Focused Information Criteria for Time Series
|
|
- Christal Rich
- 5 years ago
- Views:
Transcription
1 Focused Information Criteria for Time Series Gudmund Horn Hermansen with Nils Lid Hjort University of Oslo May 10, /22
2 Do we really need more model selection criteria? There is already a wide range of criteria, e.g. AIC, AIC C, BIC, TIC, FPR, HQ, etc. The underlying motivations are not particularly well known among practitioners. Example: For stationary time series, there are two versions based on similar reasoning of the AIC, i.e. for model M we have and AIC n (M) = 2 log-likelihood max (M) 2p AIC (M) = 2 Whittle-log-likelihood max (M) 2p + 2q, where p = dim(m) and q has to be estimated, see Hermansen and Hjort (2015). There are (at least) three good selling points for the FIC: (1) allows for problem specific focus (2) clear and simple motivation (minimising estimated mse) (3) in principle as easy to use as the AIC (e.g. no tuning parameters) 2/22
3 The focused information criterion For model M let µ M be a focus parameter as a function of M, e.g. quantile, threshold probability, covariance lag, etc. It is important the µ M has the same interpretation across models. Let µ M be estimated by the plug-in principle for each M, e.g. in the case where M is specified by θ M R p we have µ M = µ( θ M ). The goal is to find best model/estimator for µ with respect to mse( µ M ) = (bias( µ M )) 2 + Var( µ M ) = sqb( µ M ) + Var( µ M ). Model selection strategy (1) Obtain a reasonable estimator for the mse and use FIC(µ, M) = mse( µ M ) = ŝqb( µ M ) + Var( µ M ). (2) Choose the model and estimator with the smallest estimated mse. 3/22
4 The focused information criterion Here, we take the common large-sample approximations approach to obtain general estimators for the mse. This work extends Claeskens and Hjort (2003). A parametric approach, where all models are nested between a wide specified by (θ, γ) and a narrow model with (θ, γ 0 ) and γ 0 known. The true generating model for Y is parametrised by (θ 0, γ 0 + δ/ n). The mse( µ M ) is based on an (unbiased) estimate for the mse of n( µm µ true ) in the limit experiment, where µ true is µ evaluated at the truth. The extension has certain time series specific challenges, e.g. - we would like to include predictions - and data-dependent foci like µ M (Y 1,... Y n ) = Pr{Y n+1 > a and Y n+2 > a Y 1,... Y n }, for a certain constant a. 4/22
5 Model and assumptions Let Y t = m β (x t ) + ɛ t, where ɛ t is a stationary Gaussian time series with E ɛ t = 0 and spectral density f η and x t are covariate vectors. Following Claeskens and Hjort (2003) the true model is specified by and m true (x t ) = m(x t ; β 0, γ 0,1 + δ 1 / n) f true (ω) = f(ω; ν 0, γ 0,2 + δ 2 / n) where θ 0 = (β 0, ν 0 ) R p1+p2 and γ 0 = (γ 0,1, γ 0,2 ) R q1+q2. In addition, we need (essentially) that 1 n [ m 0(x t )] t Σ(f 0 ) 1 [ m 0 (x t )] M exists with m 0 (x t ) = m(x t ; β 0, γ 0,1 )/ β and Σ(f 0 ) being the associated covariance matrix, and where f 0 (ω) = f(ω; ν 0, γ 0,2 ) has continuous and bounded second order derivatives. 5/22
6 Model and assumptions This setup allows for misspecification in both trend and dependency. The wide model has p + q = (p 1 + p 2 ) + (q 1 + q 2 ) parameters. The candidate models are nested between the wide model and the narrow model, where γ = γ 0. A total of 2 q1+q2 possible models obtained by including/excluding elements of γ 0. We only consider those that are judged as sufficiently plausible. There are few papers dealing with the FIC related topics for time series models, see e.g. Claeskens et al. (2007). The derived results are also valid for locally-stationary process of Dahlhaus (1997). Example: Suppose Y t = 0 + ɛ t and that we are interested in a certain (important) covariance lag h, then µ M (h) = cov fν (Y t, Y t+h ) = π π cos(ωh)f ν (ω) dω. 6/22
7 Okay, but does it really work? Simple simulation study with an AR(4) model focusing on various µ M (h) = cov fν (Y t, Y t+h ). The true model has σ = 1, ρ = (0.4, 0.4, 0.4, 0.2), and the figure is based on 50 simulated series of length n = 50. 7/22
8 The focused information criterion for time series Suppose µ = µ(θ), where θ = (β, ν), only depends on the model parameters. Then, for each submodel M we obtain a general argument for n( µm µ true ) d Λ M, where Λ M has a certain multivariate normal distribution. From this an unbiased estimator for mse(λ M ) is constructed via mse(λ M ) = ŝqb(λ M ) + Var(Λ M ) A common challenge is that ŝqb(λ M ) is itself biased and should be corrected, resulting in a robust mse(λ M ) = max{0, ŝqb(λ M ) bias(ŝqb(λ M ))} + Var(Λ M ) The FIC strategy is to use mse(λ M ) to approximate the mse for each submodel M. Here, compared to Claeskens and Hjort (2003) the general structure, arguments and formulas are quite similar. 8/22
9 The focused information criterion for time series Under the Gaussian assumption the parameters related to trend and dependency are independent in the limit. The traditional (non-robust) FIC can therefore be expressed as FIC(µ, M) = σ 2 narrow + 2( σ 2 M f + σ 2 M m ) + ( ψ wide ψ Mf ψ Mm ) 2, where the σ are related to the variance and the ψ to the bias. If µ is independent of either trend or dependency, e.g. µ = m β (1) m β (0) or µ = C(0) the FIC-scores are indifferent to changes in the excluded direction. This suggests to either detrend prior to the analysis or that scores should be estimated under the respective candidate model. Also, the formulas involved simplify if: - m β (x t ) = β - m β (x t ) = x t tβ and x t are from a well-behaved distribution - if x t is smooth in t - Y t is a locally-stationary process (cf. Dahlhaus (1997)) 9/22
10 What makes focus functions data-dependent? Some foci are more interesting in a conditional framework. Illustration: Consider the data-independent threshold probability or data-dependent µ = Pr{Y n+1 > a and Y n+2 > a} µ(h m ) = Pr{Y n+1 > a and Y n+2 > a H m } for suitable constant a, and with (recent) history H m = (Y n m+1,..., Y n ). In principle we could have H m with m = n. In practice, quite often m is independent of n and m n. Example: For AR(q) processes it will often be sufficient with m = q. Short-memory series with H m and m = m n should be effectively approximated in a fixed and recent history framework. 10/22
11 What about predictions? Predictions are essentially data-dependent focus functions. To easily see why, - let F k = (Y n+1,..., Y n+k ) represent the near future and - suppose we intend to predict g(f k ), e.g. for one-step ahead predictions g(f k ) = Y n+1, - and that µ M (H m ) is a predictor for g(f m ) - then mse( µ M (H m )) = E { µ M (H m ) E[g(F k ) H m ] + E[g(F k ) H m ] g(f k )} 2 = E { µ M (H m ) E[g(F k ) H m ]} E {E[g(F k ) H m ] g(h m )} 2 - with the conclusion that a good predictor for g(f k ) is equivalent (in term of mse) to a good estimator for µ true (H m ) = E[g(F k ) H m ]. 11/22
12 Why this interest with recent history? A data-dependent focus begs the question of mse( µ M (H m )) or mse( µ M (H m ) H m ). Use the one that best represents what is important. Does not necessarily make sense if m = n, since mse( µ M (H m ) H m ) = mse( µ M (H n ) H n ) = 0 for all unbiased submodels estimators. If m is independent of n and m n it makes sense to introduce cfic(µ, M, H m ) = mse( µ M (H m ) H m ). If the large-sample arguments hold conditionally and things are independent of H m it the limit, then: - the familiar FIC formulas remain largely unchanged (everything involving µ do now depend on H m ) - and should be interpreted in relation to conditional mse 12/22
13 Why this interest with recent history? A key step of the FIC argument depends on the delta method, i.e. n( µm (H m ) µ true (H m )) = n(µ( θ M, γ M, γ 0,M c, H m ) µ true (H m )). = µ(θ 0, γ 0 + δ/ n, H m ) t Z n where Z n = depends on H m through ( θ, γ M ). ( ) n( θ θ0 ) n( γm γ 0,M ) In the conditional framework µ(θ 0, γ 0 + δ/ n, H m ) is not random anymore, which simplifies the arguments needed. And if Z n H m d Z, with Z independent of H m, we have justified our cfic(µ, M, H m ) = mse( µ M (H m ) H m ). Is the conditional convergence of Z n generally true? Again, this will not work if m = n and H m = H n = (Y 1,..., Y n ). 13/22
14 Now, how does this play out unconditionally? It is much harder to find a simple limit experiment such that unconditionally. n( µ M (H m ) µ true (H m )) Λ M (H m ) pr 0 However, following the general idea mse(λ M (H M )) mse( n( µ M (H m ) µ true (H m ))) = E {mse( n( µ M (H m ) µ true (H m )) H m )}. E { mse(λ M (H m ) H m )} A quick (and dirty) solution resulting in explicit (but quite messy) formulas is to use FIC(µ, M, H m ) = Ê{mse( µ(h m) H m )}. 14/22
15 Illustration: The Hjort liver quality index ( ) For a individual fish HSI fish = 100 weight of liver weight of fish. The HSI is a measure of the quality of life and is e.g. related to reproduction. Understanding the dynamic of the HSI index to e.g. external factors. 15/22
16 Illustration: The Hjort liver quality index ( ) Predicting the future liver quality index. The model we consider is HSI yeari = β 0 + β 1 year i + x t iβ + ɛ i, where the intercept β 0, and σ are protected and x i contains winter Kola temperature, mortality rate (F) and food availability (capelin). 16/22
17 Illustration: The Hjort liver quality index ( ) Other foci we looked at was relative slope and the probability of two lean years in a row. More details and discussion can be found in Hermansen et al. (2015) 17/22
18 Illustration: Prototype FIC R-package Simple threshold probability simulation experiment. True model is Y t = 1 + 2(t/n) + ɛ t, where ɛ t is an AR(4) prosess with ρ = (0.3, 0.2, 0.1, 0.1) and σ = 1, and n = 50. Focus parameter is and µ 1 (H m ) = µ 1 (H m ) = Pr{Y n+1 > 0 H m }, m f mu mse bias sd psi tau.sq fic fic.b aic bic p fir.r fic.b.r aic.r bic.r tau.null = 0.4 psi.wide = /22
19 Illustration: Prototype FIC R-package Simple threshold probability simulation experiment. True model is Y t = 1 + 2(t/n) + ɛ t, where ɛ t is an AR(4) prosess with ρ = (0.3, 0.2, 0.1, 0.1) and σ = 1, and n = 50. Focus parameter is and µ 2 (H m ) = µ 2 (H m ) = Pr{Y n+1 > 0 and Y n+2 > 0 H m }, m f mu mse bias sd psi tau.sq fic fic.b aic bic p fir.r fic.b.r aic.r bic.r tau.null = 0.49 psi.wide = /22
20 Illustration: Prototype FIC R-package Simple threshold probability simulation experiment. True model is Y t = 1 + 2(t/n) + ɛ t, where ɛ t is an AR(4) prosess with ρ = (0.3, 0.2, 0.1, 0.1) and σ = 1, and n = 50. Focus parameter is and µ 10 (H m ) = µ 10 (H m ) = Pr{Y n+10 > 0 H m }, m f mu mse bias sd psi tau.sq fic fic.b aic bic p fir.r fic.b.r aic.r bic.r tau.null = 0.57 psi.wide = /22
21 Concluding remarks Some work is still needed before completion. An R-package is planned/under development. Simulation study. Model averaging and AFIC. There are no good model selection tools for the locally-stationary processes of Dahlhaus (1997). The FIC based on the Whittle approximation (cf. Whittle (1953)) is equally rational motivation. The methodology is valid for models with known change point locations. Seasonality. Nonparametric (focused) covariance estimator. 21/22
22 References Claeskens, G., Croux, C., and Van Kerckhoven, J. (2007). Prediction focussed model selection for autoregressive models. Australian & New Zealand Journal of Statistics, 49(4): Claeskens, G. and Hjort, N. L. (2003). The focused information criterion. Journal of the American Statistical Association, 98: Dahlhaus, R. (1997). Fitting time series models to nonstationary processes. Annals of Statistics, 15:1 37. Hermansen, G. and Hjort, N. L. (2015). A new approach to Akaike s information criterion and model selection issues in stationary Gaussian time series. Technical report, University of Oslo and Norwegian Computing Centre. Hermansen, G. H., Hjort, N. L., Kjesbu, O. S., and Tara Marshall, C. (2015). Recent advances in statistical methodology applied to the hjort liver index time series ( ) and associated influential factors 1. Canadian Journal of Fisheries and Aquatic Sciences, 73(999):1 17. Whittle, P. (1953). The analysis of multiple stationary time series. Journal of the Royal Statistical Society. Series B (Methodological), 15(1): /22
The Hybrid Likelihood: Combining Parametric and Empirical Likelihoods
1/25 The Hybrid Likelihood: Combining Parametric and Empirical Likelihoods Nils Lid Hjort (with Ingrid Van Keilegom and Ian McKeague) Department of Mathematics, University of Oslo Building Bridges (at
More informationModel selection using penalty function criteria
Model selection using penalty function criteria Laimonis Kavalieris University of Otago Dunedin, New Zealand Econometrics, Time Series Analysis, and Systems Theory Wien, June 18 20 Outline Classes of models.
More informationFocuStat, FIC themes, this workshop... with more to come
1/14 FocuStat, FIC themes, this workshop... with more to come Nils Lid Hjort Department of Mathematics, University of Oslo FICology, May 2016, Oslo 2/14 Outline A FocuStat (2014-2018) B FIC: a mini-history
More informationPARAMETRIC OR NONPARAMETRIC: THE FIC APPROACH
Statistica Sinica: Supplement PARAMETRIC OR NONPARAMETRIC: THE FIC APPROACH Martin Jullum and Nils Lid Hjort Department of Mathematics, University of Oslo Supplementary Material This is a supplement to
More informationStatistics 910, #5 1. Regression Methods
Statistics 910, #5 1 Overview Regression Methods 1. Idea: effects of dependence 2. Examples of estimation (in R) 3. Review of regression 4. Comparisons and relative efficiencies Idea Decomposition Well-known
More informationChapter 3: Regression Methods for Trends
Chapter 3: Regression Methods for Trends Time series exhibiting trends over time have a mean function that is some simple function (not necessarily constant) of time. The example random walk graph from
More informationThe Behaviour of the Akaike Information Criterion when Applied to Non-nested Sequences of Models
The Behaviour of the Akaike Information Criterion when Applied to Non-nested Sequences of Models Centre for Molecular, Environmental, Genetic & Analytic (MEGA) Epidemiology School of Population Health
More informationLikelihood-Based Methods
Likelihood-Based Methods Handbook of Spatial Statistics, Chapter 4 Susheela Singh September 22, 2016 OVERVIEW INTRODUCTION MAXIMUM LIKELIHOOD ESTIMATION (ML) RESTRICTED MAXIMUM LIKELIHOOD ESTIMATION (REML)
More informationModel Selection Tutorial 2: Problems With Using AIC to Select a Subset of Exposures in a Regression Model
Model Selection Tutorial 2: Problems With Using AIC to Select a Subset of Exposures in a Regression Model Centre for Molecular, Environmental, Genetic & Analytic (MEGA) Epidemiology School of Population
More information9. Model Selection. statistical models. overview of model selection. information criteria. goodness-of-fit measures
FE661 - Statistical Methods for Financial Engineering 9. Model Selection Jitkomut Songsiri statistical models overview of model selection information criteria goodness-of-fit measures 9-1 Statistical models
More informationRegression I: Mean Squared Error and Measuring Quality of Fit
Regression I: Mean Squared Error and Measuring Quality of Fit -Applied Multivariate Analysis- Lecturer: Darren Homrighausen, PhD 1 The Setup Suppose there is a scientific problem we are interested in solving
More informationElements of Multivariate Time Series Analysis
Gregory C. Reinsel Elements of Multivariate Time Series Analysis Second Edition With 14 Figures Springer Contents Preface to the Second Edition Preface to the First Edition vii ix 1. Vector Time Series
More informationOn the Behavior of Marginal and Conditional Akaike Information Criteria in Linear Mixed Models
On the Behavior of Marginal and Conditional Akaike Information Criteria in Linear Mixed Models Thomas Kneib Department of Mathematics Carl von Ossietzky University Oldenburg Sonja Greven Department of
More informationFall 2017 STAT 532 Homework Peter Hoff. 1. Let P be a probability measure on a collection of sets A.
1. Let P be a probability measure on a collection of sets A. (a) For each n N, let H n be a set in A such that H n H n+1. Show that P (H n ) monotonically converges to P ( k=1 H k) as n. (b) For each n
More informationCh 6. Model Specification. Time Series Analysis
We start to build ARIMA(p,d,q) models. The subjects include: 1 how to determine p, d, q for a given series (Chapter 6); 2 how to estimate the parameters (φ s and θ s) of a specific ARIMA(p,d,q) model (Chapter
More informationSwitching Regime Estimation
Switching Regime Estimation Series de Tiempo BIrkbeck March 2013 Martin Sola (FE) Markov Switching models 01/13 1 / 52 The economy (the time series) often behaves very different in periods such as booms
More informationOn the Behavior of Marginal and Conditional Akaike Information Criteria in Linear Mixed Models
On the Behavior of Marginal and Conditional Akaike Information Criteria in Linear Mixed Models Thomas Kneib Institute of Statistics and Econometrics Georg-August-University Göttingen Department of Statistics
More informationA time series is called strictly stationary if the joint distribution of every collection (Y t
5 Time series A time series is a set of observations recorded over time. You can think for example at the GDP of a country over the years (or quarters) or the hourly measurements of temperature over a
More informationAn Introduction to Mplus and Path Analysis
An Introduction to Mplus and Path Analysis PSYC 943: Fundamentals of Multivariate Modeling Lecture 10: October 30, 2013 PSYC 943: Lecture 10 Today s Lecture Path analysis starting with multivariate regression
More informationNonstationary spatial process modeling Part II Paul D. Sampson --- Catherine Calder Univ of Washington --- Ohio State University
Nonstationary spatial process modeling Part II Paul D. Sampson --- Catherine Calder Univ of Washington --- Ohio State University this presentation derived from that presented at the Pan-American Advanced
More informationBIO5312 Biostatistics Lecture 13: Maximum Likelihood Estimation
BIO5312 Biostatistics Lecture 13: Maximum Likelihood Estimation Yujin Chung November 29th, 2016 Fall 2016 Yujin Chung Lec13: MLE Fall 2016 1/24 Previous Parametric tests Mean comparisons (normality assumption)
More informationOn the equivalence of confidence interval estimation based on frequentist model averaging and least-squares of the full model in linear regression
Working Paper 2016:1 Department of Statistics On the equivalence of confidence interval estimation based on frequentist model averaging and least-squares of the full model in linear regression Sebastian
More informationEfficient Estimation of Population Quantiles in General Semiparametric Regression Models
Efficient Estimation of Population Quantiles in General Semiparametric Regression Models Arnab Maity 1 Department of Statistics, Texas A&M University, College Station TX 77843-3143, U.S.A. amaity@stat.tamu.edu
More informationAn Introduction to Path Analysis
An Introduction to Path Analysis PRE 905: Multivariate Analysis Lecture 10: April 15, 2014 PRE 905: Lecture 10 Path Analysis Today s Lecture Path analysis starting with multivariate regression then arriving
More informationTime Series Forecasting: A Tool for Out - Sample Model Selection and Evaluation
AMERICAN JOURNAL OF SCIENTIFIC AND INDUSTRIAL RESEARCH 214, Science Huβ, http://www.scihub.org/ajsir ISSN: 2153-649X, doi:1.5251/ajsir.214.5.6.185.194 Time Series Forecasting: A Tool for Out - Sample Model
More informationR&D Research Project: Scaling analysis of hydrometeorological time series data
R&D Research Project: Scaling analysis of hydrometeorological time series data Extreme Value Analysis considering Trends: Methodology and Application to Runoff Data of the River Danube Catchment M. Kallache,
More informationGeographically Weighted Regression as a Statistical Model
Geographically Weighted Regression as a Statistical Model Chris Brunsdon Stewart Fotheringham Martin Charlton October 6, 2000 Spatial Analysis Research Group Department of Geography University of Newcastle-upon-Tyne
More informationRegression, Ridge Regression, Lasso
Regression, Ridge Regression, Lasso Fabio G. Cozman - fgcozman@usp.br October 2, 2018 A general definition Regression studies the relationship between a response variable Y and covariates X 1,..., X n.
More informationOutlier detection in ARIMA and seasonal ARIMA models by. Bayesian Information Type Criteria
Outlier detection in ARIMA and seasonal ARIMA models by Bayesian Information Type Criteria Pedro Galeano and Daniel Peña Departamento de Estadística Universidad Carlos III de Madrid 1 Introduction The
More informationMASM22/FMSN30: Linear and Logistic Regression, 7.5 hp FMSN40:... with Data Gathering, 9 hp
Selection criteria Example Methods MASM22/FMSN30: Linear and Logistic Regression, 7.5 hp FMSN40:... with Data Gathering, 9 hp Lecture 5, spring 2018 Model selection tools Mathematical Statistics / Centre
More informationModel Selection. Frank Wood. December 10, 2009
Model Selection Frank Wood December 10, 2009 Standard Linear Regression Recipe Identify the explanatory variables Decide the functional forms in which the explanatory variables can enter the model Decide
More informationTIME SERIES ANALYSIS AND FORECASTING USING THE STATISTICAL MODEL ARIMA
CHAPTER 6 TIME SERIES ANALYSIS AND FORECASTING USING THE STATISTICAL MODEL ARIMA 6.1. Introduction A time series is a sequence of observations ordered in time. A basic assumption in the time series analysis
More informationARIMA Modelling and Forecasting
ARIMA Modelling and Forecasting Economic time series often appear nonstationary, because of trends, seasonal patterns, cycles, etc. However, the differences may appear stationary. Δx t x t x t 1 (first
More informationWEIGHTED QUANTILE REGRESSION THEORY AND ITS APPLICATION. Abstract
Journal of Data Science,17(1). P. 145-160,2019 DOI:10.6339/JDS.201901_17(1).0007 WEIGHTED QUANTILE REGRESSION THEORY AND ITS APPLICATION Wei Xiong *, Maozai Tian 2 1 School of Statistics, University of
More informationCovariance function estimation in Gaussian process regression
Covariance function estimation in Gaussian process regression François Bachoc Department of Statistics and Operations Research, University of Vienna WU Research Seminar - May 2015 François Bachoc Gaussian
More informationOn the Behavior of Marginal and Conditional Akaike Information Criteria in Linear Mixed Models
On the Behavior of Marginal and Conditional Akaike Information Criteria in Linear Mixed Models Thomas Kneib Department of Mathematics Carl von Ossietzky University Oldenburg Sonja Greven Department of
More informationEcon 582 Nonparametric Regression
Econ 582 Nonparametric Regression Eric Zivot May 28, 2013 Nonparametric Regression Sofarwehaveonlyconsideredlinearregressionmodels = x 0 β + [ x ]=0 [ x = x] =x 0 β = [ x = x] [ x = x] x = β The assume
More informationSubject-specific observed profiles of log(fev1) vs age First 50 subjects in Six Cities Study
Subject-specific observed profiles of log(fev1) vs age First 50 subjects in Six Cities Study 1.4 0.0-6 7 8 9 10 11 12 13 14 15 16 17 18 19 age Model 1: A simple broken stick model with knot at 14 fit with
More informationEcon 623 Econometrics II Topic 2: Stationary Time Series
1 Introduction Econ 623 Econometrics II Topic 2: Stationary Time Series In the regression model we can model the error term as an autoregression AR(1) process. That is, we can use the past value of the
More informationChapter 9: Forecasting
Chapter 9: Forecasting One of the critical goals of time series analysis is to forecast (predict) the values of the time series at times in the future. When forecasting, we ideally should evaluate the
More informationStat/F&W Ecol/Hort 572 Review Points Ané, Spring 2010
1 Linear models Y = Xβ + ɛ with ɛ N (0, σ 2 e) or Y N (Xβ, σ 2 e) where the model matrix X contains the information on predictors and β includes all coefficients (intercept, slope(s) etc.). 1. Number of
More informationFORECASTING SUGARCANE PRODUCTION IN INDIA WITH ARIMA MODEL
FORECASTING SUGARCANE PRODUCTION IN INDIA WITH ARIMA MODEL B. N. MANDAL Abstract: Yearly sugarcane production data for the period of - to - of India were analyzed by time-series methods. Autocorrelation
More informationISyE 691 Data mining and analytics
ISyE 691 Data mining and analytics Regression Instructor: Prof. Kaibo Liu Department of Industrial and Systems Engineering UW-Madison Email: kliu8@wisc.edu Office: Room 3017 (Mechanical Engineering Building)
More informationOptimizing forecasts for inflation and interest rates by time-series model averaging
Optimizing forecasts for inflation and interest rates by time-series model averaging Presented at the ISF 2008, Nice 1 Introduction 2 The rival prediction models 3 Prediction horse race 4 Parametric bootstrap
More informationHow the mean changes depends on the other variable. Plots can show what s happening...
Chapter 8 (continued) Section 8.2: Interaction models An interaction model includes one or several cross-product terms. Example: two predictors Y i = β 0 + β 1 x i1 + β 2 x i2 + β 12 x i1 x i2 + ɛ i. How
More informationIntegrated Likelihood Estimation in Semiparametric Regression Models. Thomas A. Severini Department of Statistics Northwestern University
Integrated Likelihood Estimation in Semiparametric Regression Models Thomas A. Severini Department of Statistics Northwestern University Joint work with Heping He, University of York Introduction Let Y
More informationF9 F10: Autocorrelation
F9 F10: Autocorrelation Feng Li Department of Statistics, Stockholm University Introduction In the classic regression model we assume cov(u i, u j x i, x k ) = E(u i, u j ) = 0 What if we break the assumption?
More informationBooth School of Business, University of Chicago Business 41914, Spring Quarter 2013, Mr. Ruey S. Tsay. Midterm
Booth School of Business, University of Chicago Business 41914, Spring Quarter 2013, Mr. Ruey S. Tsay Midterm Chicago Booth Honor Code: I pledge my honor that I have not violated the Honor Code during
More informationWeek 5 Quantitative Analysis of Financial Markets Modeling and Forecasting Trend
Week 5 Quantitative Analysis of Financial Markets Modeling and Forecasting Trend Christopher Ting http://www.mysmu.edu/faculty/christophert/ Christopher Ting : christopherting@smu.edu.sg : 6828 0364 :
More informationConstruction of PoSI Statistics 1
Construction of PoSI Statistics 1 Andreas Buja and Arun Kumar Kuchibhotla Department of Statistics University of Pennsylvania September 8, 2018 WHOA-PSI 2018 1 Joint work with "Larry s Group" at Wharton,
More informationNEW ESTIMATORS FOR PARALLEL STEADY-STATE SIMULATIONS
roceedings of the 2009 Winter Simulation Conference M. D. Rossetti, R. R. Hill, B. Johansson, A. Dunkin, and R. G. Ingalls, eds. NEW ESTIMATORS FOR ARALLEL STEADY-STATE SIMULATIONS Ming-hua Hsieh Department
More informationStatistica Sinica Preprint No: SS R3
Statistica Sinica Preprint No: SS-2015-0364R3 Title Parametric or nonparametric: The FIC approach Manuscript ID SS-2015-0364 URL http://www.stat.sinica.edu.tw/statistica/ DOI 10.5705/ss.202015.0364 Complete
More informationSTAT 518 Intro Student Presentation
STAT 518 Intro Student Presentation Wen Wei Loh April 11, 2013 Title of paper Radford M. Neal [1999] Bayesian Statistics, 6: 475-501, 1999 What the paper is about Regression and Classification Flexible
More informationAn Akaike Criterion based on Kullback Symmetric Divergence in the Presence of Incomplete-Data
An Akaike Criterion based on Kullback Symmetric Divergence Bezza Hafidi a and Abdallah Mkhadri a a University Cadi-Ayyad, Faculty of sciences Semlalia, Department of Mathematics, PB.2390 Marrakech, Moroco
More informationBiostatistics-Lecture 16 Model Selection. Ruibin Xi Peking University School of Mathematical Sciences
Biostatistics-Lecture 16 Model Selection Ruibin Xi Peking University School of Mathematical Sciences Motivating example1 Interested in factors related to the life expectancy (50 US states,1969-71 ) Per
More informationStatistics 262: Intermediate Biostatistics Model selection
Statistics 262: Intermediate Biostatistics Model selection Jonathan Taylor & Kristin Cobb Statistics 262: Intermediate Biostatistics p.1/?? Today s class Model selection. Strategies for model selection.
More information{ } Stochastic processes. Models for time series. Specification of a process. Specification of a process. , X t3. ,...X tn }
Stochastic processes Time series are an example of a stochastic or random process Models for time series A stochastic process is 'a statistical phenomenon that evolves in time according to probabilistic
More informationMLR Model Selection. Author: Nicholas G Reich, Jeff Goldsmith. This material is part of the statsteachr project
MLR Model Selection Author: Nicholas G Reich, Jeff Goldsmith This material is part of the statsteachr project Made available under the Creative Commons Attribution-ShareAlike 3.0 Unported License: http://creativecommons.org/licenses/by-sa/3.0/deed.en
More informationEconometric Forecasting
Robert M. Kunst robert.kunst@univie.ac.at University of Vienna and Institute for Advanced Studies Vienna October 1, 2014 Outline Introduction Model-free extrapolation Univariate time-series models Trend
More informationTesting Restrictions and Comparing Models
Econ. 513, Time Series Econometrics Fall 00 Chris Sims Testing Restrictions and Comparing Models 1. THE PROBLEM We consider here the problem of comparing two parametric models for the data X, defined by
More informationarxiv: v2 [stat.me] 15 Jan 2018
Robust Inference under the Beta Regression Model with Application to Health Care Studies arxiv:1705.01449v2 [stat.me] 15 Jan 2018 Abhik Ghosh Indian Statistical Institute, Kolkata, India abhianik@gmail.com
More informationAdditive Outlier Detection in Seasonal ARIMA Models by a Modified Bayesian Information Criterion
13 Additive Outlier Detection in Seasonal ARIMA Models by a Modified Bayesian Information Criterion Pedro Galeano and Daniel Peña CONTENTS 13.1 Introduction... 317 13.2 Formulation of the Outlier Detection
More informationLecture 4: Dynamic models
linear s Lecture 4: s Hedibert Freitas Lopes The University of Chicago Booth School of Business 5807 South Woodlawn Avenue, Chicago, IL 60637 http://faculty.chicagobooth.edu/hedibert.lopes hlopes@chicagobooth.edu
More informationBayesian nonparametric modelling of covariance functions, with application to time series and spatial statistics. Gudmund Horn Hermansen
Bayesian nonparametric modelling of covariance functions, with application to time series and spatial statistics by Gudmund Horn Hermansen THESIS presented for the degree of MASTER OF SCIENCE under the
More informationIntroduction to Statistical modeling: handout for Math 489/583
Introduction to Statistical modeling: handout for Math 489/583 Statistical modeling occurs when we are trying to model some data using statistical tools. From the start, we recognize that no model is perfect
More informationSTATISTICS 174: APPLIED STATISTICS FINAL EXAM DECEMBER 10, 2002
Time allowed: 3 HOURS. STATISTICS 174: APPLIED STATISTICS FINAL EXAM DECEMBER 10, 2002 This is an open book exam: all course notes and the text are allowed, and you are expected to use your own calculator.
More informationTesting for Regime Switching in Singaporean Business Cycles
Testing for Regime Switching in Singaporean Business Cycles Robert Breunig School of Economics Faculty of Economics and Commerce Australian National University and Alison Stegman Research School of Pacific
More informationTopic 4: Model Specifications
Topic 4: Model Specifications Advanced Econometrics (I) Dong Chen School of Economics, Peking University 1 Functional Forms 1.1 Redefining Variables Change the unit of measurement of the variables will
More informationEstimation of Parameters in ARFIMA Processes: A Simulation Study
Estimation of Parameters in ARFIMA Processes: A Simulation Study Valderio Reisen Bovas Abraham Silvia Lopes Departamento de Department of Statistics Instituto de Estatistica and Actuarial Science Matematica
More informationGeneralized Linear Models
Generalized Linear Models Lecture 3. Hypothesis testing. Goodness of Fit. Model diagnostics GLM (Spring, 2018) Lecture 3 1 / 34 Models Let M(X r ) be a model with design matrix X r (with r columns) r n
More informationMISCELLANEOUS TOPICS RELATED TO LIKELIHOOD. Copyright c 2012 (Iowa State University) Statistics / 30
MISCELLANEOUS TOPICS RELATED TO LIKELIHOOD Copyright c 2012 (Iowa State University) Statistics 511 1 / 30 INFORMATION CRITERIA Akaike s Information criterion is given by AIC = 2l(ˆθ) + 2k, where l(ˆθ)
More informationDifferencing Revisited: I ARIMA(p,d,q) processes predicated on notion of dth order differencing of a time series {X t }: for d = 1 and 2, have X t
Differencing Revisited: I ARIMA(p,d,q) processes predicated on notion of dth order differencing of a time series {X t }: for d = 1 and 2, have X t 2 X t def in general = (1 B)X t = X t X t 1 def = ( X
More informationEconometría 2: Análisis de series de Tiempo
Econometría 2: Análisis de series de Tiempo Karoll GOMEZ kgomezp@unal.edu.co http://karollgomez.wordpress.com Segundo semestre 2016 IX. Vector Time Series Models VARMA Models A. 1. Motivation: The vector
More informationMS&E 226: Small Data
MS&E 226: Small Data Lecture 6: Model complexity scores (v3) Ramesh Johari ramesh.johari@stanford.edu Fall 2015 1 / 34 Estimating prediction error 2 / 34 Estimating prediction error We saw how we can estimate
More informationAkaike Information Criterion
Akaike Information Criterion Shuhua Hu Center for Research in Scientific Computation North Carolina State University Raleigh, NC February 7, 2012-1- background Background Model statistical model: Y j =
More informationLasso Maximum Likelihood Estimation of Parametric Models with Singular Information Matrices
Article Lasso Maximum Likelihood Estimation of Parametric Models with Singular Information Matrices Fei Jin 1,2 and Lung-fei Lee 3, * 1 School of Economics, Shanghai University of Finance and Economics,
More information401 Review. 6. Power analysis for one/two-sample hypothesis tests and for correlation analysis.
401 Review Major topics of the course 1. Univariate analysis 2. Bivariate analysis 3. Simple linear regression 4. Linear algebra 5. Multiple regression analysis Major analysis methods 1. Graphical analysis
More informationThis model of the conditional expectation is linear in the parameters. A more practical and relaxed attitude towards linear regression is to say that
Linear Regression For (X, Y ) a pair of random variables with values in R p R we assume that E(Y X) = β 0 + with β R p+1. p X j β j = (1, X T )β j=1 This model of the conditional expectation is linear
More informationForecast comparison of principal component regression and principal covariate regression
Forecast comparison of principal component regression and principal covariate regression Christiaan Heij, Patrick J.F. Groenen, Dick J. van Dijk Econometric Institute, Erasmus University Rotterdam Econometric
More informationEcon 423 Lecture Notes: Additional Topics in Time Series 1
Econ 423 Lecture Notes: Additional Topics in Time Series 1 John C. Chao April 25, 2017 1 These notes are based in large part on Chapter 16 of Stock and Watson (2011). They are for instructional purposes
More informationLinear models. Linear models are computationally convenient and remain widely used in. applied econometric research
Linear models Linear models are computationally convenient and remain widely used in applied econometric research Our main focus in these lectures will be on single equation linear models of the form y
More informationOn Modifications to Linking Variance Estimators in the Fay-Herriot Model that Induce Robustness
Statistics and Applications {ISSN 2452-7395 (online)} Volume 16 No. 1, 2018 (New Series), pp 289-303 On Modifications to Linking Variance Estimators in the Fay-Herriot Model that Induce Robustness Snigdhansu
More informationA strategy for modelling count data which may have extra zeros
A strategy for modelling count data which may have extra zeros Alan Welsh Centre for Mathematics and its Applications Australian National University The Data Response is the number of Leadbeater s possum
More informationGLS and FGLS. Econ 671. Purdue University. Justin L. Tobias (Purdue) GLS and FGLS 1 / 22
GLS and FGLS Econ 671 Purdue University Justin L. Tobias (Purdue) GLS and FGLS 1 / 22 In this lecture we continue to discuss properties associated with the GLS estimator. In addition we discuss the practical
More informationLesson 15: Building ARMA models. Examples
Lesson 15: Building ARMA models. Examples Dipartimento di Ingegneria e Scienze dell Informazione e Matematica Università dell Aquila, umberto.triacca@ec.univaq.it Examples In this lesson, in order to illustrate
More informationWorking Paper Series. 2/2013 Empirical prediction intervals revisited Y.S. Lee and S. Scholtes Forthcoming in the International Journal of Forecasting
Working Paper Series 2/2013 Empirical prediction intervals revisited Y.S. Lee and S. Scholtes Forthcoming in the International Journal of Forecasting Cambridge Judge Business School Working Papers These
More informationOn the Power of Tests for Regime Switching
On the Power of Tests for Regime Switching joint work with Drew Carter and Ben Hansen Douglas G. Steigerwald UC Santa Barbara May 2015 D. Steigerwald (UCSB) Regime Switching May 2015 1 / 42 Motivating
More informationS-GSTAR-SUR Model for Seasonal Spatio Temporal Data Forecasting ABSTRACT
Malaysian Journal of Mathematical Sciences (S) March : 53-65 (26) Special Issue: The th IMT-GT International Conference on Mathematics, Statistics and its Applications 24 (ICMSA 24) MALAYSIAN JOURNAL OF
More informationForecasting using R. Rob J Hyndman. 2.4 Non-seasonal ARIMA models. Forecasting using R 1
Forecasting using R Rob J Hyndman 2.4 Non-seasonal ARIMA models Forecasting using R 1 Outline 1 Autoregressive models 2 Moving average models 3 Non-seasonal ARIMA models 4 Partial autocorrelations 5 Estimation
More informationVariable Selection for Highly Correlated Predictors
Variable Selection for Highly Correlated Predictors Fei Xue and Annie Qu arxiv:1709.04840v1 [stat.me] 14 Sep 2017 Abstract Penalty-based variable selection methods are powerful in selecting relevant covariates
More informationEfficient and Robust Scale Estimation
Efficient and Robust Scale Estimation Garth Tarr, Samuel Müller and Neville Weber School of Mathematics and Statistics THE UNIVERSITY OF SYDNEY Outline Introduction and motivation The robust scale estimator
More informationLecture Stat Information Criterion
Lecture Stat 461-561 Information Criterion Arnaud Doucet February 2008 Arnaud Doucet () February 2008 1 / 34 Review of Maximum Likelihood Approach We have data X i i.i.d. g (x). We model the distribution
More informationTopic 4 Unit Roots. Gerald P. Dwyer. February Clemson University
Topic 4 Unit Roots Gerald P. Dwyer Clemson University February 2016 Outline 1 Unit Roots Introduction Trend and Difference Stationary Autocorrelations of Series That Have Deterministic or Stochastic Trends
More informationX random; interested in impact of X on Y. Time series analogue of regression.
Multiple time series Given: two series Y and X. Relationship between series? Possible approaches: X deterministic: regress Y on X via generalized least squares: arima.mle in SPlus or arima in R. We have
More informationISSN Article. Selection Criteria in Regime Switching Conditional Volatility Models
Econometrics 2015, 3, 289-316; doi:10.3390/econometrics3020289 OPEN ACCESS econometrics ISSN 2225-1146 www.mdpi.com/journal/econometrics Article Selection Criteria in Regime Switching Conditional Volatility
More informationCh 5. Models for Nonstationary Time Series. Time Series Analysis
We have studied some deterministic and some stationary trend models. However, many time series data cannot be modeled in either way. Ex. The data set oil.price displays an increasing variation from the
More informationTuning Parameter Selection in L1 Regularized Logistic Regression
Virginia Commonwealth University VCU Scholars Compass Theses and Dissertations Graduate School 2012 Tuning Parameter Selection in L1 Regularized Logistic Regression Shujing Shi Virginia Commonwealth University
More informationChoice is Suffering: A Focused Information Criterion for Model Selection
Choice is Suffering: A Focused Information Criterion for Model Selection Peter Behl, Holger Dette, Ruhr University Bochum, Manuel Frondel, Ruhr University Bochum and RWI, Harald Tauchmann, RWI Abstract.
More informationProblem Set 2: Box-Jenkins methodology
Problem Set : Box-Jenkins methodology 1) For an AR1) process we have: γ0) = σ ε 1 φ σ ε γ0) = 1 φ Hence, For a MA1) process, p lim R = φ γ0) = 1 + θ )σ ε σ ε 1 = γ0) 1 + θ Therefore, p lim R = 1 1 1 +
More informationEconometrics Honor s Exam Review Session. Spring 2012 Eunice Han
Econometrics Honor s Exam Review Session Spring 2012 Eunice Han Topics 1. OLS The Assumptions Omitted Variable Bias Conditional Mean Independence Hypothesis Testing and Confidence Intervals Homoskedasticity
More information