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Inference in Time Series Regression When the Order of Integration of a Regressor Is Unknown.
Abstract
It is well known that the distribution of statistics testing restrictions on the coefficients in time series regressions can depend on the order of integration of the regressors. In practice the order of integration is rarely blown. This paper examines two conventional approaches to this problem, finds them unsatisfactory, and proposes a new procedure. The two conventional approaches- simply to ignore unit root problems or to use unit root pretests to determine the critical values for second-stage inference - both often induce substantial size distortions. In the case of unit root pretests, this arises because type I and II pretest errors produce incorrect second-stage critical values and because, in many empirically plausible situations, the first stage test (the unit root test) and the second stage test (the exclusion restriction test) are dependent. Monte Carlo simulations reveal size distortions even if the regressor is stationary but has a large autoregressive root, a case that might arise for example in a regression of excess stock returns against the dividend yield. In the proposed alternative procedure, the second-stage test is conditional on a first-stage "unit root" statistic developed in Stock (1992); the second-stage critical values vary continuously with the value of the first-stage statistic. The procedure is shown to have the correct size asymptotically and to have good local asymptotic power against Granger-causality alternatives.
... For the special case s = 1, equation (8) shows that S XX converges in probability to the identity matrix. By virtue of Theorem 3.2 and Proposition 8 in [16], the proof of Theorem 3.1 in the stationary regime is complete if we can show that, for any (θ n ) n∈N ⊂ R n,0 , ...
... One application of the above result is robust inference in the predictive regression model. For a deeper discussion of why uniformly valid inference methods are important in this setting see [2,8], which cover the case of a univariate regressor, but the same considerations hold more generally. To fix ideas, consider the following set of parameters ...
... Finally, from the inequality 1 − n log n = e log(1−log(n)/n) ≤ e − log(n)/n = 1 n 1/n and part (c) of Lemma F.1 we see that sup θ∈Rn,0 ||EB n ||= o(1). Since B n is positive semidefinite, this implies that B n converges in probability to 0 uniformly over R n,0 and therefore concludes the proof of (8). ...
Uniformly valid inference for cointegrated vector autoregressive processes has so far proven difficult due to certain discontinuities arising in the asymptotic distribution of the least squares estimator. We show how asymptotic results from the univariate case can be extended to multiple dimensions and how inference can be based on these results. Furthermore, we show that the novel instrumental variable procedure proposed by [20] (IVX) yields uniformly valid confidence regions for the entire autoregressive matrix. The results are applied to two specific examples for which we verify the theoretical findings and investigate finite sample properties in simulation experiments.
... As shown in Toda and Yamamoto (1995), Dolado and Lutkepohl (1996), and Kilian and Lutkepohl (2017), VAR models in levels are robust to alternative cointegration ranks and vector specifications. In addition, as concluded by Elliott and Stock (1994), Cavanagh et al. (1995), andCanova (2007), these models solve the controversial issue of which cointegration restrictions to impose as well as potential unit root problems. ...
This paper theoretically and empirically investigates the puzzling decade-long concurrence of expansionary monetary and fiscal policies, decreasing credit flows, fall in price levels, and sluggish real activity observed in the Euro area from the outset of the 2007–2008 financial crisis. To this end, we propose a monetary general equilibrium model that clarifies the transmission mechanisms, debt–deflation channels, and the paramount role of financial leverage decisions underlying these peculiarities. On this basis, a vector error correction model is specified which confirms the theoretical predictions and provides insights into the elements specific to the long-term relations. In addition, the estimated impulse response functions document the associated short-term dynamics outlining the debt–deflation mechanism.
... For related predictive regression literature, see, e.g.,Elliott and Stock (1994),Jansson and Moreira (2006), andWerker and Zhou (2022). ...
This paper aims to address the issue of semiparametric efficiency for cointegration rank testing in finite-order vector autoregressive models, where the innovation distribution is considered an infinite-dimensional nuisance parameter. Our asymptotic analysis relies on Le Cam's theory of limit experiment, which in this context takes the form of Locally Asymptotically Brownian Functional (LABF). By leveraging the structural version of LABF, an Ornstein-Uhlenbeck experiment, we develop the asymptotic power envelopes of asymptotically invariant tests for both cases with and without a time trend. We propose feasible tests based on a nonparametrically estimated density and demonstrate that their power can achieve the semiparametric power envelopes, making them semiparametrically optimal. We validate the theoretical results through large-sample simulations and illustrate satisfactory size control and excellent power performance of our tests under small samples. In both cases with and without time trend, we show that a remarkable amount of additional power can be obtained from non-Gaussian distributions.
... Furthermore, all the variables have a GARCH structure, which also will affect the t-test. In addition, Table 1 in [29] indicates the size distortion of the t-test for plausible parameter values. Hence, we recommend the EL test proposed in this paper over the traditional test. ...
Testing predictability is known to be an important issue for the balanced predictive regression model. Some unified testing statistics of desirable properties have been proposed, though their validity depends on a predefined assumption regarding whether or not an intercept term nevertheless exists. In fact, most financial data have endogenous or heteroscedasticity structure, and the existing intercept term test does not perform well in these cases. In this paper, we consider the testing for the intercept of the balanced predictive regression model. An empirical likelihood based testing statistic is developed, and its limit distribution is also derived under some mild conditions. We also provide some simulations and a real application to illustrate its merits in terms of both size and power properties.
This article considers predictive regressions in which a structural break is allowed on an unknown date. We establish novel testing procedures for asset return predictability using empirical likelihood (EL) methods based on weighted score equations. The theoretical results are useful in practice because our unified framework does not require distinguishing whether the predictor variables are stationary or non-stationary. Monte Carlo simulation studies show that the EL-based tests perform well in terms of size and power in finite samples. Finally, as an empirical analysis, we test the predictability of the monthly S&P 500 value-weighted log excess return using various predictor variables.
The asymptotic behavior of quantile regression inference becomes dramatically different when it involves a persistent predictor with zero or nonzero intercept. Distinguishing various properties of a predictor is empirically challenging. In this paper, we develop a unified predictability test for quantile regression regardless of the presence of intercept and persistence of a predictor. The developed test is a novel combination of data splitting, weighted inference, and a random weighted bootstrap method. Monte Carlo simulations show that the new approach displays significantly better size and power performance than other competing methods in various scenarios, particularly when the predictive regressor contains a nonzero intercept. In an empirical application, we revisit the quantile predictability of the monthly S&P 500 returns between 1980 and 2019.
We propose an econometric environment for structural break detection in nonstationary quantile predictive regressions. We establish the limit distributions for a class of Wald and fluctuation type statistics based on both the ordinary least squares estimator and the endogenous instrumental regression estimator proposed by Phillips and Magdalinos (2009a, Econometric Inference in the Vicinity of Unity. Working paper, Singapore Management University). Although the asymptotic distribution of these test statistics appears to depend on the chosen estimator, the IVX based tests are shown to be asymptotically nuisance parameter-free regardless of the degree of persistence and consistent under local alternatives. The finite-sample performance of both tests is evaluated via simulation experiments. An empirical application to house pricing index returns demonstrates the practicality of the proposed break tests for regression quantiles of nonstationary time series data.
How synchronized are short sellers? We examine a unique data set on the distribution of profits across a stock’s short sellers and find evidence of substantial dispersion in the initiation of their positions. Consistent with this dispersion reflecting “synchronization risk,” that is, uncertainty among short sellers about when others will short sell, more dispersed short selling signals i) greater stock overpricing and ii) longer delays in overpricing correction. These effects are prevalent even among stocks facing low short-selling costs or other explicit constraints. Overall, our findings provide novel cross-sectional evidence of synchronization problems among short sellers and their pricing implications.
We propose new tests for long-horizon predictability based on IVX estimation of a transformed regression which explicitly accounts for the over-lapping nature of the dependent variable in the long-horizon regression arising from temporal aggregation. To improve efficiency, we moreover incorporate the residual augmentation approach recently used in the context of short-horizon predictability testing by Demetrescu and Rodrigues (2022). Our proposed tests improve on extant tests in the literature in a number of ways. First, they allow practitioners to remain ambivalent over the strength of the persistence of the predictors. Second, they are valid under much weaker conditions on the innovations than extant long-horizon predictability tests; in particular, we allow for general forms of conditional and unconditional heteroskedasticity in the innovations, neither of which are tied to a parametric model. Third, unlike the popular Bonferroni-based methods in the literature, our proposed tests can handle multiple predictors, and can be easily implemented as either one or two-sided hypotheses tests. Monte Carlo analysis suggests that our preferred tests offer improved finite sample properties compared to the leading tests in the literature. We report results from an empirical application investigating the use of real exchange rates for predicting nominal exchange rates and inflation.
Let n observations Y 1, Y 2, ···, Y n be generated by the model Y t = pY t−1 + e t , where Y 0 is a fixed constant and {e t } t-1 n is a sequence of independent normal random variables with mean 0 and variance σ2. Properties of the regression estimator of p are obtained under the assumption that p = ±1. Representations for the limit distributions of the estimator of p and of the regression t test are derived. The estimator of p and the regression t test furnish methods of testing the hypothesis that p = 1.
This paper develops a multivariate regression theory for integrated processes which simplifies and extends much earlier work. Our framework allows for both stochastic and certain deterministic regressors, vector autoregressions and regressors with drift. The main focus of the paper is statistical inference. The presence of nuisance parameters in the asymptotic distributions of regression F-tests is explored and new transformations are introduced to deal with these dependencies. Some specializations of our theory are considered in detail. In models with strictly exogenous regressors we demonstrate the validity of conventional asymptotic theory for appropriately constructed Wald tests. These tests provide a simple and convenient basis for specification robust inferences in this context. Single equation regression tests are also studied in detail. Here it is shown that the asymptotic distribution of the Wald test is a mixture of the chi square of conventional regression theory and the standard unit root theory. The new result accommodates both extremes and intermediate cases.
This paper continues the theoretical investigation of Park and Phillips [7]. We develop an asymptotic theory of regression for multivariate linear models that accommodates integrated processes of different orders, nonzero means, drifts, time trends and cointegrated regressors. The framework of analysis is general but has a common architecture that helps to simplify and codify what would otherwise be a myriad of isolated results. A good deal of earlier research by the authors and by others comes within the new framework. Special models of some importance are considered in detail, such as VAR systems with multiple lags and cointegrated variants.
A first-order autoregressive (AR) time series Yt = βYt-1 + εt is said to be nearly nonstationary if β is close to 1. For a nearly nonstationary AR(1) model, it is shown that the limiting distribution of the least squares estimate of β obtained by Chan and Wei (1987) can be expressed as a functional of the Ornstein-Uhlenbeck process. This alternative expression provides a simple and efficient means to tabulate the percentiles of the limiting distribution that furnishes a useful procedure to test for near nonstationarity. Based on the eigenvalue-eigenfunction consideration, it is shown that the Ornstein-Uhlenbeck formulation possesses an infinite series expansion that extends the result of Dickey and Fuller (1979) to the nearly nonstationary model. Numerical calculations based on different representations of the limiting distribution are performed and compared. It is found that the Ornstein-Uhlenbeck expression provides a better algorithm for tabulating the percentiles. Applications to other time series are also considered.
Test procedures for detecting overdifferencing or a moving average unit root in Gaussian autoregressive integrated moving average (ARIMA) models are proposed. The tests can be used when an autoregressive unit root is a serious alternative but the hypothesis of primary interest implies stationarity of the observed time series. This is the case, for example, if one wishes to test the null hypothesis that a multivariate time series is cointegrated with a given theoretical cointegration vector. A priori knowledge of the mean value of the observations turns out to be crucial for the derivation of our tests. In the special case where the differenced series follows a first-order moving average process, the proposed tests are exact and can be motivated by local optimality arguments. Specifically, when the mean value of the series is a priori known, we can obtain a locally best invariant (LBI) test that is identical to a one-sided version of the Lagrange multiplier test. But when the mean value is a priori not known, this test breaks down and we derive a locally best invariant unbiased (LBIU) test. After having tests in this special case, we develop extensions to general ARIMA models. These tests are asymptotic, but under the null hypothesis they have the same limiting distributions as in the just-mentioned special case. When the mean value is a priori known, an asymptotic χ 1 distribution is obtained, when it is unknown, the limiting distribution agrees with that of the Cramer–von Mises goodness-of-fit test statistic.
We Tabulate the Limiting Cumulative Distribution and Probability Density Functions of the Least Squares Estimator in a First-Order Autoregressive Regression When the True Model Is Near-Integrated in the Sense of Phillips (1986 A). the Results Are Obtained Using an Exact Numerical Method Which Integrates the Appropriate Limiting Moment Generating Function. the Adequacy of the Approximation Is Examined by Monte Carlo Methods for Various First-Order Autogressive Processes with a Root Close to Unity.