Partial autocorrelation parameterization for subset autoregression

McLeod, A. Ian; Zhang, Y.

doi:10.1111/j.1467-9892.2006.00481.x

Cited by 28 publications

(25 citation statements)

References 36 publications

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“…This recursion can be used to define a transformation

B : false(π_{1}, \dots, π_{p} false) \to false(ϕ_{1}, \dots, ϕ_{p} false)

that is one‐to‐one, continuous and differentiable inside the admissible region. This parameterisation has the advantage that in the π ‐space, the admissible region is simply the p ‐dimensional cube with boundary surfaces corresponding to ±1, while in the ϕ ‐space it is very complicated (see for instance McLeod & Zhang ). As an illustration, for p =2, the transformation is simply

ϕ_{1} = π_{1} false(1 - π_{2} false)

and

ϕ_{2} = π_{2}

.…”

Section: Modeling and Estimationmentioning

confidence: 99%

“…Following Box, Jenkins & Reinsel (), the exact log‐likelihood function is given by

ℓ false(bold-italicθ false| boldy false) = - \frac{1}{2} [n log σ^{2} + \frac{1}{σ^{2}} {false(boldy - boldX bold-italicβ false)}^{⊤} M_{n}^{- 1} (ϕ) (y - X β) + log (||, {boldM}_{n}, false(bold-italicϕ false))] + C,

where C is a constant independent of the parameter vector θ . Considering the reparameterisation given in and dropping constant terms (McLeod & Zhang ), we can write the log‐likelihood function as

ℓ false(bold-italicθ false| boldy false) = - \frac{1}{2} [n log σ^{2} + \frac{1}{σ^{2}} S (π, β) + log g_{p}],

where π =( π 1 ,…, π p ) ⊤ ,

g_{p} = false| {boldM}_{n} (ϕ) false| = false| {boldM}_{p} (ϕ) false| = {false\prod}_{j = 1}^{p} {(1 - π_{j}^{2})}^{- j}

and

S false(bold-italicπ, bold-italicβ false) = λ^{⊤} D false(boldy, bold-italicβ false) λ,

with D ( y , β ) being the ( p +1)×( p +1) matrix with ( i , j )‐entry which is the sum of n −( i −1)−( j −1) squares and lagged products, defined by:

D_{i,}

…”

Section: Modeling and Estimationmentioning

confidence: 99%

See 1 more Smart Citation

Influence diagnostics for censored regression models with autoregressive errors

Schumacher

Lachos

Vilca

et al. 2018

Aus NZ J of Statistics

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Summary Observations collected over time are often autocorrelated rather than independent, and sometimes include observations below or above detection limits (i.e. censored values reported as less or more than a level of detection) and/or missing data. Practitioners commonly disregard censored data cases or replace these observations with some function of the limit of detection, which often results in biased estimates. Moreover, parameter estimation can be greatly affected by the presence of influential observations in the data. In this paper we derive local influence diagnostic measures for censored regression models with autoregressive errors of order p (hereafter, AR(p)‐CR models) on the basis of the Q‐function under three useful perturbation schemes. In order to account for censoring in a likelihood‐based estimation procedure for AR(p)‐CR models, we used a stochastic approximation version of the expectation‐maximisation algorithm. The accuracy of the local influence diagnostic measure in detecting influential observations is explored through the analysis of empirical studies. The proposed methods are illustrated using data, from a study of total phosphorus concentration, that contain left‐censored observations. These methods are implemented in the R package ARCensReg.

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“…This recursion can be used to define a transformation

B : false(π_{1}, \dots, π_{p} false) \to false(ϕ_{1}, \dots, ϕ_{p} false)

ϕ_{1} = π_{1} false(1 - π_{2} false)

and

ϕ_{2} = π_{2}

.…”

Section: Modeling and Estimationmentioning

confidence: 99%

“…Following Box, Jenkins & Reinsel (), the exact log‐likelihood function is given by

ℓ false(bold-italicθ false| boldy false) = - \frac{1}{2} [n log σ^{2} + \frac{1}{σ^{2}} {false(boldy - boldX bold-italicβ false)}^{⊤} M_{n}^{- 1} (ϕ) (y - X β) + log (||, {boldM}_{n}, false(bold-italicϕ false))] + C,

where C is a constant independent of the parameter vector θ . Considering the reparameterisation given in and dropping constant terms (McLeod & Zhang ), we can write the log‐likelihood function as

ℓ false(bold-italicθ false| boldy false) = - \frac{1}{2} [n log σ^{2} + \frac{1}{σ^{2}} S (π, β) + log g_{p}],

where π =( π 1 ,…, π p ) ⊤ ,

g_{p} = false| {boldM}_{n} (ϕ) false| = false| {boldM}_{p} (ϕ) false| = {false\prod}_{j = 1}^{p} {(1 - π_{j}^{2})}^{- j}

and

S false(bold-italicπ, bold-italicβ false) = λ^{⊤} D false(boldy, bold-italicβ false) λ,

with D ( y , β ) being the ( p +1)×( p +1) matrix with ( i , j )‐entry which is the sum of n −( i −1)−( j −1) squares and lagged products, defined by:

D_{i,}

…”

Section: Modeling and Estimationmentioning

confidence: 99%

Influence diagnostics for censored regression models with autoregressive errors

Schumacher

Lachos

Vilca

et al. 2018

Aus NZ J of Statistics

View full text Add to dashboard Cite

show abstract

“…Some attempts have been made to circumvent this problem by examining the problem in partial autocorrelation space (McLeod and Zhang, 2006). However, as with most forwardselection-type model selection approaches, such procedures suffer from instability (Breiman, 1996), and in the case of all subsets selection, they quickly become computationally infeasible as k grows.…”

Section: Introductionmentioning

confidence: 99%

“…The existing LASSO procedures for AR models operate in coefficient space and are based on the conditional likelihood due to the complexity of the complete data likelihood in coefficient space (Wang et al, 2007;Hsu et al, 2008;Nardi and Rinaldo, 2008). For a chosen order k, the regular LASSO procedure estimates an AR model by finding the coefficients f = (f 1 , .…”

Section: Introductionmentioning

confidence: 99%

Estimation of stationary autoregressive models with the Bayesian LASSO

Schmidt

Makalic

2013

J. Time Ser. Anal.

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a This article explores the problem of estimating stationary autoregressive models from observed data using the Bayesian least absolute shrinkage and selection operator (LASSO). By characterizing the model in terms of partial autocorrelations, rather than coefficients, it becomes straightforward to guarantee that the estimated models are stationary. The form of the negative log-likelihood is exploited to derive simple expressions for the conditional likelihood functions, leading to efficient algorithms for computing the posterior mode by coordinate-wise descent and exploring the posterior distribution by Gibbs sampling. Both empirical Bayes and Bayesian methods are proposed for the estimation of the LASSO hyper-parameter from the data. Simulations demonstrate that the Bayesian LASSO performs well in terms of prediction when compared with a standard autoregressive order selection method.

show abstract

“…The least absolute shrinkage and selection operator (LASSO) procedure (Tibshirani, 1996) was developed to overcome similar problems in the regular linear regression setting and has recently been adapted to the AR model setting. The existing LASSO procedures for AR models operate in coefficient space and are based on the conditional likelihood due to the complexity of the complete data likelihood in coefficient space (Wang et al, 2007;Hsu et al, 2008;Nardi and Rinaldo, 2008). For a chosen order k, the regular LASSO procedure estimates an AR model by finding the coefficients f = (f 1 , .…”

Section: Introductionmentioning

confidence: 99%

Estimation of stationary autoregressive models with the Bayesian LASSO

Schmidt

Makalic

2013

Journal Time Series Analysis

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This article explores the problem of estimating stationary autoregressive models from observed data using the Bayesian least absolute shrinkage and selection operator (LASSO). By characterizing the model in terms of partial autocorrelations, rather than coefficients, it becomes straightforward to guarantee that the estimated models are stationary. The form of the negative log‐likelihood is exploited to derive simple expressions for the conditional likelihood functions, leading to efficient algorithms for computing the posterior mode by coordinate‐wise descent and exploring the posterior distribution by Gibbs sampling. Both empirical Bayes and Bayesian methods are proposed for the estimation of the LASSO hyper‐parameter from the data. Simulations demonstrate that the Bayesian LASSO performs well in terms of prediction when compared with a standard autoregressive order selection method.

show abstract

Partial autocorrelation parameterization for subset autoregression

Cited by 28 publications

References 36 publications

Influence diagnostics for censored regression models with autoregressive errors

Influence diagnostics for censored regression models with autoregressive errors

Estimation of stationary autoregressive models with the Bayesian LASSO

Estimation of stationary autoregressive models with the Bayesian LASSO

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