The Autocorrelation Function (ACF) Advanced Answer.

The j-th order autocorrelation function is defined as - correct answer rho(j) = gamma(j)/gamma(0) = ([E(X(t)-mew)*E(X(t-j)-mew)])/(E[(X(t) - mew)^2]) = (Cov(X(t),X(t-j))/Var(X(t) for j = 0,1,2,... with -1 = rho(j) = 1 The plot of rho(j) against j = 0,1,2,... is called - correct answer the correlogram For a stationary ergodic process, the ACF can be estimated by the sample moments as follows - correct answer y bar = (1/T) * Summation from t=1 to T of y(t) sigma hat^2 = (1/T) * Summation from t=1 to T of (y(t) - y bar)^2 for each j: r(j) = (Summation from t=j+1 to T of (y(t)-y bar)*(y(t-j)-y bar))/(Summation from t=1 to T of (y(t)-y bar)^2) Sample variance of r(j): Var(r(j)) = T^-1 for j=1 Var(r(j)) = T^-1 * (1+2*summation from k=1 to j-1 of (r(k)^2)) for j 1 Testing for autocorrelation - correct answer Also known as testing the null hypothesis that autocorrelations are significantly different from zero H(0): r(j) = 0 for j = 1,2,3,...,k and k T **Can be done with Box and Pierce (1970) or Ljung-Box (1978) Autocorrelation - correct answer In a longitudinal design, the correlation of one variable with itself, measured at two different times. Box-Pierce (1970) statistic - correct answer Q = T * Summation from j=1 to k of r(j)^2 Q ~ X^2(k) under H(0) Ljung-Box (1978) statistic - correct answer Q = T(T+2) * Summation from j=1 to k of r(j)^2 / (T - j) Q ~ X^2(k) under H(0) Box-Pierce and Ljung-Box in R - correct answer Examining null hypothesis of independence in a given time series ('portmanteau tests') B(x, lag=1, type=c("Box-Pierce","Ljung-Box") fitdf=0) x = numeric vector or univariate time series lag = statistic based on lag autocorrelation coefficients type = test to be performed: partial matching used fitdf = number of degrees of freedom to be subtracted if x is series of residuals **Sometimes applied to the residuals from an ARMA (p, q) fit, in which case the references suggest a better approximation to the null-hypothesis distribution is obtained by setting fitdf = p+q, provided lag fitdf -- Gives value = a list with class "htest" and following components: -statistic = value of test statistic