Vector Time Series Model Representations and Analysis with XploRe


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1 01 Vector Time Series Model Representations and Analysis with plore Julius Mungo CASE  Center for Applied Statistics and Economics HumboldtUniversität zu Berlin plore MulTi
2 Motivation 11 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 12 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 13 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 21 MulTiplot.xpl Generates a MTS plot from a kdimensional 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 22 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 23 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 24 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 31 VAR modelling plore specifies a kdimensional 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 41 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 43 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 *E2 Y Y Y*E Y Modelling Time Dependent Factor Loadings 44 Betadensity.xpl Kernel density (Epanechnikov, h = ) and boxplot for levels plore MulTi
14 Y *E2 Y Modelling Time Dependent Factor Loadings Y*E Figure 1: QQ 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 46 preanalysbetas.xpl lag Y plore MulTi Figure 2: ACF and PACF of levels
16 Modelling Time Dependent Factor Loadings 47 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: ADFTest of unitroot for levels series plore MulTi
17 Modelling Time Dependent Factor Loadings 48 coeff. lag test statistic crit. values (Kwiaskowski, (1992)) 1% 5% 10% Beta1 1 const Beta2 1 const Beta3 1 const Table 4: KPSSTest of stationarity for levels series unitrootest.xpl plore MulTi
18 Modelling Time Dependent Factor Loadings 49 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, unitroot 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 410 coeff. shift suggested test statistic crit. values (Lanne et, al(2001)) function break date (shift dummy ) 1% 5% 10% Beta Beta Beta Table 5: UnitrootTest 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 411 Beta Time Series Plot Y plore MulTi Figure 3: First difference plot of Beta series
21 Modelling Time Dependent Factor Loadings 412 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 HannanQuinnn 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 413 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 414 VAR estimates (OLS) with tvalues 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 415 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 416 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 417 (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 418 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 419 (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 420 Test Q3 JB 3 MARCH LM (3) Test statistic pvalue Table 6: Diagnostic tests for AR(3) models The tests hypothesis is rejected for pvalues 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 521 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 622 References G.C Reinsel Elements of Multivariate Time Series Anylysis. Springer Verlag, New York, W. Härdle, Z. Hlávka and S. Klinke plore Application Guide SpringerVerlag, 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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