Class: Trend-Cycle Decomposition
|
|
- Blanche Pope
- 6 years ago
- Views:
Transcription
1 Class: Trend-Cycle Decomposition Macroeconometrics - Spring 2011 Jacek Suda, BdF and PSE June 1, 2011
2 Outline Outline: 1 Unobserved Component Approach 2 Beveridge-Nelson Decomposition 3 Spectral Analysis
3 Detrending Need stationary series: Y t = X t β + ε t Granger and Newbold (1974, JoE, Spurious Regressions in Econometrics ) If y t and X t are independent random walk (β = 0), ˆβ OLS non-zero random variable, and ˆt β=0 is large: spurious regression phenomenon. Taking difference instead of levels (so we get stationary series) will bring larger standard errors => cannot reject hypothesis. Detrending still allows to analyze levels. Sometimes we are interested in trend alone.
4 Trend/Cycle Observable series y t y t = τ t + c t τ t is trend, and c t is transitory component, (I(0)). If trend contains stochastic component, random walk, then if we apply HP we get spurious cycle. τ t = µ + τ t 1 + η t We have two unobserved components and if we can model the cycle we can try to use unobserved component estimation.
5 Unobserved Components Approach Watson (1986, JME), Clark (1987, QJE), Morley, Nelson, Zivot (2003, ReStat) Approach: parametric model for c t Model ( Structural) y t = τ t + c t τ t = µ + τ t 1 + η t, η t iidn(0, ση) 2 φ(l)c t = ε t, ε t iidn(0, σ 2 ε), cov(η t, ε t ) = σ εη
6 Problem: Identification We have 1 observable series and 2 unobservable components. To get 2 unobservable components, we need some identification assumptions. Identification: If c t = ε t or c t = φc t 1 + ε t, then σ εη is not identified from the data. There can be infinitely many values of σ εη that would produce the same autocovariance generating function for the first series. However, that does not mean that all values of σ εη are equal. If it is set to zero, it imposes restriction on autocovariance generating function of 1 st differences.
7 Example: AR(1) Example: AR(1) y t = τ t + c t τ t = µ + τ t 1 + η t c t = φc t 1 + ε t Structural model: 5 parameters: µ, ση, 2 σε, 2 φ, σ εη. How many parameters can be identified from data? Reduced-Form First-difference equation y t = τ t + c t (1 L)y t = (1 L)τ t + (1 L)c t y t = µ + η t + (1 L)(1 φl) 1 ε t
8 Example: AR(1) Multiply both sides by (1 φl): (1 φl) y t = (1 φl)µ + (1 φl)η t + (1 L)ε t = c + η t + φη t 1 + ε t ε t 1, c = (1 φ)µ. They are unobserved but we have a sum of two iid series η t + ε t + ( φ)η t 1 + ( 1)ε t 1 The sum of two white noise processes = white noise: same moments as MA(1). So this model is observationally equivalent to y t = c + φ y t 1 + e t + θe t 1 ARMA(1,1) = 4 parameters: c, φ, θ, σ 2 e, that s how many we can estimate. We have 5 parameters but only 4 observed. So far estimates assumes one of parameters fixed.
9 Estimation Assume σ εη = 0 (Watson, Harvey, Clark). => shocks that drive transitory movements are not correlated with those that drive long-run behavior. With this assumption the model can be estimated: 1 Find match (functional) of observed/estimated parameters with the ones from structural model, or 2 Cast the model in a state space form and estimate via Kalman Filter:
10 State-Space Form Observation equation State equation [ τt c t ] y t = [ 1 1 ] [ τ t c t ] y t = Hβ t = [ µ 0 ] [ φ ] [ τt 1 c t 1 ] [ ηt + ε t β t = ˆµ + Fβ t 1 + e t, e t N(0, Q), [ ] σ 2 Q = η 0 0 σε 2 ],
11 Kalman Filter: Results Kalman Filter does not care about how we came up with state form. KF: τ t t and c t t, τ t T, c t T. We say τ t and c t are uncorrelated with each other, by assumption. corr(η t t, ε t t ) = 1 even though we assume corr(η t, ε t ) = 0. In classical approach corr(x t, ˆε t ) = 0 by construction, even though true relationship is corr(x t, ε t ) 0. Estimates of correlation rather than sample correlation of estimates. Identification: If we estimate the model without assuming σ εη Gauss will not converge as there is many numbers of σ εη for which likelihood doesn t decrease.
12 Morley, Nelson and Zivot (2003) RW + AR(2) makes model identified. Why? AR(1) cycles is not observationally different from RW. AR(2) has this feature that cannot be proxied by RW. Morley, Nelson and Zivot (2003): σ εη identified for c t ARMA(p, q), with p q + 2.
13 Example: AR(2) Model: y t = τ t + c t τ t = µ + τ t 1 + η t c t = φ 1 c t 1 + φ 2 c t 2 + ε t 6 parameters: µ, φ 1, φ 2, σ 2 η, σ 2 ε, σ εη. Pre-multiplying both sides with (1 L): y t = (1 L)τ t + (1 L)c t = µ + η t + (1 L)(1 φ 1 L φ 2 L 2 ) 1 ε t (1 φ 1 L φ 2 L 2 ) y t = (1 φ 1 φ 2 )µ + η t φ 1 η t 1 φ 2 η t 2 + ε t ε t 1 The model is observationally equivalent to ARMA(2,2) model: y t ARMA(2, 2) with 6 parameters: c, φ 1, φ 2, θ 1, θ 2, σ 2 e.
14 Results We can map parameters of ARMA(2,2) to our structural model or estimate KF with. [ ] σ 2 Q = η σ εη σ εη For US real GDP, setting σ εη = 0 can be rejected: ρ εη = 0.9. τ t is volatile Structural model with ARMA(3) has 7 structural parameters but is observationally equivalent to reduced-form version ARMA(3,3) with 8 parameters: overidentification. σ 2 ε Not such a big problem; ρ εη < 0 still holds.
15 Trend in UC Model τ t t is equivalent to Beveridge-Nelson trend for ARMA(2,2). From Kalman Filter: τ t t E[τ t Ω t ] = lim M E[τ t + c t+m Ω t ] E[cycles] far away in future, given current information, are zero. τ t t = lim M E[τ t + c t+m Ω t ] = lim M E[τ t + M η t+j + c t+m Ω t ] j=1 = lim M E[τ t + Mµ + M η t+j + c t+m Mµ Ω t ] j=1 = lim M E[y t+m Mµ Ω t ] Beveridge Nelson trend BN: expectations about where the series is in the future. They are different for AR(1): restricted UC model, (σ ηε = 0).
16 Beveridge-Nelson Decomposition BN trend is the long-run conditional forecast (minus deterministic trend) Let y t be y t I(1) and express it as where Then T t = TD t + TS t is trend, y t = TD t + TS t + c t, TD t is deterministic part of trend, and z t = TS t + c t is stochastic component comprising both stochastic trend, TS t, and stochastic cycle, c t. z t I(0)
17 Beveridge-Nelson Decomposition Since z t is covariance-stationary then it has a Wold form representation z t = Ψ (L)e t, Ψ (L) = Ψ k L k, Ψ 0 = 1, k=0 e t iid Result: where Ψ (1) = Ψ(L) = Ψ (L) = Ψ (1) + (1 L) Ψ(L), Ψ k, = long run impact of forecast error on y t k=0 Ψ j L j, Ψj = Ψ k, j=0 k=j+1 with (1 L) Ψ(L) measuring transitory impact of forecast errors.
18 Beveridge-Nelson Decomposition Then z t = z t 1 + Ψ (L)e t i.e. z t is like random walk with innovations of Wold form t z t = z 0 + Ψ (L) j=1 t z t = z 0 + Ψ (1) e j + (1 L) Ψ(L) j=1 y t = y 0 + µ t + Ψ (1) and e j j=1 t j=1 e j t e j + (1 L) Ψ(L) t e j, TD t = y 0 + µ t, TS t = ψ (1) c t = (1 L) Ψ(L) t j=1 e j t e j j=1 j=1
19 Example MA(1) Example: MA(1) y t = µ + e t + θe t 1, e t iid y t = µ + Ψ (L)e t, Ψ (L) = 1 + θl Beveridge-Nelson decomposition: Ψ (L) = Ψ (1) + (1 L) Ψ(L).
20 Example MA(1) For MA(1) Ψ (1) = 1 + θ (1 L) Ψ(L) = (1 L) BN decomposition: Ψ k L k, j=0 Ψ k = Ψ 0 = (Ψ 1 + Ψ 2 + Ψ ) = θ Ψ 1 = (Ψ 2 + Ψ 3 + Ψ ) = 0 Ψ j = 0. y t = y 0 + µ t + (1 θ) t e j θe t, j=1 j=k+1 with BN trend = y 0 + µ t + (1 θ) t j=1 ej, and θe t is transitory, BN cycle. Note that: corr(trend, cycle) = 1 for all models not just AR(1). Ψ j
21 Example AR(1) Example: AR(1) ( y t µ) = φ( y t 1 µ) + e t E t [( y t+1 µ)] = φ( y t µ) E t [( y t+2 µ)] = φ 2 ( y t µ) E t [( y t+j µ)] = φ j ( y t µ). To calculate a forecast of how far away from the trend you will be, all you need is how far away you are today.
22 Example AR(1) Sum them up Then, lim J J E t [( y t+j µ)] = (φ 1 + φ 2 + φ φ J )( y t µ) j=1 J j=1 E t [ ] = lim J (φ1 +φ 2 +φ φ J )( y t µ) = φ 1 φ ( y t µ) Beveridge-Nelson decomposition: BN trend t = y t + φ 1 φ ( y t µ) BN cycle t = φ 1 φ ( y t µ)
23 Remarks BN decomposition vary: for different forecasting model we have different BN decomposition. E[τ t Ω t ] - true trend with unobserved component model. Trend follows random walk in both interpretations. And variability of this RW is the same under both interpretations. BN trend estimate of true trend (KF). BN is applicable to any forecasting model: linear and non-linear. BN avoids spurious cycles (unlike HP and Baxter-King).
24 Time Domain Wold Form: Y t = µ + Ψ j ε t j, j=0 ε WN The {Y t } process can be decomposed into the sum of linear combination of shocks (errors). It s time domain because we can see Y t as function of past (in time) realization of shocks.
25 Frequency Domain For covariance-stationary process Y t = µ + π 0 α(ω) cos(ωt)dω + π 0 δ(ω) sin(ωt)dω It s a weighted average (in continuous time) of periodic cycles (sin and cos). ω determines periods: how frequent the cycles are
26 Cos xd, 8x, 0, 2 Pi<, Ticks 880, Pi, 2 Pi<, 8 1, 1<<D 1 p 2 p -1 Plot@Cos@2 xd, 8x, 0, 2 Pi<, Ticks 880, Pi, 2 Pi<, 8 1, 1<<D 1 p 2 p -1 Plot@Cos@4 xd, 8x, 0, 2 Pi<, Ticks 880, Pi, 2 Pi<, 8 1, 1<<D 1 p 2 p -1
27 Auto-covariance Generating Function Autocovariance Generating Function, where z is a complex scalar j-th autocovariance g y (z) = j= γ j z j γ j = E[(Y t µ)(y t j µ)] Exist if the sequence of autocovariances {γ j} j= is absolutely summable A function of all autocovariances for covariance-stationary process. For a covariance-stationary process it is a finite number.
28 Population spectrum Population spectrum ω is a real scalar. S Y (ω) = 1 2π g y(e iω ) = 1 2π Since e iωj = cos(ωj) i sin(ωj) S Y (ω) = 1 2π γ as γ j = γ j. S Y (ω) γ j s. γ j e iωj, j= γ j cos(ωj) We capture how much variation in time sense is due to variability (cycles) at cos and sin at different frequencies. j=1
29 Spectral Representation Theorem For a covariance-stationary process Y t = µ + π 0 [α(ω) cos(ωt) + δ(ω) sin(ωt)] dω for any frequencies < 0 < ω 1 < ω 2 <... < ω n < π and ω2 ω 1 ω2 ω 1 α(ω)dω uncorrelated with δ(ω)dω uncorrelated with ω4 ω 3 ω4 ω 3 α(ω)dω, δ(ω)dω. Decomposition of covariance stationary series into orthogonal components due to cycles at different frequencies.
30 Example: White Noise Y t WN SyHwL σ 2 ê2π p w Flat spectrum: 2 area = σ 2 = var(y t ). This defines white noise process there is equal weight on cycles for each frequency: variation is divided equally by cycles with different frequencies. In general, 2 area under spectral density = variance.
31 Example: AR(1), MA(1) E.g. MA(1), AR(1), ARMA(1,1) SyHwL MAH1L, q=0.5 SyHwL 0.7 ARH1L, f= p 2 p w 0 0 p 2 p w Low ω corresponds to low frequency and long cycles. The area under the curve depicts how much variability corresponds to given frequency of fluctuations. Hight is just as important as shape of the S Y (ω).
32 In the short horizon I can t see as much variability as over the longer time. For AR(1): var(y t Ω t 1 ) = σ 2, var(y t ) = σ2 1 φ 2, vary t > var(y t Ω t 1 ). Spectrum only for covariance stationary processes. For AR process there is more variation lower frequency than higher. E.g. AR(2): There are some frequencies that account for a lot of variation in the process. Variation in the process is driven by some middle frequencies. Peak in spectral density could be evidence that RBC are indeed cycles.
33 Randomness Cycles at frequency zero (it s not a cycle): how much of the movement are due to shocks that never occur cyclically. Spectral density at frequency zero tells about persistence of series (long-run variance). Suppose X t is log GDP and Y t GDP growth: Then Y t = X t S Y (0) = S X (0) extent to which a shock to X has permanent effect on X and is not just transitory cycle. If X t (level) is covariance-stationary then S X (0) = 0 no mass at 0 frequency because no permanent movement in it.
34 Unit Root If S X (0) 0 then X t is not covariance stationary. For unit root, S X (0) = : accumulation of shocks that never dies out. You can calculate sample spectrum. For non-stationary process we will not get, as we would for population, but a number (we can see it as downward bias).
35 Filtering It is easy to apply filter to data when you think about spectral representation of the process. Example: GDP has important seasonal component (1-4 quarters). The long-run variability might be swamped by the short-term seasonal variation. The variation in the series might be due to seasonality while we might be more interested in relative lower frequencies. Regressing C on Y might produce wacky results as they might be driven purely by seasonal behavior.
36 Filtering Filtering: remove or isolate movements in covariance-stationary series at different horizons: e.g. remove seasonality. Filter is set of weights to be applied to different frequencies: Y t = h(l)x t, S Y (ω) = h(e iω ) 2 S X (ω) h(l) is a filter. E.g. h(l) = 1 L 12 : seasonal filter for monthly data. Note: do not apply spectral analysis to integrated time series processes (not covariance stationary). If filter is applied to non-stationary series and then it s differenced the filter is distorted. We may get spurious cycle: cycles in the place where there are no cycles.
Class 4: Non-stationary Time Series
DF Phillips-Perron Stationarity Tests Variance Ratio Structural Break UC Approach Homework Jacek Suda, BdF and PSE January 24, 2013 DF Phillips-Perron Stationarity Tests Variance Ratio Structural Break
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 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 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 informationCENTRE FOR APPLIED MACROECONOMIC ANALYSIS
CENTRE FOR APPLIED MACROECONOMIC ANALYSIS The Australian National University CAMA Working Paper Series May, 2005 SINGLE SOURCE OF ERROR STATE SPACE APPROACH TO THE BEVERIDGE NELSON DECOMPOSITION Heather
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 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 informationFrequency Domain and Filtering
3 Frequency Domain and Filtering This is page i Printer: Opaque this 3. Introduction Comovement and volatility are key concepts in macroeconomics. Therefore, it is important to have statistics that describe
More informationB y t = γ 0 + Γ 1 y t + ε t B(L) y t = γ 0 + ε t ε t iid (0, D) D is diagonal
Structural VAR Modeling for I(1) Data that is Not Cointegrated Assume y t =(y 1t,y 2t ) 0 be I(1) and not cointegrated. That is, y 1t and y 2t are both I(1) and there is no linear combination of y 1t and
More informationLecture 19 - Decomposing a Time Series into its Trend and Cyclical Components
Lecture 19 - Decomposing a Time Series into its Trend and Cyclical Components It is often assumed that many macroeconomic time series are subject to two sorts of forces: those that influence the long-run
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 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 informationEquivalence of several methods for decomposing time series into permananent and transitory components
Equivalence of several methods for decomposing time series into permananent and transitory components Don Harding Department of Economics and Finance LaTrobe University, Bundoora Victoria 3086 and Centre
More informationSpectral Analysis. Jesús Fernández-Villaverde University of Pennsylvania
Spectral Analysis Jesús Fernández-Villaverde University of Pennsylvania 1 Why Spectral Analysis? We want to develop a theory to obtain the business cycle properties of the data. Burns and Mitchell (1946).
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 informationThe MIT Press Journals
The MIT Press Journals http://mitpress.mit.edu/journals This article is provided courtesy of The MIT Press. To join an e-mail alert list and receive the latest news on our publications, please visit: http://mitpress.mit.edu/e-mail
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 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 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 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 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 informationThe Clark Model with Correlated Components
The Clark Model with Correlated Components Kum Hwa Oh and Eric Zivot January 16, 2006 Abstract This paper is an extension of Why are the Beveridge-Nelson and Unobserved- Components Decompositions of GDP
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 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 informationTime-Varying Parameters
Kalman Filter and state-space models: time-varying parameter models; models with unobservable variables; basic tool: Kalman filter; implementation is task-specific. y t = x t β t + e t (1) β t = µ + Fβ
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 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 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 informationTime Series Econometrics 4 Vijayamohanan Pillai N
Time Series Econometrics 4 Vijayamohanan Pillai N Vijayamohan: CDS MPhil: Time Series 5 1 Autoregressive Moving Average Process: ARMA(p, q) Vijayamohan: CDS MPhil: Time Series 5 2 1 Autoregressive Moving
More informationTAKEHOME FINAL EXAM e iω e 2iω e iω e 2iω
ECO 513 Spring 2015 TAKEHOME FINAL EXAM (1) Suppose the univariate stochastic process y is ARMA(2,2) of the following form: y t = 1.6974y t 1.9604y t 2 + ε t 1.6628ε t 1 +.9216ε t 2, (1) where ε is i.i.d.
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 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 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 informationCointegration, Stationarity and Error Correction Models.
Cointegration, Stationarity and Error Correction Models. STATIONARITY Wold s decomposition theorem states that a stationary time series process with no deterministic components has an infinite moving average
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 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 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 informationWhat are the Differences in Trend Cycle Decompositions by Beveridge and Nelson and by Unobserved Component Models?
What are the Differences in Trend Cycle Decompositions by Beveridge and Nelson and by Unobserved Component Models? April 30, 2012 Abstract When a certain procedure is applied to extract two component processes
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 informationAdvanced Econometrics
Advanced Econometrics Marco Sunder Nov 04 2010 Marco Sunder Advanced Econometrics 1/ 25 Contents 1 2 3 Marco Sunder Advanced Econometrics 2/ 25 Music Marco Sunder Advanced Econometrics 3/ 25 Music Marco
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 informationVolatility. Gerald P. Dwyer. February Clemson University
Volatility Gerald P. Dwyer Clemson University February 2016 Outline 1 Volatility Characteristics of Time Series Heteroskedasticity Simpler Estimation Strategies Exponentially Weighted Moving Average Use
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 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 informationModel-based trend-cycle decompositions. with time-varying parameters
Model-based trend-cycle decompositions with time-varying parameters Siem Jan Koopman Kai Ming Lee Soon Yip Wong s.j.koopman@ klee@ s.wong@ feweb.vu.nl Department of Econometrics Vrije Universiteit Amsterdam
More informationX t = a t + r t, (7.1)
Chapter 7 State Space Models 71 Introduction State Space models, developed over the past 10 20 years, are alternative models for time series They include both the ARIMA models of Chapters 3 6 and the Classical
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 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 informationIS THE NORTH ATLANTIC OSCILLATION A RANDOM WALK? A COMMENT WITH FURTHER RESULTS
INTERNATIONAL JOURNAL OF CLIMATOLOGY Int. J. Climatol. 24: 377 383 (24) Published online 11 February 24 in Wiley InterScience (www.interscience.wiley.com). DOI: 1.12/joc.13 IS THE NORTH ATLANTIC OSCILLATION
More informationEC402: Serial Correlation. Danny Quah Economics Department, LSE Lent 2015
EC402: Serial Correlation Danny Quah Economics Department, LSE Lent 2015 OUTLINE 1. Stationarity 1.1 Covariance stationarity 1.2 Explicit Models. Special cases: ARMA processes 2. Some complex numbers.
More informationNotes on Time Series Modeling
Notes on Time Series Modeling Garey Ramey University of California, San Diego January 17 1 Stationary processes De nition A stochastic process is any set of random variables y t indexed by t T : fy t g
More informationDo the Hodrick-Prescott and Baxter-King Filters Provide a Good Approximation of Business Cycles?
ANNALES D ÉCONOMIE ET DE STATISTIQUE. N 77 2005 Do the Hodrick-Prescott and Baxter-King Filters Provide a Good Approximation of Business Cycles? Alain GUAY*, Pierre ST-AMANT** ABSTRACT. The authors assess
More informationReliability and Risk Analysis. Time Series, Types of Trend Functions and Estimates of Trends
Reliability and Risk Analysis Stochastic process The sequence of random variables {Y t, t = 0, ±1, ±2 } is called the stochastic process The mean function of a stochastic process {Y t} is the function
More informationForecasting with ARMA
Forecasting with ARMA Eduardo Rossi University of Pavia October 2013 Rossi Forecasting Financial Econometrics - 2013 1 / 32 Mean Squared Error Linear Projection Forecast of Y t+1 based on a set of variables
More informationNon-Stationary Time Series and Unit Root Testing
Econometrics II Non-Stationary Time Series and Unit Root Testing Morten Nyboe Tabor Course Outline: Non-Stationary Time Series and Unit Root Testing 1 Stationarity and Deviation from Stationarity Trend-Stationarity
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 informationClass 4: VAR. Macroeconometrics - Fall October 11, Jacek Suda, Banque de France
VAR IRF Short-run Restrictions Long-run Restrictions Granger Summary Jacek Suda, Banque de France October 11, 2013 VAR IRF Short-run Restrictions Long-run Restrictions Granger Summary Outline Outline:
More informationNon-Stationary Time Series and Unit Root Testing
Econometrics II Non-Stationary Time Series and Unit Root Testing Morten Nyboe Tabor Course Outline: Non-Stationary Time Series and Unit Root Testing 1 Stationarity and Deviation from Stationarity Trend-Stationarity
More informationThe Asymmetric Business Cycle
The Asymmetric Business Cycle James Morley Washington University in St. Louis Jeremy Piger University of Oregon February 24, 29 ABSTRACT: The business cycle is a fundamental, yet elusive concept in macroeconomics.
More informationHeteroskedasticity in Time Series
Heteroskedasticity in Time Series Figure: Time Series of Daily NYSE Returns. 206 / 285 Key Fact 1: Stock Returns are Approximately Serially Uncorrelated Figure: Correlogram of Daily Stock Market Returns.
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 informationFACULTEIT ECONOMIE EN BEDRIJFSKUNDE. TWEEKERKENSTRAAT 2 B-9000 GENT Tel. : 32 - (0) Fax. : 32 - (0)
FACULTEIT ECONOMIE EN BEDRIJFSKUNDE TWEEKERKENSTRAAT 2 B-9000 GENT Tel. : 32 - (0)9 264.34.61 Fax. : 32 - (0)9 264.35.92 WORKING PAPER Estimating Long-Run Relationships between Observed Integrated Variables
More informationTrend and Cycles: A New Approach and Explanations of Some Old Puzzles
Trend and Cycles: A New Approach and Explanations of Some Old Puzzles Pierre Perron Boston University Tatsuma Wada Boston University This Version: January 2, 2005 Abstract Recent work on trend-cycle decompositions
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 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 informationConsider the trend-cycle decomposition of a time series y t
1 Unit Root Tests Consider the trend-cycle decomposition of a time series y t y t = TD t + TS t + C t = TD t + Z t The basic issue in unit root testing is to determine if TS t = 0. Two classes of tests,
More informationIf we want to analyze experimental or simulated data we might encounter the following tasks:
Chapter 1 Introduction If we want to analyze experimental or simulated data we might encounter the following tasks: Characterization of the source of the signal and diagnosis Studying dependencies Prediction
More information9) Time series econometrics
30C00200 Econometrics 9) Time series econometrics Timo Kuosmanen Professor Management Science http://nomepre.net/index.php/timokuosmanen 1 Macroeconomic data: GDP Inflation rate Examples of time series
More informationIntuitive and Reliable Estimates of the Output Gap from a Beveridge-Nelson Filter
Intuitive and Reliable Estimates of the Output Gap from a Beveridge-Nelson Filter Güneş Kamber, James Morley, and Benjamin Wong The Beveridge-Nelson decomposition based on autoregressive models produces
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 informationChapter 2. Some basic tools. 2.1 Time series: Theory Stochastic processes
Chapter 2 Some basic tools 2.1 Time series: Theory 2.1.1 Stochastic processes A stochastic process is a sequence of random variables..., x 0, x 1, x 2,.... In this class, the subscript always means time.
More informationUniversità 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 informationTime Series Analysis -- An Introduction -- AMS 586
Time Series Analysis -- An Introduction -- AMS 586 1 Objectives of time series analysis Data description Data interpretation Modeling Control Prediction & Forecasting 2 Time-Series Data Numerical data
More informationECON3327: Financial Econometrics, Spring 2016
ECON3327: Financial Econometrics, Spring 2016 Wooldridge, Introductory Econometrics (5th ed, 2012) Chapter 11: OLS with time series data Stationary and weakly dependent time series The notion of a stationary
More informationEC408 Topics in Applied Econometrics. B Fingleton, Dept of Economics, Strathclyde University
EC408 Topics in Applied Econometrics B Fingleton, Dept of Economics, Strathclyde University Applied Econometrics What is spurious regression? How do we check for stochastic trends? Cointegration and Error
More informationTime Series Examples Sheet
Lent Term 2001 Richard Weber Time Series Examples Sheet This is the examples sheet for the M. Phil. course in Time Series. A copy can be found at: http://www.statslab.cam.ac.uk/~rrw1/timeseries/ Throughout,
More information11. Further Issues in Using OLS with TS Data
11. Further Issues in Using OLS with TS Data With TS, including lags of the dependent variable often allow us to fit much better the variation in y Exact distribution theory is rarely available in TS applications,
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 informationESTIMATION OF DSGE MODELS WHEN THE DATA ARE PERSISTENT
ESTIMATION OF DSGE MODELS WHEN THE DATA ARE PERSISTENT Yuriy Gorodnichenko 1 Serena Ng 2 1 U.C. Berkeley 2 Columbia University May 2008 Outline Introduction Stochastic Growth Model Estimates Robust Estimators
More informationOn the Correlations of Trend-Cycle Errors
On the Correlations of Trend-Cycle Errors Tatsuma Wada Wayne State University This version: December 19, 11 Abstract This note provides explanations for an unexpected result, namely, the estimated parameter
More informationBayesian Inference of State Space Models with Flexible Covariance Matrix Rank: Applications for Inflation Modeling Luis Uzeda
Bayesian Inference of State Space Models with Flexible Covariance Matrix Rank: Applications for Inflation Modeling Luis Uzeda A thesis submitted for the degree of Doctor of Philosopy at The Australian
More informationMultivariate State Space Models: Applications
Multivariate State Space Models: Applications Sebastian Fossati University of Alberta Application I: Clark (1989) Clark (1987) considered the UC-ARMA(2,0) model y t = µ t + C t µ t = d t 1 + µ t 1 + ε
More informationLecture 16: State Space Model and Kalman Filter Bus 41910, Time Series Analysis, Mr. R. Tsay
Lecture 6: State Space Model and Kalman Filter Bus 490, Time Series Analysis, Mr R Tsay A state space model consists of two equations: S t+ F S t + Ge t+, () Z t HS t + ɛ t (2) where S t is a state vector
More informationUNSW Business School Working Paper
UNSW Business School Working Paper UNSW Business School Research Paper No. 2016 ECON 09 Intuitive and Reliable Estimates of the Output Gap from a Beveridge-Nelson Filter Gunes Kamber James Morley Benjamin
More informationLecture 5: Unit Roots, Cointegration and Error Correction Models The Spurious Regression Problem
Lecture 5: Unit Roots, Cointegration and Error Correction Models The Spurious Regression Problem Prof. Massimo Guidolin 20192 Financial Econometrics Winter/Spring 2018 Overview Stochastic vs. deterministic
More informationCHAPTER 21: TIME SERIES ECONOMETRICS: SOME BASIC CONCEPTS
CHAPTER 21: TIME SERIES ECONOMETRICS: SOME BASIC CONCEPTS 21.1 A stochastic process is said to be weakly stationary if its mean and variance are constant over time and if the value of the covariance between
More informationEconometrics Summary Algebraic and Statistical Preliminaries
Econometrics Summary Algebraic and Statistical Preliminaries Elasticity: The point elasticity of Y with respect to L is given by α = ( Y/ L)/(Y/L). The arc elasticity is given by ( Y/ L)/(Y/L), when L
More informationA Critical Note on the Forecast Error Variance Decomposition
A Critical Note on the Forecast Error Variance Decomposition Atilim Seymen This Version: March, 28 Preliminary: Do not cite without author s permisson. Abstract The paper questions the reasonability of
More informationIt is easily seen that in general a linear combination of y t and x t is I(1). However, in particular cases, it can be I(0), i.e. stationary.
6. COINTEGRATION 1 1 Cointegration 1.1 Definitions I(1) variables. z t = (y t x t ) is I(1) (integrated of order 1) if it is not stationary but its first difference z t is stationary. It is easily seen
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 informationEconomics Department LSE. Econometrics: Timeseries EXERCISE 1: SERIAL CORRELATION (ANALYTICAL)
Economics Department LSE EC402 Lent 2015 Danny Quah TW1.10.01A x7535 : Timeseries EXERCISE 1: SERIAL CORRELATION (ANALYTICAL) 1. Suppose ɛ is w.n. (0, σ 2 ), ρ < 1, and W t = ρw t 1 + ɛ t, for t = 1, 2,....
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 informationARIMA Models. Jamie Monogan. January 16, University of Georgia. Jamie Monogan (UGA) ARIMA Models January 16, / 27
ARIMA Models Jamie Monogan University of Georgia January 16, 2018 Jamie Monogan (UGA) ARIMA Models January 16, 2018 1 / 27 Objectives By the end of this meeting, participants should be able to: Argue why
More informationNon-Stationary Time Series and Unit Root Testing
Econometrics II Non-Stationary Time Series and Unit Root Testing Morten Nyboe Tabor Course Outline: Non-Stationary Time Series and Unit Root Testing 1 Stationarity and Deviation from Stationarity Trend-Stationarity
More informationFiltering for Current Analysis
Filtering for Current Analysis Ergodic Quantitative Consulting Inc. HEC (Montréal) CIRANO CRDE Abstract: This paper shows how existing band-pass filtering techniques and their extensions may be applied
More informationVolatility, Information Feedback and Market Microstructure Noise: A Tale of Two Regimes
Volatility, Information Feedback and Market Microstructure Noise: A Tale of Two Regimes Torben G. Andersen Northwestern University Gökhan Cebiroglu University of Vienna Nikolaus Hautsch University of Vienna
More information1 Introduction to Generalized Least Squares
ECONOMICS 7344, Spring 2017 Bent E. Sørensen April 12, 2017 1 Introduction to Generalized Least Squares Consider the model Y = Xβ + ɛ, where the N K matrix of regressors X is fixed, independent of the
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 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 informationEmpirical Macroeconomics
Empirical Macroeconomics Francesco Franco Nova SBE April 5, 2016 Francesco Franco Empirical Macroeconomics 1/39 Growth and Fluctuations Supply and Demand Figure : US dynamics Francesco Franco Empirical
More information