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Consider some ARCH() process, say ARCH(),wherewith a Gaussian (strong) white noise .&#62; n=500&#62; a1=0.8&#62; a2=0.0&#62; w= 0.2&#62; set.seed(1)&#62; eta=rnorm(n)&#62; epsilon=rnorm(n)&#62; sigma2=rep(w,n)&#62; for(t in 3:n){+ sigma2[t]=w+a1*epsilon[t-1]^2+a2*epsilon[t-2]^2+ epsilon[t]=eta[t]*sqrt(sigma2[t])+ }&#62; par(mfrow=c(1,1))&#62; plot(epsilon,type="l",ylim=c(min(epsilon)-.5,max(epsilon)))&#62; lines(min(epsilon)-1+sqrt(sigma2),col="red")(the red line is the conditional variance process).&#62; par(mfrow=c(1,2))&#62; acf(epsilon,lag=50,lwd=2)&#62; acf(epsilon^2,lag=50,lwd=2)We did mention in class that if  a ARCH(), then  is an AR() process. So a first idea is to consider a regression, as we did for Gaussian AR()&#62; db=data.frame(Y=epsilon[2:n]^2,X1=epsilon[1:(n-1)]^2)&#62; summary(lm(Y~X1,data=db))Call:lm(formula = Y ~ X1, data = db)Residuals:    Min      1Q  Median      3Q     Max -2.4538 -0.3618 -0.2626  0.0935  9.3667 Coefficients:            Estimate Std. Error t value Pr(&#62;|t|)    (Intercept)  0.34963    0.04342   8.052 6.08e-15 ***X1           0.31123    0.04262   7.303 1.13e-12 ***---Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Residual standard error: 0.8413 on 497 degrees of freedomMultiple R-squared:  0.0969,Adjusted R-squared:  0.09508 F-statistic: 53.33 on 1 and 497 DF,  p-value: 1.129e-12There is some significant autocorrelation here. But since our vectors cannot be considered as Gaussian, using least squares is perhaps not the best strategy. Actually, if our series is not Gaussian, it is still conditionally Gaussian, since we assumed that  is a Gaussian (strong) white noise,The likelihood is thenand the log-likelihood isAnd a natural idea is to defineThe code is simply&#62; X=epsilon&#62; loglik=function(param){+ w=exp(param[1])+ a1=exp(param[2])+ s2=rep(w,n)+ for(t in 2:length(X)){s2[t]=w+a1*X[t-1]^2}+ logL=-.5*sum(log(s2))-.5*sum(X^2/s2)+ return(-logL)+ }&#62; OPT=optim(par=+ coefficients(lm(Y~X1,data=db)),fn=loglik)&#62; exp(OPT$par)(Intercept)          X1   0.2482241   0.5858578(since the parameters have to be positive, we assume here that they can be written as the exponential of some real values). Observe that those values are closer to the one used to generate our time series.If we use R functions to estimate those parameters, we get&#62; library(tseries)&#62; summary(garch(epsilon,c(0,1)))...Call:garch(x = epsilon, order = c(0, 1))Model:GARCH(0,1)Residuals:     Min       1Q   Median       3Q      Max -2.87023 -0.60836 -0.03426  0.66648  3.48443 Coefficient(s):    Estimate  Std. Error  t value Pr(&#62;|t|)    a0   0.24959     0.02470   10.104  &#60; 2e-16 ***a1   0.58306     0.09737    5.988 2.13e-09 ***---Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1so that the confidence interval for  is&#62; summary(garch(epsilon,c(0,1)))$coef[2,1]++ c(-1.96,1.96)*summary(garch(epsilon,c(0,1)))$coef[2,2][1] 0.3922088 0.7739088Actually, since our main interest is this   parameter, it is possible to use profile likelihood techniques,&#62; proflik=function(a){+ loglik=function(w){+ s2=rep(w,n)+ for(t in 2:length(X)){s2[t]=w+a*X[t-1]^2}+ logL=-.5*sum(log(s2))-.5*sum(X^2/s2)+ return(-logL)}+ return(-optim(par=.3,fn=loglik)$value)}&#62; A=seq(0,2,by=.05)&#62; P=Vectorize(proflik)(A)&#62; par(mfrow=c(1,1))&#62; plot(A,P,type="l")&#62; OPT=optimize(function(x) -proflik(x), interval=c(0,2))&#62; t=-OPT$objective-qchisq(.95,df=1)&#62; abline(h=t,col="red")&#62; ainf=uniroot(function(x) proflik(x)-t,c(0,OPT$minimum))$root&#62; asup=uniroot(function(x) proflik(x)-t,c(OPT$minimum,2))$root&#62;  abline(v=ainf,lty=2)&#62;  abline(v=asup,lty=2)Of course, all those techniques can be extended to higher order ARCH processes. For instance, if we assume that we have a ARCH() time serieswhere nowwith a Gaussian (strong) white noise . The log-likelihood is stilland we can defineThe code above can be changed, to take into account this additional component,&#62; db=data.frame(Y=epsilon[3:n]^2,+ X1=epsilon[2:(n-1)]^2,+ X2=epsilon[1:(n-2)]^2)&#62; X=epsilon&#62; loglik=function(param){+ w=exp(param[1])+ a1=exp(param[2])+ a2=exp(param[3])+ s2=rep(w,n)+ for(t in 3:length(X)){s2[t]=w+a1*X[t-1]^2+a2*X[t-2]^2}+ logL=-.5*sum(log(s2))-.5*sum(X^2/s2)+ return(-logL)+ }&#62; OPT=optim(par=+ coefficients(lm(Y~X1+X2,data=db)),fn=loglik)&#62; exp(OPT$par)(Intercept)          X1          X2  0.22710526  0.59475474  0.04741294We can also consider some Generalized ARCH process, e.g. a GARCH(,),where nowAgain, maximum likelihood techniques can be used. Actually, we can also code Fisher-Scoring algorithm, since (in a very general context)with here . Using a standard gradient descent algorithm, we get the following estimate for our GARCH process,&#62; X=epsilon&#62; theta=c(.2,.2,.2)&#62; G=rep(1,3)&#62; n=length(X)&#62; j=1&#62; while(sum(G^2)&#62;1e-12){+ s2=rep(theta[1],n)+ for (i in 2:n){s2[i]=theta[1]+theta[2]*X[(i-1)]^2+theta[3]*s2[(i-1)]}+ z=(X^2-s2)/s2^2+ V=cbind(z[2:n],z[2:n]*X[1:(n-1)]^2,z[2:n]*s2[1:(n-1)])+ H=(t(V)%*%V)+ G=apply(V,2,sum)+ theta=theta+solve(H)%*%G+ j=j+1}&#62; as.numeric(theta)[1] 0.20372918 0.59183911 0.08936159The interesting point, here, is that we also derive the (asymptotic) variance&#62; (stdev=sqrt(diag(solve(H))))[1] 0.01849067 0.04950477 0.02937233