive uses individual eects, year dummies, age dummies and one digit occupation dummies. Table 1: Distribution o area eects ( ˆ ) Controls: min-max minmean meanmax p90-p10 p75-p5 Year dummies 61.7% 18.% 36.8%.0% 10.6% + age, gender, occ % 9.0% 5.4% 1.7% 6.4% +age, gender, occ-, sic-3, public, union, parttime 9.0% 6.5% 1.0% 9.7% 4.% Individual ixed eects and year dummies 0.3% 7.6% 11.8% 8.9% 4.4% + age, occ1 17.3% 5.9% 10.8% 7.3% 3.8% Notes: Authors own calculations based on NES/ASHE. Speciications are described in detail in the text (occ- 1, occ- signiy one and two digit occupation; sic-3 signiies three digit industry code). Results or 157 areas. Based on 1,511,163 observations (305, 75 individuals) 8 Theoretically, we could still allow or such year on year changes in place speciic eects, but identiying them requires movers in and out o all areas in every year which turns out to be too demanding given our sample sizes. 14

15 I we ignore the role o sorting (i.e. do not control or i and x it ) then the dierences in area average wages look quite large. Moving rom the worst to the best area, average wages increase by just over 60%; rom the minimum to the mean by a little under 0%; and rom the mean to the maximum by just over 35%. O course the minimum and maximum represent extremes o the distribution. The move rom the 10 th to the 90 th percentile sees average wages increase by %, rom the 5 th to 75 th percentile by just over 10%. Introducing a limited set o observable characteristics (row ) to control or sorting substantially reduces estimated area dierences. A larger set o individual characteristics (row 3) reduces the estimated dierences urther as does allowing or individual ixed eects, with or without additional observable characteristics (rows 4 and 5). Interpreting observed area dierences as area eects considerably overstates the impact on wages that occurs as individuals move rom bad to good areas. Speciically, across the comparisons we report, ignoring the role o sorting overstates area eects by a actor o three. As discussed above, the contribution o area eects to overall (and area) wage disparities depends not only on the size o speciic eects but also on the overall distribution o area eects and on the distribution o individuals across those areas. We suggested our alternative variance decompositions or measuring this contribution. Table reports results or the contribution o area dierences to overall and area wage disparities using these our measures. We calculate them rom regression speciications using the same individual characteristics as in Table 1. For comparison, the second panel o the table reports the contribution o the individual characteristics using the dierent measures while the third panel reports the contribution o area eects to area disparities. Table : Variance decomposition Area variance share BVS( ˆ ) Year dummies only (RVS) + age, gender, occ age, gender, occ-, sic-3, public, union, pt Individual ixed eects and year dummies + age, occ-1 (1) () (3) (4) (5) 6.0% 3.%.4% 1.1% 0.9% CVS( ˆ ) 6.0%.8%.0% 0.7% 0.6% UVS( ˆ ) 6.0%.7% 1.3% 0.1% 0.1% Individual variance share BVS( ˆ ) 57.7% 75.9% 86.8% 88.3% CVS( ˆ ) 57.4% 75.5% 86.4% 87.9% UVS( ˆ ) 54.0% 73.0% 83.6% 84.9%

16 Area share o area disparities BVS( ˆ ) / RVS 53.% 40.4% 18.3% 15.9% CVS( ˆ ) / RVS 47.5% 34.1% 1.6% 10.4% UVS( ˆ ) / RVS 45.8%.3% 1.3% 1.1% Notes: Authors own calculations based on NES/ASHE. Speciications are described in detail in the text (occ-1, occ- signiy one and two digit occupation; sic-3 signiies three digit industry code; pt signiies parttime). Based on 1,511,163 observations (305, 75 individuals). The top panel o Table, column 1 shows that even i we ignore the eects o sorting, and simply consider raw area dierences in mean log wages, areas only explain 6% o the overall variation in wages across individuals (remember that in the absence o controls or any individual characteristics the variance share o area eects is the same whichever way it is calculated). Moving across the columns in Table, panel 1, we progressively add controls or individual characteristics, culminating in a speciication with individual ixed eects in columns 4 and 5. Looking down rows within the top panel shows how the variance share changes according to the way we estimate it Balanced Variance Share (BVS), Correlated Variance Share (CVS) and Uncorrelated Variance Share (UVS). As discussed in Section 3.1, the Correlated Variance Share provides an upper bound to the combined contribution o exogenous area eects plus those induced by area composition (i.e. the area mean individual characteristics). The Uncorrelated Variance Share provides a lower bound to the contribution o exogenous area eects alone. The Balanced Variance Share could be larger or smaller than the Correlated Variance Share, depending on whether there is positive or negative assortative matching o individuals to areas. It is evident rom the act that BVS> CVS > UVS in Table, that we have positive sorting across areas. On any o these measures, the contribution o area eects is around 3% or less when controlling or basic observable characteristics (columns and 3) and around 1% or less when controlling or unobservable individual characteristics (columns 4 and 5). In short, the most striking inding rom Table is that area eects only play a small role in explaining overall individual wage disparities. We have shown (in Section 3) that it is not, in general, possible to determine rom these variance shares the exact contribution o sorting, versus the inluence o area composition on area eects, versus the inluence o area eects on area composition. However, comparison o the CVS( ˆ ) and UVS( ˆ ) measures is partly revealing. Looking at the gap between the UVS and CVS in our preerred speciications with individual ixed eects in columns 4 and 5 o panel 1 suggests that area eects arising rom area composition account or over hal o everything that can be attributed to area eects. To see this, note that CVS( ˆ ) UVS( ˆ ) is more than 50% o either CVS( ˆ ) (or 16

17 even BVS( ˆ )), in columns 4 and 5. Even so, this is still a trivial share o the overall individual disparity in wages - roughly 0.5% in our preerred speciication (CVS( ˆ ) UVS( ˆ ) in column 5. In contrast to these results on area eects, the contribution o individual characteristics shown in the second panel o Table is very large. Age, gender and occupation variables alone account or 54-58% o the individual variation in wages. Adding in individual ixed eects drives this share up to between 85% and 88% in the last column, implying that the contribution o individual characteristics is between 100 times (comparing BVS) and 850 times (comparing UVS) bigger than that o area eects. Comparison o CVS( ˆ ) and UVS( ˆ ) implies that area eects also contribute very little to the variance o individual eects on wages. The inal panel o the table shows that area eects as deined here play a somewhat more important role in explaining the area disparities captured by the simple between-area variance share (the RVS in Table ). When controlling or basic characteristics, sorting accounts or a little over hal o the observed area disparities leaving area eects to account or around 48%. Once controlling or individual ixed eects the upper bound estimate o the contribution o area eects to area disparities is considerably smaller at a little over 10% with the lower bound estimate around 1%. Even so, the dominant contribution to area disparities (one minus the share reported in the inal panel o Table ) must come rom individual characteristics, and the way these are distributed across areas. Table 3: Area eects: Results by year RVS BVS CVS UVS Pooled 6.0%.4%.0% 1.3% %.3% 1.9% 0.8% %.% 1.9% 0.8% %.% 1.9% 0.8% %.3%.0% 0.9% %.7%.3% 1.% %.7%.3% 1.3% %.7%.3% 1.3% %.5%.1% 1.% %.4%.1% 1.% %.5%.1% 1.1% %.5%.1% 1.1% Notes: Based on 1,511,163 observations (305, 75 individuals). Contribution or pooled lower than average o years because pooled regressions impose time invariant area eects and coeicients on individual characteristics. The pooled estimate o Column 1 controls or year eects. All estimates in Columns and 3 control or age, gender, -digit occupation, 3-digit SIC, public sector, union and 17

18 part-time. Table has shown these decompositions based on pooling all years in our dataset. The results in Table 3 investigate the stability o the contribution o area eects over time, and shows that this contribution has been stable. For each year, the table reports the contribution o the our area variance shares controlling or the ullest possible set o individual characteristics. For comparison, the irst row reports results when pooling across years (taken rom Table ). Repeating other results rom Table by year give igures that are similarly stable across years. As discussed above, it is this stability o area eects across time, combined with the importance o individual eects which leads us to view the individual ixed eects constant-across-time area eects speciications reported in the inal two columns o table as our preerred speciication. 5 Conclusions This paper has reviewed a number o ways to decompose variance in wages into the contribution rom individual and area speciic eects. The paper collates the methods present in the literature, discusses how dierent variance decompositions are related to each other and how they should be interpreted, and inally, highlights that a clear-cut division o variation into components is strictly speaking impossible, and that whatever method a researcher chooses, assumptions and caveats will remain. In particular, we have discussed decompositions that we named Raw Variance Share (RVS), Balanced Variance Share (BVS), Correlated Variance Share (CVS) and Uncorrelated Variance Share (UVS). We have shown why disentangling the contribution o area eects to wage distribution is complex, and why an approach based on bounds such as RVS, BVS, CVS and UVS can only provide guidance to the relative importance o area eects. The RVS is the simple between-area share o the total variance and provides an overall upper bound or the contribution o area eect on total wage variation. It counts the average dierences in worker characteristics across areas as part o the area eect. For Britain, this measure has persistently accounted or roughly 6% o the overall wage variation or over a decade. The CVS excludes the direct contribution o sorting, but captures the eect that sorting may have on the variance o the area eect. We interpreted the CVS as an upper bound to the combined contribution o exogenous area eects plus interactions-based spillovers that are linked to the mean characteristics o individuals in each area. CVS in Britain has been around %, but less than 1% i individual ixed unobservable eects are controlled or. The alternative BVS decomposition that 18

19 has appeared in previous work (e.g. Abowd, Kramarz, Margolis 1999, Combes Duranton and Gobillon 008) yields a slightly higher estimate o the contribution o area eects in our setting, because there is positive assortative matching o individuals to areas (i.e. individual wage eects are correlated with area wage eects). However, in general, this measure is quite hard to interpret, since it mixes up the variance attributable to areas with the covariance between area eects and area-mean individual eects. We argue that CVS is a more natural indicator. Finally, compared to CVS, UVS urther excludes the eects that sorting may have on the variation o the area eects, and such provides a lower bound to the contribution o exogenous area eects. With this measure, the area eects explain 1.5% o wage variation in Britain. I individual ixed eects are included, the eect shrinks to less than 0.1%, and this should be considered as the lower bound o the eect o area on variation o wages. Whichever estimate we use, our general inding is that most o the observed regional inequality in average wage in Britain is explained by sorting or people rather than places. Our preerred estimates, which include the individual ixed eects, suggest that the contribution o individual characteristics to variation in wages is between 100 to 850 times larger than the contribution o area eects. 6 Reerences Abowd, John, Francis Kramarz, and David Margolis (1999). High wage workers and high wage irms. Econometrica 67(), Abowd, John., Robert Creecy, and Francis Kramarz (00), Computing Person and Firm Eects Using Linked Longitudinal Employer-Employee Data, Cornell University Working Paper Fisher, Ronald A. (1933) in a letter to L.Hogben 5 February. Printed in Natural Selection, Heredity, and Eugenics, p.18, J.H.Bennett, Oxord: Clarendon Press, 1983 quoted on (accessed February 01) Borcard, Daniel and Legendre, Pierre (00). All-scale spatial analysis o ecological data by means o principal coordinates o neighbour matrices. Ecological Modelling, 153(1-), Combes, Pierre Philppe, Gilles Duranton and Laurent Gobillon. (008). Spatial wage disparities: Sorting matters! Journal o Urban Economics, 63(), Dalmazzo, Alberto and Guido de Blassio (007). Social Returns to Education in Italian Local Labour Markets, The Annals o Regional Science, 41(1), Dickey, Heather (007). Regional Earnings Inequality In Great Britain: Evidence From Quantile Regressions," Journal o Regional Science, 47(4),

20 Duranton, Gilles. and Vassilis. Monastiriotis (00). Mind the Gaps: The Evolution o Regional Earnings Inequalities in the U.K., , Journal o Regional Science, 4(), Gibbons, Robert. S. and Lawrence. Katz (199). Does Unmeasured Ability Explain Inter-Industry Wage Dierences? Review o Economic Studies, 59, Goldstein, Harvey. (010) Multilevel Statistical Models, 4th edition, Wiley-Blackwell Kramarz, Francis, Stephen Machin, and Amine Ouazad, (009) What Makes a Test Score? The Respective Contributions o Pupils, Schools and Peers in Achievement in English Primary Education London School o Economics, Centre or Economics o Education CEEDP010: Krueger, Alan and Lawrence. H. Summers (1988). Eiciency Wages and the Inter-industry Wage Structure, Econometrica, 56(), Mion, Giordano and Paolo Naticchioni (009). The spatial sorting and matching o skills and irms, Canadian Journal o Economics, 4(1), Oice or National Statistics, Annual Survey o Hours and Earnings, : Secure Data Service Access [computer ile]. Colchester, Essex: UK Data Archive [distributor], March 011. SN: Ouazad, Amine (008) AREG: Stata module to estimate models with two ixed eects," Statistical Sotware Components. S45694, Boston College Department o Economics. Taylor, Karl. (006). UK Wage Inequality: An Industry and Regional Perspective, Labour: Review o Labour Economics and Industrial Relations, 0(1), Appendix A: Extension to individual and area ixed eects All the methods can be generalised to allow or multiple eects h, where and are particular cases o, such that equation () becomes: 9 (A1) ln w h Then the raw variance share or the contribution o any one o these eects is: Var( ) R Var(ln w) (ln w, h ) 9 The stata routine areg or two way ixed eects (Ouazad 008), based on Abowd, Creecy and Kramarz (00), is useul or practical implementation in applications including two groups o eects (or example individual and area eects). See 0

21 where h, is the coeicient vector and R (ln w, h ) the R-squared rom the regression o ln w on dummies (or other variables) h, with no control or ~, where the notation ~ means the set o all eects not including. The variance decomposition o ln wages is: Var ˆ ˆ ˆ Cov, g Var ˆ 1 (A) Var ln w Var ln w Var ln w g g Where ˆ h ˆ, ˆ are the coeicients rom OLS estimation o (A11) and the correlated variance share = Var ˆ Var ln w The uncorrelated variance share or any one o these eects is R ln w; h ~, h R ln w; h~. Appendix B: Partial R-squared Another similar measure to UVS is the partial R-squared. This measures the share o the components o area eects that are uncorrelated with x i, in the components o ln w i that are uncorrelated with x i. The partial R-squared is: ln ;, ln ; UVS d R w x d R w x 1 ln ; 1 ln ; R w x R w x (A3) Appendix C: Proo that three ways o calculating UVS are equivalent The equivalence o the R-squared and ANOVA methods ollows directly rom the deinition o the partial sum o squares. We now show that the R-squared method and partitioned regression methods are equivalent. Let the sample regression o ˆ ˆ on ˆ be: d i x i ˆ ˆˆ ˆ (A4) Which implies, by substitution in to the Mincerian wage equation (1): ln w ˆ ˆ ˆ ˆ ˆ (A5) Then: 1

22 R ln w; x, d ˆ ˆ ˆ, ˆ Var Var Cov Var ln w Var ln w Var ln w ˆ Var ˆ Var ˆ Var ˆ Var ln w Var ln w Var ln w (A6) where the irst line ollows rom the ull variance decomposition (4) and the second line ollows rom equation A4. Now, rom equation (A4): R ln w; x 1 Var Var 1 ˆ ˆ ˆ ˆ ˆ (A7) Var ln w Var ln w So rom (A5) and (A6) we have: ln ;, ln ; R w x d R w x ˆ ˆ Var ˆ Var (A8) Var ln w Now in the Partitioned regression method: R ˆ Var ln w; ˆ (A9) Var ln w Where the numerator is: ˆ ˆ ˆ ˆ ˆ ˆ, ˆ Var ˆ ˆ Var ˆ Var ˆ ˆ Var ˆ ˆ Var ˆ Var Var Var Cov (A10) Hence rom (A7)-(A9) the uncorrelated variance share is: ln ; ˆ ln ;, ln ; R w R w x d R w x ˆ ˆ Var ˆ Var (A11) Var ln w and so the R-squared and partitioned regression methods are equivalent.

23 Appendix D: Variance decomposition in terms o UVS As discussed in the main text, it is possible to provide a decomposition o the total variance based on the UVS as well as the more common CVS decomposition in equation 4. Deine: ln ;, ln ; ln ;, ln ; UVSd R w x d R w x UVSx R w x d R w d ln ;, ln ;, R w x d R w x d UVSd UVSx (A1) Then: ln ;, d u ln ;, R w x d UVS UVS R w x d UVSd UVSu R w x d ˆ ln w Var ln ;, 1 Var (A13) Appendix E: Derivation o reduced orm or wages Substituting the area average o equation (15) or in equation (14) and solving or gives: ( ) v ( v ) / (1 ) (A14) Similarly, taking the area average o equation (15), substituting or and solving gives: v ( ) v ( v ) / (1 ) (A15) Thereore Var( ) ( Var( v) Var( ) Cov( v, ) / (1 ) (A16) Var( ) Var( v) Var( ) Cov( v, ) / (1 ) (A17) Cov(, ) Var ( ) Var ( v) (1 ) Cov, v / (1 ) (A18) Which correspond to equations (16)-(18) in the text. 3

24 Appendix F: Basic descriptive statistics The table provides basic descriptive statistics or variables o interest. Statistics or two digit SIC and three digit occupation dummies available on request. Variable Mean Std. Dev. (log) wage. 0.5 Female Age Part-time Collective Agreement Public Sector Mover (annual) Authors own calculations using NES/ASHE. Based on 1,54,3 observations. 4