... DAX¹

Raw DAX version according to Statistisches Bundesamt (federal office of statistics). Other data are from the same source (except for business sentiment). Collected by Christian Puritscher, for a diploma thesis in industrial management at LMU, Munich.

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....²

We have $- \frac{\partial \ln Z(D_0,\beta)}{\partial \beta} = <_{D_0} \! E(D_0,.)\! >$ and $- \frac{\partial \ln Z_{D_0}(D,\beta)}{\partial \beta} = <_{D} \! E(D,.) \!>$. Furthermore, $\frac{\partial^2 \ln Z(D_0,\beta)}{\partial \beta^2} = <_{D_0} \! ( E(D_0,.) - <_{D_0} \! E(D_0,.) \!> )^2 \!>$ and $\frac{\partial^2 \ln Z_{D_0}(D,\beta)}{\partial \beta^2} = <_{D} \! ( E(D,.) - <_{D} \! E(D,.) \! > )^2 \!>$. See also Levin et al. (1990). Using these expressions, it can be shown: by increasing $\beta$ (starting from $\beta = 0$), we will find a $\beta$ that minimizes $\frac{1}{\beta} \ln \frac{Z_{D_0}(D,\beta)}{Z(D_0,\beta)}<0$. Increasing $\beta$ further makes this expression go to $0$.

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