Forecasting with ARMA
|
|
- Mitchell Riley
- 5 years ago
- Views:
Transcription
1 Forecasting with ARMA Eduardo Rossi University of Pavia October 2013 Rossi Forecasting Financial Econometrics / 32
2 Mean Squared Error Linear Projection Forecast of Y t+1 based on a set of variables observed at date t, X t : Yt+1 t. The loss function MSE(Yt+1 t ) = E[Y t+1 Yt+1 t ]2 The forecast with the smallest MSE is Y t+1 t = E[Y t+1 X t ] Suppose Y t+1 t is a linear function of X t: Ŷ t+1 t = α X t if E[(Y t+1 α X t )X t] = 0 then α X t is the linear projection of Y t+1 on X t. Rossi Forecasting Financial Econometrics / 32
3 Linear Projection Linear Projection The LP projection produces the smallest MSE among the class of linear forecasting rule P(Y t+1 X t ) = α X t using MSE[ P(Y t+1 X t )] MSE[E(Y t+1 X t )] E[(Y t+1 α X t )X t] = 0 E[Y t+1 X t] = α E[X t X t] α = E[Y t+1 X t]e[x t X t] 1 Rossi Forecasting Financial Econometrics / 32
4 LP and OLS Linear Projection LP is closely related to OLS regression [ 1 β = T y t+1 = β X t + u t ] 1 [ ] T X t X 1 T t X t Y t+1 T t=1 β is constructed from the sample moments, while α is constructed from population moments. If {X t, Y t+1 } is covariance stationary and ergodic for second moments, then the sample moments will converge to the population moments as the sample size T goes to infinity 1 T 1 T t=1 t=1 T X t X p t E[X t X t] T p X t y t+1 E[Xt Y t+1 ] t=1 Rossi Forecasting Financial Econometrics / 32
5 LP and OLS Linear Projection implying β p α β is consistent for the LP coefficient. Rossi Forecasting Financial Econometrics / 32
6 Forecasting based on lagged ɛ s Infinite moving average (Y t µ) = ψ(l)ɛ t ɛ t WN(0, σ 2 ) ψ 0 = 1, j=0 ψ j <. An infinite number of obs on ɛ through date t: {ɛ t, ɛ t 1,...}. We know the values of µ and {ψ 1, ψ 2,...} Y t+s = µ + ɛ t+s + ψ 1 ɛ t+s 1 + ψ 2 ɛ t+s ψ s ɛ t + ψ s+1 ɛ t The optimal linear forecast is: Ê[Y t+s ɛ t, ɛ t 1,...] = µ + ψ s ɛ t + ψ s+1 ɛ t where Ê[Y t+s X t ] P(Y t+s 1, X t ). The unknown future ɛ s are set to their expected value of zero. The forecast error is Y t+s Ê[Y t+s ɛ t, ɛ t 1,...] = ɛ t+s + ψ 1 ɛ t+s 1 + ψ 2 ɛ t+s ψ s 1 ɛ t+1 Rossi Forecasting Financial Econometrics / 32
7 Forecasting based on lagged ɛ s E[(Y t+s Ê[Y t+s ɛ t, ɛ t 1,...]) 2 ] = (1 + ψ ψ 2 s 1)σ 2 when s the MSE converges to the unconditional variance σ 2 j=0 ψ2 j. MA(q): ψ(l) = 1 + θ 1 L θ q L q Y t+s = µ + ɛ t+s + θ 1 ɛ t+s θ t+s q ɛ t+s q The optimal linear forecast is { µ + θs ɛ Ê[Y t+s ɛ t, ɛ t 1,...] = t + θ s+1 ɛ t θ q ɛ t q+s s = 1,..., q µ s = q + 1,... Rossi Forecasting Financial Econometrics / 32
8 Forecasting based on lagged ɛ s MSE: σ 2 s = 1 (1 + θ θs 1 2 )σ2 s = 2, 3,..., q (1 + θ θ2 q)σ 2 s = q + 1, q + 2,... The MSE increases with the forecast horizon up until s = q. For s > q the forecast is the unconditional mean and the MSE is the unconditional variance of the series. Rossi Forecasting Financial Econometrics / 32
9 Forecasting based on lagged ɛ s Compact lag operator ψ(l) L s = L s + ψ 1 L 1 s + ψ 2 L 2 s ψ s 1 L 1 + ψ s L 0 + ψ s+1 L 1 + ψ s+2 L the annihilation operator replaces negative powers of L by zero [ ] ψ(l) = ψ s L 0 + ψ s+1 L 1 + ψ s+2 L L s + [ ] ψ(l) Ê[Y t+s ɛ t, ɛ t 1,...] = µ + L s ɛ t + Rossi Forecasting Financial Econometrics / 32
10 Forecasting based on lagged Y s In the usual forecasting situation we have obs on lagged Y s. Suppose the infinite MA process has an Infinite AR representation η(l)(y t µ) = ɛ t η(l) = j=0 η jl j, η 0 = 1 and j=0 η j < A c.s. AR(p) satisfies η(l) = [ψ(l)] 1. (1 φ 1 L φ 2 L φ p L p )(Y t µ) = ɛ t φ(l)(y t µ) = ɛ t η(l) = φ(l) ψ(l) = [φ(l)] 1 Rossi Forecasting Financial Econometrics / 32
11 Forecasting based on lagged Y s For an MA(q): Y t µ = (1 + θ 1 L θ q L q )ɛ t Y t µ = θ(l)ɛ t ψ(l) = θ(l) η(l) = [θ(l)] 1 provided that is based on an invertible representation. Rossi Forecasting Financial Econometrics / 32
12 Forecasting based on lagged Y s ARMA(p,q) can be represented as an AR( ) with ψ(l) = θ(l) φ(l) provided that the roots of φ(z) and θ(z) lie outside the unit circle. When the restrictions are satisfied obs on {Y t, Y t 1, Y t 2,... } will be sufficient to construct {ɛ t, ɛ t 1,...}. Rossi Forecasting Financial Econometrics / 32
13 Forecasting based on lagged Y s For example for an AR(1): (1 φl)(y t µ) = ɛ t given φ and µ and Y t,y t 1, the value of ɛ t can be constructed from ɛ t = (Y t µ) φ(y t 1 µ) For an invertible MA(1): (1 + θl) 1 (Y t µ) = ɛ t given an infinite number of obs on Y, we can compute: ɛ t = (Y t µ) θ(y t 1 µ) + θ 2 (Y t 2 µ) θ 3 (Y t 3 µ) +... Rossi Forecasting Financial Econometrics / 32
14 Forecasting based on lagged Y s Under the conditions [ ] ψ(l) Ê[Y t+s Y t, Y t 1,...] = µ + L s η(l)(y t µ) + the forecast of Y t+s as a function of lagged Y s. Using η(l) = [ψ(l)] 1 [ ] ψ(l) Ê[Y t+s Y t, Y t 1,...] = µ + L s [ψ(l)] 1 (Y t µ) + Wiener-Kolmogorov prediction formula. Rossi Forecasting Financial Econometrics / 32
15 Wiener-Kolmogorov prediction formula - AR(1) For example for an AR(1): ψ(l) = (1 φl)(y t µ) = ɛ t 1 1 φl = 1 + φl + φ2 L φ s L s +... the annihilation operator is: [ ] ψ(l) L s = φ s + φ s+1 L 1 + φ s+2 L = φs 1 φl [ ] ψ(l) Ê[Y t+s Y t, Y t 1,...] = µ + where ɛ t = (1 φl)(y t µ). + L s η(l)(y t µ) = µ + + Ê[Y t+s Y t, Y t 1,...] = µ + φ s (Y t µ) φs 1 φl (1 φl)(y t µ) the forecast decays geometrically from (Y t µ) toward µ as s increases. Rossi Forecasting Financial Econometrics / 32
16 Wiener-Kolmogorov prediction formula - AR(1) Given that ψ j = φ j, from the MSE of a MA( ), we have that the MSE s-period-ahead forecast error is: [1 + φ φ 2(s 1) ]σ 2 as s MSE = σ2 1 φ 2 Rossi Forecasting Financial Econometrics / 32
17 Wiener-Kolmogorov prediction formula - AR(p) Stationary AR(p) process Y t+s µ = f (s) 11 (Y t µ) + f (s) 12 (Y t 1 µ) f (s) 1p (Y t p+1 µ) + ɛ t+s + ψ 1 ɛ t+s ψ s 1 ɛ t+1 the optimal s-period-ahead forecast is forecast error ψ j = f (j) 11 Ŷ t+s t = µ + f (s) 11 (Y t µ) f (s) 1p (Y t p+1 µ) Y t+s t Ŷt+s t = ɛ t+s + ψ 1 ɛ t+s ψ s 1 ɛ t+1 Rossi Forecasting Financial Econometrics / 32
18 Wiener-Kolmogorov prediction formula - AR(p) To calculate the optimal forecast we use a recursion. Start with the forecast Ŷt+1 t Ŷ t+1 t µ = φ 1 (Y t µ) φ p (Y t p+1 µ) Ŷ t+2 t+1 : Ŷ t+2 t+1 µ = φ 1 (Y t+1 µ) φ p (Y t p+2 µ) Law of Iterated Projections: Forecast Ŷt+2 t+1 projected on date t information set then we obtain Ŷt+2 t: Ŷ t+2 t µ = φ 1 (Ŷt+1 t µ) φ p (Y t p+2 µ) substituting Ŷt+1 t Ŷ t+2 t µ = φ 1 [φ 1 (Y t µ) φ p (Y t p+1 µ)] + φ 2 (Y t µ) φ p (Y t p+2 µ) Rossi Forecasting Financial Econometrics / 32
19 Wiener-Kolmogorov prediction formula - AR(p) Ŷ t+2 t µ = (φ φ 2 )(Y t µ) + (φ 1 φ 2 + φ 3 )(Y t 1 µ) (φ 1 φ p 1 + φ p )(Y t p+2 µ) + φ 1 φ p (Y t p+1 µ) The s-period-ahead forecast of an AR(p) process can be obtained by iterating on Ŷ t+j t µ = φ 1 (Ŷt+j 1 t µ) φ p (Ŷt+j p t µ) Rossi Forecasting Financial Econometrics / 32
20 Wiener-Kolmogorov prediction formula - MA(1) Invertible MA(1) (Y t µ) = (1 + θl)ɛ t with θ < 1. Wiener-Kolmogorov formula [ ] ψ(l) Ŷ t+s t = µ + L s (1 + θl) 1 (Y t µ) + Forecast s = 1 [ ] (1 + θl) = θ Alternatively in practice L 1 Ŷ t+1 t = µ + θ 1 + θl (Y t µ) + = µ + θ(y t µ) θ 2 (Y t 1 µ) + θ 3 (Y t 2 µ) +... ɛ t = (1 + θl) 1 (Y t µ) ɛ t = (Y t µ) θ ɛ t 1. For s = 2, 3,... [ ] Rossi Forecasting Financial Econometrics / 32
21 Wiener-Kolmogorov prediction formula - MA(q) (Y t µ) = θ(l)ɛ t θ(l) = (1 + θ 1 L + θ 2 L θ q L q ) [ 1 + θ1 L θ q L q ] 1 Ŷ t+s t = µ + L s + θ(l) (Y t µ) [ 1 + θ1 L θ q L q ] { 1 + θs L + θ = s+1 L θ q L q s s = 1,..., q 0 s = q + 1,... For L s + Ŷ t+s t = µ + (θ s + θ s+1 L θ q L q s ) ɛ t ɛ t = (Y t µ) θ 1 ɛ t 1... θ q ɛ t q Rossi Forecasting Financial Econometrics / 32
22 Wiener-Kolmogorov prediction formula - ARMA(1,1) (1 φl)(y t µ) = (1 + θl)ɛ t Stationarity: φ < 1. Invertibility: θ < 1. [ ] 1 + θl Ŷ t+s t = µ + (1 φl)l s [ 1 + θl ] (1 φl)l s + = = + 1 φl 1 + θl (Y t µ) 1 (1 φl) = 1 + φl + φ2 L [ ] 1 (1 φl)l s + θl (1 φl)l s + [ (1 + φl + φ 2 L ) L s + θl(1 + φl + φ2 L 2 ] +...) L s = (φ s + φ s+1 L + φ s+2 L ) + θ(φ s 1 + φ s L + φ s+1 L ) + Rossi Forecasting Financial Econometrics / 32
23 Wiener-Kolmogorov prediction formula - ARMA(1,1) [ 1 + θl ] (1 φl)l s + = φ s (1 + φl + φ 2 L ) + θφ s 1 (1 + φl + φ 2 L ) = (φ s + θφ s 1 )(1 + φl + φ 2 L ) = φs + θφ s 1 1 φl Rossi Forecasting Financial Econometrics / 32
24 Wiener-Kolmogorov prediction formula - ARMA(1,1) For s = 2, 3,... the forecast [ ] 1 + θl 1 φl Ŷ t+s t = µ + (1 φl)l s θl (Y t µ) = µ + φs + θφ s 1 1 φl 1 φl 1 + θl (Y t µ) = µ + φs + θφ s θl (Y t µ) Ŷ t+s t µ = φ(ŷt+s 1 t µ) the forecast decays geometrically at the rate φ toward the unconditional mean µ. The one-period-ahead forecast (s=1) is given by Ŷ t+1 t = µ + φ + θ 1 + θl (Y t µ) Rossi Forecasting Financial Econometrics / 32
25 Wiener-Kolmogorov prediction formula - ARMA(1,1) φ(1 + θl) + θ(1 φl) Ŷ t+1 t = µ + (Y t µ) 1 + θl = µ + φ(y t µ) + 1 φl 1 + θl (Y t µ) ɛ t = 1 φl 1 + θl (Y t µ) = (Y t µ) φ(y t 1 µ) θ ɛ t 1 ɛ t = Y t Ŷt t 1 Rossi Forecasting Financial Econometrics / 32
26 Wiener-Kolmogorov prediction formula - ARMA(1,1) s = 2, Ŷ t+2 t = µ + φ2 + θφ 1 + θl (Y t µ) = µ + φ φ + θ 1 + θl (Y t µ) = µ + φ(φ + θ)(1 θl + θ 2 L 2 θ 3 L )(Y t µ) = µ + φ(φ + θ)(y t µ) φ(φ + θ)θ(y t 1 µ) +... Rossi Forecasting Financial Econometrics / 32
27 Wiener-Kolmogorov prediction formula - ARMA(p,q) ARMA(p,q): φ(l)(y t µ) = θ(l)ɛ t Ŷ t+1 t µ = φ 1 (Y t µ) φ p (Y t p+1 µ) + θ 1 ɛ t θ q ɛ t q+1 ɛ t = Y t Y t t 1 Ŷ τ t = Y τ τ t φ 1 (Ŷt+s 1 t µ) φ p (Y t+s p t µ) + θ 1 ɛ t θ q ɛ t+s q for s = 1,..., q Ŷ t+s t µ = φ 1 (Ŷt+s 1 t µ) φ p (Y t+s p t µ) for s = q + 1,... Rossi Forecasting Financial Econometrics / 32
28 Forecasts based on a finite number of observations Exact Finite-sample Properties {Y t, Y t 1,..., Y t m+1 } observations. Presample ɛ s all equal to 0. Approximation Ê[Y t+s Y t, Y t 1,...] = Ê[Y t+s Y t,..., Y t m+1, ɛ t m = 0, ɛ t m 1 = 0,...] MA(q): ɛ t m = ɛ t m 1 =... = ɛ t m q+1 = 0 ɛ t m+1 = Y t m+1 µ The values are to be replaced in For s = q = 1: ɛ t m+2 = Y t m+2 µ θ 1 ɛ t m+1 ɛ t m+3 = Y t m+3 µ θ 1 ɛ t m+2 θ 2 ɛ t m+1 Ŷ t+s t = µ + (θ s + θ s+1 L + θ s+2 L θ q L q s ) ɛ t Ŷ t+s t = µ + θ(y t µ) θ 2 (Y t 1 µ) ( 1) m 1 θ m (Y t m+1 µ) truncated infinite AR. For m and θ small we have a good approximation. For θ = 1 the approximation may be poorer. Rossi Forecasting Financial Econometrics / 32
29 Forecasts based on a finite number of observations Exact Finite-sample Properties Alternative approach: Exact projection of Y t+1 on its most recent values 1 Y t X t =. Y t m+1 Linear Forecast α (m) X t = α m 0 + α m 1 Y t α m my t m+1 Rossi Forecasting Financial Econometrics / 32
30 Forecasts based on a finite number of observations Exact Finite-sample Properties If Y t is c.s. implies E[Y t Y t j ] = γ j + µ 2 X t = [1, Y t,..., Y t m+1 ] α (m) = [ µ (γ 1 + µ 2 )... (γ m + µ 2 ) ] 1 µ... µ µ (γ 0 + µ 2 )... (γ m 1 + µ 2 )... µ (γ m 1 + µ 2 )... (γ 0 + µ 2 ) when a constant term is included in X t it is more convenient to express variables in deviations from the mean. 1 Rossi Forecasting Financial Econometrics / 32
31 Forecasts based on a finite number of observations Calculate the projection of (Y t+1 µ) on (Y t µ), (Y t 1 µ),..., (Y t m+1 µ) α (m) = s-period-ahead forecast γ 0 γ 1... γ m 1... γ m 1 γ m 2... γ 0 1 γ 1. γ m Ŷ t+s t = µ + α (m,s) 1 (Y t µ) α (m,s) m (Y t m+s µ) α (m,s) 1. α (m,s) m = γ 0 γ 1... γ m 1... γ m 1 γ m 2... γ 0 1 γ s. γ s+m 1 Rossi Forecasting Financial Econometrics / 32
32 Forecasts based on a finite number of observations Inversion of an (m m) matrix. Two algorithms: 1 Kalman Filter to compute finite-sample forecast. 2 Triangular Factorization. Rossi Forecasting Financial Econometrics / 32
Università di Pavia. Forecasting. Eduardo Rossi
Università di Pavia Forecasting Eduardo Rossi Mean Squared Error Forecast of Y t+1 based on a set of variables observed at date t, X t : Yt+1 t. The loss function MSE(Y t+1 t ) = E[Y t+1 Y t+1 t ]2 The
More informationCh. 15 Forecasting. 1.1 Forecasts Based on Conditional Expectations
Ch 15 Forecasting Having considered in Chapter 14 some of the properties of ARMA models, we now show how they may be used to forecast future values of an observed time series For the present we proceed
More informationIntroduction to ARMA and GARCH processes
Introduction to ARMA and GARCH processes Fulvio Corsi SNS Pisa 3 March 2010 Fulvio Corsi Introduction to ARMA () and GARCH processes SNS Pisa 3 March 2010 1 / 24 Stationarity Strict stationarity: (X 1,
More informationECON/FIN 250: Forecasting in Finance and Economics: Section 6: Standard Univariate Models
ECON/FIN 250: Forecasting in Finance and Economics: Section 6: Standard Univariate Models Patrick Herb Brandeis University Spring 2016 Patrick Herb (Brandeis University) Standard Univariate Models ECON/FIN
More informationIntroduction to Stochastic processes
Università di Pavia Introduction to Stochastic processes Eduardo Rossi Stochastic Process Stochastic Process: A stochastic process is an ordered sequence of random variables defined on a probability space
More information3. ARMA Modeling. Now: Important class of stationary processes
3. ARMA Modeling Now: Important class of stationary processes Definition 3.1: (ARMA(p, q) process) Let {ɛ t } t Z WN(0, σ 2 ) be a white noise process. The process {X t } t Z is called AutoRegressive-Moving-Average
More informationEmpirical Market Microstructure Analysis (EMMA)
Empirical Market Microstructure Analysis (EMMA) Lecture 3: Statistical Building Blocks and Econometric Basics Prof. Dr. Michael Stein michael.stein@vwl.uni-freiburg.de Albert-Ludwigs-University of Freiburg
More informationECON/FIN 250: Forecasting in Finance and Economics: Section 7: Unit Roots & Dickey-Fuller Tests
ECON/FIN 250: Forecasting in Finance and Economics: Section 7: Unit Roots & Dickey-Fuller Tests Patrick Herb Brandeis University Spring 2016 Patrick Herb (Brandeis University) Unit Root Tests ECON/FIN
More informationECONOMETRICS Part II PhD LBS
ECONOMETRICS Part II PhD LBS Luca Gambetti UAB, Barcelona GSE February-March 2014 1 Contacts Prof.: Luca Gambetti email: luca.gambetti@uab.es webpage: http://pareto.uab.es/lgambetti/ Description This is
More informationMidterm Suggested Solutions
CUHK Dept. of Economics Spring 2011 ECON 4120 Sung Y. Park Midterm Suggested Solutions Q1 (a) In time series, autocorrelation measures the correlation between y t and its lag y t τ. It is defined as. ρ(τ)
More informationCh. 14 Stationary ARMA Process
Ch. 14 Stationary ARMA Process A general linear stochastic model is described that suppose a time series to be generated by a linear aggregation of random shock. For practical representation it is desirable
More informationECON 616: Lecture 1: Time Series Basics
ECON 616: Lecture 1: Time Series Basics ED HERBST August 30, 2017 References Overview: Chapters 1-3 from Hamilton (1994). Technical Details: Chapters 2-3 from Brockwell and Davis (1987). Intuition: Chapters
More informationPermanent Income Hypothesis (PIH) Instructor: Dmytro Hryshko
Permanent Income Hypothesis (PIH) Instructor: Dmytro Hryshko 1 / 36 The PIH Utility function is quadratic, u(c t ) = 1 2 (c t c) 2 ; borrowing/saving is allowed using only the risk-free bond; β(1 + r)
More informationCovariance Stationary Time Series. Example: Independent White Noise (IWN(0,σ 2 )) Y t = ε t, ε t iid N(0,σ 2 )
Covariance Stationary Time Series Stochastic Process: sequence of rv s ordered by time {Y t } {...,Y 1,Y 0,Y 1,...} Defn: {Y t } is covariance stationary if E[Y t ]μ for all t cov(y t,y t j )E[(Y t μ)(y
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 informationAutoregressive Moving Average (ARMA) Models and their Practical Applications
Autoregressive Moving Average (ARMA) Models and their Practical Applications Massimo Guidolin February 2018 1 Essential Concepts in Time Series Analysis 1.1 Time Series and Their Properties Time series:
More informationForecasting and Estimation
February 3, 2009 Forecasting I Very frequently the goal of estimating time series is to provide forecasts of future values. This typically means you treat the data di erently than if you were simply tting
More informationChapter 4: Models for Stationary Time Series
Chapter 4: Models for Stationary Time Series Now we will introduce some useful parametric models for time series that are stationary processes. We begin by defining the General Linear Process. Let {Y t
More informationDiscrete time processes
Discrete time processes Predictions are difficult. Especially about the future Mark Twain. Florian Herzog 2013 Modeling observed data When we model observed (realized) data, we encounter usually the following
More information5 Transfer function modelling
MSc Further Time Series Analysis 5 Transfer function modelling 5.1 The model Consider the construction of a model for a time series (Y t ) whose values are influenced by the earlier values of a series
More informationClass 1: Stationary Time Series Analysis
Class 1: Stationary Time Series Analysis Macroeconometrics - Fall 2009 Jacek Suda, BdF and PSE February 28, 2011 Outline Outline: 1 Covariance-Stationary Processes 2 Wold Decomposition Theorem 3 ARMA Models
More informationLecture 2: ARMA(p,q) models (part 2)
Lecture 2: ARMA(p,q) models (part 2) 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 informationEconometrics II Heij et al. Chapter 7.1
Chapter 7.1 p. 1/2 Econometrics II Heij et al. Chapter 7.1 Linear Time Series Models for Stationary data Marius Ooms Tinbergen Institute Amsterdam Chapter 7.1 p. 2/2 Program Introduction Modelling philosophy
More informationPrinciples of forecasting
2.5 Forecasting Principles of forecasting Forecast based on conditional expectations Suppose we are interested in forecasting the value of y t+1 based on a set of variables X t (m 1 vector). Let y t+1
More informationCh 9. FORECASTING. Time Series Analysis
In this chapter, we assume the model is known exactly, and consider the calculation of forecasts and their properties for both deterministic trend models and ARIMA models. 9.1 Minimum Mean Square Error
More informationUnivariate Time Series Analysis; ARIMA Models
Econometrics 2 Fall 24 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 informationCh. 19 Models of Nonstationary Time Series
Ch. 19 Models of Nonstationary Time Series In time series analysis we do not confine ourselves to the analysis of stationary time series. In fact, most of the time series we encounter are non stationary.
More informationState-space Model. Eduardo Rossi University of Pavia. November Rossi State-space Model Financial Econometrics / 49
State-space Model Eduardo Rossi University of Pavia November 2013 Rossi State-space Model Financial Econometrics - 2013 1 / 49 Outline 1 Introduction 2 The Kalman filter 3 Forecast errors 4 State smoothing
More informationLecture 3: Autoregressive Moving Average (ARMA) Models and their Practical Applications
Lecture 3: Autoregressive Moving Average (ARMA) Models and their Practical Applications Prof. Massimo Guidolin 20192 Financial Econometrics Winter/Spring 2018 Overview Moving average processes Autoregressive
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 informationUnivariate Nonstationary Time Series 1
Univariate Nonstationary Time Series 1 Sebastian Fossati University of Alberta 1 These slides are based on Eric Zivot s time series notes available at: http://faculty.washington.edu/ezivot Introduction
More information1 Linear Difference Equations
ARMA Handout Jialin Yu 1 Linear Difference Equations First order systems Let {ε t } t=1 denote an input sequence and {y t} t=1 sequence generated by denote an output y t = φy t 1 + ε t t = 1, 2,... with
More informationChapter 1. Basics. 1.1 Definition. A time series (or stochastic process) is a function Xpt, ωq such that for
Chapter 1 Basics 1.1 Definition A time series (or stochastic process) is a function Xpt, ωq such that for each fixed t, Xpt, ωq is a random variable [denoted by X t pωq]. For a fixed ω, Xpt, ωq is simply
More informationLecture 1: Stationary Time Series Analysis
Syllabus Stationarity ARMA AR MA Model Selection Estimation Lecture 1: Stationary Time Series Analysis 222061-1617: Time Series Econometrics Spring 2018 Jacek Suda Syllabus Stationarity ARMA AR MA Model
More information4. MA(2) +drift: y t = µ + ɛ t + θ 1 ɛ t 1 + θ 2 ɛ t 2. Mean: where θ(l) = 1 + θ 1 L + θ 2 L 2. Therefore,
61 4. MA(2) +drift: y t = µ + ɛ t + θ 1 ɛ t 1 + θ 2 ɛ t 2 Mean: y t = µ + θ(l)ɛ t, where θ(l) = 1 + θ 1 L + θ 2 L 2. Therefore, E(y t ) = µ + θ(l)e(ɛ t ) = µ 62 Example: MA(q) Model: y t = ɛ t + θ 1 ɛ
More informationSingle Equation Linear GMM with Serially Correlated Moment Conditions
Single Equation Linear GMM with Serially Correlated Moment Conditions Eric Zivot October 28, 2009 Univariate Time Series Let {y t } be an ergodic-stationary time series with E[y t ]=μ and var(y t )
More information2.5 Forecasting and Impulse Response Functions
2.5 Forecasting and Impulse Response Functions Principles of forecasting Forecast based on conditional expectations Suppose we are interested in forecasting the value of y t+1 based on a set of variables
More informationARMA Estimation Recipes
Econ. 1B D. McFadden, Fall 000 1. Preliminaries ARMA Estimation Recipes hese notes summarize procedures for estimating the lag coefficients in the stationary ARMA(p,q) model (1) y t = µ +a 1 (y t-1 -µ)
More informationLecture 1: Stationary Time Series Analysis
Syllabus Stationarity ARMA AR MA Model Selection Estimation Forecasting Lecture 1: Stationary Time Series Analysis 222061-1617: Time Series Econometrics Spring 2018 Jacek Suda Syllabus Stationarity ARMA
More informationLesson 9: Autoregressive-Moving Average (ARMA) models
Lesson 9: Autoregressive-Moving Average (ARMA) models Dipartimento di Ingegneria e Scienze dell Informazione e Matematica Università dell Aquila, umberto.triacca@ec.univaq.it Introduction We have seen
More informationForecasting. This optimal forecast is referred to as the Minimum Mean Square Error Forecast. This optimal forecast is unbiased because
Forecasting 1. Optimal Forecast Criterion - Minimum Mean Square Error Forecast We have now considered how to determine which ARIMA model we should fit to our data, we have also examined how to estimate
More informationLecture 1: Fundamental concepts in Time Series Analysis (part 2)
Lecture 1: Fundamental concepts in Time Series Analysis (part 2) Florian Pelgrin University of Lausanne, École des HEC Department of mathematics (IMEA-Nice) Sept. 2011 - Jan. 2012 Florian Pelgrin (HEC)
More informationBasic concepts and terminology: AR, MA and ARMA processes
ECON 5101 ADVANCED ECONOMETRICS TIME SERIES Lecture note no. 1 (EB) Erik Biørn, Department of Economics Version of February 1, 2011 Basic concepts and terminology: AR, MA and ARMA processes This lecture
More informationLecture 2: Univariate Time Series
Lecture 2: Univariate Time Series Analysis: Conditional and Unconditional Densities, Stationarity, ARMA Processes Prof. Massimo Guidolin 20192 Financial Econometrics Spring/Winter 2017 Overview Motivation:
More informationCh 4. Models For Stationary Time Series. Time Series Analysis
This chapter discusses the basic concept of a broad class of stationary parametric time series models the autoregressive moving average (ARMA) models. Let {Y t } denote the observed time series, and {e
More informationTime Series Analysis. James D. Hamilton PRINCETON UNIVERSITY PRESS PRINCETON, NEW JERSEY
Time Series Analysis James D. Hamilton PRINCETON UNIVERSITY PRESS PRINCETON, NEW JERSEY & Contents PREFACE xiii 1 1.1. 1.2. Difference Equations First-Order Difference Equations 1 /?th-order Difference
More informationProblem Set 1 Solution Sketches Time Series Analysis Spring 2010
Problem Set 1 Solution Sketches Time Series Analysis Spring 2010 1. Construct a martingale difference process that is not weakly stationary. Simplest e.g.: Let Y t be a sequence of independent, non-identically
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 informationAutoregressive and Moving-Average Models
Chapter 3 Autoregressive and Moving-Average Models 3.1 Introduction Let y be a random variable. We consider the elements of an observed time series {y 0,y 1,y2,...,y t } as being realizations of this randoms
More informationState-space Model. Eduardo Rossi University of Pavia. November Rossi State-space Model Fin. Econometrics / 53
State-space Model Eduardo Rossi University of Pavia November 2014 Rossi State-space Model Fin. Econometrics - 2014 1 / 53 Outline 1 Motivation 2 Introduction 3 The Kalman filter 4 Forecast errors 5 State
More informationat least 50 and preferably 100 observations should be available to build a proper model
III Box-Jenkins Methods 1. Pros and Cons of ARIMA Forecasting a) need for data at least 50 and preferably 100 observations should be available to build a proper model used most frequently for hourly or
More information1 Class Organization. 2 Introduction
Time Series Analysis, Lecture 1, 2018 1 1 Class Organization Course Description Prerequisite Homework and Grading Readings and Lecture Notes Course Website: http://www.nanlifinance.org/teaching.html wechat
More informationTrend-Cycle Decompositions
Trend-Cycle Decompositions Eric Zivot April 22, 2005 1 Introduction A convenient way of representing an economic time series y t is through the so-called trend-cycle decomposition y t = TD t + Z t (1)
More informationTime Series 3. Robert Almgren. Sept. 28, 2009
Time Series 3 Robert Almgren Sept. 28, 2009 Last time we discussed two main categories of linear models, and their combination. Here w t denotes a white noise: a stationary process with E w t ) = 0, E
More informationEASTERN MEDITERRANEAN UNIVERSITY ECON 604, FALL 2007 DEPARTMENT OF ECONOMICS MEHMET BALCILAR ARIMA MODELS: IDENTIFICATION
ARIMA MODELS: IDENTIFICATION A. Autocorrelations and Partial Autocorrelations 1. Summary of What We Know So Far: a) Series y t is to be modeled by Box-Jenkins methods. The first step was to convert y t
More informationCointegrated VAR s. Eduardo Rossi University of Pavia. November Rossi Cointegrated VAR s Financial Econometrics / 56
Cointegrated VAR s Eduardo Rossi University of Pavia November 2013 Rossi Cointegrated VAR s Financial Econometrics - 2013 1 / 56 VAR y t = (y 1t,..., y nt ) is (n 1) vector. y t VAR(p): Φ(L)y t = ɛ t The
More informationProblem Set 2 Solution Sketches Time Series Analysis Spring 2010
Problem Set 2 Solution Sketches Time Series Analysis Spring 2010 Forecasting 1. Let X and Y be two random variables such that E(X 2 ) < and E(Y 2 )
More informationCovariances of ARMA Processes
Statistics 910, #10 1 Overview Covariances of ARMA Processes 1. Review ARMA models: causality and invertibility 2. AR covariance functions 3. MA and ARMA covariance functions 4. Partial autocorrelation
More informationSingle Equation Linear GMM with Serially Correlated Moment Conditions
Single Equation Linear GMM with Serially Correlated Moment Conditions Eric Zivot November 2, 2011 Univariate Time Series Let {y t } be an ergodic-stationary time series with E[y t ]=μ and var(y t )
More informationLINEAR STOCHASTIC MODELS
LINEAR STOCHASTIC MODELS Let {x τ+1,x τ+2,...,x τ+n } denote n consecutive elements from a stochastic process. If their joint distribution does not depend on τ, regardless of the size of n, then the process
More informationESSE Mid-Term Test 2017 Tuesday 17 October :30-09:45
ESSE 4020 3.0 - Mid-Term Test 207 Tuesday 7 October 207. 08:30-09:45 Symbols have their usual meanings. All questions are worth 0 marks, although some are more difficult than others. Answer as many questions
More informationChapter 6: Model Specification for Time Series
Chapter 6: Model Specification for Time Series The ARIMA(p, d, q) class of models as a broad class can describe many real time series. Model specification for ARIMA(p, d, q) models involves 1. Choosing
More informationTime Series Analysis. James D. Hamilton PRINCETON UNIVERSITY PRESS PRINCETON, NEW JERSEY
Time Series Analysis James D. Hamilton PRINCETON UNIVERSITY PRESS PRINCETON, NEW JERSEY PREFACE xiii 1 Difference Equations 1.1. First-Order Difference Equations 1 1.2. pth-order Difference Equations 7
More informationReview Session: Econometrics - CLEFIN (20192)
Review Session: Econometrics - CLEFIN (20192) Part II: Univariate time series analysis Daniele Bianchi March 20, 2013 Fundamentals Stationarity A time series is a sequence of random variables x t, t =
More informationUniversity of Oxford. Statistical Methods Autocorrelation. Identification and Estimation
University of Oxford Statistical Methods Autocorrelation Identification and Estimation Dr. Órlaith Burke Michaelmas Term, 2011 Department of Statistics, 1 South Parks Road, Oxford OX1 3TG Contents 1 Model
More informationLecture on ARMA model
Lecture on ARMA model Robert M. de Jong Ohio State University Columbus, OH 43210 USA Chien-Ho Wang National Taipei University Taipei City, 104 Taiwan ROC October 19, 2006 (Very Preliminary edition, Comment
More informationTIME SERIES AND FORECASTING. Luca Gambetti UAB, Barcelona GSE Master in Macroeconomic Policy and Financial Markets
TIME SERIES AND FORECASTING Luca Gambetti UAB, Barcelona GSE 2014-2015 Master in Macroeconomic Policy and Financial Markets 1 Contacts Prof.: Luca Gambetti Office: B3-1130 Edifici B Office hours: email:
More informationDefine y t+h t as the forecast of y t+h based on I t known parameters. The forecast error is. Forecasting
Forecasting Let {y t } be a covariance stationary are ergodic process, eg an ARMA(p, q) process with Wold representation y t = X μ + ψ j ε t j, ε t ~WN(0,σ 2 ) j=0 = μ + ε t + ψ 1 ε t 1 + ψ 2 ε t 2 + Let
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 informationWeek 5 Quantitative Analysis of Financial Markets Characterizing Cycles
Week 5 Quantitative Analysis of Financial Markets Characterizing Cycles Christopher Ting http://www.mysmu.edu/faculty/christophert/ Christopher Ting : christopherting@smu.edu.sg : 6828 0364 : LKCSB 5036
More informationVector Auto-Regressive Models
Vector Auto-Regressive Models Laurent Ferrara 1 1 University of Paris Nanterre M2 Oct. 2018 Overview of the presentation 1. Vector Auto-Regressions Definition Estimation Testing 2. Impulse responses functions
More informationSTAD57 Time Series Analysis. Lecture 8
STAD57 Time Series Analysis Lecture 8 1 ARMA Model Will be using ARMA models to describe times series dynamics: ( B) X ( B) W X X X W W W t 1 t1 p t p t 1 t1 q tq Model must be causal (i.e. stationary)
More informationVAR Models and Applications
VAR Models and Applications Laurent Ferrara 1 1 University of Paris West M2 EIPMC Oct. 2016 Overview of the presentation 1. Vector Auto-Regressions Definition Estimation Testing 2. Impulse responses functions
More information5: MULTIVARATE STATIONARY PROCESSES
5: MULTIVARATE STATIONARY PROCESSES 1 1 Some Preliminary Definitions and Concepts Random Vector: A vector X = (X 1,..., X n ) whose components are scalarvalued random variables on the same probability
More informationLECTURE 10 LINEAR PROCESSES II: SPECTRAL DENSITY, LAG OPERATOR, ARMA. In this lecture, we continue to discuss covariance stationary processes.
MAY, 0 LECTURE 0 LINEAR PROCESSES II: SPECTRAL DENSITY, LAG OPERATOR, ARMA In this lecture, we continue to discuss covariance stationary processes. Spectral density Gourieroux and Monfort 990), Ch. 5;
More informationLesson 2: Analysis of time series
Lesson 2: Analysis of time series Time series Main aims of time series analysis choosing right model statistical testing forecast driving and optimalisation Problems in analysis of time series time problems
More information18.S096 Problem Set 4 Fall 2013 Time Series Due Date: 10/15/2013
18.S096 Problem Set 4 Fall 2013 Time Series Due Date: 10/15/2013 1. Covariance Stationary AR(2) Processes Suppose the discrete-time stochastic process {X t } follows a secondorder auto-regressive process
More informationARMA (and ARIMA) models are often expressed in backshift notation.
Backshift Notation ARMA (and ARIMA) models are often expressed in backshift notation. B is the backshift operator (also called the lag operator ). It operates on time series, and means back up by one time
More informationFE570 Financial Markets and Trading. Stevens Institute of Technology
FE570 Financial Markets and Trading Lecture 5. Linear Time Series Analysis and Its Applications (Ref. Joel Hasbrouck - Empirical Market Microstructure ) Steve Yang Stevens Institute of Technology 9/25/2012
More informationE 4101/5101 Lecture 6: Spectral analysis
E 4101/5101 Lecture 6: Spectral analysis Ragnar Nymoen 3 March 2011 References to this lecture Hamilton Ch 6 Lecture note (on web page) For stationary variables/processes there is a close correspondence
More informationStationary Stochastic Time Series Models
Stationary Stochastic Time Series Models When modeling time series it is useful to regard an observed time series, (x 1,x,..., x n ), as the realisation of a stochastic process. In general a stochastic
More informationEstimating Moving Average Processes with an improved version of Durbin s Method
Estimating Moving Average Processes with an improved version of Durbin s Method Maximilian Ludwig this version: June 7, 4, initial version: April, 3 arxiv:347956v [statme] 6 Jun 4 Abstract This paper provides
More information2. An Introduction to Moving Average Models and ARMA Models
. An Introduction to Moving Average Models and ARMA Models.1 White Noise. The MA(1) model.3 The MA(q) model..4 Estimation and forecasting of MA models..5 ARMA(p,q) models. The Moving Average (MA) models
More informationSTAT Financial Time Series
STAT 6104 - Financial Time Series Chapter 4 - Estimation in the time Domain Chun Yip Yau (CUHK) STAT 6104:Financial Time Series 1 / 46 Agenda 1 Introduction 2 Moment Estimates 3 Autoregressive Models (AR
More information7. MULTIVARATE STATIONARY PROCESSES
7. MULTIVARATE STATIONARY PROCESSES 1 1 Some Preliminary Definitions and Concepts Random Vector: A vector X = (X 1,..., X n ) whose components are scalar-valued random variables on the same probability
More informationChapter 3 - Temporal processes
STK4150 - Intro 1 Chapter 3 - Temporal processes Odd Kolbjørnsen and Geir Storvik January 23 2017 STK4150 - Intro 2 Temporal processes Data collected over time Past, present, future, change Temporal aspect
More informationTMA4285 December 2015 Time series models, solution.
Norwegian University of Science and Technology Department of Mathematical Sciences Page of 5 TMA4285 December 205 Time series models, solution. Problem a) (i) The slow decay of the ACF of z t suggest that
More informationMultivariate ARMA Processes
LECTURE 8 Multivariate ARMA Processes A vector y(t) of n elements is said to follow an n-variate ARMA process of orders p and q if it satisfies the equation (1) A 0 y(t) + A 1 y(t 1) + + A p y(t p) = M
More informationCalculation of ACVF for ARMA Process: I consider causal ARMA(p, q) defined by
Calculation of ACVF for ARMA Process: I consider causal ARMA(p, q) defined by φ(b)x t = θ(b)z t, {Z t } WN(0, σ 2 ) want to determine ACVF {γ(h)} for this process, which can be done using four complementary
More informationEconometrics of financial markets, -solutions to seminar 1. Problem 1
Econometrics of financial markets, -solutions to seminar 1. Problem 1 a) Estimate with OLS. For any regression y i α + βx i + u i for OLS to be unbiased we need cov (u i,x j )0 i, j. For the autoregressive
More informationTime Series 2. Robert Almgren. Sept. 21, 2009
Time Series 2 Robert Almgren Sept. 21, 2009 This week we will talk about linear time series models: AR, MA, ARMA, ARIMA, etc. First we will talk about theory and after we will talk about fitting the models
More informationSome Time-Series Models
Some Time-Series Models Outline 1. Stochastic processes and their properties 2. Stationary processes 3. Some properties of the autocorrelation function 4. Some useful models Purely random processes, random
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 III. Stationary models 1 Purely random process 2 Random walk (non-stationary)
More informationPart III Example Sheet 1 - Solutions YC/Lent 2015 Comments and corrections should be ed to
TIME SERIES Part III Example Sheet 1 - Solutions YC/Lent 2015 Comments and corrections should be emailed to Y.Chen@statslab.cam.ac.uk. 1. Let {X t } be a weakly stationary process with mean zero and let
More informationDynamic Regression Models
Università di Pavia 2007 Dynamic Regression Models Eduardo Rossi University of Pavia Data Generating Process & Models Setup y t denote an (n 1) vector of economic variables generated at time t. The collection
More informationClassic Time Series Analysis
Classic Time Series Analysis Concepts and Definitions Let Y be a random number with PDF f Y t ~f,t Define t =E[Y t ] m(t) is known as the trend Define the autocovariance t, s =COV [Y t,y s ] =E[ Y t t
More informationProf. Dr. Roland Füss Lecture Series in Applied Econometrics Summer Term Introduction to Time Series Analysis
Introduction to Time Series Analysis 1 Contents: I. Basics of Time Series Analysis... 4 I.1 Stationarity... 5 I.2 Autocorrelation Function... 9 I.3 Partial Autocorrelation Function (PACF)... 14 I.4 Transformation
More informationAdvanced Econometrics
Based on the textbook by Verbeek: A Guide to Modern Econometrics Robert M. Kunst robert.kunst@univie.ac.at University of Vienna and Institute for Advanced Studies Vienna May 2, 2013 Outline Univariate
More informationExercises - Time series analysis
Descriptive analysis of a time series (1) Estimate the trend of the series of gasoline consumption in Spain using a straight line in the period from 1945 to 1995 and generate forecasts for 24 months. Compare
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 information