Search Results

Inexpensive and/or readily available auxiliary variable, x, values may often be available at little or no cost. If these variables are highly correlated with the target variable, y, then use of ratio or regression estimators may greatly reduce sampling variance. These estimators are not unbiased, but bias is generally small compared to the target of estimation and contributes a very small proportion of overall mean square error, the relevant measure of accuracy for biased estimators. Ratio estimation can also be incorporated in the context of stratified designs, again possibly offering a reduction in overall sampling variance. Model-based prediction offers an alternative to the design-based ratio and regression estimators and we present an overview of this approach. In model-based prediction, the y values associated with population units are viewed as realizations of random variables which are assumed to be related to auxiliary variables according to specified models. The realized values of the target variable are known for the sample, but must be predicted using an assumed model dependency on the auxiliary variable for the non-sampled units in the population. Insights from model-based thinking may assist the design-based sampling theorist in selection of an appropriate estimator. Similarly, we show that insights from design-based estimation may improve estimation of uncertainty in model-based mark-recapture estimation.

Chapter VII has two purposes. One is to study the methodological problems that arise in analysing positively valued time series in foreign exchange. The other is to contrast the analysis of time series in formal econometrics with the analysis of such data in present-day econometrics. The chapter presents an axiomatic data confrontation of a theory of spot and forward exchange in foreign currency markets. In the formulation of the axioms, actual and auxiliary theory and data variables interact in such a way that the problem that usually arise in the analysis of positively valued time series disappears. The data for the empirical analysis comprise observations on spot and forward exchange rates in the market for Swiss Francs and US Dollars. In the empirical analysis, the given data are analysed, first, with the prescriptions of formal econometrics and, then, with the prescriptions on which present-day econometric time-series analysis insist. The statistical results yield different descriptions of the dynamics of foreign exchange and different inferences about the economics of social reality. In doing that the two contrasting empirical analyses provide interesting ingredients for the discussion of how best to incorporate economic theory in empirical analyses.

Sampling theory concerns itself with development of procedures for random selection of a subset of units, a sample, from a larger finite population, and with how to best use sample data to make scientifically and statistically sound inferences about the population as a whole. The Introduction orients the reader to the essential content and structure of the text, including (a) an overview of the nature and objectives of design-based sampling theory as well an explanation for why we emphasize the design-based approach to sampling, (b) organization of content by chapters, (c) summary of notational conventions adopted throughout, (d) suggestions for instructors using the text for a class, (e) assumed level of mathematics and statistics background expected from readers, and (f) discussion of how this text differs from other published sampling texts. The chapter concludes with a brief review of the history and development of sampling theory, including both the design-based theory (that is stressed in the text), and model-based prediction theory (that is motivated and illustrated in Chapter 7). The Introduction notes that the text presents small sample space numerical illustrations of the performance of many sampling strategies, and that the text adopts a logical and consistent notation which clarifies presentation of sampling theory concepts. The ecological/environmental/natural resources orientation of the text is most evident in problem exercises associated with most chapters, many of which have been motivated by practical resource applications.

Chapter VI begins with a discussion of the axioms of a formal theory-data confrontation in which the data appear as vector-valued sequences of observations of a vector-valued random process. Then it describes important characteristics of I(1) ARIMA processes, one of which is their tendency to display long positive and negative sojourns. This property of the process can be used to carry out meaningful empirical analyses of positively valued time series; e.g., spot and forward exchange rate: Let the observations of the exchange rates be observations of an auxiliary I(1) ARIMA process, analyse them with currently available software programs, and check if long sequences of the exchange rates have the characteristics of an I(1) ARIMA process. A second characteristic of an I(1) ARIMA process is that any multivariate version of such a process can be written as an error correction model. In present-day econometrics, statistical analyses of error correction models of vector-valued time series are used to determine the degree of cointegration of the time series. The import of such an analysis hinges on the theoretical meaningfulness of the pertinent error correction model. The chapter demonstrates that an empirically relevant error correction model need not be theoretically meaningful.