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I have two random variables $X$ (price of a commodity) and $Y$ (default rate of that commodity), that are correlated through $\rho$. $X$ follows a log-normal distribution and $Y$ has not enough datapoints to affirm that follows a normal distribution or two perform a linear model of $X$ on $Y$. I estimate $X$ multiple times with a geometric brownian motion. I want generate $Y$ from the outcome of $X$. Is there a way in which i can estimate $Y$ with the results of $X$?

Daniel Gutierrez
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  • Is "$\rho$" another variable or is it a correlation coefficient? If it's the latter, how could you possibly know its value if you don't even know the general shape of the distribution of $Y$ or even whether $(X,Y)$ are linearly related? – whuber Apr 21 '22 at 15:20
  • @whuber $\rho$ is the correlation coefficient provided by an outside source. The idea is to perform several simulations changing the value of $\rho$. – Daniel Gutierrez Apr 21 '22 at 15:34
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    The best way to estimate $Y$ from $X$ would be to ask your source for all their information and assumptions about the joint distribution of $(X,Y).$ Otherwise you just have to make assumptions wholesale with little assurance your estimates are any good (and no way even to assess how poor they might be). – whuber Apr 21 '22 at 15:54
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    I will ask for the distribution of Y, in the meanwhile this previous post helps me to make an approach making some assumptions of normality in both variables. https://stats.stackexchange.com/questions/15011/generate-a-random-variable-with-a-defined-correlation-to-an-existing-variables – Daniel Gutierrez Apr 21 '22 at 19:10

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