Quantitative Methods, Basic Assumptions

Assumptions, Models, and Estimators

A statistical model is a mathematical representation of the actual process in the world that generated the data (known as a data-generating process, or DGP). The point of creating a statistical model is both to describe the data produced by the DGP and to make inferences about features of the DGP that are unknown. As noted above, the model itself consists of a set of assumptions. This account makes it clear that assumptions are characteristics of models and not characteristics of data, and the question to ask of a model is not whether it is true or false but how descriptively useful is it. Models are more or less useful in describing a given data set, and when an assumption fails to be useful in the [Page 2177]process of description, the error lies with the model, not with the data.

Once a model has been selected, then an estimator (a function of the sample data that provides an estimated value for an unknown parameter) is chosen. Many common models can be estimated by any number of estimators, and the choice among estimators is driven by the model's assumptions, which imbue the estimators with various properties. An estimator with good (or better) properties under the assumptions of the model is chosen over an estimator with bad (or worse) properties under the assumptions of the model. The reference to “good estimators” and not to “good estimates” is by design. In frequentist statistics, a good estimate, by definition, is one produced by a good estimator. Properties of estimators that are considered good include unbiasedness (the estimator is on average neither high nor low), efficiency (the estimator has a small variance around the true value), and consistency (the estimator is near the true value almost all the time when the sample size is large). Often these properties are assessed in the aggregate, as a slightly biased estimator with a small variance may be preferred over an unbiased estimator with a larger variance.

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