3 The Specification
Consider an income tax change between periods 1 and 2, involving changes in marginal rates, tj,k, for periods j = 1,2, and tax brackets. The income thresholds may also change from period 1 to 2. Information is available about the taxable income, yj,i of i = 1,...,N individuals in each period. Let τj,i denote the marginal tax rate actually faced by individual i in period j. This is the appropriate rate, depending on yj,i, from the set of rates tj,k. Define:
which is an approximation for the proportional change in the net-of-tax rate, 1 - τ. The proportional change in taxable income, qi, is approximated by:
The 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 ui is a random variable and η is the elasticity of taxable income. There are inevitably income changes which would occur in the absence of tax changes. The challenge is thus to avoid attributing those exogenous income changes to the tax rate changes.
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 approach in which the instrument, zi, is used, defined as:
where τ2,i* is the marginal tax rate that would be faced by the individual in period 2 if taxable income were to remain constant at y1,i.
The first stage involves a linear regression of xi on zi 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:
The second stage then involves estimating the elasticity of taxable income, η using ordinary least squares on:
The question examined here is whether this is a reliable approach, bearing in mind that for z to be a good instrument, it must be correlated with x but independent of the errors, and uncorrelated with q other than via any effects on x.[8]
Notes
- [8] The regression of x on z implies (in the absence of other exogenous variables) that for zi = 0,
. These different cases thus have values of qi aligned along a straight line at .
