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Econ 240C. Lecture 14. ARCH-GARCH Structure?. Outline. Part I: Conditional Heteroskedasticity: example Part II: Detecting ARCH Part III: Modeling ARCH & GARCH Part IV: Estimating GARCH. Part I. Conditional Heteroskedasticity. An Example. Producer Price Index for Finished Goods

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## Econ 240C

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**Econ 240C**Lecture 14**Outline**• Part I: Conditional Heteroskedasticity: example • Part II: Detecting ARCH • Part III: Modeling ARCH & GARCH • Part IV: Estimating GARCH**Part I. Conditional Heteroskedasticity**• An Example**Producer Price Index for Finished Goods**• April 1947-April 2007 • 1982=100 • Seasonally adjusted rate (SAR)**Transformations**• PPI is evolutionary • Take logarithms • Then difference • Obtain the fractional changes, i.e. the inflation rate for producer goods**Modeling dlnppi**• Try an arthree**Modeling dlnppi**• Try an ARMA(1,1)**ARMA(1,1) Model of Producer Goods Inflation**• Residuals from ARMA(1, 1) model are approximately orthogonal but not normal • Are we done?**Part II. Examine Residuals**• Trace of residuals • Trace of square of residuals**Episodic variance**• Not homoskedastic • So call heteroskedastic, conditional on dates or episodes when the variance kicks up • Hence name “conditional heteroskedaticity”**Clues**• Check trace of residuals squared • Check correlogram of residuals squared • Equation window: View menu, residuals • Check ARCH Lagrange Multiplier Test**Clues**• Check trace of residuals squared • can get residuals from Actual, fitted, residuals table • Check correlogram of residuals squared • EVIEWS option along with correlogram of residuals • Heteroskedasticity of residuals • Histogram of residuals • kurtotic residuals are a clue**Part III: Modeling Conditional Heteroskedasticity**• Robert Engle: UCSD • Autoregressive error variance model**Modeling the error**• Model the error e(t) as the product of two independent parts, wn(t) and h(t) • WN(t) ~N(0,1)**Modeling the error**• Assume that WN(t) is independent of • So density f{wn(t)*[h(t)]1/2} is the product of two densities, g and k: • f =g[wn(t)]*k{[h(t)]1/2} • And expectations can be written as products of expectations • This is related to writing the Probability of P(A and B) as P(A)*P(B) if events A • And B are independent**Modeling the error**• We would like the error, e(t) to have the usual properties of mean zero, orthogonality, and unconditional variance constant • E e(t) = E {[h(t)]1/2*WN(t)} = E{[h(t)]1/2}*E[WN(t)] , the product of expectations because of independence • We may not know E{[h(t)]1/2}, but we know E[WN(t)] =0 so Ee(t)=0

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