5 Construction of Alternative Instruments
Subsection 5.1 describes the model of income dynamics used to construct the alternative instruments discussed in the previous section. Subsection 5.2 presents 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
5.1 Income Dynamics
The model used here is 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 processes are captured by the following autoregressive form, where μj is the arithmetic mean of log-income in period j and u is a random error term with variance, σu2:
In addition, if the age-profile of μj is thought to be quadratic, then letting si denote i's age at time, j, (8) can be replaced by:
Thus (3) can be augmented by adding 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.
Following the 2001 tax policy change the tax structure remained unchanged for a number of years. Hence estimation of equation (7) involves regressions based on post-reform years 2003 to 2005. Thus, the model regresses log y05,i -μ05 on log y04,i - μ04 and log y03,i - μ03, for the same individuals as used in the estimation of the elasticity of taxable income. The evidence in Claus et al. (2012), who examine taxpayer income share changes, suggests strongly that responses to the 2001 tax reform did not persist into the 2003-05 period.[14] The data are described in Section 6. The parameter estimates of α2 and α3 are 0.6677 and 0.1988, with t-values of 145.49 and 43.41, with σu = 2.62144.[15]
The above specification is consistent with a dynamic process with regression towards the mean of β, where log yj,i - μj = β (log yj-1,i - μj-1) + ui,j, and first-order serial correlation of γ, where ui,j = γui,j-1 + εi,j. Hence, α2 = γ + β and α3 = -γβ; see Creedy (1985). It can be shown that β = {α2 + (α22 + 4α3)0.5} /2 = 0.891 and γ = {α2 - (α22 + 4α3)0.5} α2 - α 2 + 4α3∕2 = -0.223. 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).
Notes
- [14] See Claus et al. (2012, Figures 1 and 2). Some individuals experience marginal tax rate changes resulting from fiscal drag. However, with low inflation, the majority of income changes over this period can reasonably be thought to reflect non-tax related income movements.
- [15] The mean of logarithms of income in 2003, 2004 and 2005 are 10.311, 10.367 and 10.367, with standard deviation of logarithms of 0.9194, 0.9110 and 0.9651 respectively.
