"Frontmatter". In: Analysis of Financial Time Series

354 VECTOR TIME SERIES3. Σ 12 = 0 if and only if x 1 and x 2 are independent.4. The random variable y = (x − µ) ′ Σ −1 (x − µ) follows a chi-squared distributionwith m degrees of freedom.5. The conditional distribution of x 1 given x 2 = b is also normally distributed as(x 1 | x 2 = b) ∼ N p [µ 1 + Σ 12 Σ −122 (b − µ 2), Σ 11 − Σ 12 Σ −122 Σ 21].The last property is useful in many scientific areas. For instance, it forms the basisfor time series forecasting under the normality assumption and for recursive leastsquares estimation.EXERCISES1. Consider the monthly log stock returns, in percentages and including dividends,of Merck & Company, Johnson & Johnson, General Electric, General Motors,Ford Motor Company, and value-weighted index from January 1960 to December1999; see the file “m-mrk2vw.dat,” which has six columns in the order listedbefore.(a) Compute the sample mean, covariance matrix, and correlation matrix of thedata.(b) Test the hypothesis H 0 : ρ 1 = ··· = ρ 6 = 0, whereρ i is the lag-i crosscorrelationmatrix of the data. Draw conclusion based on the 5% significancelevel.(c) Is there any lead-lag relationship among the six return series?(d) Perform a principal component analysis of the data using the sample covariancematrix.(e) Perform a principal component analysis of the data using the sample correlationmatrix.(f) Perform a factor analysis on the data. Identify the number of common factors.Obtain estimates of factor loadings using both the principal component andmaximum likelihood methods.2. The Federal Reserve Bank of St Louis publishes selected interest rates and U.S.financial data on its Web site:http://www.stls.frb.org/fred/index.htmlConsider the monthly 1-year and 10-year Treasury constant maturity rates fromApril 1953 to October 2000 for 571 observations; see the file “m-gs1n10.dat.”The rates are in percentages.(a) Let c t = r t − r t−1 be the change series of the monthly interest rate r t . Builda bivariate autoregressive model for the two change series. Discuss the implicationsof the model. Transform the model into a structural form.