stiff gulch
I am trying to model interval responses using MLE. I have absolutely no clue how to solve this problem. Here is the explanation of my problem:
My data stems from a survey. In this survey each individual is asked two questions about their expectations of the 1 year ahead returns of the Dow Jones. Assume that when making an investment decision individual $i$ thinks of 1-year-ahead returns as $R_i$ with mean $\mu_i$ and standard deviation $\sigma_i$. $R_i$ is assumed to be normally distributed. The first question is "By next year at this time, what is the percent chance that the Dow Jones will be worth more than today?". The answer to this question is called $P_0$. The second question is "By next year at this time, what is the chance it will have [gained/fallen] by x percent or more?". For this question individuals are randomly assigned to either get the gained or fallen version. Also, x represents a value from the set {10, 20, 30, 40}. The answer to this question is called ${P_x}+$ for positive growth questions and ${P_x}-$ for negative growth questions.
Because a lot of individuals round their answer, the answers to the two questions are put in 10 percentage point wide intervals: [0, 5); [5, 15); ... [95, 100).
In addition, The survey answers of individual $i$ are based on a noisy version of $R_i$ that is denoted as $R_{ji}$ (where $j$ denotes the question so that $j$ = 0, $x+$ or $x-$). The noise is assumed to be additive: the mean of $R_{ji}$ is $\mu_i$ + $v_{ji}$, where $v_{ji}$ is a mean-zero noise variable specific to question and individual. The noise terms are allowed to be different for questions $P_0$ and $P_x$ but are correlated across questions: corr($v_{0i}$, $v_{xi}$) = $\rho$.
These assumptions are then combined equation (1) – (3). Furthermore, equations (4) – (7) also need to hold. Lastly, $u$ $v_0$ and $v_x$ are jointly normally distributed and $u$ is independent of survey noise.
I want to estimate all the dependent variables from equations (4)–(7)
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