Preparation

Цена в USD

library(readr)
library(dplyr)
library(ggplot2)
library(readr)
library(ggplot2)
library(dplyr)
library(forecast)
library(smooth)
library(tidyquant)
library(plotly)
library(AER)
library(dynlm)
library(forecast)
library(readxl)
library(stargazer)
library(scales)
library(quantmod)
library(urca)
library(TSA)
library(quantmod)
library(tseries)
SP <- read_delim("SP.csv", ";", 
    escape_double = FALSE, trim_ws = TRUE)
#SP$Price_SP=SP$Price_SP*1000
RTC<- read_delim("rtc.csv", ";", escape_double = FALSE, 
    trim_ws = TRUE)
Gas <- read_delim("Gas.csv", ";", escape_double = FALSE, 
    trim_ws = TRUE)
USD <- read_delim("USD.csv", ";", escape_double = FALSE, 
    trim_ws = TRUE)
Brent <- read_delim("Brent.csv", 
    ";", escape_double = FALSE, trim_ws = TRUE)
names(SP)[1] <- "Date"
names(RTC)[1] <- "Date"
names(Gas)[1] <- "Date"
names(USD)[1] <- "Date"
names(Brent)[1] <- "Date"
RTC$Цена[RTC$Цена>500] <- 222.39
SP =as.data.frame(SP)
SP$Price_SP=as.numeric(SP$Price_SP)
SP =as.data.frame(SP)
Brent$Цена[Brent$Date=="2021-03-17"] <- 60

Визуализация

library(lubridate)

plot(SP$Price_SP,type="l")

plot(Brent$Цена,type="l")

plot(Gas$Цена,type="l")

plot(USD$Цена,type="l")

plot(RTC$Цена,type="l")

SandP <- plot_ly(SP, type = 'scatter', mode = 'lines')%>%
  add_trace(x = ~ Date, y = ~ Price_SP) %>%
  layout(showlegend = F, title='S&P',
         xaxis = list(rangeslider = list(visible = F)))
SandP

Brents <- plot_ly(Brent, type = 'scatter', mode = 'lines')%>%
  add_trace(x = ~ Date, y = ~ Цена) %>%
  layout(showlegend = F, title='Brent',
         xaxis = list(rangeslider = list(visible = T)))
Brents

RTCs <- plot_ly(RTC, type = 'scatter', mode = 'lines')%>%
  add_trace(x = ~ Date, y = ~ Цена) %>%
  layout(showlegend = F, title='RTC',
         xaxis = list(rangeslider = list(visible = T)))
RTCs

USDs <- plot_ly(USD, type = 'scatter', mode = 'lines')%>%
  add_trace(x = ~ Date, y = ~ Цена) %>%
  layout(showlegend = F, title='Курс',
         xaxis = list(rangeslider = list(visible = T)))
USDs

Gass <- plot_ly(Gas, type = 'scatter', mode = 'lines')%>%
  add_trace(x = ~ Date, y = ~ Цена) %>%
  layout(showlegend = F, title='Gas',
         xaxis = list(rangeslider = list(visible = T)))
Gass

Все в одном

full_data=inner_join(Brent,Gas,by="Date")
full_data=inner_join(full_data,RTC,by="Date")
full_data=inner_join(full_data,SP,by="Date")
full_data=inner_join(full_data,USD,by="Date")
full_data=full_data %>% select(Date,Цена.x,Цена.y,Цена.x.x,Price_SP,Цена.y.y)
colnames(full_data) <- c("Date","Brent", "Gas","RTC","SP","USD")

library(tidyquant)
library(plotly)

fig <- plot_ly(full_data, type = 'scatter', mode = 'lines')%>%
  add_trace(x = ~Date, y = ~ Brent*1000, name = 'Brent')%>%
  add_trace(x = ~Date, y = ~Gas*10000, name = 'Gas')%>%
  add_trace(x = ~Date, y = ~RTC*1000, name = 'RTC')%>%
  add_trace(x = ~Date, y = ~SP*10000, name = 'SP')%>%
  add_trace(x = ~Date, y = ~USD*1000, name = 'USD')%>%
  layout(title = 'Данные по индексам',yaxis = list(title = 'Стоимость в тыс.$'))
fig

fig <- plot_ly(full_data, type = 'scatter', mode = 'lines')%>%
  add_trace(x = ~Date, y = ~ Brent*1000, name = 'Brent')%>%
  add_trace(x = ~Date, y = ~Gas*10000, name = 'Gas')%>%
  add_trace(x = ~Date, y = ~RTC*1000, name = 'RTC')%>%
  add_trace(x = ~Date, y = ~SP*1000, name = 'SP')%>%
  add_trace(x = ~Date, y = ~USD*1000, name = 'USD')%>%
  layout(title = 'Данные по индексам',yaxis = list(title = 'y'),
         xaxis = list(dtick = "M3", tickformat="%b<br>%Y"), width = 1100)
options(warn = -1)
fig <- fig %>%
  layout(
         xaxis = list(zerolinecolor = '#ffff',
                      zerolinewidth = 1,
                      gridcolor = 'ffff'),
         yaxis = list(zerolinecolor = '#ffff',
                      zerolinewidth = 1,
                      gridcolor = 'ffff'),
         plot_bgcolor='#e5ecf6')

fig <- fig %>% layout(legend = TRUE)
fig

Проверка на стационарность: unit-root тест;AD

library(urca)
ur.df(SP$Price_SP, lags = 0, 
      type = c("none"), 
      selectlags = c("AIC"))

## 
## ############################################################### 
## # Augmented Dickey-Fuller Test Unit Root / Cointegration Test # 
## ############################################################### 
## 
## The value of the test statistic is: 0.4504

ur.df(RTC$Цена, lags = 0, 
      type = c("none"), 
      selectlags = c("AIC"))

## 
## ############################################################### 
## # Augmented Dickey-Fuller Test Unit Root / Cointegration Test # 
## ############################################################### 
## 
## The value of the test statistic is: -0.6254

ur.df(Gas$Цена, lags = 0, 
      type = c("none"), 
      selectlags = c("AIC"))

## 
## ############################################################### 
## # Augmented Dickey-Fuller Test Unit Root / Cointegration Test # 
## ############################################################### 
## 
## The value of the test statistic is: -1.2999

ur.df(Brent$Цена, lags = 0, 
      type = c("none"), 
      selectlags = c("AIC"))

## 
## ############################################################### 
## # Augmented Dickey-Fuller Test Unit Root / Cointegration Test # 
## ############################################################### 
## 
## The value of the test statistic is: -1.1418

ur.df(USD$Цена, lags = 0, 
      type = c("none"), 
      selectlags = c("AIC"))

## 
## ############################################################### 
## # Augmented Dickey-Fuller Test Unit Root / Cointegration Test # 
## ############################################################### 
## 
## The value of the test statistic is: 0.7633

qnorm(c(.1,.05,.01)/2)

## [1] -1.644854 -1.959964 -2.575829

Все данные превышает допустимый максимальный порог в 10%, поэтому мы можем утверждать, что изучаемые ряды нестационарны.

ADF-тест.

#perform augmented Dickey-Fuller test 
adf.test(SP$Price_SP)

## 
##  Augmented Dickey-Fuller Test
## 
## data:  SP$Price_SP
## Dickey-Fuller = -2.4594, Lag order = 10, p-value = 0.3839
## alternative hypothesis: stationary

adf.test(RTC$Цена)

## 
##  Augmented Dickey-Fuller Test
## 
## data:  RTC$Цена
## Dickey-Fuller = -2.8351, Lag order = 10, p-value = 0.2248
## alternative hypothesis: stationary

adf.test(Gas$Цена)

## 
##  Augmented Dickey-Fuller Test
## 
## data:  Gas$Цена
## Dickey-Fuller = -2.1383, Lag order = 11, p-value = 0.5198
## alternative hypothesis: stationary

adf.test(Brent$Цена)

## 
##  Augmented Dickey-Fuller Test
## 
## data:  Brent$Цена
## Dickey-Fuller = -2.4304, Lag order = 10, p-value = 0.3961
## alternative hypothesis: stationary

adf.test(USD$Цена)

## 
##  Augmented Dickey-Fuller Test
## 
## data:  USD$Цена
## Dickey-Fuller = -3.3849, Lag order = 10, p-value = 0.05604
## alternative hypothesis: stationary

Так же как и стестом на единичный корень-мы не можем отвергнуть нулевую гипотезу о нестационарности рядов, так как p-value> 0.05.

The Augmented Engle-Granger test: θ is unknown

Необходимо оценить коинтегрирующий коэффициент с помощью EG-ADF теста. Для начала предположим, что θ=1:

Выберем период, где визуально наблюдается самая сильная зависимость 2019:Q4-2020:Q4

library(dplyr)
times=full_data %>% filter(full_data$Date>"2019-01-01" & full_data$Date < "2020-12-31")
ur.df(window(times$RTC), 
      lags = 10, 
      selectlags = "AIC", 
      type = "drift")

## 
## ############################################################### 
## # Augmented Dickey-Fuller Test Unit Root / Cointegration Test # 
## ############################################################### 
## 
## The value of the test statistic is: -1.5337 1.1787

ur.df(window(times$SP), 
      lags = 10, 
      selectlags = "AIC", 
      type = "drift")

## 
## ############################################################### 
## # Augmented Dickey-Fuller Test Unit Root / Cointegration Test # 
## ############################################################### 
## 
## The value of the test statistic is: -1.1174 0.7969

ur.df(window(times$Brent), 
      lags = 11, 
      selectlags = "AIC", 
      type = "drift")

## 
## ############################################################### 
## # Augmented Dickey-Fuller Test Unit Root / Cointegration Test # 
## ############################################################### 
## 
## The value of the test statistic is: -1.2769 0.8289

ur.df(window(times$Gas), 
      lags = 10, 
      selectlags = "AIC", 
      type = "drift")

## 
## ############################################################### 
## # Augmented Dickey-Fuller Test Unit Root / Cointegration Test # 
## ############################################################### 
## 
## The value of the test statistic is: -1.8814 1.7812

ur.df(window(times$USD), 
      lags = 10, 
      selectlags = "AIC", 
      type = "drift")

## 
## ############################################################### 
## # Augmented Dickey-Fuller Test Unit Root / Cointegration Test # 
## ############################################################### 
## 
## The value of the test statistic is: -1.569 1.2776

ur.df(window(times$RTC) - window(times$SP), 
      lags = 6, 
      selectlags = "AIC", 
      type = "drift")

## 
## ############################################################### 
## # Augmented Dickey-Fuller Test Unit Root / Cointegration Test # 
## ############################################################### 
## 
## The value of the test statistic is: -1.5992 1.2861

ur.df(window(times$RTC) - window(times$Brent), 
      lags = 6, 
      selectlags = "AIC", 
      type = "drift")

## 
## ############################################################### 
## # Augmented Dickey-Fuller Test Unit Root / Cointegration Test # 
## ############################################################### 
## 
## The value of the test statistic is: -1.7242 1.4921

ur.df(window(times$RTC) - window(times$Gas), 
      lags = 6, 
      selectlags = "AIC", 
      type = "drift")

## 
## ############################################################### 
## # Augmented Dickey-Fuller Test Unit Root / Cointegration Test # 
## ############################################################### 
## 
## The value of the test statistic is: -1.5944 1.2806

ur.ers(window(times$RTC), 
      model="constant")

## 
## ############################################################### 
## # Elliot, Rothenberg and Stock Unit Root / Cointegration Test # 
## ############################################################### 
## 
## The value of the test statistic is: -1.3709

ur.ers(window(times$SP), 
       model="constant")

## 
## ############################################################### 
## # Elliot, Rothenberg and Stock Unit Root / Cointegration Test # 
## ############################################################### 
## 
## The value of the test statistic is: -0.1049

ur.ers(window(times$Brent), 
       model="constant")

## 
## ############################################################### 
## # Elliot, Rothenberg and Stock Unit Root / Cointegration Test # 
## ############################################################### 
## 
## The value of the test statistic is: -1.2773

ur.ers(window(times$Gas), 
       model="constant")

## 
## ############################################################### 
## # Elliot, Rothenberg and Stock Unit Root / Cointegration Test # 
## ############################################################### 
## 
## The value of the test statistic is: -1.797

ur.ers(window(times$USD), 
       model="constant")

## 
## ############################################################### 
## # Elliot, Rothenberg and Stock Unit Root / Cointegration Test # 
## ############################################################### 
## 
## The value of the test statistic is: -0.8621

#Spread
ur.ers(window(times$RTC) - window(times$SP),
       model = "constant", 
       lag.max = 10)

## 
## ############################################################### 
## # Elliot, Rothenberg and Stock Unit Root / Cointegration Test # 
## ############################################################### 
## 
## The value of the test statistic is: -1.736

ur.ers(window(times$RTC) - window(times$Brent),
       model = "constant", 
       lag.max = 10)

## 
## ############################################################### 
## # Elliot, Rothenberg and Stock Unit Root / Cointegration Test # 
## ############################################################### 
## 
## The value of the test statistic is: -1.6491

ur.ers(window(times$RTC) - window(times$Gas),
       model = "constant", 
       lag.max = 10)

## 
## ############################################################### 
## # Elliot, Rothenberg and Stock Unit Root / Cointegration Test # 
## ############################################################### 
## 
## The value of the test statistic is: -1.725

ur.ers(window(times$RTC) - window(times$USD),
       model = "constant", 
       lag.max = 10)

## 
## ############################################################### 
## # Elliot, Rothenberg and Stock Unit Root / Cointegration Test # 
## ############################################################### 
## 
## The value of the test statistic is: -1.6282

critical values for the levels
10%, 5% and 1% are −3.12, −3.41 and − 3.96.

Таким образом гипотеза о нестационарности разности (spread) не отвергается, для всех пар с индексом RTC.