This paper presents a general minimum distance estimation method for nonstationary time series models with evolutionary spectral representations. The method extends Whittle's approach, which is used for stationary processes, to nonstationary ones. The asymptotic properties of the estimator are analyzed under model misspecification. The paper compares the Whittle estimator to the least squares estimator for autoregressive processes with time-varying coefficients and discusses the behavior of estimators when a stationary model is applied to a nonstationary process. The paper introduces the concept of locally stationary processes, which are processes that exhibit stationary behavior over short time intervals. It defines a class of nonstationary processes with a time-varying spectral representation and provides a framework for asymptotic statistical inference. The paper also discusses the estimation of parameters for such processes, using a generalization of the Whittle function with local periodograms. The paper proves the asymptotic normality of the estimator under model misspecification and shows that the estimator is efficient. It also discusses the behavior of the estimator when a stationary model is fitted to a nonstationary process. The paper provides a detailed analysis of autoregressive models with time-varying coefficients, showing that the estimator is consistent and asymptotically normal. It compares the minimum distance estimator to the least squares estimator and shows that the minimum distance estimator is more efficient in the heteroscedastic case. The paper concludes with a discussion of model selection and the use of graphical tools for diagnostics. It also discusses the computational aspects of the estimator and its efficiency in different scenarios. The paper provides a comprehensive analysis of the estimation of nonstationary time series models and their asymptotic properties.This paper presents a general minimum distance estimation method for nonstationary time series models with evolutionary spectral representations. The method extends Whittle's approach, which is used for stationary processes, to nonstationary ones. The asymptotic properties of the estimator are analyzed under model misspecification. The paper compares the Whittle estimator to the least squares estimator for autoregressive processes with time-varying coefficients and discusses the behavior of estimators when a stationary model is applied to a nonstationary process. The paper introduces the concept of locally stationary processes, which are processes that exhibit stationary behavior over short time intervals. It defines a class of nonstationary processes with a time-varying spectral representation and provides a framework for asymptotic statistical inference. The paper also discusses the estimation of parameters for such processes, using a generalization of the Whittle function with local periodograms. The paper proves the asymptotic normality of the estimator under model misspecification and shows that the estimator is efficient. It also discusses the behavior of the estimator when a stationary model is fitted to a nonstationary process. The paper provides a detailed analysis of autoregressive models with time-varying coefficients, showing that the estimator is consistent and asymptotically normal. It compares the minimum distance estimator to the least squares estimator and shows that the minimum distance estimator is more efficient in the heteroscedastic case. The paper concludes with a discussion of model selection and the use of graphical tools for diagnostics. It also discusses the computational aspects of the estimator and its efficiency in different scenarios. The paper provides a comprehensive analysis of the estimation of nonstationary time series models and their asymptotic properties.