Our second approach is to estimate versions of the Euler equation given by equation (5) for the real return on shares for the total sample and for the two groups defined on the basis of the conditional probability of share ownership. Results are reported in Table 4. As mentioned in Section 2, in the absence of measurement error, any variable dated t-1 is a valid instrument As we use grouped data, however, small sample variability induces MA(1) errors and therefore one has to lag consumption growth twice to avoid getting inconsistent estimates.17 Furthermore, one needs to correct the standard errors for the presence of such an error structure. Finally, one can improve the efficiency of the estimates relative to a simple IV procedure by using a GMM estimator.
Because the estimated variance covariance matrix of the residuals after the first step is not always positive definite,18 we cannot always use a standard GMM estimator. Therefore, we develop a procedure, which is slighdy different from the standard one. The estimator we use, whose details are given in the Appendix, consists of three steps. A first step that gives consistent estimates; a second that corrects for the presence of MA residuals and a third that computes an efficient GMM estimation on the transformed model.
As before, consumption growth is measured as the log change in de-seasonalised consumption per adult equivalent in the various groups considered. The instruments, which include the second lag of consumption growth and several financial variables, are listed in the notes to the table. The only instrument that deserves a mention is a financial market liberalization dummy, which is meant to captures the process of transformation of British financial markets in the second half of the 1980s. Results are not greatly affected by the use of this particular instrument.