Income Mobility of Owners of Small Businesses when Boundaries between Occupations are Vague

where Φ is the standard normal cumulative distribution function. Similarly, we can also see the
placement in period t for individuals who are non-rich in period t-1 as following from the same type of
modeling:

(5)         y_i^*_t =z_i^'_tγ+α_i+ξ_it,

where z_i^'_t is a vector of explanatory variables explaining outcome in t, given that the individual is non-
rich at t-1, and ξ_it is an normally distributed error term. This leads to a joint probability for observing
both being rich in period t and being non-rich in period t-1:

(⁶)       P [ У_и = 1, У_и-1 = 0 ]=^φ2( zit-1Y ^{+ α}i^{, -} xtβ ^{- α}i),

where Φ₂ is the cumulative distribution function of the bivariate standard normal. Unobserved
heterogeneity in equations (3) and (4) is represented by individual specific time invariant effects, α_i ,
plus error terms, τ_it-1 and ξ_it in the two equations, respectively. Letting u_it-1 = α_i +τ_it-1

and ε_it = α_i +ξ_it , the approach taken in this study, i.e., focusing on non-causal results, means that the
relationship between error terms, (τ_it-1,ξ_it), is not explicitly addressed. Obviously, unobservables
determining the base year probability to be non-rich are correlated with the unobservables determining
the conditional transfer into being rich in period t. This is a version of the initial condition problem of
dynamic discrete choice models, see Heckman (1981). However, endogeneity problems in this
analysis are also due to the key explanatory variable, business ownership, being clearly endogenous
with the income hierarchy placement in period t-1 and with the placement in period t. Thus, given
these measurement problems, we interpret results in terms of non-causal relationships, focusing on
correlations between income hierarchy movements and business ownership. If we neglect the
dependency between error terms, the conditional probability can be seen as,

Equation (7) defines the probability of being rich in at t, conditional on being non-rich at t-1, as related
to a number of explanatory variables, including being an owner of a small business. With information
from 10 waves of transitions, {1993/94,1994/95,...,2002/2003}, we estimate an average correlation
between business ownership and transitions into “rich” for the period 1993-2003, when the dual
income tax system was in force. A similar type of reasoning generates a corresponding equation for
the probability of staying rich, given that the individual was rich in period t-1. We use an identical set

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