In this paper, we develop an efficient estimation method and an asymptotic chi-square testing procedure in regression models with errors having conditional heteroskedasticity generated by an integrated covariate in both stationary regression and cointegrating regression. In the presence of nonstationary volatility in the regression errors, it is known that the least squares estimator suffers from the second order biases generated by heteroskedasticity and endogeneity, and the standard chi-square test becomes invalid. It is shown that the efficient estimator proposed in the paper is asymptotically unbiased and follows a mixed normal distribution, and the test based on the estimator has chi-square limit distribution. Finite sample performances also confirm our theoretical findings.