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Article Excerpt
1. INTRODUCTION
Measurement error is a ubiquitous in many statistical problems and has received considerable attention in various regression contexts (e.g., Fuller 1987; Carroll, Ruppert, and Stefanski 1995). Time series data are no exception when it comes to the presence of measurement error; the variable of interest is often estimated rather than observed exactly. Ecological examples include the estimation of animal densities through capture/recapture techniques and estimation of food abundance over a large area by spatial subsampling. In environmental studies, chemical concentrations in the air or water at a particular time are typically obtained from collections at various locations, whereas many economic, labor, and public health variables are estimated on a regular basis over time, often through a complex sampling scheme. Examples of the latter include medical data indices and results of national opinion polls (Scott and Smith 1974; Scott, Smith, and Jones 1977), estimation of the number of households in Canada and labor force statistics in Israel (Pfeffermann, Feder, and Signorelli 1998), and retail sales figures (Bell and Hillmer 1990; Bell and Wilcox 1993).
To frame the general problem, consider a times series {[Y.sub.t]}, where [Y.sub.t] is a random variable and t indexes time. The realized true values are denoted by {[y.sub.t], t = 1,..., T}, but instead of [y.sub.t] one observes the outcome of [W.sub.t], where E([W.sub.t]|[y.sub.t]) = [y.sub.t], with "|[y.sub.t]" shorthand for "given [Y.sub.t] = [y.sub.t]." Hence [W.sub.t] is a conditionally unbiased estimator of [y.sub.t] or, equivalently, [W.sub.t] = [y.sub.t] + [u.sub.t], where [u.sub.t], which represents measurement/survey error, has E([u.sub.t]|[y.sub.t]) = 0. In Section 2 we show that this last assumption implies that cov([u.sub.t], [Y.sub.t]) = but allows the conditional measurement error variance to depend on [Y.sub.t].
The focus of this article is on estimation of the parameters in an autoregressive (AR) model for [Y.sub.t] from observations of [W.sub.t] when the measurement error is additive, uncorrelated, and possibly heteroscedastic. AR models are a popular choice in many disciplines for the modeling of time series. This is especially true of population dynamics where AR(1) or AR(2) models are often used with Y equal to the log of population abundance (or density) (see, e.g., Williams and Liebhold 1995; Dennis and Taper 1994, eq. 7). That work does not consider measurement error, however.
To motivate some aspects of the problem, Table 1 presents estimated mouse densities (mice per hectare on the log scale to which AR models are usually fit for abundance data) and associated standard errors over a 9-year period from one stand in the Quabbin Reservoir in Western Massachusetts. This is one part of the dataset used by Elkinton et al. (1996). The stand density and associated standard errors were obtained using estimates from subplots. The sampling effort consists of two subplots in 1987, 1989, and 1990 and three subplots in the other years. The estimated standard errors for the log densities were obtained using the delta method. The purpose of this example is to illustrate that the estimated standard errors vary considerably (with 1989 and 1990 values differing by a factor of about 11). This can occur for various reasons, including the changes in sampling effort and changes of the nature of the population being sampled at each time point, and indicate the need to consider a heteroscedastic measurement error model. Of course, if the estimated density is unbiased for true density, then (by Jensen's inequality) the same will not be true after a log transformation. In this example, estimates of the approximate biases were negligible, and the assumption of additive measurement error is reasonable. If this were not the case, then a nonlinear time series/measurement model would be required.
The article is organized as follows. First, we spell out the models in Section 2, paying particular interest to allowing for heteroscedasticity in the measurement error variances and the case where estimates of the measurement error variances are available. This problem has not yet received close attention in the literature. (We review and discuss existing methods in Secs. 4.2-4.4.) Following that, in Section 3 we derive the asymptotic bias of estimators of parameters in the AR(p) model when the measurement error is ignored, the bias of so-called "naive" estimators. These results also do not appear to be in the current literature. For the AR(1) model, the bias has a simple, explicit, and somewhat surprising "attenuation" form. For p > 1, the biases have simple matrix forms, and we illustrate the pernicious effects of measurement error in the AR(2) case. The naive estimator of the error variance has positive asymptotic bias for all p. Next, in Section 4.1 we present a simple new estimator for the problem of fitting AR models in the presence of measurement error. This new estimator is based on corrected asymptotic estimating equations and does not require the specification of a likelihood or the assumption of constant measurement error variance. In Section 4.1 we also develop the asymptotic properties of that estimator under various assumptions. We believe that this is the first article to develop asymptotic results for this problem when the measurement error is heteroscedastic. After that, in Section 4.2 we discuss a pseudo-maximum likelihood (pML) approach that assumes normality and uses estimated measurement error variances, and in Sections 4.3 and 4.4 we review two other existing methods that do not require knowledge of the measurement error variance. In Section 5 we compare our new method to the existing methods both asymptotically and with a small simple simulation study. We conclude with discussion in Section 6.
In addition to estimating the parameters in the dynamic model, a number of authors have addressed forecasting [Y.sub.t] for t > T or estimating [y.sub.t] for t [less than or equal to] T in the presence of measurement error. The latter has been the focus of much of the previous work in this area (e.g., Scott and Smith 1974; Scott et al. 1977; Bell and Hillmer 1990; Pfeffermann 1991; Pfeffermann et al. 1998; Feder 2001). Certain aspects of forecasting have been addressed by Scott and Smith (1974), Scott et al. (1977), Wong and Miller (1990), and Koreisha and Fang (1999). We do not address forecasting or estimation of [y.sub.t] directly in this article.
2. MODEL FOR OBSERVED DATA
Assuming that [Y.sub.t] is centered to have mean 0, the AR model specifies that [Y.sub.t] is a linear combination of previous values, plus noise,
[Y.sub.t] = [[phi].sub.1][Y.sub.t-1] + ... + [[phi].sub.p][Y.sub.t-p] + [e.sub.t], t = p + 1,..., T, (1)
where the [e.sub.t] are iid with mean and variance [[sigma].sub.e.sup.2]. Let Y refer to the entire series. We focus on the case where [[phi].sub.1],..., [[phi].sub.p] are such that the process [Y.sub.t] is stationary (e.g., Box, Jenkins, and Reinsel 1994, chap. 3).
As in Section 1, Y is unobserved; we observe [W.sub.t] = [Y.sub.t] + [u.sub.t] with E([u.sub.t]|[y.sub.t]) = or, equivalently, E([W.sub.t]|[y.sub.t]) = [y.sub.t]. The [u.sub.t] represents additive measurement or survey error. We assume that the measurement errors are uncorrelated, that is, cov([u.sub.t], [u.sub.t']|[y.sub.t], [y.sub.t']) = for t [not equal to] t'. Note that the assumption E([u.sub.t]|[y.sub.t]) = is sufficient by itself to imply that cov([u.sub.t], [Y.sub.t]) = because cov([u.sub.t], [Y.sub.t]) = E{cov([u.sub.t], [Y.sub.t]|[Y.sub.t])} + cov{E([u.sub.t]|[Y.sub.t]), E([Y.sub.t]|[Y.sub.t])} = E(0) + cov(0, [Y.sub.t]) = 0. Interestingly, this occurs even if var([u.sub.t]|[y.sub.t]) is a function of [Y.sub.t], in which case [u.sub.t] and [Y.sub.t] are dependent but uncorrelated.
One of the techniques discussed later relies on the following well-known result (see Box et al. 1994, sec. A4.3; Granger and Morris 1976; Pagano 1974).
Lemma 1. If [Y.sub.t] is an AR(p) process with AR parameters [phi] and [U.sub.t] is white noise with constant variance [[sigma].sub.u.sup.2], then [W.sub.t] = [Y.sub.t] + [U.sub.t] follows an autoregressive moving average (ARMA) process, ARMA(p, p). This can be written as [phi](B)[W.sub.t] = [theta](B)[b.sub.t], where [b.sub.t] is white noise with mean and variance [[sigma].sub.b.sup.2], [phi](z) = 1 - [[phi].sub.1]z - ... - [[phi].sub.p][z.sup.p], [theta](z) = 1 + [[theta].sub.1]z + ... + [[theta].sub.p][z.sup.p], and B is the backward shift operator with [B.sup.j][W.sub.t] = [W.sub.t-j].
The AR parameters ([[phi].sub.k]'s) in this model are the same as in the model for Y. The moving average ([[theta].sub.k]'s) and conditional variance ([[sigma].sub.b.sup.2]'s) parameters can...
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