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The Basics: Outline

The Basics: Outline. What is a time series? What is a financial time series? What is the purpose of our analysis?  Classification of Time Series. Correlation Autocorrelation Partial Autocorrelation Cross Correlation Basic transformation to stationarity Differencing. Review

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The Basics: Outline

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  1. The Basics: Outline • What is a time series? What is a financial time series? • What is the purpose of our analysis?  • Classification of Time Series. • Correlation • Autocorrelation • Partial Autocorrelation • Cross Correlation • Basic transformation to stationarity • Differencing K. Ensor, STAT 421

  2. Review Random variable Distribution (cdf, pdf) Moments Mean Variance Covariance Correlation Skewness Kurtosis Time Series Random process – random variable is a function of time Distribution? Moments Mean Variance Covariance Correlation Skewness Kurtosis What is a time series? K. Ensor, STAT 421

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  5. Further examples of a time series • Anything observed sequentially (by time?) • Returns, volatility, interest rates, exchange rates, bond yields, … • Hourly temperature, hourly ozone levels • ??? K. Ensor, STAT 421

  6. What is different? • The observations are not independent. • There is correlation from observation to observation. • Consider the log of the J&J series. • Is there correlation in the observations over time? K. Ensor, STAT 421

  7. What are our objectives? • Making decisions based on the observed realization requires: • Descriptive: Estimating summary measures (e.g. mean) • Inferential: Understanding / Modeling • Prediction / Forecasting • Control of the process • If correlation is present between the observations then our typical approaches are not correct (as they assume iid samples). K. Ensor, STAT 421

  8. Dimension of T Time, space, space-time Nature of T Discrete Equally Unequally spaced Continuous Observed continuously Observed by some random process Dimension of X Univariate Multivariate State spce Discrete Continuous Memory types Stationary No memory Short memory Long memory Nonstationary Classification of a Time Series K. Ensor, STAT 421

  9. Stationarity Strictly Stationary All finite dimensional distributions are the same. What does stationarity provide? Covariance Stationary First and second moment structure does not change with time. K. Ensor, STAT 421

  10. Autocorrelation K. Ensor, STAT 421

  11. Autocorrelation Function for a CSTS • In theory… • How to estimate this quantity? K. Ensor, STAT 421

  12. Autocorrelation? How would you determine or show correlation over time? K. Ensor, STAT 421

  13. Sample ACF and PACF • Sample ACF – sample estimate of the autocorrelation function. • Substitute sample estimates of the covariance between X(t) and X(t+h). Note: We do not have “n” pairs but “n-h” pairs. • Subsitute sample estimate of variance. • Sample PACF – correlation between observations X(t) and X(t+h) after removing the linear relationship of all observations in that fall between X(t) and X(t+h). K. Ensor, STAT 421

  14. Summary Plots K. Ensor, STAT 421

  15. Cross Correlation K. Ensor, STAT 421

  16. How can we study the relationship between 2 or more time series? U.S. weekly interest rate series measured in percentages Time: From 1/5/1962 to 9/10/1999. Variables: r1(t) = The 1-year Treasury constant maturity rate r2(t) = The 3-year Treasury constant maturity rate And the corresponding change series c1(t)=(1-B)r1(t) c2(t)=(1-B)r2(t) Multivariate Series K. Ensor, STAT 421

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  18. Scatterplots • between series simultaneous in time • and the change in each series. • The two series are highly correlated. K. Ensor, STAT 421

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  20. What is the cross-correlation between the two series? K. Ensor, STAT 421

  21. Differencing to achieve Stationarity K. Ensor, STAT 421

  22. Detrending by taking first difference. Y(t)=X(t) – X(t-1) What happens to the trend? Suppose X(t)=a+bt+Z(t) Z(t) is a random variable. K. Ensor, STAT 421

  23. Sumary Plots of Detrended J&J log earnings per share. K. Ensor, STAT 421

  24. Removing Seasonal Trend – one way to proceed. • Suppose Y(t)=g(t)+W(t) where g(t)=g(t-s) where s is our “season” for all t. W(t) is again a new random variable • Form a new series U(t) by taking the “s” difference U(t)=Y(t)-Y(t-s) =g(t)-g(t-s) + W(t)-W(t-s) =W(t)-W(t-s) again a random variable K. Ensor, STAT 421

  25. Summary of Transformed J&J Series K. Ensor, STAT 421

  26. Summary of Transformations: • X(t) = log (Q(t)) • Y(t)=X(t)-X(t-1) = (1-B)X(t) • U(t)= (1-B4)Y(t) • U(t)=(1-B4) (1-B)X(t) K. Ensor, STAT 421

  27. An example of Forecasting K. Ensor, STAT 421

  28. What is the next step? • U(t) is a time series process called a moving average of order 1 (or possibly a MA(1) plus a seasonal MA(1)) • U(t)=q e(t-1) + e(t) • Proceed to estimate q and then we can estimate summary information about the earnings per share as well as predict the future earnings per share. K. Ensor, STAT 421

  29. Forecast of J&J series K. Ensor, STAT 421

  30. Wrap up • Basics of distribution theory. • Classification of time series. • Basics of stationarity. • Correlation functions • Autocorrelation • Partial autocorrelation • Cross correlation • Transformations to a stationary series • differencing K. Ensor, STAT 421

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