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

Size: px
Start display at page:

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

Transcription

1 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 of a stationary ARMA(p,q) model. Problem: To t an ARMA(p,q) model to a vector of time series fy ; y ; :::; y T g with zero unconditional mean. An ARMA(p,q) process is given by y t = y t + ::: + p y t p + " t + " t + ::: + q " t q ; where " t is an i.i.d. shock normally distributed with mean zero and variance. If the original P series do not have zero mean, we rst construct ~y t = y T t s= y s=t and then t the ARMA model to ~y t. Usage: results = arma_mle(y,p,q,[info]) Arguments: y = vector of observed time series with mean zero. p = length of the autoregressive part (AR) of the ARMA model (integer) q = length of the moving average part (MA) of the ARMA model (integer) info = [optional] If info is not zero, the program prints information about the convergence of the optimization algorithm. The default value is zero. Output: A structure with the following elements: results.ar = h^ ; ^ ; :::; ^ i p results.ma = h^ ; ^ ; :::; ^ i q : estimated coe cients of the AR part. : estimated coe cients of the MA part. results.sigma =^ : estimated standard deviation of " t.

2 The le test_arma_mle.m performs a Montecarlo experiment using the function arma_mle.m. The user inputs a theoretical ARMA model. The program runs a large number of simulations and then estimates the parameters for each simulation. Finally, the histograms of the estimates are shown. Algorithm In this section I describe the algorithm used to compute the maximum likelihood estimates of the ARMA(p,q) process. Suppose that we want to t the (mean zero) time series fy t g T t= the the following ARMA(p,q) model y t = y t + ::: + p y t p + " t + " t + ::: + q " t q ; () where " t is an i.i.d. shock normally distributed with mean zero and variance. Let r = max (p; q + ), and rewrite the model as y t = y t + ::: + r y t r + " t + " t + ::: + r " t r+ : () We interpret j = for j > p and j = for j > q. The estimation procedure is based on the Kalman lter (see Hamilton (994) for the derivation of the lter). To use the Kalman lter we need to write the model in the following (state-space) form x t+ = Ax t + R" t+ (3) y t = Z x t (4) where x t is an r state vector, A is an r r matrix, and R and Z are r vectors. These matrices and vectors are de ned as follows A = ; R = r 5 4 r. r 3 3 ; Z = To see that the system (3) and (4) is equivalent to (), write the last row of (3) as x r;t+ = r x ;t + r " t+

3 Lagging this equation r periods we nd x r;t r+ = r L r x ;t + r L r " t+ (5) where we de ne L r x t = x t row implies r as the r lag operator for any integer r. The second to last x r ;t+ = r x ;t + x r;t + r " t+ Lagging r periods we obtain x r ;t r+3 = r L r x ;t + x r;t r+ + r L r " t+ Introducing (5) into the previous equation we nd x r ;t r+3 = r L r x ;t + r L r x ;t + r L r " t+ + r L r " t+ or x r ;t r+3 = r L r + r L r x ;t + r L r + r L r " t+ (6) Take now row r, x r ;t+ = r x ;t + x r ;t + r 3 " t+ Lagging r 3 periods we nd x r ;t r+4 = r L r 3 x ;t + x r ;t r+3 + r 3 L r 3 " t+ Plugging (6) into the previous equation we obtain x r ;t r+4 = r L r 3 + r L r + r L r x ;t + r L r + r L r + r 3 L r 3 " t+ Following this iterative procedure until row r we nd x ;t+ = + ::: + r L r 3 + r L r + r L r x ;t + r L r + r L r + r 3 L r 3 + ::: + " t+ or L L ::: r L r x ;t+ = r L r + r L r + r 3 L r 3 + ::: + " t+ (7) 3

4 Now, the observation equation (4) and the de nition of Z imply y t = x ;t Using (7) evaluated at t we arrive at the ARMA representation (), L L ::: r L r y t = r L r + r L r + r 3 L r 3 + ::: + " t which proves that the system (3), (4) is equivalent to (). Denote by ^x t+jt = E t [x t+ jy ; :::; y t ; x ] the expected value of x t+ conditional on the history of observations (y ; :::; y t ). The Kalman lter provides an algorithm for computing recursively ^x t+jt given an initial value ^x j =. (Note that is the unconditional mean of x t ). Associated with each of these forecasts is a mean squared error matrix, de ned as h P t+jt = E x t+ ^x t+jt xt+ ^x t+jt i : Given the estimate ^x tjt, we use (4) to compute the innovations a t = y t E [y t jy ; :::; y t ; x ] = y t Z ^x tjt The innovation variance, denoted by! t, satis es! t = E y t Z ^x tjt yt Z ^x tjt = E Z x t Z ^x tjt Z x t Z ^x tjt = Z P tjt Z: (8) In addition to the estimates ^x t+jt, the Kalman lter equations imply the following evolution of the matrices P t+jt P t+jt = A P tjt P tjt ZZ P tjt =! t A + RR : (9) Given an initial matrix P j = E (x t x t) and the initial value ^x j =, the likelihood function of the observation vector fy ; y ; :::; y T g is given by L = TY (! t ) = exp a t! t 4

5 Taking logarithms, dropping the constant, and multiplying by we obtain ln (!t ) + a t =! t () In principle, to nd the MLE estimates we maximize () with respect to the parameters j, j, and. However, the following trick allows us to concentrate-out the term, and maximize only with respect to the parameters j, j. Suppose we initialize the lter with the matrix ~P j = P j. Then, from (9) it follows that each P t+jt is proportional to, and from (8) it follows that the innovation variance is also proportional to. This implies that we can optimize rst with respect to by hand, replace the result into the objective function, and then optimize the resulting expression (called the concentrated log-likelihood ) with respect to the parameters j, j : To see this, note that () becomes ln a! t + t! t () and is cancelled out in the evolution equations of P t+jt and in the projections ^x t+jt. So we can directly optimize () with respect to to obtain = T a t =! t : Replacing this result into () we obtain the concentrated log-likelihood function = = ln + ln! t + a t! t "! a P # T t ln : + ln! t + a t =! t P T! T t n a t =! t : " # T ln (=T ) + T + T ln a t =! t + ln! t or, dropping irrelevant constants, " T ln a t =! t + # ln! t () Because the innovations a t and the variances! t are nonlinear functions of the parameters [; ], 5

6 we use numerical methods to maximize (). The Matlab function arma_mle.m performs this task using the optimization routine fminunc.m from the Matlab optimization package. The initial condition for the parameters are based on the two-step regression procedure described in Hannan and McDougall (984). The rst step consists in running a (relatively) long autoregression and computing the tted residuals. The second steps computes an OLS regression of y t on its p lagged values, and on q lagged values of the tted residuals obtained in the rst step. REFERENCES [] James D. Hamilton Time Series Analysis, 994. Princeton University Press. [] E.J. Hannan and A.J. McDougall. Regression Procedures for ARMA Estimation, Journal of the American Statistical Association, Vol 83, No 49, June

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

Time Series Analysis

Time 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 Identification of univariate time series models, cont.:

More information

Topic 5: Stochastic Growth and Real Business Cycles

Topic 5: Stochastic Growth and Real Business Cycles Topic 5: Stochastic Growth and Real Business Cycles Yulei Luo SEF of HKU October 1, 2015 Luo, Y. (SEF of HKU) Macro Theory October 1, 2015 1 / 45 Lag Operators The lag operator (L) is de ned as Similar

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

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 2. Dynamic panel data models

Chapter 2. Dynamic panel data models Chapter 2. Dynamic panel data models Master of Science in Economics - University of Geneva Christophe Hurlin, Université d Orléans Université d Orléans April 2010 Introduction De nition We now consider

More information

Impulse Response Functions

Impulse Response Functions Impulse Response Functions Wouter J. Den Haan University of Amsterdam April 28, 2011 General definition IRFs The IRF gives the j th -period response when the system is shocked by a one-standard-deviation

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

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

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

General Framework for an Iterative Solution of Ax b. Jacobi s Method

General Framework for an Iterative Solution of Ax b. Jacobi s Method 2.6 Iterative Solutions of Linear Systems 143 2.6 Iterative Solutions of Linear Systems Consistent linear systems in real life are solved in one of two ways: by direct calculation (using a matrix factorization,

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

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

160 CHAPTER 4. VECTOR SPACES

160 CHAPTER 4. VECTOR SPACES 160 CHAPTER 4. VECTOR SPACES 4. Rank and Nullity In this section, we look at relationships between the row space, column space, null space of a matrix and its transpose. We will derive fundamental results

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

1.2 Solving a System of Linear Equations

1.2 Solving a System of Linear Equations 1.. SOLVING A SYSTEM OF LINEAR EQUATIONS 1. Solving a System of Linear Equations 1..1 Simple Systems - Basic De nitions As noticed above, the general form of a linear system of m equations in n variables

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

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 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

CAPM, Arbitrage, and Linear Factor Models

CAPM, Arbitrage, and Linear Factor Models CAPM, Arbitrage, and Linear Factor Models CAPM, Arbitrage, Linear Factor Models 1/ 41 Introduction We now assume all investors actually choose mean-variance e cient portfolios. By equating these investors

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

Statistics in Retail Finance. Chapter 6: Behavioural models

Statistics in Retail Finance. Chapter 6: Behavioural models Statistics in Retail Finance 1 Overview > So far we have focussed mainly on application scorecards. In this chapter we shall look at behavioural models. We shall cover the following topics:- Behavioural

More information

Representation of functions as power series

Representation of functions as power series Representation of functions as power series Dr. Philippe B. Laval Kennesaw State University November 9, 008 Abstract This document is a summary of the theory and techniques used to represent functions

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

1 Teaching notes on GMM 1.

1 Teaching notes on GMM 1. Bent E. Sørensen January 23, 2007 1 Teaching notes on GMM 1. Generalized Method of Moment (GMM) estimation is one of two developments in econometrics in the 80ies that revolutionized empirical work in

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

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

INDIRECT INFERENCE (prepared for: The New Palgrave Dictionary of Economics, Second Edition)

INDIRECT INFERENCE (prepared for: The New Palgrave Dictionary of Economics, Second Edition) INDIRECT INFERENCE (prepared for: The New Palgrave Dictionary of Economics, Second Edition) Abstract Indirect inference is a simulation-based method for estimating the parameters of economic models. Its

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

Bias in the Estimation of Mean Reversion in Continuous-Time Lévy Processes

Bias in the Estimation of Mean Reversion in Continuous-Time Lévy Processes Bias in the Estimation of Mean Reversion in Continuous-Time Lévy Processes Yong Bao a, Aman Ullah b, Yun Wang c, and Jun Yu d a Purdue University, IN, USA b University of California, Riverside, CA, USA

More information

Lecture 2: ARMA(p,q) models (part 3)

Lecture 2: ARMA(p,q) models (part 3) Lecture 2: ARMA(p,q) models (part 3) Florian Pelgrin University of Lausanne, École des HEC Department of mathematics (IMEA-Nice) Sept. 2011 - Jan. 2012 Florian Pelgrin (HEC) Univariate time series Sept.

More information

Package EstCRM. July 13, 2015

Package EstCRM. July 13, 2015 Version 1.4 Date 2015-7-11 Package EstCRM July 13, 2015 Title Calibrating Parameters for the Samejima's Continuous IRT Model Author Cengiz Zopluoglu Maintainer Cengiz Zopluoglu

More information

Detekce změn v autoregresních posloupnostech

Detekce změn v autoregresních posloupnostech Nové Hrady 2012 Outline 1 Introduction 2 3 4 Change point problem (retrospective) The data Y 1,..., Y n follow a statistical model, which may change once or several times during the observation period

More information

Chapter 5: The Cointegrated VAR model

Chapter 5: The Cointegrated VAR model Chapter 5: The Cointegrated VAR model Katarina Juselius July 1, 2012 Katarina Juselius () Chapter 5: The Cointegrated VAR model July 1, 2012 1 / 41 An intuitive interpretation of the Pi matrix Consider

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

Chapter 7 Nonlinear Systems

Chapter 7 Nonlinear Systems Chapter 7 Nonlinear Systems Nonlinear systems in R n : X = B x. x n X = F (t; X) F (t; x ; :::; x n ) B C A ; F (t; X) =. F n (t; x ; :::; x n ) When F (t; X) = F (X) is independent of t; it is an example

More information

14.451 Lecture Notes 10

14.451 Lecture Notes 10 14.451 Lecture Notes 1 Guido Lorenzoni Fall 29 1 Continuous time: nite horizon Time goes from to T. Instantaneous payo : f (t; x (t) ; y (t)) ; (the time dependence includes discounting), where x (t) 2

More information

Adaptive Demand-Forecasting Approach based on Principal Components Time-series an application of data-mining technique to detection of market movement

Adaptive Demand-Forecasting Approach based on Principal Components Time-series an application of data-mining technique to detection of market movement Adaptive Demand-Forecasting Approach based on Principal Components Time-series an application of data-mining technique to detection of market movement Toshio Sugihara Abstract In this study, an adaptive

More information

I. Basic concepts: Buoyancy and Elasticity II. Estimating Tax Elasticity III. From Mechanical Projection to Forecast

I. Basic concepts: Buoyancy and Elasticity II. Estimating Tax Elasticity III. From Mechanical Projection to Forecast Elements of Revenue Forecasting II: the Elasticity Approach and Projections of Revenue Components Fiscal Analysis and Forecasting Workshop Bangkok, Thailand June 16 27, 2014 Joshua Greene Consultant IMF-TAOLAM

More information

Cardiff Economics Working Papers

Cardiff Economics Working Papers Cardiff Economics Working Papers Working Paper No. E015/8 Comparing Indirect Inference and Likelihood testing: asymptotic and small sample results David Meenagh, Patrick Minford, Michael Wickens and Yongdeng

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

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

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

Applications of Fermat s Little Theorem and Congruences

Applications of Fermat s Little Theorem and Congruences Applications of Fermat s Little Theorem and Congruences Definition: Let m be a positive integer. Then integers a and b are congruent modulo m, denoted by a b mod m, if m (a b). Example: 3 1 mod 2, 6 4

More information

Lecture 3: Linear methods for classification

Lecture 3: Linear methods for classification Lecture 3: Linear methods for classification Rafael A. Irizarry and Hector Corrada Bravo February, 2010 Today we describe four specific algorithms useful for classification problems: linear regression,

More information

State Space Time Series Analysis

State Space Time Series Analysis State Space Time Series Analysis p. 1 State Space Time Series Analysis Siem Jan Koopman http://staff.feweb.vu.nl/koopman Department of Econometrics VU University Amsterdam Tinbergen Institute 2011 State

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

Chapter 5. Analysis of Multiple Time Series. 5.1 Vector Autoregressions

Chapter 5. Analysis of Multiple Time Series. 5.1 Vector Autoregressions Chapter 5 Analysis of Multiple Time Series Note: The primary references for these notes are chapters 5 and 6 in Enders (2004). An alternative, but more technical treatment can be found in chapters 10-11

More information

Maximum likelihood estimation of mean reverting processes

Maximum likelihood estimation of mean reverting processes Maximum likelihood estimation of mean reverting processes José Carlos García Franco Onward, Inc. jcpollo@onwardinc.com Abstract Mean reverting processes are frequently used models in real options. For

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

How To Solve A Minimum Set Covering Problem (Mcp)

How To Solve A Minimum Set Covering Problem (Mcp) Measuring Rationality with the Minimum Cost of Revealed Preference Violations Mark Dean and Daniel Martin Online Appendices - Not for Publication 1 1 Algorithm for Solving the MASP In this online appendix

More information

Technical Analysis and the London Stock Exchange: Testing Trading Rules Using the FT30

Technical Analysis and the London Stock Exchange: Testing Trading Rules Using the FT30 Technical Analysis and the London Stock Exchange: Testing Trading Rules Using the FT30 Terence C. Mills* Department of Economics, Loughborough University, Loughborough LE11 3TU, UK This paper investigates

More information

Master s Theory Exam Spring 2006

Master s Theory Exam Spring 2006 Spring 2006 This exam contains 7 questions. You should attempt them all. Each question is divided into parts to help lead you through the material. You should attempt to complete as much of each problem

More information

Portfolio selection based on upper and lower exponential possibility distributions

Portfolio selection based on upper and lower exponential possibility distributions European Journal of Operational Research 114 (1999) 115±126 Theory and Methodology Portfolio selection based on upper and lower exponential possibility distributions Hideo Tanaka *, Peijun Guo Department

More information

Multivariate Normal Distribution

Multivariate Normal Distribution Multivariate Normal Distribution Lecture 4 July 21, 2011 Advanced Multivariate Statistical Methods ICPSR Summer Session #2 Lecture #4-7/21/2011 Slide 1 of 41 Last Time Matrices and vectors Eigenvalues

More information

Investment and Financial Constraints: Empirical Evidence for Firms in Brazil and China

Investment and Financial Constraints: Empirical Evidence for Firms in Brazil and China Investment and Financial Constraints: Empirical Evidence for Firms in Brazil and China Stephen R. Bond Nu eld College and Department of Economics, University of Oxford and Institute for Fiscal Studies

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

DATA ANALYSIS II. Matrix Algorithms

DATA ANALYSIS II. Matrix Algorithms DATA ANALYSIS II Matrix Algorithms Similarity Matrix Given a dataset D = {x i }, i=1,..,n consisting of n points in R d, let A denote the n n symmetric similarity matrix between the points, given as where

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

Optimal linear-quadratic control

Optimal linear-quadratic control Optimal linear-quadratic control Martin Ellison 1 Motivation The lectures so far have described a general method - value function iterations - for solving dynamic programming problems. However, one problem

More information

CCNY. BME I5100: Biomedical Signal Processing. Linear Discrimination. Lucas C. Parra Biomedical Engineering Department City College of New York

CCNY. BME I5100: Biomedical Signal Processing. Linear Discrimination. Lucas C. Parra Biomedical Engineering Department City College of New York BME I5100: Biomedical Signal Processing Linear Discrimination Lucas C. Parra Biomedical Engineering Department CCNY 1 Schedule Week 1: Introduction Linear, stationary, normal - the stuff biology is not

More information

Section IV.1: Recursive Algorithms and Recursion Trees

Section IV.1: Recursive Algorithms and Recursion Trees Section IV.1: Recursive Algorithms and Recursion Trees Definition IV.1.1: A recursive algorithm is an algorithm that solves a problem by (1) reducing it to an instance of the same problem with smaller

More information

Abstract: We describe the beautiful LU factorization of a square matrix (or how to write Gaussian elimination in terms of matrix multiplication).

Abstract: We describe the beautiful LU factorization of a square matrix (or how to write Gaussian elimination in terms of matrix multiplication). MAT 2 (Badger, Spring 202) LU Factorization Selected Notes September 2, 202 Abstract: We describe the beautiful LU factorization of a square matrix (or how to write Gaussian elimination in terms of matrix

More information

Lecture Notes on Polynomials

Lecture Notes on Polynomials Lecture Notes on Polynomials Arne Jensen Department of Mathematical Sciences Aalborg University c 008 Introduction These lecture notes give a very short introduction to polynomials with real and complex

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

LOGISTIC REGRESSION. Nitin R Patel. where the dependent variable, y, is binary (for convenience we often code these values as

LOGISTIC REGRESSION. Nitin R Patel. where the dependent variable, y, is binary (for convenience we often code these values as LOGISTIC REGRESSION Nitin R Patel Logistic regression extends the ideas of multiple linear regression to the situation where the dependent variable, y, is binary (for convenience we often code these values

More information

1 Solving LPs: The Simplex Algorithm of George Dantzig

1 Solving LPs: The Simplex Algorithm of George Dantzig Solving LPs: The Simplex Algorithm of George Dantzig. Simplex Pivoting: Dictionary Format We illustrate a general solution procedure, called the simplex algorithm, by implementing it on a very simple example.

More information

AUTOMATION OF ENERGY DEMAND FORECASTING. Sanzad Siddique, B.S.

AUTOMATION OF ENERGY DEMAND FORECASTING. Sanzad Siddique, B.S. AUTOMATION OF ENERGY DEMAND FORECASTING by Sanzad Siddique, B.S. A Thesis submitted to the Faculty of the Graduate School, Marquette University, in Partial Fulfillment of the Requirements for the Degree

More information

Logistic Regression. Jia Li. Department of Statistics The Pennsylvania State University. Logistic Regression

Logistic Regression. Jia Li. Department of Statistics The Pennsylvania State University. Logistic Regression Logistic Regression Department of Statistics The Pennsylvania State University Email: jiali@stat.psu.edu Logistic Regression Preserve linear classification boundaries. By the Bayes rule: Ĝ(x) = arg max

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

Large Time-Varying Parameter VARs

Large Time-Varying Parameter VARs Large Time-Varying Parameter VARs Gary Koop University of Strathclyde Dimitris Korobils University of Glasgow February 22, 2012 Abstract In this paper, we develop methods for estimation and forecasting

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

Forecasting of Economic Quantities using Fuzzy Autoregressive Model and Fuzzy Neural Network

Forecasting of Economic Quantities using Fuzzy Autoregressive Model and Fuzzy Neural Network Forecasting of Economic Quantities using Fuzzy Autoregressive Model and Fuzzy Neural Network Dušan Marček 1 Abstract Most models for the time series of stock prices have centered on autoregressive (AR)

More information

Direct Methods for Solving Linear Systems. Matrix Factorization

Direct Methods for Solving Linear Systems. Matrix Factorization Direct Methods for Solving Linear Systems Matrix Factorization Numerical Analysis (9th Edition) R L Burden & J D Faires Beamer Presentation Slides prepared by John Carroll Dublin City University c 2011

More information

How To Price Garch

How To Price Garch 2011 3rd International Conference on Information and Financial Engineering IPEDR vol.12 (2011) (2011) IACSIT Press, Singapore A Study on Heston-Nandi GARCH Option Pricing Model Suk Joon Byun KAIST Business

More information

Lecture 1: Systems of Linear Equations

Lecture 1: Systems of Linear Equations MTH Elementary Matrix Algebra Professor Chao Huang Department of Mathematics and Statistics Wright State University Lecture 1 Systems of Linear Equations ² Systems of two linear equations with two variables

More information

Maximum Likelihood Estimation

Maximum Likelihood Estimation Math 541: Statistical Theory II Lecturer: Songfeng Zheng Maximum Likelihood Estimation 1 Maximum Likelihood Estimation Maximum likelihood is a relatively simple method of constructing an estimator for

More information

7 Gaussian Elimination and LU Factorization

7 Gaussian Elimination and LU Factorization 7 Gaussian Elimination and LU Factorization In this final section on matrix factorization methods for solving Ax = b we want to take a closer look at Gaussian elimination (probably the best known method

More information

Forecasting Geographic Data Michael Leonard and Renee Samy, SAS Institute Inc. Cary, NC, USA

Forecasting Geographic Data Michael Leonard and Renee Samy, SAS Institute Inc. Cary, NC, USA Forecasting Geographic Data Michael Leonard and Renee Samy, SAS Institute Inc. Cary, NC, USA Abstract Virtually all businesses collect and use data that are associated with geographic locations, whether

More information

MATLAB and Big Data: Illustrative Example

MATLAB and Big Data: Illustrative Example MATLAB and Big Data: Illustrative Example Rick Mansfield Cornell University August 19, 2014 Goals Use a concrete example from my research to: Demonstrate the value of vectorization Introduce key commands/functions

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

Introduction to Logistic Regression

Introduction to Logistic Regression OpenStax-CNX module: m42090 1 Introduction to Logistic Regression Dan Calderon This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License 3.0 Abstract Gives introduction

More information

Business Cycles, Theory and Empirical Applications

Business Cycles, Theory and Empirical Applications Business Cycles, Theory and Empirical Applications Seminar Presentation Country of interest France Jan Krzyzanowski June 9, 2012 Table of Contents Business Cycle Analysis Data Quantitative Analysis Stochastic

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

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

The PageRank Citation Ranking: Bring Order to the Web

The PageRank Citation Ranking: Bring Order to the Web The PageRank Citation Ranking: Bring Order to the Web presented by: Xiaoxi Pang 25.Nov 2010 1 / 20 Outline Introduction A ranking for every page on the Web Implementation Convergence Properties Personalized

More information

1 Another method of estimation: least squares

1 Another method of estimation: least squares 1 Another method of estimation: least squares erm: -estim.tex, Dec8, 009: 6 p.m. (draft - typos/writos likely exist) Corrections, comments, suggestions welcome. 1.1 Least squares in general Assume Y i

More information

Performing Unit Root Tests in EViews. Unit Root Testing

Performing Unit Root Tests in EViews. Unit Root Testing Página 1 de 12 Unit Root Testing The theory behind ARMA estimation is based on stationary time series. A series is said to be (weakly or covariance) stationary if the mean and autocovariances of the series

More information

Comparing Neural Networks and ARMA Models in Artificial Stock Market

Comparing Neural Networks and ARMA Models in Artificial Stock Market Comparing Neural Networks and ARMA Models in Artificial Stock Market Jiří Krtek Academy of Sciences of the Czech Republic, Institute of Information Theory and Automation. e-mail: krtek@karlin.mff.cuni.cz

More information

MISSING DATA TECHNIQUES WITH SAS. IDRE Statistical Consulting Group

MISSING DATA TECHNIQUES WITH SAS. IDRE Statistical Consulting Group MISSING DATA TECHNIQUES WITH SAS IDRE Statistical Consulting Group ROAD MAP FOR TODAY To discuss: 1. Commonly used techniques for handling missing data, focusing on multiple imputation 2. Issues that could

More information

Notes on Factoring. MA 206 Kurt Bryan

Notes on Factoring. MA 206 Kurt Bryan The General Approach Notes on Factoring MA 26 Kurt Bryan Suppose I hand you n, a 2 digit integer and tell you that n is composite, with smallest prime factor around 5 digits. Finding a nontrivial factor

More information

MAN-BITES-DOG BUSINESS CYCLES ONLINE APPENDIX

MAN-BITES-DOG BUSINESS CYCLES ONLINE APPENDIX MAN-BITES-DOG BUSINESS CYCLES ONLINE APPENDIX KRISTOFFER P. NIMARK The next section derives the equilibrium expressions for the beauty contest model from Section 3 of the main paper. This is followed by

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

Package SCperf. February 19, 2015

Package SCperf. February 19, 2015 Package SCperf February 19, 2015 Type Package Title Supply Chain Perform Version 1.0 Date 2012-01-22 Author Marlene Silva Marchena Maintainer The package implements different inventory models, the bullwhip

More information

Credit Ratings and Bank Monitoring Ability

Credit Ratings and Bank Monitoring Ability Credit Ratings and Bank Monitoring Ability Leonard I. Nakamura Federal Reserve Bank of Philadelphia Kasper Roszbach Sveriges Riksbank April 13, 2005 Abstract In this paper we use detailed credit data from

More information

Statistics in Retail Finance. Chapter 2: Statistical models of default

Statistics in Retail Finance. Chapter 2: Statistical models of default Statistics in Retail Finance 1 Overview > We consider how to build statistical models of default, or delinquency, and how such models are traditionally used for credit application scoring and decision

More information

by the matrix A results in a vector which is a reflection of the given

by the matrix A results in a vector which is a reflection of the given Eigenvalues & Eigenvectors Example Suppose Then So, geometrically, multiplying a vector in by the matrix A results in a vector which is a reflection of the given vector about the y-axis We observe that

More information

IDENTIFICATION IN A CLASS OF NONPARAMETRIC SIMULTANEOUS EQUATIONS MODELS. Steven T. Berry and Philip A. Haile. March 2011 Revised April 2011

IDENTIFICATION IN A CLASS OF NONPARAMETRIC SIMULTANEOUS EQUATIONS MODELS. Steven T. Berry and Philip A. Haile. March 2011 Revised April 2011 IDENTIFICATION IN A CLASS OF NONPARAMETRIC SIMULTANEOUS EQUATIONS MODELS By Steven T. Berry and Philip A. Haile March 2011 Revised April 2011 COWLES FOUNDATION DISCUSSION PAPER NO. 1787R COWLES FOUNDATION

More information

Option Valuation. Chapter 21

Option Valuation. Chapter 21 Option Valuation Chapter 21 Intrinsic and Time Value intrinsic value of in-the-money options = the payoff that could be obtained from the immediate exercise of the option for a call option: stock price

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