Weak Theories and Parameter Instability: Using Flexible Least Squares to Take Time Varying Relationships Seriously

Abstract

A common assumption by time-series analysts is that the estimated coefficients remain fixed through time. Yet this strong assumption often has little grounds in substantive theory or empirical tests. If the true coefficients change over time, but are estimated with fixed-coefficient methods such as Ordinary Least Squares (OLS) or its many offshoots, then this can lead to significant information loss, as well as errors of inference. This article demonstrates a method, Flexible Least Squares (FLS), for exploring the relative stability of time-series coefficients. FLS is superior to other such methods in that it enables the analyst to diagnose the magnitude of coefficient variation and detect which particular coefficients are changing. FLS also provides an estimated vector of time-varying coefficients for exploratory or descriptive purposes. FLS properties are demonstrated through simulation analysis and an evaluation of the time-varying characteristics of explanations of presidential approval from 1978 to 1997. s with many statistical methods used by political scientists, regression analysis was borrowed from the physical sciences where relationships are often invariant across both experimental observations and through time.1 For example, if we drop an object from an altitude of 100 feet, we can easily calculate the velocity with which the object will hit the ground, and with great accuracy. Using regression analysis and multiple experiments, we can also discover the effect of wind speed, friction, and other factors on this velocity. We know with some degree of certainty that controlling for these other factors, the velocity will be the same across all similar experiments, next year, in ten years, or in a hundred years. Many such invariant relationships exist in the physical sciences (e.g., the rate of decay of a radioactive isotope, the relationship between pressure and temperature in contained gases, the rate of growth of biological populations, the change in crop yields as a function of fertilizer, etc.). However, one is hard pressed to find such invariant relationships in the science of politics. Political science theories are weak in that they identify few such steadfast relationships, yet political scientists who use regression analysis and its many offshoots commonly assume parameter invariance when they restrict the coefficients to a single K x 1 vector. Political scientists rarely give the parameter invariance assumption any serious consideration either through substantive theory or their choice of estimation methods. The purpose of this article is to demonstrate that this is a mistake and to illustrate a relatively new method for evaluating the parameter-invariance assumption. In particular we shall be concerned with the parameter-invariance assumption with respect to time.