In this appendix, we give the SCA commands used in Section 2.9. The 1-year maturity interest rates are in the file w-gs1yr.txt and the 3-year rates are in the file w-gs3yr.txt.
-- load the data into SCA, denote the data by rate1 and rate3.
input year,mom,day,rate1. file ‘w-gs1yr.txt’
--
input year,mon,day,rate3. file ‘w-gs3yr.txt’
-- specify a simple linear regression model.
tsm m1. model rate3=b0+(b1)rate1+noise.
-- estimate the specified model and store residual in r1.
estim m1. hold resi(r1).
-- compute 10 lags of residual acf.
acf r1. maxl 10.
-- difference the two series, denote the new series by c1t
and c3t
diff old rate1,rate3. new c1t, c3t. compress.
-- specify a linear regression model for the differenced data
tsm m2. model c3t=h0+(h1)c1t+noise.
-- estimation
estim m2. hold resi(r2).
-- compute residual acf.
acf r2. maxl 10.
-- specify a regression model with time series errors.
tsm m3. model c3t=g0+(g1)c1t+(1)noise.
-- estimate the model using the exact likelihood method.
estim m3. method exact. hold resi(r3).
-- compute residual acf.
acf r3. maxl 10.
-- refine the model to include more MA lags.
tsm m4. model c3t=g0+(g1)c1t+(1,4,6,7)noise.
-- estimation
estim m4. method exact. hold resi(r4).
-- compute residual acf.
acf r4. maxl 10.
-- exit SCA
stop
Exercises
If not specifically specified, use 5% significance level to draw conclusions in the exercises.
2.1 Suppose that the simple return of a monthly bond index follows the MA(1) model
Assume that a100 = 0.01. Compute the 1-step- and 2-step-ahead forecasts of the return at the forecast origin t = 100. What are the standard deviations of the associated forecast errors? Also compute the lag-1 and lag-2 autocorrelations of the return series.
2.2 Suppose that the daily log return of a security follows the model
where {at} is a Gaussian white noise series with mean zero and variance 0.02. What are the mean and variance of the return series rt? Compute the lag-1 and lag-2 autocorrelations of rt. Assume that r100 = − 0.01, and r99 = 0.02. Compute the 1- and 2-step-ahead forecasts of the return series at the forecast origin t = 100. What are the associated standard deviations of the forecast errors?
2.3 Consider the monthly U.S. unemployment rate from January 1948 to March 2009 in the file m-unrate.txt. The data are seasonally adjusted and obtained from the Federal Reserve Bank of St Louis. Build a time series model for the series and use the model to forecast the unemployment rate for the April, May, June, and July of 2009. In addition, does the fitted model imply the existence of business cycles? Why? (Note that there are more than one model fits the data well. You only need an adequate model.)
2.4 Consider the monthly simple returns of the Decile 1, Decile 2, Decile 9, and Decile 10 of NYSE/AMEX/NASDAQ based on market capitalization. The data span is from January 1970 to December 2008, and the data are obtained from CRSP.
a. For the return series of Decile 2 and Decile 10, test the null hypothesis that the first 12 lags of autocorrelations are zero at the 5% level. Draw your conclusion.
b. Build an ARMA model for the return series of Decile 2. Perform model checking and write down the fitted model.
c. Use the fitted ARMA model to produce 1- to 12-step-ahead forecasts of the series and the associated standard errors of forecasts.
2.5 Consider the daily simple returns of IBM stock from 1970 to 2008 in the file d-ibm3dx7008.txt. Compute the first 100 lags of ACF of the absolute series of daily simple returns of IBM stock. Is there evidence of long-range dependence? Why?
2.6 Consider the demand of electricity of a manufacturing sector in the United States. The data are logged, denote the demand of a fixed day of each month, and are in power6.txt. Build a time series model for the series and use the fitted model to produce 1- to 24-step-ahead forecasts.
2.7 Consider the daily simple returns of IBM stock, CRSP value-weighted index, CRSP equal-weighted index, and the S&P composite index from January 1980 to December 2008. The index returns include dividend distributions. The data file is d-ibm3dxwkdays8008.txt, which has 12 columns. The columns are (year, month, day, IBM, VW, EW, SP, M, T, W, H, F), where M, T, W, R, and F denotes indicator variables for Monday to Friday, respectively. Use a regression model to study the effects of trading days on the equal-weighted index returns. What is the fitted model? Are the weekday effects significant in the returns at the 5% level? Use the HAC estimator of the covariance matrix to obtain the t ratio of regression estimates. Does the HAC estimator change the conclusion of weekday effects? Are there serial correlations in the regression residuals? If yes, build a regression model with time series error to study weekday effects.
2.8 Consider the data set of the previous question, but focus on the daily simple returns of the S&P composite index. Perform the necessary data analysis and statistical tests using the 5% significance level to answer the following questions:
a. Is there any weekday effect on the daily simple returns of the S&P composite index? You may employ a linear regression model to answer this question. Estimate the model, check its validity, and test the hypothesis that there is no Friday effect. Draw your conclusion.
b. Check the residual serial correlations using Q(12) statistic. Are there any significant serial correlations in the residuals? If yes, build a regression model with time series errors for the data.
2.9 Now consider similar questions of the previous exercise for the IBM stock returns.
a. Is there any weekday effect on the daily simple returns of IBM stock? Estimate your model and test the hypothesis that there is no Friday effect. Draw your conclusion.
b. Are there serial correlations in the residuals? Use Q(12) to perform the test. Draw your conclusion.
c. Refine the above model by using the technique of regression model with time series errors. In there a significant weekday effect based on the refined model?
2.10 Consider the weekly yields of Moody's Aaa and Baa seasoned bonds from January 5, 1962, to April 10, 2009. The data are obtained from the Federal Reserve Bank of St Louis. Weekly yields are averages of daily yields. Obtain the summary statistics (sample mean, standard deviation, skewness, excess kurtosis, minimum, and maximum) of the two yield series. Are the bond yields skewed? Do they have heavy tails? Answer the questions using 5% significance level.
2.11 Consider the monthly Aaa bond yields of the prior problem. Build a time series model for the series.
2.12 Again, consider the two bond yield series, that is, Aaa and Baa. What is the relationship between the two series? To answer this question, build a time series model using yields of Aaa bonds as the dependent variable and yields of Baa bonds as independent variable.
2.13 Consider the monthly log returns of CRSP equal-weighted index from January 1962 to December 1999 for 456 observations. You may obtain the data from CRSP directly or from the file m-ew6299.txt on the Web.
a. Build an AR model for the series and check the fitted model.
b. Build an MA model for the series and check the fitted model.
c. Compute 1- and 2-step-ahead forecasts of the AR and MA models built in the previous two questions.
d. Compare the fitted AR and MA models.
2.14 This problem is concerned with the dynamic relationship between the spot and futures prices of the S&P 500 index. The data file sp5may.dat has three columns: log(futures price), log(spot price), and cost-of-carry ( × 100). The data were obtained from the Chicago Mercantile Exchange for the S&P 500 stock index in May 1993 and its June futures contract. The time interval is 1 minute (intraday). Several authors used the data to study index futures arbitrage. Here we focus on the first two columns. Let ft and st be the log prices of futures and spot, respectively. Consider yt = ft − ft−1 and xt = st − st−1. Build a regression model with time series errors between {yt} and {xt}, with yt being the dependent variable.
2.15 The quarterly gross domestic product implicit price deflator is often used as a measure of inflation. The file q-gdpdef.txt contains the data for the United States from the first quarter of 1947 to the last quarter of 2008. Data format is year, month, day, and deflator. The data are seasonally adjusted and equal to 100 for year 2000. Build an ARIMA model for the series and check the validity of the fitted model. Use the fitted model to predict the inflation for each quarter of 2009. The data are obtained from the Federal Reserve Bank of St Louis.
Akaike, H. (1973). Information theory and an extension of the maximum likelihood principle. InB. N. Petrov andF. Csaki, (eds.). 2nd International Symposium on Information Theory, pp. 267–281. Akademia Kiado, Budapest.
Box, G. E. P. and Pierce, D. (1970). Distribution of residual autocorrelations in autoregressive-integrated moving average time series models. Journal of the American Statistical Association 65: 1509–1526.
Box, G. E. P., Jenkins, G. M., and Reinsel, G. C. (1994). Time Series Analysis: Forecasting and Control,3rd ed. Prentice Hall, Englewood Cliffs, NJ.
Brockwell, P. J. and Davis, R. A. (1991). Time Series: Theory and Methods,2nd ed. Springer, New York.
Brockwell, P. J. and Davis, R. A. (1996). Introduction to Time Series and Forecasting. Springer, New York.
Chan, N. H. and Wei, C. Z. (1988). Limiting distributions of least squares estimates of unstable autoregressive processes. Annals of Statistics 16: 367–401.
Dickey, D. A. and Fuller, W. A. (1979). Distribution of the estimates for autoregressive time series with a unit root. Journal of the American Statistical Association 74: 427–431.
Ding, Z., Granger, C. W. J., and Engle, R. F. (1993). A long memory property of stock returns and a new model. Journal of Empirical Finance 1: 83–106.
Eicker, F. (1967). Limit theorems for regression with unequal and dependent Errors. InL. LeCam andJ. Neyman (eds.). Proceedings of the 5th Berkeley Symposium on Mathematical Statistics and Probability. University of California Press, Berkeley.
Fuller, W. A. (1976). Introduction to Statistical Time Series, Wiley, New York.
Greene, W. H. (2003). Econometric Analysis,5th ed. Prentice-Hall, Upper Saddle River, NJ.
Hosking, J. R. M. (1981). Fractional differencing. Biometrika 68: 165–176.
Ljung, G. and Box, G. E. P. (1978). On a measure of lack of fit in time series models. Biometrika 66: 67–72.
Newey, W. and West, K. (1987). A simple positive semidefinite, heteroscedasticity and autocorrelation consistent covariance matrix. Econometrica 55: 863–898.
Phillips, P. C. B. (1987). Time series regression with a unit root. Econometrica 55: 277–301.
Shumway, R. H. and Stoffer, D. S. (2000). Time Series Analysis and Its Applications. Springer, New York.
Tiao, G. C. and Tsay, R. S. (1983). Consistency properties of least squares estimates of autoregressive parameters in ARMA models. Annals of Statistics 11: 856–871.
Tsay, R. S. and Tiao, G. C. (1984). Consistent estimates of autoregressive parameters and extended sample autocorrelation function for stationary and nonstationary ARMA models. Journal of the American Statistical Association 79: 84–96.
White, H. (1980). A heteroscedasticity consistent covariance matrix estimator and a direct test for heteroscedasticity. Econometrica 48: 827–838.