TP No2 d’Econométrie des Séries temporelles
Ndao
2026-01-22
=========================================================
================= Travail à faire ========================
=========================================================
Simuler deux Séries de taille 1000 à partir d’un processsus ARMA (1, 1) à coefficient phi = 0.3 et Têta = 0.5
Etudier la stationnarité
# Importation des library
library(tseries) # adf.test et kpss.test## Registered S3 method overwritten by 'quantmod':
## method from
## as.zoo.data.frame zoolibrary(vars)## Warning: le package 'vars' a été compilé avec la version R 4.5.2## Le chargement a nécessité le package : MASS## Le chargement a nécessité le package : strucchange## Warning: le package 'strucchange' a été compilé avec la version R 4.5.2## Le chargement a nécessité le package : zoo##
## Attachement du package : 'zoo'## Les objets suivants sont masqués depuis 'package:base':
##
## as.Date, as.Date.numeric## Le chargement a nécessité le package : sandwich## Warning: le package 'sandwich' a été compilé avec la version R 4.5.2## Le chargement a nécessité le package : urca## Le chargement a nécessité le package : lmtest# ==============================================================================
# ===================== Le travail =============================================
# ==============================================================================
# 1) Simulation des données
### Fixer la graine pour la reproductibilité
set.seed(123)
### Paramètres
n = 1000
phi = 0.3
theta = 0.5
### Simulation des deux séries ARMA(1,1)
# serie1 = arima.sim(n = n,
# model = list(ar = phi, ma = theta))
serie2 = arima.sim(n = n,
model = list(ar = phi, ma = theta))
serie1 = 0.5 * serie2 + rnorm(1000)
### Faisons une petite visualisation
par(mfrow = c(2, 1))
plot(serie1, main = "Série ARMA(1,1) - Série 1")
plot(serie2, main = "Série ARMA(1,1) - Série 2")
# 2) Etude de la stationnarité
## Méthode de Dickey-Fuller (ADF))
adf_serie1 <- adf.test(serie1)## Warning in adf.test(serie1): p-value smaller than printed p-valueadf_serie2 <- adf.test(serie2)## Warning in adf.test(serie2): p-value smaller than printed p-value## Méthode de KPSS
kpss_serie1 <- kpss.test(serie1)## Warning in kpss.test(serie1): p-value greater than printed p-valuekpss_serie2 <- kpss.test(serie2)## Warning in kpss.test(serie2): p-value greater than printed p-value## Affichage des résultats
adf_serie1##
## Augmented Dickey-Fuller Test
##
## data: serie1
## Dickey-Fuller = -10.391, Lag order = 9, p-value = 0.01
## alternative hypothesis: stationaryadf_serie2##
## Augmented Dickey-Fuller Test
##
## data: serie2
## Dickey-Fuller = -8.9692, Lag order = 9, p-value = 0.01
## alternative hypothesis: stationarykpss_serie1##
## KPSS Test for Level Stationarity
##
## data: serie1
## KPSS Level = 0.036098, Truncation lag parameter = 7, p-value = 0.1kpss_serie2##
## KPSS Test for Level Stationarity
##
## data: serie2
## KPSS Level = 0.048007, Truncation lag parameter = 7, p-value = 0.1D’après la méthode de Dickey-Fuller, les p-value étant iférieurs à 0,5, ainsi, on rejette l’hypothèse H0 de non satationnarité des séries. Ce résultat est confirmé par la méthode de KPSS avec une p-value supérieur à 0, 5
# Définition de la série
serie = cbind(serie1, serie2)
serie## Time Series:
## Start = 1
## End = 1000
## Frequency = 1
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## 7 0.763950135 0.589836359
## 8 -0.373021353 -0.323548869
## 9 0.893015101 1.411927909
## 10 1.134985438 1.814885419
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## 12 -0.031370702 -0.633920564
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## 19 -0.995423425 -2.447960797
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## 896 -0.309337215 -0.553357526
## 897 2.448178096 1.622841493
## 898 0.866455735 2.390995411
## 899 -2.286518344 0.635039654
## 900 0.866059317 0.704151061
## 901 -0.598648803 0.575638416
## 902 -0.773677906 0.449721000
## 903 1.693784122 0.545934617
## 904 3.882703454 2.869410894
## 905 -0.558152872 0.963259340
## 906 0.929651809 -0.202746744
## 907 -0.989318529 -1.790341382
## 908 -0.276820109 -0.837255136
## 909 1.792411648 1.140349893
## 910 1.381710241 2.336071445
## 911 -1.102111703 -0.501492698
## 912 -0.820534344 -0.700250938
## 913 1.480974965 1.589679419
## 914 -1.906160863 0.859572938
## 915 1.014601091 -0.161286582
## 916 -1.662649330 -0.190998463
## 917 0.166656364 0.289450601
## 918 0.276893238 0.934504432
## 919 2.258445562 1.930752151
## 920 0.760173563 1.142648497
## 921 0.347603535 0.491328799
## 922 2.064903579 2.733535540
## 923 0.420564708 2.495158529
## 924 0.380526210 1.152850132
## 925 0.726578648 -0.902011531
## 926 0.214039134 -0.938868974
## 927 -1.526276118 -0.501558822
## 928 0.501868091 -0.272176553
## 929 -1.708886386 -0.662593522
## 930 -1.184962069 -1.173302530
## 931 0.502834275 -1.416172217
## 932 -2.750916674 -0.999742996
## 933 -1.298535445 0.960957491
## 934 2.312092772 2.021440198
## 935 -0.429292657 0.771000255
## 936 0.426405050 -1.634603937
## 937 -1.257211619 -2.177922837
## 938 -0.338938871 -1.533429099
## 939 0.520643394 -0.585270992
## 940 -1.044396655 -0.786369571
## 941 0.415142168 1.439466182
## 942 0.166359163 1.162912665
## 943 2.315482057 -1.001204438
## 944 -0.398150593 -1.048595324
## 945 -0.222493106 -1.390587057
## 946 -0.722608141 -0.763709201
## 947 -0.200491289 0.082598708
## 948 1.627346684 0.497184031
## 949 -0.986338292 -1.294909244
## 950 -0.469209003 -0.978690612
## 951 0.440561435 0.127198542
## 952 -1.045585667 -1.227723844
## 953 1.897311287 -0.123505737
## 954 -0.816058623 1.224801978
## 955 3.540865677 3.059145394
## 956 0.760536254 2.224256286
## 957 1.753300600 0.792361907
## 958 -1.820346826 0.357661170
## 959 1.068759371 0.225357499
## 960 0.003440060 -1.745982410
## 961 -2.830796056 -3.119158169
## 962 -1.692748852 -1.848849947
## 963 -0.393721079 -1.174491848
## 964 0.849763069 -0.588144720
## 965 -1.896141067 -2.260283527
## 966 -1.840319749 -3.232387499
## 967 0.161072527 -2.820548816
## 968 -1.332771312 -0.410848143
## 969 0.583436311 -0.728715340
## 970 -0.334918590 -1.567373559
## 971 -1.500862141 -0.790095226
## 972 -0.927786852 -0.519838020
## 973 0.697823806 -0.170907408
## 974 1.412458607 2.326998349
## 975 1.788111690 0.726026824
## 976 -0.926765676 -0.649963428
## 977 -2.146729345 -0.864483298
## 978 1.545162283 0.994670713
## 979 -0.466064616 0.285114004
## 980 -0.070307055 -0.381295164
## 981 2.034892620 0.631701613
## 982 -0.488170675 -0.475513245
## 983 2.041168615 0.983226574
## 984 -0.197980502 1.798318296
## 985 0.827096280 -0.196829913
## 986 0.176011977 -0.141574470
## 987 -0.751016829 -0.028490580
## 988 -1.180994926 0.198028026
## 989 0.139667462 0.126047597
## 990 0.786836121 1.063342717
## 991 0.029027041 -0.496839552
## 992 -0.136753356 -1.347218756
## 993 -0.917817893 -0.914664653
## 994 -1.209000452 -1.394793460
## 995 -0.741208343 -1.956292444
## 996 -1.104011919 -1.124845496
## 997 0.979918196 -0.478618902
## 998 -1.205372131 -2.759016012
## 999 -1.146169684 -1.061802735
## 1000 -1.581157142 0.451471643Une autre méthode serait
Matrice = matrix(rep(0, 1), 1000, 2)
Matrice## [,1] [,2]
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## [1000,] 0 0# serie = Matrice# Reprsésentation graphique de la matrice
plot.ts(serie)
# Détermination de l'ordre du modèle
VARselect(serie, lag.max = 4)## $selection
## AIC(n) HQ(n) SC(n) FPE(n)
## 3 3 2 3
##
## $criteria
## 1 2 3 4
## AIC(n) 0.09772915 0.01805163 0.007561659 0.01076156
## HQ(n) 0.10895884 0.03676778 0.033764271 0.04445063
## SC(n) 0.12726979 0.06728604 0.076489833 0.09938349
## FPE(n) 1.10266413 1.01821572 1.007590787 1.01082066# Estimation des modèles avec l'ordre 1,, 2, 3 et 4
### Estimation avec l'ordre 1
est_var1 = VAR(serie, p = 1)
est_var1##
## VAR Estimation Results:
## =======================
##
## Estimated coefficients for equation serie1:
## ===========================================
## Call:
## serie1 = serie1.l1 + serie2.l1 + const
##
## serie1.l1 serie2.l1 const
## -0.04084429 0.29489500 0.04871146
##
##
## Estimated coefficients for equation serie2:
## ===========================================
## Call:
## serie2 = serie1.l1 + serie2.l1 + const
##
## serie1.l1 serie2.l1 const
## -0.001539104 0.564790731 0.010048044### Estimatio avec l'ordre 2
est_var2 = VAR(serie, p = 2)
summary(est_var2)##
## VAR Estimation Results:
## =========================
## Endogenous variables: serie1, serie2
## Deterministic variables: const
## Sample size: 998
## Log Likelihood: -2833.457
## Roots of the characteristic polynomial:
## 0.5334 0.5334 0.2013 0.2013
## Call:
## VAR(y = serie, p = 2)
##
##
## Estimation results for equation serie1:
## =======================================
## serie1 = serie1.l1 + serie2.l1 + serie1.l2 + serie2.l2 + const
##
## Estimate Std. Error t value Pr(>|t|)
## serie1.l1 -0.05215 0.03675 -1.419 0.15628
## serie2.l1 0.38340 0.04106 9.338 < 2e-16 ***
## serie1.l2 -0.05519 0.03660 -1.508 0.13185
## serie2.l2 -0.11573 0.04044 -2.861 0.00431 **
## const 0.05301 0.03681 1.440 0.15016
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 1.16 on 993 degrees of freedom
## Multiple R-Squared: 0.09894, Adjusted R-squared: 0.09531
## F-statistic: 27.26 on 4 and 993 DF, p-value: < 2.2e-16
##
##
## Estimation results for equation serie2:
## =======================================
## serie2 = serie1.l1 + serie2.l1 + serie1.l2 + serie2.l2 + const
##
## Estimate Std. Error t value Pr(>|t|)
## serie1.l1 -0.01696 0.03179 -0.533 0.594
## serie2.l1 0.73449 0.03552 20.680 < 2e-16 ***
## serie1.l2 -0.02933 0.03166 -0.927 0.354
## serie2.l2 -0.27041 0.03499 -7.729 2.65e-14 ***
## const 0.01608 0.03184 0.505 0.614
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 1.004 on 993 degrees of freedom
## Multiple R-Squared: 0.3743, Adjusted R-squared: 0.3718
## F-statistic: 148.5 on 4 and 993 DF, p-value: < 2.2e-16
##
##
##
## Covariance matrix of residuals:
## serie1 serie2
## serie1 1.3464 0.5865
## serie2 0.5865 1.0076
##
## Correlation matrix of residuals:
## serie1 serie2
## serie1 1.0000 0.5036
## serie2 0.5036 1.0000### Estimation avec l'ordre 3
est_var3 = VAR(serie, p = 3)
summary(est_var3)##
## VAR Estimation Results:
## =========================
## Endogenous variables: serie1, serie2
## Deterministic variables: const
## Sample size: 997
## Log Likelihood: -2822.065
## Roots of the characteristic polynomial:
## 0.5024 0.4991 0.4991 0.315 0.315 0.2332
## Call:
## VAR(y = serie, p = 3)
##
##
## Estimation results for equation serie1:
## =======================================
## serie1 = serie1.l1 + serie2.l1 + serie1.l2 + serie2.l2 + serie1.l3 + serie2.l3 + const
##
## Estimate Std. Error t value Pr(>|t|)
## serie1.l1 -0.05108 0.03677 -1.389 0.165145
## serie2.l1 0.40092 0.04247 9.441 < 2e-16 ***
## serie1.l2 -0.04837 0.03678 -1.315 0.188683
## serie2.l2 -0.16441 0.04910 -3.348 0.000844 ***
## serie1.l3 0.04557 0.03661 1.245 0.213497
## serie2.l3 0.03667 0.04170 0.879 0.379390
## const 0.05017 0.03685 1.362 0.173611
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 1.159 on 990 degrees of freedom
## Multiple R-Squared: 0.1023, Adjusted R-squared: 0.09682
## F-statistic: 18.8 on 6 and 990 DF, p-value: < 2.2e-16
##
##
## Estimation results for equation serie2:
## =======================================
## serie2 = serie1.l1 + serie2.l1 + serie1.l2 + serie2.l2 + serie1.l3 + serie2.l3 + const
##
## Estimate Std. Error t value Pr(>|t|)
## serie1.l1 -0.01719 0.03159 -0.544 0.58648
## serie2.l1 0.77181 0.03648 21.158 < 2e-16 ***
## serie1.l2 -0.01912 0.03159 -0.605 0.54512
## serie2.l2 -0.36887 0.04218 -8.745 < 2e-16 ***
## serie1.l3 0.04743 0.03145 1.508 0.13189
## serie2.l3 0.10170 0.03582 2.839 0.00462 **
## const 0.01182 0.03165 0.373 0.70897
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.996 on 990 degrees of freedom
## Multiple R-Squared: 0.3852, Adjusted R-squared: 0.3815
## F-statistic: 103.4 on 6 and 990 DF, p-value: < 2.2e-16
##
##
##
## Covariance matrix of residuals:
## serie1 serie2
## serie1 1.3444 0.5782
## serie2 0.5782 0.9921
##
## Correlation matrix of residuals:
## serie1 serie2
## serie1 1.0000 0.5006
## serie2 0.5006 1.0000### Estimation avec l'ordre 4
est_var4 = VAR(serie, p = 4)
summary(est_var4)##
## VAR Estimation Results:
## =========================
## Endogenous variables: serie1, serie2
## Deterministic variables: const
## Sample size: 996
## Log Likelihood: -2813.885
## Roots of the characteristic polynomial:
## 0.5311 0.5311 0.4831 0.4831 0.3626 0.3626 0.3331 0.1117
## Call:
## VAR(y = serie, p = 4)
##
##
## Estimation results for equation serie1:
## =======================================
## serie1 = serie1.l1 + serie2.l1 + serie1.l2 + serie2.l2 + serie1.l3 + serie2.l3 + serie1.l4 + serie2.l4 + const
##
## Estimate Std. Error t value Pr(>|t|)
## serie1.l1 -0.050227 0.036695 -1.369 0.17138
## serie2.l1 0.402068 0.042666 9.424 < 2e-16 ***
## serie1.l2 -0.050439 0.036730 -1.373 0.17000
## serie2.l2 -0.163548 0.051095 -3.201 0.00141 **
## serie1.l3 0.037699 0.036722 1.027 0.30486
## serie2.l3 0.047761 0.050992 0.937 0.34918
## serie1.l4 0.009177 0.036600 0.251 0.80207
## serie2.l4 -0.015513 0.041785 -0.371 0.71053
## const 0.047031 0.036817 1.277 0.20175
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 1.157 on 987 degrees of freedom
## Multiple R-Squared: 0.1036, Adjusted R-squared: 0.09633
## F-statistic: 14.26 on 8 and 987 DF, p-value: < 2.2e-16
##
##
## Estimation results for equation serie2:
## =======================================
## serie2 = serie1.l1 + serie2.l1 + serie1.l2 + serie2.l2 + serie1.l3 + serie2.l3 + serie1.l4 + serie2.l4 + const
##
## Estimate Std. Error t value Pr(>|t|)
## serie1.l1 -0.01794 0.03156 -0.568 0.569893
## serie2.l1 0.77908 0.03670 21.228 < 2e-16 ***
## serie1.l2 -0.01790 0.03159 -0.566 0.571228
## serie2.l2 -0.39012 0.04395 -8.876 < 2e-16 ***
## serie1.l3 0.04260 0.03159 1.349 0.177721
## serie2.l3 0.14963 0.04386 3.411 0.000673 ***
## serie1.l4 0.02229 0.03148 0.708 0.479100
## serie2.l4 -0.07277 0.03594 -2.025 0.043183 *
## const 0.01077 0.03167 0.340 0.733766
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.995 on 987 degrees of freedom
## Multiple R-Squared: 0.3882, Adjusted R-squared: 0.3833
## F-statistic: 78.29 on 8 and 987 DF, p-value: < 2.2e-16
##
##
##
## Covariance matrix of residuals:
## serie1 serie2
## serie1 1.338 0.576
## serie2 0.576 0.990
##
## Correlation matrix of residuals:
## serie1 serie2
## serie1 1.0000 0.5005
## serie2 0.5005 1.0000coef(est_var1)## $serie1
## Estimate Std. Error t value Pr(>|t|)
## serie1.l1 -0.04084429 0.03685864 -1.108133 2.680718e-01
## serie2.l1 0.29489500 0.03550303 8.306192 3.201894e-16
## const 0.04871146 0.03705583 1.314542 1.889664e-01
##
## $serie2
## Estimate Std. Error t value Pr(>|t|)
## serie1.l1 -0.001539104 0.03298908 -0.04665495 9.627976e-01
## serie2.l1 0.564790731 0.03177579 17.77424762 1.333990e-61
## const 0.010048044 0.03316557 0.30296613 7.619789e-01Commentaires
Les coefficients des modèles sont singificatifs pour un modèle var d’ordre 2 et 3
# Analyses des résidus
resid2 = resid(est_var2)
resid3 = resid(est_var3)resultat_test2 = serial.test(est_var2, lags.pt = 10, type = "PT.asymptotic")
resultat_test3 = serial.test(est_var3, lags.pt = 10, type = "PT.asymptotic")D’après les résultats, le test d’autocorrélation montre que le modèle d’ordre 3 est celui à retenir.
plot(resultat_test3)
# Test de normalité des résidus
resultat_normalites = normality.test(est_var2)# Prévision
previsions = predict(est_var2, n.ahead = 20, colors = "red")
### Visualisation
plot(previsions, xlim = c(950, 1100))
# Une autre méthide de prévision
fanchart(previsions, colors = "red")
# Test de causalité des séries
causality(est_var2, cause = "serie1")## $Granger
##
## Granger causality H0: serie1 do not Granger-cause serie2
##
## data: VAR object est_var2
## F-Test = 0.55203, df1 = 2, df2 = 1986, p-value = 0.5759
##
##
## $Instant
##
## H0: No instantaneous causality between: serie1 and serie2
##
## data: VAR object est_var2
## Chi-squared = 201.87, df = 1, p-value < 2.2e-16causality(est_var2, cause = "serie2")## $Granger
##
## Granger causality H0: serie2 do not Granger-cause serie1
##
## data: VAR object est_var2
## F-Test = 44.494, df1 = 2, df2 = 1986, p-value < 2.2e-16
##
##
## $Instant
##
## H0: No instantaneous causality between: serie2 and serie1
##
## data: VAR object est_var2
## Chi-squared = 201.87, df = 1, p-value < 2.2e-16# Chocs /// Fonctions de réponse
chocs = irf(est_var2, impute = c("serie1", "serie2"), reponse = c("serie1", "seie2"), n.ahead = 10)
plot(chocs)
# Décomposition de la variance
fevd_results = fevd(est_var2)
plot(fevd_results)