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The $\lambda$ value used in the original paper is arbitrary, but you can estimate that by assuming (in the simplest case) 2 assets and running the following model: $\sigma^2_{12,t+1}$ $=$ $\lambda$$*$$\sigma^2_{12,t-1}$$+$$(1-\lambda)$$r_{1,t}$$*$$r_{2,t}$; given $r_{1,t}$ and $r_{2,t}$ respectively as the returns for the asset 1 and 2 and ...

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Approach 1 is parametric regression, whereas approach 2 is non-parametric regression. How are they related: non-parametric regression models the entire distribution of all possible function forms, and then do the integration to calculate a single value E[Y|X]. It is function-form free. In contrast, parametric linear regression ASSUMES that the function ...

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You can apply the Kolmogorov-Smirnov test. I simply quote from the entry: "The two-sample K–S test is one of the most useful and general nonparametric methods for comparing two samples, as it is sensitive to differences in both location and shape of the empirical cumulative distribution functions of the two samples." There is an R-implementation too.

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