4 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.

4.1 The Standard Instrument

Figure 1 shows a segment of a multi-step tax function. An individual in the kth tax bracket with initial (period 1) taxable income of y₁, faces the (pre-reform) marginal tax rate, t_k. A reform which decreases t_k 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, y₂ or y₂^′ respectively. Thus, where exogenous income increases and the expected tax effect operate in the same direction, observed income increases to y₂. 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 y₂^′.

Figure 1: Income and Tax Rate Changes

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 t_k, 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. There can be a resulting bias in standard instrument estimates of the income response to a tax structure change in the presence of exogenous income changes for which allowance is not fully made. For example, an increase in τ_k 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.[10] 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, estimates based on reforms involving tax cuts (increases in 1 - τ_k), in association with average nominal or real incomes increases (independently of tax reform) suggests the possibility of negative biases.

Hence it is important first to model 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. 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 used directly in the construction of a tax rate instrument.