Vector Time Series Model Representations and Analysis with XploRe

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1 0-1 Vector Time Series Model Representations and Analysis with plore Julius Mungo CASE - Center for Applied Statistics and Economics Humboldt-Universität zu Berlin plore MulTi

2 Motivation 1-1 Multiple time series analysis approach involves a frame work for analyzing time series systems and the possible cross relationships among its levels. Modelling such systems entails investigating whether some variables in the system have a tendency to lead others. there is feedbacks between the variables, the question of contemporaneous movements, impulses (shocks, innovations) transfer from one time series to another plore MulTi

3 Motivation 1-2 The modelling procedure in plore uses the quantlet library MulTi to model a system of multiple time series. how plore MulTi is used to empirically investigate and modell various MTS systems. attention is on Vector Autoregressive (VAR) and the Vector Equilibrium Correction (ECM) models representations and modelling. Granger & Newbold ( 1986) plore MulTi

4 Motivation 1-3 Outline 1. Motivation 2. plore Quantlib MulTi 3. Modelling Time Dependent Factor Loadings from a DSFM for IV String Dynamics 4. Summary 5. References plore MulTi

5 plore Quantlib MulTi 2-1 MulTiplot.xpl Generates a MTS plot from a k-dimensional time series data, allowing for the series transformation, with graphics, plots and their properties to be investigated MulTiplot01.xpl plore MulTi

6 plore Quantlib MulTi 2-2 MulTifr.xpl For general analysis of the Full VAR Model; VAR order selection criteria, parameter estimation, Residual Analysis, Structural Analysis and Forecasting. MulTifr02.xpl plore MulTi

7 plore Quantlib MulTi 2-3 MulTiira.xpl For VAR impulse response analysis 1 library (" multi ") 2 x= read (" mts. dat ") 3 MulTiira (x,4,"m " " Y " " I") MulTiira01.xpl plore MulTi

8 plore Quantlib MulTi 2-4 MulTirr.xpl For Reduced Rank VAR analysis MulTiss.xpl General analysis for a Subset VAR Model MulTici.xpl General analysis for cointegration plore MulTi

9 VAR modelling with plore 3-1 VAR modelling plore specifies a k-dimensional VAR(p) model of the form Y t = υ + A 1 Y t 1 + A 2 Y t 2 +,..., +A p Y t p + ε t (1) Y t = (Y 1t,..., Y kt ) are observable vectors of k endogenous variables υ = (υ 1,..., υ k ) is a vector of intercept terms, A i are (K K) coefficient matrices ε t is a white noise with covariance matrix Σ ε > 0 plore MulTi

10 Modelling Time Dependent Factor Loadings 4-1 Modelling Time Dependent Factor Loadings from a DSFM for IV String Dynamics The DSFM is represented by Y i,j = m o ( i,j ) + l β i,l m l ( i,j ) + ɛ i,j l=1 m l are smooth basis function (l = 0, 1,..., L) i,j are two dimensional covariables β i,l are weights of m l depending on time i β i = (β i,1, β i,2,..., β i,l ) t (Fengler et al (2004)) form an observed MTS plore MulTi

11 time time time Time series plots for the beta coeff. series ( ) MTSplot.xpl Beta1 coeff. time plot number sold(thousands) Beta2 coeff. time plot number sold(thousands) Beta3 coeff. time plot number sold(thousands)*e

12 Modelling Time Dependent Factor Loadings 4-3 Min. Max. Mean Median Stdd. Skewn. Kurt. beta beta beta Table 1: Summary statistics for Beta coeff. series beta1 beta2 beta3 beta beta beta3 1 Table 2: Contemp. correlation Betasummary.xpl plore MulTi

13 Y Y Distribution: Beta Distribution: Beta Distribution: Beta *E-2 Y Y Y*E Y Modelling Time Dependent Factor Loadings 4-4 Betadensity.xpl Kernel density (Epanechnikov, h = ) and boxplot for levels plore MulTi

14 Y *E-2 Y Modelling Time Dependent Factor Loadings Y*E Figure 1: Q-Q plots of the normal against the emprical quanttiles for the Beta series plore MulTi

15 acf acf acf Sample autocorrelation function (acf) lag Sample autocorrelation function (acf) lag Sample autocorrelation function (acf) lag pacf pacf pacf Sample partial autocorrelation function (pacf) lag Sample partial autocorrelation function (pacf) lag Sample partial autocorrelation function (pacf) Modelling Time Dependent Factor Loadings 4-6 preanalysbetas.xpl lag Y plore MulTi Figure 2: ACF and PACF of levels

16 Modelling Time Dependent Factor Loadings 4-7 Testing β i levels for random walk Coeff. Test Deterministic lags testvalue asymptotic crit. values term (David & Mackinnon, (1993)) 1% 5% 10% Beta1 ADF constant Beta2 ADF constant Beta3 ADF constant Table 3: ADF-Test of unitroot for levels series plore MulTi

17 Modelling Time Dependent Factor Loadings 4-8 coeff. lag test statistic crit. values (Kwiaskowski, (1992)) 1% 5% 10% Beta1 1 const Beta2 1 const Beta3 1 const Table 4: KPSS-Test of stationarity for levels series unitrootest.xpl plore MulTi

18 Modelling Time Dependent Factor Loadings 4-9 Modelling Beta series Results: At 1% significant level, unit root exist for Beta1 and beta3 at all lags considered Beta2 indicates of some kind of misspecification Beta3 do not reject at 1% level, unit-root null hypothesis. Even at 5% or 10%, rejecting unit root will be marginal. KPSS clearly rejects its null hypothesis of stationarity around a constant plore MulTi

19 Modelling Time Dependent Factor Loadings 4-10 coeff. shift suggested test statistic crit. values (Lanne et, al(2001)) function break date (shift dummy ) 1% 5% 10% Beta Beta Beta Table 5: Unitroot-Test of stationarity for levels series in the presence of structural break We specify a stationary model with first differences and consider fitting an VAR model, t = ( Beta1, Beta2, Beta3) and determine the autoregressive order for the model plore MulTi

20 Modelling Time Dependent Factor Loadings 4-11 Beta Time Series Plot Y plore MulTi Figure 3: First difference plot of Beta series

21 Modelling Time Dependent Factor Loadings 4-12 Order Selection Criteria Final Prediction Error Akaike Information Criterion AIC = ln (n) ε = ln FPE(n) = T + kn + 1 T kn 1 ˆ ˆ (n) ε + 2nK 2 T Schwarz Information Criterion k det( ˆ ε (n)) + 2(the number of freely estimated parameters) T SIC = ln Hannan-Quinnn Information Criterion plore MulTi HQ = ln ˆ (n) ε + lnt T nk 2 ˆ (n) ε + 2ln(lnT ) T nk 2

22 Modelling Time Dependent Factor Loadings 4-13 order ln(fpe) AIC HQ SC We choose to apply the order 3 as indicated by HQ. HQ and HC have been justified as consistent, (see, Paulsen(1984) and Tsay(1984)) plore MulTi

23 Modelling Time Dependent Factor Loadings 4-14 VAR estimates (OLS) with t-values in parenthesis 2 4 Beta1t Beta2 t Beta3 t 3 5 = (3.24) 0.19(2.30) 0.06( 0.40) 0.09(6.35) 0.07( 17.17) 0.07(1.03) 0.02(2.25) 0.02(1.42) 0.26( 8.24) ( 2.40) 0.04( 0.79) 0.11(0.64) 0.03( 1.58) 0.04(7.94) 0.05( 0.66) 0.00( 0.63) 0.00(0.16) 0.06( 1.89) 0.02( 0.58) 0.13( 1.53) 0.14(+0.83) 0.00(+0.03) 0.12( 3.42) 0.02( 0.26) 0.01( 0.53) 0.02( 1.36) 0.05( 1.52) 2 4 ˆε 1,t ˆε 2,t ˆε 3,t Beta1 t 1 Beta2 t 1 Beta3 t Beta1 t 2 Beta2 t 2 Beta3 t 2 4 Beta1 t 3 Beta2 t 3 Beta3 t plore MulTi

24 Modelling Time Dependent Factor Loadings 4-15 Covariance matrix of residuals ˆΣ ε = Correlation matrix of residuals ˆ Corr(ε t ) = The correlation matrix indicates that there is some contemporaneous correlation structure in the residual vector. Not all elements of the parameter matrices are significantly different from zero. Especially the coefficients for Beta1 t 3. plore MulTi

25 Modelling Time Dependent Factor Loadings 4-16 Model Validation (i) Multivariate Portmanteau test for autocorrelation H 0 : E(ε t ε t i) = 0, i = 1,..., h H 1 : at least one autocovariance (autocorrelation) is non zero Test statistic: (Ljung & Box (1978)) h Qp = T 2 1 { T i tr C i C 1 i=1 C i = T 1 0 C i T t=i+1 ε t ε t i } C0 1 χ 2 k 2 (h p) C 0 and C i are the contemporaneous correlations and autocovariance of residuals respectively plore MulTi

26 Modelling Time Dependent Factor Loadings 4-17 (ii) Testing for ARCH effects Test for neglected conditional heteroscedasticity (ARCH) based on fitting ARCH(q) model to the estimated residuals. ˆε 2 t = β 0 + β 1ˆε 2 t β q ˆε 2 t q + error t H 0 : β 1 = = β q = 0, (no ARCH effects) H 1 : β 1 0 or... or β q 0 plore MulTi

27 Modelling Time Dependent Factor Loadings 4-18 Lagrange Multiplier (LM) statistic: (see, Engle (1982)) The R 2 form, test statistic: ARCH LM = 1 2 ˆε t ˆε χ 2 q T R 2 χ 2 q R 2 is the squared multiple R 2 value of the regression of ˆε 2 t on an intercept and q lagged values of ˆε 2 t ARCHtest.xpl plore MulTi

28 Modelling Time Dependent Factor Loadings 4-19 (iii) Testing for Nonnormality H 0 : E(µ s t) 3 = 0 & E(µ s t) 3 = 0 H 1 : E(µ s t) 3 0 & E(µ s t) 3 0 Test statistic: (Jarque and Bera (1987)) JB = T 6 ( T ) 2 ( T 1 (ˆµ s t) 3 + T T T 1 (ˆµ s 24 t) 4 3 t=1 The test displays the χ 2 -statistics associated with the skewness and kurtosis of the standardized residuals for testing nonnormality. t=1 ) 2 plore MulTi

29 Modelling Time Dependent Factor Loadings 4-20 Test Q3 JB 3 MARCH LM (3) Test statistic p-value Table 6: Diagnostic tests for AR(3) models The tests hypothesis is rejected for p-values smaller than Results show some autocorrelation in the residuals and the presence of heteroscedastic effects in the conditional variance. We therefore maintain that there is some ARCH effects in model residuals. plore MulTi

30 Summary 5-21 Summary Testing for ARCH effects reveal neglected conditional heteroscedasticity. This gives an indication of fitting an ARCH or ARCH type model. Observing that not all elements of the estimated VAR parameter matrices are significantly different from zero, we could choose a subset VAR model where single elements of the estimated coefficient matrices are restricted to zero. plore MulTi

31 References 6-22 References G.C Reinsel Elements of Multivariate Time Series Anylysis. Springer Verlag, New York, W. Härdle, Z. Hlávka and S. Klinke plore Application Guide Springer-Verlag, Heidelberg, H. Lütkepohl Introduction to Multiple Time Series Analysis. Springer Verlag,1993. K. Patterson An Introduction to Applied Econometrics a time series approach. Macmillan Press Ltd, plore MulTi

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