Peter C. B. Phillips Publications

Edgeworth expansions are developed for the finite sample distribution of the least squares estimator in a time series parametric first order autoregression with Hilbert space curves of cross section data. The main result extends to this functional data environment the Edgeworth expansion in the corresponding scalar time series AR(1). In doing so, the results show how function-valued cross section data, and hence general forms of cross section dependence, affect the finite sample distribution of the serial correlation coefficient. Autoregressions with functional fixed effect intercepts are included and the results therefore relate to dynamic panel autoregression with individual effects. The primary impact of the use of high-dimensional curved cross section data is to reduce the variation in scalar regression estimation and provide some improvement in the accuracy of the usual asymptotic approximation to the finite sample distribution. Limit results for the expansions under full cross section dependence matching the scalar time series case and independence matching the dynamic panel case are given as special cases. The findings are supported by numerical computations of the exact distributions and the approximations.

Optimal estimation is explored in long run relations that are modeled within a semiparametric triangular multicointegrated system. In nonsingular cointegrated systems, where there is no multicointegration, optimal estimation is well understood (Phillips, 1991a). This paper establishes corresponding optimal results for singular systems, thereby accommodating a wide class of multicointegrated nonstationary time series with nonparametric transient dynamics. The optimality and sub-optimality of existing estimators are considered and new optimal estimators of both the cointegrating and multicointegrating coefficients are introduced that are based on spectral regression.

This paper studies high-dimensional curve time series with common stochastic trends. A dual functional factor model structure is adopted with a high-dimensional factor model for the observed curve time series and a low-dimensional factor model for the latent curves with common trends. A functional PCA technique is applied to estimate the common stochastic trends and functional factor loadings. Under some regularity conditions we derive the mean square convergence and limit distribution theory for the developed estimates, allowing the dimension and sample size to jointly diverge to infinity. We propose an easy-to-implement criterion to consistently select the number of common stochastic trends and further discuss model estimation when the nonstationary factors are cointegrated. Extensive Monte-Carlo simulations and two empirical applications to large-scale temperature curves in Australia and log-price curves of S&P 500 stocks are conducted, showing finite-sample performance and providing practical implementations of the new methodology.

This paper is part of a joint study of parametric autoregression with cross section curve time series, focussing on unit root (UR) nonstationary curve data autoregression. The Hilbert space setting extends scalar UR and local UR models to accommodate high dimensional cross section dependent data under very general conditions. New limit theory is introduced that involves two parameter Gaussian processes that generalize the standard UR and local UR asymptotics. Bias expansions provide extensions of the well-known results in scalar autoregression and fixed effect dynamic panels to functional dynamic regressions. Semiparametric and ADF-type UR tests are developed with corresponding limit theory that enables time series inference with high dimensional curve cross section data, allowing also for functional fixed effects and deterministic trends. The asymptotics reveal the effects of general forms of cross section dependence in wide nonstationary panel data modeling and show dynamic panel regression limit theory as a special limiting case of curve time series asymptotics. Simulations provide evidence of the impact of curve cross section data on estimation and test performance and the adequacy of the asymptotics. An empirical illustration of the methodology is provided to assess the presence of time series nonstationarity in household Engel curves among ageing seniors in Singapore using the Singapore life panel dataset.

This paper develops and applies new asymptotic theory for estimation and inference in parametric autoregression with function valued cross section curve time series. The study provides a new approach to dynamic panel regression with high dimensional dependent cross section data. Here we deal with the stationary case and provide a full set of results extending those of standard Euclidean space autoregression, showing how function space curve cross section data raises efficiency and reduces bias in estimation and shortens confidence intervals in inference. Methods are developed for high-dimensional covariance kernel estimation that are useful for inference. The findings reveal that function space models with wide-domain and narrow-domain cross section dependence provide insights on the effects of various forms of cross section dependence in discrete dynamic panel models with fixed and interactive fixed effects. The methodology is applicable to panels of high dimensional wide datasets that are now available in many longitudinal studies. An empirical illustration is provided that sheds light on household Engel curves among ageing seniors in Singapore using the Singapore life panel longitudinal dataset.

To safeguard economic and financial stability policymakers regularly take actions designed to increase resilience to systemic risks and curb speculative market behavior. To assess the effectiveness of such mitigation policies, we introduce a counterfactual approach tailored to accommodate the mildly explosive dynamics that occur during speculative bubbles. We derive asymptotics of the estimated treatment effect under a common factor structure that allows for explosive, I(1), and stationary factors, thereby having applicability to a wide range of prevailing economic conditions. An inferential procedure is proposed for the policy treatment effect that has asymptotic validity and demonstrates satisfactory finite sample performance. An empirical analysis examines the monetary policy of interest rate hikes implemented by the Reserve Bank of New Zealand, beginning in October 2021.This policy exerted a statistically significant cooling effect on all regional housing markets in New Zealand. Our findings show that this policy led to 20%-33% reductions in house prices in five out of six regions seven months after the enactment of the interest rate hike.

Cointegrating rank selection is studied in a function space reduced rank regression where the data are time series of cross section curves. A semiparametric approach to rank selection is employed using information criteria suitably modified to take account of the function space context, extending the linear cointegrating model to accommodate cross section data under general forms of dependence. A parametric formulation is employed analogous to recent work on cross section curve autoregression and cointegrating regression. Consistent cointegrating rank estimation is developed by the use of information criteria methods that are extended to the curve time series environment. The asymptotic theory involves two parameter Gaussian processes that generalize the standard limit processes involved in cointegrating regressions with conventional multiple time series. Simulations provide evidence of the effectiveness of consistent rank selection by the BIC criterion and the tendency of AIC to overestimate order as it does in standard lag order selection in autoregression as well as in reduced rank regression with multiple time series.

Predictive regression models are often used to evaluate the predictive capability of economic fundamentals on bond and equity returns. Inferential procedures in these regressions typically employ parameter constancy or piecewise constancy in slope coefficients. Such formulations are prone to misspecification, more especially during periods of disturbance or evolution in prevailing economic and financial conditions, which can lead to size distortion and spurious evidence of predictability. To address these issues the present work proposes a semiparametric predictive regression model with mixed-root regressors and time-varying coefficients that allow for smooth evolution in the generating mechanism over time. For estimation and inference a novel variant of the self-generated instrument approach called Sieve-IVX is introduced, giving a robust approach to inference concerning time-varying predictability that is applicable irrespective of the degrees of persistence. Asymptotic theory of the Sieve-IVX approach is provided together with both pointwise and uniform inference procedures for testing predictability and model specification. Simulations show excellent performance characteristics of these statistics in finite samples. An empirical exercise is conducted to examine excess S&P 500 returns, applying Sieve-IVX regression coupled with pointwise and uniform tests to reveal evidence of time-varying patterns in the predictive capability of commonly used fundamental variables.

The recent artificial intelligence (AI) boom covers a period of rapid innovation and wide adoption of AI intelligence technologies across diverse industries. These developments have fueled an unprecedented frenzy in the Nasdaq, with AI-focused companies experiencing soaring stock prices that raise concerns about speculative bubbles and real-economy consequences. Against this background the present study investigates the formation of speculative bubbles in the Nasdaq stock market with a specific focus on the so-called ‘Magnificent Seven’ (Mag-7) individual stocks during the AI boom, spanning the period January 2017 to January 2025. We apply the real time PSY bubble detection methodology of Phillips et al. (2015a,b), while controlling for market and industry factors for individual stocks. Confidence intervals to assess the degree of speculative behavior in asset price dynamics are calculated using the near-unit root approach of Phillips (2023). The findings reveal the presence of speculative bubbles in the Nasdaq stock market and across all Mag-7 stocks. Nvidia and Microsoft experience the longest speculative periods over January 2017 – December 2021, while Nvidia and Tesla show the fastest rates of explosive behavior. Speculative bubbles persist in the market and in six of the seven stocks (excluding Apple) from December 2022 to January 2025. Near-unit-root inference indicates mildly explosive dynamics for Nvidia and Tesla (2017–2021) and local-to-unity near explosive behavior for all assets in both periods.

This note shows that the mixed normal asymptotic limit of the trend IV estimator with a fixed number of deterministic instruments (fTIV) holds in both singular (multicointegrated) and nonsingular cointegration systems, thereby relaxing the exogeneity condition in (Phillips and Kheifets, 2024, Theorem 1(ii)). The mixed normality of the limiting distribution of fTIV allows for asymptotically pivotal F tests about the cointegration parameters and for simple efficiency comparisons of the estimators for different numbers K of instruments, as well as comparisons with the trend IV estimator when K → ∞ with the sample size.

In GMM estimation, it is well known that if the moment dimension grows with the sample size, the asymptotics of GMM differ from the standard finite dimensional case. The present work examines the asymptotic properties of infinite dimensional GMM estimation when the weight matrix is formed by inverting Brownian motion or Brownian bridge covariance kernels. These kernels arise in econometric work such as minimum Cramer-von Mises distance estimation when testing distributional specification. The properties of GMM estimation are studied under different environments where the moment conditions converge to a smooth Gaussian or non-differentiable Gaussian process. Conditions are also developed for testing the validity of the moment conditions by means of a suitably constructed J-statistic. In case these conditions are invalid we propose another test called the U-test. As an empirical application of these infinite dimensional GMM procedures the evolution of cohort labor income inequality indices is studied using the Continuous Work History Sample database. The findings show that labor income inequality indices are maximized at early career years, implying that economic policies to reduce income inequality should be more effective when designed for workers at an early stage in their career cycles.

This paper builds on methodology that corrects for irregular spacing between realizations of unevenly spaced time series and provides appropriately corrected estimates of autoregressive model parameters. Using these methods for dealing with missing data, we develop time series tools for forecasting and estimation of autoregressions with cyclically varying parameters in which periodicity is assumed. To illustrate the robustness and flexibility of the methodology, an application is conducted to model daily temperature data. The approach helps to uncover cyclical (daily as well as annual) patterns in the data without imposing restrictive assumptions. Using the Central England Temperature (CET) time series (1772 - present) we find with a high level of accuracy that temperature intra-year averages and persistence have increased in the later sample 1850-2020 compared to 1772 - 1850, especially for the winter months, whereas the estimated variance of the random shocks in the autoregression seems to have decreased over time.

New limit theory is provided for a wide class of sample variance and covariance functionals involving both nonstationary and stationary time series. Sample functionals of this type commonly appear in regression applications and the asymptotics are particularly relevant to estimation and inference in nonlinear nonstationary regressions that involve unit root, local unit root or fractional processes. The limit theory is unusually general in that it covers both parametric and nonparametric regressions. Self normalized versions of these statistics are considered that are useful in inference. Numerical evidence reveals interesting strong bimodality in the finite sample distributions of conventional self normalized statistics similar to the bimodality that can arise in t-ratio statistics based on heavy tailed data. Bimodal behavior in these statistics is due to the presence of long memory innovations and is shown to persist for very large sample sizes even though the limit theory is Gaussian when the long memory innovations are stationary. Bimodality is shown to occur even in the limit theory when the long memory innovations are nonstationary. To address these complications new self normalized versions of the test statistics are introduced that deliver improved approximations that can be used for inference.

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.

Financial econometrics is a dynamic discipline that began to take on its present form around the turn of the century. Since then it has found a permanent position as a popular course sequence in both undergraduate and graduate teaching programs in economics, finance, and business schools. Because of the breadth of the subject’s foundations, its extensive coverage in applications and because these courses attract a wide range of students with accompanying interests and skill sets that cover diverse areas and technical capabilities, teaching financial econometrics presents many challenges to the university educator. This chapter addresses some of these challenges, provides helpful guidelines to educators, and draws on the combined experience of the authors as teachers and researchers of modern financial econometrics as well as their recent textbook Financial Econometric Modeling (Hurn et al., 2021). The focus is on students converting to finance and econometrics with limited technical background