Nevertheless, since we aim to a different target than \citet*{Ghosh_2016}, we will explore this equation in different way than they did.  We first notice that, in Lucas setting, we can have a twofold interpretation of analysts' forecasts like it's shown below:
1. First Interpretation
In the first interpretation we suppose analysts' forecasts are in agents' information set which implies strong correlation and even causality in the sense of \citet{Granger_1969} between forecasts and actual prices.   Formally, the interpretation means:
 \(p_{t}= E_{t}\left\{ \beta. \frac{u'(c_{t+1})}{u'(c_{t})}.\left(p_{t+1} + d_{t+1}\right) | \left\{p^{T}\right\} \cup I_{t} \right\}\)
where \(p^{T}\) means analysts' forecast for prices, that is, target prices and \(I_{t}\) means the set which elements are any other variable which helps but not target prices .
2. Second Interpretation
The second interpretation rests on the fact all agents are equal in Lucas setting and then the analysts are themselves the agents who is solving Lucas problem.  Consequently, in the second interpretation, actual prices should converge in distribution to analysts' forecasts.  Mathematically, we can express it as:
\(p_{t}= E_{t}\left\{ \beta. \frac{u'(c_{t+1})}{u'(c_{t})}.\left(p^{T}_{t+1} + d_{t+1}\right) | I_{t} \right\}\)
In the following we'll show how we'll use both interpretations above to test for the relation of analysts' forecast  and expected stock returns.

Methods and Data

The empirical strategy consists in two steps.  The first one is to collect actual prices and target prices from analysts of Brendan E. Cryan & Co., Citadel Securities LLC, GTS Securities LLC, IMC Financial Markets, and Virtu Financial Capital Markets LLC, which are market makers in New York Stock Exchange (NYSE).  Then I will investigate the first interpretation of Lucas model, that is, we will consider target prices as the result of an estimation of Lucas asset pricing model and test for causality in the sense of \citet{Granger_1969} between those actual and target prices.