Timeseries_presentation.ppt

Timeseries_presentation.ppt

• 1.
  An Introduction toTime Series Ginger Davis VIGRE Computational Finance Seminar Rice University November 26, 2003
• 2.
  What is aTime Series? • Time Series – Collection of observations indexed by the date of each observation • Lag Operator – Represented by the symbol L • Mean of Yt = μt   T y y y , , , 2 1  1   t t x Lx
• 3.
  White Noise Process •Basic building block for time series processes         0 0 2 2              t t t t t E E E
• 4.
  White Noise Processes,cont. • Independent White Noise Process – Slightly stronger condition that and are independent • Gaussian White Noise Process   2 , 0 ~   N t t   
• 5.
  Autocovariance • Covariance ofYt with its own lagged value • Example: Calculate autocovariances for:    j t j t t t jt Y Y E              j t t j t t jt t t E Y Y E Y               
• 6.
  Stationarity • Covariance-stationary orweakly stationary process – Neither the mean nor the autocovariances depend on the date t      j j t t t Y Y E Y E         
• 7.
  Stationarity, cont. • 2processes – 1 covariance stationary, 1 not covariance stationary t t t t t Y Y        
• 8.
  Stationarity, cont. • Covariancestationary processes – Covariance between Yt and Yt-j depends only on j (length of time separating the observations) and not on t (date of the observation) j j    
• 9.
  Stationarity, cont. • Strictstationarity – For any values of j1, j2, …, jn, the joint distribution of (Yt, Yt+j1 , Yt+j2 , ..., Yt+jn ) depends only on the intervals separating the dates and not on the date itself
• 10.
  Gaussian Processes • Gaussianprocess {Yt} – Joint density is Gaussian for any • What can be said about a covariance stationary Gaussian process?   n n j t j t j t j t t Y Y Y y y y f     , , , 1 1 1 , , ,   n j j j , , , 2 1 
• 11.
  Ergodicity • A covariance-stationaryprocess is said to be ergodic for the mean if converges in probability to E(Yt) as    T t t y T y 1 1   T
• 12.
  Describing the dynamics ofa Time Series • Moving Average (MA) processes • Autoregressive (AR) processes • Autoregressive / Moving Average (ARMA) processes • Autoregressive conditional heteroscedastic (ARCH) processes
• 13.
  Moving Average Processes •MA(1): First Order MA process • “moving average” – Yt is constructed from a weighted sum of the two most recent values of . 1     t t t Y    
• 14.
  Properties of MA(1)                     0 1 2 2 2 1 2 2 2 1 1 2 1 1 1 2 2 2 1 2 1 2 2 1 2                                                                j t t t t t t t t t t t t t t t t t t t t t t t Y Y E E E Y Y E E E Y E Y E for j>1
• 15.
  MA(1) • Covariance stationary –Mean and autocovariances are not functions of time • Autocorrelation of a covariance-stationary process • MA(1) 0    j j      2 2 2 2 1 1 1          
• 16.
  Autocorrelation Function forWhite Noise: 0.0 0.2 0.4 0.6 0.8 1.0 0 5 10 15 20 Lag Autocorrelation t t Y  
• 17.
  Autocorrelation Function forMA(1): 1 8 . 0    t t t Y   0.0 0.2 0.4 0.6 0.8 1.0 0 5 10 15 20 Lag Autocorrelation
• 18.
  Moving Average Processes ofhigher order • MA(q): qth order moving average process • Properties of MA(q) q t q t t t t Y                   2 2 1 1     q j q j j j q q j j j j q                 , 0 , , 2 , 1 , 1 2 2 2 1 1 2 2 2 2 2 1 0                  
• 19.
  Autoregressive Processes • AR(1):First order autoregression • Stationarity: We will assume • Can represent as an MA t t t Y c Y      1 1                                    2 2 1 2 2 1 1 t t t t t t t c c c c Y           : ) (
• 20.
  Properties of AR(1)          2 2 2 4 2 2 2 2 1 2 0 1 1 1                                t t t t E Y E c
• 21.
  Properties of AR(1),cont.              j j j j j j j j j t j t j t j t j t t t j t t j E Y Y E                                                                       0 2 2 2 4 2 2 4 2 2 2 1 2 2 1 1 1     
• 22.
  Autocorrelation Function forAR(1): t t t Y Y    1 8 . 0 0.0 0.2 0.4 0.6 0.8 1.0 0 5 10 15 20 Lag Autocorrelation
• 23.
  Autocorrelation Function forAR(1): t t t Y Y     1 8 . 0 -0.5 0.0 0.5 1.0 0 5 10 15 20 Lag Autocorrelation
• 24.
  Gaussian White Noise 020 40 60 80 100 -2 -1 0 1 2
• 25.
  AR(1), 0 20 4060 80 100 -3 -2 -1 0 1 2 5 . 0  
• 26.
  AR(1), 0 20 4060 80 100 -2 0 2 4 9 . 0  
• 27.
  AR(1), 0 20 4060 80 100 -4 -2 0 2 4 9 . 0   
• 28.
  Autoregressive Processes of higherorder • pth order autoregression: AR(p) • Stationarity: We will assume that the roots of the following all lie outside the unit circle. t p t p t t t Y Y Y c Y               2 2 1 1 0 1 2 2 1      p p z z z    
• 29.
  Properties of AR(p) •Can solve for autocovariances / autocorrelations using Yule-Walker equations   p c           2 1 1
• 30.
  Mixed Autoregressive Moving AverageProcesses • ARMA(p,q) includes both autoregressive and moving average terms q t q t t t p t p t t t Y Y Y c Y                             2 2 1 1 2 2 1 1
• 31.
  Time Series Models forFinancial Data • A Motivating Example – Federal Funds rate – We are interested in forecasting not only the level of the series, but also its variance. – Variance is not constant over time
• 32.
  U. S. FederalFunds Rate Time 1955 1960 1965 1970 1975 2 4 6 8 10 12
• 33.
  Modeling the Variance •AR(p): • ARCH(m) – Autoregressive conditional heteroscedastic process of order m – Square of ut follows an AR(m) process – wt is a new white noise process t p t p t t t u y y y c y              2 2 1 1 t m t m t t t w u u u u          2 2 2 2 2 1 1 2     
• 34.
  References • Investopia.com • Economagic.com •Hamilton, J. D. (1994), Time Series Analysis, Princeton, New Jersey: Princeton University Press.