"Frontmatter". In: Analysis of Financial Time Series

CORRELATION AND AUTOCORRELATION FUNCTION 25result can be used to perform the hypothesis testing of H o : ρ l = 0vsH a : ρ l ̸= 0.For more information about the asymptotic distribution of sample autocorrelations,see Fuller (1976, Chapter 6) and Brockwell and Davis (1991, Chapter 7).In finite samples, ˆρ l is a biased estimator of ρ l . The bias is in the order of 1/T ,which can be substantial when the sample size T is small. In most financial applications,T is relatively large so that the bias is not serious.Portmanteau TestFinancial applications often require to test jointly that several autocorrelations of r tare zero. Box and Pierce (1970) propose the Portmanteau statisticQ ∗ (m) = Tm∑ˆρ l2l=1as a test statistic for the null hypothesis H o : ρ 1 = ··· = ρ m = 0 against thealternative hypothesis H a : ρ i ̸= 0forsomei ∈{1,...,m}. Under the assumptionthat {r t } is an iid sequence with certain moment conditions, Q ∗ (m) is asymptoticallya chi-squared random variable with m degrees of freedom.Ljung and Box (1978) modify the Q ∗ (m) statistic as below to increase the powerof the test in finite samples,Q(m) = T (T + 2)m∑l=1ˆρ 2 lT − l . (2.3)In practice, the selection of m may affect the performance of the Q(m) statistic.Several values of m are often used. Simulation studies suggest that the choice ofm ≈ ln(T ) provides better power performance.The function ˆρ 1 , ˆρ 2 ,...is called the sample autocorrelation function (ACF) of r t .It plays an important role in linear time series analysis. As a matter of fact, a lineartime series model can be characterized by its ACF, and linear time series modelingmakes use of the sample ACF to capture the linear dynamic of the data. Figure 2.1shows the sample autocorrelation functions of monthly simple and log returns ofIBM stock from January 1926 to December 1997. The two sample ACFs are veryclose to each other, and they suggest that the serial correlations of monthly IBM stockreturns are very small, if any. The sample ACFs are all within their two standard-errorlimits, indicating that they are not significant at the 5% level. In addition, for thesimple returns, the Ljung–Box statistics give Q(5) = 5.4 andQ(10) = 14.1, whichcorrespond to p value of 0.37 and 0.17, respectively, based on chi-squared distributionswith 5 and 10 degrees of freedom. For the log returns, we have Q(5) = 5.8andQ(10) = 13.7 with p value 0.33 and 0.19, respectively. The joint tests confirm thatmonthly IBM stock returns have no significant serial correlations. Figure 2.2 showsthe same for the monthly returns of the value-weighted index from the Center forResearch in Security Prices (CRSP), University of Chicago. There are some significantserial correlations at the 5% level for both return series. The Ljung–Box statisticsgive Q(5) = 27.8 andQ(10) = 36.0 for the simple returns and Q(5) = 26.9