Standard error correction in two-stage estimation with nested samples

Data at different levels of aggregation are often used in two-stage estimation, with estimates obtained at the higher level of aggregation entering the estimation at the lower level of aggregation. An example is customers within markets: first-stage estimates on market data provide variables that enter the second-stage model on customers. We derive the asymptotic covariance matrix of the second-stage estimates for situations such as these. We implement the formulae in the Petrin–Train application of households’ choice of TV reception and compare the calculated standard errors with those obtained without correction. In this application, ignoring the sampling variance in the first-stage estimates would be seriously misleading.