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This paper provides an analytical study of linear regressions involving the levels of economic time series. An asymptotic theory is developed for regressions that relate quite general integrated random processes. This includes the spurious regressions of Granger and Newbold (1974) and the recent cointegrating regressions of Granger and Engle (1985). An asymptotic theory is developed for the regression coefficients and for conventional significance tests. It is shown that the usual t- and F-ratio test statistics do not possess limiting distributions in this context but actually diverge as the sample size T ↑ ∞. The limiting behavior of regression diagnostics such as the Durbin–Watson statistic, the coefficient of determination and the Box–Pierce statistic is also analyzed. The theoretical results that we present explain many of the earlier simulation findings of Granger and Newbold, 1974, Granger and Newbold, 1977.