Models of Feedback: Interpretation and Discovery

 This thesis is about feedback models in which it is possible that A causes B, and simultaneously, that B causes A. More particularly a kind of statistical model which takes  this form, called a non-recursive structural equation model. Since models of this sort were originally introduced in econometrics, in Chapter One I  discuss the historical background which led to their creation. I consider it to be particularly  important to examine the historical origins in order to try to establish what were the goals  of those who first used these models, and why they considered that these models helped  them to achieve these goals. This is necessary because the original motivation was later  submerged as technical questions began to dominate the research agenda. I also describe  some of the disciplinary factors which led to almost universal acceptance of this model  form, at least among those building macro-economic models. A classic example of feedback is provided by the economic theory of a market: the price of  a good may be a function of the quantity either demanded or supplied, while these  quantities themselves may be influenced by the price or the expectation of price that  consumers or suppliers may have. Thus studies of supply and demand might be considered  the proper area of application for a feedback model. It is therefore somewhat surprising that  some of the most successful studies of demand used models which made no allowance for  simultaneity. I investigate why these empiricists chose not to include feedback in their  models. 'Reciprocal' causation of the kind suggested by the structure of a non-recursive model does  not fit easily with our intuitive notion of cause and effect. Indeed some statisticians have  questioned the meaningfulness and applicability of such non-recursive models. Similar  concerns were voiced by economists and econometricians when non-recursive models were  first introduced. In Chapter Two I describe this debate in detail since I think it raises  interesting issues concerning the relationship between dynamic systems and static models  that approximate them. In addition to quite specific questions about the correct way to  model the world statistically, the debate covered very general questions about causality and  the nature of explanation. The second major theme of this thesis centers upon the inference of causal structure from  observational statistical data, given various kinds of background knowledge. Inferences of this kind are made frequently in the social sciences (economics, sociology, psychology,  epidemiology) where often only observational data are available. It is common in these  fields to find very litde (if any) justified consensus about the causal processes that may  have generated the data. For this reason the traditional method of postulating a "model" and  then seeing whether it is rejected by data is inappropriate. Even if the available data does  not reject the model it is quite possible that there are a large number of others that are also  compatible with the data. Perhaps in one model changes in A (e.g. the interest rate) bring  about changes in B (e.g. the money supply), while in another model the reverse is true.  Suppose one is a policy maker trying to influence B by manipulating A. If the first model is  true the policy may very well be effective, but if the second is true it will be completely  futile, treating the symptoms rather than the causes of the variable B. Evidently it is of little  use to be told that a model is compatible with given data unless one knows all of the other  models that are similarly compatible. Although we may be unable to advise the policy maker about which of many competing  candidates is the 'true' model, we may still be able to infer that certain causal relations are  common to all models compatible with the data. If it turns out that in all such models A is a  cause of B then this may suffice for a policy decision. In order to produce such a list we  must systematically characterize the way in which statistical data underdetermines causal  theories. I have constructed an efficient and correct algorithm which produces a set of features  common to all linear feedback models compatible with data provided as input (assuming  that there are no unmeasured common causes or 'correlated errors'). This algorithm,  presented in Chapter Three, makes causal inferences on the basis of conditional  independence tests. This is an extension of the theory developed by Spirtes, Glymour and  Scheines in their book Causation, Prediction and Search, where it is assumed that no  feedback is present. The output representation of the algorithm, which I call a Partial  Ancestral Graph or PAG, allows for the easy incorporation of background knowledge. In  addition, though I do not discuss this here, PAGs can also be used to represent features  common to a very broad class of recursive models, including latent variables, correlated  errors, and selection bias. Spirtes (1994) also considers the more general class of non-linear, non-recursive structural  equation models. In Chapter Four I present an algorithm for carrying out causal inference,  given certain assumptions, from data generated from a non-linear non-recursive structural  equation model