This chapter’s objectives are to: Examine the so-called stylized facts concerning the properties of economic timeseries data, introduce the basic ARCH and GARCH models, show how ARCH and GARCH models have been used to estimate inflation rate volatility,. | Chapter 3: Modeling Volatility Applied Econometric Time Series 3rd ed. 1 ECONOMIC TIME SERIES: THE STYLIZED FACTS Section 1 5 2. ARCH and GARCH PROCESSES ARCH Processes The GARCH Model Other Methods Let et = demeanded daily return. One method is to use 30-day moving average /30 Implicit volatility Logs can stabilize volatility One simple strategy is to model the conditional variance as an AR(q) process using squares of the estimated residuals In contrast to the moving average, here the weights need not equal1/30 (or 1/N). The forecasts are: Properties of the Simple ARCH Model Since vt and et-1 are independent: Eet = E[ vt(a0 + a1et-12 )]1/2 ] = 0 Et-1et = Et-1vtEt-1 [a0 + a1et-12 ]1/2 ] = 0 Eet et-i = 0 ( i ≠ 0) Eet 2= E[ vt 2(a0 + a1et-12 )] = a0 + a1E(et-1) 2 = a0/( 1 - a1 ) Et-1et 2= Et-1 [ vt 2(a0 + a1(et-1 ) 2 )] = a0 + a1(et-1) 2 13 ARCH Interactions with the Mean Consider: yt =a0 + a1yt–1 + et Var(yt yt–1, yt–2, ) = Et–1(yt – a0 – a1yt–1)2 = Et–1(et)2 = a0 + a1(et–1)2 Unconditional Variance: Since: 14 Figure : Simulated ARCH Processes yt = + εt yt = + εt White Noise Process vt 15 Other Processes ARCH(q) GARCH(p, q) The benefits of the GARCH model should be clear; a high-order ARCH model may have a more parsimonious GARCH representation that is much easier to identify and estimate. This is particularly true since all coefficients must be positive. 16 Testing For ARCH Step 1: Estimate the {yt} sequence using the "best fitting" ARMA model (or regression model) and obtain the squares of the fitted errors . Consider the regression equation: If there are no ARCH effects a1 = a2 = = 0 All the coefficients should be statistically significant No simple way to distinguish between various ARCH and GARCH models 17 Testing for ARCH II Examine the ACF of the squared residuals: Calculate and plot the sample autocorrelations of the squared residuals Ljung–Box Q-statistics can be used to test for groups of significant coefficients. Q has an asymptotic | Chapter 3: Modeling Volatility Applied Econometric Time Series 3rd ed. 1 ECONOMIC TIME SERIES: THE STYLIZED FACTS Section 1 5 2. ARCH and GARCH PROCESSES ARCH Processes The GARCH Model Other Methods Let et = demeanded daily return. One method is to use 30-day moving average /30 Implicit volatility Logs can stabilize volatility One simple strategy is to model the conditional variance as an AR(q) process using squares of the estimated residuals In contrast to the moving average, here the weights need not equal1/30 (or 1/N). The forecasts are: Properties of the Simple ARCH Model Since vt and et-1 are independent: Eet = E[ vt(a0 + a1et-12 )]1/2 ] = 0 Et-1et = Et-1vtEt-1 [a0 + a1et-12 ]1/2 ] = 0 Eet et-i = 0 ( i ≠ 0) Eet 2= E[ vt 2(a0 + a1et-12 )] = a0 + a1E(et-1) 2 = a0/( 1 - a1 ) Et-1et 2= Et-1 [ vt 2(a0 + a1(et-1 ) 2 )] = a0 + a1(et-1) 2 13 ARCH Interactions with the Mean Consider: yt =a0 + a1yt–1 + et Var(yt yt–1, yt–2, ) = Et–1(yt – a0 – a1yt–1)2 = Et–1(et)2 = a0 + a1(et–1)2 .
