Bartlett Correction in the Stable Ar(1) Model With Intercept and Trend

Giersbergen, N.P.A. van

doi:10.1017/s0266466609090690

Cited by 13 publications

(11 citation statements)

References 23 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In this paper, we analyze the stable AR(2) model. Hence, we generalize the results for the AR(1) model that was investigated by TANIGUCHI (1988TANIGUCHI ( , 1991 without deterministic terms and VAN GIERSBERGEN (2009) with intercept and trend. Although the *N.P.…”

Section: Introductionsupporting

confidence: 53%

See 1 more Smart Citation

Bartlett correction in the stable second‐order autoregressive model with intercept and trend

Giersbergen

2013

Statistica Neerlandica

Self Cite

View full text Add to dashboard Cite

This paper derives the Bartlett factors that can be used to obtain higher‐order improvements for testing hypotheses about the autoregressive (AR) parameters in the stable AR(2) model with possible intercept and linear trend. The factors are obtained for testing hypotheses about individual parameters (φ1 and φ2) as well as their sum. Moreover, the effect of deterministic terms on the correction factors is found explicitly. All corrections are non‐decreasing in the AR parameters. Furthermore, the Bartlett corrections for φ1 and φ2 tend to infinity as φ2 approaches 1, whereas the correction for φ1 + φ2 tends to infinity as φ1 + φ2 is close to 1. The effectiveness of these Bartlett corrections in finite samples is evaluated by simulations.

show abstract

Section: Introductionsupporting

confidence: 53%

“…This is similar to the AR(1) model ( φ 2 = 0), in which the correction for φ 1 in the model without deterministic regressors is also constant, viz. 1 − 1/(2 T ); see van Giersbergen (). For φ 1 , there appears to be an additive term to the Bartlett correction of φ 1 + φ 2 .…”

Section: The Ar(2) Models Without Trendmentioning

confidence: 99%

Bartlett correction in the stable second‐order autoregressive model with intercept and trend

Giersbergen

2013

Statistica Neerlandica

Self Cite

View full text Add to dashboard Cite

show abstract

“…The empirical densities of LRT α (the zero intercept case) are seen to be remarkably well approximated by the limiting χ 2 1 distribution, both when α is far from the unit root and when α is close to unity, whereas those of LRT α,μ (the intercept case) are far moved from that of the χ 2 1 . Simulation results in Figure 1 of van Giersbergen (2006) further confirm the accuracy of the standard χ 2 1 approximation for LRT α , even when α is close to unity. Not surprisingly, van Garderen (1999) has found that for the univariate AR(1) model in (2) with μ known to be zero, the Efron (1975) curvature (one of the standard measures of curvature of the likelihood) is very small, 4 being of the order O(n −2 ) and converging to zero as α → 1 for every fixed n.…”

Section: Restricted Maximum Likelihood Estimationmentioning

confidence: 55%

“…As a result, we can use the calculations in van Garderen (1999) obtained for the zero-mean AR(1) model and establish that the Efron curvature properties of the restricted likelihood for the AR(1) are the same as those of the AR(1) model with known intercept, up to order O(n −1 ). In addition, we also get the following theorem, which provides a formal expansion for the distribution of the RLRT in the model in (2) by arguments identical to those for Theorem 1 of van Giersbergen (2006), established for the zero-mean AR(1) model (see also note 2 for the expansion in (4)). THEOREM 1.…”

Section: Restricted Maximum Likelihood Estimationmentioning

confidence: 70%

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

Chen

Deo

2009

Econom. Theory

View full text Add to dashboard Cite

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.

show abstract

“…This is due to the low Efron curvature property (Efron, 1975) of the likelihood function in the case of zero intercept; analytically, the leading error term in the approximation of the LRT distribution is proportional to 0.25/ n which, unlike that of the t ‐statistic, is free of α (Chen and Deo, 2009a). Unfortunately, the LRT does not have the same superior property when an intercept is included in the model since the leading error term becomes proportional to n −1 (1 − α ) −1 , see van Giersbergen (2009). The above discussion leads to the motivation of finding a likelihood function with low curvature, while also allowing for intercept/trend.…”

Section: Introductionmentioning

confidence: 99%

The restricted likelihood ratio test for autoregressive processes

Deo

2011

Journal Time Series Analysis

View full text Add to dashboard Cite

The restricted likelihood is known to produce estimates with significantly less bias in AR(p) models with intercept and/or trend. In AR(1) models, the corresponding restricted likelihood ratio test (RLRT), unlike the t-statistic or the usual LRT, has been recently shown to be well approximated by the chi-square distribution even close to the unit root, thus yielding confidence intervals with good coverage properties. In this article, we extend this result to AR(p) processes of arbitrary order p by obtaining the expansion of the RLRT distribution around that of the limiting chi-squared and showing that the error is bounded even as the unit root is approached. Next, we investigate the correspondence between the AR coefficients and the partial autocorrelations, which is well known in the stationary region, and extend to the more general situation of potentially multiple unit roots. In the case of one positive unit root, which is of most practical interest, the resulting parameter space is shown to be the bounded p-dimensional hypercube (À1, 1] · (À1, 1) pÀ1 . This simple parameter space, in addition with a stable algorithm that we provide for computing the restricted likelihood, allows its easy computation and optimization as well as construction of confidence intervals for the sum of the AR coefficients. In simulations, the RLRT intervals are shown to have not only near exact coverage in keeping with our theoretical results, but also shorter lengths and significantly higher power against stationary alternatives than other competing interval procedures. An application to the well-known Nelson-Plosser data yields RLRT based intervals that can be markedly different from those in the literature.

show abstract

Bartlett Correction in the Stable Ar(1) Model With Intercept and Trend

Cited by 13 publications

References 23 publications

Bartlett correction in the stable second‐order autoregressive model with intercept and trend

Bartlett correction in the stable second‐order autoregressive model with intercept and trend

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

The restricted likelihood ratio test for autoregressive processes

Contact Info

Product

Resources

About