Introduction to time series analysis ppt

components of time series analysis ppt and time series analysis and its applications ppt
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Time Series Analysis Lecture 8: Time Series Analysis MIT 18.S096 Dr. Kempthorne Fall 2013 MIT 18.S096 Time Series Analysis 1Stationarity and Wold Representation Theorem Autoregressive and Moving Average (ARMA) Models Time Series Analysis Accommodating Non-Stationarity: ARIMA Models Estimation of Stationary ARMA Models Tests for Stationarity/Non-Stationarity Outline 1 Time Series Analysis Stationarity and Wold Representation Theorem Autoregressive and Moving Average (ARMA) Models Accommodating Non-Stationarity: ARIMA Models Estimation of Stationary ARMA Models Tests for Stationarity/Non-Stationarity MIT 18.S096 Time Series Analysis 2Stationarity and Wold Representation Theorem Autoregressive and Moving Average (ARMA) Models Time Series Analysis Accommodating Non-Stationarity: ARIMA Models Estimation of Stationary ARMA Models Tests for Stationarity/Non-Stationarity Stationarity and Wold Representation Theorem A stochastic processf:::;X ;X ;X ;:::g consisting of random t1 t t+1 variables indexed by time index t is a time series. The stochastic behavior offXg is determined by specifying the t probability density/mass functions (pdf's) p(x ;x ;:::;x ) t t t 1 2 m for all nite collections of time indexes f(t ;t ;:::;t ); m1g 1 2 m i.e., all nite-dimensional distributions offXg. t De nition: A time seriesfXg is Strictly Stationary if t p(t +;t +;:::;t +) = p(t ;t ;:::;t ); 1 2 m 1 2 m 8;8m,8(t ;t ;:::;t ). 1 2 m (Invariance under time translation) MIT 18.S096 Time Series Analysis 3Stationarity and Wold Representation Theorem Autoregressive and Moving Average (ARMA) Models Time Series Analysis Accommodating Non-Stationarity: ARIMA Models Estimation of Stationary ARMA Models Tests for Stationarity/Non-Stationarity De nitions of Stationarity De nition: A time seriesfXg is Covariance Stationary if t E (X ) =  t 2 Var(X ) =  t X Cov(X ;X ) = () t t+ (all constant over time t) The auto-correlation function offXg is t p () = Cov(X ;X )= Var(X )Var(X ) t t+ t t+ = ()= (0) MIT 18.S096 Time Series Analysis 4Stationarity and Wold Representation Theorem Autoregressive and Moving Average (ARMA) Models Time Series Analysis Accommodating Non-Stationarity: ARIMA Models Estimation of Stationary ARMA Models Tests for Stationarity/Non-Stationarity Representation Theorem Wold Representation Theorem: Any zero-mean covariance stationary time seriesfXg can be decomposed as X = V +S t t t t where fVg is a linearly deterministic process, i.e., a linear t combination of past values of V with constant coecients. t P 1 S =  is an in nite moving average process of t i ti i=0 error terms, where P 1 2 = 1; 1 0 i=0 i fg is linearly unpredictable white noise, i.e., t 2 2 E ( ) = 0; E ( ) = , E (  ) = 08t,8s =6 t, t t s t andfg is uncorrelated withfVg : t t E ( V ) = 0,8t;s t s MIT 18.S096 Time Series Analysis 5Stationarity and Wold Representation Theorem Autoregressive and Moving Average (ARMA) Models Time Series Analysis Accommodating Non-Stationarity: ARIMA Models Estimation of Stationary ARMA Models Tests for Stationarity/Non-Stationarity Intuitive Application of the Wold Representation Theorem Suppose we want to specify a covariance stationary time series fXg to model actual data from a real time series t fx ;t = 0; 1;:::;Tg t Consider the following strategy: Initialize a parameter p, the number of past observations in the linearly deterministic term of the Wold Decomposition of fXg t Estimate the linear projection of X on (X ;X ;:::;X ) t t1 t2 tp Consider an estimation sample of size n with endpoint t  T . 0 Letfj =(p 1);:::; 0; 1; 2;:::ng index the subseries of ft = 0; 1;:::;Tg corresponding to the estimation sample and de nefy : y = x g, (with t  n +p) j j tn+j 0 0 De ne the vector Y (T  1) and matrix Z (T  p + 1) as: MIT 18.S096 Time Series Analysis 6Stationarity and Wold Representation Theorem Autoregressive and Moving Average (ARMA) Models Time Series Analysis Accommodating Non-Stationarity: ARIMA Models Estimation of Stationary ARMA Models Tests for Stationarity/Non-Stationarity Estimate the linear projection of X on (X ;X ;:::;X ) t t1 t2 tp (continued) 0 1 2 3 1 y y  y y 1 0 1 (p1) B C 6 7 y 1 y y  y 2 1 0 (p2) B C 6 7 y =B C Z =6 7 . . . . . . . . . . . . A 4 5 . . . . . . y 1 y y  y n n1 n2 np Apply OLS to specify the projection: T 1 y = Z(Z Z) Zy = P(Y j Y ;Y ;:::Y ) t t1 t2 tp (p) = y Compute the projection residual (p) (p)  = y y MIT 18.S096 Time Series Analysis 7Stationarity and Wold Representation Theorem Autoregressive and Moving Average (ARMA) Models Time Series Analysis Accommodating Non-Stationarity: ARIMA Models Estimation of Stationary ARMA Models Tests for Stationarity/Non-Stationarity Apply time series methods to the time series of residuals (p) f g to specify a moving average model: j P (p) 1  =  t j ti i=0 yieldingf g andf g; estimates of parameters and j t innovations. Conduct a case analysis diagnosing consistency with model assumptions (p) Evaluate orthogonality of  to Y ; s p. ts If evidence of correlation, increase p and start again. Evaluate the consistency off g with the white noise t assumptions of the theorem. If evidence otherwise, consider revisions to the overall model Changing the speci cation of the moving average model. Adding additional `deterministic' variables to the projection model. MIT 18.S096 Time Series Analysis 8Stationarity and Wold Representation Theorem Autoregressive and Moving Average (ARMA) Models Time Series Analysis Accommodating Non-Stationarity: ARIMA Models Estimation of Stationary ARMA Models Tests for Stationarity/Non-Stationarity Note: Theoretically, (p) lim y = y = P(Y j Y ;Y ;:::) p1 t t1 t2 but if p1 is required, then n1 while p=n 0. Useful models of covariance stationary time series have Modest nite values of p and/or include Moving average models depending on a parsimonious number of parameters. MIT 18.S096 Time Series Analysis 9Stationarity and Wold Representation Theorem Autoregressive and Moving Average (ARMA) Models Time Series Analysis Accommodating Non-Stationarity: ARIMA Models Estimation of Stationary ARMA Models Tests for Stationarity/Non-Stationarity Lag Operator L() De nition The lag operator L() shifts a time series back by one time increment. For a time seriesfXg : t L(X ) = X : t t1 Applying the operator recursively we de ne: 0 L (X ) = X t t 1 L (X ) = X t t1 2 L (X ) = L(L(X )) = X t t t2  n n1 L (X ) = L(L (X )) = X t t tn Inverses of these operators are well de ned as: n L (X ) = X , for n = 1; 2;::: t t+n MIT 18.S096 Time Series Analysis 10Stationarity and Wold Representation Theorem Autoregressive and Moving Average (ARMA) Models Time Series Analysis Accommodating Non-Stationarity: ARIMA Models Estimation of Stationary ARMA Models Tests for Stationarity/Non-Stationarity Wold Representation with Lag Operators The Wold Representation for a covariance stationary time series fXg can be expressed as t P 1 X =  +V t i ti t i=0 P 1 i = L ( ) +V i t t i=0 = (L) +V t t P 1 i where (L) = L . i i=0 De nition The Impulse Response Function of the covariance stationary processfXg is t X t IR(j) = = : j  tj The long-run cumulative response offXg is t P P 1 1 IR(j) = = (L) with L = 1. i i=0 i=0 MIT 18.S096 Time Series Analysis 11Stationarity and Wold Representation Theorem Autoregressive and Moving Average (ARMA) Models Time Series Analysis Accommodating Non-Stationarity: ARIMA Models Estimation of Stationary ARMA Models Tests for Stationarity/Non-Stationarity Equivalent Auto-regressive Representation Suppose that the operator (L) is invertible, i.e., P 1 1  i (L) = L such that i=0 i 1 0 (L) (L) = I = L : Then, assuming V = 0 (i.e., X has been adjusted to t t  X = X V ), we have the following equivalent expressions of the t t t time series model forfXg t X = (L) t t 1 (L)X =  t t 1 De nition When (L) exists, the time seriesfXg is Invertible t and has an auto-regressive representation: P 1  X = ( X ) + t ti t i=0 i MIT 18.S096 Time Series Analysis 12Stationarity and Wold Representation Theorem Autoregressive and Moving Average (ARMA) Models Time Series Analysis Accommodating Non-Stationarity: ARIMA Models Estimation of Stationary ARMA Models Tests for Stationarity/Non-Stationarity Outline 1 Time Series Analysis Stationarity and Wold Representation Theorem Autoregressive and Moving Average (ARMA) Models Accommodating Non-Stationarity: ARIMA Models Estimation of Stationary ARMA Models Tests for Stationarity/Non-Stationarity MIT 18.S096 Time Series Analysis 13Stationarity and Wold Representation Theorem Autoregressive and Moving Average (ARMA) Models Time Series Analysis Accommodating Non-Stationarity: ARIMA Models Estimation of Stationary ARMA Models Tests for Stationarity/Non-Stationarity ARMA(p,q) Models De nition: The times seriesfXg follows the ARMA(p;q) Model t with auto-regressive order p and moving-average order q if X =  + (X ) + (X ) + (X ) t 1 t1 2 t1 p tp + +  +  +  t 1 t1 2 t2 q tq 2 wherefg is WN(0; ), \White Noise" with t E ( ) = 0; 8t t 2 2 E ( ) =  1; 8t ; and E (  ) = 0; 8t6= s t s t With lag operators 2 P (L) = (1 L L  L ) and 1 2 p 2 q (L) = (1 + L + L + + L ) 1 2 q we can write (L) (X ) =(L) t t and the Wold decomposition is 1 X = + (L) ; where (L) = (L)) (L) t t MIT 18.S096 Time Series Analysis 14Stationarity and Wold Representation Theorem Autoregressive and Moving Average (ARMA) Models Time Series Analysis Accommodating Non-Stationarity: ARIMA Models Estimation of Stationary ARMA Models Tests for Stationarity/Non-Stationarity AR(p) Models Order-p Auto-Regression Model: AR(p) (L) (X ) = where t t 2 fg is WN(0; ) and t 2 p (L) = 1 L L  + L 1 2 p Properties: 2 Linear combination offX ;X ;:::X g is WN(0; ): t t1 tp X follows a linear regression model on explanatory variables t (X ;X ;:::;X ), i.e t1 t2 tp P p X = c +  X + t j tj t j=1 where c =(1); (replacing L by 1 in (L)). MIT 18.S096 Time Series Analysis 15Stationarity and Wold Representation Theorem Autoregressive and Moving Average (ARMA) Models Time Series Analysis Accommodating Non-Stationarity: ARIMA Models Estimation of Stationary ARMA Models Tests for Stationarity/Non-Stationarity AR(p) Models Stationarity Conditions Consider (z) replacing L with a complex variable z: 2 p (z) = 1 z z  z . 1 2 p Let  ; ;::: be the p roots of (z) = 0: 1 2 p 1 1 1 (L) = (1 L) (1 L) (1 L)    1 2 p Claim:fXg is covariance stationary if and only if all the roots of t (z) = 0 (the\characteristic equation") lie outside the unit circle fz :jzj 1g, i.e.,jj 1; j = 1; 2;:::;p j For complex number :jj 1; 1 1 1 1 1 2 2 3 3 (1 L) = 1 + ( )L + ( ) L + ( ) L +     P 1 1 i i = ( ) L i=0      1 Q p 1 1  (L) = 1 L j=1  j MIT 18.S096 Time Series Analysis 16Stationarity and Wold Representation Theorem Autoregressive and Moving Average (ARMA) Models Time Series Analysis Accommodating Non-Stationarity: ARIMA Models Estimation of Stationary ARMA Models Tests for Stationarity/Non-Stationarity AR(1) Model SupposefXg follows the AR(1) process, i.e., t X  =(X ) + ; t = 1; 2;::: t t1 t 2 where   WN(0; ). t The characteristic equation for the AR(1) model is (1z) = 0 1 with root  = :  The AR(1) model is covariance stationary if (and only if) jj 1 (equivalentlyjj 1) The rst and second moments offXg are t E (X ) =  t 2 2 Var(X ) =  = =(1) (= (0)) t X 2 Cov(X ;X ) =  t t1 X j 2 Cov(X ;X ) =   (= (j)) t tj X j Corr(X ;X ) =  =(j) (= (j)= (0)) t tj MIT 18.S096 Time Series Analysis 17Stationarity and Wold Representation Theorem Autoregressive and Moving Average (ARMA) Models Time Series Analysis Accommodating Non-Stationarity: ARIMA Models Estimation of Stationary ARMA Models Tests for Stationarity/Non-Stationarity AR(1) Model For  :jj 1; the Wold decomposition of the AR(1) model P 1 j is: X = +  t tj j=0 For  : 0 1; the AR(1) process exhibits exponential mean-reversion to  For  : 01; the AR(1) process exhibits oscillating exponential mean-reversion to  For  = 1; the Wold decomposition does not exist and the process is the simple random walk (non-stationary). For  1; the AR(1) process is explosive. Examples of AR(1) Models (mean reverting with 0 1) Interest rates (Ornstein Uhlenbeck Process; Vasicek Model) Interest rate spreads Real exchange rates Valuation ratios (dividend-to-price, earnings-to-price) MIT 18.S096 Time Series Analysis 18Stationarity and Wold Representation Theorem Autoregressive and Moving Average (ARMA) Models Time Series Analysis Accommodating Non-Stationarity: ARIMA Models Estimation of Stationary ARMA Models Tests for Stationarity/Non-Stationarity Yule Walker Equations for AR(p) Processes Second Order Moments of AR(p) Processes From the speci cation of the AR(p) model: (X ) =  (X ) + (X ) + + (X ) + t 1 t1 2 t1 p tp t we can write the Yule-Walker Equations (j = 0; 1;:::) E (X )(X ) =  E (X )(X ) t tj 1 t1 tj +  E (X )(X )+ 2 t1 tj  +  E (X )(X ) p tp tj + E  (X ) t tj (j) =  (j 1) + (j 2)+ 1 2 2  + (jp) +   p 0;j Equations j = 1; 2;:::p yield a system of p linear equations in  : j MIT 18.S096 Time Series Analysis 19Stationarity and Wold Representation Theorem Autoregressive and Moving Average (ARMA) Models Time Series Analysis Accommodating Non-Stationarity: ARIMA Models Estimation of Stationary ARMA Models Tests for Stationarity/Non-Stationarity Yule-Walker Equations 0 1 2 30 1 (1) (0) (1) (2)  ((p1))  1 B C 6 7B C (2) (1) (0) (1)  ((p2))  2 B C 6 7B C B . C =6 . . . . 7B . C . . . . . . . . A 4 5 A . . . . . . . (p) (p1) (p2) (p3)  (0))  p Given estimates (j);j = 0;:::;p (and  ) the solution of these equations are the Yule-Walker estimates of the  ; using the property (j) = (+j),8j j Using these in equation 0 2 (0) =  (1)+ (2)++ (p) +   1 2 p 0;0 2 provides an estimate of  P p 2  = (0)  (j) j j1 When all the estimates (j) and  are unbiased, then the Yule-Walker estimates apply the Method of Moments Principle of Estimation. MIT 18.S096 Time Series Analysis 20

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