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


 Isaac Hoover
 11 months ago
 Views:
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 (statespace) 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 concentrateout 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 loglikelihood ) 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 loglikelihood 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 twostep 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
Handout 7: Business Cycles
University of British Columbia Department of Economics, Macroeconomics (Econ 502) Prof. Amartya Lahiri Handout 7: Business Cycles We now use the methods that we have introduced to study modern business
More informationChapter 4. Simulated Method of Moments and its siblings
Chapter 4. Simulated Method of Moments and its siblings Contents 1 Two motivating examples 1 1.1 Du e and Singleton (1993)......................... 1 1.2 Model for nancial returns with stochastic volatility...........
More informationChapter 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 informationTime Series Analysis
Time Series Analysis hm@imm.dtu.dk Informatics and Mathematical Modelling Technical University of Denmark DK2800 Kgs. Lyngby 1 Outline of the lecture Identification of univariate time series models, cont.:
More informationTopic 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 informationOverview 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 informationChapter 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 informationImpulse 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 onestandarddeviation
More informationCentre 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 informationUnivariate 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 informationGeneral 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 informationUnivariate 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 informationChapter 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 informationForecasting methods applied to engineering management
Forecasting methods applied to engineering management Áron SzászGábor Abstract. This paper presents arguments for the usefulness of a simple forecasting application package for sustaining operational
More information1. The Classical Linear Regression Model: The Bivariate Case
Business School, Brunel University MSc. EC5501/5509 Modelling Financial Decisions and Markets/Introduction to Quantitative Methods Prof. Menelaos Karanasos (Room SS69, Tel. 018956584) Lecture Notes 3 1.
More informationSome 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. Nonseasonal
More informationJava Modules for Time Series Analysis
Java Modules for Time Series Analysis Agenda Clustering Nonnormal distributions Multifactor modeling Implied ratings Time series prediction 1. Clustering + Cluster 1 Synthetic Clustering + Time series
More informationThe 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 informationAn EM algorithm for the estimation of a ne statespace systems with or without known inputs
An EM algorithm for the estimation of a ne statespace systems with or without known inputs Alexander W Blocker January 008 Abstract We derive an EM algorithm for the estimation of a ne Gaussian statespace
More informationAnalysis of Financial Time Series with EViews
Analysis of Financial Time Series with EViews Enrico Foscolo Contents 1 Asset Returns 2 1.1 Empirical Properties of Returns................. 2 2 Heteroskedasticity and Autocorrelation 4 2.1 Testing for
More informationTime Series Analysis
Time Series Analysis Forecasting with ARIMA models Andrés M. Alonso Carolina GarcíaMartos Universidad Carlos III de Madrid Universidad Politécnica de Madrid June July, 2012 Alonso and GarcíaMartos (UC3MUPM)
More information160 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 informationTime 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 information1.2 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 informationLecture 6. Inverse of Matrix
Lecture 6 Inverse of Matrix Recall that any linear system can be written as a matrix equation In one dimension case, ie, A is 1 1, then can be easily solved as A x b Ax b x b A 1 A b A 1 b provided that
More informationEstimation 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 informationUnivariate 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 informationSolutions Problem Set 2 Macro II (14.452)
Solutions Problem Set 2 Macro II (4.452) Francisco A. Gallego 4/22 We encourage you to work together, as long as you write your own solutions. Intertemporal Labor Supply Consider the following problem.
More information4.6 Null Space, Column Space, Row Space
NULL SPACE, COLUMN SPACE, ROW SPACE Null Space, Column Space, Row Space In applications of linear algebra, subspaces of R n typically arise in one of two situations: ) as the set of solutions of a linear
More informationARMA, 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 informationSystems of simultaneous equations
Systems of simultaneous equations JeanMarc Robin January 29 References Je rey M. Wooldridge, Econometric Analysis of Cross Section and Panel Data, the MI Press, October 2. ISBN 26223297. William Green,
More information1 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 informationINDIRECT 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 simulationbased method for estimating the parameters of economic models. Its
More informationStatistics 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 informationEstimating 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 informationFinancial Econometrics Jeffrey R. Russell Final Exam
Name Financial Econometrics Jeffrey R. Russell Final Exam You have 3 hours to complete the exam. Use can use a calculator. Try to fit all your work in the space provided. If you find you need more space
More informationCardiff 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 informationDetekce 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 informationCAPM, 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 meanvariance e cient portfolios. By equating these investors
More informationPackage EstCRM. July 13, 2015
Version 1.4 Date 2015711 Package EstCRM July 13, 2015 Title Calibrating Parameters for the Samejima's Continuous IRT Model Author Cengiz Zopluoglu Maintainer Cengiz Zopluoglu
More informationBias in the Estimation of Mean Reversion in ContinuousTime Lévy Processes
Bias in the Estimation of Mean Reversion in ContinuousTime 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 informationExercise 1. 1) Using Eviews (file named return.wf1), plot the ACF and PACF function for the series returns. Identify the series.
Exercise 1 1) Using Eviews (file named return.wf1), plot the ACF and PACF function for the series returns. Identify the series. 2) Read the paper Do we really know that financial markets are efficient?
More informationCHAPTER 9: SERIAL CORRELATION
Serial correlation (or autocorrelation) is the violation of Assumption 4 (observations of the error term are uncorrelated with each other). Pure Serial Correlation This type of correlation tends to be
More informationTIME SERIES ANALYSIS
TIME SERIES ANALYSIS L.M. BHAR AND V.K.SHARMA Indian Agricultural Statistics Research Institute Library Avenue, New Delhi0 02 lmb@iasri.res.in. Introduction Time series (TS) data refers to observations
More informationAdvanced Linear Modeling
Ronald Christensen Advanced Linear Modeling Multivariate, Time Series, and Spatial Data; Nonparametric Regression and Response Surface Maximization Second Edition Springer Preface to the Second Edition
More informationThe GMWM: A New Framework for Inertial Sensor Calibration
The GMWM: A New Framework for Inertial Sensor Calibration Roberto Molinari Research Center for Statistics (GSEM) University of Geneva joint work with S. Guerrier (UIUC), J. Balamuta (UIUC), J. Skaloud
More informationRegime Switching Models: An Example for a Stock Market Index
Regime Switching Models: An Example for a Stock Market Index Erik Kole Econometric Institute, Erasmus School of Economics, Erasmus University Rotterdam April 2010 In this document, I discuss in detail
More informationChapter 6. Econometrics. 6.1 Introduction. 6.2 Univariate techniques Transforming data
Chapter 6 Econometrics 6.1 Introduction We re going to use a few tools to characterize the time series properties of macro variables. Today, we will take a relatively atheoretical approach to this task,
More informationChapter 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 informationChapter 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 informationRepresentation 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 informationNote: We will use EViews 5.1 in this tutorial. There may be minor differences in EViews 6.
ECON 604 EVIEWS TUTORIAL #2 IDENTIFYING AND ESTIMATING ARMA MODELS Note: We will use EViews 5.1 in this tutorial. There may be minor differences in EViews 6. Preliminary Analysis In this tutorial we will
More informationLecture 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 (IMEANice) Sept. 2011  Jan. 2012 Florian Pelgrin (HEC) Univariate time series Sept.
More informationAdaptive DemandForecasting Approach based on Principal Components Timeseries an application of datamining technique to detection of market movement
Adaptive DemandForecasting Approach based on Principal Components Timeseries an application of datamining technique to detection of market movement Toshio Sugihara Abstract In this study, an adaptive
More informationITSMR Reference Manual
ITSMR 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 informationNormalization 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 EMail: r.j.rossana@wayne.edu
More informationTime Series Graphs. Model ACF PACF. White Noise All zeros All zeros. AR(p) Exponential Decay P significant lags before dropping to zero
Time Series Graphs Model ACF PACF White Noise All zeros All zeros AR(p) Exponential Decay P significant lags before dropping to zero MA(q) q significant lags before dropping to zero Exponential Decay ARMA(p,q)
More informationNotes on Chapter 1, Section 2 Arithmetic and Divisibility
Notes on Chapter 1, Section 2 Arithmetic and Divisibility August 16, 2006 1 Arithmetic Properties of the Integers Recall that the set of integers is the set Z = f0; 1; 1; 2; 2; 3; 3; : : :g. The integers
More informationI. 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 IMFTAOLAM
More informationEC 6310: Advanced Econometric Theory
EC 6310: Advanced Econometric Theory July 2008 Slides for Lecture on Bayesian Computation in the Nonlinear Regression Model Gary Koop, University of Strathclyde 1 Summary Readings: Chapter 5 of textbook.
More informationAnalysis 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 informationMATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix.
MATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix. Inverse matrix Definition. Let A be an n n matrix. The inverse of A is an n n matrix, denoted
More informationState 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 informationEconomics 140A Identification in Simultaneous Equation Models Simultaneous Equation Models
Economics 140A Identification in Simultaneous Equation Models Simultaneous Equation Models Our second extension of the classic regression model, to which we devote two lectures, is to a system (or model)
More informationSYSTEMS 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 information1 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 informationy 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 informationMaximum 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 information14.451 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 informationLimitations of regression analysis
Limitations of regression analysis Ragnar Nymoen Department of Economics, UiO 8 February 2009 Overview What are the limitations to regression? Simultaneous equations bias Measurement errors in explanatory
More informationMultivariate Normal Distribution
Multivariate Normal Distribution Lecture 4 July 21, 2011 Advanced Multivariate Statistical Methods ICPSR Summer Session #2 Lecture #47/21/2011 Slide 1 of 41 Last Time Matrices and vectors Eigenvalues
More informationAmath 546/Econ 589 Multivariate GARCH Models
Amath 546/Econ 589 Multivariate GARCH Models Eric Zivot May 15, 2013 Lecture Outline Exponentially weighted covariance estimation Multivariate GARCH models Prediction from multivariate GARCH models Reading
More informationMaster 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 informationPortfolio 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 informationMeasuring Rationality with the Minimum Cost of Revealed Preference Violations. Mark Dean and Daniel Martin. Online Appendices  Not for Publication
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 informationEviews Tutorial. File New Workfile. Start observation End observation Annual
APS 425 Professor G. William Schwert Advanced Managerial Data Analysis CS3110L, 5852752470 Fax: 5854615475 email: schwert@schwert.ssb.rochester.edu Eviews Tutorial 1. Creating a Workfile: First you
More informationTechnical 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 informationSales forecasting # 2
Sales forecasting # 2 Arthur Charpentier arthur.charpentier@univrennes1.fr 1 Agenda Qualitative and quantitative methods, a very general introduction Series decomposition Short versus long term forecasting
More informationGeneralized Linear Models. Today: definition of GLM, maximum likelihood estimation. Involves choice of a link function (systematic component)
Generalized Linear Models Last time: definition of exponential family, derivation of mean and variance (memorize) Today: definition of GLM, maximum likelihood estimation Include predictors x i through
More informationTime 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 informationA Multiplicative Seasonal BoxJenkins Model to Nigerian Stock Prices
A Multiplicative Seasonal BoxJenkins 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 informationLecture 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 informationCCNY. 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 informationAbstract: 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(a) The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular.
Theorem.7.: (Properties of Triangular Matrices) (a) The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular. (b) The product
More informationMath 407A: Linear Optimization
Math 407A: Linear Optimization Lecture 4: LP Standard Form 1 1 Author: James Burke, University of Washington LPs in Standard Form Minimization maximization Linear equations to linear inequalities Lower
More informationTime Series Laboratory
Time Series Laboratory Computing in Weber Classrooms 205206: 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 informationChapter 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 1011
More informationDATA 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 informationEconometria dei mercati finanziari c.a. A.A. 20112012. 4. AR, MA and ARMA Time Series Models. Luca Fanelli. University of Bologna
Econometria dei mercati finanziari c.a. A.A. 20112012 4. AR, MA and ARMA Time Series Models Luca Fanelli University of Bologna luca.fanelli@unibo.it For each class of models considered in these slides
More informationLogistic 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 informationGeneralized Linear Model Theory
Appendix B Generalized Linear Model Theory We describe the generalized linear model as formulated by Nelder and Wedderburn (1972), and discuss estimation of the parameters and tests of hypotheses. B.1
More informationSystem 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 informationLecture 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 informationTime Series Analysis
Time Series Analysis Autoregressive, MA and ARMA processes Andrés M. Alonso Carolina GarcíaMartos Universidad Carlos III de Madrid Universidad Politécnica de Madrid June July, 212 Alonso and GarcíaMartos
More informationLOGISTIC 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 informationForecasting 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 informationDirect 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 information1111: Linear Algebra I
1111: Linear Algebra I Dr. Vladimir Dotsenko (Vlad) Lecture 3 Dr. Vladimir Dotsenko (Vlad) 1111: Linear Algebra I Lecture 3 1 / 12 Vector product and volumes Theorem. For three 3D vectors u, v, and w,
More informationECONOMETRIC MODELS. The concept of Data Generating Process (DGP) and its relationships with the analysis of specification.
ECONOMETRIC MODELS The concept of Data Generating Process (DGP) and its relationships with the analysis of specification. Luca Fanelli University of Bologna luca.fanelli@unibo.it The concept of Data Generating
More information