Consider an infinitely lived economy (i.e., \(T=\infty\) ) and suppose that the dividend process of a risky security satisfies the following dynamics:

\[\begin{equation*}d_{t+1}=(1+g) d_{t}+\sum_{k=0}^{N} \gamma_{k}\left(E_{t-k+1}-(1+g) E_{t-k}\right), \quad\text { for all } t \in \mathbb{N}, \tag{7.17}\end{equation*}\]

for some \(N \in \mathbb{N}\), with \(d_{0}=0\) and where \(\left(E_{t}\right)_{t \in \mathbb{N}}\) denotes the earnings process and \(\left\{\gamma_{k}\right\}_{k=0,1 \ldots, N}\) is a family of non-negative weight factors. This model represents the situation where the dividend process is set according to a growth rate equal to \(g\), but managers deviate from this long run growth path in response to changes in earnings that deviate from their long run growth path. Define by \(\bar{d}_{t}:=d_{t} /(1+g)^{t}\) the detrended dividend and, similarly, \(\bar{E}_{t}:=E_{t} /(1+g)^{t}\), for all \(t \in \mathbb{N}\). Suppose that the earnings process \(\left(E_{t}\right)_{t \in \mathbb{N}}\) is related to the firm value process \(\left(V_{t}\right)_{t \in \mathbb{N}}\) by \(E_{t}=r_{f} V_{t}\), for all \(t \in \mathbb{N}\), where \(r_{f}\) is the constant risk free rate and suppose also that stocks are priced rationally, i.e., \(s_{t}=V_{t}\), for all \(t \in \mathbb{N}\). Show that, under the present assumption, the rational ex-post price as defined in (7.10) (with the terminal value \(\bar{s}_{T}^{e}\) being defined as \(\bar{s}_{T}^{e}:=\sum_{t=0}^{T-1} \bar{s}_{t} / T\), with \(\left(\bar{s}_{t}\right)_{t \in \mathbb{N}}\) denoting the de-trended observed price process) admits the representation

\[\begin{equation*}\hat{s}_{t}^{e}=\sum_{k=-N}^{T-1} w_{t k} \bar{s}_{k}, \quad \text { for all } 0 \leq t \leq T \tag{7.18}\end{equation*}\]

for a suitable family \(\left\{w_{t k}\right\}\) of weight factors.