Multivariate TS Analysis

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require(tseries)
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require(chron)
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require(directlabels)
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require(zoo)
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require(lmtest)
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require(TTR)
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require(MTS)
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require(lattice)
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require(grid)
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set.seed(100)
x = rnorm(50, 1)
y = rnorm(100, 51)
veri = data.frame(x, y)
class(veri)
## [1] "data.frame"
İnflation_and_GDP = ts(veri,
           frequency = 12,
           start = c(1972, 4))
plot(İnflation_and_GDP)

theme_set(theme_bw())
autoplot(İnflation_and_GDP) +
  ggtitle("Time Series Plot of the `inflation_and_GDP' Time-Series") +
  theme(plot.title = element_text(hjust = 0.5))

Veri setinin ilk 6 satırını göstermek için head fonksiyonunu kullanıyoruz.

head(İnflation_and_GDP)
##                  x        y
## Apr 1972 0.4978076 50.55294
## May 1972 1.1315312 49.26140
## Jun 1972 0.9210829 51.17886
## Jul 1972 1.8867848 52.89747
## Aug 1972 1.1169713 48.72807
## Sep 1972 1.3186301 51.98046

Kullandığım veri setinin yapısını anlamak için class fonksiyonunu kullandım

class(İnflation_and_GDP)
## [1] "mts"    "ts"     "matrix"
plot(İnflation_and_GDP) 

autoplot(İnflation_and_GDP) +
  ggtitle("Time Series Plot of the `İnflation_and_GDP' Time-Series") +
  theme(plot.title = element_text(hjust = 0.5))

Veri çerçevesini veya matrisi girdi olarak almak ve vektör, liste veya dizi halinde çıktı verilmesi için apply fonksiyonunu kullandım. Apply işlevi, öncelikle döngü yapılarının açık kullanımlarından kaçınmak için kullanılır.

apply(İnflation_and_GDP, 2, adf.test)
## Warning in FUN(newX[, i], ...): p-value smaller than printed p-value

## Warning in FUN(newX[, i], ...): p-value smaller than printed p-value
## $x
## 
##  Augmented Dickey-Fuller Test
## 
## data:  newX[, i]
## Dickey-Fuller = -7.1564, Lag order = 4, p-value = 0.01
## alternative hypothesis: stationary
## 
## 
## $y
## 
##  Augmented Dickey-Fuller Test
## 
## data:  newX[, i]
## Dickey-Fuller = -5.2529, Lag order = 4, p-value = 0.01
## alternative hypothesis: stationary
apply(İnflation_and_GDP, 2, adfTest, 
      lags=0,
      type="c", 
      title = "ADF Test for İnflation_and_GDP Data")
## Warning in FUN(newX[, i], ...): p-value smaller than printed p-value

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## $x
## 
## Title:
##  ADF Test for İnflation_and_GDP Data
## 
## Test Results:
##   PARAMETER:
##     Lag Order: 0
##   STATISTIC:
##     Dickey-Fuller: -13.2893
##   P VALUE:
##     0.01 
## 
## Description:
##  Tue Dec 20 21:19:51 2022 by user: User1
## 
## 
## $y
## 
## Title:
##  ADF Test for İnflation_and_GDP Data
## 
## Test Results:
##   PARAMETER:
##     Lag Order: 0
##   STATISTIC:
##     Dickey-Fuller: -12.3434
##   P VALUE:
##     0.01 
## 
## Description:
##  Tue Dec 20 21:19:51 2022 by user: User1

stnry fonksiyonu, kullanılan verinin durağan olup olmadığını test etmek için kullanılır.

diffm fonksiyonu, çok değişkenli zaman serisinin farkına bakıldığı zaman kullanılır.

stnry = diffM(İnflation_and_GDP)

Kullanılan veri setinin zaman serisi grafiği

plot.ts(İnflation_and_GDP)

Kullandığım veri setini görselleştirmek için autoplot fonksiyonunu kullandım

autoplot(ts(İnflation_and_GDP,
            start = c(1972,50),
            frequency = 100)) +
  ggtitle("Time Series Plot of the stationary `İnflation_and_GDP' Time-Series")

VAR MODELİ

library(vars)
varmodeli<-var(2:length(İnflation_and_GDP),)

Uygun bir gecikme seçmeme yardımcı olabilmesi için VARselect fonksiyonunu kullandım

VARselect(İnflation_and_GDP, 
          type = "none", 
          lag.max = 10) 
## $selection
## AIC(n)  HQ(n)  SC(n) FPE(n) 
##      5      5      3      5 
## 
## $criteria
##                1           2           3           4          5           6
## AIC(n) 0.3082514 -0.02417370 -0.18939313 -0.27896542 -0.3500788 -0.32390608
## HQ(n)  0.3530545  0.06543254 -0.05498377 -0.09975294 -0.1260632 -0.05508736
## SC(n)  0.4193541  0.19803160  0.14391482  0.16544519  0.2054344  0.34270983
## FPE(n) 1.3610631  0.97623054  0.82778898  0.75727826  0.7059321  0.72563261
##                   7           8          9         10
## AIC(n) -0.322082463 -0.28256713 -0.2032457 -0.2043306
## HQ(n)  -0.008460623  0.07585783  0.1999823  0.2437006
## SC(n)   0.455636101  0.60625409  0.7966781  0.9066959
## FPE(n)  0.728338354  0.75962511  0.8250465  0.8275802
var.a <- vars::VAR(İnflation_and_GDP,
                   lag.max = 10, 
                   ic = "AIC",
                   type = "none")

R’deki bir vektör, veri çerçevesi, regresyon modelindeki değerleri hızlı bir şekilde özetlemek için summary fonksiyonunu kullandım

summary(var.a)
## 
## VAR Estimation Results:
## ========================= 
## Endogenous variables: x, y 
## Deterministic variables: none 
## Sample size: 95 
## Log Likelihood: -231.854 
## Roots of the characteristic polynomial:
##     1 0.8707 0.8651 0.8651 0.775 0.775 0.7098 0.7098 0.677 0.677
## Call:
## vars::VAR(y = İnflation_and_GDP, type = "none", lag.max = 10, 
##     ic = "AIC")
## 
## 
## Estimation results for equation x: 
## ================================== 
## x = x.l1 + y.l1 + x.l2 + y.l2 + x.l3 + y.l3 + x.l4 + y.l4 + x.l5 + y.l5 
## 
##       Estimate Std. Error t value Pr(>|t|)  
## x.l1 -0.275181   0.109452  -2.514   0.0138 *
## y.l1 -0.056934   0.071610  -0.795   0.4288  
## x.l2  0.098886   0.110553   0.894   0.3736  
## y.l2 -0.004228   0.066418  -0.064   0.9494  
## x.l3 -0.255254   0.114895  -2.222   0.0290 *
## y.l3  0.052462   0.065520   0.801   0.4255  
## x.l4 -0.272833   0.115330  -2.366   0.0203 *
## y.l4  0.071553   0.065730   1.089   0.2794  
## x.l5 -0.286365   0.117363  -2.440   0.0168 *
## y.l5 -0.020389   0.068712  -0.297   0.7674  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## 
## Residual standard error: 0.7261 on 85 degrees of freedom
## Multiple R-Squared: 0.7447,  Adjusted R-squared: 0.7146 
## F-statistic: 24.79 on 10 and 85 DF,  p-value: < 2.2e-16 
## 
## 
## Estimation results for equation y: 
## ================================== 
## y = x.l1 + y.l1 + x.l2 + y.l2 + x.l3 + y.l3 + x.l4 + y.l4 + x.l5 + y.l5 
## 
##      Estimate Std. Error t value Pr(>|t|)   
## x.l1 -0.20083    0.15711  -1.278  0.20465   
## y.l1  0.02043    0.10279   0.199  0.84296   
## x.l2 -0.47607    0.15869  -3.000  0.00354 **
## y.l2  0.22983    0.09534   2.411  0.01808 * 
## x.l3 -0.23600    0.16493  -1.431  0.15612   
## y.l3  0.24821    0.09405   2.639  0.00989 **
## x.l4  0.20012    0.16555   1.209  0.23010   
## y.l4  0.21643    0.09435   2.294  0.02427 * 
## x.l5 -0.16206    0.16847  -0.962  0.33879   
## y.l5  0.30415    0.09863   3.084  0.00276 **
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## 
## Residual standard error: 1.042 on 85 degrees of freedom
## Multiple R-Squared: 0.9996,  Adjusted R-squared: 0.9996 
## F-statistic: 2.269e+04 on 10 and 85 DF,  p-value: < 2.2e-16 
## 
## 
## 
## Covariance matrix of residuals:
##         x       y
## x 0.52722 0.09188
## y 0.09188 1.08635
## 
## Correlation matrix of residuals:
##        x      y
## x 1.0000 0.1214
## y 0.1214 1.0000

serial.test fonksiyonu, seri olarak ilişkili hatalar için çok değişkenli Portmanteau- ve Breusch-Godfrey testini hesaplar.

Stabil bir VAR(p)’de hh mertebesine kadar seri olarak ilişkili bozulmaların yokluğunu test etmek için Portmanteau istatistiği şu şekilde tanımlanır:

serial.test(var.a)
## 
##  Portmanteau Test (asymptotic)
## 
## data:  Residuals of VAR object var.a
## Chi-squared = 55.987, df = 44, p-value = 0.1062

Özellikle veri setinin son n satırını görüntülemek için tail fonksiyonunu kullandım

tail(İnflation_and_GDP)
##                   x        y
## Feb 1980  0.4777166 50.01817
## Mar 1980  2.3222310 49.88636
## Apr 1980  0.6365597 50.56265
## May 1980  2.3190657 50.48389
## Jun 1980  1.0437791 51.41900
## Jul 1980 -0.8786559 51.13416
x = cumsum(x) + 5473.72
par(mar = c(2.5,2.5,1,2.5))
plot.ts(x)

DAXinv =ts(c(İnflation_and_GDP[,1], x),
           start = c(1972,50), 
           frequency = 100)
plot(DAXinv)