Spring 2005 Lab Nine 1 Econ240CI. ARCH-GARCH, An Example: The Producer Price Index, 1982=100, seasonally adjusted, for finished goods.A. The Producer Price IndexOpen EViewsFile Menu/Open: ppifgs.wf1( in the Econ 240C Folder) available at FREDWorkfile Window: Sample: 1947:04 2005:04 (can be updated at FRED)Workfile Window: Select ppifgsWorkfile Menu: VIEW: • open selectionGroup Window: VIEW: • line graphNote: The producer price index appears to be trending upward. I took logarithms and differenced the series, i.e. GENR lnppfgsi=log(ppifgs), Genr dlnppifgs=lnppifgs-lnppifgs(-1), or fractional changes in the index of producer prices. Multiplying by 100 would give the monthly percent increase, and multiplying again by 12 would give the annual rate of percentage increase for each month.B. Identifying and Modeling Fractional Changes in the Producer Price IndexWorkfile Window: Select dlnppiWorkfile Menu: VIEW: • open selectionGroup Window: VIEW: • line graphNote: the series appears to be stationary, and has a cycleGroup Window: VIEW: • histogram and statsNote: the series is kurtotic, i.e. the inflation rate in producer prices is subject to big changes.Group Window: VIEW: • correlogram: level, 36 lagsNote:the autocorrelation function shows a cycle, like the series, and the partial autocorrelation function has spikes at lags one and two so an AR(2) model seems like a place to start.Group Window: VIEW: •unit root test: constant, no trend, 1 lagNote: the time series dlnppifgs is stationary, with the augmented Dickey-Fuller statistic clearly indicating rejection of a unit root.Spring 2005 Lab Nine 2 Econ240CObject Menu/New/Equationdlnppifgs c ar(1) ar(2)Equation Object Window Menu: VIEW: • actual, fitted, residuals: graphEquation Object Window Menu: VIEW: • residual tests: correlogram, 36 lagsNote: the standard error in the autocorrelations is about 0.04. The autocorrelations at lags 2 and 3 are beyond two standard deviations from zero. The high Q statistic begins at lag 2 or 3. I decided to try an ARMA(1,1) model.Equation Object Window Menu: Estimate: dlnppifgs c ar(1) ma(1)Equation Object Window Menu: VIEW: • actual, fitted, residuals: graphEquation Object Window Menu: VIEW: • residual tests: correlogram, 36 lagsNote: There is no patterned structure in the ACF and PACF although there appears to be low lying noise. This PPI series has been deseasonalized This is probably the origin of this noise. The standard error of the regression is marginally better forthe ARMA(1,1) compared to the AR(2). So the residual from this model is approximately orthogonal.Equation Object Window Menu: VIEW: • residual tests: histogram-normality testNote: the residual is kurtotic, a signal to look at the residual squared Workfile Menu: GENR: residsq=resid*residWorkfile Window: Select residsqWorkfile Menu: VIEW: • open selection; you will see the spreadsheet viewSeries Window: VIEW: • graphNote: the time series for the square of the residual is very episodic showing periods of high volatility.Series Window: VIEW: • histogram and statsSpring 2005 Lab Nine 3 Econ240CNote: the distribution looks geometric, approximatelySeries Window: VIEW: • correlogram: level, 36 lagsNote: the correlogram shows some autoregressive structure.C. A Garch(1,1) Model for the Inflation Rate in Wholesale PricesQUICK: Estimate Equation, Method ArchThe mean equation: dlnppifgs c AR(1) MA(1) The variance equation: cARCH 1GARCH 1Equation Object Window Menu: VIEW: • estimation outputEquation Object Window Menu: VIEW: • actual, fitted, residuals: graphNote: these are the residuals, error(t), from the ARMA(1,1) model for dlnppiEquation Object Window Menu: VIEW: • residual tests: correlogram, 36 lagsNote: This is the correlogram for the standardized residuals from the ARMA(1, 1) model for dlnppi, where the standardized residual is error(t)/√h(t) using the estimates for both error(t) and the conditional variance h(t), and so if the models for both dlnppifgs and h(t) are correct, then this is an estimate of wn(t) = error(t)/√h(t).Equation Object Window Menu: VIEW: • residual tests: correlogram-square residsNote: this is the correlogram for the square of the standardized residuals and if the equation for h(t) is correct, there should be no significant Q statistics.Equation Object Window Menu: VIEW: • residual tests: histogram-normality testNote: this is the histogram of the standardized residuals and if the models for the conditional mean and the conditional variance are correct, should be white noise.Equation Object Window Menu: VIEW: • residual tests: ARCH LM TestSpring 2005 Lab Nine 4 Econ240CNote: this is a Lagrange multiplier test on the standardized residuals to see if there is any ARCH component left in the standardized residuals.The estimated conditional variance is:h(t) = 0 + 1 [e(t-1)]2 + 1 h(t-1) = 0.00000209+ 0.206 [Error(t-1)]2 + 0.699 h(t-1), with a standard error of 0.00000057 in 0 , a standard error of 0.034 in 1 , and a standard error of 0.052 in 1 .so E2005.04 h(2005.05) = 0.00000209+ 0.206 (0.00211)2 +0.699( 2.04x10-5) = 0.00001727and √[ E2005.04 h(2005.05)] = 0.00416 is the standard error in the one period ahead forecast. Equation Object Window Menu: VIEW: • conditional SD graphNote: this is the trace of the square root of the estimated series for h(t). Equation Object Window Menu: Procs: • make GARCH Variance seriesNote: the series garch01 is now in the workfile window. This is an estimate of the series h(t), and is the source of h(2005.04) = 2.04x10-5 in the calculation of E2005.04 h(2005.05) above.II. Simulation of a GARCH(0,1), i.e. ARCH(1) ModelOpen EviewsObject Menu: NewWorkfile Object: “ARCH(1)”Frequency: • undatedobservations: 1 250Workfile Menu: GENR“WN=NRND”Workfile Window: Select WNWorkfile Menu: VIEW: • open selection; you will see the spreadsheet viewSeries Window: VIEW: • graphSeries Window: VIEW: • histogram and statsSpring 2005 Lab Nine 5 Econ240CSeries Window: VIEW: • correlogram: level, 36 lagsWorkfile Menu: GENR“ERROR = WN”sample 1 1Workfile Menu: GENR“ERROR = WN*SQR(1+0.7*error(-1)*error(-1))”sample 2 250Note: This is the generation of the error, e(t) = WN(t) √h(t), where h(t) is the conditional variance, i.e. Et-1 [e(t)e(t)] = Et-1 [WN(t)WN(t)] Et-1 [√h(t)√h(t)] = 1* Et-1 h(t) = h(t),where √h(t) = √1+ 0.7*e(t-1)*e(t-1), and so Et-1 h(t) = 1+ 0.7*e(t-1)*e(t-1).Workfile Window: Select