Nonstationary Regression Research Articles

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.

Fully modified semiparametric GLS estimation for regressions with nonstationary seasonal regressors

Abstract Regression models with seasonally integrated and possibly endogenous regressors and serially correlated regression errors are studied. Spectral decompositions of generalized sums of cross products of regressors and regression errors are used to develop a feasible generalized least squares estimator (FGLSE) which does not require parametric specifications for error processes. Using the FGLSE and following the spirit of “Fully Modified estimation” of Phillips and Hansen (Rev. Econ. Stud. 57 (1990) 99), a fully modified GLSE (FM-GLSE) and inference procedures are constructed. The distribution of the FM-GLSE is shown to be asymptotically a mixed normal distribution which validates standard inference based on the FM-GLSE with normal theory. A Monte-Carlo simulation shows that the FM-GLSE is more efficient than the ordinary least squares estimator (OLSE) in the cases of endogeneity or serial correlation and more efficient than the FM-estimator based on the OLSE in the case of serial correlation.

Asset pricing dynamics

This paper is concerned with the issue of dynamics in financial data and asset pricing models such as the CAPM. A literature review in this area is undertaken and highlights the need for a modern time series econometric approach in asset pricing. Such an approach is discussed and deals with problems related to structural breaks and microstructures, dynamics in the mean and variance process, and non-stationary regressions and cointegration. An empirical application using UK stock market data demonstrates the merit of the proposed methodology in correcting market model regressions.

Do Property‐Casualty Insurance Underwriting Margins Have Unit Roots?

AbstractA growing literature analyzes determinants of insurance prices using time series data on insurer underwriting margins. If the variables analyzed are stationary, conventional regression models may be appropriately used to test hypotheses. Based on pretests for a unit root, several studies have instead used co‐integration analysis to analyze the long‐run relationship between purportedly nonstationary underwriting margins and macroeconomic variables. We apply a battery of unit root tests to investigate whether underwriting margins are stationary under different assumptions concerning deterministic components in the data generating process (DGP). When linear and/or quadratic trends are included in the assumed DGPs, the tests reject the null hypothesis of a unit root for loss ratios, expense ratios, combined ratios, and economic loss ratios from 1953 through 1998 for many of the individual lines examined and for all lines combined. Consistent with prior work on whether macroeconomic variables have unit roots, a simulation of test power for underwriting margins during the sample period demonstrates that nonrejections of the null hypothesis of a unit root could easily reflect low power. The overall findings suggest that conventional regression methods can be used appropriately to analyze underwriting margins after controlling for deterministic influences and transforming any nonstationary regressors.

Instrumental variables estimation of a nearly nonstationary, heterogeneous error component model

This paper studies instrumental variables (IV) estimation for an error component model with stationary and nearly nonstationary regressors. It is assumed that the numbers of cross section and time series observations are infinite. Furthermore, autoregressive disturbances are assumed for the error component model, the structure of which may vary with individuals. The estimators considered are the Within-IV-OLS, Within-IV-GLS and IV-GLS estimators. The GLS estimators use Gohberg's formula, which is particularly useful when autoregressive structures are imposed on the disturbance terms. Sequential limit theories for the estimators are derived, and it is shown that all of the estimators have normal distributions in the limit. Additionally, Wald tests for coefficient vectors are shown to have chi-square distributions in the limit. Simulation results regarding the estimator efficiency and the size of the Wald tests are also reported. The results show that the Within-IV-GLS and IV-GLS estimators are more efficient than the Within-IV-OLS estimator in most cases and that the Wald tests keep nominal size reasonably well. The relation between the trade and budget deficits of 23 OECD nations is examined using the panel IV estimators. The empirical results support the view that the budget and trade deficits move in the same direction.

Reduced rank regression in cointegrated models

The coefficient matrix of a cointegrated first-order autoregression is estimated by reduced rank regression (RRR), depending on the larger canonical correlations and vectors of the first difference of the observed series and the lagged variables. In a suitable coordinate system the components of the least-squares (LS) estimator associated with the lagged nonstationary variables are of order 1/ T, where T is the sample size, and are asymptotically functionals of a Brownian motion process; the components associated with the lagged stationary variables are of the order T −1/2 and are asymptotically normal. The components of the RRR estimator associated with the stationary part are asymptotically the same as for the LS estimator. Some components of the RRR estimator associated with nonstationary regressors have zero error to order 1/ T and the other components have a more concentrated distribution than the corresponding components of the LS estimator.

Nonlinear econometric models with cointegrated and deterministically trending regressors

Summary This paper develops an asymptotic theory for a general class of nonlinear nonstationary regressions, extending earlier work by Phillips and Hansen (1990) on linear cointegrating regressions.The model considered accommodates a linear time trend and stationary regressors, as well as multiple I(1) regressors. We establish consistency and derive the limit distribution of the nonlinear least squares estimator. The estimator is consistent under fairly general conditions but the convergence rate and the limiting distribution are critically dependent upon the type of the regression function. For integrable regression functions, the parameter estimates converge at a reduced n 1/4 rate and have mixed normal limit distributions. On the other hand, if the regression functions are homogeneous at infinity, the convergence rates are determined by the degree of the asymptotic homogeneity and the limit distributions are non-Gaussian. It is shown that nonlinear least squares generally yields inefficient estimators and invalid tests, just as in linear nonstationary regressions. The paper proposes a methodology to overcome such difficulties. The approach is simple to implement, produces efficient estimates and leads to tests that are asymptotically chi-square. It is implemented in empirical applications in much the same way as the fully modified estimator of Phillips and Hansen.

Consistency of the maximum likelihood estimators for nonstationary ARMA regressions with time trends

Abstract For autoregressive moving averages with time trends and autoregressive unit roots, the consistency of the maximum likelihood estimators is established. General uniform approximations are established for the quadratic forms which appear in the Gaussian likelihood.

On the Gibson Paradox

This paper uses recent developments in the theory of nonstationary regressors to investigate empirical relationships previously taken to support the Gibson paradox, using quarterly data over the 1957:1–1994:4 period on nominal interest rates and prices for eight European Union countries—Belgium, Denmark, England, France, Germany, Ireland, Italy, and The Netherlands. Using the methodology suggested by Kydland and Prescott, it is shown that the (relevant) cyclical nominal interest rate—price level contemporaneous correlations are weak, thereby punching a hole in the Gibson paradox. Evidence is also presented, based on the integration properties of the data, that standard Gibson paradox regressions are spurious.

Estimation of a Change Point in Multiple Regression Models

This paper studies the least squares estimation of a change point in multiple regressions. Consistency, rate of convergence, and asymptotic distributions are obtained. The model allows for lagged dependent variables and trending regressors. The error process can be dependent and heteroskedastic. For nonstationary regressors or disturbances, the asymptotic distribution is shown to be skewed. The analytical density function and the cumulative distribution function for the general skewed distribution are derived. The analysis applies to both pure and partial changes. The method is used to analyze the response of market interest rates to discount rate changes.

Fully modified IV, GIVE and GMM estimation with possibly non-stationary regressors and instruments

This paper develops a general theory of instrumental variables (IV) estimation that allows for both I(1) and I(0) regressors and instruments. The main goal of this paper is to develop a theory in which one does not need to know the integration properties of the regressors in order to obtain efficient estimators. The estimation techniques involve an extension of the fully modified (FM) regression procedure that was introduced in earlier work by Phillips and Hansen (1990). FM versions of the generalized instrumental variable estimation (GIVE) method and the generalized method of moments (GMM) estimator are developed. In models with both stationary and nonstationary components, the FM-GIVE and FM-GMM techniques provide efficiency gains over FM-IV in the estimation of the stationary components of a model that has both stationary and non-stationary regressors. The paper exploits a result of Phillips (1991a) that we can apply FM techniques in models with cointegrated regressors and even in stationary regression models without losing the method's good asymptotic properties. The present paper shows how to take advantage jointly of the good asymptotic properties of FM estimators with respect to the non-stationary elements of a model and the good asymptotic properties of the GIVE and GMM estimators with respect to the stationary components. The theory applies even when there is no prior knowledge of the number of unit roots in the system or the dimension or the location of the cointegration space. An FM extension of the Sargan (1958) test for the validity of the instruments is proposed.

Explaining stationary variables with non-stationary regressors

When variables included in an OLS regression are stationary, conventional statistical measures such as t-statistics and R 2's – in addition to a priori information from economic theory – are the standard indicators used to assess the performance of the hypothesized model. However, if the variables under consideration are non-stationary, such conventional measures no longer have the usual interpretation. With recent developments in time-series analysis, namely cointegration, researchers are able to deal with models containing non-stationary variables effectively. A standard cointegration model, however, requires all variables included in the regression to be of the same order of integration. In this paper we consider a regression in which the dependent variable is integrated of order zero, I(0), while the explanatory variables are integrated of order one, I(1). Conventional statistical measures are inapplicable because the regressors are not stationary. On the other hand, cointegration statistics are inapplicable because the variables are not of the same order of integration. This letter proposes a methodology on how to evaluate the performance of such a model.

Nonstationary regression models with a lagged dependent variable

This paper studies regression models with a lagged dependent variable when both the dependent and independent variables are nonstationary, and the regression model is misspecified in some dimension. In particular, we discuss the limiting properties of leastsquares estimates of the parameters in such regression models, and the limiting distributions of their test statistics. We show that the estimate of the lagged dependent variable tends to unity asymptotically independent of its true value, while the estimates of the independent variables tend to zero. The limiting distributions of their test statistics are shown to diverge with sample size.

Robust Nonstationary Regression

This paper provides a robust statistical approach to nonstationary time series regression and inference. Fully modified extensions of traditional robust statistical procedures are developed that allow for endogeneities in the nonstationary regressors and serial dependence in the shocks that drive the regressors and the errors that appear in the equation being estimated. The suggested estimators involve semiparametric corrections to accommodate these possibilities, and they belong to the same family as the fully modified least-squares (FM-OLS) estimator of Phillips and Hansen (1990, Review of Economic Studies 57,99–125). Specific attention is given to fully modified least absolute deviation (FM-LAD) estimation and fully modified M (FM-M) estimation. The criterion function for LAD and some M-estimators is not always smooth, and this paper develops generalized function methods to cope with this difficulty in the asymptotics. The results given here include a strong law of large numbers and some weak convergence theory for partial sums of generalized functions of random variables. The limit distribution theory for FM-LAD and FM-M estimators that is developed includes the case of finite variance errors and the case of heavytailed (infinite variance) errors. Some simulations and a brief empirical illustration are reported.

Efficient IV Estimation in Nonstationary Regression

A limit theory for instrumental variables (IV) estimation that allows for possibly nonstationary processes was developed in Kitamura and Phillips (1992, Fully Modified IV, GIVE, and GMM Estimation with Possibly Non-stationary Regressors and Instruments, mimeo, Yale University). This theory covers a case that is important for practitioners, where the nonstationarity of the regressors may not be of full rank, and shows that the fully modified (FM) regression procedure of Phillips and Hansen (1990) is still applicable. FM. versions of the generalized method of moments (GMM) estimator and the generalized instrumental variables estimator (GIVE) were also developed, and these estimators (FM-GMM and FM-GIVE) were designed specifically to take advantage of potential stationarity in the regressors (or unknown linear combinations of them). These estimators were shown to deliver efficiency gains over FM-IV in the estimation of the stationary components of a model.This paper provides an overview of the FM-IV, FM-GMM, and FM-GIVE procedures and investigates the small sample properties of these estimation procedures by simulations. We compare the following five estimation methods: ordinary least squares, crude (conventional) IV, FM-IV, FM-GMM, and FM-GIVE. Our findings are as follows, (i) In terms of overall performance in both stationary and nonstationary cases, FM-IV is more concentrated and better centered than OLS and crude IV, though it has a higher root mean square error than crude IV due to occasional outliers, (ii) Among FM-IV, FM-GMM, and FM-GIVE, (a) when applied to the stationary coefficients, FM-GIVE generally outperforms FM-IV and FM-GMM by a wide margin, whereas the difference between the latter two is quite small when the AR roots of the stationary processes are rather large; and (b) when applied to the nonstationary coefficients, the three estimators are numerically very close. The performance of the FM-GIVE estimator is generally very encouraging.

The CUSUM test based on least squares residuals in regressions with integrated variables

Ploberger and Krämer Ecometrica, 1992, 60, 271–285 constructed a CUSUM test for structural stability based on OLS residuals which imposes strong assumptions on the regressors, ruling out non-stationary regressors. We derive the CUSUM test using OLS residuals in a regression on time and in the unit root autoregression with drift.

Ricardian Equivalence, Rational Expectations and the Permanent Income Hypothesis

The Ricardian equivalence proposition and the permanent income hypothesis are tested in an intertemporal consumption model with rational expectations. The representative consumer incorporates the government budget constraint. An alternative hypothesis of incomplete tax discounting is nested within this model. A deterministic time trend is rejected for the variables of the model. Variables exhibit instead a stochastic trend. The permanent income model is not rejected by annual U.S. data (i.e., no excess sensitivity is found). The evidence with respect to Ricardian equivalence is mixed. The empirical study employs theorems from the cointegration literature and specifically addresses issues of nonstationary regressors. Copyright 1990 by Ohio State University Press.

The P-Star Approach to the Link Between Money and Prices

This paper examines several specification errors in the M2-based P* model and develops an M1-based estimate of this model. The apparent statistical significance of M2 is shown to arise from a spurious regression that uses a non-stationary regressor and because the significance test for M2 is biased by including the influence of a lagged dependent variable whose coefficient is not normally distributed. When these problems are addressed, M2 is not statistically significant related to the price level. The M1-based P* model exhibits a significant relationship between M1 and the price level, however.

An orthogonal series estimate of time-varying regression

Let (X1,Y1), (X2,Y2),... be independent pairs of random variables according to the modelYn=tn(Xn)R(Xn)+Zn,n=1,2,..., wheretn andR are unknown functions.Zn's are i.i.d. random variables with zero mean and finite variance. The marginal density ofXn is independent ofn. In the paper nonparametric estimates of a nonstationary regression function E{Yn|Xn=x}=tn(x)R(x) are proposed and their asymptotic properties are investigated.

Stochastic estimation of signal parameters with a non-stationary regression function†

Abstract Stochastic approximation techniques have been applied to the estimation of signal parameters by Sakrison (1966) and the authors (1971). This paper is concerned with an algorithm having a non-stationary regression function. Convergence conditions and a mean-square error bound are presented. Its application to feedback communication and coherent estimation of Doppler frequency is discussed in some detail.