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This paper considers the inference problems in nonlinear quantile regressions with both stationary and nonstationary covariates. The nonparametric local constant quantile estimator is proposed to estimate the unknown quantile regression function, whose asymptotic properties are established under quite general conditions. Specification testing of the quantile regression function is further considered through a statistic constructed based on the integrated squared distance between the parametric and the nonparametric estimators for the regression function. The test statistic is shown to converge to a random variable related to the local time of an Ornstein–Uhlenbeck process under the parametric null. The power of the test against local alternatives is also investigated. Additional asymptotic results on the null parametric quantile estimators and a bootstrap test are developed as well. Numerical results demonstrate that the proposed nonparametric estimator and the specification test enjoy attractive finite sample performance.

Functional-coefficient cointegrating models have become popular to model nonlinear nonstationarity in econometrics (Cai et al., 2009; Xiao, 2009). However, there is rare study on testing the existence of functional-coefficient cointegration. Consequently, functional-coefficient regressions involving nonstationary regressors may be spurious. This paper investigates the effect that spurious functional-coefficient regression has on the model diagnostics. We find that common characteristics of spurious regressions are manifest, including divergent local significance tests, random local goodness-of-fit, and local Durbin–Watson ratio converging to zero, complementing those discovered in spurious linear and nonparametric regressions (Phillips, 1986, 2009). In addition, spuriousness causes the divergences of the global significance tests proposed by Xiao (2009) and Sun et al. (2016), which are likely to produce misleading conclusions for practitioners when cointegration tests fail to reject a spurious regression. To resolve the problems, we propose a simple-to-implement inference procedure based on a semiparametric balanced regression, by augmenting regressors of the original spurious regression with lagged dependent variable and independent variables, with the aid of the marginal integration. This procedure achieves spurious regression detection via standard nonparametric inferential asymptotics, and is found robust to the true relationship between the integrated processes. The theoretical results are also corroborated by simulations.

Robust M-estimation uses loss functions, such as least absolute deviation (LAD), quantile loss and Huber's loss, to construct its objective function, in order to for example eschew the impact of outliers, whereas the difficulty in analysing the resultant estimators rests on the nonsmoothness of these losses. Generalized functions have advantages over ordinary functions in several aspects, especially generalized functions possess derivatives of any order. Generalized functions incorporate local integrable functions, the so-called regular generalized functions, while the so-called singular generalized functions (e.g. Dirac delta function) can be obtained as the limits of a sequence of sufficient smooth functions, so-called regular sequence in generalized function context. This makes it possible to use these singular generalized functions through approximation. Nevertheless, a significant contribution of this paper is to establish the convergence rate of regular sequence to nonsmooth loss that answers a call from the relevant literature. For parameter estimation where objective function may be nonsmooth, this paper first shows as a general paradigm that how generalized function approach can be used to tackle the nonsmooth loss functions in Section two using a very simple model. This approach is of general interest and applicability. We further use the approach in robust M-estimation for additive single-index cointegrating time series models; the asymptotic theory is established for the proposed estimators. We evaluate the finite-sample performance of the proposed estimation method and theory by both simulated data and an empirical analysis of predictive regression of stock returns., arXiv admin note: text overlap with arXiv:2111.02023

This paper studies spurious regressions involving processes moderately deviated from a unit root (PMDURs), and establishes the limiting distributions for the least squares estimator, the associated t -statistic, the coefficient of determination R 2 and the Durbin–Watson statistic. We find that these limiting distributions critically depend on nuisance parameters that characterize the local deviations from unity, making inference for spurious regressions practically impossible using the conventional t -statistic. As a cure, we propose robust inference based on the balanced regression model, where the lagged regressor and the lagged dependent variable are augmented to the original regression. The induced t -statistic via such an augmentation is shown to be asymptotically standard normal and is therefore free of nuisance parameters, which turns out to be a robust and simple-to-implement tool for spurious regressions inference. Moreover, the limiting properties of other statistics are investigated. The balanced regression based inference is further shown to continue to work for cointegration models with PMDURs, which is therefore robust to whether the PMDURs are spuriously related or cointegrated. Finally, the finite sample properties of the robust method are demonstrated through both Monte Carlo and real data examples.

In recent years statistical inference for nonlinear cointegration has attracted attention from both academics and practitioners. This paper proposes a new type of cointegration in the sense that two univariate time series yt and xt are cointegrated via two (unknown) smooth nonlinear transformations, further generalizing the notion of cointegration initially revealed by Box and Tiao (1977), and more systematically studied by Engle and Granger (1987). More precisely, it holds that G(yt,β0)=g(xt)+ut, where G(⋅,β0) is strictly increasing and known up to an unknown parameter β0, g(⋅) is unknown and smooth, xt is I(1), and ut is the stationary disturbance. This setting nests the nonlinear cointegration model of Wang and Phillips (2009b) as a special case with G(y,β0)=y. It extends the model of Linton et al. (2008) to the cases with a unit-root nonstationary regressor. Sieve approximations to the smooth nonparametric function g are applied, leading to an extremum estimator for β and a plugging-in estimator for g(⋅). Asymptotic properties of the estimators are established, revealing that both the convergence rates and the limiting distributions depend intimately on the properties of the two nonlinear transformation functions. Simulation studies demonstrate that the estimators perform well even with small samples. A real data example on the environmental Kuznets curve portraying the nonlinear impact of per-capita GDP on air-pollution illustrates the practical relevance of the proposed double-nonlinear cointegration.

We provide a large sample theory for the Dickey-Fuller unit root test when the volatility process is driven by a nonlinear transformation of nonstationary time series. Our theory allows the dynamic...

In a predictive regression, a less persistent return series is regressed on the first lag of some highly persistent predictors. Therefore, predictability could often be missed due to the persistence imbalance. The aim of this paper is to balance the predictive regression by augmenting it with an additional lag of the predictors. This second lag generally reduces the persistence level on the right-hand side of the equation to achieve balance. We then propose a simple test procedure for univariate and multivariate predictive regressions, based on least squares estimation. Empirically, we reexamine the popular predictors in the literature and find quite different results.

This paper proposes a class of parametric multiple-index time series models that involve linear combinations of time trends, stationary variables and unit root processes as regressors. The inclusion of the three different types of time series, along with the use of a multiple-index structure for these variables to circumvent the curse of dimensionality, is due to both theoretical and practical considerations. The M-type estimators (including OLS, LAD, Huber's estimator, quantile and expectile estimators, etc.) for the index vectors are proposed, and their asymptotic properties are established, with the aid of the generalized function approach to accommodate a wide class of loss functions that may not be necessarily differentiable at every point. The proposed multiple-index model is then applied to study the stock return predictability, which reveals strong nonlinear predictability under various loss measures. Monte Carlo simulations are also included to evaluate the finite-sample performance of the proposed estimators.

This paper proposes a simultaneous test for the specification of the conditional mean and conditional variance functions as well as the error distribution in nonlinear time series models. Constructed by comparing two density estimators for the response variable, the proposed test has a Gumbel‐limiting distribution under the null hypothesis and is consistent against a general class of alternative hypotheses. A parametric bootstrap procedure is proposed for practical implementation, and is shown to perform well in extensive simulations. The application to the continuous time diffusion model is illustrated via an analysis on the U.S. Federal fund rate data.

This paper analyses functional coefficient cointegration models with both stationary and non‐stationary covariates, allowing time‐varying (unconditional) volatility of a general form. The conventional kernel weighted least squares (KLS) estimator is subject to potential efficiency loss, and can be improved by an adaptive kernel weighted least squares (AKLS) estimator that adapts to heteroscedasticity of unknown form. The AKLS estimator is shown to be as efficient as the oracle generalized kernel weighted least squares estimator asymptotically, and can achieve significant efficiency gain relative to the KLS estimator in finite samples. An illustrative example is provided by investigating the Purchasing Power Parity hypothesis.

This article studies the adaptive estimation of the heteroskedastic functional-coefficient regressions. The motivation for such a theoretical study originates from the empirical analysis of Jansen et al., where the role of fiscal policy on the U.S. asset markets (treasury bonds) is evaluated via the functional-coefficient model. It is found that this model is subject to time-varying heteroskedasticity. As a result, the local least square (LLS) estimator suffers from efficiency loss. To overcome this problem, we propose an adaptive LLS (ALLS) estimator, which can adapt to heteroskedasticity of unknown form asymptotically. Simulation studies confirm that the ALLS estimator can achieve significant efficiency gain in finite samples, compared to the LLS estimator. Real data analysis reveals that the heteroskedastic functional-coefficient model provides adequate fit and better out-of-sample forecasting.

This study explores the spurious effects in linear regressions with moderately explosive processes. Asymptotic results are developed for the least square estimator, the typical t‐statistic, the Durbin–Watson statistic, and the coefficient of determination. The typical t‐statistic is unable to detect the presence of a spurious relationship, due to the presence of nuisance parameters that characterize deviations from unity. Moreover, the t‐statistic for common explosive processes has different asymptotics compared to that for distinct explosive processes. Such differences further complicate the use of the t‐statistic. We demonstrate that two popular methods available in the literature are incapable for this purpose due to similar difficulties. To overcome these limitations, we propose a t‐test based upon balanced regressions that induces asymptotic inference based on the standard normal distribution, which is therefore robust to deviations from unity. These results are further generalized to spurious regressions with multivariate mildly explosive processes. Simulation results confirm that our test is effective in finite samples, while other alternatives are not. An empirical example that demonstrates the phenomenon of spurious correlation between the NASDAQ stock index and crude oil price in the US is provided to show the practical merit of our proposed method.

In this chapter, I propose a model averaging estimation of nonparametric models based on Shannon’s entropy measure. The choice of weights in the averaging estimator is implemented bya maximizing the Shannon’s entropy measure which aggregates both model uncertainty and data uncertainty. Finite sample simulation studies show that the proposed averaging estimator outperforms the local linear least square estimator in terms of mean-squared errors and outperforms the Mallows averaging estimator of Hansen (2007) when the signal-to-noise ratio is low. An empirical example to apply the proposed estimator is provided to study the wage equation and illustrates its superiority in out-of-sample forecasts.

Cointegration inferences often rely on a correct specification for the short-run dynamic vector autoregression. However, this specification is unknown, a priori. A lag length that is too small leads to an erroneous inference as a result of the misspecification. In contrast, using too many lags leads to a dramatic increase in the number of parameters, especially when the dimension of the time series is high. In this paper, we develop a new methodology which adds an error-correction term for the long-run equilibrium to a latent factor model in order to model the short-run dynamic relationship. The inferences use the eigenanalysis-based methods to estimate the cointegration and latent factor process. The proposed error-correction factor model does not require an explicit specification of the short-run dynamics, and is particularly effective for high-dimensional cases, in which the standard error-correction suffers from overparametrization. In addition, the model improves the predictive performance of the pure factor model. The asymptotic properties of the proposed methods are established when the dimension of the time series is either fixed or diverging slowly as the length of the time series goes to infinity. Lastly, the performance of the model is evaluated using both simulated and real data sets.

Spatial autocorrelation is a parameter of importance for network data analysis. To estimate spatial autocorrelation, maximum likelihood has been popularly used. However, its rigorous implementation requires the whole network to be observed. This is practically infeasible if network size is huge (e.g., Facebook, Twitter, Weibo, WeChat, etc). In that case, one has to rely on sampled network data to infer about spatial autocorrelation. By doing so, network relationships (i.e., edges) involving unsampled nodes are overlooked. This leads to distorted network structure and underestimated spatial autocorrelation. To solve the problem, we propose here a novel solution. By temporarily assuming that the spatial autocorrelation is small, we are able to approximate the likelihood function by its first order Taylor's expansion. This leads to the method of approximate maximum likelihood estimator (AMLE), which further inspires the development of paired maximum likelihood estimator (PMLE). Compared with AMLE, PMLE is computationally superior and thus is particularly useful for large scale network data analysis. Under appropriate regularity conditions (without assuming a small spatial autocorrelation), we show theoretically that PMLE is consistent and asymptotically normal. Numerical studies based on both simulated and real datasets are presented for illustration purpose.

This paper considers spurious regressions with partial unit root and near partial unit root processes. Three models are investigated via simulations to study the rejection probability of the associated t-statistics in detecting the spurious correlation. The results reveal that the spurious regression is a not-so-spurious problem, which cannot be eliminated by adding AR(1) errors in re-estimation. Moreover, the advocated balanced regression approach is found effective in detecting the non-existent relationship.

In cointegrated vector autoregressive (VAR) models, error correction terms often have indeterminate effects on forecasting, thus we are concerned with inclusion or exclusion of the cointegration relation in forecast. This paper considers the model averaging strategies for cointegrated VAR models with heterogeneous variance or variance breaks. The estimated cointegration rank along with other data information are used to formulate the model averaging weights. This specific but unknown pattern of time-varying variances has nontrivial effects on the choices of model weights. Our numerical results strongly advocate the Mallows averaging estimator, but caution against the commonly used pre-testing approach.

This paper focuses on the role of the quantity theory in improving inflation forecasts. We find that the cointegration-based quantity theory does not hold for the period after 1995 for the U.S. data. However, that period is well explained by an adaptive quantity theory based on a functional-coefficient cointegration that adapts to the unemployment rate. The forecasting exercises show that the adaptive quantity theory has superior predictive power for targeting future inflation.

The authors propose an estimator for the density of the response variable in the parametric mean regression model where the error density is left unspecified. With the application of empirical process theory, they derive its n -consistency and asymptotical normality. This result is further extended to models which allow possible parametric misspecification on the regression function and a special location-scale model. However, it is found that n -consistency breaks down in the presence of endogeneity. Monte Carlo simulations show that the proposed estimators have superior performance in finite sample compared to other density estimators available in the literature. Two real data illustrations reveal the advantage of the proposed density estimator over the Rosenblatt-Parzen kernel density estimator.

This paper develops a testing procedure to simultaneously check (i) the independence between the error and the regressor(s), and (ii) the parametric specification in nonlinear regression models. This procedure generalizes the existing work of Sen and Sen [“Testing Independence and Goodness-of-fit in Linear Models,” Biometrika, 101, 927–942.] to a regression setting that allows any smooth parametric form of the regression function. We establish asymptotic theory for the test procedure under both conditional homoscedastic error and heteroscedastic error. The derived tests are easily implementable, asymptotically normal, and consistent against a large class of fixed alternatives. Besides, the local power performance is investigated. To calibrate the finite sample distribution of the test statistics, a smooth bootstrap procedure is proposed and found work well in simulation studies. Finally, two real data examples are analyzed to illustrate the practical merit of our proposed tests.

Numerous studies have demonstrated the significant role of environmental factors on trading behaviors. Air pollution, one of the most critical environmental stressors, may also trigger collective mood swings and influence investment decisions. Few and inconsistent results have been found regarding the relationship between air pollution and stock returns. We claim that public awareness may be a missing key, which is a vital antecedent of mood, playing an intermediate role in the link between air pollution and stock returns. The present study explores and verifies the underlying mechanism of the relationship between air pollution and stock returns. We demonstrate the joint effect of the current day’s air condition and consecutive days of air pollution on stock returns through awareness.

This paper proposes a transformed linear cointegration model of the form Λ ( y t , β 0 ) = x t ⊤ θ 1 0 + z t ⊤ θ 2 0 + u t , where Λ is a monotonic function, x t is the nonstationary vector regressor, z t is the stationary regressor, u t is the regression error, and β 0 , θ 1 0 , θ 2 0 are unknown parameters. This model offers a flexible nonlinear cointegration via the monotonic transformation of the dependent variable, and includes the conventional linear cointegration model as a special case. Asymptotic properties of the proposed estimators are established. Simulations demonstrate that the estimators perform well in small samples. A real data example on the purchasing power parity illustrates the practical merits of our model.

Expectile regression is a useful tool for modeling data with heterogeneous conditional distributions. This paper develops the jackknife model averaging method for expectile regressions. The asymptotic properties of expectile estimator under misspecification with increasing dimension of parameters have been studied. The model averaging expectile estimator using the leave-one-out cross-validated weight is shown to be asymptotically optimal in the sense of out-of-sample final prediction error. Numerical results demonstrate the nice performance of the averaging estimators.

This paper defines a new procedure to efficiently estimate non parametric simultaneous equations models. The proposed estimation procedure exploits the additive structure and achieves oracle efficiency without the knowledge of unobserved error terms. Furthermore, simulation results show that our new estimator outperforms the existing estimator in terms of mean squared error.

This paper considers the estimation of a semiparametric transformation model, Λ ( y t , β 0 ) = g ( x t ) + u t , where Λ ( ⋅ , β 0 ) is a strictly increasing function known up to an l -dimensional parameter β 0 , g is an unknown link function. Hermite polynomial expansion is used to approximate the link function g , which leads to an extreme estimator for β 0 and a plug-in estimator for g . Asymptotic properties of the estimators are established. Simulation results demonstrate that the estimators perform well in finite samples. An example on Canadian occupation prestige is provided to illustrate the practical value of the proposed model.

The equity premium, return on equity minus return on risk-free asset, is expected to be positive. We consider imposing such positivity constraint in local historical average (LHA) in nonparametric kernel regression framework. It is also extended to the semiparametric single index model when multiple predictors are used. We construct the constrained LHA estimator via an indicator function which operates as `model-selection' between the unconstrained LHA and the bound of the constraint (zero for the positivity constraint). We smooth the indicator function by bagging (Breiman 1996a), which operates as `model-averaging' and yields a combined forecast of unconstrained LHA forecasts and the bound of the constraint. The local combining weights are determined by the probability that the constraint is binding. Asymptotic properties of the constrained LHA estimators without and with bagging are established, which show how the positive constraint and bagging can help reduce the asymptotic variance and mean squared errors. Monte Carlo simulations are conducted to show the finite sample behavior of the asymptotic properties. In predicting U.S. equity premium, we show that substantial nonlinearity can be captured by LHA and that the local positivity constraint can improve out-of-sample prediction of the equity premium.

This paper considers testing additive error structure in nonparametric structural models, against the alternative hypothesis that the random error term enters the nonparametric model non-additively. We propose a test statistic under a set of identification conditions considered by Hoderlein, Su and White (2012), which require the existence of a control variable such that the regressor is independent of the error term given the control variable. The test statistic is motivated from the observation that, under the additive error structure, the partial derivative of the nonparametric structural function with respect to the error term is one under identification. The asymptotic distribution of the test is established and a bootstrap version is proposed to enhance its finite sample performance. Monte Carlo simulations show that the test has proper size and reasonable power in finite samples.

In a nonparametric multiplicative error model, this paper considers estimation of the regression mean function when the error is conditional homoscedastic. To exploit the conditional homoscedasticity information, we propose to estimate the conditional mean function using nonparametric generalized method of moments. The resulted estimator is shown to be asymptotically normal and enjoy superior finite sample properties in Monte Carlo simulations.

This article establishes the functional coefficient moving average model (FMA), which allows the coefficient of the classical moving average model to adapt with a covariate. The functional coefficient is identified as a ratio of two conditional moments. Local linear estimation technique is used for estimation and asymptotic properties of the resulting estimator are investigated. Its convergence rate depends on whether the underlying function reaches its boundary or not, and asymptotic distribution could be nonstandard. A model specification test in the spirit of Hardle-Mammen (1993) is developed to check the stability of the functional coefficient. Intensive simulations have been conducted to study the finite sample performance of our proposed estimator, and the size and the power of the test. The real data example on CPI data from China Mainland shows the efficacy of FMA. It gains more than 20% improvement in terms of relative mean squared prediction error compared to moving average model.

This paper considers nonparametric and semiparametric regression models subject to monotonicity constraint. We use bagging as an alternative approach to Hall and Huang (2001). Asymptotic properties of our proposed estimators and forecasts are established. Monte Carlo simulation is conducted to show their nite sample performance. An application to predicting equity premium is taken to illustrate the merits of the proposed approach. We introduce a new forecasting evaluation criterion based on the second order stochastic dominance in the size of forecast errors, which enables us to compare the competing forecasting models over dierent sizes of forecast errors. We show that imposing monotonicity constraint can mitigate the chance of making large size forecast errors.

The bulk of the literature on retail location looks at the topic from the perspective of either the retail firm or the individual shopper. Another branch of the literature examines the spatial distribution of retail activities within a city or region, drawing on either central place theory or the Lowry model, neither of which incorporates either markets or agglomeration economies. This paper looks at retail location from the perspective of a general equilibrium model of location and land use, with agglomeration economies in retailing. In particular, drawing on the Fujita-Ogawa (1982) model of non- monocentric cities, it develops a model of retail location, assuming that retail firms behave competitively, subject to spatial agglomeration economies. Locations are distinguished according to the effective variety of retail goods they offer. Shoppers are willing to pay more for goods at locations with greater effective variety, and in their choice of where to shop trade off retail price, product variety, and accessibility to home. Retail prices and land rents at different locations adjust to achieve spatial equilibrium.