4.3 Correlations among Measures

To explore the merits of these alternative instruments, consider their correlations with observed income changes. Of interest here is the observed changes in income, Δy, and tax rates, Δτ, and the change in the relevant tax rate instrument, Δτ^*. To simplify the exposition at this stage, changes in tax rates, τ, rather than net-of-tax rates, 1 - τ, are considered.

Table 1 shows, for any individual, the possible combinations of Δy, changes in the actual tax rate Δτ = (τ₂ -τ₁), and the instrumented tax rate Δτ^* = (τ₂^*-τ₁) 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 1, leaving 18 possible cases.[11] 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 Δy, Δτ, and Δτ^* in Table 1, Table 2 shows the unconditional partial correlations, ρ, between the income change and each of the three tax instruments, where Δτ₁^* is the (change in the) standard instrument discussed above, but applied to Δτ rather than Δ^{(1 -τ)}. Tax changes for the new proposed instruments are shown as Δτ_E(y)^* and Δτ_E(τ)^*.

The table identifies, with a tick (✓), those categories where the correlation between the income change and each tax instrument takes the expected, ceteris paribus, negative sign: ρ < 0. Other entries ('incorrect' zero or positive correlation: ρ ≥ 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 (cases 4 and 8) or because they are not feasible for the particular tax instrument in question. Table 2 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 property of the standard approach whereby 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 3 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.[12] These represent approximate percentage changes in the three reformed tax rates (using log differences) of -3.5, -13.4 and +16.7 per cent.[13] 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 2 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 4 shows the numbers of taxpayers in each category, separated into those categories where ρ ≥ 0 (columns 1–4) and ρ < 0 (columns 5–8), where ρ 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 4 shows that the total numbers of correlations involving the expected tax rate, τ_E(τ)^*, yield quite different outcomes from those involving the other two instruments: τ^*(y₁) and τ_E(y)^*. In particular, there is a much higher ratio of incorrectly signed correlations (ρ ≥ 0) to correctly signed correlations (ρ < 0) for the standard and expected income instruments. Around 54 per cent of correlations, 431 and 433 out of 803 respectively for standard and expected income instruments reveal ρ ≥ 0.However, for τ_E(τ)^* 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 4 reveals that, across instruments, the numbers of positive or negative correlations can be different within each category. For example, there is a 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.