Measurement Error in Dynamic Models

Abstract

Many time series contain measurement (often sampling) error and the problem of assessing the impacts of such errors and accounting for them has been receiving increasing attention of late. This paper provides a survey of this problem with an emphasis on estimating the coefficients of the underlying dynamic model, primarily in the context of fitting linear and nonlinear autoregressive models. An overview is provided of the biases induced by ignoring the measurement error and of methods that have been proposed to correct for it, and remaining inferential challenges are outlined.

1 Appendix

Assessing bias via estimating equations. Suppose $\mathbf{Y} \sim N(\mu \mathbf{1},\boldsymbol{\Sigma }_{Y })$. The estimating equations for the ML estimators of the parameters in $\boldsymbol{\Sigma }_{Y }$ (say ψ ₁, …, ψ _J ) can be written McCulloch et al. (2008, p. 165) as $tr(\boldsymbol{\Sigma }_{Y }^{-1}\mathbf{G}_{j}) - (\mathbf{y} -\mu \mathbf{1})^\prime\boldsymbol{\Sigma }_{Y }^{-1}\mathbf{G}_{j}\boldsymbol{\Sigma }_{Y }^{-1}(\mathbf{y} -\mu \mathbf{1}) = 0$, for j = 1 to J, where J is the number of parameters in $\boldsymbol{\Sigma }_{Y }$ and $\mathbf{G}_{j} = \partial \boldsymbol{\Sigma }_{Y }/\partial \psi _{j}$. Replacing y with W and taking the expected value, but denoting the arguments of the estimating equations denoted with a ∗ leads to an expected value of the jth estimating equation of $E_{j} = tr(\boldsymbol{\Sigma }_{Y }^{{\ast}-1}\mathbf{G}_{j}^{{\ast}}) - tr(\boldsymbol{\Sigma }_{Y }^{{\ast}-1}\mathbf{G}_{j}^{{\ast}}\boldsymbol{\Sigma }_{Y }^{{\ast}-1}\boldsymbol{\Sigma }_{W})$. If we can find $\boldsymbol{\Sigma }_{Y }^{{\ast}}$ of the same form as $\boldsymbol{\Sigma }_{Y }$ so that each E _j is 0, then the naive estimators of the parameters in $\boldsymbol{\Sigma }_{Y }$ are consistent for the parameters in $\boldsymbol{\Sigma }_{Y }^{{\ast}}$. Obviously E _j is 0 if $\boldsymbol{\Sigma }_{Y }^{{\ast}} = \boldsymbol{\Sigma }_{W}$ but this only provides the asymptotic bias immediately if $\boldsymbol{\Sigma }_{W}$ is of the same form as $\boldsymbol{\Sigma }_{Y }$. If the measurement errors are additive with constant (unconditional) variance $\sigma _{u}^{2}$, then $\boldsymbol{\Sigma }_{W} = \boldsymbol{\Sigma }_{Y } +\sigma _{ u}^{2}\mathbf{I}$, and we can take $\boldsymbol{\Sigma }_{Y }^{{\ast}} = \boldsymbol{\Sigma }_{W}$. This means the naive estimator or $\sigma _{Y }^{2}$ asymptotically estimates $\sigma _{Y }^{2} +\sigma _{ u}^{2}$ while the naive estimators of the off-diagonal covariance terms in Y are correct and the ML estimators are asymptotically like the YW estimators.

Allowing unequal unconditional variances (as can occur with changing sampling effort) $\boldsymbol{\Sigma }_{W} = \boldsymbol{\Sigma }_{Y } + Diag(\sigma _{u1}^{2},\ldots,\sigma _{uT}^{2})$ the question, which we have not investigated, is whether E _j  → 0 as T increases if we take $\boldsymbol{\Sigma }_{Y }^{{\ast}} = \boldsymbol{\Sigma }_{Y } +\sigma _{ u}^{2}$, where $\sigma _{u}^{2} =\sum _{ t=1}^{T}\sigma _{ut}^{2}/T$.