Regression Estimates of the Elasticity of Taxable Income and the Choice of Instrument

This paper examines estimation of the elasticity of taxable income using instrumental variable regression methods. It is argued that the ‘standard instrument’ for the net-of-tax rate − the rate that would be applicable post-reform but with unchanged income levels − is unsatisfactory in contexts where there are substantial exogenous changes in taxable income. Two alternative tax rate instruments are proposed, using estimates of the dynamics of taxable income for a panel of taxpayers over a period that involves no tax changes. The parameters derived from this procedure are then used to construct hypothetical (or counterfactual) post-reform incomes that would be expected in the absence of reform. The first method is based on the tax rate each individual would face if income were equal to ‘expected income’, conditional on income in two periods before the tax change. The second alternative uses the form of the conditional distribution of income for each taxpayer to obtain an instrument based on the ‘expected tax rate’. The methods are applied to the tax change in New Zealand in 2001.


Introduction
The 'elasticity of taxable income' (ETI) was proposed by Feldstein (1995) as a way of capturing the combined impact of various economic responses to changes in marginal income tax rates. 1 The elasticity is defined in terms of the response of taxable income to variations in the net-of-tax rate, 1 −  , rather than the marginal tax rate,  , and is therefore expected to be positive. There has been a plethora of empirical estimates of the elasticity, mainly for the US and using a variety of methods. However, as the recent review by Saez et al. (2012) points out, estimation presents a number of challenges. In particular, Saez et al. (2012, p. 18, emphasis added) point out that, 'in order to isolate the effects of the net-of-tax rate, one would want to compare observed reported incomes after the tax rate change to the incomes that would have been reported had the tax change not taken place. Obviously, the latter are not observed and must be estimated'. This paper has two main objectives. First, it examines the use of instrumental variable regression methods. It is argued that the 'standard instrument' for the net-oftax rate − the rate that would be applicable post-reform but with unchanged income levels − is unsatisfactory in contexts where there are substantial exogenous changes in taxable income. 2 This is in addition to acknowledged problems associated with controlling for income changes as part of the regression specification. Two alternative tax rate instruments are proposed that, it is suggested, better approximate the desired tax rate. The approach advocated here to deal with the challenge posed by Saez et al. involves estimating the dynamics of taxable income for a panel of taxpayers, using data over a period that involves no tax changes. The parameters derived from this procedure are then used to construct hypothetical (or counterfactual) post-reform incomes that would be expected in the absence of reform. From the resulting probability distribution of income for each taxpayer, two alternative net-of-tax rate instruments 1 A key property of the elasticity is that it captures all responses to a change in the tax rate in a simple reduced-form specification and, under certain conditions, provides a convenient method of calculating the welfare effects of tax changes. See, for example, Seaz et al. (2012) for further discussion and Creedy (2010) for a technical introduction. 2 Studies using the standard approach include, for example, Moffitt and Wilhelm (1998), Auten and Carroll (1999), Goolsbee (2000), Sillamaa and Veall (2000), Aarbu and Thoresen (2001), Gruber and Saez (2002), Selen (2002), Giertz (2004Giertz ( , 2007Giertz ( , 2010, Hansson (2004), Kopczuk (2005), Thomas (2007), Auten et al. (2008), Heim (2009). Carroll (2008) is based on the tax rate evaluated at the average taxable income over a seven year period. may be obtained. One instrument is based on the tax rate each individual would face if their income were equal to 'expected income', conditional on income in the previous two periods and knowledge of the process of relative income dynamics. The preferred alternative uses the form of the conditional distribution of income for each taxpayer to obtain an instrument based on their 'expected tax rate'. The second objective is to use the proposed new instruments to estimate the elasticity of taxable income in New Zealand, using information about taxable incomes for a sample of taxpayers before and after the income tax rate changes in 2001. This reform provides an especially useful context in which to examine the performance of the three instruments, given the nature of that reform and the availability of suitable data to estimate 'no reform' income dynamics.
Following a brief review of existing estimates obtained using instrumental variable and other methods in section 2, section 3 summarises the basic instrumental variable specification. Section 4 compares some key properties of the 'standard instrument' and the two proposed alternatives. The construction of these alternatives is described in detail in Section 5. Section 6 applies the various instruments to a tax policy change in New Zealand in 2001 and discusses the resulting estimates of the elasticity of taxable income. Brief conclusions are provided in section 7.

Approaches to Estimation
In examining the elasticity of taxable income using regression methods, a constant elasticity specification is ubiquitous in the literature, whereby the logarithm of taxable income is expressed as a linear function of the logarithm of the net-of-tax rate. Fixed effects are generally eliminated by taking first-differences, so the form of equation to be estimated has the change in the logarithm of taxable income related to the change in the logarithm of the net-of-tax rate (these log-changes also providing approximations to the proportional changes), along with other available exogenous variables such as age. The approach therefore requires information about taxable income of a sample of individuals in at least two years (before and after a tax structure change), and the regression is cross-sectional. 3 In addition, some measure of initial or lagged income is often added as a regressor, to capture any tendency for proportional income changes to depend on income levels (that is, the existence of any regression towards or away from the mean). All the observed change in income is attributed to the tax change and the exogenous variables included in the regression.
The reduced-form specification faces the well known problem that, with a nonlinear income tax function reflecting marginal rate progression, the change in the net-of-tax rate is itself endogenous. To overcome this problem a number of authors have used an instrumental variable (equivalent to two-stage least squares) approach, in which the instrument is, for each individual, the marginal tax rate which would be faced in the second period if there were no change in income. 4 The first stage involves a regression of the change in the actual log-net-of-tax rate on the change in the log-net-of-tax rate that would apply with no change in income, and other exogenous variables. This is used to obtain 'predicted' values of the log-change in the net of tax rate. The latter is then used in the second stage regression (with the change in the logarithm of income as dependent variable) instead of the actual change. Hence, the most commonly adopted 'standard instrument' involves using the tax rate that would apply post-reform to the taxpayer's pre-reform income. Where comparisons involve a number of years, annual incomes are often adjusted for inflation. 5 Alternatively, in determining the individual's reformonly change in marginal tax rates, some studies have adopted a common intermediate income between pre-and post-reform levels. 6 As is well-known, following Feldstein's (1995) relatively large estimates for the ETI (he found values between 1 and 3 for the 1986 and 1993 US tax reforms), subsequent studies have tended to find lower values. 'Typical' values are increasingly reported in the 02-06 range. These are often claimed to be more plausible, at least for the contexts in which they are estimated -mainly US taxpayers, often with high incomes. However, recent reviews of this literature by Giertz (2009) 5 However, this is not an innocuous adjustment for estimates of ETI responses. Since tax liability is defined in nominal terms, a nominal income increase involving no real income change could nevertheless be associated with a tax-induced income response where nominal fiscal drag pushes the taxpayer into a higher tax bracket. Indeed this is the identification method adopted by Saez (2003) to obtain ETI estimates from 'bracket creep'. 6 For example, Auten and Carroll (1999) use an average of pre-and post-reform incomes, while Blomquist and Selin (2009) use an intermediate year.
have suggested that even the more rigorous recent studies, including those employing a variety of income controls, obtain a wide range of statistically significant ETI estimates including negative values.   ) and other methods. These reform episodes include both tax increases and tax cuts, though the vast majority of ETI studies of tax reforms over the last 50-60 years involve marginal tax rate reductions. Table 1 shows that, with or without income controls, negative ETI estimates are not uncommon, and some large positive values continue to be reported.
These results serve to emphasise that, even where small positive and plausible ETI estimates are reported, they generally form part of a suite of results that include a much wider range of values, including 'wrongly' signed estimates. Partly in response to the varied findings, Giertz (2010) concludes that 'it is incredibly difficult to isolate responses to changes in tax rates from income changes due to a myriad of other complex factors. While flexible income controls are intended to control for both mean reversion and divergence within the income distribution, it is impossible to conclude that these problems are adequately mitigated'.
It is shown here that using the standard instrument is likely to involve a number of biases, the sign and size of which depend on the nature of the tax reform and the exogenous income changes that occur in association with it. Indeed, the new instruments proposed below were stimulated by finding a completely unrealistic value of the elasticity using the standard approach in the New Zealand context.

The Specification
Consider an income tax change between periods 1 and 2, involving changes in marginal rates,   , for periods  = 1 2, and tax brackets  = 1    . The number of thresholds, in addition to various rates, may change from period 1 to 2. Information is available about the taxable income,   of  = 1   individuals in each period. Let   denote the marginal tax rate actually faced by individual  in period . This is the appropriate rate, depending on   , from the set of rates   . Define: which is an approximation for the proportional change in the net-of-tax rate, 1 −  . The proportional change in taxable income,   , is approximated by: A simple constant elasticity relationship between taxable income and the net-of-tax marginal rate, ignoring for the moment exogenous variables which may influence income changes, is: where   is a random variable and  is the elasticity of taxable income. There are inevitably income changes which would occur in the absence of tax changes. In general it is difficult to isolate groups of individuals who would otherwise experience the same pattern of income changes but where only one group faces a tax change. The challenge is thus to avoid attributing those exogenous income changes to the tax rate changes. 7 Estimation of the form in (3), augmented by further exogenous variables, presents a fundamental problem because of the endogeneity of the change in the log-tax-rate. This means that ordinary least squares estimates are biased and inconsistent. In order to avoid this problem, researchers have used the following instrumental variable (or equivalently, a two-stage least squares) approach in which the instrument,   , is used, defined as: where  * 2 is the marginal tax rate that would be faced by the individual in period 2 if taxable income were to remain constant at  1 .
The first stage involves a linear regression of   on   and all the exogenous variables in the model, and the calculation of the 'predicted' values,  , using the parameter estimates (also indicated by 'hats') so that, again ignoring the other exogenous variables discussed above for convenience: The second stage then involves estimating the elasticity of taxable income,  using ordinary least squares on: The question examined here is whether this is a reliable approach, bearing in mind that for  to be a good instrument, it must be correlated with  but independent of the errors, and uncorrelated with  other than via any effects on . 8 7 The difference-in-difference approach, whether regression-based or not, essentially relies on being able to isolate 'treatment' and 'control' groups that can reasonably be regarded as otherwise identical (at least with respect to aspects relevant to the tax response under investigation). 8 The regression of  on  implies (in the absence of other exogenous variables) that for   = 0,   =. These different cases thus have values of   aligned along a straight line at  =.

Alternative Instruments
This section considers problems associated with the standard instrument and proposes two alternatives. It shows that with non-tax related income changes -a ubiquitous feature of the income dynamics of most taxpayers -it is not surprising that a tax rate instrument that ignores those changes may perform poorly. Typically, exogenous variables are added to the basic specification in (6) to capture elements of income dynamics. However, the alternative approaches suggested here also involve the use of an independently estimated process of earnings dynamics in the construction of the instrument itself. Figure 1 shows a segment of a multi-step tax function. Consider an individual in the th tax bracket with initial (period 1) taxable income of  1 , facing the (pre-reform) marginal tax rate,   . A reform which decreases   in period 2 with thresholds unchanged, would be expected to increase taxable income. However, other exogenous influences on income, ceteris paribus, could either raise or lower period 2 income, yielding observed income in period 2,  2 or  0 2 respectively. Thus, where exogenous income increases and the expected tax effect operate in the same direction, observed income increases to  2 . For an exogenous income fall, the expected tax response operates in the opposite direction to the income change, compensating for the exogenous income fall, as shown by the arrows around  0 2 . Therefore tax cuts are expected to be negatively correlated with observed income changes (the a priori relationship) for exogenous sources of income increase, but positively correlated for exogenous income decreases. 9 For a tax reform involving an increase, rather than a decrease, in   , these correlations are reversed. For any given reform, the problem for the empirical investigator therefore is to separate the two unobservable components of each taxpayer's observed income changes from periods 1 to 2. Table 2 shows the resulting bias that can be expected in standard instrument estimates of the income response to a tax structure change in the presence of exogenous

The Standard Instrument
income changes for which allowance is not fully made; as before   refers to the marginal rate facing the individual, depending on the income level and thus the tax bracket into which the individual falls. 10 For example, consider the final row of Table 2, which shows that an increase in   due to a reform is expected a priori to reduce income. The standard instrument avoids attributing to the tax reform any observed tax rate reduction induced by the income fall. However any exogenous increases in income would raise the taxpayer's marginal rate where that income change involves crossing a tax threshold. Failure to accommodate this second effect within the instrumented tax rate therefore risks attributing some of this positive association between exogenous income change and tax rates to the tax instrument. That is, the parameter on the tax rate term in an instrumental variable regression, attempting to capture the behavioural response to tax reform, is likely to be positively biased (less negative, or more positive).
As mentioned above, a number of existing studies of taxable income elasticities do attempt to control for other sources of income change, such as regression towards the mean, though this income change process is generally not allowed to affect the measurement of the tax instrument. 11 Since imperfect controls for exogenous income changes could result in either over-or under-estimates of their effects, the resulting biases could be in the opposite direction to the 'no income control' cases above. However, the tendency for US elasticity estimates to be based on reforms involving tax cuts (increases in 1−  ), in association with average nominal or real incomes increases (independently of tax reform) suggests the possibility of negative biases in Table 2 predominating in these cases.
This simple illustration therefore demonstrates the importance, for accurate estimation of tax responses, of first modelling non-tax induced changes in incomes as accurately as possible and without systematic bias. Second, where exogenous income changes are imperfectly captured, it is important to be aware of the different biases associated with estimates of tax-induced income responses. As Table 2 shows, these depend on the type of tax reform and the exogenous income changes experienced. To the extent that income changes, in the absence of tax reforms, follow a systematic pattern, rather than being purely random, the biases can be substantial. The following subsection suggests how information about income dynamics can be directly used in the construction of a tax rate instrument.

Incorporating Income Dynamics
A partial solution to deal with potential biases was suggested by Saez et al. (2012, p. 27-8), whereby, 'in situations with mean reversion, it is useful to include episodes of both increases and decreases in tax rates for identification, as mean reversion creates biases in opposite directions in the case of tax increases versus tax decreases'. Saez et al. (2012, p. 28) also find that 'panel regression estimates of taxable income responses are sensitive to the choice of instrument for the marginal tax rate', such that 'standard methods do not control adequately for mean reversion'. Indeed, as argued further below, these standard methods cannot capture more general features of income dynamics, such that their ability to separate 'tax reform only' from 'non-tax induced' changes in income is questionable, especially given the potential for various, reform-specific, biases described above.
The key problem with the standard instrument is that it represents the simplest approximation of income dynamics, namely no change in (real) reported incomes in the absence of tax reform. The alternative approach proposed here provides more sophisticated modelling of taxpayers' income dynamics, using annual income data that, by construction, are unaffected by tax reform.
The method captures any exogenous regression to the mean and serial correlation in relative income changes over a number of years during which there are no tax changes, to yield predicted values of future incomes, given current and past income levels. This yields a conditional probability distribution of income for each future year and taxpayer. Using this information allows construction of two possible marginal tax rate instruments. First, the mean income from the conditional income distribution, given initial income for each taxpayer, for any post-reform year, , (  ), can be obtained. For this expected mean income, the associated tax rate is obtained from the postreform tax code. This instrument is labelled  * () . Here individual subscripts have been suppressed. Alternatively, the complete probability distribution of incomes for year  for each taxpayer can be used in conjunction with the post-reform tax code to obtain the set of tax rates associated with each income level. Using the full income distribution to weight each tax rate appropriately yields an 'expected marginal tax rate' after reform which more fully incorporates information on income dynamics. This expected tax rate instrument is labelled  * ( ) . In terms of Figure 1, a probability distribution of income, centred for example on  2 , potentially includes a wide dispersion of incomes with associated tax rates   and  −1 , as well as  +1 and any other higher or lower rates. Hence, whereas the set of 'expected income' based tax rates,  * () , includes only the discrete set of rates specified in the tax schedule, 'expected tax rates',  * ( ) , can take a wide range of values, reflecting the income-weighting of each discrete rate.

Correlations among Measures
To explore the merits of these alternative instruments, and given the earlier discussion of potential biases, it is first useful to consider their correlations with observed income changes. Of interest here is the observed changes in income, ∆, and tax rates, ∆ , and the change in the relevant tax rate instrument, ∆ * . To simplify the exposition at this stage, changes in tax rates,  , rather than net-of-tax rates, 1 −  , are considered. Table 3 shows, for any individual, the possible combinations of ∆, changes in the actual tax rate ∆ = ( 2 −  1 ), and the instrumented tax rate ∆ * = ( * 2 −  1 ) between the pre-and post-reform periods (1 and 2 respectively). There are 3 × 3 × 3 = 27 possible combinations of negative, zero or positive change. The zero income change cases are excluded from Table 3, leaving 18 possible cases. 12 Of the 18 cases, 2 are not feasible with a tax schedule with marginal rate progression everywhere; for example, a positive income change cannot be associated with an actual marginal tax rate decrease.
Given the 16 possible combinations of values for ∆ ∆ and ∆ * in Table 3, Table discussed above, but applied to ∆ rather than ∆ (1 −  ). Tax changes for the new proposed instruments are shown as ∆ * () and ∆ * ( ) . The table identifies, with a tick (X), those categories where the correlation between the income change and each tax instrument takes the expected, ceteris paribus, negative sign:   0. Other entries ('incorrect' zero or positive correlation:  > 0) are shown by a cross (×). There are also several 'not feasible' cases. These arise either because of the increasing marginal rate nature of the tax schedule as discussed above (cases 4 and 8) or because they are not feasible for the particular tax instrument in question.  Table 4 also reveals that there are four cases (7,11,12,15) for the standard instrument, which are not feasible, but which can be accommodated by the other two instruments. This reflects the relative inflexibility of the standard instrument's property that the instrumented tax rate is always that which applies to initial income. Section 6 below compares the regression-based performance of these three instruments in the context of the year 2001 tax reforms in New Zealand. But it is useful here to consider the numbers of New Zealand taxpayers who fall into each of the above categories. Table 5 shows the pre-and post-reform New Zealand tax rates. Of the four marginal rates in the tax schedule in 1999, the reform involved 0.75 and 3 percentage point decreases in two middle tax rates respectively (from 21.75 and 24 per cent to a common 21 per cent rate) and a 6 percentage point increase in the top rate (from 33 to 39 per cent) for incomes above $60,000. 13 These represent approximate percentage changes in the three reformed tax rates (using log differences) of −35, −134 and +167 per cent. 14 This makes the New Zealand reform a particularly helpful one to analyse in this context because of the mixture of tax rate increases and decreases (and no change) across a wide range of incomes.
Based on pre-and post-reform years of 1999 and 2002, the framework in Table 4 can be used to compare each taxpayer's observed change in income with changes in their actual and instrumented tax rates; these years are chosen to avoid effects of income shifting between the announcement and implementation of the tax change; see Claus et al. (2012) for further discussion of this phenomenon. Table 6 shows the numbers of taxpayers in each category, separated into those categories where  > 0 (columns 1-4) and   0 (columns 5-8), where  refers to the unconditional correlation between the income change and the change in the relevant tax instrument. The correlation of interest to identify behavioural responses to tax rate reform is the conditional correlation between the tax instrument and reform-related income change. However, since all three tax instruments considered here attempt, in their different ways, to control for income changes in defining each instrument, the sign on the unconditional correlation involving the total income change might be expected to provide a useful guide to the prospects of finding a similarly signed conditional correlation.
The final row of Table 6 shows that the total numbers of correlations involving the expected tax rate,  * ( ) , yield quite different outcomes from those involving the other two instruments:  * ( 1 ) and  * () . In particular, there is a much higher ratio of incorrectly signed correlations ( > 0) to correctly signed correlations (  0) for the standard and expected income instruments. Around 54 per cent (431/803) of  correlations for those two instruments reveal  > 0, whereas for  * ( ) the ratio is only 25 per cent (204/803).
However, the expected tax rate does not out-perform in all categories, in the sense of having more negative correlations than the other instruments. Table 6 reveals that, across instruments, the numbers of positive or negative correlations can be quite different within each category. For example, there is a very high number (122,000) of positive correlations in category 1 using the expected tax rate, whereas the alternative instruments have lower numbers: 53,000 and 27,000. An opposite ranking of instruments is observed for (positive) correlation category 18.
The main reason for the strong correlation performance of the expected tax rate instrument arises from its ability to reclassify 101,000 and 152,000 taxpayers in the positive correlation categories 5 and 18 respectively into negative correlation categories 6 and 16. These numbers provide a clue as to why regression results reported below appear strongly to favour the expected tax rate instrument over the alternatives. To recap, the expected tax rate instrument was constructed to identify changes between the tax rate applicable to actual 1999 income and the rate applicable to 2002 income if the latter had been unaffected by the tax reform. This tax rate instrument appears to be much more highly negatively correlated with observed income changes over those years than the alternatives.

Construction of Alternative Instruments
This section explains the model of income dynamics used to construct the alternative instruments discussed in the previous section. First, income dynamics are described in subsection 5.1. Subsection 5.2 describes the instrument based on each individual's conditional expected income. Subsection 5.3 presents the expected tax rate instrument. The tax rate distributions using alternative instruments are examined in subsection 5.4

Income Dynamics
The model used here is essentially a stochastic model which identifies two types of relative income change, arising from non-tax related factors. These are 'regression towards the mean' and serial correlation in relative income changes. From Creedy (1985), the two process are captured by the following autoregressive form, where   is the arithmetic mean of log-income in period  and  is a random error term with variance,  2  : This can be rearranged as: In addition, if the age-profile of   is thought to be quadratic, then letting   denote 's age at time, , (8) can be replaced by: Thus (3) can be augmented by adding the terms on the right hand side of (9). Alternatively, , so this is consistent with having terms equal to the base period log-income and the previous period's log-income change. In the empirical analysis below, as in Giertz (2009), these variables are used as additional exogenous variables. However, as explained above, the independently estimated parameters  2 and  3 can also be used, along with  2  , in the construction of an instrument. The following subsections show how the alternative instruments can be constructed using estimates of these income dynamics parameters.
Following the tax policy change in 2001, the tax structure remained unchanged for a number of years. Estimation of the earnings dynamics model of equation (7) involves regressing log  05 −  05 on log  04 −  04 and log  03 −  03 , for the same individuals as used in the estimation of the elasticity of taxable income. It is nevertheless not possible completely to rule out marginal tax rate changes related to income growth for some individuals, since fiscal drag takes those who are near the upper income threshold of their tax bracket into a higher-rate bracket. But if inflation is low, the vast majority of income changes can reasonably be thought to reflect non-tax related income movements. The data are described in Section 6. The parameter estimates of  2 and  3 are 0.6677 and 0.1988, with -values of 145.49 and 43.41, with   = 262144. 15 The above specification is consistent with a dynamic process with regression towards the mean of , where log Creedy (1985). It can be shown that  = These values imply a high degree of regression towards the mean along with negative serial correlation whereby, for example, those who experience a large income increase are more likely to have a subsequent decrease. These results are consistent with those obtained using New Zealand incomes from the early 1990s; see Creedy (1998, pp. 188-191).

Tax Rate Applied to Expected Income
The instrument based on expected income uses the estimates from the income dynamics model to construct for each person an expected income in period 2 (the post-change year being considered), by projecting forward for the required number of years. An instrument for the tax rate change can then be constructed using the tax rate applicable to the 'expected' income which would arise from the dynamic process alone (rather than, in the standard instrument, the rate that would apply to an unchanged income). Formally, the procedure can be described as follows.
Use the parameter estimates from (7) to obtain values of 2 by projecting forward from period 1. This requires the values of   for the relevant years (period 1 and period 2, as well as the two years before period 1). (For the tax change considered below, it is necessary to project several years ahead, as discussed in the following subsection.) The values of  ( 2 ) give the expected income values in the second period under consideration if the above process of income change were to apply in the absence of any tax changes. Then obtain the tax rate,  *  [ ( 2 )] that would be faced by the individual, given ) and, as before, define   as   = log (1 −  1 ) − log (1 −  0 ), and let   denote age. Carry out a regression of the form: The exogenous variables other than age can be added to the right hand side of (10). Finally, using the parameter estimates from (10) to obtain the  values, carry out a regression of the form: One problem arising from this method is that in projecting forward, it is necessary to use the value of  2 , that is, the mean log-income in the post-change year. Thus the estimated parameters are used with the autoregressive form to obtain, say, Φ  , defined as the appropriate right hand side of (7) obtained by setting the stochastic term equal to zero and projecting forward from the base year. In the intermediate stages only the terms ¡ log   −   ¢ are needed, rather than separate values of   and log   . But in the final step, 2 = exp ( 2 + Φ  ), and the desired value of  is that which would occur in the absence of tax changes, but of course cannot be known. Hence, using the actual  for year 2 is tantamount to assuming that it was not substantially affected by the tax change; that is, in aggregate the tax effects -from some individuals experiencing higher rates while others experience lower rates -are small relative to the other influences on aggregate income growth. This approximation is unavoidable.

An Instrument Based on Expected Tax Rate
This subsection examines the construction of the alternative instrument which makes use of the conditional distribution of income that is predicted in the absence of tax changes, rather than only the expected value. Given a distribution of income for each individual, conditional on income in the two years preceding the tax change, it is possible to calculate an expected tax rate. Clearly there is no reason to expect this rate to be the same as that applied to the conditional expected value of income in the post-change period. Similarly, the average will not generally correspond to a statutory marginal rate in the multi-rate structure.
As before, let   denote individual 's income at time , and let   denote arithmetic mean log income at time, . The process of relative income change is the same as described in (7) above, which allows for serial correlation and regression towards the mean in the process of relative income change. Rearranging this equation gives: Hence, assuming that  (  ) = 0 and  (  ) =  2  are the constant mean and variance of   for all , taking expectations gives: (13) and the variance of logarithms of conditional income is: As explained earlier, in the context of the tax change in New Zealand, it is necessary to obtain values relating to 2002, given incomes in 1999 and 1998. Hence, moving forward one year gives: with a variance of logarithms of: Finally, moving a further year forward gives: with a conditional variance of logarithms of: These last two expressions can be used to give the mean and variance of log-income in 2002 conditional on income in 1999 and 1998. The variance is of course the same for each individual. It is possible to find the expected tax rate for the individual in period  + 2, given a set of tax thresholds and rates, as follows. Suppose the income tax function has rates   for  = 1   applying between income thresholds   and  +1 where  1 = 1 and On the assumption that the  are normally distributed, log-income is normally distributed and the probability that the individual falls into the th bracket is: where  (| 0 1) is the area to the left of  of a standard normal distribution. Here  ¡  +2+1¯0  1 ¢ = 1 and  ¡  +21¯0  1 ¢ = 0. The expected tax rate for the individual,  ( +2 ) is thus: This gives the expected tax rate instrument,  *  [ ( 2 )] =  ( +2 ), for each individual.

The Distribution of the Tax Rate Instruments
In section 6, these three tax rate instruments are used to calculate the instrumented change in the net-of-tax rate. First it is useful to compare the distributions of  *  for the three cases. That is, each of the three methods yields a predicted tax rate in 2002 for each taxpayer. For the standard and expected income instruments these are represented by the four statutory tax rates in the 2002 schedule. For the expected tax rate instrument, being an income-weighted average, in principle these may take any number of possible values between the lowest and highest statutory rates, of 15   The top half of Figure 2 shows the percentage of taxable income associated with taxpayers facing the respective marginal rates based on the first two instrumental variable measures. This 'taxable income share' distribution is more relevant for behavioural responses than the equivalent share of taxpayers. As can be seen in Figure 2, the share of income facing the four different rates is quite similar. However, whereas the standard instrument produces an increasing share of income across the 21, 33 and 39 per cent rates, the reverse is true for the expected income tax instrument. Taxpayers facing the 39 per cent marginal rate (based on these instruments) account for almost 40 per cent of taxable income using the standard instrument, but the corresponding proportion is less than 20 per cent based on the expectd income instrument. This probably reflects the ability of the expected income instrument to capture the likelihood that some of those observed (pre-reform) in the top tax bracket, experience an income fall that pushes them into a lower tax bracket. (Those experiencing an expected income increase, on the other hand, cannot shift into a higher tax bracket). The standard instrument cannot accommodate this aspect.
The lower half of Figure 2 shows the equivalent histogram for the expected tax rate instrument. Both the share of taxpayers, and the share of taxable income, are included for comparison. In each case the expected tax rate shown on the horizontal axis, for example, 24 and 25 per cent, represents the share of income, or taxpayers, having a tax rate instrument lying between 24.00 and 24.99 per cent, 25.00 and 25.99 per cent and so on.
The resulting range of expected tax rates is, in fact, much narrower than for the other two instruments, lying between rates of 24 and 33. This reflects the weighting process across the probability distribution of possible tax rates with a minimum and maximum rate respectively of 15 and 39 per cent. The distribution of taxable income shares associated with these tax rates can be seen to be slightly asymmetric around a mode of 28 per cent, with a more-skewed distribution using the share of taxpayers which has a mode of 27 per cent. 16 6 Applications: The 2001 Tax Change Table 5 in section 4 shows the New Zealand income tax structure reforms in 2001. After a period with relatively few changes, the 2001 reforms represented a significant policy change, involving a number of tax rate changes, but especially an increase in the top marginal rate from 0.33 to 0.39 above $60,000. This policy change is examined using comparisons of top income shares by Claus et al. (2012). They show that the announcement of the tax changes led to a certain amount of income shifting between periods, so that a comparison between incomes in 2000 and those immediately after the change gives highly misleading results. Using a longer interval allows for these inter-temporal shifts in income to settle down. The income dynamics model discussed above allows for the possible effect of regression towards the mean in generating relative income changes that are independent of tax changes, along with serially correlated changes; that is, proportional changes may depend on previous proportional changes as well as relative income position. In examining the 2001 New Zealand tax change, period 2 refers to 2002. Period 1 refers to 1999, so that the use of lagged income terms requires information on incomes in 1998.
In addition to the age terms and income terms in the regression, a dummy variable was added to allow for the composition of income. This dummy was set equal to zero if the individual received only wage or salary income in 1998, 1999 and 2002, the three years used in the regressions. The dummy was set equal to 1 if the individual received, either in addition to or instead of wage and salary income, any 'other income'. Other income includes the following official income categories: dividends, trust and estate income, partnership, rental income, business or other income, shareholder employee income, and overseas income. Subsection 6.1 describes the special dataset used here. Subsection 6.2 presents the regression results, while subsection 6.3 considers in further detail the characteristics of those who responded to the tax changes.

The Data
The database used here was constructed by randomly sampling the Inland Revenue Department's individual taxpayer population, and covers the period 1994-2009. The number of taxpayers in the random sample rises from 138,464 in 1999 to 139,420 in 2002. The sample is weighted to match the individual taxpayer population, which increased from 2,800,528 taxpayers in 1999 to 2,962,200 in 2002. For the regressions outlined below, various restrictions are imposed on the data. Age restrictions are imposed in order to remove those taxpayers likely to be in the very early stages of their careers as well as those becoming eligible for New Zealand superannuation. Only taxpayers aged 25-64 across the entire period are included. Income restrictions are also imposed, in order to remove very high income earners (over $1 million in 1999) and those likely to be receiving some form of government benefit (under $16,000). The latter face abatements rates which mean that their effective marginal tax rates differ significantly from those of a standard taxpayer. Finally, those without sufficient income data across all relevant years (1998, 1999, 2002, 2003, 2004 and 2005) are necessarily excluded. As a result, the sample size is reduced to 38,744, which, when weighted up to reflect the population, represents 803,920 individual taxpayers. Further details of the data, the restrictions and the sampling process are given in Appendix B.

Regression Results
The results of applying the three alternative instruments to a sample of over 800,000 taxpayers are reported in Table 7. The standard instrument gives an estimated elasticity of a huge negative number, −175 (-value = −011) which is obviously meaningless. Also, none of the coefficients on the age and income variables is significantly different from zero.
Introducing the first of the alternative instruments, the tax rate associated with expected income in 2002, in Table 7, radically changes the parameter estimates. In particular, using the expected income instrument, the estimate of the elasticity of taxable income becomes 0.575, (-value = 199) and all variables are significantly different from zero at standard levels of significance. Furthermore, this value is in the range obtained by Claus et al. (2012), using non-regression methods.
Using the expected tax rate instrument has a modest impact on the point estimate of the elasticity of taxable income (0.676 compared with 0.575) but more than halves the standard error, resulting in a -value of 5.4. Similarly all variables in the regression now have higher coefficient -values. The use of the expected tax rate instrument therefore appears to substantially improve the robustness of the estimated marginal tax rate effect on taxable income, with a plausible mean value.
The specification in Table 7 only includes an intercept shift dummy, allowing for observed income changes to differ for taxpayers with other income from those with only wage and salary income. However, it cannot capture the potential for different tax rate responsiveness by those with other income; for example, if other income is easier to shift, re-classify or evade for tax purposes. To allow for the possibility that the elasticity coefficient on ∆ log (1 −  ) depends on the composition of income, Table 8 adds an  interaction term equal to the product of the dummy variable and ∆ log (1 −  ). 17 The results suggest that the elasticity for those without other income is smaller, at 0.414 (-ratio = 239) , while the coefficient on the interaction term is 0.495 and significantly different from zero (-ratio = 213). That is, the estimated elasticity of taxable income for those who have only wage and salary income, at 0.414, is around half of the value for those who have income from other sources, at 0.909 (= 0414 + 0495). This latter value is similar to values found by Claus et al. for the higher income groups. It is consistent with the findings for the US by Saez (2004) that taxpayers' non-salary income appears to be especially responsive, via income shifting, to marginal tax rates and other tax parameters. 18 However, Saez's (2004) evidence for the 1986 US tax reform was less clear regarding whether the observed growth of wage and salary income after reform represented a tax response.
To test this aspect further, the total sample was decomposed into taxpayers with and without at least one of the categories of non-wage-or-salary income sources (dividends, trust income and so on). If it is either easier to alter other income than salary income, or taxpayers have a greater propensity to do so, in response to tax changes, then it might be expected that those taxpayers with non-salary income would demon- 17 Here the term ∆ log (1 −  ) in the table denotes the difference, log (1 − 02 ) − log (1 −  99 )  18 Although there is a considerable overlap in the income distributions of those with a zero dummy and those with a dummy equal to 1, income at the 90th percentile of the two distributions is $53,704 and $87,714 respectively. That is, the richest 10% (in terms of total taxable income) of taxpayers with positive other income have substantially higher income compared to the richest 10% of taxpayers with no other income. strate a larger elasticity than taxpayers with no other income. Secondly, among the subset of taxpayers with other income, the responsiveness of their other income to a change in marginal tax rates might be expected to be greater than the equivalent response for salary income. This is likely to be important in the case of New Zealand's 2001 reform. Though all types of personal income (salary, dividends and so on) above the new $60,000 threshold after the reform were taxed at 39 per cent, rather than 33 per cent, income received by trusts and companies continued to be taxed at 33 per cent. Diversion of income to trusts, and incorporation, are relatively easy (with a low cost) in New Zealand, and the new 6 percentage point gap provided a strong incentive to shift income away from the personal tax code to those alternatives. This can be expected to have further induced reductions in other income received by individual taxpayers in the current sample in response to the tax rise. 19 Table 9 shows the regression parameters on ∆ log (1 −  ) for the specification in Table 7 but for those taxpayer/income sub-samples. The estimate for all taxpayers (0.676) is repeated from Table 7. Splitting the sample into taxpayers who had other income in at least one of the three years (1998,1999,2002), and those with only salary income gives a much larger parameter estimate of 0.514 for those with other income compared with the estimate of 0.19 for those with no other income. Only the former is statistically significant at conventional levels with much more noise associated with the 0.19 estimate.
Testing the sub-set of taxpayers with non-salary income both before and after reform (1999 and 2002), Table 9 confirms that the responsiveness of other income is substantially greater, with a parameter estimate on ∆ log (1 −  ) of 2.48 (-ratio = 728) for other income, compared with 0.22 (-ratio = 153) for salary income.
The importance of the other income component of total taxable income, and its response to the 2001 reform, can be seen in Figure 3. This shows the percentage distribution of all other income across ($5000) income bands both in 1999 and 2002. The first point to note is that 'other income' is not especially concentrated among taxpayers with high taxable incomes; the bulk of other income is received by taxpayers in the $30-70 taxable income range. This tends to suggest that the estimate above of high responsiveness of other income to tax rate changes is not exclusively a highincome earner phenomenon. Secondly, the clearest difference between the 1999 and 20002 distributions of other income is the new large spike in 2002 around the new $60 threshold introduced in the 2001 reform. That is, a much larger fraction of other income in 2002 is accounted for by taxpayers with income around $60 than was the case in 1999, with a compensating decrease in other income received by taxpayers with taxable incomes around $25-35.
Unfortunately, the available data do not allow investigation of intra-household transfers of income in response to the 2001 tax reform. However, the evidence in Figure 3 may be indicative, in part, of previously low-income earners within a household (where the higher earner has income in excess of $60) taking a greater share of declared household income after reform. That is, both household members' incomes move towards the $60 threshold in opposite directions. Section 6.3 below discusses in more detail how different types of individuals responded to the tax reform.

Who Responded to the 2001 Reform?
The above results suggest that taxpayers in receipt of non-wage and salary income responded especially strongly to the 2001 reform and, in particular, by altering the declared 'other income' component of their taxable income. This subsection considers whether these were exclusively, or mainly, those on higher incomes facing the 33 to 39 per cent tax rate change, or whether this response applied more generally.
Furthermore, the 2001 New Zealand tax reform involved a combination of constant, increasing and decreasing tax rates, so it is possible to identify the categories of taxpayer shown in Table 6 who contributed most to the observed responses. Table 6 shows that it was mainly taxpayers in categories 3, 6, 7, 15 and 16 whose incomes responded in the expected direction. These categories account for 75 per cent of all taxpayers in the sample. From the combinations of ∆, ∆ , and ∆ * which each category represents, it is possible to identify those tax brackets within the New Zealand tax system in which those taxpayers are located. Figure 4 shows the tax schedules for 1999 and 2002, with tax rates rising for incomes above $60,000, remaining constant between $38,000 and $60,000 and falling for taxpayers between $14,000 and $38,000. The Figure also shows the five categories of taxpayer of interest. The unbroken arrows indicate the observed movement in those taxpayers' incomes and marginal tax rates between 1999 and 2002; the broken arrows indicate the predicted movement in their 'expected tax rate' in the absence of reform based on the income dynamics described earlier. For example, consider category 16, involving 152,000 taxpayers. Those taxpayers experienced a fall in their income and actual tax rate, while their predicted tax rate rose. This included two groups: taxpayers between $38 and $60 in 1999 whose incomes fell to below the $38 threshold in 2002 but who were (in the absence of the reform) expected to move above the $60 threshold. It also includes taxpayers with incomes above $60 in 1999 whose 2002 income fell to less than $38. Their responses are discussed further below.
Category 7 is another large sub-set of 194,000 taxpayers. The experienced income increases took them towards the $34.2 (1999) or $38 (1999 and 2002) thresholds, but their predicted tax rate increase implies that they were expected to experience an income increase to above the $38 threshold. Thus the higher jump in marginal rates at $38 after 2001 (from 21 to 33 per cent instead of from 24 to 33 per cent) may have persuaded some taxpayers to declare lower income than otherwise expected, keeping their 2002 declared income below $38.
In addition to the categories listed above, category 1 in Table 6 captures a large number of taxpayers (122,000) where ∆, ∆ , and ∆ * are all positive. This includes taxpayers for whom their 'no reform' predicted income increase exceeds their actual income increase; that is, their response is consistent with a smaller declared income increase in response to the tax rate change from 33 to 39 per cent. This includes taxpayers below $60 in 1999 and 2002 who would otherwise have crossed that threshold by 2002, and those above the $60 threshold in both years, as shown in Figure 4. The standard instrument cannot account for the former group (below $60 in 1999 and 2002) because of the restriction that the instrumented tax rate is based on unchanged income levels (∆ * = 0 for incomes in the range $38 and $60).
In summary, actual and expected taxpayer income movements, as depicted in Figure 4, suggest a large amount of crossing, and bunching around, the $38 and $60 thresholds. This is confirmed by an examination of the distribution of taxable income in 1999 and 2002. Figure 5 shows the two distributions of aggregate taxable income by $1000 income band over the $16 to $100 range (the range relevant to the analysis here). In addition to the general tendency for taxable incomes to rise over the three years (the 2002 profile generally lies above the 1999 version), a large spike can be seen to emerge around $60 in 2002 which did not exist in 1999. Further, the small spike evident around $38 in 1999 is considerable larger by 2002. Figure 5 also suggests an increased concentration of taxable income in the $38 to $60 range in 2002 compared with 1999. The percentage of taxpayers and taxable income in this range rose from respectively 12 and 23 per cent to 15 and 26 per cent. Almost all of this reflected a net movement out of the $9.5 to $38 bracket. While the marginal tax rate in this bracket remained unchanged at 33 per cent before and after reform, the increased concentration here is consistent with expected behavioural responses to the combination of a reduced tax rate in the bracket below (24 to 21 per cent) and the increased rate in the bracket above (33 to 39 per cent).
Finally, actual Inland Revenue data on numbers of taxpayers in various income groups, and their share of income, in 2002 after the tax reform, can be compared with those predicted by the income dynamics modelled. This is especially useful for taxpayers with incomes above/below the $60 threshold where the new 39% tax rate was introduced. Actual 2002 data for all taxpayers shows that 14.2 per cent of all taxpayers with incomes over $16 (the equivalent group to that used above to model Figure 5: Distribution of Taxable Income dynamics), had incomes in excess of $60. They accounted for 33.2 per cent of total taxable income. 21 Using expected income and the associated tax rate as instrument, the equivalent percentages for predicted taxpayers and expected taxable income in excess of $60 are 17.3 and 38.0 per cent. These values suggest plausible responses by taxpayers to the higher marginal rate; that is, around 3 per cent of taxpayers, and 5 per cent of total taxable income shifted to below the $60 threshold after the reform, relative to what would otherwise have been expected.
By itself, the increased bunching of taxable income around these two thresholds in Figure 5, and income growth within the $38 to $60 bracket, might be considered merely suggestive of responses to the 2001 reform. However, the regression evidence and the income movements identified in Table 6 offer strong confirmation that this reflects the predicted causal behavioural responses to tax reform when those predictions are based on modelling income changes that occur both with and without that reform.

Conclusion
This paper has examined estimation of the elasticity of taxable income using instrumental variable regression methods. It has argued that the 'standard instrument' for the net-of-tax rate − the rate that would be applicable post-reform but with unchanged income levels − is unsatisfactory in contexts where there are large numbers of taxpayers with exogenous changes in their taxable income. Two alternative tax rate instruments were proposed, based estimates of the dynamics of taxable income for a panel of taxpayers over a period that involved no tax changes.
The parameters derived from that procedure were then used to construct hypothetical (or counterfactual) post-reform incomes that would be expected in the absence of reform. The first method is based on the tax rate each individual would face if their income were equal to 'expected income', conditional on income in two periods before the tax change. The second alternative uses the form of the conditional distribution of income for each taxpayer to obtain an instrument based on their 'expected tax rate'.
These methods were applied to the 2001 tax reform in New Zealand. This involved a convenient mix of marginal tax rate increases, decreases and no change across a wide range of incomes. Comparing taxable incomes in 1999 and 2002, the paper first examined taxpayer responses in terms of observed correlations between income change and changes in the actual and instrumented tax rates. Secondly, instrumental variable regressions were examined. Thirdly, these results were compared with observed and predicted changes in key parts of the taxable income distribution between 1999 and 2002. All three approaches suggest that observed income changes after reform reflect the causal behavioural responses to tax reform predicted by the elasticity of taxable income literature. However, an instrument based on the standard approach, of assuming unchanged income levels after reform, performed poorly. Instruments that are based on a model of income dynamics, estimated using extraneous information on incomes over a three-year period without any tax structure changes, performed much better, particularly the instrument based on an expected tax rate for each individual. Importantly, they produced estimated behavioural responses that are both plausible and consistent with earlier estimates obtained using different methods by Claus et al.

Appendix A: 1980s Tax Reforms in New Zealand
The first application of the instrumental variable approach to New Zealand was by Thomas (2007), who examined the1980s income tax reforms using IRD data. The same dataset is used here to consider the properties of the instrument. The tax structures for 1986 and 1988 are shown in Table 10. The reforms involved a reduction in income tax rates and a change in the direct-indirect tax mix as a result of the introduction of a Goods and Services Tax (GST). The analysis is restricted to individuals aged 25 to 65. The number of those in each of the possible outcomes, adjusted to population values using sample weights, is shown in Table 11, where the number indicated in the left hand column refers to the corresponding case in Table 3. The small number in category 18 is not surprising, since this relates to people who reported no change in their taxable income. The large number of individuals in category 18 have decreases in their taxable incomes despite the fact that they have a reduction in both the tax rate they face and the rate they would face without any change in income. A substantial component of the income change must therefore arise from non-tax related factors. In carrying out regressions using (3), Thomas (2007) used the following exogenous variables: age; age-squared; 1986 taxable income; 1986 capital income; an entrepreneurship dummy. He also reported results separately for taxable income and labour income, and for unweighted and weighted values. 22 For present purposes, it is useful to concentrate on taxable income, and only the age and 1986 income variables are included. 23 The income term is the most important exogenous variable: when only the terms involving age are included, both the ordinary least squares and instrumental variable estimates of the elasticity of taxable income were found to be negative (at around −2 and −08 respectively). With all three exogenous variables, the instrumental variable estimate of  was 0.61, and with only taxable income in 1986 included it was 0.58. 24 22 The elasticity in the latter case was found to be higher than for the former. Thomas argued that this demonstrated a higher elasticity for higher-income groups. 23 The inclusion of a term in age-squared implies an age-log-income profile that is cubic in form, so it is not expected to have much influence on the results, given that most profiles are closer to quadratic. 24 The -value when all three exogenous variables are included is 10.79. Excluding those in category 27 increases the instrumental variable estimate to 0.84, with a -value of 19.77.

Appendix B: The Inland Revenue Data
The data used in this paper are personal income information sourced from the New Zealand Inland Revenue Department's (IRD's) tax returns and employer PAYE records. The database is a stratified random sample, including 2 per cent of all wage and salary earners (which in turn includes people in receipt of taxable welfare benefits) and 10 per cent of all other individual taxpayers, such as the self-employed. The database omits individuals with no personal taxable income (unless they filed a tax return), and those whose only income was from investments with the correct amount of tax deducted at source and no requirement to file a tax return. The former group are not of interest for this study, and the latter are expected to be a fairly small group representing a very small proportion of total taxable income. The database does not include income not attributed to natural persons, for example income held in companies or trusts.
Randomness is ensured by sampling taxpayers based on the last two digits of their unique 'IRD number', which are issued broadly sequentially and not reflective of the characteristics of the specific individual. In order to ensure these are representative of the total individual taxpayer population, weights are applied to each observation in the sample according to the characteristics of the individual. For 1999, the database includes a total sample of 138,464 individual taxpayers, representing a total population of 2,800,528 taxpayers. For 2002 the sample size increases to 139,420, representing a taxpayer population of 2,962,200.
The database covers the years 1994 to 2009, and allows users to follow individuals across time by use of their IRD number. Because filing requirements have changed across time, the dataset contains a number of structural breaks. These include a break across the 1999-2002 period considered here, when the pre-populated personal tax summary (PTS) replaced the old IR5 tax return. This had a minor impact on some income tax data collected, particularly with regards to dividend and interest income below a small threshold. Aside from salary and wage income data, the database also includes data on business income, trust income, interest, dividends, rental income, shareholderemployee salary, partnership income and other income. Expenses and losses claimed (including those through LAQCs) are also recorded, as well as information on demographic characteristics such as date of birth and gender. These data are taken from a range of sources, largely tax returns submitted to the IRD.
For the regressions in this study, various restrictions are applied to the data. Firstly, in recognition that various unrelated behavioural changes may bias the results, those taxpayers who were younger than 25 in 1999, or older than 64 in 2002, are removed from the sample. This fairly common restriction removes those taxpayers likely to be in the very early stages of a career, as well as those likely to have retired at the age of 65 (the age of eligibility for New Zealand superannuation). Secondly, those with 1999 taxable income less than $16,000 or greater than $1,000,000 are excluded from the sample. The first of these restrictions is particularly important in order to remove a significant segment of the population who received some form of government benefit, as abatement rates mean that these individuals face different effective marginal tax rates to standard taxpayers. Finally, the sample is necessarily reduced to only those individuals who have sufficient data in all six relevant income years (ending 1998, 1999, 2002, 2003, 2004 and 2005). Some taxpayers either entered or exited the tax system over this time, which means that their income dynamics cannot be estimated. A number of smaller, less significant restrictions are also imposed, such as the removal of zero or negative taxable income values and data entry errors (such as negative ages). Combined, these restrictions reduce the sample size to 38,744, which, when weighted up to reflect the population, represents 803,920 individual taxpayers (29 per cent of the original 1999 weighted sample).