G2870 g4671g2019 g2871 g2026 g3047g2879g2870 g2870 8

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g2870 g4671+g2019 g2871 g2026 g3047g2879g2870 g2870 (8) Substituting for g2026 g3047g2879g2870 g2870 and prior lag orders of the conditional variance forecasts until time g1866−1 periods previous; a pattern emerges such that: g2026 g3047g2878g2869 g2870 =g46661−g2019g4667g4670g2013 g3047 g2870 +g2019g2013 g3047g2879g2869 g2870 +g2019 g2870 g2013 g3047g2879g2870 g2870 +⋯+g2019 g3041g2879g2869 g2013 g3047g2879g3041g2878g2869 g2870 g4671+g2019 g3041 g2026 g3047g2879g3041g2878g2869 g2870 (9) Therefore, re-writing Equation (9) in ‘sigma’ notation: g2026 g3047g2878g2869 g2870 =g46661−g2019g4667g3533g2019 g3037g2879g2869 g3041 g3037g2880g2869 g2013 g3047g2879g3037g2878g2869 g2870 +g2019 g3041 g2026 g3047g2879g3041g2878g2869 g2870 (10)
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BUSE4008 – AFRM [2] YS 2015 Now considering that 0<g2019<1 as g1866→∞ , g2019 g3041 →0 . Therefore, the final term of Equation (10) will tend towards zero as g1866→∞ . Hence, as g1866→∞ , Equation (10) will become: g2026 g3047g2878g2869 g2870 =g46661−g2019g4667g3533g2019 g3037g2879g2869 g3041 g3037g2880g2869 g2013 g3047g2879g3037g2878g2869 g2870 (11) Substituting back for the original definition of the residual (error) term: g2026 g3047g2878g2869 g2870 =g46661−g2019g4667g3533g2019 g3037g2879g2869 g3041 g3037g2880g2869 g3435g1844 g3047g2879g3037g2878g2869 −g1844 g3364 g3439 g2870 (12) Q.E.D. Now, Equation (12) is precisely the standard formula for the EWMA. Hence, it is proven that as g1866→∞ , the recursive formula for the EWMA reverts to the standard formula for the EWMA. 2. EWMA as a Special Case of a GARCH(1,1) Prove that the exponentially-weighted moving average (EWMA) can be seen as a special case of a generalised autoregressive conditional heteroscedastic, GARCH(1,1) model. Starting with the specification for the conditional variance under a GARCH(1,1) model: g3047 =g2009 g2868 +g2009 g2869 g2013 g3047g2879g2869 g2870 +g2010 g2869 g3047g2879g2869 (1) Replacing with the more familiar notation for variance g4666g2026 g2870 g4667 : g2026 g3047 g2870 =g2009 g2868
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  • Fall '19
  • GARCH, Notation, EWMA

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