Top incomes and inequality in the UK: reconciling estimates from household survey and tax return data

We provide the ﬁrst systematic comparison of UK inequality estimates derived from tax data (World Wealth and Income Database) and household survey data (the Households Below Average Income [HBAI] subﬁle of the Family Resources Survey). We document by how much existing survey data underestimate top income shares relative to tax data. Exploiting the ﬂexibility that access to unit-record survey data provides, we then derive new top-income-adjusted data. These data enable us to: better track tax-data-estimated top income shares; change the deﬁnitions of income, income-sharing unit, and unit of analysis used and thereby undertake more comparable cross-national comparisons (we provide a UK-US illustration); and examine UK inequality levels and trends using four summary indices. Our estimates reveal a greater rise in the inequality of equivalized gross household income among all per-sons between the mid-1990s and late 2000s than shown by the corresponding HBAI series, especially between 2004/05 and 2007/08.


Introduction
Income inequality is increasingly a subject of public discussion and analysis. Major books about it receive public attention (e.g. Atkinson, 2015;Piketty, 2014). Productive debate about what is happening to inequality requires reliable estimates. Yet the two main sources of information-household survey and personal tax return data-provide very different estimates of inequality trends. We address this problem, demonstrating how to reconcile and combine survey-based and tax-based data in a manner that exploits the strengths of each, thus providing better answers to questions such as: what has been happening to income inequality since the mid-1990s?
Household survey data are the usual source for monitoring income inequality in most countries. The UK and the USA use them to derive official statistics. They are the basis of OECD's (2008OECD's ( , 2011OECD's ( , 2015 cross-national comparative studies and the source for most other distributional analysis. Their definitions provide best-practice measures of personal living standards (as we discuss below). Their income-sharing unit is virtually always the household, and their income definition is disposable (post-tax post-transfer) income, adjusted for differences in household size and composition using an equivalence scale. Their unit of analysis is the individual (regardless of age). Inequality is typically summarized using indices based on the entire income distribution such as the Gini coefficient.
A long-standing challenge to survey-based estimates is that survey under-coverage of top incomes can cause an incomplete picture of inequality. The personal tax return data that top incomes researchers use do a much better job of capturing top incomes, but at a cost. Its income, income-sharing unit, and unit of analysis definitions are based on each country's tax administration (definitions that differ from the survey-based ones), and are restricted to summary inequality measures (top income shares) that do not incorporate differences across the full income range. Figure 1 demonstrates the need to reconcile and combine data. It shows UK income inequality trends over the last five decades using commonly used measures derived from tax data (the share of total income received by the richest 1%) and from household survey data (the Gini coefficient and the ratio of the 90 th to the 10 th percentile, p90/p10). We express each measure relative to its 1962 value so that proportionate changes in the measures are comparable. The survey data estimates are the same as those provided by the UK's official income distribution statistics (the annual 'Households Below Average Income' reports), and Inequality (indexed 1962 = 100) 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 0 1 p / 0 9 p i n i G Share top 1%

Fig. 1. Trends in UK income inequality since 1961
Notes: Household survey-based Gini and p90/p10 measures use the same definitions as employed by the UK's official income distribution statistics (source: spreadsheet accompanying Belfield et al. [2015]). Tax return-based top 1% share measure derived by the authors from Alvaredo et al. [2015]). See Section 2 for further details. and lead to lower Gini estimates than do the DWP's series: see Burkhauser et al. (2017)). These same concerns motivate us, but our combination procedure accounts for survey under-coverage of top incomes in a more extensive manner and we benchmark them against WID-based top income shares.
Sections 3-5 provide results. First, we show how poorly existing survey-based data estimates of top income shares track their tax-based data counterparts. Second, we show how with our combined data, we can derive estimates of top income shares that are almost identical to WID's. Then, exploiting the flexibility of our survey data, we employ our combined data to analyse the implications of changing definitions of income, income-sharing unit, and unit of analysis, and compare top income share estimates in the UK and the USA using more harmonized definitions. We show that conclusions about transatlantic differences in top income shares depend on which income group one considers. However, the UK trends are similar whichever distributional definitions we employ.
Finally, we use our combined data to examine inequality trends based on standard survey-based definitions with summary inequality measures that consider income differences over the full income range (Gini and generalized entropy indices). For the period spanning fiscal years 2004/05 and 2007/08, we find Gini increases that are greater in the combined data than the survey data. Using more top-sensitive inequality measures shows an even sharper differential. These trends differ from those implied by the survey-based data underlying the UK official income distribution statistics (Fig. 1).

Data sources and variables: reconciliation and combination
Here we explain how we reconcile tax and survey data on incomes for each year using common definitions and then combine them. Figure 2 contains (2014), and corresponding documentation for previous years' data.
The principal SPI income variable is gross income, i.e. total taxable income from the market ('market income') plus taxable government transfers, and before income tax deductions. Atkinson (2005) used these data as well as published tabulations from the SPI and from supertax and surtax returns for earlier years in his pioneering analysis of trends in UK top income shares since 1908. Atkinson et al. (2011) survey the methodology, main findings, and perspectives emerging from the research using personal income tax data. WID data for many countries, with top income share estimates and associated metadata such as control totals, are freely downloadable (Alvaredo et al., 2015).
Top income share estimates cannot be derived from SPI unit record data directly because the SPI does not cover all individuals in the UK population or all of their income. To derive top income shares estimates of 'shares in total income among all individuals aged 15 years or more', WID uses external population and income control totals for each year, i.e. estimates of all individuals aged 15 or more, and of the total income held by them. For 2009 and later years, WID derives its control totals for income from the national accounts (Atkinson, 2012). For earlier years, WID derives total income from income tax statistics and estimates of income from non-filers (Atkinson, 2007;Atkinson and Ooms, 2015). WID control totals for the population aged 15 and over come from the Office for National Statistics mid-year population estimates (Atkinson, 2012). We use WID control totals throughout this paper.

Household survey data: the Family Resources Survey, and Households Below Average Income subfiles
The usual source for analysis of the UK income distribution is the Family Resources Survey (FRS), a large cross-sectional household survey, and the accompanying subfiles of derived income variables called the Households Below Average Income (HBAI) dataset. The DWP administers the FRS, and DWP staff produce the HBAI subfile that they use to derive the UK's official income distribution statistics. 2 Despite its label, the HBAI provides information about the entire income distribution.
(The fiscal year runs from April to the following March.) The Institute for Fiscal Studies, whom the A major shortcoming of household surveys in general, and the FRS in particular, is their inability to capture income in the top ranges fully. Atkinson et al. (2011) provide a striking US example of such under-coverage: 'the top percentile plays a major role in the increase in the Gini over the last three decades and [Current Population Survey] data that do not measure top incomes fail to capture about half of this increase in overall inequality ' (2011, p. 32).
Under-coverage at the top has long exercised the producers of the UK's HBAI statistics. 3 Since 1992, the derived variables in the HBAI subfile include an 'SPI-adjustment' to 'improve the quality of data on very high incomes and combat spurious volatility' (Department of Social Security, 1996, p. 23), as we discuss in more detail later. We use the term HBAI-SPI data to refer to these DWP-produced variables and to distinguish them from HBAI-SPI2 data, which we derive using an alternative SPI adjustment procedure (Fig.  2). Later, we compare level and trend estimates of top income shares and other inequality measures using the FRS, HBAI, HBAI-SPI, and HBAI-SPI2 series.
We address the problem that the HBAI-SPI and the SPI-based WID series use different definitions of income, income-sharing unit, and unit of analysis. The principal HBAI-SPI measures refer to household net ('disposable') income, i.e. the sum over all individuals within a household of all income from the market plus government cash transfers and after the deduction of personal income taxes, national insurance contributions, and some other deductions. The SPI definitions reflect the administration of the UK personal income tax system, which has been individual-based since 1990: the WID tax-based data capture the taxable gross incomes of adult individuals.
The public-release FRS files contain a large number of income variables, but the separately released HBAI subfiles are the primary data source of analysis of the UK income distribution and its trends. The HBAI subfiles' income variables incorporate a number of DWP staff adjustments to improve their validity (we refer to these as the HBAI variables). In addition, the HBAI subfiles contain another set of variables incorporating a further modification-the SPI adjustment. These 'HBAI-SPI' variables are the source of the official income distribution statistics and the ones most researchers use (cf. Fig. 1).
The income statistics that the DWP publish (e.g. Department for Work and Pensions, 2015) are based on a specific income distribution definition: household net income adjusted for size and composition using the modified-OECD equivalence scale, with each individual (adult or child) attributed the equivalized net income of the household to which s/he belongs. However, access to unit record data allows researchers like us to change the definitions of income and income-sharing unit, though we are constrained by which derived variables are released. Such variables change over time. Table 1 lists the income variables we use. There are three income definitions (market, gross, and net income) and three income-sharing units (the individual, the family, and the household). 4 We distinguish between variables made available to researchers-those DWP contract to check its HBAI calculations, produces their own annual report based on the data (e.g. Belfield et al., 2015). 3 The quality of survey data on the very lowest incomes is also a concern (e.g. Brewer et al., 2017, and references therein). We do not consider this issue. 4 Our definition of 'family' follows the use in UK household surveys and assessments of benefit eligibility. A family is either a single adult living alone, or a couple with or without dependent children. See Department for Work and Pensions (2015, Annex 5). A household refers to the group of people residing at a particular address and may include multiple families who may or may not be related coming from the survey directly or from DWP's calculations-and variables we construct. We discuss the available variables in the next subsection and our constructed ones in Section 2.4.

Households Below Average Income data and the SPI-adjustment
The yearly SPI-adjustment to HBAI data consists of two elements. First, to include more top income, the DWP replaces the individual incomes of a small number of very rich individuals in the FRS with individual incomes they derive from the SPI. Second, to improve representation of top income individuals per se, the DWP modifies survey weights slightly.
The new weights use the SPI-derived numbers of very rich individuals as control totals alongside the other control totals used to create the weights for FRS and non-SPI-adjusted HBAI data (Department for Work and Pensions, 2010, Appendix 2).
To implement the first element, DWP staff derive each adult's income variables to match the SPI annual income definition. They use four different high-income thresholds to determine an FRS individual's eligibility: whether the individual is a pensioner or not, and whether resident in Great Britain or Northern Ireland. Importantly, this implies that individuals subject to SPI adjustments are not necessarily those with the highest incomes. (e.g. a couple and a co-resident adult son or daughter count as two families). A couple refers to two adults in a de jure or a de facto cohabiting union.
The DWP adjusts thresholds annually and has changed the adjustment rule slightly over time. Most notably, a change was introduced in 2009/10 to base 'the SPI adjustment methodology . . . on adjusting a fixed fraction of the population rather than on adjusting the incomes of all those individuals with incomes above a fixed cash terms level. This should prevent an increasing fraction of the dataset being adjusted' (Department for Work and Pensions, 2010, p. 242). The DWP introduced a further change in 2009/10, basing the threshold for non-pensioners on net rather than gross income. SPI adjustment rules were also modified from 2002/03 when the FRS expanded its coverage to include respondents from Northern Ireland. Because of these changes, the number of individuals in the HBAI subject to adjustment varies from year to year. Focusing on individuals aged 15 years or more (the population included in WID analyses), between 45 and 175 individuals are subject to the SPI adjustment each year from 1994/95 to 2012/13, representing between 0.12% and 0.52% of the weighted adult population. For further details, see Burkhauser et al. (2017).
Once an FRS individual eligible for the SPI adjustment is identified, his or her individual income is replaced by the average SPI income above the high-income threshold among SPI respondents with the same pensioner-location status as the FRS individual. Because SPI data take longer to become available than the survey data, SPI income values used for a given year's adjustment are based on HMRC staff projections using SPI data from one or two years earlier. (Details of the projection method are undocumented.) Because the age variable in the public-use SPI data is too broadly grouped to accurately identify female pensioners, we cannot recreate the SPI-adjustment cell-means on an out-turn basis.
After the individual-level income imputations, DWP statisticians recalculate incomes at the family and household levels. The official income distribution statistics use these SPIadjusted household income variables in their HBAI-SPI series (e.g. Department for Work and Pensions, 2015).

Our construction of missing individual-level HBAI and HBAI-SPI variables
Our analysis requires individual-level income variables. With them, we can also derive family-and household-level variables. However, the individual-level variables we need are not available in the HBAI and HBAI-SPI public-use datasets: see Table 1, top panel. The exception is HBAI-SPI individual-level gross income, which is available but only from 2005/06. Here we explain how we constructed a full set of individual-level market and gross income variables in the HBAI and HBAI-SPI datasets for the whole period. We do not do so for net income, as we explain below.
The FRS documentation lists all income components that enter into the computation of each FRS income variable. By contrast, the HBAI subfile documentation is limited and does not explain the construction of all its new income variables. 5 However, in some cases the files contain enough income components that when added up allow us to reconstruct some HBAI and HBAI-SPI income variables. For example, government transfers are provided for each income series. Therefore, we can construct market income variables at all three income-sharing unit levels. To obtain HBAI individual-level income variables, we proceed in two steps. From the FRS, we extract the share of family income accruing to each adult family member. 6 We do this separately for gross and market incomes. We then apply these shares to the HBAI family gross and market incomes to obtain our new HBAI individual gross and market income variables. Assuming that these adjustments do not change individuals' income shares within each family, our derived HBAI individual-income series are fully consistent with the HBAI family and household income data. 7 Once we obtain our HBAI individual-income series, we derive HBAI-SPI individual-income series from it. For all individuals in families not subject to the SPI adjustments, we set individual gross and market incomes equal to the corresponding HBAI values for years before 2005/06.
For individuals in a family subject to the SPI-adjustment, we start from the raw FRS individual gross and market income series rather than the HBAI series. This is because the HBAI-SPI individual-income series available from 2005/06 onwards reveal that members of couples who are not subject to the SPI-adjustment themselves, but who belong to an SPIadjusted family, are assigned their FRS income rather than their HBAI income.
We calculate the income change associated with the SPI-adjustment as the difference between HBAI-SPI and FRS incomes at the family level for gross and market incomes, respectively. We then identify individuals subject to the SPI-adjustments and add the income change to their individual income.
Before 1997/98, this is straightforward because individuals subject to the SPIadjustment within each family are identified in the HBAI subfiles. For 1997/98 onwards, we know only the number of individuals within each family subject to the SPI-adjustment; and whether these individuals are pensioners or non-pensioners. If this is insufficient information to identify the individual subject to the SPI-adjustment uniquely, we adjust the income of the individual with the largest FRS individual gross income. In the few cases where both members of a couple are subject to the SPI-adjustment, we assume they share it equally. We ignore the post-SPI-adjustment modifications made to individual incomes by DWP statisticians.
Comparisons of our derived variables and the actual SPI-adjustment variables, an exercise possible since 2005/06, show that we can accurately identify the individuals subject to the SPI-adjustment with few exceptions, and our methods yield a close approximation to the actual SPI-adjustment variables. For the 1,105 individuals subject to SPI-adjustments between 2005/06 and 2012/13 and their 939 partners, the average difference between our derived gross income variables and their HBAI-SPI counterpart is £0.01 per week. The difference is greater than £4 in 66 cases only. In addition, we obtain almost exactly the same mean and median for gross and market incomes (Appendix Fig. B1).
Although we construct individual-level market and gross incomes, we do not do so for net incomes. The DWP makes individual-level FRS net income variables available in the public-use files, from which we could derive HBAI and HBAI-SPI individual-level net income variables as described above. We did not do so for two reasons. First, family net 6 We never change children's incomes because they are identical in the FRS and HBAI data. 7 Although our assumption that income shares of individuals within families are unaffected by HBAI adjustments is often violated, the deviations from this assumption only have minor effects: for years when they can be compared, our SPI-adjusted income variables are very close to those created by the DWP (see below).
income is not available in the FRS, so it is not possible to compute the size of the HBAI and SPI income adjustments at the family level and reallocate them to individual family members as we do for gross income. Second, and more importantly, the tax-benefit calculator the DWP uses to determine net incomes is not publicly accessible and there are no comparable alternatives available for the years we consider.

Our HBAI-SPI2 adjustment
We show below that HBAI-SPI-based estimates of top income shares do not match their WID tax-based counterparts. This motivates our 'SPI2' adjustment to improve the top income coverage of the HBAI data.
With respect to DWP's goals of improving the quality of data on top incomes and combatting spurious volatility (see the Introduction) with its SPI-adjustment, we focus on the former. However, as we show below, our approach also reduces year-to-year volatility.
The essential problem is survey under-coverage of top incomes. We assume this is the result of income-under-reporting among the very richest rather than unit non-response, and we provide evidence consistent with this assumption. Choosing how to adjust survey data depends on the nature of the under-reporting. In particular, at what income level does it begin and where within the range above this threshold are incomes under-represented? We answer these questions first, as it helps explain the principles underlying our SPI2 adjustment method, which we then describe in detail. Figure 3 shows that survey under-coverage of top incomes is a serious problem. We take survey-based top income groups and their WID tax-based counterparts each year and  compare their group mean incomes. Ratios below 100% represent under-coverage. Undercoverage is particularly acute at the extreme top of the distribution (at the 99 th percentile or above). There are ratios of 90% or smaller. This is especially so for the top 0.1%, but here the few respondents underlying the survey-based estimates lead to great year-on-year volatility as well. However, under-coverage also occurs further down the distribution, especially in the mid-to late 2000s. Importantly, it occurs below the DWP's SPI-adjustment range (approximately the top 1 =2% in recent years). Under-coverage is much less of a problem for the 'top 10% to 5%' group for most of the period (ratios around 100%), though more so later. 8 Under-coverage means that we observe an insufficiently large fraction of the population in the top income ranges. Our SPI2-adjustment works by using the information that tax data provide to place more people in the critical ranges. It does so more extensively than the DWP's SPI-adjustment. Figure 4 illustrates this with kernel density estimates of the upper tail of three distributions for 2010/11-HBAI (i.e. unadjusted), HBAI-SPI (with the DWP's adjustment), and HBAI-SPI2 (our adjustment using the variant based on adjusting the top 5% of survey incomes-see below). A higher density at a given income value means a greater concentration of people. We present estimates for distributions of log(income) rather than income itself to focus attention on the upper tail and use a small bandwidth to bring out the fine structure. The dashed vertical lines mark the HBAI distribution's 90 th , 95 th , 99 th , and 99.5 th percentiles. The HBAI distribution (grey density line) has a long right- hand tail with a very tiny amount of density mass. In 2010/11, the maximum observed value is approximately £3.54M. The next 48 highest values range down to £323,321. The DWP's SPI-adjustment replaces the sparse upper tail in the HBAI distribution with only two clumps of density mass (for the non-pensioner groups): look at the dashed density line. The effect of changing the incomes of individuals in the two 'very rich' pensioner groups is imperceptible.
Our SPI2-adjustment places density mass over a greater income range than the SPIadjustment, and the concentration in the top income range is greater in total: look at the solid black line. (By contrast with the HBAI distribution, the highest 49 observations in the raw HBAI-SPI2 distribution each have an income value of around £971,000.) The reduction in sparsity of coverage of top incomes by the SPI2-adjustment also reduces the chances of non-robustness in inequality estimates due to 'high leverage' outliers in the survey data (Cowell and Flachaire, 2007) and the likelihood of spurious volatility in estimates of inequality trends. 9 Seam problem minimization is another of our design principles. When we adjust the incomes of the top p percent in the survey data, we do not want to have big differences in the density of the HBAI-SPI2 and HBAI distributions around the 'join' (the 100(1-p) th percentile of the HBAI distribution). Figure 4 demonstrates that our SPI2-adjustment (focusing on the top 5%) does not introduce seam problems around the 95 th percentile of the HBAI distribution: the density functions for the HBAI-SPI2 and HBAI distributions are very similar around that point. The fact that 95 th percentile estimates are very close in the HBAI and in SPI data provides evidence consistent with our assumption that HBAI under-reporting rather than unit non-response is responsible for under-coverage at the top. 10 We analysed SPI2 variants with adjustments made to the incomes of the top 10, 5, 4, 3, 2, 1, or 1 =2%, but we prefer the 5% adjustment. It best addresses under-coverage (and potential seam effects) while also being a suitable compromise given the changes in undercoverage over the entire period. The SPI2 analysis we report henceforth uses this variant. However, our SPI2-based estimates are insensitive to variants that include at least the top 2%. 11 Here follow the details of the derivation of our HBAI-SPI2 income variables for each fiscal year. First, we rank individuals in the SPI unit record data by their taxable gross income. Second, we allocate individuals to income groups, with the size of each group equal to 1/1000 of the total adult population shown by the WID control total for the relevant year. Third, we calculate the average income for each income group. Next, we repeat the first and second steps with the HBAI data for the same year using our derived measure of (pre-SPI-adjustment) HBAI individual gross income and the FRS grossing-up weights. 12 Finally, 9 A parametric functional form such as a Pareto distribution fitted to tax data (Jenkins, 2017) is an alternative to using discrete mass points to approximate the density function. 10 In support of this, Jenkins (2017, Appendix I) shows that close to 5% of the SPI distribution have incomes greater than the 95 th percentile calculated from the corresponding year's survey data. 11 Our findings are almost identical to those in the first version of this paper using a 10% variant . Results based on other variants are available on request. 12 Re-estimating our inequality estimates using HBAI-SPI grossing-up weights instead of FRS weights has virtually no effect because the differences in the weights are small. Estimates are available on request.
for each of the 50 top HBAI income groups (i.e. the top 5%), we replace the individuallevel HBAI incomes with the mean income of the corresponding group in the SPI. 13 Our procedures ensure that total taxable gross income for the top 5%-and for each of the 50 centile groups of the top 5%-is the same in HBAI and SPI/WID data. We also apply the same approach to derive SPI2-adjustment market income variables using SPI group mean incomes.
Because the same SPI unit record data underlie both our HBAI-based data imputations and the WID estimates, we ensure consistency in the total income of individuals in the top 5% of the distribution. Hence, we obtain the same top income shares from the two sources (for groups within and including the top 5%), provided we use the same income and population control totals. For gross income, we can use either the WID income control total or the internal HBAI control total (i.e. the weighted sum of individual incomes) and assess the difference it makes. However, when considering different income definitions-for instance, market income-we cannot use the WID, which does not contain suitable UK income control totals. Therefore, we use internal HBAI control totals for these other income definitions.
By using the WID estimates of the total adult population, we ensure that the HBAI and WID data contain the same number of individuals in each top income group. However, WID population controls are unavailable for alternative definitions of the income-receiving unit such as the family or the household. Instead, we rely on the internal HBAI population control totals (the sum of the relevant grossing-up weights in each year). This approach slightly underestimates the size of the top income groups, and thus top income shares, since HBAI fails to capture between 1% (in recent years) and 6% (in early years) of the full UK adult population. This is because the FRS excludes homeless and institutionalized people as well as households containing a married adult whose spouse is temporarily absent and, before 2002/03, individuals living in Northern Ireland.

How well do survey-based estimates of top income shares match those from the WID?
Our cross-walk of income and income-sharing unit definitions from standard HBAI ones to those WID uses (individual-level gross income for those aged 15 years or more) allows us to compare like with like. Here we analyse how well our various survey data estimates of top income shares match those WID tax-based data provide. Figure 5 summarizes our first comparison. It shows estimates for the income shares of the top 1% for 1995/96 through 2010/11 derived using multiple data sources and definitions. 14 ( Figure 2 defines series acronyms used here and elsewhere.) 13 For the other SPI2 variants, we change the number of income groups appropriately (maintaining each group's size at 1/1000 of the total adult population). E.g. to adjust the incomes of the top 10%, we use 100 groups. We also repeated the analysis for a 'top 5%' variant using 10 equal-sized groups rather than 50. Coarsening the grouping affects the results little. We could use group sizes smaller than 1/1000 of the total adult population but chose not to because the discrete approximation to the density worked well and increasing the number of groups would substantially increase computational intensity. 14 We discuss point estimates, assessing differences associated with changing definitions and sources (non-sampling variation) rather than sampling variability. However, estimates of standard errors for all inequality measures appear in Appendices E and F.
The benchmark in Fig. 5 is the WID series, the uppermost one. Its top 1%'s share increases from around 11% in 1995/96 to 13.5% in 2000/01, and is slightly smaller over the next four years, before increasing to around 15.5% in 2007/08.
There is a sharp fall between 2009/10 and 2010/11. The Great Recession and its aftermath provide one obvious explanation, but this conclusion is suspect. A 50% top marginal income tax rate was introduced in April 2010 (up from 40%). Its announcement and introduction provided incentives for high income tax payers to bring forward income to 2009/ 10 that they would otherwise report in 2010/11 or later. Seely (2014) discusses this 'forestalling' process in detail. Forestalling and the unavailability of SPI data for 2008/09 make assessing recent UK income distribution trends problematic for any measure using data on top incomes, especially top income share statistics. We therefore focus our discussion through to 2007/08.
The series shown in grey in Fig. 5 refer to FRS and HBAI survey-based estimates, i.e. without SPI-adjustments. These provide broadly similar estimates of levels and trends to each other, but compared to the WID tax-based series, the yearly estimates are not only substantially lower but also fluctuate appreciably between years. This fluctuation is the spurious volatility the HBAI's producers aimed to offset with their SPI-adjustment.
The SPI-adjustment is successful in this respect: the HBAI-SPI series is much smoother than the two non-adjusted series, and its estimates are generally larger in each year. However, HBAI-SPI series estimates are distinctly smaller than WID series estimates. Differences range from 1 1 =2 to 3 percentage points, with the larger differences more apparent in the mid-to late 2000s. This UK evidence is consistent with Atkinson et al.'s (2011) findings for the USA that household survey estimates do not fully capture the rise in inequality shown by tax data (see the Introduction). Do our HBAI-SPI2 estimates of the share of the top 1% do better? Look at Fig. 6, which shows the SPI (WID control), which we hereafter label as WID, and HBAI-SPI series (as in Fig. 5) plus two HBAI-SPI2 series. The first uses the WID population and income controls, and SPI-adjusted values in the HBAI as discussed in Section 2.2. This is the series labelled 'HBAI-SPI2 (WID controls)'. The second series uses the same SPI-adjusted values in the HBAI, but uses HBAI-estimated population and gross income control totals. The HBAI-SPI2 series with WID controls is virtually identical to the WID series (by construction). The HBAI-SPI2 series using HBAI control totals does less well but much better than the HBAI-SPI series: it is much closer in levels and trends to the WID series. Put differently, the use of the internal HBAI control totals rather than the WID ones leads to an under-estimation of the income share of the top 1% by around 1 percentage point, but the trends are similar. 15 We also find this close match between estimates based on HBAI-SPI2 data with WID controls and the WID estimates for other top income share groups. See Appendix Figs D1, D2, and D3 for comparisons of series for the top 0.1, 1 to 0.1, 10 to 5, and 5 to 1% income groups.
Our HBAI-SPI2 data almost exactly capture WID levels and trends based on the SPI unit record data when we use WID controls and do a reasonable job even when we use internal HBAI controls. It is the latter we use in the next two sections, for the reasons given earlier. 15 Appendix C provides more information on WID and internal HBAI control total differences. Fig. C1 summarizes differences in denominators. The increasing gap between WID and HBAI-SPI2 total income between 2001/02 and 2004/05 largely explains the increasing gap between the top 1%'s income share based on WID and HBAI control totals. In contrast, differences in population control totals in the WID and HBAI adult populations do not appear to play an important role (Fig. C2).

Using our combined (HBAI-SPI2) data for better cross-national comparisons of top incomes in the UK and USA
Top incomes researchers employ tax-based data and are therefore limited to looking at distributions of taxable income among tax units. Because the definitions of the tax unit and taxable income differ across countries, there are potential comparability problems for cross-national comparisons of top income shares. By exploiting the flexibility of household survey data and using cross-walked definitions, we can address this issue. We illustrate this lesson with the UK-US comparison summarized in Fig. 7. The first top 1% income share series for the USA comes from Piketty and Saez (2003) and their updates found in WID. Reflecting US personal tax administration, these estimates refer to taxable market income and the income-sharing unit and unit of analysis is the family. We focus on the US top income series that excludes taxable realized capital gains, since the UK series does not include realized capital gains. 16 Figure 7 shows that the share of market income held by the top 1% of US families-the series labelled 'US family market income'-increased from around 13.5% in 1995 to around 16.5% in 2000 before falling over the recession years of the early 2000s. The share then reached a peak of just over 18% in 2007, just before the Great Recession hit the USA. Shares then declined over the next two years but started to recover in 2010/11. Top incomes researchers often compare these US estimates of levels and trends with WID estimates for the UK (e.g. Atkinson et al., 2011;Roine and Waldenströ m, 2015). According to the WID series, UK top 1% income shares are substantially lower than in the   16 Armour et al. (2013Armour et al. ( , 2014 and Larrimore et al. (2016) discuss the merits of including taxable realized capital gains.
USA, but follow approximately the same trend through 2007. However, this is not a comparison of like to like. The UK series refers to gross income and its sharing unit and unit of analysis is the individual (adult). When we cross-walk our UK survey data from a gross income definition to a market income definition (moving from the series labelled 'individual gross income' to the one labelled 'individual market income'), the estimated top 1% share increases by around 2 percentage points each year, a relatively large change. 17 Alternatively, if we shift to the family but retain the gross income definition, estimated top 1% shares fall by about 1 percentage point or slightly more: look at the 'family gross income' series.
The UK definition that is most like the US one is the distribution of market income among families (the 'family market income' series). The net result of using this series, rather than the 'individual gross income' series, is a relatively small increase in the share of income held by the top 1%. However, this is the coincidental result of the two definitional changes going in opposite directions and almost completely offsetting one another.
This robustness in the cross-national comparison is not necessarily the general rule. See the series for the top 0.1, 1 to 0.1, 5 to 1, and 10 to 5% shares presented in figs 8-11 in the same format as Fig. 7. The further up the distribution the shares refer to, the more important is the shift from an individual to a family definition of the tax unit (unit of analysis) and the less important is the shift from gross to market income (at the very highest top income ranges almost all income is market income).
For the top 0.1%, the shift downward from individual to family more than offsets the upward shift from gross to market income. The result is that our like-to-like comparison of USA and UK 'family market income' shows an even greater difference in shares than the conventional top income literature comparisons (Fig. 8). However, this quickly turns 17 The UK estimate is slightly larger than is its US counterpart in 2009/10 most likely because of the forestalling problem discussed earlier.
around. When we focus on the top 1% to 0.1% (Fig. 9), the difference in top income shares between the two countries almost completely disappears. When we move to lower top income ranges, not only are the estimates from the 'UK family market income' series greater than the 'UK individual gross income' estimates, they are greater than the corresponding estimates for the USA (Figs 10 and 11). The other two series in Fig. 7 show what happens to top income shares if we employ the standard definitions of household survey-based income distribution analysis. 'UK household gross income' uses the household as both the income-sharing unit and unit of analysis. 'UK equivalized household gross income' shifts to a more appropriate definition of income: household income is adjusted using an equivalence scale depending on the number of adults and children and with each person assigned his/her household's value. Its distributional statistics relate to all individuals in the population, including dependents. Gottschalk and Smeeding (1997), d' Ercole and Fö rster (2012), and the influential Canberra Group (2011) make the case for this standard set of survey definitions. Burtless (2015) criticizes the definitions used in the top incomes literature for not following these standards. Figure 7 shows that once these factors are taken into consideration, the share of equivalized income held by the top 1% of all individuals is substantially reduced relative to the 'UK individual gross income' series which employs top incomes literature definitions. This finding is reassuring since households are institutions that enable individuals to pool and share income, and larger households benefit from economies of scale relative to smallersized ones. We have summarized the impact of different definitional choices in more detail (using the top 1% share data in Appendix Table E1), regressing top 1% shares on sets of binary indicator variables that characterize differences in survey year, data source, income definition, income sharing unit, and types of control total. Relative to baseline, using a market income rather than a gross income definition is associated with an increase in the top 1% share of 2 percentage points, other things being equal. By contrast, using household equivalized income reduces the top 1% share by 2.6 percentage points. Using the family (household) as income-sharing unit rather than the individual tax unit is associated with a top 1% share some 1.4 (2.3) percentage points lower.
Although estimates of top income share levels are sensitive to the choice of definition of income and income-sharing unit, figs 7-11 show that estimates of trends are similar. All of the UK series move closely in parallel with each other. This finding echoes that of Lakner (2014), who showed that US top income share trends were similar across various tax-unit size adjustments.

Using our combined (HBAI-SPI2) data to summarize inequality differently
Using tax-based data requires that inequality be summarized using top income share estimates. In this literature, the share usually refers to people with incomes above the 90 th percentile or an even higher threshold. By contrast, most of the survey-based income distribution literature summarizes inequality with indices that account for differences throughout the full income range. A further advantage of our survey-based approach is that we can calculate these measures in addition to top income shares, and see the extent to which they move in tandem with top income shares. 18 As in our 'UK equivalized household gross income' series, the income-sharing unit is the household; the unit of analysis is the individual (adults and children); and we equivalize household incomes in Fig. 12 and the two tables below. In Fig. 12, we compare Gini coefficient estimates based on HBAI-SPI and HBAI-SPI2 data, and for both gross and market income definitions. Reassuringly, market income is more unequally distributed than gross income (the latter includes redistributive government transfers), regardless of the series employed, and the Gini coefficients are larger for the HBAI-SPI2 series than for the HBAI-SPI in corresponding years, since the HBAI-SPI2 data better capture top incomes.
Note also the marked increase in the Gini coefficient between 2004/05 and 2007/08reflecting better capture of the rise in top income shares over this period. The bottom series,  18 Leigh (2007) finds a positive but far from perfect correlation between movements in top income shares and Gini coefficients using a 13-country panel dataset. Our UK case study examines in more detail when and how trends differ for these two measures and the sources of any divergence in trends.
in grey, is the HBAI-SPI series for household net income. This corresponds to the series reported in the official HBAI statistics (see also Fig. 1). 19 These Gini estimates are 4 to 5 percentage points lower than their gross income counterparts, but the trends are similar. Table 2 extends our comparisons of the HBAI-SPI2 and HBAI-SPI data. It compares the growth in equivalized household gross income across the distribution. Reassuringly, there is very little difference between datasets in estimated income growth for the bottom 90% of the distribution regardless of the comparison years chosen, since the SPI-adjustments in both series focus on the richest individuals. For the top 10%, the patterns are more complex. Over the entire period from 1995/96 to 2007/08, HBAI-SPI2 data show higher income growth than HBAI-SPI data for individuals between the 91 st and 99 th percentiles, and substantially greater income growth for the top 1% (102% versus 76%). However, most of this difference in growth occurs over the period 2004/05 to 2007/08. Importantly, and with regard to the credibility of the DWP's current adjustment procedure for the HBAI (focusing on the prevalence of people with incomes below 60% of median income), there is virtually no difference in estimates of the change in the median across the datasets. However, there are slightly more pronounced differences for the mean (34% versus 31%), and these differences also appear to be driven by a divergence in income growth in the final sub-period (4% versus 6% between 2004/05 and 2007/08). Notes and sources: As for Fig. 5. Household income is equivalized using the modified-OECD scale. The individual is the unit of analysis. Estimates are based on the full population (aged less than 15 as well as aged 15þ). The price index used is the all items RPI excluding Council tax (agg4111), a bespoke index created for the Department for Work and Pensions by the Office for National Statistics. Table 3 expands our comparisons between datasets by, in addition to the Gini coefficient, considering the mean logarithmic deviation (MLD), the Theil index, and half the squared coefficient of variation (HSCV). These are generalized entropy indices with sensitivity parameters 0, 1, and 2, respectively. The larger the parameter, the more sensitive is the index to income differences at the top of the distribution relative to those in the middle or at the bottom-the MLD (and Gini coefficient) are middle-sensitive indices. Table 3 shows inequality levels for four years between 1995/96 and 2007/08 and changes over periods between those years. Period-specific changes are quite similar for HBAI-SPI gross and net definitions; the main contrast is between the differently adjusted gross income series. The exception is the period 1995/96 to 2001/02 when, for each index, all income definitions show similar inequality increases.
The Gini coefficient for the HBAI-SPI2 data is greater than the HBAI-SPI data for both 1995/96 and 2007/08, as is the percentage change between those two years (10% versus 7%). For the other three indices, the level of inequality is again greater in the HBAI-SPI2 data. However, the more top-sensitive the index (as we move down the panels in the table), the larger is the measured inequality increase. For example, the increase between 1995/96 and 2007/08 in the HBAI-SPI2-based HSCV is much larger than in its HBAI-SPI-based counterpart (141% versus 52%).
Regardless of the index chosen, most of the increase in its value over the period 1995/96 to 2007/08 occurs between 2004/05 and 2007/08. We know from the WID estimates that the concentration of incomes at the very top increased substantially over the period as a whole but especially over this latter period, according to the top income shares (cf. Figs 1 and 5), and the more comprehensive top-sensitive indices pick this up. (The sub-period changes-inequality rise, fall, and rise again-are also similar.) In short, the more we adjust the survey data to better account for survey under-coverage of top incomes, the greater is the observed effect on inequality.

Summary and conclusions
Exploiting the flexibility provided by access to unit record household survey data, notably the ability to cross-walk between different sets of definitions so that like is compared with like, we provide the most extensive comparison of UK inequality estimates derived from survey and income tax data. Using our reconciled definitions, we combine these two data sources to yield better estimates of UK inequality levels and trends, accounting for survey under-coverage of top incomes. The magnitude of our estimate of the increase in inequality, especially between 2004/05 and 2007/08, depends on whether the index is middle-or topsensitive. Nonetheless, all our indices show a greater increase in inequality than do the headline income distribution statistics based on the DWP's HBAI-SPI data.
The flexibility of unit record survey data also allows us to employ different definitions of income, income-sharing unit, and unit of analysis to improve the comparability of crossnational comparisons. With UK definitions cross-walked to the definitions underlying the US top income share series, we show that conclusions about transatlantic differences in top income shares depend on which top income group one compares.
More generally, we show that it is possible to adjust household survey data to take better account of their non-coverage of top incomes using tax data for top income groups. Official UK income distribution statistics have long done so using their SPI-adjustment. Our SPI2-adjustment better addresses the problem of under-coverage of top incomes. R.V. BURKHAUSER ET AL.
One advantage of our data combination approach relative to the DWP's is that we exploit WID estimates to provide a benchmark. However, their accuracy is conditional on the reliability of the estimates of the population and income control totals that WID employs. Assessment of these would be a valuable topic for further research, with pay-offs for the top incomes literature as well as for survey-based approaches like ours.
Our SPI2-adjustment comes entirely from comparisons of unconditional distributions of income. Unlike the DWP's SPI-adjustment procedure, we do not use other personal characteristics to construct stratification groups. We improve overall inequality estimates by seeking better coverage of high incomes, regardless of who has them. The DWP's SPIadjustment procedure applies to the incomes of individuals outside the entire population's top income group. If the primary goal is to improve estimates of the distributions of income for specific subgroups of the population (e.g. men/women; individuals in families with/ without children), then modifications of our approach are necessary. However, implementation depends on the personal characteristics of interest being in both the tax and survey data. For example, for individuals in UK SPI data, we can identify men and women but know nothing about marital status or family size.
Arguably, the DWP SPI-adjustment is fit for purpose because the official HBAI reports' principal focus is the prevalence of low income with assessments based on low-income cutoffs equal to fractions of median income (not fractions of mean income). Estimates of median-based cut-offs and of low-income prevalence are insensitive to the SPI-adjustment variation used.
However, this argument fails to recognise that the DWP's HBAI data are the most often used source for assessing UK inequality levels and trends. Researchers in most other countries also employ survey data for this purpose, and for cross-national comparisons. Yet household surveys in all countries are subject to under-coverage of top incomes. We therefore hope that our reconciled and combined data approach will stimulate further research on how to take better account of top incomes in inequality assessments. What can be done for different countries and periods will depend on the availability of comparable data from surveys and tax return data or other sources that can be used to provide suitable benchmarks regarding survey under-coverage of top incomes.

Supplementary material
The FRS, HBAI, and SPI data used in this paper are available from the UK Data Service to researchers who register with them. The Stata code used to create derived variables and for the analysis is available online at the OUP website, along with the Appendix.

Funding
This work was supported by an Australian Research Council Discovery Grant (number DP150102409), and core funding of the Research Centre on Micro-Social Change at the Institute for Social and Economic Research by the University of Essex and the UK Economic and Social Research Council (grant number ES/L009153). For helpful discussions and perceptive comments on an earlier version, we thank Facundo Alvaredo, Tony Atkinson, and Peter Matejic (DWP), three referees, and audiences at the IARIW 2016 conference, Nuffield College, York, Alicante, and Clermont Ferrand.