Exact Geometry of Autoregressive Models

Garderen, Kees Jan van

doi:10.1111/1467-9892.00122

Cited by 26 publications

(6 citation statements)

References 6 publications

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“…For the case of observation coming from a time series, Ravishanker et al (1990), considered the Gaussian ARMA models as members of the curved exponential family (Efron, 1975). Taniguchi (1991a) found the corrections for the AR(1) and MA(1) models without disturbance parameters, while van Garderen (1999) used differential geometry and suggested how to compute the Bartlett correction factor geometrically, for the case of no disturbance parameter.…”

Section: Introductionmentioning

confidence: 99%

Improvement of the Likelihood Ratio Test Statistic in ARMA Models

Lagos

Morettin

2004

Journal Time Series Analysis

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In this paper, we develop a Bartlett correction for the likelihood ratio statistic used to test hypotheses about parameters of a Gaussian stationary and invertible model belonging to the ARMA (autoregressive moving average) family. Alternative hypotheses with and without disturbance parameters are considered. The correction formulae are written in matrix form with the advantage of being easily implemented with the aid of some symbolic or numerical matrix language. Some simulation results are also presented.

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Section: Introductionmentioning

confidence: 99%

Improvement of the Likelihood Ratio Test Statistic in ARMA Models

Lagos

Morettin

2004

Journal Time Series Analysis

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“…4. Interestingly and somewhat surprisingly, van Garderen (1999) found that if the innovation variance σ 2 v is known the Efron curvature of the model is 2n −1 + O(n −2 ), which, though still small, is larger than when σ 2 v is unknown, and he provided a geometrical explanation for this phenomenon. Correspondingly, results in van Giersbergen (2006) imply that the coefficient of the leading term in (5) increases from −0.25n −1 to −1n −1 when σ 2 v is known, indicating that the LRT is better approximated by the limiting chi-square distribution when the innovation variance is unknown than when it is known, though, of course, both approximations are very good by themselves.…”

mentioning

confidence: 99%

Bias Reduction and Likelihood-Based Almost Exactly Sized Hypothesis Testing in Predictive Regressions Using the Restricted Likelihood

Chen

Deo

2009

Econom. Theory

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Difficulties with inference in predictive regressions are generally attributed to strong persistence in the predictor series. We show that the major source of the problem is actually the nuisance intercept parameter, and we propose basing inference on the restricted likelihood, which is free of such nuisance location parameters and also possesses small curvature, making it suitable for inference. The bias of the restricted maximum likelihood (REML) estimates is shown to be approximately 50% less than that of the ordinary least squares (OLS) estimates near the unit root, without loss of efficiency. The error in the chi-square approximation to the distribution of the REML-based likelihood ratio test (RLRT) for no predictability is shown to be $({\textstyle{3 \over 4}} - \rho ^2)n^{ - 1} (G_3 (\cdot) - G_1 (\cdot)) + O(n^{ - 2}),$ where |ρ| < 1 is the correlation of the innovation series and Gs(·) is the cumulative distribution function (c.d.f.) of a $\chi _s^2 $ random variable. This very small error, free of the autoregressive (AR) parameter, suggests that the RLRT for predictability has very good size properties even when the regressor has strong persistence. The Bartlett-corrected RLRT achieves an O(n−2) error. Power under local alternatives is obtained, and extensions to more general univariate regressors and vector AR(1) regressors, where OLS may no longer be asymptotically efficient, are provided. In simulations the RLRT maintains size well, is robust to nonnormal errors, and has uniformly higher power than the Jansson and Moreira (2006, Econometrica 74, 681–714) test with gains that can be substantial. The Campbell and Yogo (2006, Journal of Financial Econometrics 81, 27–60) Bonferroni Q test is found to have size distortions and can be significantly oversized.

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“…Abadir, 1993b;Abadir and Lucas, 2004). In spite of work by Van Garderen (1999, 2000, the underlying geometry of non-stationary autoregressive models is still not fully characterized. The shared characteristics between the four-step random walk problem and least-squares estimation in autoregressive models could motivate new approaches towards the latter that embrace machinery currently commonplace in Analytic Number Theory and 26 J. R. McCrorie…”

Section: Discussionmentioning

confidence: 99%

Moments in Pearson's Four-Step Uniform Random Walk Problem and Other Applications of Very Well-Poised Generalized Hypergeometric Series

McCrorie

2020

Sankhya B

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This paper considers the representation of odd moments of the distribution of a four-step uniform random walk in even dimensions, which are based on both linear combinations of two constants representable as contiguous very well-poised generalized hypergeometric series and as even moments of the square of the complete elliptic integral of the first kind. Neither constants are currently available in closed form. New symmetries are found in the critical values of the L-series of two underlying cusp forms, providing a sense in which one of the constants has a formal counterpart. The significant roles this constant and its counterpart play in multidisciplinary contexts is described. The results unblock the problem of representing them in terms of lower-order generalized hypergeometric series, offering progress towards identifying their closed forms. The same approach facilitates a canonical characterization of the hypergeometry of the parbelos, adding to the characterizations outlined by Campbell, D'Aurozio and Sondow (2020, The American Mathematical Monthly127(1), 23-32). The paper also connects the econometric problem of characterizing the bias in the canonical autoregressive model under the unit root hypothesis to very well-poised generalized hypergeometric series. The confluence of ideas presented reflects a multidisciplinarity that accords with the approach and philosophy of Prasanta Chandra Mahalanobis.

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Exact Geometry of Autoregressive Models

Cited by 26 publications

References 6 publications

Improvement of the Likelihood Ratio Test Statistic in ARMA Models

Improvement of the Likelihood Ratio Test Statistic in ARMA Models

Bias Reduction and Likelihood-Based Almost Exactly Sized Hypothesis Testing in Predictive Regressions Using the Restricted Likelihood

Moments in Pearson's Four-Step Uniform Random Walk Problem and Other Applications of Very Well-Poised Generalized Hypergeometric Series

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