State Space Time Series Analysis

Size: px
Start display at page:

Download "State Space Time Series Analysis"

Transcription

1 State Space Time Series Analysis p. 1 State Space Time Series Analysis Siem Jan Koopman Department of Econometrics VU University Amsterdam Tinbergen Institute 2011

2 State Space Time Series Analysis p. 2 Classical Decomposition A basic model for representing a time series is the additive model y t = µ t +γ t +ε t, t = 1,...,n, also known as the Classical Decomposition. y t = observation, µ t = slowly changing component (trend), γ t = periodic component (seasonal), ε t = irregular component (disturbance). In a Structural Time Series Model (STSM) or Unobserved Components Model (UCM), the RHS components are modelled explicitly as stochastic processes.

3 State Space Time Series Analysis p. 3 Local Level Model Components can be deterministic functions of time (e.g. polynomials), or stochastic processes; Deterministic example: y t = µ+ε t with ε t NID(0,σ 2 ε). Stochastic example: the Random Walk plus Noise, or Local Level model: y t = µ t +ε t, ε t NID(0,σ 2 ε) µ t+1 = µ t +η t, η t NID(0,σ 2 η), The disturbances ε t, η s are independent for all s,t; The model is incomplete without a specification for µ 1 (note the non-stationarity): µ 1 N(a,P)

4 State Space Time Series Analysis p. 4 Local Level Model y t = µ t +ε t, ε t NID(0,σ 2 ε) µ t+1 = µ t +η t, η t NID(0,σ 2 η), µ 1 N(a,P) The level µ t and the irregular ε t are unobserved; Parameters: σ 2 ε,σ 2 η; Trivial special cases: σ 2 η = 0 = y t NID(µ 1,σ 2 ε) (WN with constant level); σ 2 ε = 0 = y t+1 = y t +η t (pure RW); Local Level is a model representation for EWMA forecasting.

5 State Space Time Series Analysis p. 5 Local Linear Trend Model The LLT model extends the LL model with a slope: y t = µ t +ε t, ε t NID(0,σ 2 ε), µ t+1 = β t +µ t +η t, η t NID(0,σ 2 η), β t+1 = β t +ξ t, ξ t NID(0,σ 2 ξ). All disturbances are independent at all lags and leads; Initial distributions β 1,µ 1 need to specified; If σξ 2 = 0 the trend is a random walk with constant drift β 1; (For β 1 = 0 the model reduces to a LL model.) If additionally ση 2 = 0 the trend is a straight line with slope β 1 and intercept µ 1 ; If σξ 2 > 0 but σ2 η = 0, the trend is a smooth curve, or an Integrated Random Walk;

6 State Space Time Series Analysis p. 6 Trend and Slope in LLT Model 5.0 µ β

7 State Space Time Series Analysis p. 7 Trend and Slope in Integrated Random Walk Model 10 µ β

8 State Space Time Series Analysis p. 8 Local Linear Trend Model Reduced form of LLT is ARIMA(0,2,2); LLT provides a model for Holt-Winters forecasting; Smooth LLT provides a model for spline-fitting; Smoother trends: higher order Random Walks d µ t = η t

9 State Space Time Series Analysis p. 9 Seasonal Effects We have seen specifications for µ t in the basic model y t = µ t +γ t +ε t. Now we will consider the seasonal term γ t. Let s denote the number of seasons in the data: s = 12 for monthly data, s = 4 for quarterly data, s = 7 for daily data when modelling a weekly pattern.

10 State Space Time Series Analysis p. 10 Dummy Seasonal The simplest way to model seasonal effects is by using dummy variables. The effect summed over the seasons should equal zero: γ t+1 = s 1 j=1 γ t+1 j. To allow the pattern to change over time, we introduce a new disturbance term: γ t+1 = s 1 j=1 γ t+1 j +ω t, ω t NID(0,σ 2 ω). The expectation of the sum of the seasonal effects is zero.

11 State Space Time Series Analysis p. 11 Trigonometric Seasonal Defining γ jt as the effect of season j at time t, an alternative specification for the seasonal pattern is γ t = [s/2] j=1 γ jt, γ j,t+1 = γ jt cosλ j +γ jtsinλ j +ω jt, γ j,t+1 = γ jt sinλ j +γ jtcosλ j +ω jt, ω jt,ω jt NID(0,σ 2 ω), λ j = 2πj/s. Without the disturbance, the trigonometric specification is identical to the deterministic dummy specification. The autocorrelation in the trigonometric specification lasts through more lags: changes occur in a smoother way;

12 State Space Time Series Analysis p. 12 Seatbelt Law

13 State Space Time Series Analysis p. 13 State Space Model Linear Gaussian state space model (LGSSM) is defined in three parts: State equation: Observation equation: α t+1 = T t α t +R t ζ t, ζ t NID(0,Q t ), y t = Z t α t +ε t, ε t NID(0,H t ), Initial state distribution α 1 N(a 1,P 1 ). Notice that ζ t and ε s independent for all t,s, and independent from α 1 ; observation y t can be multivariate; state vector α t is unobserved; matrices T t,z t,r t,q t,h t determine structure of model.

14 State Space Time Series Analysis p. 14 State Space Model state space model is linear and Gaussian: therefore properties and results of multivariate normal distribution apply; state vector α t evolves as a VAR(1) process; system matrices usually contain unknown parameters; estimation has therefore two aspects: measuring the unobservable state (prediction, filtering and smoothing); estimation of unknown parameters (maximum likelihood estimation); state space methods offer a unified approach to a wide range of models and techniques: dynamic regression, ARIMA, UC models, latent variable models, spline-fitting and many ad-hoc filters; next, some well-known model specifications in state space form...

15 State Space Time Series Analysis p. 15 Regression with Time Varying Coefficients General state space model: α t+1 = T t α t +R t ζ t, ζ t NID(0,Q t ), y t = Z t α t +ε t, ε t NID(0,H t ). Put regressors in Z t, T t = I, R t = I, Result is regression model with coefficient α t following a random walk.

16 State Space Time Series Analysis p. 16 ARMA in State Space Form Example: AR(2) model y t+1 = φ 1 y t +φ 2 y t 1 +ζ t, in state space: α t+1 = T t α t +R t ζ t, ζ t NID(0,Q t ), y t = Z t α t +ε t, ε t NID(0,H t ). with 2 1 state vector α t and system matrices: [ ] Z t = 1 0, H t = 0 [ ] [ ] φ T t =, R t =, Q t = σ 2 φ Z t and H t = 0 imply that α 1t = y t ; First state equation implies y t+1 = φ 1 y t +α 2t +ζ t with ζ t NID(0,σ 2 ); Second state equation implies α 2,t+1 = φ 2 y t ;

17 State Space Time Series Analysis p. 17 ARMA in State Space Form Example: MA(1) model y t+1 = ζ t +θζ t 1, in state space: α t+1 = T t α t +R t ζ t, ζ t NID(0,Q t ), y t = Z t α t +ε t, ε t NID(0,H t ). with 2 1 state vector α t and system matrices: [ ] Z t = 1 0, H t = 0 [ ] [ ] T t =, R t =, Q t = σ θ Z t and H t = 0 imply that α 1t = y t ; First state equation implies y t+1 = α 2t +ζ t with ζ t NID(0,σ 2 ); Second state equation implies α 2,t+1 = θζ t ;

18 State Space Time Series Analysis p. 18 ARMA in State Space Form Example: ARMA(2,1) model in state space form y t = φ 1 y t 1 +φ 2 y t 2 +ζ t +θζ t 1 [ y t ] α t = φ 2 y t 1 +θζ t [ ] Z t = 1 0, H t = 0, [ ] [ ] φ T t =, R t =, Q t = σ 2 φ 2 0 θ All ARIMA(p, d, q) models have a (non-unique) state space representation.

19 State Space Time Series Analysis p. 19 UC models in State Space Form State space model: α t+1 = T t α t +R t ζ t, y t = Z t α t +ε t. LL model µ t+1 = η t and y t = µ t +ε t : α t = µ t, T t = 1, R t = 1, Q t = σ 2 η, Z t = 1, H t = σ 2 ε. LLT model µ t+1 = β t +η t, β t+1 = ξ t and y t = µ t +ε t : α t = [ µ t β t ] [ ] [ 1 1, T t =, R t = 0 1 [ ] Z t = 1 0, H t = σε ], Q t = [ ] ση σξ 2,

20 State Space Time Series Analysis p. 20 UC models in State Space Form State space model: α t+1 = T t α t +R t ζ t, y t = Z t α t +ε t. LLT model with season: µ t+1 = β t +η t, β t+1 = ξ t, S(L)γ t+1 = ω t and y t = µ t +γ t +ε t : α t = [ µ t β t γ t γ t 1 γ t 2 ], ση T t = , Q t = 0 σξ 2 0, R t = 0 0 1, σω [ ] Z t = , H t = σε. 2

21 State Space Time Series Analysis p. 21 Kalman Filter The Kalman filter calculates the mean and variance of the unobserved state, given the observations. The state is Gaussian: the complete distribution is characterized by the mean and variance. The filter is a recursive algorithm; the current best estimate is updated whenever a new observation is obtained. To start the recursion, we need a 1 and P 1, which we assumed given. There are various ways to initialize when a 1 and P 1 are unknown, which we will not discuss here.

22 State Space Time Series Analysis p. 22 Kalman Filter The unobserved state α t can be estimated from the observations with the Kalman filter: v t = y t Z t a t, F t = Z t P t Z t +H t, K t = T t P t Z tf 1 t, a t+1 = T t a t +K t v t, P t+1 = T t P t T t +R t Q t R t K t F t K t, for t = 1,...,n and starting with given values for a 1 and P 1. Writing Y t = {y 1,...,y t }, a t+1 = E(α t+1 Y t ), P t+1 = var(α t+1 Y t ).

23 State Space Time Series Analysis p. 23 Kalman Filter State space model: α t+1 = T t α t +R t ζ t, y t = Z t α t +ε t. Writing Y t = {y 1,...,y t }, define a t+1 = E(α t+1 Y t ), P t+1 = var(α t+1 Y t ); The prediction error is v t = y t E(y t Y t 1 ) = y t E(Z t α t +ε t Y t 1 ) = y t Z t E(α t Y t 1 ) = y t Z t a t ; It follows that v t = Z t (α t a t )+ε t and E(v t ) = 0; The prediction error variance is F t = var(v t ) = Z t P t Z t +H t.

24 State Space Time Series Analysis p. 24 Lemma The proof of the Kalman filter uses a lemma from multivariate Normal regression theory. Lemma Suppose x, y and z are jointly Normally distributed vectors with E(z) = 0 and Σ yz = 0. Then E(x y,z) = E(x y)+σ xz Σ 1 zz z, var(x y,z) = var(x y) Σ xz Σ 1 zz Σ xz,

25 State Space Time Series Analysis p. 25 Multivariate local level model Seemingly Unrelated Time Series Equations model: y t = µ t +ε t, ε t NID(0,Σ ε ), µ t+1 = µ t +η t, η t NID(0,Σ η ). Observations are p 1 vectors; The disturbances ε t, η s are independent for all s,t; The p different time series are related through correlations in the disturbances. For a full discussion, see Harvey and Koopman (1997). A pdf version (scanned) at under section Publications and subsection Published articles as contributions to books.

26 State Space Time Series Analysis p. 26 Multivariate LL Model The multivariate LL model is given by y t = µ t +ε t, ε t NID(0,Σ ε ), µ t+1 = µ t +η t, η t NID(0,Σ η ). First difference y t = η t 1 + ε t is stationary; Reduced form: y t is VMA(1) or VAR( );

27 State Space Time Series Analysis p. 27 Multivariate LL Model Stochastic properties are multivariate analogous of univariate case: Γ 0 = E( y t y t) = Σ η +2Σ ε Γ 1 = E( y t y t 1) = Σ ε Γ τ = E( y t y t τ) = 0, τ 2, The unrestricted vector MA(1) process has p 2 +p(p+1)/2 parameters, the SUTSE has p (p+1); Such multivariate reduced form representations can also be established for general models.

28 State Space Time Series Analysis p. 28 Homogeneous Multivariate LL Model The homogeneous multivariate LL model is given by y t = µ t +ε t, ε t NID(0,Σ ε ), µ t+1 = µ t +η t, η t NID(0,qΣ ε ), where q is a non-negative scalar. This implies that Σ η = qσ ε. The model is restricted, all series in y t have the same dynamic properties (the same acf). Not so relevant in practical work apart from forecasting. It is the model representation for exponentially weighted moving average (EWMA) forecasting of multiple time series. This can be generalised to more general components models. Easy to estimate, only a set of univariate Kalman filters are required.

29 State Space Time Series Analysis p. 29 Common Levels The common local level model is given by y t = µ t +ε t, ε t NID(0,Σ ε ), µ t+1 = µ t +η t, η t NID(0,Σ η ), where rank(σ η ) = r < p. the model can be described by r underlying level components, the common levels; Σ η = AΣ c A, A is p r, Σ c is r r of full rank; interpretation of A: factor loading matrix.

30 State Space Time Series Analysis p. 30 Common Levels The common local level model y t = µ t +ε t, ε t NID(0,Σ ε ), µ t+1 = µ t +η t, η t NID(0,AΣ c A ), can be rewritten in terms of underlying levels: y t = a+aµ c t +ε t, µ c t+1 = µ c t +η c t, η c t NID(0,Σ c ), so that µ t = a+aµ c t, η t = Aη c t.

31 State Space Time Series Analysis p. 31 Common Levels For the common local level model y t = µ t +ε t, ε t NID(0,Σ ε ), µ t+1 = µ t +η t, η t NID(0,AΣ c A ), notice that decomposition Σ η = AΣ c A is not unique; identification restrictions: Σ c is diagonal, Choleski decomposition, principal compoments (based on economic theory); more interesting interpretation can be obtained by factor rotations; can be interpreted as dynamic factor analysis, see later.

32 State Space Time Series Analysis p. 32 Common components Common dynamic factors: are useful for interpretation cointegration; have consequence for inference and forecasting (dimension of parameter space reduces as a result). common local level model can be generally represented as a VAR( ) or VECM models, details can be provided upon request.

33 State Space Time Series Analysis p. 33 Multivariate components So far, we have concentrated on multivariate variants of the local level model; Similar considerations can be applied to other components such as the slope of the trend, seasonal and cycle components and other time-varying features in the multiple time series. Harvey and Koopman (1997) review such extensions. In particular, they define the similar cycle component, see Exercises.

34 State Space Time Series Analysis p. 34 Common and idiosyncratic factors Multiple trends can also be decomposed into a one common factor and multiple idiosyncratic factors: y t = µ t +ε t, ε t NID(0,Σ ε ), µ t+1 = µ t +η t, η t NID(0,Σ η ), where Σ η = δδ +D η with vector δ and diagonal matrix D η. This implies that the level can be represented by µ t = δµ c t +µ t, η t = δη c t +η t with common level (scalar) µ c t and independent level µ it generated by µ c t+1 = η c t NID(0,1), µ t+1 = η t NID(0,D η ).

35 State Space Time Series Analysis p. 35 Mulitvariate Kalman filter The Kalman filter is valid for the general multivariate state space model. Computationally it is not convenient when p becomes large, very large. Each step of the Kalman filter requires the inversion of the p p matrix F t. This is no problem when p = 1 (univariate) but when p > 20, say, it will slow down the Kalman filter considerably. However, we can treat each element in the p 1 observation vector y t as a single realisation. In other words, we can "update" each single element of y t within the Kalman filter. The arguments are given in DK book 6.4. The same applies to smoothing.

36 State Space Time Series Analysis p. 36 Univariate treatment of Kalman filter Consider standard model: y t = Z t α t +ε t and α t+1 = T t α t +R t η t where Var(ε t ) = H t is diagonal. Observation vector y t = (y t,1,...,y t,pt ) is treated and we view observation model as a set of p t separate equations. We then have, y t,i = Z t,i α t,i +ε t,i with α t,i = α t for i = 1,...,p t. The associated transition equations become α t,i+1 = α t,i for i = 1,...,p t and α t+1,1 = T t α t,pt +R t η t for t = 1,...,n. This disentangled model can be treated by the Kalman filter and smoother equations straightforwardly. Innovations are now relative to the past and the previous observations inside y t,pt! Non-diagonal matrix H t can be treated by data-transformation or by including ε t in the state vector α t. More details in DK book 6.4.

37 State Space Time Series Analysis p. 37 Exercise 1 Consider the common trends model of Harvey and Koopman (1997, and 9.4.2). 1. Put the common trends model with (possibly common) stochastic slopes and based on equations (21)-(23) in state space form. 2. Put the common trends model with (possibly common) stochastic slopes and based on equations (24)-(26) in state space form. Define all vectors and matrices precisely. 3. Discuss the generalisation of Σ η 0 and the consequences for the state space formulation of the model as in 2.

38 State Space Time Series Analysis p. 38 Exercise 2 Consider the multivariate trend model of Harvey and Koopman (1997). 1. Consider a multiple data set of N time series y t. The aim is to decompose the time series into trend and stationary components. It is further required that the multiple trend can be decomposed into a common single trend (common to all N time series) and idiosyncratic trends (specific to the individual time series). Formulate a model for such a decomposition. Discuss the identification of the different trends. Express the model in state space form. 2. Once multiple trend models are expressed in state space form, we need to estimate the parameter coefficients of the model. Please describe shortly some relevant issues of maximum likelihood estimation. Is it feasible? What problems can you expect? Any recommendations for a successful implementation?

39 State Space Time Series Analysis p. 39 Exercise 3 Consider the similar cycle model of Harvey and Koopman (1997) with observation equation y t = ψ t +ε t, ε t N(0,Σ ε ), where y t is a 3 1 observation vector. Cycle ψ t represents a common similar cycle component of rank Please provide the state space representation of this model. 2. Comment on the restrictive nature of the similar cycle model. 3. How would you modify the similar cycle model so that each time series in y t has a different cycle frequency λ. 4. Can you apply the univariate Kalman filter of DK 6.4 in case Σ ε is diagonal? What if Σ ε is not diagonal? Give details.

Lecture 4: Seasonal Time Series, Trend Analysis & Component Model Bus 41910, Time Series Analysis, Mr. R. Tsay

Lecture 4: Seasonal Time Series, Trend Analysis & Component Model Bus 41910, Time Series Analysis, Mr. R. Tsay Lecture 4: Seasonal Time Series, Trend Analysis & Component Model Bus 41910, Time Series Analysis, Mr. R. Tsay Business cycle plays an important role in economics. In time series analysis, business cycle

More information

Analysis of Historical Time Series with Messy Features: The Case of Commodity Prices in Babylonia

Analysis of Historical Time Series with Messy Features: The Case of Commodity Prices in Babylonia Analysis of Historical Time Series with Messy Features: The Case of Commodity Prices in Babylonia Siem Jan Koopman and Lennart Hoogerheide VU University Amsterdam, The Netherlands Tinbergen Institute,

More information

Financial TIme Series Analysis: Part II

Financial TIme Series Analysis: Part II Department of Mathematics and Statistics, University of Vaasa, Finland January 29 February 13, 2015 Feb 14, 2015 1 Univariate linear stochastic models: further topics Unobserved component model Signal

More information

11. Time series and dynamic linear models

11. Time series and dynamic linear models 11. Time series and dynamic linear models Objective To introduce the Bayesian approach to the modeling and forecasting of time series. Recommended reading West, M. and Harrison, J. (1997). models, (2 nd

More information

1 Short Introduction to Time Series

1 Short Introduction to Time Series ECONOMICS 7344, Spring 202 Bent E. Sørensen January 24, 202 Short Introduction to Time Series A time series is a collection of stochastic variables x,.., x t,.., x T indexed by an integer value t. The

More information

Univariate and Multivariate Methods PEARSON. Addison Wesley

Univariate and Multivariate Methods PEARSON. Addison Wesley Time Series Analysis Univariate and Multivariate Methods SECOND EDITION William W. S. Wei Department of Statistics The Fox School of Business and Management Temple University PEARSON Addison Wesley Boston

More information

Rob J Hyndman. Forecasting using. 11. Dynamic regression OTexts.com/fpp/9/1/ Forecasting using R 1

Rob J Hyndman. Forecasting using. 11. Dynamic regression OTexts.com/fpp/9/1/ Forecasting using R 1 Rob J Hyndman Forecasting using 11. Dynamic regression OTexts.com/fpp/9/1/ Forecasting using R 1 Outline 1 Regression with ARIMA errors 2 Example: Japanese cars 3 Using Fourier terms for seasonality 4

More information

Chapter 4: Vector Autoregressive Models

Chapter 4: Vector Autoregressive Models Chapter 4: Vector Autoregressive Models 1 Contents: Lehrstuhl für Department Empirische of Wirtschaftsforschung Empirical Research and und Econometrics Ökonometrie IV.1 Vector Autoregressive Models (VAR)...

More information

SYSTEMS OF REGRESSION EQUATIONS

SYSTEMS OF REGRESSION EQUATIONS SYSTEMS OF REGRESSION EQUATIONS 1. MULTIPLE EQUATIONS y nt = x nt n + u nt, n = 1,...,N, t = 1,...,T, x nt is 1 k, and n is k 1. This is a version of the standard regression model where the observations

More information

Time Series Analysis

Time Series Analysis Time Series Analysis Forecasting with ARIMA models Andrés M. Alonso Carolina García-Martos Universidad Carlos III de Madrid Universidad Politécnica de Madrid June July, 2012 Alonso and García-Martos (UC3M-UPM)

More information

Time Series Analysis

Time Series Analysis Time Series Analysis Identifying possible ARIMA models Andrés M. Alonso Carolina García-Martos Universidad Carlos III de Madrid Universidad Politécnica de Madrid June July, 2012 Alonso and García-Martos

More information

The VAR models discussed so fare are appropriate for modeling I(0) data, like asset returns or growth rates of macroeconomic time series.

The VAR models discussed so fare are appropriate for modeling I(0) data, like asset returns or growth rates of macroeconomic time series. Cointegration The VAR models discussed so fare are appropriate for modeling I(0) data, like asset returns or growth rates of macroeconomic time series. Economic theory, however, often implies equilibrium

More information

Introduction to time series analysis

Introduction to time series analysis Introduction to time series analysis Margherita Gerolimetto November 3, 2010 1 What is a time series? A time series is a collection of observations ordered following a parameter that for us is time. Examples

More information

Introduction to Time Series Analysis. Lecture 1.

Introduction to Time Series Analysis. Lecture 1. Introduction to Time Series Analysis. Lecture 1. Peter Bartlett 1. Organizational issues. 2. Objectives of time series analysis. Examples. 3. Overview of the course. 4. Time series models. 5. Time series

More information

Analysis of algorithms of time series analysis for forecasting sales

Analysis of algorithms of time series analysis for forecasting sales SAINT-PETERSBURG STATE UNIVERSITY Mathematics & Mechanics Faculty Chair of Analytical Information Systems Garipov Emil Analysis of algorithms of time series analysis for forecasting sales Course Work Scientific

More information

Software Review: ITSM 2000 Professional Version 6.0.

Software Review: ITSM 2000 Professional Version 6.0. Lee, J. & Strazicich, M.C. (2002). Software Review: ITSM 2000 Professional Version 6.0. International Journal of Forecasting, 18(3): 455-459 (June 2002). Published by Elsevier (ISSN: 0169-2070). http://0-

More information

Java Modules for Time Series Analysis

Java Modules for Time Series Analysis Java Modules for Time Series Analysis Agenda Clustering Non-normal distributions Multifactor modeling Implied ratings Time series prediction 1. Clustering + Cluster 1 Synthetic Clustering + Time series

More information

Estimation and Inference in Cointegration Models Economics 582

Estimation and Inference in Cointegration Models Economics 582 Estimation and Inference in Cointegration Models Economics 582 Eric Zivot May 17, 2012 Tests for Cointegration Let the ( 1) vector Y be (1). Recall, Y is cointegrated with 0 cointegrating vectors if there

More information

3. Regression & Exponential Smoothing

3. Regression & Exponential Smoothing 3. Regression & Exponential Smoothing 3.1 Forecasting a Single Time Series Two main approaches are traditionally used to model a single time series z 1, z 2,..., z n 1. Models the observation z t as a

More information

Time Series Analysis

Time Series Analysis JUNE 2012 Time Series Analysis CONTENT A time series is a chronological sequence of observations on a particular variable. Usually the observations are taken at regular intervals (days, months, years),

More information

Centre for Central Banking Studies

Centre for Central Banking Studies Centre for Central Banking Studies Technical Handbook No. 4 Applied Bayesian econometrics for central bankers Andrew Blake and Haroon Mumtaz CCBS Technical Handbook No. 4 Applied Bayesian econometrics

More information

Chapter 6: Multivariate Cointegration Analysis

Chapter 6: Multivariate Cointegration Analysis Chapter 6: Multivariate Cointegration Analysis 1 Contents: Lehrstuhl für Department Empirische of Wirtschaftsforschung Empirical Research and und Econometrics Ökonometrie VI. Multivariate Cointegration

More information

Advanced Forecasting Techniques and Models: ARIMA

Advanced Forecasting Techniques and Models: ARIMA Advanced Forecasting Techniques and Models: ARIMA Short Examples Series using Risk Simulator For more information please visit: www.realoptionsvaluation.com or contact us at: admin@realoptionsvaluation.com

More information

7 Time series analysis

7 Time series analysis 7 Time series analysis In Chapters 16, 17, 33 36 in Zuur, Ieno and Smith (2007), various time series techniques are discussed. Applying these methods in Brodgar is straightforward, and most choices are

More information

Time Series Analysis III

Time Series Analysis III Lecture 12: Time Series Analysis III MIT 18.S096 Dr. Kempthorne Fall 2013 MIT 18.S096 Time Series Analysis III 1 Outline Time Series Analysis III 1 Time Series Analysis III MIT 18.S096 Time Series Analysis

More information

Temporal Disaggregation Using Multivariate Structural Time Series Models

Temporal Disaggregation Using Multivariate Structural Time Series Models Temporal Disaggregation Using Multivariate Structural Time Series Models Filippo Moauro Istituto Nazionale di Statistica, Istat Via A. Depretis 74/B, 00184 Roma, Italia (moauro@istat.it) Giovanni Savio

More information

Time Series - ARIMA Models. Instructor: G. William Schwert

Time Series - ARIMA Models. Instructor: G. William Schwert APS 425 Fall 25 Time Series : ARIMA Models Instructor: G. William Schwert 585-275-247 schwert@schwert.ssb.rochester.edu Topics Typical time series plot Pattern recognition in auto and partial autocorrelations

More information

2.2 Elimination of Trend and Seasonality

2.2 Elimination of Trend and Seasonality 26 CHAPTER 2. TREND AND SEASONAL COMPONENTS 2.2 Elimination of Trend and Seasonality Here we assume that the TS model is additive and there exist both trend and seasonal components, that is X t = m t +

More information

Trend and Seasonal Components

Trend and Seasonal Components Chapter 2 Trend and Seasonal Components If the plot of a TS reveals an increase of the seasonal and noise fluctuations with the level of the process then some transformation may be necessary before doing

More information

A Regime-Switching Model for Electricity Spot Prices. Gero Schindlmayr EnBW Trading GmbH g.schindlmayr@enbw.com

A Regime-Switching Model for Electricity Spot Prices. Gero Schindlmayr EnBW Trading GmbH g.schindlmayr@enbw.com A Regime-Switching Model for Electricity Spot Prices Gero Schindlmayr EnBW Trading GmbH g.schindlmayr@enbw.com May 31, 25 A Regime-Switching Model for Electricity Spot Prices Abstract Electricity markets

More information

Advanced time-series analysis

Advanced time-series analysis UCL DEPARTMENT OF SECURITY AND CRIME SCIENCE Advanced time-series analysis Lisa Tompson Research Associate UCL Jill Dando Institute of Crime Science l.tompson@ucl.ac.uk Overview Fundamental principles

More information

y t by left multiplication with 1 (L) as y t = 1 (L) t =ª(L) t 2.5 Variance decomposition and innovation accounting Consider the VAR(p) model where

y t by left multiplication with 1 (L) as y t = 1 (L) t =ª(L) t 2.5 Variance decomposition and innovation accounting Consider the VAR(p) model where . Variance decomposition and innovation accounting Consider the VAR(p) model where (L)y t = t, (L) =I m L L p L p is the lag polynomial of order p with m m coe±cient matrices i, i =,...p. Provided that

More information

Time Series Analysis in Economics. Klaus Neusser

Time Series Analysis in Economics. Klaus Neusser Time Series Analysis in Economics Klaus Neusser May 26, 2015 Contents I Univariate Time Series Analysis 3 1 Introduction 1 1.1 Some examples.......................... 2 1.2 Formal definitions.........................

More information

A Multiplicative Seasonal Box-Jenkins Model to Nigerian Stock Prices

A Multiplicative Seasonal Box-Jenkins Model to Nigerian Stock Prices A Multiplicative Seasonal Box-Jenkins Model to Nigerian Stock Prices Ette Harrison Etuk Department of Mathematics/Computer Science, Rivers State University of Science and Technology, Nigeria Email: ettetuk@yahoo.com

More information

Univariate Time Series Analysis; ARIMA Models

Univariate Time Series Analysis; ARIMA Models Econometrics 2 Spring 25 Univariate Time Series Analysis; ARIMA Models Heino Bohn Nielsen of4 Outline of the Lecture () Introduction to univariate time series analysis. (2) Stationarity. (3) Characterizing

More information

Chapter 1. Vector autoregressions. 1.1 VARs and the identi cation problem

Chapter 1. Vector autoregressions. 1.1 VARs and the identi cation problem Chapter Vector autoregressions We begin by taking a look at the data of macroeconomics. A way to summarize the dynamics of macroeconomic data is to make use of vector autoregressions. VAR models have become

More information

Analysis of Bayesian Dynamic Linear Models

Analysis of Bayesian Dynamic Linear Models Analysis of Bayesian Dynamic Linear Models Emily M. Casleton December 17, 2010 1 Introduction The main purpose of this project is to explore the Bayesian analysis of Dynamic Linear Models (DLMs). The main

More information

Lecture 1: Univariate Time Series B41910: Autumn Quarter, 2008, by Mr. Ruey S. Tsay

Lecture 1: Univariate Time Series B41910: Autumn Quarter, 2008, by Mr. Ruey S. Tsay Lecture 1: Univariate Time Series B41910: Autumn Quarter, 2008, by Mr. Ruey S. Tsay 1 Some Basic Concepts 1. Time Series: A sequence of random variables measuring certain quantity of interest over time.

More information

Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

More information

TIME SERIES ANALYSIS

TIME SERIES ANALYSIS TIME SERIES ANALYSIS L.M. BHAR AND V.K.SHARMA Indian Agricultural Statistics Research Institute Library Avenue, New Delhi-0 02 lmb@iasri.res.in. Introduction Time series (TS) data refers to observations

More information

Sales and operations planning (SOP) Demand forecasting

Sales and operations planning (SOP) Demand forecasting ing, introduction Sales and operations planning (SOP) forecasting To balance supply with demand and synchronize all operational plans Capture demand data forecasting Balancing of supply, demand, and budgets.

More information

TIME SERIES ANALYSIS

TIME SERIES ANALYSIS TIME SERIES ANALYSIS Ramasubramanian V. I.A.S.R.I., Library Avenue, New Delhi- 110 012 ram_stat@yahoo.co.in 1. Introduction A Time Series (TS) is a sequence of observations ordered in time. Mostly these

More information

Time Series Analysis of Aviation Data

Time Series Analysis of Aviation Data Time Series Analysis of Aviation Data Dr. Richard Xie February, 2012 What is a Time Series A time series is a sequence of observations in chorological order, such as Daily closing price of stock MSFT in

More information

Ã'HSWRIÃ(FRQRPHWULFVÃ)DFXOW\ÃRIÃ(FRQRPLFVÃDQGÃ(FRQRPHWULFVÃ8QLYHUVLW\ÃRIÃ$PVWHUGDPÃ Ã7LQEHUJHQÃ,QVWLWXWHÃ

Ã'HSWRIÃ(FRQRPHWULFVÃ)DFXOW\ÃRIÃ(FRQRPLFVÃDQGÃ(FRQRPHWULFVÃ8QLYHUVLW\ÃRIÃ$PVWHUGDPÃ Ã7LQEHUJHQÃ,QVWLWXWHÃ 7, 7LQEHUJHQ,QVWLWXWH'LVFXVVLRQ3DSHU,QWHUYHQWLRQ7LPH6HULHV$QDO\VLV RI&ULPH5DWHV 6DQMHHY6ULGKDUDQ 6XQýLFD9XMLF 6LHP-DQ.RRSPDQ Ã:HVWDWÃ5RFNYLOOHÃ0'Ã86$Ã Ã'HSWRIÃ(FRQRPHWULFVÃ)DFXOW\ÃRIÃ(FRQRPLFVÃDQGÃ(FRQRPHWULFVÃ8QLYHUVLW\ÃRIÃ$PVWHUGDPÃ

More information

COMP6053 lecture: Time series analysis, autocorrelation. jn2@ecs.soton.ac.uk

COMP6053 lecture: Time series analysis, autocorrelation. jn2@ecs.soton.ac.uk COMP6053 lecture: Time series analysis, autocorrelation jn2@ecs.soton.ac.uk Time series analysis The basic idea of time series analysis is simple: given an observed sequence, how can we build a model that

More information

Estimating an ARMA Process

Estimating an ARMA Process Statistics 910, #12 1 Overview Estimating an ARMA Process 1. Main ideas 2. Fitting autoregressions 3. Fitting with moving average components 4. Standard errors 5. Examples 6. Appendix: Simple estimators

More information

Time Series Analysis. 1) smoothing/trend assessment

Time Series Analysis. 1) smoothing/trend assessment Time Series Analysis This (not surprisingly) concerns the analysis of data collected over time... weekly values, monthly values, quarterly values, yearly values, etc. Usually the intent is to discern whether

More information

Forecasting methods applied to engineering management

Forecasting methods applied to engineering management Forecasting methods applied to engineering management Áron Szász-Gábor Abstract. This paper presents arguments for the usefulness of a simple forecasting application package for sustaining operational

More information

9th Russian Summer School in Information Retrieval Big Data Analytics with R

9th Russian Summer School in Information Retrieval Big Data Analytics with R 9th Russian Summer School in Information Retrieval Big Data Analytics with R Introduction to Time Series with R A. Karakitsiou A. Migdalas Industrial Logistics, ETS Institute Luleå University of Technology

More information

Normalization and Mixed Degrees of Integration in Cointegrated Time Series Systems

Normalization and Mixed Degrees of Integration in Cointegrated Time Series Systems Normalization and Mixed Degrees of Integration in Cointegrated Time Series Systems Robert J. Rossana Department of Economics, 04 F/AB, Wayne State University, Detroit MI 480 E-Mail: r.j.rossana@wayne.edu

More information

Some useful concepts in univariate time series analysis

Some useful concepts in univariate time series analysis Some useful concepts in univariate time series analysis Autoregressive moving average models Autocorrelation functions Model Estimation Diagnostic measure Model selection Forecasting Assumptions: 1. Non-seasonal

More information

FULLY MODIFIED OLS FOR HETEROGENEOUS COINTEGRATED PANELS

FULLY MODIFIED OLS FOR HETEROGENEOUS COINTEGRATED PANELS FULLY MODIFIED OLS FOR HEEROGENEOUS COINEGRAED PANELS Peter Pedroni ABSRAC his chapter uses fully modified OLS principles to develop new methods for estimating and testing hypotheses for cointegrating

More information

Time Series Analysis 1. Lecture 8: Time Series Analysis. Time Series Analysis MIT 18.S096. Dr. Kempthorne. Fall 2013 MIT 18.S096

Time Series Analysis 1. Lecture 8: Time Series Analysis. Time Series Analysis MIT 18.S096. Dr. Kempthorne. Fall 2013 MIT 18.S096 Lecture 8: Time Series Analysis MIT 18.S096 Dr. Kempthorne Fall 2013 MIT 18.S096 Time Series Analysis 1 Outline Time Series Analysis 1 Time Series Analysis MIT 18.S096 Time Series Analysis 2 A stochastic

More information

Maximum Likelihood Estimation of an ARMA(p,q) Model

Maximum Likelihood Estimation of an ARMA(p,q) Model Maximum Likelihood Estimation of an ARMA(p,q) Model Constantino Hevia The World Bank. DECRG. October 8 This note describes the Matlab function arma_mle.m that computes the maximum likelihood estimates

More information

Sales forecasting # 2

Sales forecasting # 2 Sales forecasting # 2 Arthur Charpentier arthur.charpentier@univ-rennes1.fr 1 Agenda Qualitative and quantitative methods, a very general introduction Series decomposition Short versus long term forecasting

More information

Time Series Analysis and Forecasting Methods for Temporal Mining of Interlinked Documents

Time Series Analysis and Forecasting Methods for Temporal Mining of Interlinked Documents Time Series Analysis and Forecasting Methods for Temporal Mining of Interlinked Documents Prasanna Desikan and Jaideep Srivastava Department of Computer Science University of Minnesota. @cs.umn.edu

More information

System Identification for Acoustic Comms.:

System Identification for Acoustic Comms.: System Identification for Acoustic Comms.: New Insights and Approaches for Tracking Sparse and Rapidly Fluctuating Channels Weichang Li and James Preisig Woods Hole Oceanographic Institution The demodulation

More information

FIXED EFFECTS AND RELATED ESTIMATORS FOR CORRELATED RANDOM COEFFICIENT AND TREATMENT EFFECT PANEL DATA MODELS

FIXED EFFECTS AND RELATED ESTIMATORS FOR CORRELATED RANDOM COEFFICIENT AND TREATMENT EFFECT PANEL DATA MODELS FIXED EFFECTS AND RELATED ESTIMATORS FOR CORRELATED RANDOM COEFFICIENT AND TREATMENT EFFECT PANEL DATA MODELS Jeffrey M. Wooldridge Department of Economics Michigan State University East Lansing, MI 48824-1038

More information

Factor analysis. Angela Montanari

Factor analysis. Angela Montanari Factor analysis Angela Montanari 1 Introduction Factor analysis is a statistical model that allows to explain the correlations between a large number of observed correlated variables through a small number

More information

ITSM-R Reference Manual

ITSM-R Reference Manual ITSM-R Reference Manual George Weigt June 5, 2015 1 Contents 1 Introduction 3 1.1 Time series analysis in a nutshell............................... 3 1.2 White Noise Variance.....................................

More information

Overview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model

Overview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model Overview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model 1 September 004 A. Introduction and assumptions The classical normal linear regression model can be written

More information

BayesX - Software for Bayesian Inference in Structured Additive Regression

BayesX - Software for Bayesian Inference in Structured Additive Regression BayesX - Software for Bayesian Inference in Structured Additive Regression Thomas Kneib Faculty of Mathematics and Economics, University of Ulm Department of Statistics, Ludwig-Maximilians-University Munich

More information

4.3 Moving Average Process MA(q)

4.3 Moving Average Process MA(q) 66 CHAPTER 4. STATIONARY TS MODELS 4.3 Moving Average Process MA(q) Definition 4.5. {X t } is a moving-average process of order q if X t = Z t + θ 1 Z t 1 +... + θ q Z t q, (4.9) where and θ 1,...,θ q

More information

Time Series Analysis

Time Series Analysis Time Series Analysis Time series and stochastic processes Andrés M. Alonso Carolina García-Martos Universidad Carlos III de Madrid Universidad Politécnica de Madrid June July, 2012 Alonso and García-Martos

More information

Econometrics Simple Linear Regression

Econometrics Simple Linear Regression Econometrics Simple Linear Regression Burcu Eke UC3M Linear equations with one variable Recall what a linear equation is: y = b 0 + b 1 x is a linear equation with one variable, or equivalently, a straight

More information

ELEC-E8104 Stochastics models and estimation, Lecture 3b: Linear Estimation in Static Systems

ELEC-E8104 Stochastics models and estimation, Lecture 3b: Linear Estimation in Static Systems Stochastics models and estimation, Lecture 3b: Linear Estimation in Static Systems Minimum Mean Square Error (MMSE) MMSE estimation of Gaussian random vectors Linear MMSE estimator for arbitrarily distributed

More information

Studying Achievement

Studying Achievement Journal of Business and Economics, ISSN 2155-7950, USA November 2014, Volume 5, No. 11, pp. 2052-2056 DOI: 10.15341/jbe(2155-7950)/11.05.2014/009 Academic Star Publishing Company, 2014 http://www.academicstar.us

More information

MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1.

MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1. MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0-534-40596-7. Systems of Linear Equations Definition. An n-dimensional vector is a row or a column

More information

Chapter 3: The Multiple Linear Regression Model

Chapter 3: The Multiple Linear Regression Model Chapter 3: The Multiple Linear Regression Model Advanced Econometrics - HEC Lausanne Christophe Hurlin University of Orléans November 23, 2013 Christophe Hurlin (University of Orléans) Advanced Econometrics

More information

A Model for Hydro Inow and Wind Power Capacity for the Brazilian Power Sector

A Model for Hydro Inow and Wind Power Capacity for the Brazilian Power Sector A Model for Hydro Inow and Wind Power Capacity for the Brazilian Power Sector Gilson Matos gilson.g.matos@ibge.gov.br Cristiano Fernandes cris@ele.puc-rio.br PUC-Rio Electrical Engineering Department GAS

More information

Chapter 7 The ARIMA Procedure. Chapter Table of Contents

Chapter 7 The ARIMA Procedure. Chapter Table of Contents Chapter 7 Chapter Table of Contents OVERVIEW...193 GETTING STARTED...194 TheThreeStagesofARIMAModeling...194 IdentificationStage...194 Estimation and Diagnostic Checking Stage...... 200 Forecasting Stage...205

More information

Matrices and Polynomials

Matrices and Polynomials APPENDIX 9 Matrices and Polynomials he Multiplication of Polynomials Let α(z) =α 0 +α 1 z+α 2 z 2 + α p z p and y(z) =y 0 +y 1 z+y 2 z 2 + y n z n be two polynomials of degrees p and n respectively. hen,

More information

Understanding and Applying Kalman Filtering

Understanding and Applying Kalman Filtering Understanding and Applying Kalman Filtering Lindsay Kleeman Department of Electrical and Computer Systems Engineering Monash University, Clayton 1 Introduction Objectives: 1. Provide a basic understanding

More information

Time Series Laboratory

Time Series Laboratory Time Series Laboratory Computing in Weber Classrooms 205-206: To log in, make sure that the DOMAIN NAME is set to MATHSTAT. Use the workshop username: primesw The password will be distributed during the

More information

Analysis and Computation for Finance Time Series - An Introduction

Analysis and Computation for Finance Time Series - An Introduction ECMM703 Analysis and Computation for Finance Time Series - An Introduction Alejandra González Harrison 161 Email: mag208@exeter.ac.uk Time Series - An Introduction A time series is a sequence of observations

More information

Time Series Analysis: Basic Forecasting.

Time Series Analysis: Basic Forecasting. Time Series Analysis: Basic Forecasting. As published in Benchmarks RSS Matters, April 2015 http://web3.unt.edu/benchmarks/issues/2015/04/rss-matters Jon Starkweather, PhD 1 Jon Starkweather, PhD jonathan.starkweather@unt.edu

More information

Vector autoregressions, VAR

Vector autoregressions, VAR 1 / 45 Vector autoregressions, VAR Chapter 2 Financial Econometrics Michael Hauser WS15/16 2 / 45 Content Cross-correlations VAR model in standard/reduced form Properties of VAR(1), VAR(p) Structural VAR,

More information

Section A. Index. Section A. Planning, Budgeting and Forecasting Section A.2 Forecasting techniques... 1. Page 1 of 11. EduPristine CMA - Part I

Section A. Index. Section A. Planning, Budgeting and Forecasting Section A.2 Forecasting techniques... 1. Page 1 of 11. EduPristine CMA - Part I Index Section A. Planning, Budgeting and Forecasting Section A.2 Forecasting techniques... 1 EduPristine CMA - Part I Page 1 of 11 Section A. Planning, Budgeting and Forecasting Section A.2 Forecasting

More information

ARMA, GARCH and Related Option Pricing Method

ARMA, GARCH and Related Option Pricing Method ARMA, GARCH and Related Option Pricing Method Author: Yiyang Yang Advisor: Pr. Xiaolin Li, Pr. Zari Rachev Department of Applied Mathematics and Statistics State University of New York at Stony Brook September

More information

I. Introduction. II. Background. KEY WORDS: Time series forecasting, Structural Models, CPS

I. Introduction. II. Background. KEY WORDS: Time series forecasting, Structural Models, CPS Predicting the National Unemployment Rate that the "Old" CPS Would Have Produced Richard Tiller and Michael Welch, Bureau of Labor Statistics Richard Tiller, Bureau of Labor Statistics, Room 4985, 2 Mass.

More information

Univariate Time Series Analysis; ARIMA Models

Univariate Time Series Analysis; ARIMA Models Econometrics 2 Fall 25 Univariate Time Series Analysis; ARIMA Models Heino Bohn Nielsen of4 Univariate Time Series Analysis We consider a single time series, y,y 2,..., y T. We want to construct simple

More information

Notes for STA 437/1005 Methods for Multivariate Data

Notes for STA 437/1005 Methods for Multivariate Data Notes for STA 437/1005 Methods for Multivariate Data Radford M. Neal, 26 November 2010 Random Vectors Notation: Let X be a random vector with p elements, so that X = [X 1,..., X p ], where denotes transpose.

More information

15.062 Data Mining: Algorithms and Applications Matrix Math Review

15.062 Data Mining: Algorithms and Applications Matrix Math Review .6 Data Mining: Algorithms and Applications Matrix Math Review The purpose of this document is to give a brief review of selected linear algebra concepts that will be useful for the course and to develop

More information

Forecasting in STATA: Tools and Tricks

Forecasting in STATA: Tools and Tricks Forecasting in STATA: Tools and Tricks Introduction This manual is intended to be a reference guide for time series forecasting in STATA. It will be updated periodically during the semester, and will be

More information

Review Jeopardy. Blue vs. Orange. Review Jeopardy

Review Jeopardy. Blue vs. Orange. Review Jeopardy Review Jeopardy Blue vs. Orange Review Jeopardy Jeopardy Round Lectures 0-3 Jeopardy Round $200 How could I measure how far apart (i.e. how different) two observations, y 1 and y 2, are from each other?

More information

Traffic Safety Facts. Research Note. Time Series Analysis and Forecast of Crash Fatalities during Six Holiday Periods Cejun Liu* and Chou-Lin Chen

Traffic Safety Facts. Research Note. Time Series Analysis and Forecast of Crash Fatalities during Six Holiday Periods Cejun Liu* and Chou-Lin Chen Traffic Safety Facts Research Note March 2004 DOT HS 809 718 Time Series Analysis and Forecast of Crash Fatalities during Six Holiday Periods Cejun Liu* and Chou-Lin Chen Summary This research note uses

More information

Module 6: Introduction to Time Series Forecasting

Module 6: Introduction to Time Series Forecasting Using Statistical Data to Make Decisions Module 6: Introduction to Time Series Forecasting Titus Awokuse and Tom Ilvento, University of Delaware, College of Agriculture and Natural Resources, Food and

More information

Forecasting Web Page Views: Methods and Observations

Forecasting Web Page Views: Methods and Observations Journal of Machine Learning Research 9 (28) 2217-225 Submitted 6/8; Revised 8/8; Published 1/8 Forecasting Web Page Views: Methods and Observations Jia Li Visiting Scientist Google Labs, 472 Forbes Avenue

More information

Vector Time Series Model Representations and Analysis with XploRe

Vector Time Series Model Representations and Analysis with XploRe 0-1 Vector Time Series Model Representations and Analysis with plore Julius Mungo CASE - Center for Applied Statistics and Economics Humboldt-Universität zu Berlin mungo@wiwi.hu-berlin.de plore MulTi Motivation

More information

4. Simple regression. QBUS6840 Predictive Analytics. https://www.otexts.org/fpp/4

4. Simple regression. QBUS6840 Predictive Analytics. https://www.otexts.org/fpp/4 4. Simple regression QBUS6840 Predictive Analytics https://www.otexts.org/fpp/4 Outline The simple linear model Least squares estimation Forecasting with regression Non-linear functional forms Regression

More information

Econometric Analysis of Cross Section and Panel Data Second Edition. Jeffrey M. Wooldridge. The MIT Press Cambridge, Massachusetts London, England

Econometric Analysis of Cross Section and Panel Data Second Edition. Jeffrey M. Wooldridge. The MIT Press Cambridge, Massachusetts London, England Econometric Analysis of Cross Section and Panel Data Second Edition Jeffrey M. Wooldridge The MIT Press Cambridge, Massachusetts London, England Preface Acknowledgments xxi xxix I INTRODUCTION AND BACKGROUND

More information

Observing the Changing Relationship Between Natural Gas Prices and Power Prices

Observing the Changing Relationship Between Natural Gas Prices and Power Prices Observing the Changing Relationship Between Natural Gas Prices and Power Prices The research views expressed herein are those of the author and do not necessarily represent the views of the CME Group or

More information

NEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS

NEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS NEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS TEST DESIGN AND FRAMEWORK September 2014 Authorized for Distribution by the New York State Education Department This test design and framework document

More information

Analysis of Financial Time Series

Analysis of Financial Time Series Analysis of Financial Time Series Analysis of Financial Time Series Financial Econometrics RUEY S. TSAY University of Chicago A Wiley-Interscience Publication JOHN WILEY & SONS, INC. This book is printed

More information

Trends and Breaks in Cointegrated VAR Models

Trends and Breaks in Cointegrated VAR Models Trends and Breaks in Cointegrated VAR Models Håvard Hungnes Thesis for the Dr. Polit. degree Department of Economics, University of Oslo Defended March 17, 2006 Research Fellow in the Research Department

More information

Introduction to time series analysis

Introduction to time series analysis Introduction to time series analysis Jean-Marie Dufour First version: December 1998 Revised: January 2003 This version: January 8, 2008 Compiled: January 8, 2008, 6:12pm This work was supported by the

More information

Random Vectors and the Variance Covariance Matrix

Random Vectors and the Variance Covariance Matrix Random Vectors and the Variance Covariance Matrix Definition 1. A random vector X is a vector (X 1, X 2,..., X p ) of jointly distributed random variables. As is customary in linear algebra, we will write

More information

Time Series Analysis

Time Series Analysis Time Series Analysis Autoregressive, MA and ARMA processes Andrés M. Alonso Carolina García-Martos Universidad Carlos III de Madrid Universidad Politécnica de Madrid June July, 212 Alonso and García-Martos

More information

CS229 Lecture notes. Andrew Ng

CS229 Lecture notes. Andrew Ng CS229 Lecture notes Andrew Ng Part X Factor analysis Whenwehavedatax (i) R n thatcomesfromamixtureofseveral Gaussians, the EM algorithm can be applied to fit a mixture model. In this setting, we usually

More information

Chapter 9: Univariate Time Series Analysis

Chapter 9: Univariate Time Series Analysis Chapter 9: Univariate Time Series Analysis In the last chapter we discussed models with only lags of explanatory variables. These can be misleading if: 1. The dependent variable Y t depends on lags of

More information