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By Alexander J. W.

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On the possibility of excess kurtosis, instead. For fourth moments, we have that E{XlXjt) = ^. E(XlXjtXkt) = 0, E{XitXjtXktXit) = 0, i¥^j. i, jf, k are all different, i, j , fc, / are all different. 100) Positive expressions only result from index combinations of the form E{Xf^) and E{XlX'^jt), We have and n\E{Xl)E{u%) + K]E{X%)E{UI) K^n] + E{ulu%) i^j. 102) This is the point where the moments of a VAR(l) with Gaussian innovations and the moments of a VAR(l) with innovations from a Gaussian mixture (that has the same second moments) differ.

For skewness and kurtosis we obtain skew{Ay^) = ^J<}0^^skew{u,) and ^^ See the appendix for the computation. ^^ This follows already from the fact that the stationary distribution of yf is Gaussian. ^^ Again, moments for differenced yields are computed in the appendix. 3 AflSne Multifactor Gaussian Mixture Models kurt{Ay^) = ^ ^ 4(1 + ^2) 39 '-kurt{ut) respectively. Again, both measures do not depend on time to maturity and are proportional to the corresponding quantity of Ut. Against the background of the stylized facts, it is also instructive to compare the higher moments of yields in levels and yields in first differences.

56) ( ^a;5e-^^+^^^^^+* (A+^ovA In the next section, we will consider the afRne mult if actor mixture model of which the one-factor mixture model is a special case. Thus, the proof given there also holds for this proposition here^^ and we omit proving the special case. Again, the intercept parameter S can be chosen in such a way that the state variable Xt equals the short rate. We must have Ai =0 which implies S = \n ( ^a;6e-^^'>+^^'^' J . e. the expected excess return divided by its standard deviation, one obtains^^ 1 Bn-icr In (j2uj,e^^^^'^^'A - I n (^j^^bC^^^^-^^^^'^^^^^-^^'A] .

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An Example of a Simply Connected Surface Bounding a Region which is not Simply Connected by Alexander J. W.


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