g2009 g2869 g2013 g3047g2879g2869 g2870 g2010 g2869 g2026 g3047g2879g2869 g2870

G2009 g2869 g2013 g3047g2879g2869 g2870 g2010 g2869

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+g2009 g2869 g2013 g3047g2879g2869 g2870 +g2010 g2869 g2026 g3047g2879g2869 g2870 (2) Moving one period forward: g2026 g3047g2878g2869 g2870 =g2009 g2868 +g2009 g2869 g2013 g3047 g2870 +g2010 g2869 g2026 g3047 g2870 (3) Now consider the recursive formula for EWMA: g2026 g3047g2878g2869|g3047 g2870 =g2019g2026 g3047|g3047g2879g2869 g2870 +g46661−g2019g4667g4666g1844 g3047 −g1844 g3364 g4667 g2870 (4) Removing the conditionality: g2026 g3047g2878g2869 g2870 =g2019g2026 g3047 g2870 +g46661−g2019g4667g4666g1844 g3047 −g1844 g3364 g4667 g2870 (5) Under the GARCH(1,1) specification (Equation (3)), when g2009 g2868 =0 , g2009 g2869 =1−g2019 and g2010 g2869 =g2019 ; the GARCH(1,1) specification reverts to the recursive formula for the EWMA (Equation (5)). Therefore, the EWMA proves to be a special case of the GARCH(1,1) specification when g2009 g2868 =0 , g2009 g2869 =1−g2019 and g2010 g2869 =g2019 .

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BUSE4008 – AFRM [3] YS 2015 3. Estimate of the Long-Run Variance Using a GARCH(1,1) specification for the conditional variance, derive an estimate for the ‘true’ long- run variance. Explain clearly the steps you undertake in arriving at your estimate. 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 +g2009 g2869 g2013 g3047g2879g2869 g2870 +g2010 g2869 g2026 g3047g2879g2869 g2870 (2) Moving one period forward: g2026 g3047g2878g2869 g2870 =g2009 g2868 +g2009 g2869 g2013 g3047 g2870 +g2010 g2869 g2026 g3047 g2870 (3) As Equation (3) is set to describe the complete nature of the variance in determining a forecast for the conditional variance; it should therefore include both the short-run and long-run dynamics of the variance. It is assumed that if we can observe over an infinite horizon, the long-run dynamics of the variance it would be explained by a constant long-run variance. Under the conditional variance forecast for time g1872+1 , the previous value of the error (ARCH) term and the previous value of the conditional variance (GARCH) term, are set to describe the short-run stochastic changes in the conditional variance. The contribution that these two terms make towards the overall variance forecast is dictated by their relative weightings. In this case, the relative
• Fall '19
• GARCH, Notation, EWMA

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