Statistics 910, #9
1
Covariances of ARMA Processes
Overview
1. Review ARMA models: causality and invertibility
2. AR covariance functions
3. MA and ARMA covariance functions
4. Partial autocorrelation function
5. Discussion
Review of ARMA processes
ARMA p
Statistics 910, #14
1
State-Space Models
Overview
1. State-space models (a.k.a., dynamic linear models, DLM)
2. Regression Examples
3. AR, MA and ARMA models in state-space form
See S&S Chapter 6, which emphasizes tting state-space models to data via
the
Statistics 910, #15
1
Kalman Filter
Overview
1. Summary of Kalman lter
2. Derivations
3. ARMA likelihoods
4. Recursions for the variance
Summary of Kalman lter
Simplications To make the derivations more direct, assume that the
two noise processes are unco
Statistics 910, #12
1
Estimating an ARMA Process
Overview
1. Main ideas
2. Fitting autoregressions
3. Fitting with moving average components
4. Standard errors
5. Examples
6. Appendix: Simple estimators for autoregressions
Main ideas
Eciency Maximum likel
Statistics 910, #13
1
Resampling Methods for Time Series
Overview
1. Bootstrap resampling
2. Eects of dependence
3. Subsampling
4. Further directions
Bootstrap resampling
Main idea Estimate the sampling distribution of a statistic, with particular emphasi
Statistics 910, #8
1
Introduction to ARMA Models
Overview
1. Modeling paradigm
2. Review stationary linear processes
3. ARMA processes
4. Stationarity of ARMA processes
5. Identiability of ARMA processes
6. Invertibility of ARMA processes
7. ARIMA process
Statistics 910, #10
1
Predicting ARMA Processes
Overview
Prediction of ARMA processes resembles in many ways prediction in regression models, at least in the case of AR models. We focus on linear predictors,
those that express the prediction as a weighted
Statistics 910, #11
1
Asymptotic Distributions in Time Series
Overview
Standard proofs that establish the asymptotic normality of estimators constructed from random samples (i.e., independent observations) no longer
apply in time series analysis. The usua
Statistics 910, #6
1
Harmonic Regression
Overview
1. Example: periodic data
2. Regression at Fourier frequencies
3. Discrete Fourier transform (periodogram)
4. Examples of the DFT
Example: Periodic Data
Magnitude of variable star This integer time series
Statistics 910, #5
1
Regression Methods
Overview
1. Idea: eects of dependence
2. Examples of estimation (in R)
3. Review of regression
4. Comparisons and relative eciencies
Idea
Decomposition Well-known way to approach time series analysis is to
decompose
Statistics 910, #4
1
Properties of Descriptive Estimators
Overview
1. Properties of X
2. Simulation of estimator compared to
3. Properties of (h)
4. Simulation of pointwise and sequence-wide properties
See S&S, Appendix A, for further details on the prop
Statistics 910, #2
1
Examples of Stationary Time Series
Overview
1. Stationarity
2. Linear processes
3. Cyclic models
4. Nonlinear models
Stationarity
Strict stationarity (Defn 1.6) Probability distribution of the stochastic
process cfw_Xt is invariant un
Statistics 910, #1
1
Introduction
Overview
1. Data
2. Concepts
3. Models, methods
Text Examples of Time Series
JJ earnings Choice between complicated polynomial with changing variance versus percentage change modeled as constant.
Transformations often sim
Statistics 910, #3
1
Descriptive Estimators
Overview
1. Moment estimators for , (h), and correlations (h).
2. Simulate estimators using R.
See S&S, Appendix A, for further details on the properties of these estimators that well cover in the next class.
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