1、Analyzing Time Series & Signals,Professor Melvin J. Hinich hinichmail.la.utexas.edu,2,Useful Models Should be Derived from Science,Linear autoregressive (AR) and vector AR models are the most widely applied model in contemporary time series methodology They are examples of discrete-time linear dynam
2、ical systems They almost never have a firm scientific foundation What are linear dynamical systems?,3,Linear Dynamical Systems,Continuous time System of first order differential equations System of pth order differential equations,Discrete-time System of first order difference equations System of pt
3、h order difference equations,They are almost equivalent,4,First Order Linear Differential Equation,Solution Exponential Trend: x(t)=x(0)ert,Forced First Order DE,5,Convolution,6,Discrete-time Convolution,7,1st Order Linear Discrete-Time Eqn.,Forced First Linear Difference Equation.,8,Linear Harmonic
4、 Oscillator,9,Harmonic Solution,c is called the damping parameter. Since it is positive the solution goes to zero as,10,2nd Order Discrete-time Equation,11,Example Impulse Response,12,Constant Coefficient Linear Dynamical System,13,Homogeneous Linear Dynamical System,14,Uncoupled System,15,Uncoupled
5、 System,The system is stable if all the rm 0,If wm=0 then the solution is an exponential.,damped oscillation,exploding exponential oscillation,unstable equilibrium,16,Impulse Response,17,Forced Linear Dynamical System,Uncoupled linear system,18,Complex Transfer Function,19,Autoregressive AR(p) Model
6、,20,Discrete-time Linear System,Written as a first order homogeneous system,21,Discrete & Continuous-time Linear System,22,Discrete-time Solution,If there is no aliasing!,damped oscillation,unstable equilibrium,23,Aliasing,Aliasing is an identification problem,24,Exponential Trend,Trend only depends
7、 on the rate r & the initial value x(0) logx(t)-logx(0)=r t How should we model evolution of the rate r? Model rate using covariates and stochastic shocks Estimate trend from data using the model Analyze the residual process to test the model,25,Deterministic Trend Plus AR(p),Fit the trend using lea
8、st squares,Subtract estimated trend from y(tn),Estimate AR(p) from the residuals,Use orthogonal polynomials for curvilinear trend,26,Stochastic Trend,is a pure white noise process,creates a unit root in the implicit model,27,Implicit AR(p+1) Model,with the convention that,The z transform of the syst
9、em is,28,Stochastic Trend with Drift=0.2 se=5 AR(10),29,Identifying Linear Systems,Harder problem that is commonly believed There are many specifications that give similar correlations across variables Forecasting is the goal of this enterprise,30,Linear Dynamical System Plus Noise,31,A Simple First
10、 Order Nonlinear System,32,Example of a Simple Nonlinear Model,33,Nonlinear Model - Uniform Input s = 0.5, c = 0.05 , w = 0.2p , d = 0.5,34,Phase Plot of the Nonlinear Impulse Response,35,Phase Plot of the Linear Impulse Response,36,Results of a Least Squares AR Fit to the Data,Data: Skew = 0.288e-0
11、2 Kurtosis = 0.988,AR( 6) parameters & t values,a4 = 0.02 a6 = -0.01,166.9 60.5 22.8,Adjusted R Square = 0.401,a1= - 0.75 a2 = - 0.34 a3 = - 0.13,3.6 2.2,37,Moving Frame Detection Method,Data is prewhitened using an AR(20) fit Statistics from residuals of an AR( 7) fit to each frame Frame Length = 5
12、0 No. of frames = 2000 100 Bootstraps Sizes:H =0.00476 C =0.0276,38,Analyzing the Residuals - Whiteness,Standardize the data,Correlation Test Statistic for e 0.5,39,Analyzing the Residuals - Nonlinearity,Bicorrelation for lags r , s where 0 s r,Bicorrelation Test Statistic for e 0.5,40,C - Correlati
13、on Statistics,41,H - Bicorrelation Statistics,42,Standard Deviations & R2,43,One Approach to Estimating this Nonlinear Model,Divide the sample into overlapping frames,Estimate a linear model for each frame,Compute the eigenvalues for each frame model,Estimate the nonlinear parameter by least squares,Compute the log of each eigenvalue,
