Structural Spurious Regressions and A Hausman-Wu-type Cointegration Test∗

Economic models often imply that certain variables are cointegrated. However, tests often fail to reject the null hypothesis of no cointegration for these variables. One possible explanation of these test results is that the error is unit root nonstationary due to a nonstationary measurement error in one variable. For example, currency held by domestic economic agents for legitimate transactions is very hard to measure due to currency held by foreign residents and black market transactions. Therefore, money may be measured with a nonstationary error. If the money demand function is stable in the long-run, we have a cointegrating regression when money is measured with a stationary measurement error but have a spurious regression when money is measured with a nonstationary measurement error. We can still recover structural parameters under certain conditions for the nonstationary measurement error. This paper proposes two estimators based on asymptotic theory to estimate structural parameters with spurious regressions involving unit root nonstationary variables. This approach motivates a Hausman-Wu-type test for the null hypothesis of cointegration for dynamic Ordinary Least Squares estimation using one of our estimators for spurious regressions. We show that this test can be used to test for cointegration even when the spurious regression is not structural under the alternative hypothesis.