Here we present a method of genome-wide inferred study (GWIS) that provides an approximation of genome-wide association study (GWAS) summary statistics for a variable that is a function of phenotypes for which GWAS summary statistics, phenotypic means, and covariances are available. A GWIS can be performed regardless of sample overlap between the GWAS of the phenotypes on which the function depends. Because a GWIS provides association estimates and their standard errors for each SNP, a GWIS can form the basis for polygenic risk scoring, LD score regression, Mendelian randomization studies, biological annotation, and other analyses. GWISs can also be used to boost power of a GWAS meta-analysis where cohorts have not measured all constituent phenotypes in the function. We demonstrate the accuracy of a BMI GWIS by performing power simulations and type I error simulations under varying circumstances, and we apply a GWIS by reconstructing a body mass index (BMI) GWAS based on a weight GWAS and a height GWAS. Furthermore, we apply a GWIS to further our understanding of the underlying genetic structure of bipolar disorder and schizophrenia and their relation to educational attainment. Our analyses suggest that the previously reported genetic correlation between schizophrenia and educational attainment is probably induced by the observed genetic correlation between schizophrenia and bipolar disorder and the previously reported genetic correlation between bipolar disorder and educational attainment.