Using administrative and survey data I show that survey misreporting leads to biases in common statistical analyses. Standard corrections for measurement error cannot remove these biases. I describe a method to obtain consistent estimates by combining parameter estimates from the linked data with publicly available data. This solves the problem that access to linked administrative data is usually restricted. I apply this method to study Food Stamp Program (now SNAP) receipt. I examine the degree to which this approach can be used to extrapolate across time and geography, in order to solve a common limitation of validation data. Administrative data on SNAP receipt and amounts linked to American Community Survey data from New York State show that survey data can misrepresent the program in important ways. For example, more than one billion dollars received are not reported in New York State alone. 40 percent of dollars received by households with annual income above the poverty line are not reported in the survey data, while only 13 percent are missing below the poverty line. The conditional density method I describe recovers the correct estimates using public use data only. I present evidence from within New York State that the extent of heterogeneity is small enough to make extrapolation work well across both time and geography. Extrapolation to the entire U.S. yields substantive differences to survey data and reduces deviations from official aggregates by a factor of 4 to 8 compared to survey aggregates.