This paper proposes new, simple, and more accurate statistical tests in a cointegrated system that allows for endogenous regressors and serially dependent errors. The approach involves first transforming the time series using orthonormal basis functions in L2[0, 1], which have energy concentrated at low frequencies, and then running an augmented regression based on the transformed data and constructing the test statistics in the usual way. The approach is essentially the same as the trend instrumental variable approach of Phillips (2014), but the number of orthonormal basis functions is held fixed for the development of the standard F and t asymptotic theory. The tests are extremely simple to implement, as they can be carried out in exactly the same way as if the transformed regression is a classical linear normal regression. In particular, critical values are from the standard F or t distribution. The proposed F and t tests are robust in that they are asymptotically valid regardless of whether the number of basis functions is held fixed or allowed to grow with the sample size. The F and t tests have more accurate size in finite samples than existing tests such as the asymptotic chi-squared and normal tests based on the fully modified OLS estimator of Phillips and Hansen (1990) and can be made as powerful as the latter test.

We prove the long standing conjecture of Ding and Granger (1996) about the existence of a stationary Long Memory ARCH model with finite fourth moment. This result follows from the necessary and sufficient conditions for the existence of covariance stationary integrated AR(∞), ARCH(∞), and FIGARCH models obtained in the present article. We also prove that such processes always have long memory.

Expansion and collapse are two key features of a financial asset bubble. Bubble expansion may be modeled using a mildly explosive process. Bubble implosion may take several different forms depending on the nature of the collapse and therefore requires some flexibility in modeling. This paper first strengthens the theoretical foundation of the real time bubble monitoring strategy proposed in Phillips, Shi and Yu (2015a,b, PSY) by developing analytics and studying the performance characteristics of the testing algorithm under alternative forms of bubble implosion which capture various return paths to market normalcy. Second, we propose a new reverse sample use of the PSY procedure for detecting crises and estimating the date of market recovery. Consistency of the dating estimators is established and the limit theory addresses new complications arising from the alternative forms of bubble implosion and the endogeneity effects present in the reverse regression. A real-time version of the strategy is provided that is suited for practical implementation. Simulations explore the finite sample performance of the strategy for dating market recovery. The use of the PSY strategy for bubble monitoring and the new procedure for crisis detection are illustrated with an application to the Nasdaq stock market.

We derive uniform convergence results of lag-window spectral density estimates for a general class of multivariate stationary processes represented by an arbitrary measurable function of iid innovations. Optimal rates of convergence, that hold as both the time series and the cross section dimensions diverge, are obtained under mild and easily verifiable conditions. Our theory complements earlier results, most of which are univariate, which primarily concern in-probability, weak or distributional convergence, yet under a much stronger set of regularity conditions, such as linearity in iid innovations. Based on cross spectral density functions, we then propose a new test for independence between two stationary time series. We also explain the extent to which our results provide the foundation to derive the double asymptotic results for estimation of generalized dynamic factor models.

This article revisits the asymptotic inference for nonstationary AR(1) models of Phillips and Magdalinos (2007a) by incorporating a structural change in the AR parameter at an unknown time k0. Consider the model ${y_t} = {\beta _1}{y_{t - 1}}I\{ t \le {k_0}\} + {\beta _2}{y_{t - 1}}I\{ t > {k_0}\} + {\varepsilon _t},t = 1,2, \ldots ,T$, where I{·} denotes the indicator function, one of ${\beta _1}$ and ${\beta _2}$ depends on the sample size T, and the other is equal to one. We examine four cases: Case (I): ${\beta _1} = {\beta _{1T}} = 1 - c/{k_T}$, ${\beta _2} = 1$; (II): ${\beta _1} = 1$, ${\beta _2} = {\beta _{2T}} = 1 - c/{k_T}$; (III): ${\beta _1} = 1$, ${\beta _2} = {\beta _{2T}} = 1 + c/{k_T}$; and case (IV): ${\beta _1} = {\beta _{1T}} = 1 + c/{k_T}$, ${\beta _2} = 1$, where c is a fixed positive constant, and kT is a sequence of positive constants increasing to ∞ such that kT = o(T). We derive the limiting distributions of the t-ratios of ${\beta _1}$ and ${\beta _2}$ and the least squares estimator of the change point for the cases above under some mild conditions. Monte Carlo simulations are conducted to examine the finite-sample properties of the estimators. Our theoretical findings are supported by the Monte Carlo simulations.

The intercept of the binary response model is not regularly identified (i.e., $\sqrt n$ consistently estimable) when the support of both the special regressor V and the error term ε are the whole real line. The estimator of the intercept potentially has a slower than $\sqrt n$ convergence rate, which can result in a large estimation error in practice. This paper imposes additional tail restrictions which guarantee the regular identification of the intercept and thus the $\sqrt n$-consistency of its estimator. We then propose an estimator that achieves the $\sqrt n$ rate. Last, we extend our tail restrictions to a full-blown model with endogenous regressors.

We consider nonparametric identification and estimation of truncated regression models with unknown conditional heteroskedasticity. The existing methods (e.g., Chen (2010, Review of Economic Studies 77, 127–153)) that ignore heteroskedasticity often result in inconsistent estimators of regression functions. In this paper, we show that both the regression and heteroskedasticity functions are identified in a location-scale setting. Based on our constructive identification results, we propose kernel-based estimators of regression and heteroskedasticity functions and show that the estimators are asymptotically normally distributed. Our simulations demonstrate that our new method performs well in finite samples. In particular, we confirm that in the presence of heteroskedasticity, our new estimator of the regression function has a much smaller bias than Chen’s (2010, Review of Economic Studies 77, 127–153) estimator.

Limit theory is developed for the dynamic panel IV estimator in the presence of an autoregressive root near unity. In the unit root case, Anderson–Hsiao lagged variable instruments satisfy orthogonality conditions but are well known to be irrelevant. For a fixed time series sample size (T) IV is inconsistent and approaches a shifted Cauchy-distributed random variate as the cross-section sample size n → ∞. But when T → ∞, either for fixed n or as n → ∞, IV is $\sqrt T$ consistent and its limit distribution is a ratio of random variables that converges to twice a standard Cauchy as n → ∞. In this case, the usual instruments are uncorrelated with the regressor but irrelevance does not prevent consistent estimation. The same Cauchy limit theory holds sequentially and jointly as (n, T) → ∞ with no restriction on the divergence rates of n and T. When the common autoregressive root $\rho = 1 + c/\sqrt T$ the panel comprises a collection of mildly integrated time series. In this case, the IV estimator is $\sqrt n$ consistent for fixed T and $\sqrt {nT}$ consistent with limit distribution N (0, 4) when (n, T) → ∞ sequentially or jointly. These results are robust for common roots of the form ρ = 1+c/Tγ for all γ ∈ (0, 1) and joint convergence holds. Limit normality holds but the variance changes when γ = 1. When γ > 1 joint convergence fails and sequential limits differ with different rates of convergence. These findings reveal the fragility of conventional Gaussian IV asymptotics to persistence in dynamic panel regressions.

This paper proposes two simple and new specification tests based on the use of an orthogonal series for a considerable class of bivariate nonlinearly cointegrated time series models with endogeneity and nonstationarity. The first test is proposed for the case where the regression function is integrable, which fills a gap in the literature, and the second test, which nests the first one, deals with regression functions in a quite large function space that is sufficient for both theoretical and practical use. As a starting point of our asymptotic theory, the first test is studied initially and then the theory is extended to the second test. Endogeneity in two general forms is allowed in the models to be tested. The finite sample performance of the tests is examined through several simulated examples. Our experience generally shows that the proposed tests are easily implementable and also have stable sizes and good power properties even when the ‘distance’ between the null hypothesis and a sequence of local alternatives is asymptotically negligible.

We propose nonparametric estimators for conditional value-at-risk (CVaR) and conditional expected shortfall (CES) associated with conditional distributions of a series of returns on a financial asset. The return series and the conditioning covariates, which may include lagged returns and other exogenous variables, are assumed to be strong mixing and follow a nonparametric conditional location-scale model. First stage nonparametric estimators for location and scale are combined with a generalized Pareto approximation for distribution tails proposed by Pickands (1975, Annals of Statistics 3, 119–131) to give final estimators for CVaR and CES. We provide consistency and asymptotic normality of the proposed estimators under suitable normalization. We also present the results of a Monte Carlo study that sheds light on their finite sample performance. Empirical viability of the model and estimators is investigated through a backtesting exercise using returns on future contracts for five agricultural commodities.

We provide nonparametric methods for stochastic volatility modeling. Our methods allow for the joint evaluation of return and volatility dynamics with nonlinear drift and diffusion functions, nonlinear leverage effects, and jumps in returns and volatility with possibly state-dependent jump intensities, among other features. In the first stage, we identify spot volatility by virtue of jump-robust nonparametric estimates. Using observed prices and estimated spot volatilities, the second stage extracts the functions and parameters driving price and volatility dynamics from nonparametric estimates of the bivariate process’ infinitesimal moments. For these infinitesimal moment estimates, we report an asymptotic theory relying on joint in-fill and long-span arguments which yields consistency and weak convergence under mild assumptions.

This paper examines the ordinary least squares (OLS) estimator of the structural parameters in a simple macroeconomic model in which agents are boundedly rational and use an adaptive learning rule to form expectations of the endogenous variable. The popularity of learning models has recently increased amongst applied economists and policy makers who seek to estimate them empirically. Yet the econometrics of learning models is largely uncharted territory. We consider two prominent learning algorithms, namely constant gain and decreasing gain learning. For each of the two learning rules, our analysis proceeds in two stages. First, the paper derives the asymptotic properties of agents’ expectations. At the second stage, the paper derives the asymptotics of OLS in the structural model, taking the first stage learning dynamics as given. In the case of constant gain learning, the structural model effectively amounts to a stationary, cointegrating, or co-explosiveness regression. With decreasing gain learning, the regressors are asymptotically collinear such that OLS does not satisfy, in general, the Grenander conditions for consistent estimability. Nevertheless, this paper shows that the OLS estimator remains consistent in all models considered. It also shows, however, that its asymptotic distribution, and hence any inference based upon it, may be nonstandard.

This paper studies the GMM estimation and inference problem that occurs when the Jacobian of the moment conditions is rank deficient of known forms at the true parameter values. Dovonon and Renault (2013) recently raised a local identification issue stemming from this type of degenerate Jacobian. The local identification issue leads to a slow rate of convergence of the GMM estimator and a nonstandard asymptotic distribution of the over-identification test statistics. We show that the known form of rank-deficient Jacobian matrix contains nontrivial information about the economic model. By exploiting such information in estimation, we provide GMM estimator and over-identification tests with standard properties. The main theory developed in this paper is applied to the estimation of and inference about the common conditionally heteroskedastic (CH) features in asset returns. The performances of the newly proposed GMM estimators and over-identification tests are investigated under the similar simulation designs used in Dovonon and Renault (2013).

In this article, we propose a general method for testing inequality restrictions on nonparametric functions. Our framework includes many nonparametric testing problems in a unified framework, with a number of possible applications in auction models, game theoretic models, wage inequality, and revealed preferences. Our test involves a one-sided version of Lp functionals of kernel-type estimators (1 ≤ p < ∞) and is easy to implement in general, mainly due to its recourse to the bootstrap method. The bootstrap procedure is based on the nonparametric bootstrap applied to kernel-based test statistics, with an option of estimating “contact sets.” We provide regularity conditions under which the bootstrap test is asymptotically valid uniformly over a large class of distributions, including cases where the limiting distribution of the test statistic is degenerate. Our bootstrap test is shown to exhibit good power properties in Monte Carlo experiments, and we provide a general form of the local power function. As an illustration, we consider testing implications from auction theory, provide primitive conditions for our test, and demonstrate its usefulness by applying our test to real data. We supplement this example with the second empirical illustration in the context of wage inequality.

Many empirical studies estimate the structural effect of some variable on an outcome of interest while allowing for many covariates. We present inference methods that account for many covariates. The methods are based on asymptotics where the number of covariates grows as fast as the sample size. We find a limiting normal distribution with variance that is larger than the standard one. We also find that with homoskedasticity this larger variance can be accounted for by using degrees-of-freedom-adjusted standard errors. We link this asymptotic theory to previous results for many instruments and for small bandwidth(s) distributional approximations.

This paper investigates whether there can exist regular estimators in models characterized by nonparametric instrumental variable (NPIV). We show by a number of examples that regular estimation is impossible in general for nonlinear functionals.

We propose a characteristic function based test for conditional independence, applicable to both cross-sectional and time series data. We also derive a class of derivative tests, which deliver model-free tests for such important hypotheses as omitted variables, Granger causality in various moments and conditional uncorrelatedness. The proposed tests have a convenient asymptotic null N (0, 1) distribution, and are asymptotically locally more powerful than a variety of related smoothed nonparametric tests in the literature. Unlike other smoothed nonparametric tests for conditional independence, we allow nonparametric estimators for both conditional joint and marginal characteristic functions to jointly determine the asymptotic distributions of the test statistics. Monte Carlo studies demonstrate excellent power of the tests against various alternatives. In an application to testing Granger causality, we document the existence of nonlinear relationships between money and output, which are missed by some existing tests.

This paper considers nonparametric instrumental variable regression when the endogenous variable is contaminated with classical measurement error. Existing methods are inconsistent in the presence of measurement error. We propose a wavelet deconvolution estimator for the structural function that modifies the generalized Fourier coefficients of the orthogonal series estimator to take into account the measurement error. We establish the convergence rates of our estimator for the cases of mildly/severely ill-posed models and ordinary/super smooth measurement errors. We characterize how the presence of measurement error slows down the convergence rates of the estimator. We also study the case where the measurement error density is unknown and needs to be estimated, and show that the estimation error of the measurement error density is negligible under mild conditions as far as the measurement error density is symmetric.