Nonstationary Regression Research Articles

Data-Driven Bandwidth Selection for Nonparametric Nonstationary Regressions

We provide a solution to the open problem of bandwidth selection for the nonparametric estimation of potentially non-stationary regressions, a setting in which the popular method of cross-validation has not been justified theoretically. Our procedure is based on minimizing moment conditions involving nonparametric residuals and applies to β-recurrent Markov chains, stationary processes being a special case, as well as nonlinear functions of integrated processes. Local and uniform versions of the criterion are proposed. The selected bandwidths are rate-optimal up to a logarithmic factor, a typical cost of adaptation in other contexts. We further show that the bias induced by (near-) minimax optimality can be removed by virtue of a simple randomized procedure. In a Monte Carlo exercise, we find that our proposed bandwidth selection method, and its subsequent bias correction, fare favorably relative to cross-validation, even in stationary environments.

Corruption and the environmental Kuznets Curve: Empirical evidence for sulfur

We investigate how corruption influences the income level at the turning point of the relationship between sulfur emissions and income, using a wide cross-national panel of countries, at different levels of development and with different degrees of corruption. Our results support the Environmental Kuznets Curve hypothesis for sulfur. We find evidence that the higher the country's degree of corruption, the higher the per capita income at the turning point, suggesting different income–pollution paths across countries due to corruption. We build upon a new specification for the EKC developed by Bradford et al. (2005) that avoids using nonlinear transformations of potentially nonstationary regressors in panel estimation. Also, we account for the indirect impact of corruption on emissions through its impact on per capita income. Our main findings remain unchanged when we investigate additional heterogeneity allowing for different income slopes across richer and poorer countries.

Non-stationary structural model with time-varying demand elasticities

The paper considers local linear regression of a time series model with non-stationary regressors and errors. Asymptotic property of the local linear estimator is derived under a new dependence measure of non-stationary time series. We apply the local linear regression method to estimate the “time-varying” coefficients of an economic-causal model for the industrial sector of the U.S. economy. Nonparametric bootstrap test on the time-varying coefficients strongly suggests that the price/income elasticities of the U.S. durable goods demand are time-varying.

NONPARAMETRIC SPECIFICATION TESTING FOR NONLINEAR TIME SERIES WITH NONSTATIONARITY

This paper considers a nonparametric time series regression model with a nonstationary regressor. We construct a nonparametric test for whether the regression is of a known parametric form indexed by a vector of unknown parameters. We establish the asymptotic distribution of the proposed test statistic. Both the setting and the results differ from earlier work on nonparametric time series regression with stationarity. In addition, we develop a bootstrap simulation scheme for the selection of suitable bandwidth parameters involved in the kernel test as well as the choice of simulated critical values. An example of implementation is given to show that the proposed test works in practice.

LOCAL LIMIT THEORY AND SPURIOUS NONPARAMETRIC REGRESSION

A local limit theorem is proved for sample covariances of nonstationary time series and integrable functions of such time series that involve a bandwidth sequence. The resulting theory enables an asymptotic development of nonparametric regression with integrated or fractionally integrated processes that includes the important practical case of spurious regressions. Some local regression diagnostics are suggested for forensic analysis of such regresssions, including a localR2and a local Durbin–Watson (DW) ratio, and their asymptotic behavior is investigated. The most immediate findings extend the earlier work on linear spurious regression (Phillips, 1986,Journal of Econometrics33, 311–340) showing that the key behavioral characteristics of statistical significance, lowDWratios and moderate to highR2continue to apply locally in nonparametric spurious regression. Some further applications of the limit theory to models of nonlinear functional relations and cointegrating regressions are given. The methods are also shown to be applicable in partial linear semiparametric nonstationary regression.

A Non-Stationary Approach for Financial Returns with Nonparametric Heteroscedasticity

A non-stationary regression model for financial returns is examined theoretically. Volatility dynamics are modeled by nonparametric curve estimation on equidistant return vectors. We prove consistency and asymptotic normality of symmetric estimators and of one-sided estimators for variances and covariance matrices analytically, and derive remarks on kernels and bandwidths. Further attention is paid to asymmetry and heavy tails of returns, captured by an asymmetric Pearson type VII distribution for random residuals. Using a method of moments for its parameter estimation and a Student-t connection, a factor-based VaR implementation is derived. The approximation quality of the non-stationary approach is supported by simulation studies.

The monitoring test for the stability of regression models with nonstationary regressors

In this paper, we consider the monitoring process in time series regression models with nonstationary regressors. To this end, we propose a monitoring process based on a modified square of residuals. Simulation results are provided for illustration.

Long-Run PPP under the Presence of Near-to-Unit Roots: The Case of the British Pound-US Dollar Rate

Empirical tests typically provide evidence that the British pound–US dollar exchange rate and the relative wholesale price index contain exact unit roots and exhibit cointegration. However, the cointegrating vector is significantly different from [1, −1], thus raising doubts on the validity of the purchasing power parity (PPP) hypothesis. Following Elliott (1998), we show that if the exchange rate and relative price series contain near-to-unit roots in the context of a bivariate system, then any inference on the “cointegrating” vector and consequently on PPP, which is based on standard cointegration estimation methods, will be misleading. We then argue that the existing evidence against the PPP hypothesis in the British pound–US dollar market can be attributed to the finite sample bias of the standard cointegration estimators, arising from an endogenous and “nearly” nonstationary regressor. We also show that when robust procedures are employed the evidence favors the PPP hypothesis.

Adaptive multiple subtraction using regularized nonstationary regression

Stationary regression is the backbone of seismic data-processing algorithms including match filtering, which is commonly applied for adaptive multiple subtraction. However, the assumption of stationarity is not always adequate for describing seismic signals. I have developed a general method of nonstationary regression and that applies to nonstationary match filtering. The key idea is the use of shaping regularization to constrain the variability of nonstationary regression coefficients. Simple computational experiments demonstrate advantages of shaping regularization over classic Tikhonov's regularization, including a more intuitive selection of parameters and a faster iterative convergence. Using benchmark synthetic data examples, I have successfully applied this method to the problem of adaptive subtraction of multiple reflections.

Asymptotic properties of estimators for the linear panel regression model with random individual effects and serially correlated errors: the case of stationary and non-stationary regressors and residuals

Summary This paper studies the asymptotic properties of standard panel data estimators in a simple panel regression model with random error component disturbances. Both the regressor and the remainder disturbance term are assumed to be autoregressive and possibly non-stationary. Asymptotic distributions are derived for the standard panel data estimators including ordinary least squares (OLS), fixed effects (FE), first-difference (FD) and generalized least squares (GLS) estimators when both T and n are large. We show that all the estimators have asymptotic normal distributions and have different convergence rates dependent on the non-stationarity of the regressors and the remainder disturbances. We show using Monte Carlo experiments that the loss in efficiency of the OLS, FE and FD estimators relative to true GLS can be substantial.

New results for nonstationary panel regression

This article derives the limiting distributions of the Ordinary Least Squares (OLS) and Least Square Dummy Variable (LSDV) estimators in both spurious and cointegrated panel regressions. The limit theories employed in this article are different from those of Kao (1999) and Phillips and Moon (1999), in which the time dimension of the panel is fixed.

The limit distribution of the estimates in cointegrated regression models with multiple structural changes

We study estimation and inference in cointegrated regression models with multiple structural changes allowing both stationary and integrated regressors. Both pure and partial structural change models are analyzed. We derive the consistency, rate of convergence and the limit distribution of the estimated break fractions. Our technical conditions are considerably less restrictive than those in Bai et al. [Bai, J., Lumsdaine, R.L., Stock, J.H., 1998. Testing for and dating breaks in multivariate time series. Review of Economic Studies 65, 395–432] who considered the single break case in a multi-equations system, and permit a wide class of practically relevant models. Our analysis is, however, restricted to a single equation framework. We show that if the coefficients of the integrated regressors are allowed to change, the estimated break fractions are asymptotically dependent so that confidence intervals need to be constructed jointly. If, however, only the intercept and/or the coefficients of the stationary regressors are allowed to change, the estimates of the break dates are asymptotically independent as in the stationary case analyzed by Bai and Perron [Bai, J., Perron, P., 1998. Estimating and testing linear models with multiple structural changes. Econometrica 66, 47–78]. We also show that our results remain valid, under very weak conditions, when the potential endogeneity of the non-stationary regressors is accounted for via an increasing sequence of leads and lags of their first-differences as additional regressors. Simulation evidence is presented to assess the adequacy of the asymptotic approximations in finite samples.

The CUSUM of squares test for the stability of regression models with non-stationary regressors

In this paper, we propose a modified CUSUM of squares test in time series regression models with a non-stationary regressor and show that the limiting distribution of this test is the sup of the absolute value of a Brownian bridge.

Nonstationary discrete choice: A corrigendum and addendum

Abstract We correct the limit theory presented in an earlier paper by Hu and Phillips [2004a. Nonstationary discrete choice. Journal of Econometrics 120, 103–138] for nonstationary time series discrete choice models with multiple choices and thresholds. The new limit theory shows that, in contrast to the binary choice model with nonstationary regressors and a zero threshold where there are dual rates of convergence ( n 1 / 4 and n 3 / 4 ), all parameters including the thresholds converge at the rate n 3 / 4 . The presence of nonzero thresholds therefore materially affects rates of convergence. Dual rates of convergence reappear when stationary variables are present in the system. Some simulation evidence is provided, showing how the magnitude of the thresholds affects finite sample performance. A new finding is that predicted probabilities and marginal effect estimates have finite sample distributions that manifest a pile-up, or increasing density, towards the limits of the domain of definition.

Asymptotic Properties of Estimators for the Linear Panel Regression Model with Individual Effects and Serially Correlated Errors: The Case of Stationary and Non-Stationary Regressors and Residuals

This paper studies the asymptotic properties of standard panel data estimators in a simple panel regression model with error component disturbances. Both the regressor and the remainder disturbance term are assumed to be autoregressive and possibly non-stationary. Asymptotic distributions are derived for the standard panel data estimators including ordinary least squares, fixed effects, first-difference, and generalized least squares (GLS) estimators when both T and n are large. We show that all the estimators have asymptotic normal distributions and have different convergence rates dependent on the non-stationarity of the regressors and the remainder disturbances. We show using Monte Carlo experiments that the loss in efficiency of the OLS, FE and FD estimators relative to true GLS can be substantial.

Exploring the environmental Kuznets hypothesis: Theoretical and econometric problems

Focussing on the prime example of CO 2 emissions, we discuss several important theoretical and econometric problems that arise when studying environmental Kuznets curves (EKCs). An EKC refers to an inverse U-shaped relationship between a pollutant and economic activity. The dominant theoretical approach is given by integrated assessment modelling, which consists of economic models that are combined with environmental impact models. We critically evaluate the aggregation, model dynamics and calibration aspects and their implications for the validity of the results. We then turn to a discussion of several important econometric problems that go almost unnoticed in the literature. The most fundamental problems relate to nonlinear transformations of nonstationary regressors and, in a nonstationary panel context, to neglected cross-sectional dependence. We discuss the implications of these two major and some minor problems that arise in the econometric analysis of Kuznets curves. Our discussion shows that EKC modelling as performed to date is subject to major drawbacks at both the theoretical and the econometric level.

The Environmental Kuznets Curve: Exploring a Fresh Specification

The objective of this paper is primarily methodological. Using a new specification, we reanalyze the data on worldwide environmental quality investigated by Gene Grossman and Alan Krueger in their well-known paper on the environmental Kuznets curve (which postulates an inverse U-shaped relationship between income level and pollution). This new specification avoids using nonlinear transformations of potentially nonstationary regressors in panel estimation, which is a major unresolved econometric problem plaguing much of the existing literature. We furthermore draw conclusions from fixed effects estimation, which had eluded Grossman and Krueger. Our estimation results indicate the presence of an EKC for only six of the fourteen pollutants, whereas Grossman and Krueger find support for all but one pollutant.

Behaviour in small samples of some tests of non-nested hypotheses in non-stationary regressions and their bootstrap versions

In this paper, we use simulation methods to assess, in small samples, the performance of the Davidson and MacKinnon (1981) J-test when it is used to discriminate between two non-nested models with non-stationary regressors. We distinguish two cases: first, we assume that the sets of regressors of the two models are cointegrated; secondly, we consider the case where the regressors are not cointegrated. We also compare the behaviour of this test with that of the Fisher McAleer JA-type test and the bootstrap-adjusted J-tests, in order to assess its relative performance.

Non-Stationarity Effects in Causal Sales Forecasting Models

One fundamental assumption of classical econometrics that is gaining a lot of attention in econometric modeling is that of stationarity of variables used to specify a regression model. We analyze some of the benefits and drawbacks in applying econometric models of nonstationary behavior to forecasts involving marketing variables. We use standard residual-based unit root and cointegration tests to identify non-stationary variables in the sales and marketing data of two CPG brands and any potential long-run equilibrium. The out-of-sample forecast Mean Absolute Percent Errors (MAPE) from single-equation causal forecast models of sales volume for the brands, with the identified non-stationary regressors is compared to that from benchmark univariate models, to gauge the effect of nonstationarity on forecast quality. The model is then re-specified after appropriately stationarizing these models. The forecasts from the stationarized models are compared to the benchmark models to analyze potential improvements in forecast quality. The results show that it may not be necessary to difference nonstationary, non-cointegrating regressors, when the dependent variable is stationary, and when the dependent variable is nonstationary, it might be possible to obtain a stable regression model with levels variable by using an autoregressive error model.

Environmental and biological controls on the seasonal variations in latent heat fluxes derived from flux data for three forest sites

One difficulty that arises when predicting canopy‐scale energy fluxes is that the parameterization of complex (bio)physical soil vegetation atmosphere transfer schemes is often only partially conditioned by the information content of the eddy covariance data commonly used for calibration, rendering subsequent predictions and extrapolations somewhat uncertain. Here we derive a functional description for daily evaporative fluxes directly from observations at the canopy scale using a nonstationary regression framework. This method is applied to 3 year blocks of eddy covariance and micrometeorological data from three different FLUXNET forest sites: Harvard Forest, USA, University of Michigan Biological Station, USA, and Hyytiälä, Finland, covering a variety of climate and vegetation conditions. The approach yields a simple three‐parameter model which is based on partitioning latent heat between equilibrium latent heat fluxes and fluxes that are under strong stomatal control and hence are related to CO2 fluxes. Despite being well defined and able to account for much of the observed variations in latent heat, predictive validation of the model emphasizes the need to account for surface and subsurface water balance in such descriptions.