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Hi: This is an incomplete answer but I needed room. The Wald statistic for testing a linear constraint , $Rb = r$ is , $(Rb - r )^{\prime}[R(X^{\prime}X)^{-1} R^{\prime}]^{-1}(Rb - r)/s^2$ $X^{\prime}X$ can be obtained from P4 and, in your case, $R = 1$ and $b = \alpha$. But I still don't see how the expression for $(X^\prime X)^{-1}$ results in what you ...

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There are famous proofs by Arrow & Debreu and others, based on the Kakutani Fixed Point Theorem, but they are at a very abstract and general level. I am not sure the details of market mechanism matter, as long as excess demand affects prices, and prices affect excess demand, there will be a convergence to a price equilibrium.

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There is no mathematical proof of EMH. You would need all market participants to agree on a singular pricing model for that to be possible. Without a singular, agreed-upon model, what you are asking for is proof that people's collective opinions represent people's collective opinions. This is called a tautology. Tautology There is an endless number of ...

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