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.