High-dimensional tensor-valued data have recently gained attention from researchers in economics and finance. We consider the estimation and inference of high-dimensional tensor factor models, where each dimension of the tensor diverges. Our focus is on a factor model that admits CP-type tensor decomposition, which allows for non-orthogonal loading vectors. Based on the contemporary covariance matrix, we propose an iterative simultaneous projection estimation method. Our estimator is robust to weak dependence among factors and weak correlation across different dimensions in the idiosyncratic shocks. We establish an inferential theory, demonstrating both consistency and asymptotic normality under relaxed assumptions. Within a unified framework, we consider two eigenvalue ratio-based estimators for the number of factors in a tensor factor model and justify their consistency. Simulation studies confirm the theoretical results and an empirical application to sorted portfolios reveals three important factors: a market factor, a long-short factor, and a volatility factor.

In this paper, we consider diffusion index forecasting with both tensor and non-tensor predictors, where the tensor structure is preserved with a Canonical Polyadic (CP) tensor factor model. When the number of non-tensor predictors is small, we study the asymptotic properties of the least squares estimator in this tensor factor-augmented regression, allowing for factors with different strengths. We derive an analytical formula for prediction intervals that accounts for the estimation uncertainty of the latent factors. In addition, we propose a novel thresholding estimator for the high-dimensional covariance matrix that is robust to cross-sectional dependence. When the number of non-tensor predictors exceeds or diverges with the sample size, we introduce a multi-source factor-augmented sparse regression model and establish the consistency of the corresponding penalized estimator. Simulation studies validate our theoretical results and an empirical application to U.S. trade flows demonstrates the advantages of our approach over other popular methods in the literature.

Matrix-variate data of high dimensions are frequently observed in finance and economics, spanning extended time periods, such as the long-term data on international trade flows among numerous countries. To address potential structural shifts and explore the matrix structure's informational context, we propose a time-varying matrix factor model. This model accommodates changing factor loadings over time, revealing the underlying dynamic structure through nonparametric principal component analysis and facilitating dimension reduction. We establish the consistency and asymptotic normality of our estimators under general conditions that allow for weak correlations across time, rows, or columns of the noise. A novel approach is introduced to overcome rotational ambiguity in the estimators, enhancing the clarity and interpretability of the estimated loading matrices. Our simulation study highlights the merits of the proposed estimators and the effective of the smoothing operation. In an application to international trade flow, we investigate the trading hubs, centrality, patterns, and trends in the trading network.

Detecting structural changes in economic relationships has been a longstanding challenge in econometrics. Most of the literature on structural breaks has considered abrupt structural breaks. Existing tests for detecting smooth structural change typically rely on kernel estimation. In this article, we introduce a novel tuning-parameter-free test that minimizes a criterion function over all possible nondecreasing or nonincreasing structural change functions. This test is pivotal (after appropriate scaling) in the scalar case and remains computationally simple even in multivariate settings. Compared to existing nonparametric tests, our method offers superior power against local monotonic structural changes and does not involve the choice of a bandwidth parameter. A simulation study and two empirical examples highlight the merits of the proposed test relative to some popular tests for structural changes in the literature.

We examine how many factors out of a wide range of 207 that have incremental information in explaining cross-sectional stock returns. First, we find that the significance of each factor changes drastically over time. After accounting for false discovery rate (FDR), only 157 out of 207 factors are significant from 1967 to 2021, and only 56 from 2000 to 2021. Second, from 2000 to 2021, we find strikingly that only 3 clusters of factors that have incremental information. We further propose a new flexible time-varying latent factor model, and test in an alternative way on the number of factors that capture the information of the 56 significant factors while controlling for FDR, and find only 3, the market plus 2 latent ones, a number much fewer than widely believed.

In this paper, we propose a new nonparametric estimator of time-varying forecast combination weights. When the number of individual forecasts is small, we study the asymptotic properties of the local linear estimator. When the number of candidate forecasts exceeds or diverges with the sample size, we consider penalized local linear estimation with the group SCAD penalty. We show that the estimator exhibits the oracle property and correctly selects relevant forecasts with probability approaching one. Simulations indicate that the proposed estimators outperform existing combination schemes when structural changes exist. An empirical application on inflation and unemployment forecasting highlights the merits of our approach relative to other popular methods in the literature.

We consider a new nonparametric test for serial correlation of unknown form in the estimated residuals of a panel regression model, where individual and time effects can be fixed or random, and the panel data can be balanced or unbalanced. Our test is robust against potential weak error cross-sectional dependence and error serial dependence in higher-order moments. This is in contrast to existing tests for serial correlation in panel data models, which assume error components to be cross-sectionally and serially independent. Our test has an asymptotic N(0, 1) distribution under the null hypothesis and is consistent against serial correlation of unknown form. No common alternative is assumed and hence our test allows for substantial inhomogeneity in serial correlation across individuals. A simulation study highlights the merits of the proposed test relative to a variety of existing tests in the literature. We apply the new test to the empirical study of Wolfers on the relationship between unilateral divorce laws and divorce rates and find strong evidence against serial uncorrelatedness even controlling for the fixed effect.

Detecting and modeling structural changes in time series models have attracted great attention. However, relatively little effort has been paid to the testing of structural changes in panel data models despite their increasing importance in economics and finance. In this paper, we propose a new approach to testing structural changes in panel data models. Unlike the bulk of the literature on structural changes, which focuses on detection of abrupt structural changes, we consider smooth structural changes for which model parameters are unknown deterministic smooth functions of time except for a finite number of time points. We use nonparametric local smoothing method to consistently estimate the smooth changing parameters and develop two consistent tests for smooth structural changes in panel data models. The first test is to check whether all model parameters are stable over time. The second test is to check potential time-varying interaction while allowing for a common trend. Both tests have an asymptotic N(0,1) distribution under the null hypothesis of parameter constancy and are consistent against a vast class of smooth structural changes as well as abrupt structural breaks with possibly unknown break points alternatives. Simulation studies show that the tests provide reliable inference in finite samples and two empirical examples with respect to a cross-country growth model and a capital structure model are discussed.

Modeling and detecting parameter stability of econometric models is a long standing problem. Most existing estimation and testing methods are designed for models without endogeneity. Little attention has been paid to models with endogeneous regressors, which may arise in many scenarios in economics. In this paper, we first consider a time-varying coefficient time series model with potential time-varying endogeneity. A local linear two stage least squared estimation is developed to estimate coefficient functions. The consistency and asymptotic normality of the estimator are derived. Furthermore, a nonparametric test is proposed to check smooth structural changes as well as abrupt structural breaks with possibly unknown change points in regression models with potential endogeneity. The idea is to compare the fitted values of the unrestricted nonparametric time-varying coefficient model and the restricted constant parameter model. The test has an asymptotic N(0,1) distribution and does not require any prior information about the alternatives. A simulation study highlights the merits of the proposed estimator and test. In an application, we estimate the New Keynesian Phillips Curve for the US nonparametrically and find strong evidence against its stability.

Detecting and modeling structural changes in GARCH processes have attracted increasing attention in time series econometrics. In this paper, we propose a new approach to testing structural changes in GARCH models. The idea is to compare the log likelihood of a time-varying parameter GARCH model with that of a constant parameter GARCH model, where the time-varying GARCH parameters are estimated by a local quasi-maximum likelihood estimator (QMLE) and the constant GARCH parameters are estimated by a standard QMLE. The test does not require any prior information about the alternatives of structural changes. It has an asymptotic N(0,1) distribution under the null hypothesis of parameter constancy and is consistent against a vast class of smooth structural changes as well as abrupt structural breaks with possibly unknown break points. A consistent parametric bootstrap is employed to provide a reliable inference in finite samples and a simulation study highlights the merits of our test.

We propose a test for invertibility or fundamentalness of structural vector autoregressive moving average models generated by non-Gaussian independent and identically distributed structural shocks. We prove that in these models and under some regularity conditions the Wold innovations are a martingale difference sequence (mds) if and only if the structural shocks are fundamental. This simple but powerful characterization suggests an empirical strategy to assess invertibility. We propose a test based on a generalized spectral density to check for the mds property of the Wold innovations. This approach does not require the specification and estimation of the economic agent’s information flows or the identification and estimation of the structural parameters and the noninvertible roots. Moreover, the proposed test statistic uses all lags in the sample and it has a convenient asymptotic N(0 1) distribution under the null hypothesis of invertibility, and hence, it is straightforward to implement. In case of rejection, the test can be further used to check if a given set of additional variables provides sufficient informational content to restore invertibility. A Monte Carlo study is conducted to examine the finite-sample performance of our test. Finally, the proposed test is applied to two widely cited works on the effects of fiscal shocks by Blanchard and Perotti (2002) and Ramey (2011).

We develop a nonparametric test to check whether a process can be represented by a stochastic differential equation driven only by a Brownian motion. Our testing procedure utilizes the infinitesimal operator-based martingale characterization combined with a generalized spectral approach. Such a testing procedure is feasible and convenient because the infinitesimal operator of the diffusion process has a closed-form expression. The proposed test is applicable to both univariate and multivariate processes and has an N(0,1) limit distribution under the diffusion hypothesis. Simulation and empirical studies show that the proposed test has reasonable performance in small samples.

Modeling conditional distributions in time series has attracted increasing attention in economics and finance. We develop a new class of generalized Cramer–von Mises (GCM) specification tests for time series conditional distribution models using a novel approach, which embeds the empirical distribution function in a spectral framework. Our tests check a large number of lags and are therefore expected to be powerful against neglected dynamics at higher order lags, which is particularly useful for non-Markovian processes. Despite using a large number of lags, our tests do not suffer much from loss of a large number of degrees of freedom, because our approach naturally downweights higher order lags, which is consistent with the stylized fact that economic or financial markets are more affected by recent past events than by remote past events. Unlike the existing methods in the literature, the proposed GCM tests cover both univariate and multivariate conditional distribution models in a unified framework. They exploit the information in the joint conditional distribution of underlying economic processes. Moreover, a class of easy-to-interpret diagnostic procedures are supplemented to gauge possible sources of model misspecifications. Distinct from conventional CM and Kolmogorov–Smirnov (KS) tests, which are also based on the empirical distribution function, our GCM test statistics follow a convenient asymptotic N ( 0 , 1 ) distribution and enjoy the appealing “nuisance parameter free” property that parameter estimation uncertainty has no impact on the asymptotic distribution of the test statistics. Simulation studies show that the tests provide reliable inference for sample sizes often encountered in economics and finance.

The Markov property is a fundamental property in time series analysis and is often assumed in economic and financial modeling. We develop a new test for the Markov property using the conditional characteristic function embedded in a frequency domain approach, which checks the implication of the Markov property in every conditional moment (if it exists) and over many lags. The proposed test is applicable to both univariate and multivariate time series with discrete or continuous distributions. Simulation studies show that with the use of a smoothed nonparametric transition density-based bootstrap procedure, the proposed test has reasonable sizes and all-around power against several popular non-Markov alternatives in finite samples. We apply the test to a number of financial time series and find some evidence against the Markov property.

Checking parameter stability of econometric models is a long-standing problem. Almost all existing structural change tests in econometrics are designed to detect abrupt breaks. Little attention has been paid to smooth structural changes, which may be more realistic in economics. We propose a consistent test for smooth structural changes as well as abrupt structural breaks with known or unknown change points. The idea is to estimate smooth time-varying parameters by local smoothing and compare the fitted values of the restricted constant parameter model and the unrestricted time-varying parameter model. The test is asymptotically pivotal and does not require prior information about the alternative. A simulation study highlights the merits of the proposed test relative to a variety of popular tests for structural changes. In an application, we strongly reject the stability of univariate and multivariate stock return prediction models in the postwar and post-oil-shocks periods.

We develop an omnibus specification test for multivariate continuous-time models using the conditional characteristic function, which often has a convenient closed-form or can be accurately approximated for many multivariate continuous-time models in finance and economics. The proposed test fully exploits the information in the joint conditional distribution of underlying economic processes and hence is expected to have good power in a multivariate context. A class of easy-to-interpret diagnostic procedures is supplemented to gauge possible sources of model misspecification. Our tests are also applicable to discrete-time distribution models. Simulation studies show that the tests provide reliable inference in finite samples.

We develop a nonparametric regression-based goodness-of-fit test for multifactor continuous-time Markov models using the conditional characteristic function, which often has a convenient closed form or can be approximated accurately for many popular continuous-time Markov models in economics and finance. An omnibus test fully utilizes the information in the joint conditional distribution of the underlying processes and hence has power against a vast class of continuous-time alternatives in the multifactor framework. A class of easy-to-interpret diagnostic procedures is also proposed to gauge possible sources of model misspecification. All the proposed test statistics have a convenient asymptotic N (0, 1) distribution under correct model specification, and all asymptotic results allow for some data-dependent bandwidth. Simulations show that in finite samples, our tests have reasonable size, thanks to the dimension reduction in nonparametric regression, and good power against a variety of alternatives, including misspecifications in the joint dynamics, but the dynamics of each individual component is correctly specified. This feature is not attainable by some existing tests. A parametric bootstrap improves the finite-sample performance of proposed tests but with a higher computational cost.

Modelling and detecting structural changes in GARCH processes have attracted a great amount of attention in econometrics over the past few years. We generalize Dahlhaus and Rao (2006)’s time varying ARCH processes to time varying GARCH processes and show the consistency of the weighted quasi maximum likelihood estimator. A class of generalized likelihood ratio tests are proposed to check smooth structural changes as well as abrupt structural breaks with known or unknown change points in GARCH models. The idea is to compare the log likelihood of the unrestricted nonparametric time-varying GARCH model and the restricted constant parameter GARCH model, which can be viewed as a generalization of likelihood ratio tests from the parametric framework to the nonparametric framework. The tests have a convenient asymptotic N(0,1) distribution and do not require any prior information about the alternatives. A simulation study highlights the merits of the proposed tests. JEL Classi…cations: C1, C4, E0. Key words: GARCH, Kernel, Model stability, Parameter constancy, QMLE, Smooth structural change 1. INTRODUCTION Since the seminal works by Engle (1982) and Bollerslev (1986), Generalized Autoregressive conditional heteroskedasticity (GARCH) type models have been commonly used to capture volatility dynamics of …nancial time series. However, underlying all these models is the assumption of stationarity. Given the changing pace of the underlying economic mechanism, modeling …nancial variables over a long time horizon may not be suitable. It is quite plausible that structural changes have occurred, causing the time series to deviate from stationarity. Indeed, various economic factors may lead to structural changes detected in …nancial time series. For example, one driving force for structural changes are“shocks” induced by institutional changes, such as changes of exchange rate systems from the …xed exchange rate mechanism to the ‡oating exchange rate mechanism, or the introduction of Euro. As Lucas (1976) points out, “any change in policy will systematically alter the structure of econometric models”, given that the structure of an econometric model depends crucially on agents’expectations, which in turn vary systematically with changes in the structure of time series relevant to decision makers. The prevalence of structural instability in …nancial time series has been con…rmed by numerous empirical studies. For example, Andreou and Ghysels (2002) examine the change-point hypothesis in volatility dynamics of international stock market indices and foreign exchange returns and …nd multiple breaks associated with the Asian and Russian …nancial crisis; Mikosch and Starica (2004) apply their goodness-of-…t test to the S&P500 returns and detect structural changes related to shifts of unconditional variance. Model stability is crucial for statistical inference, out-of-sample forecasts, and any policy implications drawn from the model. In particular, ignoring structural changes in …nancial time series can easily lead to spurious persistence in the conditional volatility parameters. Diebold (1986), Hendry (1986) and Lamoureux and Lastrapes (1990) are among the …rst to suggest that structural changes unaccounted for can yield Integrated GARCH or long memory e¤ects. More recently, Mikosch and Starica (2004) and Hillebrand (2005) provide some theoretical explanation for this phenomenon. The spurious IGARCH e¤ects imply that shocks have a permanent impact on volatility so current information remains relevant when forecasting the conditional variance for all horizons while for the short memory process, shocks to variance do decay over time. Moreover, model instability may a¤ect asset allocation or lead to large errors in pricing, hedging and managing risk. Pettenuzzo and Timmerman (2005) show that the possibility of future breaks has its largest e¤ect at long investment horizons, but historical breaks can signi…cantly change investment decisions even at short horizons through its e¤ect on current parameter estimates. Some tests have been proposed to test structural breaks in GARCH models. For example, Chu (1995) generalizes Andrews’(1993) supremum Lagrange multiplier (LM) test to GARCH framework. However, the test just considers one-time shift as the alternative so does not have 1 good power against multiple breaks. Berkes, Gombay, Horvath and Kokoszka (2004) develop a sequential likelihood-ratio (LR) based test for evaluating the stability of the GARCH parameters. Their test can be used to check which parameter of a GARCH model has a change point and hence is more informative than some existing tests. However, it is computationally intensive as it involves the calculation of quasilikelihood scores. Kulperger and Yu (2005) derive the properties of structural break tests based on the partial sums of residuals of GARCH models. Almost all existing change-point tests for GARCH models are to detect abrupt changes. One exception is the test of Amado and Ter :: asvirta (2008), who considers testing for a smooth time varying structure of GARCH models. In fact, smooth changes may be more realistic because volatility usually evolves over time in a continuous manner and volatility jumps are rare. Empirical evidences show that various economic events, such as liberalization of emerging markets, integration of world equity markets, changes in exchange rate or interest rate regimes, may lead to structural changes in volatility models. This paper proposes a class of consistent tests for smooth structural changes as well as abrupt structural breaks in GARCH models with known or unknown change points. The idea is to estimate the smooth time-varying parameters of GARCH model by weighted Quasi maximum likelihood estimation (WQMLE) and compare them with the QMLE parameter estimator. The tests compare the log likelihood of the unrestricted nonparametric time-varying GARCH model and the restricted constant parameter GARCH model, which can be viewed as a generalization of LR tests from the parametric framework to the nonparametric framework. Compared with the existing tests for structural breaks in GARCH models in the literature, the proposed tests have a number of appealing features. First, the proposed tests are consistent against a large class of smooth time-varying parameter alternatives. They are also consistent against multiple sudden structural breaks in GARCH models with unknown break points. Second, no prior information on a structural change GARCH alternative is needed. In particular, we do not need to know whether the structural changes are smooth or abrupt, and in the cases of abrupt structural breaks, we do not need to know the dates or the number of breaks. Third, unlike most tests for structural breaks in GARCH models in the literature, which often have nonstandard asymptotic distributions, the proposed tests have a convenient null asymptotic N(0,1) distribution. The only inputs required are the QMLE andWQMLE parameter estimators. Hence, any standard econometric software can carry out computational implementation easily. Fourth, the nonparametric time-varying parameter estimator is sensitive to the local behavior of time-varying parameters. Because only local information is employed in estimating parameters at each time point, the proposed tests have symmetric power against structural breaks that occur either in the …rst or second half of the sample period. This is di¤erent from some existing tests

Checking parameter stability of economic models is a long-standing problem in time series econometrics. A classical econometric procedure is Chow’s (1960) test, which checks for the existence of a structural change on a known date. Various extensions have been made to test multiple changes with known or unknown change points. However, almost all existing structural change tests in econometrics are deigned to detect abrupt breaks. Little attention has been paid to smooth structural changes, which may be more realistic in economics. This paper proposes two consistent tests for smooth structural changes as well as abrupt structural breaks with known or unknown change points. The idea is to estimate the smooth time-varying parameters by local smoothing and compare them with the OLS parameter estimator. The …rst test compares the sums of squared residuals of the restricted constant parameter model and the unrestricted nonparametric time-varying parameter model, in a spirit similar to Chow’s (1960) F test. The second test compares the …tted values of the restricted and unrestricted models, which can be viewed as a generalization of Hausman’s (1978) test. Both tests have a convenient asymptotic N(0,1) distribution and do not require any prior information about the alternatives. Interestingly, unlike Chow’s (1960) test, the generalized Chow test is no longer optimal; it is asymptotically less powerful than the generalized Hausman test. A simulation study highlights the merits of the proposed tests in comparison with a variety of popular tests for structural changes. JEL Classi…cations: C1, C4, E0.