Analysis of Integrated and Cointegrated Time Series Pfaff
Analysis of Integrated and Cointegrated Time Series
Univariate Time Series Definitions Representation / Models Nonstationary Processes Statistical tests
Dr. Bernhard Pfaff
[email protected]
Multivariate Time Series VAR SVAR Cointegration
Invesco Asset Management Deutschland GmbH, Frankfurt am Main
SVEC
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The 1st International R/Rmetrics User and Developer Workshop 8–12 July 2007, Meielisalp, Lake Thune, Switzerland
Monographies R packages
Contents Univariate Time Series Definitions Representation / Models Nonstationary Processes Statistical tests Multivariate Time Series VAR SVAR Cointegration SVEC
Analysis of Integrated and Cointegrated Time Series Pfaff Univariate Time Series Definitions Representation / Models Nonstationary Processes Statistical tests
Multivariate Time Series VAR SVAR Cointegration SVEC
Topics left out Monographies R packages
Topics left out Monographies R packages
Contents Univariate Time Series Definitions Representation / Models Nonstationary Processes Statistical tests Multivariate Time Series VAR SVAR Cointegration SVEC
Analysis of Integrated and Cointegrated Time Series Pfaff Univariate Time Series Definitions Representation / Models Nonstationary Processes Statistical tests
Multivariate Time Series VAR SVAR Cointegration SVEC
Topics left out Monographies R packages
Topics left out Monographies R packages
Contents Univariate Time Series Definitions Representation / Models Nonstationary Processes Statistical tests Multivariate Time Series VAR SVAR Cointegration SVEC
Analysis of Integrated and Cointegrated Time Series Pfaff Univariate Time Series Definitions Representation / Models Nonstationary Processes Statistical tests
Multivariate Time Series VAR SVAR Cointegration SVEC
Topics left out Monographies R packages
Topics left out Monographies R packages
Contents Univariate Time Series Definitions Representation / Models Nonstationary Processes Statistical tests Multivariate Time Series VAR SVAR Cointegration SVEC
Analysis of Integrated and Cointegrated Time Series Pfaff Univariate Time Series Definitions Representation / Models Nonstationary Processes Statistical tests
Multivariate Time Series VAR SVAR Cointegration SVEC
Topics left out Monographies R packages
Topics left out Monographies R packages
Contents Univariate Time Series Definitions Representation / Models Nonstationary Processes Statistical tests Multivariate Time Series VAR SVAR Cointegration SVEC
Analysis of Integrated and Cointegrated Time Series Pfaff Univariate Time Series Definitions Representation / Models Nonstationary Processes Statistical tests
Multivariate Time Series VAR SVAR Cointegration SVEC
Topics left out Monographies R packages
Topics left out Monographies R packages
Univariate Time Series Overview
Analysis of Integrated and Cointegrated Time Series Pfaff Univariate Time Series Definitions Representation / Models Nonstationary Processes Statistical tests
Definitions Representations / Models Nonstationary processes Statistical tests
Multivariate Time Series VAR SVAR Cointegration SVEC
Topics left out Monographies R packages
Analysis of Integrated and Cointegrated Time Series
Definitions Stochastic Process
Pfaff Univariate Time Series Definitions
Time Series
Representation / Models
A discrete time series is defined as an ordered sequence of random numbers with respect to time. More formally, such a stochastic process can be written as:
Nonstationary Processes Statistical tests
Multivariate Time Series VAR SVAR
{y (s, t), s ∈ S, t ∈ T} ,
Cointegration
(1)
SVEC
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where for each t ∈ T, y (·, t) is a random variable on the sample space S and a realization of this stochastic process is given by y (s, ·) for each s ∈ S with regard to a point in time t ∈ T.
Monographies R packages
Analysis of Integrated and Cointegrated Time Series
Definitions Stochastic Process – Examples
Pfaff
7.0
25
Univariate Time Series Definitions
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Nonstationary Processes
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Statistical tests
Multivariate Time Series
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6.0
VAR SVAR 5
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logarithm of real gnp
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unemployment rate in percent
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5.0
Cointegration SVEC 1920
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Figure: U.S. GNP > > > > > >
Figure: U.S. unemployment rate
library(urca) , xlab = "") abline(a=5, b=0.5, col = "red")
Representation / Models Nonstationary Processes Statistical tests
80 60
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SVEC
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y.tsar2
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> > + > >
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Figure: Trend-stationary series
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Analysis of Integrated and Cointegrated Time Series
Nonstationary Processes Difference Stationary Time Series
Pfaff
Definition The series yt is difference stationary if φ(z) = 0 has one root on the unit circle and the others are outside the unit circle.
Univariate Time Series Definitions Representation / Models Nonstationary Processes Statistical tests
θ(L) can be factored as
Multivariate Time Series VAR
φ(L) = (1 − L)φ∗ (L) whereby
(11)
SVAR Cointegration SVEC
φ∗ (z) = 0 has all p − 1 roots outside the unit circle. ∆Zt is stationary and has an ARMA(p-1, q) representation. If Zt is difference stationary, then Zt is integrated of order one: Zt ∼ I (1). P Recursive substitution yields: yt = y0 + tj=1 uj .
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Analysis of Integrated and Cointegrated Time Series
Nonstationary Processes Difference Stationary Time Series – Example
Pfaff Univariate Time Series Definitions
I(1) process without drift 80
Representation / Models
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Nonstationary Processes Statistical tests
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set.seed(12345) u.ar2 > > > + > + >
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Figure: Difference-stationary series
Note: If ut ∼ IWN(0, σ 2 ), then yt is a random walk.
Analysis of Integrated and Cointegrated Time Series
Statistical tests Unit Root vs. Stationarity Tests
Pfaff Univariate Time Series
General Remarks
Definitions
Consider, the following trend-cycle decomposition of a time series yT : yt = TDt + Zt = TDT + TSt + Ct with
(12)
Representation / Models Nonstationary Processes Statistical tests
Multivariate Time Series VAR SVAR Cointegration
TDt signifies the deterministic trend, TSt is the stochastic trend and Ct is a stationary component. Unit root tests: H0 : TSt 6= 0 vs. H1 : TSt = 0, that is yt ∼ I (1) vs. yt ∼ I (0). Stationarity tests: H0 : TSt = 0 vs. H1 : TSt 6= 0, that is yt ∼ I (0) vs. yt ∼ I (0).
SVEC
Topics left out Monographies R packages
Analysis of Integrated and Cointegrated Time Series
Autoregressive unit root tests General Remarks
Pfaff Univariate Time Series
Tests are based on the following framework:
Definitions Representation / Models
yt = φyt−1 + ut , ut ∼ I (0)
(13)
Nonstationary Processes Statistical tests
Multivariate Time Series VAR
H0 : φ = 1, H1 : |φ| < 1
SVAR Cointegration SVEC
Tests: ADF- and PP-test. ADF: Serial correlation in ut is captured by autoregressive parametric structure of test. PP: Non-parametric correction based on estimated long-run variance of ∆yt .
Topics left out Monographies R packages
Analysis of Integrated and Cointegrated Time Series
Autoregressive unit root tests Augmented Dickey-Fuller Test, I
Pfaff
Test Regression
Univariate Time Series
0
p X
0
j=1 p X
yt = β Dt + φyt−1 +
Definitions Representation / Models
ψj ∆yt−j + ut ,
(14)
Nonstationary Processes Statistical tests
∆yt = β Dt + πyt−1 +
Multivariate Time Series
ψj ∆yt−j + ut with π = φ − 1
(15)
VAR SVAR
j=1
Cointegration SVEC
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Test Statistic
Monographies R packages
φˆ − 1 , SE (φ) π ˆ = . SE (π)
ADFt : tφ=1 =
(16)
ADFt : tπ=0
(17)
Autoregressive unit root tests Augmented Dickey-Fuller Test, II
Analysis of Integrated and Cointegrated Time Series Pfaff
R Resources
Univariate Time Series Definitions
Function ur.df in package urca.
Representation / Models Nonstationary Processes
Function ADF.test in package uroot. Function adf.test in package tseries. Function urdfTest in package fSeries.
Statistical tests
Multivariate Time Series VAR SVAR Cointegration SVEC
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Literature Dickey, D. and W. Fuller, Distribution of the Estimators for Autoregressive Time Series with a Unit Root, Journal of the American Statistical Society, 74 (1979), 427–341. Dickey, D. and W. Fuller, Likelihood Ratio Statistics for Autoregressive Time Series with a Unit Root, Econometrica, 49, 1057–1072. Fuller, W., Introduction to Statistical Time Series, 2nd Edition, 1996, New York: John Wiley. MacKinnon, J., Numerical Distribution Functions for Unit Root and Cointegration Tests, Journal of Applied Econometrics, 11 (1996), 601-618.
Monographies R packages
Analysis of Integrated and Cointegrated Time Series
Autoregressive unit root tests Augmented Dickey-Fuller Test, III
Pfaff
R code
Univariate Time Series
library(urca) y1.adf.nc.2 + > + >
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Partial ACF
0.8
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−0.10
10pct −1.62 −1.62
ACF
5pct −1.95 −1.95
0.4
1pct −2.58 −2.58
Partial Autocorrelations of Residuals 0.10
Autocorrelations of Residuals
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Statistic 0.85 −8.14
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Table: ADF-test results Figure: Residual plot of y 1 ADF-regression
Note: Use critical values of Dickey & Fuller, Fuller or MacKinnon.
Analysis of Integrated and Cointegrated Time Series
Autoregressive unit root tests Phillips & Perron Test, I
Pfaff Univariate Time Series
Test Regression
Definitions Representation / Models
∆yt = β 0 Dt + πyt−1 + ut , ut ∼ I (0)
(18)
Nonstationary Processes Statistical tests
Multivariate Time Series VAR
Test Statistic
SVAR Cointegration SVEC
Zt =
σ ˆ2 ˆ2 λ
1/2
Zπ = T π ˆ−
1 · tπ=0 − 2
ˆ2 − σ λ ˆ2 ˆ2 λ
! T · SE (ˆ π) , · σ ˆ2
T 2 · SE (ˆ π) ˆ2 · (λ − σ ˆ2) . 2ˆ σ2
ˆ and σ λ ˆ signify consistent estimates of the error variance.
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(19)
Monographies R packages
(20)
Autoregressive unit root tests Phillips & Perron Test, II
Analysis of Integrated and Cointegrated Time Series Pfaff Univariate Time Series
R Resources
Definitions Representation / Models Nonstationary Processes
Function ur.pp in package urca. Function pp.test in package tseries.
Statistical tests
Multivariate Time Series VAR
Function urppTest in package fSeries.
SVAR
Function PP.test in package stats.
SVEC
Cointegration
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Literature Phillips, P.C.B., Time Series Regression with a Unit Root, Econometrica, 55, 227–301. Phillips, P.C.B. and P. Perron, Testing for Unit Roots in Time Series Regression, Biometrika, 75, 335–346.
R packages
Analysis of Integrated and Cointegrated Time Series
Autoregressive unit root tests Phillips & Perron Test, III
Pfaff
R code
Univariate Time Series Definitions
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Diagram of fit for model with intercept and trend
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library(urca) y1.pp.ts + > + >
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0.6
10pct −3.14 −3.14
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5pct −3.43 −3.43
Partial ACF
1pct −4.00 −4.00
Partial Autocorrelations of Residuals
−0.2
Statistic −2.04 −7.19
ACF
y1 ∆y1
0.0 0.4 0.8
Autocorrelations of Residuals
Monographies 5
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Table: PP-test results Figure: Residual plot of y 1 PP-regression
Note: Same asymptotic distribution as ADF-Tests.
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Autoregressive unit root tests Remarks
Analysis of Integrated and Cointegrated Time Series Pfaff Univariate Time Series Definitions Representation / Models Nonstationary Processes
ADF and PP test are asymptotically equivalent. PP has better small sample properties than ADF.
Statistical tests
Multivariate Time Series VAR
Both have low power against I (0) alternatives that are close to being I (1) processes.
SVAR Cointegration SVEC
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Power of the tests diminishes as deterministic terms are added to the test regression.
Monographies R packages
Analysis of Integrated and Cointegrated Time Series
Efficient unit root tests Elliot, Rothenberg & Stock, I
Pfaff
Model
Univariate Time Series Definitions
yt = dt + ut , ut = aut−1 + vt
(21) (22)
Representation / Models Nonstationary Processes Statistical tests
Multivariate Time Series VAR SVAR Cointegration
Test Statistics
SVEC
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Point optimal test: PT =
Monographies
S(a = ¯a) − ¯aS(a = 1) , ω ˆ2
R packages
(23)
DF-GLS test: d d d ∆ytd = α0 yt−1 + α1 ∆yt−1 + . . . + αp ∆yt−p + εt
(24)
Efficient unit root tests Elliot, Rothenberg & Stock, II
Analysis of Integrated and Cointegrated Time Series Pfaff Univariate Time Series Definitions Representation / Models
R Resources Function ur.ers in package urca.
Nonstationary Processes Statistical tests
Multivariate Time Series VAR
Function urersTest in package fSeries.
SVAR Cointegration SVEC
Literature Elliot, G., T.J. Rothenberg and J.H. Stock, Efficient Tests for an Autoregressive Time Series with a Unit Root, Econometrica, 64 (1996), 813–836.
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Analysis of Integrated and Cointegrated Time Series
Efficient unit root tests Elliot, Rothenberg & Stock, III
Pfaff
Definitions Representation / Models Nonstationary Processes Statistical tests
Multivariate Time Series
1500
library(urca) set.seed(12345) u.ar1 > + >
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ERS ADF
Statistic 33.80 −1.40
1pct 3.96 −3.99
5pct 5.62 −3.43
Table: ERS / ADF-tests
10pct 6.89 −3.13
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Figure: Near I (1) process
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Unit Root Tests, other Schmidt & Phillips, I
Analysis of Integrated and Cointegrated Time Series Pfaff Univariate Time Series Definitions Representation / Models
Problem of DF-type tests: nuisance parameters, i.e., the coefficients of the deterministic regressors, are either not defined or have a different interpretation under the alternative hypothesis of stationarity.
Nonstationary Processes Statistical tests
Multivariate Time Series VAR SVAR Cointegration
Solution: LM-type test, that has the same set of nuisance parameters under both the null and alternative hypothesis.
SVEC
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Higher polynomials than a linear trend are allowed.
Analysis of Integrated and Cointegrated Time Series
Unit Root Tests, other Schmidt & Phillips, II
Pfaff
Model yt = α + Zt δ + xt
with
xt = πxt−1 + εt
(25)
Univariate Time Series Definitions Representation / Models Nonstationary Processes Statistical tests
Test Regression
Multivariate Time Series VAR
∆yt = ∆Zt γ + φS˜t−1 + vt
(26)
SVAR Cointegration SVEC
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Test Statistics
R packages
ρ˜ T φ˜ = 2 2 ω ˆ ω ˆ τ˜ = 2 ω ˆ
Z (ρ) = Z (τ )φ=0
(27) (28)
Unit Root Tests, other Schmidt & Phillips, III
Analysis of Integrated and Cointegrated Time Series Pfaff Univariate Time Series Definitions Representation / Models
R Resources Function ur.sp in package urca.
Nonstationary Processes Statistical tests
Multivariate Time Series VAR
Function urspTest in package fSeries.
SVAR Cointegration SVEC
Literature Schmidt, P. and P.C.B. Phillips, LM Test for a Unit Root in the Presence of Deterministic Trends, Oxford Bulletin of Economics and Statistics, 54(3) (1992), 257-287.
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Analysis of Integrated and Cointegrated Time Series
Unit Root Tests, other Schmidt & Phillips, IV
Pfaff
R code
Definitions
set.seed(12345) y1 > > +
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2000
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0
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Z (τ ) Z (ρ)
Statistic −2.53 −12.70
1pct −4.08 −32.40
5pct −3.55 −24.80
Table: S & P tests
10pct −3.28 −21.00
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Figure: I(1)-process with polynomial trend
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Unit Root Tests, other Zivot & Andrews, I
Analysis of Integrated and Cointegrated Time Series Pfaff Univariate Time Series Definitions
Problem: Difficult to statistically distinguish between an I (1)–series from a stable I (0) that is contaminated by a structrual shift. If break point is known: Perron and Perron & Vogelsang tests. But risk of , season=4) lttest.1 > > > + > + + + > + + + > + > +
data(UKpppuip) attach(UKpppuip) dat1