The Basics: OutlineWhat is a time series?PowerPoint PresentationSlide 4Further examples of a time seriesWhat is different?What are our objectives?Classification of a Time SeriesStationaritySlide 10Autocorrelation Function for a CSTSAutocorrelation?Sample ACF and PACFSummary PlotsSlide 15Multivariate SeriesSlide 17Slide 18Slide 19Slide 20Slide 21Detrending by taking first difference.Sumary Plots of Detrended J&J log earnings per share.Removing Seasonal Trend – one way to proceed.Summary of Transformed J&J SeriesSummary of Transformations:Slide 27What is the next step?Forecast of J&J seriesWrap upK. Ensor, STAT 4211Spring 2005The 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–DifferencingK. Ensor, STAT 4212Spring 2005What is a time series?•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•KurtosisK. Ensor, STAT 4213Spring 20050 10 20 30-4 -2 0 2Stationary Time Series with 100 Future RealizationsK. Ensor, STAT 4214Spring 2005Time in weekspercent4 6 8 10 12 14 1601/05/1962 07/18/1969 01/28/1977 08/10/1984 02/21/1992 09/03/1999U.S. Weekly Interest Rates Red line is 1-year; Blue line is 3-yearK. Ensor, STAT 4215Spring 2005Further examples of a time series•Anything observed sequentially (by time?)•Returns, volatility, interest rates, exchange rates, bond yields, …•Hourly temperature, hourly ozone levels•???Time in quartersearning0 5 10 15Jan 60 Jan 64 Jan 68 Jan 72 Jan 76 Jan 80Quarterly Earning per shar Johnson and JohnsonK. Ensor, STAT 4216Spring 2005What 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?Time in quarterslog earning0 1 2Jan 60 Jan 64 Jan 68 Jan 72 Jan 76 Jan 80Log Quarterly Earning per share Johnson and JohnsonK. Ensor, STAT 4217Spring 2005What 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 4218Spring 2005Classification of a Time Series•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–NonstationaryK. Ensor, STAT 4219Spring 2005StationarityStrictly StationaryAll finite dimensional distributions are the same.First and second moment structure does not change with time.Covariance StationaryWhat does stationarity provide?K. Ensor, STAT 42110Spring 2005AutocorrelationK. Ensor, STAT 42111Spring 2005Autocorrelation Function for a CSTS•In theory…•How to estimate this quantity?K. Ensor, STAT 42112Spring 2005Autocorrelation?How would you determine or show correlation over time? lagged 1Series 10 1 20 1 2lagged 2Series 10 1 20 1 2lagged 3Series 10 1 20 1 2lagged 4Series 10 1 20 1 2lagged 5Series 10 1 20 1 2lagged 6Series 10 1 20 1 2lagged 7Series 10 1 20 1 2lagged 8Series 10 1 20 1 2lagged 9Series 10 1 20 1 2lag ged 10Series 10 1 20 1 2Lagged Scatterplots : xLagged Scatterplots : xK. Ensor, STAT 42113Spring 2005Sample 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 42114Spring 2005Summary PlotsTime in quarters0 1 2Jan 60 Jan 64 Jan 68 Jan 72 Jan 76 Jan 80Log Quarterly Earnings for J&J-1 0 1 2 30 5 10 15HistogramLog Quarterly Earnings for J&J LagACF0 5 10 15-1.0 -0.5 0.0 0.5 1.0ACFLagACF0 5 10 15-1.0 -0.5 0.0 0.5 1.0PACFK. Ensor, STAT 42115Spring 2005Cross CorrelationK. Ensor, STAT 42116Spring 2005Multivariate Series•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)K. Ensor, STAT 42117Spring 2005Time in weekspercent4 6 8 10 12 14 1601/05/1962 07/18/1969 01/28/1977 08/10/1984 02/21/1992 09/03/1999U.S. Weekly Interest Rates Red line is 1-year; Blue line is 3-yearK. Ensor, STAT 42118Spring 2005yr1yr34 6 8 10 12 14 164 6 8 10 12 14 16Scatterplot of U.S. Weekly Interest Ratediff(yr1)diff(yr3)-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5-1.0 -0.5 0.0 0.5 1.0 1.5Scatterplot of Change 1yr rate and 3yr rateScatterplots•between series simultaneous in time•and the change in each series.The two series are highly correlated.K. Ensor, STAT 42119Spring 2005Change in 1-year rateTime0 500 1000 1500 2000-1.5 -0.5 0.5 1.5Change in 3-year rate0 500 1000 1500 2000-1.0 0.0 0.5 1.0 1.5K. Ensor, STAT 42120Spring 2005c1c3-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5-1.0 0.0 0.5 1.0 1.5Scatterplot of Change in 1-year and 3-year rate c1 ACF0 5 10 15 20 25 300.0 0.2 0.4 0.6 0.8 1.0 c1 and c30 5 10 15 20 25 300.0 0.2 0.4 0.6 0.8 c3 and c1LagACF-30 -25 -20 -15 -10 -5 00.0 0.2 0.4 0.6 0.8 c3 Lag0 5 10 15 20 25 300.0 0.2 0.4 0.6 0.8 1.0Multivariate Series : cbind(c1, c3)What is the cross-correlation between the two series?K. Ensor, STAT 42121Spring 2005Differencing to achieve StationarityK. Ensor, STAT 42122Spring 2005Detrending by taking first difference.Time in quartersfirst difference of log earning-0.6 -0.4 -0.2 0.0 0.2 0.4Apr 60 Apr 64 Apr 68 Apr 72 Apr 76 Apr 80First Difference Log Quarterly Earning per share J&JY(t)=X(t) – X(t-1)What happens to