Limit theory for functional coefficient cointegrating regression was recently found to be considerably more complex than earlier understood. The issues were explained and correct limit theory derived for the kernel weighted local constant estimator in Phillips and Wang (2023b). The present paper provides complete limit theory for the general kernel weighted local p-th order polynomial estimator of the functional coefficient and the coefficient deriva-tives. Both stationary and nonstationary regressors are allowed. Implications for bandwidth selection are discussed. An adaptive procedure to select the fit order p is proposed and found to work well. A robust t-ratio is constructed following the new correct limit theory, which corrects and improves the usual t-ratio in the literature. Furthermore, the robust t-ratio is valid and works well regardless of the properties of the regressors, thereby providing a unified procedure to compute the t-ratio and facilitating practical inference. Testing constancy of the functional coefficient is also considered. Supportive finite sample studies are provided that corroborate the new asymptotic theory.
Functional coefficient (FC) cointegrating regressions offer empirical investigators flexibility in modeling economic relationships by introducing covariates that influence the direction and intensity of comovement among nonstationary time series. FC regression models are also useful when formal cointegration is absent, in the sense that the equation errors may themselves be nonstationary, but where the nonstationary series display well-defined FC linkages that can be meaningfully interpreted as correlation measures involving the covariates. The present paper proposes new nonparametric estimators for such FC regression models where the nonstationary series display linkages that enable consistent estimation of the correlation measures between them. Specifically, we develop √n-consistent estimators for the functional coefficient and establish their asymptotic distributions, which involve mixed normal limits that facilitate inference. Two novel features that appear in the limit theory are (i) the need for non-diagonal matrix normalization due to the presence of stationary and nonstationary components in the regression; and (ii) random bias elements that appear in the asymptotic distribution of the kernel estimators, again resulting from the nonstationary regression components. Numerical studies reveal that the proposed estimators achieve significant efficiency improvements compared to the estimators suggested in earlier work by Sun et al. (2011). Easily implementable specification tests with standard chi-square asymptotics are suggested to check for constancy of the functional coefficient. These tests are shown to have faster divergence rate under local alternatives and enjoy superior performance in simulations than tests proposed recently in Gan et al. (2014). An empirical application based on the quantity theory of money illustrates the practical use of correlated but non-cointegrated regression relations.
A general asymptotic theory is established for sample cross moments of nonstationary time series, allowing for long range dependence and local unit roots. The theory provides a substantial extension of earlier results on nonparametric regression that include near-cointegrated nonparametric regression as well as spurious nonparametric regression. Many new models are covered by the limit theory, among which are functional coefficient regressions in which both regressors and the functional covariate are nonstationary. Simulations show finite sample performance matching well with the asymptotic theory and having broad relevance to applications, while revealing how dual nonstationarity in regressors and covariates raises sensitivity to bandwidth choice and the impact of dimensionality in nonparametric regression. An empirical example is provided involving climate data regression to assess Earth’s climate sensitivity to CO2, where nonstationarity is a prominent feature of both the regressors and covariates in the model. This application is the first rigorous empirical analysis to assess nonlinear impacts of CO2 on Earth’s climate.
Limit distribution theory in the econometric literature for functional coefficient cointegrating regression is incorrect in important ways, influencing rates of convergence, distributional properties, and practical work. The correct limit theory reveals that components from both bias and variance terms contribute to variability in the asymptotics. The errors in the literature arise because random variability in the bias term has been neglected in earlier research. In stationary regression this random variability is of smaller order and can be ignored in asymptotic analysis but not without consequences for finite sample performance. Implications of the findings for rate efficient estimation are discussed. Simulations in the Online Supplement provide further evidence supporting the new limit theory in nonstationary functional coefficient regressions.
Functional coefficient (FC) regressions allow for systematic flexibility in the responsiveness of a dependent variable to movements in the regressors, making them attractive in applications where marginal effects may depend on covariates. Such models are commonly estimated by local kernel regression methods. This paper explores situations where responsiveness to covariates is locally flat or fixed. The paper develops new asymptotics that take account of shape characteristics of the function in the locality of the point of estimation. Both stationary and integrated regressor cases are examined. The limit theory of FC kernel regression is shown to depend intimately on functional shape in ways that affect rates of convergence, optimal bandwidth selection, estimation, and inference. In FC cointegrating regression, flat behavior materially changes the limit distribution by introducing the shape characteristics of the function into the limiting distribution through variance as well as centering. In the boundary case where the number of zero derivatives tends to infinity, near parametric rates of convergence apply in stationary and nonstationary cases. Implications for inference are discussed and a feasible pre-test inference procedure is proposed that takes unknown potential flatness into consideration and provides a practical approach to inference.
Functional coefficient (FC) regressions allow for systematic flexibility in the responsiveness of a dependent variable to movements in the regressors, making them attractive in applications where marginal effects may depend on covariates. Such models are commonly estimated by local kernel regression methods. This paper explores situations where responsiveness to covariates is locally flat or fixed. In such cases, the limit theory of FC kernel regression is shown to depend intimately on functional shape in ways that affect rates of convergence, optimal bandwidth selection, estimation, and inference. The paper develops new asymptotics that take account of shape characteristics of the function in the locality of the point of estimation. Both stationary and integrated regressor cases are examined. Locally flat behavior in the coefficient function has, as expected, a major effect on bias and thereby on the trade-off between bias and variance, and on optimal bandwidth choice. In FC cointegrating regression, flat behavior materially changes the limit distribution by introducing the shape characteristics of the function into the limiting distribution through variance as well as centering. Both bias and variance depend on the number of zero derivatives in the coefficient function. In the boundary case where the number of zero derivatives tends to infinity, near parametric rates of convergence apply for both stationary and nonstationary cases. Implications for inference are discussed and simulations characterizing finite sample behavior are reported.
Limit distribution theory in the econometric literature for functional coefficient cointegrating (FCC) regression is shown to be incorrect in important ways, influencing rates of convergence, distributional properties, and practical work. In FCC regression the cointegrating coefficient vector \beta(.) is a function of a covariate z_t. The true limit distribution of the local level kernel estimator of \beta(.) is shown to have multiple forms, each form depending on the bandwidth rate in relation to the sample size n and with an optimal convergence rate of n^{3/4} which is achieved by letting the bandwidth have order 1/n^{1/2}.when z_t is scalar. Unlike stationary regression and contrary to the existing literature on FCC regression, the correct limit theory reveals that component elements from the bias and variance terms in the kernel regression can both contribute to variability in the asymptotics depending on the bandwidth behavior in relation to the sample size. The trade-off between bias and variance that is a common feature of kernel regression consequently takes a different and more complex form in FCC regression whereby balance is achieved via the dual-source of variation in the limit with an associated common convergence rate. The error in the literature arises because the random variability of the bias term has been neglected in earlier research. In stationary regression this random variability is of smaller order and can correctly be neglected in asymptotic analysis but with consequences for finite sample performance. In nonstationary regression, variability typically has larger order due to the nonstationary regressor and its omission leads to deficiencies and partial failure in the asymptotics reported in the literature. Existing results are shown to hold only in scalar covariate FCC regression and only when the bandwidth has order larger than 1/n and smaller than 1/n^{1/2}. The correct results in cases of a multivariate covariate z_t are substantially more complex and are not covered by any existing theory. Implications of the findings for inference, confidence interval construction, bandwidth selection, and stability testing for the functional coefficient are discussed. A novel self-normalized t-ratio statistic is developed which is robust with respect to bandwidth order and persistence in the regressor, enabling improved testing and confidence interval construction. Simulations show superior performance of this robust statistic that corroborate the finite sample relevance of the new limit theory in both stationary and nonstationary regressions.
Behavior at the individual level in panels or at the station level in spatial models is often influenced by aspects of the system in aggregate. In particular, the nature of the interaction between individual-specific explanatory variables and an individual dependent variable may be affected by `global’ variables that are relevant in decision making and shared communally by all individuals in the sample. To capture such behavioral features, we employ a functional coefficient panel model in which certain communal covariates may jointly influence panel interactions by means of their impact on the model coefficients. Two classes of estimation procedures are proposed, one based on station averaged data the other on the full panel, and their asymptotic properties are derived. Inference regarding the functional coefficient is also considered. The finite sample performance of the proposed estimators and tests are examined by simulation. An empirical spatial model illustration is provided in which the climate sensitivity of temperature to atmospheric CO_2 concentration is studied at both station and global levels.