scholarly journals Asymptotic inference for nearly unstable INAR(1) models

2003 ◽  
Vol 40 (03) ◽  
pp. 750-765 ◽  
Author(s):  
M. Ispány ◽  
G. Pap ◽  
M. C. A. van Zuijlen
Keyword(s):  
Least Squares ◽  
Rate Of Convergence ◽  
Limiting Distribution ◽  
Least Squares Estimator ◽  
Asymptotic Inference ◽  
First Order ◽  
Conditional Least Squares ◽  
Conditional Least Squares Estimator

A sequence of first-order integer-valued autoregressive (INAR(1)) processes is investigated, where the autoregressive-type coefficient converges to 1. It is shown that the limiting distribution of the conditional least squares estimator for this coefficient is normal and the rate of convergence is n 3/2. Nearly critical Galton–Watson processes with unobservable immigration are also discussed.

scholarly journals Asymptotic inference for nearly unstable INAR(1) models

2003 ◽  
Vol 40 (3) ◽  
pp. 750-765 ◽  
Author(s):  
M. Ispány ◽  
G. Pap ◽  
M. C. A. van Zuijlen
Keyword(s):  
Least Squares ◽  
Rate Of Convergence ◽  
Limiting Distribution ◽  
Least Squares Estimator ◽  
Asymptotic Inference ◽  
First Order ◽  
Conditional Least Squares ◽  
Conditional Least Squares Estimator

A sequence of first-order integer-valued autoregressive (INAR(1)) processes is investigated, where the autoregressive-type coefficient converges to 1. It is shown that the limiting distribution of the conditional least squares estimator for this coefficient is normal and the rate of convergence is n3/2. Nearly critical Galton–Watson processes with unobservable immigration are also discussed.

scholarly journals Fluctuation limit of branching processes with immigration and estimation of the means

2005 ◽  
Vol 37 (02) ◽  
pp. 523-538 ◽  
Author(s):  
M. Ispány ◽  
G. Pap ◽  
M. C. A. van Zuijlen
Keyword(s):  
Least Squares ◽  
Critical Case ◽  
Branching Processes ◽  
Critical Value ◽  
Subcritical Case ◽  
Asymptotically Normal ◽  
Fluctuation Limit ◽  
Conditional Least Squares ◽  
Type Process ◽  
Conditional Least Squares Estimator

We investigate a sequence of Galton-Watson branching processes with immigration, where the offspring mean tends to its critical value 1 and the offspring variance tends to 0. It is shown that the fluctuation limit is an Ornstein-Uhlenbeck-type process. As a consequence, in contrast to the case in which the offspring variance tends to a positive limit, it transpires that the conditional least-squares estimator of the offspring mean is asymptotically normal. The norming factor is n 3/2, in contrast to both the subcritical case, in which it is n 1/2, and the nearly critical case with positive limiting offspring variance, in which it is n.

scholarly journals UNIT ROOT AND COINTEGRATING LIMIT THEORY WHEN INITIALIZATION IS IN THE INFINITE PAST

2009 ◽  
Vol 25 (6) ◽  
pp. 1682-1715 ◽  
Author(s):  
Peter C.B. Phillips ◽  
Tassos Magdalinos
Keyword(s):  
Normal Distribution ◽  
Least Squares ◽  
Rate Of Convergence ◽  
Unit Root ◽  
Initial Conditions ◽  
Cauchy Distribution ◽  
Least Squares Estimator ◽  
Initial Condition ◽  
Limit Theory ◽  
Regression Theory

It is well known that unit root limit distributions are sensitive to initial conditions in the distant past. If the distant past initialization is extended to the infinite past, the initial condition dominates the limit theory, producing a faster rate of convergence, a limiting Cauchy distribution for the least squares coefficient, and a limit normal distribution for the t-ratio. This amounts to the tail of the unit root process wagging the dog of the unit root limit theory. These simple results apply in the case of a univariate autoregression with no intercept. The limit theory for vector unit root regression and cointegrating regression is affected but is no longer dominated by infinite past initializations. The latter contribute to the limiting distribution of the least squares estimator and produce a singularity in the limit theory, but do not change the principal rate of convergence. Usual cointegrating regression theory and inference continue to hold in spite of the degeneracy in the limit theory and are therefore robust to initial conditions that extend to the infinite past.

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