Inversion and the Lucas critique
As pointed out by Desai (1984), the reversal of dependent and independent variables represents a continuing controversy in the literature on inflation modelling. Section 4.1.1 recounts how Lucas’s supply curve turns the causality of the conventional Phillips curve on its head. Moreover, the Lucas critique states that conditional Phillips curve models will experience structural breaks whenever agents change their expectations, for example, following a change in economic policy. In this section, we discuss both inversion and the Lucas critique, with the aim of showing how the direction of the regression and the relevance of the Lucas critique can be tested in practice.
Under the assumption of super exogeneity, the results for a conditional econometric model, for example, a conventional augmented Phillips curve, are not invariant to a re-normalisation. One way to see this is to invoke the well-known formula
/3 • 3 = r2yx, (4.16)
where ryx denotes the correlation coefficient and в is the estimated regression coefficient when y is the dependent variable and x is the regressor. в* is the estimated coefficient in the reverse regression. By definition, ‘regime shifts’ entail that correlation structures alter, hence ryx shifts. If, due to super exogeneity, в nevertheless is constant, then в* cannot be constant.
Equation (4.16) applies more generally, with ryx interpreted as the partial correlation coefficient. Hence, if (for example) the Phillips curve (4.1) is estimated by OLS, then finding that [3w1 is recursively stable entails that в**і for the re-normalised equation (on the rate of unemployment) is recursively unstable. Thus, finding a stable Phillips curve over a sample period that contains changes in the (partial) correlations, refutes any claim that the model has a Lucas supply curve interpretation. This simple procedure also applies to estimation by instrumental variables (due to endogeneity of, for example, Aqt and/or Apt) provided that the number of instrumental variables is lower than the number of endogenous variables in the Phillips curve.