RDP 1999-02: Reservation Wages and the Duration of Unemployment 5. Estimation Strategy

If the conditional expectation of the error term in Equation 6 is zero and the stationary job-search framework is appropriate, then OLS estimation is the appropriate estimation strategy. However, if omitted variables result in the conditional expectation of the error term being non-zero or, if the reservation wage is dependent on unemployment duration, it will be necessary to use instrumental variable methods to ensure that parameter estimates are not biased and inconsistent.

The assumption that the conditional expectation of the error term is zero will be violated if omitted variables are correlated with other explanatory variables. In this case, theory suggests that all the variables which affect the job-offer arrival rate and the parameters of the wage-offer distribution will also affect the reservation wage. Therefore, in the case of omitted variables, there will be a correlation between the error term and the reservation wage variable which will exist even if the sample size is very large, ensuring that the OLS parameter estimates are inconsistent.

The solution to this problem is to instrument the reservation wage. The choice of instrument must be restricted to variables which directly affect the reservation wage but which do not affect the job-offer arrival rate or the wage-offer distribution. The most likely candidates are therefore variables which affect the cost of unemployment such as the level of unemployment benefits. This is discussed below.

The second assumption which must hold for OLS estimation to be appropriate is that the reservation wage has no duration dependence. If this is not true it is necessary to think about the duration of unemployment and the reservation wage as two endogenous variables in a simultaneous system. There are several arguments for why reservation wages will decline with duration. These include:

• declining job offers because employers use duration as a screening device or because human capital diminishes with time spent out of work;
• limits to search due to fixed working life, fixed assets, limited duration of unemployment benefits; and
• learning about the wage-offer distribution.

In terms of the derivation of the duration equation above, the progression from Equation 1 to Equation 2 will no longer be valid in general. However, by making specific assumptions about the relationship between the reservation wage and the duration of unemployment, Lancaster (1985) shows that it is possible to derive a tractable expression for the reservation wage equation in a simultaneous system, where the duration equation is the same as Equation 6. In particular, by assuming that the reservation wage declines exponentially as a function of the duration of unemployment, and that its minimum level is an exponential function of background characteristics, the second equation of the simultaneous system will be log linear.

The parameter ρ will have a structural interpretation as the exponential rate of decline in reservation wages with duration, if the underlying assumptions are accepted. Regardless of whether the coefficients can be interpreted as structural estimates, the estimation of Equation 6 will still require instrumental variable techniques to deal with the endogeneity bias introduced by the log of the reservation wage. We have already argued that all the variables in X (Equation 6) must also be in Z (Equation 7). Therefore identification of our unemployment duration equation requires that there are some variables in Z and not in X. These variables are appropriate instruments for the reservation wage in estimating Equation 6. Practically, the choice of instruments will be the same as for the omitted variable single equation case discussed above.

The best candidates for instruments are variables which affect the costs of unemployment, because they affect the level of the reservation wage but are not generally considered to affect the job-offer arrival rate or the parameters of the wage-offer distribution. The most obvious candidate for an instrument is therefore the level of unemployment benefits. This is the instrument used by Jones (1988) and Gorter and Gorter (1993).

As already mentioned, the SEUP does not provide information about the level of unemployment benefits, although we have been able to construct a variable which measures eligibility. This would be a valid instrument except that it is likely to affect the degree of search intensity and hence, the job-offer arrival rate. Wadsworth (1991) provides evidence of this for the UK. Further evidence from UK data suggests that although eligibility increases the number of job-search methods used, through information and incentive effects, there is no evidence that the level of unemployment benefits has an effect (Schmitt and Wadsworth 1993). Consequently, valid instruments will be variables which affect the level of benefits rather than eligibility for benefits. The variables we have available which best fit this description are family size which will be correlated with the number of dependents and the income of the family unit. Lancaster (1985) uses the number of dependents as an instrument.