# In this case the relative weightings are given by the

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the overall variance forecast is dictated by their relative weightings. In this case, the relative weightings are given by the parameters g2009 g2869 and g2010 g2869 . Therefore, the only term that can capture the long-run dynamics is g2009 g2868 . Now defining the constant long-run variance estimate to be g1848 g3013 , g2009 g2868 should also include the relative weighting g4666g2011g4667 that describes the contribution that the constant long-run variance makes towards forecasting the conditional variance. Hence, one would define: g2009 g2868 =g2011g1848 g3013 (4) Since the conditional variance forecast specification is set describe the complete dynamics of the conditional variance, the relative weightings should sum to unity: g2011+g2009 g2869 +g2010 g2869 =1 (5) Furthermore, for a well-defined GARCH(1,1) specification: 1−g2009 g2869 −g2010 g2869 >0 (6) ⇒ g2011 > 0 From Equation (5): g2011 = 1 − g2009 g2869 − g2010 g2869 (7)

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BUSE4008 – AFRM [4] YS 2015 Rewriting Equation (4): g1848 g3013 = g2009 g2868 g2011 (8) Substituting for g2011=1−g2009 g2869 −g2010 g2869 into Equation (8), one arrives at the estimate for the constant long- run variance: g1848 g3013 = g2009 g2868 1−g2009 g2869 −g2010 g2869 (9) Q.E.D.
• Fall '19
• GARCH, Notation, EWMA

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