One of the assumptions of the classical linear regression model is that the explanatory variables are non-stochastic, or fixed in repeated samples. There are various ways of stating this condition, some of which are slightly more or less strict, but all of which have the same broad implication. It could also be stated that all of the variables contained in the X matrix are assumed to be exogenous or that the model is ‘conditioned on’ the variables in X. However, this assumption will be violated when there is feedback from the explained variable to the explanatory variable(s) – in other words, if there is a simultaneous relationship between them. This chapter first considers how to model simultaneous equations using an example on the relationship between inflation and stock returns.

Setting up a system

What is the relationship between inflation and stock returns? Holding stocks is often thought to provide a good hedge against inflation, since the payments to equity holders are not fixed in nominal terms and represent a claim on real assets (unlike the coupons on bonds, for example). However, the majority of empirical studies that have investigated the sign of this relationship have found it to be negative. Various explanations of this puzzling empirical phenomenon have been proposed, including a link through real activity, so that real activity is negatively related to inflation but positively related to stock returns so that stock returns and inflation vary negatively.