STA 6857 Cross Spectra & Linear Filters ( 4.6 & 4.7)
|
|
- Bryan Perkins
- 6 years ago
- Views:
Transcription
1 STA 6857 Cross Spectra & Linear Filters ( 4.6 & 4.7)
2 Outline 1 Midterm Exam Solutions 2 Final Project 3 From Last Time 4 Multiple Series and Cross-Spectra 5 Linear Filters Arthur Berg STA 6857 Cross Spectra & Linear Filters ( 4.6 & 4.7) 2/ 23
3 Outline 1 Midterm Exam Solutions 2 Final Project 3 From Last Time 4 Multiple Series and Cross-Spectra 5 Linear Filters Arthur Berg STA 6857 Cross Spectra & Linear Filters ( 4.6 & 4.7) 3/ 23
4 Outline 1 Midterm Exam Solutions 2 Final Project 3 From Last Time 4 Multiple Series and Cross-Spectra 5 Linear Filters Arthur Berg STA 6857 Cross Spectra & Linear Filters ( 4.6 & 4.7) 4/ 23
5 Homework 4 Homework 4 is due Wednesday, October 24 in class. Arthur Berg STA 6857 Cross Spectra & Linear Filters ( 4.6 & 4.7) 5/ 23
6 Final Project Description From the syllabus: Final Project The final project will be a topic of your choice. I will happily suggest a topic should you wish. You will be required to submit a typed report as well as give a 12 minute presentation during the class. The written portion should describe the project, introduce the statistical methodology, present results, and provide discussions. The presentation should be a vibrant powerpoint-type presentation with at most a few words on each slide. Everyone should meet with me at least once before the presentation. Quality is more important than quantity! Two Possibilities: Analyze a dataset from your own research using several techniques from time series analysis Introduce a new topic and, if feasible, use simulated or real data to motivate the new techniques Arthur Berg STA 6857 Cross Spectra & Linear Filters ( 4.6 & 4.7) 6/ 23
7 Outline 1 Midterm Exam Solutions 2 Final Project 3 From Last Time 4 Multiple Series and Cross-Spectra 5 Linear Filters Arthur Berg STA 6857 Cross Spectra & Linear Filters ( 4.6 & 4.7) 7/ 23
8 Nonparametric Spectral Density Estimation Reduce the influence of γ(h) at extreme values of h. Consider the following estimator: f (ω) = n 1 h= (n 1) λ(h) γ(h)e 2πiω jh where λ(h) starts out at 1 when h 0, but then decreases as h increases. Under general assumptions (for instance ARMA models), using a flat-top lag-window function can guarantee: ) ) ( ) 1 bias ( f (ω) = E ( f (ω) f (ω j ) = O n ) ( ) log n var ( f (ω) = O n ) ( ) log n MSE ( f (ω) = O n Arthur Berg STA 6857 Cross Spectra & Linear Filters ( 4.6 & 4.7) 8/ 23
9 Examples of Lag Windows Arthur Berg STA 6857 Cross Spectra & Linear Filters ( 4.6 & 4.7) 9/ 23
10 Outline 1 Midterm Exam Solutions 2 Final Project 3 From Last Time 4 Multiple Series and Cross-Spectra 5 Linear Filters Arthur Berg STA 6857 Cross Spectra & Linear Filters ( 4.6 & 4.7) 10/ 23
11 Cross Spectrum Recall the cross-covariance function γ xy (h) for a jointly stationary series x t and y t as defined by γ xy (h) = E [(x t+h µ x )(y t µ y )] Taking the Fourier transform, we have the cross-spectrum f xy (ω) = = h= h= γ xy (h)e 2πiωh γ xy (h) cos(2πωh) i } {{ } c xy(ω) h= γ xy (h) sin(2πωh) }{{} qxy(ω) Arthur Berg STA 6857 Cross Spectra & Linear Filters ( 4.6 & 4.7) 11/ 23
12 Squared Coherence Function The squared coherence function measures the the strength of the relationship between x t and y t in the frequency domain. Definition (Squared Coherence Function) The squared coherence function is defined as ρ 2 yx(ω) = f yx(ω) 2 f x (ω)f y (ω) Note the analogy to the conventional square correlation given by ρ 2 yx = σ2 yx σ 2 xσ 2 y Arthur Berg STA 6857 Cross Spectra & Linear Filters ( 4.6 & 4.7) 12/ 23
13 Spectral Matrix For a p-dimensional stationary vector time series with autocovariance matrix given by Γ(h) = E [ (x t+h µ)(x t µ) ] The spectral matrix is the term-by-term Fourier transform of the autocovariance matrix which can be simply written as f (ω) = h= Γ(h)e 2πiωh, 1/2 ω 1/2 Arthur Berg STA 6857 Cross Spectra & Linear Filters ( 4.6 & 4.7) 13/ 23
14 Testing for Significant Coherence Under the hypothesis ρ 2 yx(ω) = 0 the statistic ρ 2 yx(ω) 1 ρ 2 yx(ω) follows an F distribution which allows one to test for significance against the null. Arthur Berg STA 6857 Cross Spectra & Linear Filters ( 4.6 & 4.7) 14/ 23
15 Coherence Function Between SOI and Recruitment Series > x = ts(cbind(soi,rec)) > s = spec.pgram(x, kernel("daniell",9), taper=0) > s$df # df = [1] > f = qf(.999, 2, s$df-2) # f = > c = f/(18+f) # c = > plot(s, plot.type = "coh", ci.lty = 2) > abline(h = c, lwd=3, col="purple") Arthur Berg STA 6857 Cross Spectra & Linear Filters ( 4.6 & 4.7) 15/ 23
16 Outline 1 Midterm Exam Solutions 2 Final Project 3 From Last Time 4 Multiple Series and Cross-Spectra 5 Linear Filters Arthur Berg STA 6857 Cross Spectra & Linear Filters ( 4.6 & 4.7) 16/ 23
17 Linear Filter Definition A linear filter is a linear transformation of a process x t given as y t = a r x t r r= The coefficients a r are collectively called the impulse response function and it is assumed that they are absolutely summable, i.e. t= a t < Arthur Berg STA 6857 Cross Spectra & Linear Filters ( 4.6 & 4.7) 17/ 23
18 Output Spectrum of Filtered Series The impulse response function, A yx (ω), is defined as Fourier transform of the impulse response function, i.e. A yx (ω) = t= a t e 2πiωt Theorem The spectrum of the filtered output y t satsifies f y (ω) = A yx 2 f x (ω) Note the relation to classical statistics where multiplying a random variable X by a constant c changes its variance to c 2 var(x). Arthur Berg STA 6857 Cross Spectra & Linear Filters ( 4.6 & 4.7) 18/ 23
19 Two Filters of SOI Let x t represent the SOI values. Consider the two filters: The difference filter y t = x t = x t x t 1 Here a 0 = 1 and a 1 = 1. This is an example of a high-pass filter. The symmetric moving average filter y t = 1 24 (x t 6 + x t+6 ) r= 5 x t r Here a 6 = a 6 = 1/24, a k = 1/12 for 5 k 5, and a k = 0 otherwise. This is an example of a low-pass filter. Arthur Berg STA 6857 Cross Spectra & Linear Filters ( 4.6 & 4.7) 19/ 23
20 Two Filters Applied to SOI Arthur Berg STA 6857 Cross Spectra & Linear Filters ( 4.6 & 4.7) 20/ 23
21 Frequency Response Functions of the Two Examples Arthur Berg STA 6857 Cross Spectra & Linear Filters ( 4.6 & 4.7) 21/ 23
22 > w = seq(0,.5, length=1000) #-- frequency response > FR = abs(1-exp(2i*pi*w))^2 > FR2<-double(length(w)) > count<-1 > for(j in w){ + FR2[count]=abs(1/12*(1+cos(12*pi*j)+2*sum(cos(2*pi*j*1:5))))^2 + count<-count+1 + } > plot(w,fr2,type="l",lwd=3,col="blue") > windows() > plot(w, FR, type="l",lwd=3,col="orange") Arthur Berg STA 6857 Cross Spectra & Linear Filters ( 4.6 & 4.7) 22/ 23
23 Frequency Response Functions of the Two Examples Here we can see why differencing is referred to as a high-pass filter and the moving average as a low-pass filter. Arthur Berg STA 6857 Cross Spectra & Linear Filters ( 4.6 & 4.7) 23/ 23
Cross Spectra, Linear Filters, and Parametric Spectral Estimation
Cross Spectra,, and Parametric Spectral Estimation Outline 1 Multiple Series and Cross-Spectra 2 Arthur Berg Cross Spectra,, and Parametric Spectral Estimation 2/ 20 Outline 1 Multiple Series and Cross-Spectra
More informationSTA 6857 Cross Spectra, Linear Filters, and Parametric Spectral Estimation ( 4.6, 4.7, & 4.8)
STA 6857 Cross Spectra, Linear Filters, and Parametric Spectral Estimation ( 4.6, 4.7, & 4.8) Outline 1 Midterm Results 2 Multiple Series and Cross-Spectra 3 Linear Filters Arthur Berg STA 6857 Cross Spectra,
More informationOutline. STA 6857 Cross Spectra, Linear Filters, and Parametric Spectral Estimation ( 4.6, 4.7, & 4.8) Summary Statistics of Midterm Scores
STA 6857 Cross Spectra, Linear Filters, and Parametric Spectral Estimation ( 4.6, 4.7, & 4.8) Outline 1 Midterm Results 2 Multiple Series and Cross-Spectra 3 Linear Filters Arthur Berg STA 6857 Cross Spectra,
More informationThe triangular lag window, also known as the Bartlett or Fejér window, W (!) = sin2 ( r!) r sin 2 (!).
4.6 Parametric Spectral Estimation 11 The triangular lag window, also known as the Bartlett or Fejér window, given by w(x) =1 x, x apple 1 leads to the Fejér smoothing window: In this case, (4.73) yields
More informationSTA 6857 Signal Extraction & Long Memory ARMA ( 4.11 & 5.2)
STA 6857 Signal Extraction & Long Memory ARMA ( 4.11 & 5.2) Outline 1 Signal Extraction and Optimal Filtering 2 Arthur Berg STA 6857 Signal Extraction & Long Memory ARMA ( 4.11 & 5.2) 2/ 17 Outline 1 Signal
More informationSTA 6857 Autocorrelation and Cross-Correlation & Stationary Time Series ( 1.4, 1.5)
STA 6857 Autocorrelation and Cross-Correlation & Stationary Time Series ( 1.4, 1.5) Outline 1 Announcements 2 Autocorrelation and Cross-Correlation 3 Stationary Time Series 4 Homework 1c Arthur Berg STA
More informationHeteroskedasticity and Autocorrelation Consistent Standard Errors
NBER Summer Institute Minicourse What s New in Econometrics: ime Series Lecture 9 July 6, 008 Heteroskedasticity and Autocorrelation Consistent Standard Errors Lecture 9, July, 008 Outline. What are HAC
More informationD.S.G. POLLOCK: BRIEF NOTES
BIVARIATE SPECTRAL ANALYSIS Let x(t) and y(t) be two stationary stochastic processes with E{x(t)} = E{y(t)} =. These processes have the following spectral representations: (1) x(t) = y(t) = {cos(ωt)da
More informationIntroduction to Time Series Analysis. Lecture 18.
Introduction to Time Series Analysis. Lecture 18. 1. Review: Spectral density, rational spectra, linear filters. 2. Frequency response of linear filters. 3. Spectral estimation 4. Sample autocovariance
More informationLECTURE 10 LINEAR PROCESSES II: SPECTRAL DENSITY, LAG OPERATOR, ARMA. In this lecture, we continue to discuss covariance stationary processes.
MAY, 0 LECTURE 0 LINEAR PROCESSES II: SPECTRAL DENSITY, LAG OPERATOR, ARMA In this lecture, we continue to discuss covariance stationary processes. Spectral density Gourieroux and Monfort 990), Ch. 5;
More informationChapter 12 - Lecture 2 Inferences about regression coefficient
Chapter 12 - Lecture 2 Inferences about regression coefficient April 19th, 2010 Facts about slope Test Statistic Confidence interval Hypothesis testing Test using ANOVA Table Facts about slope In previous
More informationNonlinear Time Series
Nonlinear Time Series Recall that a linear time series {X t } is one that follows the relation, X t = µ + i=0 ψ i A t i, where {A t } is iid with mean 0 and finite variance. A linear time series is stationary
More informationClassic Time Series Analysis
Classic Time Series Analysis Concepts and Definitions Let Y be a random number with PDF f Y t ~f,t Define t =E[Y t ] m(t) is known as the trend Define the autocovariance t, s =COV [Y t,y s ] =E[ Y t t
More informationNonparametric Function Estimation with Infinite-Order Kernels
Nonparametric Function Estimation with Infinite-Order Kernels Arthur Berg Department of Statistics, University of Florida March 15, 2008 Kernel Density Estimation (IID Case) Let X 1,..., X n iid density
More information6.435, System Identification
System Identification 6.435 SET 3 Nonparametric Identification Munther A. Dahleh 1 Nonparametric Methods for System ID Time domain methods Impulse response Step response Correlation analysis / time Frequency
More informationCorrelation and Regression
Correlation and Regression October 25, 2017 STAT 151 Class 9 Slide 1 Outline of Topics 1 Associations 2 Scatter plot 3 Correlation 4 Regression 5 Testing and estimation 6 Goodness-of-fit STAT 151 Class
More information3. ARMA Modeling. Now: Important class of stationary processes
3. ARMA Modeling Now: Important class of stationary processes Definition 3.1: (ARMA(p, q) process) Let {ɛ t } t Z WN(0, σ 2 ) be a white noise process. The process {X t } t Z is called AutoRegressive-Moving-Average
More informationModule 29.3: nag tsa spectral Time Series Spectral Analysis. Contents
Time Series Analysis Module Contents Module 29.3: nag tsa spectral Time Series Spectral Analysis nag tsa spectral calculates the smoothed sample spectrum of a univariate and bivariate time series. Contents
More informationX random; interested in impact of X on Y. Time series analogue of regression.
Multiple time series Given: two series Y and X. Relationship between series? Possible approaches: X deterministic: regress Y on X via generalized least squares: arima.mle in SPlus or arima in R. We have
More informationLecture 18: Analysis of variance: ANOVA
Lecture 18: Announcements: Exam has been graded. See website for results. Lecture 18: Announcements: Exam has been graded. See website for results. Reading: Vasilj pp. 83-97. Lecture 18: Announcements:
More informationA Probability Review
A Probability Review Outline: A probability review Shorthand notation: RV stands for random variable EE 527, Detection and Estimation Theory, # 0b 1 A Probability Review Reading: Go over handouts 2 5 in
More informationSTAT 443 Final Exam Review. 1 Basic Definitions. 2 Statistical Tests. L A TEXer: W. Kong
STAT 443 Final Exam Review L A TEXer: W Kong 1 Basic Definitions Definition 11 The time series {X t } with E[X 2 t ] < is said to be weakly stationary if: 1 µ X (t) = E[X t ] is independent of t 2 γ X
More informationCOMPARING SPECTRAL DENSITIES IN REPLICATED TIME SERIES BY SMOOTHING SPLINE ANOVA
COMPARING SPECTRAL DENSITIES IN REPLICATED TIME SERIES BY SMOOTHING SPLINE ANOVA by Sangdae Han BS, Hallym University, 1996 MS, Michigan State University, 1998 MA, University of Pittsburgh, 2005 Submitted
More informationSTAD57 Time Series Analysis. Lecture 23
STAD57 Time Series Analysis Lecture 23 1 Spectral Representation Spectral representation of stationary {X t } is: 12 i2t Xt e du 12 1/2 1/2 for U( ) a stochastic process with independent increments du(ω)=
More informationOn 1.9, you will need to use the facts that, for any x and y, sin(x+y) = sin(x) cos(y) + cos(x) sin(y). cos(x+y) = cos(x) cos(y) - sin(x) sin(y).
On 1.9, you will need to use the facts that, for any x and y, sin(x+y) = sin(x) cos(y) + cos(x) sin(y). cos(x+y) = cos(x) cos(y) - sin(x) sin(y). (sin(x)) 2 + (cos(x)) 2 = 1. 28 1 Characteristics of Time
More informationEE538 Final Exam Fall 2007 Mon, Dec 10, 8-10 am RHPH 127 Dec. 10, Cover Sheet
EE538 Final Exam Fall 2007 Mon, Dec 10, 8-10 am RHPH 127 Dec. 10, 2007 Cover Sheet Test Duration: 120 minutes. Open Book but Closed Notes. Calculators allowed!! This test contains five problems. Each of
More informationWavelet Methods for Time Series Analysis. Motivating Question
Wavelet Methods for Time Series Analysis Part VII: Wavelet-Based Bootstrapping start with some background on bootstrapping and its rationale describe adjustments to the bootstrap that allow it to work
More informationProbability and Statistics for Final Year Engineering Students
Probability and Statistics for Final Year Engineering Students By Yoni Nazarathy, Last Updated: May 24, 2011. Lecture 6p: Spectral Density, Passing Random Processes through LTI Systems, Filtering Terms
More informationTime Series Analysis. James D. Hamilton PRINCETON UNIVERSITY PRESS PRINCETON, NEW JERSEY
Time Series Analysis James D. Hamilton PRINCETON UNIVERSITY PRESS PRINCETON, NEW JERSEY & Contents PREFACE xiii 1 1.1. 1.2. Difference Equations First-Order Difference Equations 1 /?th-order Difference
More informationA Diagnostic for Seasonality Based Upon Autoregressive Roots
A Diagnostic for Seasonality Based Upon Autoregressive Roots Tucker McElroy (U.S. Census Bureau) 2018 Seasonal Adjustment Practitioners Workshop April 26, 2018 1 / 33 Disclaimer This presentation is released
More informationStochastic Processes. M. Sami Fadali Professor of Electrical Engineering University of Nevada, Reno
Stochastic Processes M. Sami Fadali Professor of Electrical Engineering University of Nevada, Reno 1 Outline Stochastic (random) processes. Autocorrelation. Crosscorrelation. Spectral density function.
More informationLecture 4 - Spectral Estimation
Lecture 4 - Spectral Estimation The Discrete Fourier Transform The Discrete Fourier Transform (DFT) is the equivalent of the continuous Fourier Transform for signals known only at N instants separated
More informationSTAT420 Midterm Exam. University of Illinois Urbana-Champaign October 19 (Friday), :00 4:15p. SOLUTIONS (Yellow)
STAT40 Midterm Exam University of Illinois Urbana-Champaign October 19 (Friday), 018 3:00 4:15p SOLUTIONS (Yellow) Question 1 (15 points) (10 points) 3 (50 points) extra ( points) Total (77 points) Points
More informationSTA 6857 Estimation and ARIMA Models ( 3.6 cont. and 3.7)
STA 6857 Estimation and ARIMA Models ( 3.6 cont. and 3.7) Outline 1 AR Bootstrap 2 ARIMA 3 Homework 4b Arthur Berg STA 6857 Estimation and ARIMA Models ( 3.6 cont. and 3.7) 2/ 20 Outline 1 AR Bootstrap
More informationSTA 6857 Forecasting ( 3.5 cont.)
STA 6857 Forecasting ( 3.5 cont.) Outline 1 Forecasting 2 Arthur Berg STA 6857 Forecasting ( 3.5 cont.) 2/ 20 Outline 1 Forecasting 2 Arthur Berg STA 6857 Forecasting ( 3.5 cont.) 3/ 20 Best Linear Predictor
More informationSTAT 520: Forecasting and Time Series. David B. Hitchcock University of South Carolina Department of Statistics
David B. University of South Carolina Department of Statistics What are Time Series Data? Time series data are collected sequentially over time. Some common examples include: 1. Meteorological data (temperatures,
More informationWavelet Methods for Time Series Analysis. Part IX: Wavelet-Based Bootstrapping
Wavelet Methods for Time Series Analysis Part IX: Wavelet-Based Bootstrapping start with some background on bootstrapping and its rationale describe adjustments to the bootstrap that allow it to work with
More informationProbability Space. J. McNames Portland State University ECE 538/638 Stochastic Signals Ver
Stochastic Signals Overview Definitions Second order statistics Stationarity and ergodicity Random signal variability Power spectral density Linear systems with stationary inputs Random signal memory Correlation
More informationAdvanced Econometrics
Advanced Econometrics Marco Sunder Nov 04 2010 Marco Sunder Advanced Econometrics 1/ 25 Contents 1 2 3 Marco Sunder Advanced Econometrics 2/ 25 Music Marco Sunder Advanced Econometrics 3/ 25 Music Marco
More informationThe Box-Cox Transformation and ARIMA Model Fitting
The Box-Cox Transformation and ARIMA Model Fitting Outline 1 4.3: Variance Stabilizing Transformations 2 6.1: ARIMA Model Identification 3 Homework 3b Arthur Berg The Box-Cox Transformation and ARIMA Model
More informationM(t) = 1 t. (1 t), 6 M (0) = 20 P (95. X i 110) i=1
Math 66/566 - Midterm Solutions NOTE: These solutions are for both the 66 and 566 exam. The problems are the same until questions and 5. 1. The moment generating function of a random variable X is M(t)
More informationStochastic Processes. A stochastic process is a function of two variables:
Stochastic Processes Stochastic: from Greek stochastikos, proceeding by guesswork, literally, skillful in aiming. A stochastic process is simply a collection of random variables labelled by some parameter:
More informationCircle the single best answer for each multiple choice question. Your choice should be made clearly.
TEST #1 STA 4853 March 6, 2017 Name: Please read the following directions. DO NOT TURN THE PAGE UNTIL INSTRUCTED TO DO SO Directions This exam is closed book and closed notes. There are 32 multiple choice
More informationMultivariate Time Series
Multivariate Time Series Notation: I do not use boldface (or anything else) to distinguish vectors from scalars. Tsay (and many other writers) do. I denote a multivariate stochastic process in the form
More informationTime Series Analysis. James D. Hamilton PRINCETON UNIVERSITY PRESS PRINCETON, NEW JERSEY
Time Series Analysis James D. Hamilton PRINCETON UNIVERSITY PRESS PRINCETON, NEW JERSEY PREFACE xiii 1 Difference Equations 1.1. First-Order Difference Equations 1 1.2. pth-order Difference Equations 7
More informationSTA 6857 ARIMA and SARIMA Models ( 3.8 and 3.9)
STA 6857 ARIMA and SARIMA Models ( 3.8 and 3.9) Outline 1 Building ARIMA Models 2 SARIMA 3 Homework 4c Arthur Berg STA 6857 ARIMA and SARIMA Models ( 3.8 and 3.9) 2/ 34 Outline 1 Building ARIMA Models
More informationTIME SERIES AND FORECASTING. Luca Gambetti UAB, Barcelona GSE Master in Macroeconomic Policy and Financial Markets
TIME SERIES AND FORECASTING Luca Gambetti UAB, Barcelona GSE 2014-2015 Master in Macroeconomic Policy and Financial Markets 1 Contacts Prof.: Luca Gambetti Office: B3-1130 Edifici B Office hours: email:
More informationPart III Spectrum Estimation
ECE79-4 Part III Part III Spectrum Estimation 3. Parametric Methods for Spectral Estimation Electrical & Computer Engineering North Carolina State University Acnowledgment: ECE79-4 slides were adapted
More informationLecture Notes 5 Convergence and Limit Theorems. Convergence with Probability 1. Convergence in Mean Square. Convergence in Probability, WLLN
Lecture Notes 5 Convergence and Limit Theorems Motivation Convergence with Probability Convergence in Mean Square Convergence in Probability, WLLN Convergence in Distribution, CLT EE 278: Convergence and
More informationSTA 6857 ARIMA and SARIMA Models ( 3.8 and 3.9) Outline. Return Rate. US Gross National Product
STA 6857 ARIMA and SARIMA Models ( 3.8 and 3.9) Outline 1 Building ARIMA Models 2 SARIMA 3 Homework 4c Arthur Berg STA 6857 ARIMA and SARIMA Models ( 3.8 and 3.9) 2/ 34 Return Rate Suppose x t is the value
More informationStatistics 349(02) Review Questions
Statistics 349(0) Review Questions I. Suppose that for N = 80 observations on the time series { : t T} the following statistics were calculated: _ x = 10.54 C(0) = 4.99 In addition the sample autocorrelation
More informationECE302 Spring 2006 Practice Final Exam Solution May 4, Name: Score: /100
ECE302 Spring 2006 Practice Final Exam Solution May 4, 2006 1 Name: Score: /100 You must show ALL of your work for full credit. This exam is open-book. Calculators may NOT be used. 1. As a function of
More informationIntroduction to Time Series Analysis. Lecture 15.
Introduction to Time Series Analysis. Lecture 15. Spectral Analysis 1. Spectral density: Facts and examples. 2. Spectral distribution function. 3. Wold s decomposition. 1 Spectral Analysis Idea: decompose
More informationIntroduction to Time Series Analysis. Lecture 21.
Introduction to Time Series Analysis. Lecture 21. 1. Review: The periodogram, the smoothed periodogram. 2. Other smoothed spectral estimators. 3. Consistency. 4. Asymptotic distribution. 1 Review: Periodogram
More informationSimple Linear Regression
Simple Linear Regression In simple linear regression we are concerned about the relationship between two variables, X and Y. There are two components to such a relationship. 1. The strength of the relationship.
More informationLECTURE ON HAC COVARIANCE MATRIX ESTIMATION AND THE KVB APPROACH
LECURE ON HAC COVARIANCE MARIX ESIMAION AND HE KVB APPROACH CHUNG-MING KUAN Institute of Economics Academia Sinica October 20, 2006 ckuan@econ.sinica.edu.tw www.sinica.edu.tw/ ckuan Outline C.-M. Kuan,
More informationFourier Analysis of Stationary and Non-Stationary Time Series
Fourier Analysis of Stationary and Non-Stationary Time Series September 6, 2012 A time series is a stochastic process indexed at discrete points in time i.e X t for t = 0, 1, 2, 3,... The mean is defined
More informationξ t = Fξ t 1 + v t. then λ is less than unity in absolute value. In the above expression, A denotes the determinant of the matrix, A. 1 y t 1.
Christiano FINC 520, Spring 2009 Homework 3, due Thursday, April 23. 1. In class, we discussed the p th order VAR: y t = c + φ 1 y t 1 + φ 2 y t 2 +... + φ p y t p + ε t, where ε t is a white noise with
More informationESTIMATION OF DSGE MODELS WHEN THE DATA ARE PERSISTENT
ESTIMATION OF DSGE MODELS WHEN THE DATA ARE PERSISTENT Yuriy Gorodnichenko 1 Serena Ng 2 1 U.C. Berkeley 2 Columbia University May 2008 Outline Introduction Stochastic Growth Model Estimates Robust Estimators
More informationTopic 4 Unit Roots. Gerald P. Dwyer. February Clemson University
Topic 4 Unit Roots Gerald P. Dwyer Clemson University February 2016 Outline 1 Unit Roots Introduction Trend and Difference Stationary Autocorrelations of Series That Have Deterministic or Stochastic Trends
More informationEE538 Final Exam Fall :20 pm -5:20 pm PHYS 223 Dec. 17, Cover Sheet
EE538 Final Exam Fall 005 3:0 pm -5:0 pm PHYS 3 Dec. 17, 005 Cover Sheet Test Duration: 10 minutes. Open Book but Closed Notes. Calculators ARE allowed!! This test contains five problems. Each of the five
More informationBootstrap tests. Patrick Breheny. October 11. Bootstrap vs. permutation tests Testing for equality of location
Bootstrap tests Patrick Breheny October 11 Patrick Breheny STA 621: Nonparametric Statistics 1/14 Introduction Conditioning on the observed data to obtain permutation tests is certainly an important idea
More informationStochastic Modelling Solutions to Exercises on Time Series
Stochastic Modelling Solutions to Exercises on Time Series Dr. Iqbal Owadally March 3, 2003 Solutions to Elementary Problems Q1. (i) (1 0.5B)X t = Z t. The characteristic equation 1 0.5z = 0 does not have
More informationStochastic Processes
Elements of Lecture II Hamid R. Rabiee with thanks to Ali Jalali Overview Reading Assignment Chapter 9 of textbook Further Resources MIT Open Course Ware S. Karlin and H. M. Taylor, A First Course in Stochastic
More informationMAT 3379 (Winter 2016) FINAL EXAM (SOLUTIONS)
MAT 3379 (Winter 2016) FINAL EXAM (SOLUTIONS) 15 April 2016 (180 minutes) Professor: R. Kulik Student Number: Name: This is closed book exam. You are allowed to use one double-sided A4 sheet of notes.
More informationFourier Methods in Digital Signal Processing Final Exam ME 579, Spring 2015 NAME
Fourier Methods in Digital Signal Processing Final Exam ME 579, Instructions for this CLOSED BOOK EXAM 2 hours long. Monday, May 8th, 8-10am in ME1051 Answer FIVE Questions, at LEAST ONE from each section.
More informationSearching for Big Bang relicts with LIGO. Stefan W. Ballmer For the LSC/VIRGO collaboration GW2010, University of Minnesota October 16, 2010
Searching for Big Bang relicts with LIGO Stefan W. Ballmer For the LSC/VIRGO collaboration GW2010, University of Minnesota October 16, 2010 Outline The search for a stochastic GW background» Motivation»
More informationChapter 6: Nonparametric Time- and Frequency-Domain Methods. Problems presented by Uwe
System Identification written by L. Ljung, Prentice Hall PTR, 1999 Chapter 6: Nonparametric Time- and Frequency-Domain Methods Problems presented by Uwe System Identification Problems Chapter 6 p. 1/33
More information4 Spectral Analysis and Filtering
4 Spectral Analysis and Filtering 4.1 Introduction The notion that a time series exhibits repetitive or regular behavior over time is of fundamental importance because it distinguishes time series analysis
More informationAn Introduction to Wavelets with Applications in Environmental Science
An Introduction to Wavelets with Applications in Environmental Science Don Percival Applied Physics Lab, University of Washington Data Analysis Products Division, MathSoft overheads for talk available
More informationSingle Equation Linear GMM with Serially Correlated Moment Conditions
Single Equation Linear GMM with Serially Correlated Moment Conditions Eric Zivot October 28, 2009 Univariate Time Series Let {y t } be an ergodic-stationary time series with E[y t ]=μ and var(y t )
More informationwhere r n = dn+1 x(t)
Random Variables Overview Probability Random variables Transforms of pdfs Moments and cumulants Useful distributions Random vectors Linear transformations of random vectors The multivariate normal distribution
More informationIf we want to analyze experimental or simulated data we might encounter the following tasks:
Chapter 1 Introduction If we want to analyze experimental or simulated data we might encounter the following tasks: Characterization of the source of the signal and diagnosis Studying dependencies Prediction
More informationApplied Regression. Applied Regression. Chapter 2 Simple Linear Regression. Hongcheng Li. April, 6, 2013
Applied Regression Chapter 2 Simple Linear Regression Hongcheng Li April, 6, 2013 Outline 1 Introduction of simple linear regression 2 Scatter plot 3 Simple linear regression model 4 Test of Hypothesis
More informationEmpirical Market Microstructure Analysis (EMMA)
Empirical Market Microstructure Analysis (EMMA) Lecture 3: Statistical Building Blocks and Econometric Basics Prof. Dr. Michael Stein michael.stein@vwl.uni-freiburg.de Albert-Ludwigs-University of Freiburg
More informationECE 636: Systems identification
ECE 636: Systems identification Lectures 3 4 Random variables/signals (continued) Random/stochastic vectors Random signals and linear systems Random signals in the frequency domain υ ε x S z + y Experimental
More informationLevinson Durbin Recursions: I
Levinson Durbin Recursions: I note: B&D and S&S say Durbin Levinson but Levinson Durbin is more commonly used (Levinson, 1947, and Durbin, 1960, are source articles sometimes just Levinson is used) recursions
More informationCircle a single answer for each multiple choice question. Your choice should be made clearly.
TEST #1 STA 4853 March 4, 215 Name: Please read the following directions. DO NOT TURN THE PAGE UNTIL INSTRUCTED TO DO SO Directions This exam is closed book and closed notes. There are 31 questions. Circle
More informationSemester , Example Exam 1
Semester 1 2017, Example Exam 1 1 of 10 Instructions The exam consists of 4 questions, 1-4. Each question has four items, a-d. Within each question: Item (a) carries a weight of 8 marks. Item (b) carries
More information8.2 Harmonic Regression and the Periodogram
Chapter 8 Spectral Methods 8.1 Introduction Spectral methods are based on thining of a time series as a superposition of sinusoidal fluctuations of various frequencies the analogue for a random process
More informationFINANCIAL ECONOMETRICS AND EMPIRICAL FINANCE -MODULE2 Midterm Exam Solutions - March 2015
FINANCIAL ECONOMETRICS AND EMPIRICAL FINANCE -MODULE2 Midterm Exam Solutions - March 205 Time Allowed: 60 minutes Family Name (Surname) First Name Student Number (Matr.) Please answer all questions by
More informationFigure 18: Top row: example of a purely continuous spectrum (left) and one realization
1..5 S(). -.2 -.5 -.25..25.5 64 128 64 128 16 32 requency time time Lag 1..5 S(). -.5-1. -.5 -.1.1.5 64 128 64 128 16 32 requency time time Lag Figure 18: Top row: example o a purely continuous spectrum
More informationNonstationary time series models
13 November, 2009 Goals Trends in economic data. Alternative models of time series trends: deterministic trend, and stochastic trend. Comparison of deterministic and stochastic trend models The statistical
More informationAutomatic Autocorrelation and Spectral Analysis
Piet M.T. Broersen Automatic Autocorrelation and Spectral Analysis With 104 Figures Sprin ger 1 Introduction 1 1.1 Time Series Problems 1 2 Basic Concepts 11 2.1 Random Variables 11 2.2 Normal Distribution
More informationNew Introduction to Multiple Time Series Analysis
Helmut Lütkepohl New Introduction to Multiple Time Series Analysis With 49 Figures and 36 Tables Springer Contents 1 Introduction 1 1.1 Objectives of Analyzing Multiple Time Series 1 1.2 Some Basics 2
More informationAnalysis. Components of a Time Series
Module 8: Time Series Analysis 8.2 Components of a Time Series, Detection of Change Points and Trends, Time Series Models Components of a Time Series There can be several things happening simultaneously
More informationDigital Image Processing
Digital Image Processing 2D SYSTEMS & PRELIMINARIES Hamid R. Rabiee Fall 2015 Outline 2 Two Dimensional Fourier & Z-transform Toeplitz & Circulant Matrices Orthogonal & Unitary Matrices Block Matrices
More informationSTA441: Spring Multiple Regression. This slide show is a free open source document. See the last slide for copyright information.
STA441: Spring 2018 Multiple Regression This slide show is a free open source document. See the last slide for copyright information. 1 Least Squares Plane 2 Statistical MODEL There are p-1 explanatory
More information10. Time series regression and forecasting
10. Time series regression and forecasting Key feature of this section: Analysis of data on a single entity observed at multiple points in time (time series data) Typical research questions: What is the
More informationStat 5101 Notes: Algorithms (thru 2nd midterm)
Stat 5101 Notes: Algorithms (thru 2nd midterm) Charles J. Geyer October 18, 2012 Contents 1 Calculating an Expectation or a Probability 2 1.1 From a PMF........................... 2 1.2 From a PDF...........................
More informationTime Series Analysis
Time Series Analysis hm@imm.dtu.dk Informatics and Mathematical Modelling Technical University of Denmark DK-2800 Kgs. Lyngby 1 Outline of the lecture Input-Output systems The z-transform important issues
More informationProbability Distributions & Sampling Distributions
GOV 2000 Section 4: Probability Distributions & Sampling Distributions Konstantin Kashin 1 Harvard University September 26, 2012 1 These notes and accompanying code draw on the notes from Molly Roberts,
More informationE 4101/5101 Lecture 6: Spectral analysis
E 4101/5101 Lecture 6: Spectral analysis Ragnar Nymoen 3 March 2011 References to this lecture Hamilton Ch 6 Lecture note (on web page) For stationary variables/processes there is a close correspondence
More informationTime Series Examples Sheet
Lent Term 2001 Richard Weber Time Series Examples Sheet This is the examples sheet for the M. Phil. course in Time Series. A copy can be found at: http://www.statslab.cam.ac.uk/~rrw1/timeseries/ Throughout,
More information3 Theory of stationary random processes
3 Theory of stationary random processes 3.1 Linear filters and the General linear process A filter is a transformation of one random sequence {U t } into another, {Y t }. A linear filter is a transformation
More informationSignal interactions Cross correlation, cross spectral coupling and significance testing Centre for Doctoral Training in Healthcare Innovation
Signal interactions Cross correlation, cross spectral coupling and significance testing Centre for Doctoral Training in Healthcare Innovation Dr. Gari D. Clifford, University Lecturer & Director, Centre
More informationSimple and Multiple Linear Regression
Sta. 113 Chapter 12 and 13 of Devore March 12, 2010 Table of contents 1 Simple Linear Regression 2 Model Simple Linear Regression A simple linear regression model is given by Y = β 0 + β 1 x + ɛ where
More informationEcon 1123: Section 2. Review. Binary Regressors. Bivariate. Regression. Omitted Variable Bias
Contact Information Elena Llaudet Sections are voluntary. My office hours are Thursdays 5pm-7pm in Littauer Mezzanine 34-36 (Note room change) You can email me administrative questions to ellaudet@gmail.com.
More informationTime series models in the Frequency domain. The power spectrum, Spectral analysis
ime series models in the Frequency domain he power spectrum, Spectral analysis Relationship between the periodogram and the autocorrelations = + = ( ) ( ˆ α ˆ ) β I Yt cos t + Yt sin t t= t= ( ( ) ) cosλ
More informationEASTERN MEDITERRANEAN UNIVERSITY ECON 604, FALL 2007 DEPARTMENT OF ECONOMICS MEHMET BALCILAR ARIMA MODELS: IDENTIFICATION
ARIMA MODELS: IDENTIFICATION A. Autocorrelations and Partial Autocorrelations 1. Summary of What We Know So Far: a) Series y t is to be modeled by Box-Jenkins methods. The first step was to convert y t
More information