A course in Time Series Analysis

1 A course in time Series AnalysisSuhasiniSubba 17, 2021 Contents1 time Series data .. Filtering time Series .. Terminology ..172 Trends in a time Parametric trend .. squares estimation .. Differencing .. Nonparametric methods (advanced) .. windows .. estimators .. What is trend and what is noise? .. Periodic functions .. sine and cosine transform .. Fourier transform (the sine and cosine transform in disguise) .. discrete Fourier transform .. discrete Fourier transform and periodic signals .. trends and its corresponding DFT .. detection .. detection and correlated noise .. of the periodogram .. Data Analysis : EEG data .. Hertz and Frequencies .. Analysis .. Exercises.

2 583 Stationary time Preliminaries .. definition of a time Series .. The sample mean and its standard error .. variance of the estimated regressors in a linear regression modelwith correlated errors .. Stationary processes .. of stationarity .. statistical inference for time Series .. What makes a covariance a covariance? .. Spatial covariances (advanced) .. Exercises ..864 Linear time Motivation .. Linear time Series and moving average models .. sums of random variables .. The AR(p) model .. equations and back-shift operators .. of two particular AR(1) models .. solution of a general AR(p) .. an explicit solution of an AR(2) model .. of the periodogram (Part II) .. of Pseudo periodic AR(2) models.

3 Of Pseudo periodicity functions in an AR(2) .. Autoregressive models .. of the general AR( ) model (advanced) .. Simulating from an Autoregressive process .. The ARMA model .. ARFIMA models .. Unit roots, integrated and non-invertible processes .. roots .. processes .. Simulating from models .. Some diagnostics .. and PACF plots for checking for MA and AR behaviour .. for unit roots .. Appendix .. 1305 A review of some results from multivariate Preliminaries: Euclidean space and projections .. products and norms .. vectors .. in multiple stages .. of random variables .. Linear prediction .. Partial correlation.

4 Properties of the precision matrix .. of results .. of results .. Appendix .. 1496 The autocovariance and partial covariance of a stationary time The autocovariance function .. rate of decay of the autocovariance of an ARMA process .. autocovariance of an autoregressive process and the Yule-Walkerequations .. autocovariance of a moving average process .. autocovariance of an ARMA process (advanced) .. the ACF from data .. Partial correlation in time Series .. general definition .. correlation of a stationary time Series .. fitting AR(p) model .. fitting AR(p) parameters and partial correlation .. partial autocorrelation plot .. the ACF and PACF for model identification.

5 The variance and precision matrix of a stationary time Series .. matrix for AR(p) and MA(p) models .. The ACF of non-causal time Series (advanced) .. Yule-Walker equations of a non-causal process .. non-causal AR models .. 1857 Using prediction in estimation .. Forecasting for autoregressive processes .. Forecasting for AR(p) .. Forecasting for general time Series using infinite past .. : Forecasting yearly temperatures .. One-step ahead predictors based on the finite past .. algorithm .. proof of the Durbin-Levinson algorithm based on projections .. the Durbin-Levinson to obtain the Cholesky Comparing finite and infinite predictors (advanced).

6 Ahead predictors based on the finite past .. Forecasting for ARMA processes .. ARMA models and the Kalman filter .. Kalman filter .. state space (Markov) representation of the ARMA model .. using the Kalman filter .. Forecasting for nonlinear models (advanced) .. Forecasting volatility using an ARCH(p) model .. Forecasting volatility using a GARCH(1,1) model .. Forecasting using a BL(1,0,1,1) model .. Nonparametric prediction (advanced) .. The Wold Decomposition (advanced) .. Kolmogorov s formula (advanced) .. Appendix: Prediction coefficients for an AR(p) model .. Appendix: Proof of the Kalman filter .. 2398 Estimation of the mean and An estimator of the mean.

7 Sampling properties of the sample mean .. An estimator of the covariance .. properties of the covariance estimator .. asymptotic properties of the sample autocovariance and autocor-relation .. covariance of the sample autocovariance .. Checking for correlation in a time Series .. the assumptions: The robust Portmanteau test (advanced) . Checking for partial correlation .. Checking for Goodness of fit (advanced) .. Long range dependence (long memory) versus changes in the mean .. 2809 Parameter Estimation for Autoregressive models .. Yule-Walker estimator .. tapered Yule-Walker estimator .. Gaussian likelihood .. conditional Gaussian likelihood and least squares.

8 S algorithm .. properties of the AR regressive estimators .. Estimation for ARMA models .. Gaussian maximum likelihood estimator .. approximate Gaussian likelihood .. using the Kalman filter .. properties of the ARMA maximum likelihood estimator .. Hannan-Rissanen AR( ) expansion method .. The quasi-maximum likelihood for ARCH processes .. 31310 Spectral How we have used Fourier transforms so far .. The near uncorrelatedness of the DFT .. Testing for second order stationarity: An application of the near decor-relation property .. Proof of Lemma .. The DFT and complete decorrelation .. Summary of spectral representation results .. The spectral (Cramer s) representation theorem.

9 Bochner s theorem .. The spectral density and spectral distribution .. The spectral density and some of its properties .. The spectral distribution and Bochner s (Hergoltz) theorem .. The spectral representation theorem .. The spectral density functions of MA, AR and ARMA models .. The spectral representation of linear processes .. The spectral density of a linear process .. Approximations of the spectral density to AR and MA spectral Cumulants and higher order spectrums .. Extensions .. The spectral density of a time Series with randomly missing Appendix: Some proofs .. 35411 Spectral The DFT and the periodogram .. Distribution of the DFT and Periodogram under linearity.

10 Estimating the spectral density function .. The Whittle Likelihood .. Connecting the Whittle and Gaussian likelihoods .. Sampling properties of the Whittle likelihood estimator .. Ratio statistics in time Series .. Goodness of fit tests for linear time Series models .. Appendix .. 39712 Multivariate time Background .. Preliminaries 1: Sequences and functions .. Preliminaries 2: Convolution .. Preliminaries 3: Spectral representations and mean squared errors .. Multivariate time Series regression .. Conditional independence .. Partial correlation and coherency between time Series .. Cross spectral density of{ (a)t,Y, (a)t,Y}: The spectral partial coherencyfunction.