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In our study we analyze wealth inheritance over 10 generations. We have wealth data for each individual and their respective ancestors and descendents. Now I want to expand the analysis in a way that I can predict for each family their expected income. As data I just have wealth for each family z in generation t, t-1, t-2, ...t-10 for 7000 different families. Furthermore I have the years and the family names. What kind of machine learning technique would you suggest for this prediction?

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You can try performing a nonlinear regression prediction using a neural network. Common practice is to let the output node (the $y$ your predict) not use a nonlinear activation function. You can use as bootstrap estimates (1$-$the historic saving rate) as fraction of expected income.

Later on, you may adapt your model when its predictions are too linear.

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