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I have two models for prospect theory utility functions and want to compare their fits. However, I'm not sure if the models are nested or not. The first model is given by:

$U(x) = x^α$ for $x ≥ 0$ and $U(y) = −λ|y|^α$ for $y < 0$

The second model is given by:

$U(x) = x^α$ for $x ≥ 0$ and $U(y) = −λ|y|^\beta$ for $y < 0$

The first model could be a specific version of the second model where $\beta = \alpha$ but my gut tells me this isn't how nested models work. Any help identifying nested models would be appreciated.

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    $\begingroup$ Two models are nested if we can simplify the bigger model into the smaller model by imposing constraints on the parameters. In your case the constraint is $\beta=\alpha$ just as you point out, so model 1 is nested in model 2. $\endgroup$
    – dipetkov
    Commented Apr 17, 2022 at 14:46
  • $\begingroup$ See also the similar post: stats.stackexchange.com/questions/509957/nested-glm-models/… $\endgroup$ Commented Apr 17, 2022 at 17:57

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Two models are nested if we can simplify the bigger model into the smaller model by imposing constraints on the parameters.

Often this means setting a parameter to 0. For example, regression M1

$$ g(E\{Y\}) = \beta_0 + \beta_1 X_1 \color{white}{+ \beta_2 X_2} $$

is nested in regression M2

$$ g(E\{Y\}) = \beta_0 + \beta_1 X_1 + \beta_2 X_2 $$

because M2 simplifies into M1 if we set $\beta_2 = 0$.

However, the principle is more general than fixing a parameter to equal a constant. In your case the constraint is $\beta = \alpha$. So your model 1 is indeed nested in your model 2.

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