T 1 t y t 1 y t l v n 0 e u t 1 2 u t 2 e u t 1 2 u t

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t ± 1 T y t " 1 y t L v N 0, E u t " 1 2 u t 2    E u t " 1 2 u t 2   ± E £ > E u t 4   ² 1 " >  ¡ E u t 2  ¢ 2 ¤ ³ E u t " 1 2   E u t 2   > ± ¡ 1 " ) ² -   - ¢ ) 1 ² - 2 " 2 ) ² -   -
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7 OLS t statistic: t L v N 0, V 11   V 11 ³ 1 V 11 p v . as 3 ) 2 ² 2 )- ² - 2 p v 1 Asymptotic rejection probability for OLS t -test that autoregressive coefficient is zero as a function of GARCH(1,1) parameters α and δ . Note: null hypothesis is actually true and test has nominal size of 5%. Taylor rule: r t ± + 0 ² + 1 = t ² + 2 y t ² + 3 y t " 1 ² + r r t " 1 ² + 5 r t " 1 ² v t r t ± fed funds rate for quarter t = t ± inflation y t ± deviation of real GDP from potential
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8 Claim: + 1 and + 2 are higher now than in 1970s, which contributes to greater economic stability 0.24 0.13 -0.53 d t r t -1 0.08 0.08 0.05 d t r t -1 0.21 0.14 -0.55 d t y t -1 0.24 0.14 0.64 d t y t 0.16 0.09 0.26 d t π t 0.30 0.24 -0.50 d t 0.13 0.11 0.42 r t -1 0.06 0.07 -0.21 r t -1 0.07 0.08 -0.07 y t -1 0.07 0.08 0.18 y t 0.04 0.07 0.17 π t 0.19 0.19 0.37 constant Std error (White) Std error (OLS) Coefficient Regressor Taylor Rule with separate pre- and post-Volcker parameters as estimated by OLS regression ( d t = 1 for t > 1979:Q2). 0.11 -0.01 d t r t -1 0.03 -0.01 d t r t -1 0.07 0.02 d t y t -1 0.07 0.05 d t y t 0.04 0.09 d t π t 0.12 -0.03 d t 0.09 0.47 r t -1 0.03 -0.07 r t -1 0.03 -0.12 y t -1 0.03 0.14 y t 0.03 0.06 π t 0.08 0.13 constant Asymptotic std error Coefficient Regressor Taylor Rule with separate pre- and post-Volcker parameters as estimated by GARCH-t maximum likelihood ( d t = 1 for t > 1979:Q2).
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9 Change in fed funds rate, 1956:Q2-2007:Q1 date change in funds rate 1956 1959 1962 1965 1968 1971 1974 1977 1980 1983 1986 1989 1992 1995 1998 2001 2004 2007 -4 -2 0 2 4 6 8 Scatter diagram, 1979:Q2-2007:Q1 GDP deviation change in funds rate -8 -6 -4 -2 0 2 4 -4 -2 0 2 4 6 8 VII. Time-varying variances A. Introduction to ARCH models B. Extensions generalized autoregressive conditional heteroskedasticity ( GARCH ) Tim Bollerslev dissertation u t ± h t v t v t L 0,1  
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10 u t ± h t v t v t L 0,1   ARCH m   : h t ± ? ² ) L   u t 2 ) L   ± ) 1 L ² ) 2 L 2 ² C ) m L m ARCH .   : h t ± ? ² = L   u t 2 = L   ± ! j ± 0 . = j L j parsimony: = L   ± ) 1 L ² ) 2 L 2 ² C ) m L m 1 " - 1 L " - 2 L 2 " C " - r L r 1 " - 1 L " - 2 L 2 " C " - r L r   h t ± 1 " - 1 " - 2 " C " - r   ? ² ) 1 L ² ) 2 L 2 ² C ) m L m   u t 2 u t L GARCH r , m  
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11 almost all applications use GARCH 1,1   1 " - 1 L   h t ± 4 ² ) 1 Lu t 2 h t ± 4 ² - 1 h t " 1 ² ) 1 u t " 1 2 h t ± 4 ² - 1 h t " 1 ² ) 1 u t " 1 2 add u t 2 to both sides: h t ² u t 2 ± 4 ² - 1 u t " 1 2 " - 1 u t " 1 2 " h t " 1   ² ) 1 u t " 1 2 ² u t 2 u t 2 ± 4 ² - 1 ² ) 1   u t " 1 2 ² u t 2 " h t   " - 1 u t " 1 2 " h t " 1   E u t 2 | u t " 1 , u t " 2 ...   ± h t w t ± u t 2 " h t u t 2 ± 4 ² - 1 ² ) 1   u t " 1 2 ² w t " - 1 w t " 1 u t 2 ± 4 ² - 1 ² ) 1   u t " 1 2 ² w t " - 1 w t " 1 conclusion: u t L GARCH 1,1   ´ u t 2 L ARMA 1,1   AR coefficient ± - 1 ² ) 1 MA coefficient ± " - 1 stationarity requires: | ) 1
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