‘Estimation and hypothesis testing for collections of autoregressive time series’

Abstract

Formulae for ψ ^ ( α ) and σ ^ 2 ( α ) on p. 87 do not hold for all α , since they may not satisfy the constraint σ ^ 2 ( α ) ≤ ψ ^ ( α ) induced by the parameterization. Then the expressions do not give maximum likelihood estimates for given α . When the inequality fails, the maximum likelihood estimates are on the boundary of the parameter space, namely where σ 2 = ψ ⁠ . The maximum then occurs for σ ^ 2 ( α ) = ψ ^ ( α ) both equal to σ ^ 2 ( α ) given at p. 88, shortly after (7). Formula (6) still applies, provided the appropriate expressions for σ ^ 2 ( α ) and ψ ^ ( α ) are used. In the definition of d rs , near (2), the subscript t − r + s should read t + r − s .