广义自回归过程的统一推理（英）

Tassos Magdalinos|Katerina Petrova UniformInference withGeneralAutoregressiveProcessesTassos MagdalinosandKaterina Petrova FederalReserve Bank of New York Staff Reports, no.1151April2025https://doi.org/10.59576/sr.1151 Abstract A unified theory of estimation and inference is developed for an autoregressive processwith root in(-∞, ∞)that includes thestationary, local-to-unity, explosive and all intermediate regions. Thediscontinuity of the limit distribution of the t-statistic outside thestationary region and its dependence onthe distribution of the innovations in the explosive regions(-∞,-1)∪(1,∞)are addressed simultaneously.A novel estimation procedure,based on a data-driven combination of a near-stationary and a mildlyexplosive artificiallyconstructed instrument, delivers mixed-Gaussian limit theory and gives rise to anasymptotically standard normal t-statistic across all autoregressive regions. The resulting hypothesistests and confidence intervals are shown to have correct asymptotic size (uniformly over thespace ofautoregressive parameters and the space of innovation distribution functions) in autoregressive, predictiveregression and local projection models, thereby establishing a generaland unified framework forinference with autoregressive processes. Extensive Monte Carlosimulation shows that the proposedmethodology exhibits very goodfinite sample propertiesover the entire autoregressive parameter space(-∞, ∞)and comparesfavorablyto existingmethods within their parametric(-1,1]validity range. Wedemonstrate how our procedurecan be used to construct valid confidence intervals in standardepidemiological models aswell as to test in real-time for speculative bubbles in the price of theMagnificent Seven techstocks. JEL classification:C12, C22Keywords:uniform inference,central limit theory,autoregression,predictive regression,instrumentation,mixed-Gaussianity, t-statistic,confidence intervals This paper presents preliminary findings and is being distributed to economists and other interestedreaders solely to stimulate discussion and elicit comments. The views expressed in this paper are those ofthe author(s) and do not necessarily reflect theposition of the Federal Reserve Bank of New York or theFederal Reserve System. Any errors or omissions are the responsibility of the author(s). 1Introduction Imposing short memory assumptions in macroeconomic and Önancial models is convenientsince it delivers standard econometric inference on the modelsí parameters with conventionalasymptotic distributions. However, such stationarity assumptions are often empirically unrealisticand a variety of stochastic trends has been found in many macroeconomic and Önancial time series. Whenever a nonstationary regressor is included in a regression model, the additional signalfrom the strong time dependence present in the regressor, while facilitating more precise pointestimates, usually makes the series non-ergodic, thus invalidating standard central limit theory(CLT). Consequently, OLS-based test statistics have di§erent distributional limits depending onthe persistence degree of the regressor and the critical values required by practitioners to constructconÖdence intervals (CIs) or run hypothesis tests on the modelís parameters are vastly di§erent.As a result, misspecifying the type of regressor invalidates inference resulting in size distortionswhich do not improve with the sample size and lead to erroneous empirical conclusions. In this paper, we focus on a single regressor generated by the prototypical time series model,a Örst-order autoregressive (AR) process with root in(1;1). We consider inference in severalregression models where this regressor enters on the right-hand side:a pure autoregression, apredictive regression and a local projection model.Even in these simple setups, di§erent per-sistence degrees of the regressor create discontinuities in the OLS limit distributions with limitsinvolving stochastic integrals in the vicinity of (negative and positive) unit root as well as limitsof unknown distributional form which depend on the distributions of the innovations in explo-sive regions. The discontinuities in the limits of OLS-based test statistics, well-documented in aseries of classical papers (e.g. Mann and Wald (1943) for autoregressions with stationary roots,Anderson (1959) for explosive roots, Phillips (1987a) for unit-roots and Chan and Wei (1987) forlocal-to-unity roots) present a major challenge for inference.Even robust standard proceduressuch as bootstrap have been shown to be invalid in presence of unit roots (Basawa et al. (1991)).Given the inference challenges arising from nonstationarity, a strand of the literature is dedicatedto designing screening procedures for researchers to detect possible nonstationarities in the data(e.g.unit root and cointegration testing) and account for or remove them, e.g.by di§erencingor detrending.Pre-testing not only leads to size pile-up but, as Cavanagh et al.(1