Kointegrace

Priklad 1

consum=read.csv("C:/Users/Lukas/Desktop/Ekonometrie/AKM/NEW!!!/Cviceni/kointegrace/consum.csv")

Consumer Price Index for All Urban Consumers: All Items, Index 1982-1984=100, Monthly, Seasonally Adjusted
M2 Money Stock, Billions of Dollars, Monthly, Seasonally Adjusted

\(log(CPI)_t=\beta_0+\beta_1 log(M2)_t+\epsilon_t\)

par(mfrow=c(2,1))
plot.ts(GDP, main="M2")
plot.ts(Cons, main="CPI")

adf.test(con_per)

## 
##  Augmented Dickey-Fuller Test
## 
## data:  con_per
## Dickey-Fuller = -1.9524, Lag order = 2, p-value = 0.5905
## alternative hypothesis: stationary

adf.test(gdp_per)

## 
##  Augmented Dickey-Fuller Test
## 
## data:  gdp_per
## Dickey-Fuller = -2.8191, Lag order = 2, p-value = 0.2604
## alternative hypothesis: stationary

Zkusíme jeste dalsí test na stacionaritu. KPSS test. POZOR u KPSS testu je alternativní hypotéza unit roor!

kpss.test(gdp_per, null = "Trend")

## Warning in kpss.test(gdp_per, null = "Trend"): p-value greater than printed
## p-value

## 
##  KPSS Test for Trend Stationarity
## 
## data:  gdp_per
## KPSS Trend = 0.10581, Truncation lag parameter = 1, p-value = 0.1

kpss.test(con_per, null = "Trend")

## 
##  KPSS Test for Trend Stationarity
## 
## data:  con_per
## KPSS Trend = 0.15399, Truncation lag parameter = 1, p-value =
## 0.04334

Prikloníme se k hypotéze, ze CPI není trendove stacionární casová rada. Obsahuje tedy minimálne jeden jednotkový koren. Musíme zjisti, kolik jich má.

Abychom se vyznali v kodu, tak si radeji jiz zadefinujeme logaritmus. Zároven se casto setkáte v makroekonomii, ze jsou malými písmeny znaceny logaritmus promenných.

adf.test(diff(gdp_per))

## 
##  Augmented Dickey-Fuller Test
## 
## data:  diff(gdp_per)
## Dickey-Fuller = -3.2431, Lag order = 2, p-value = 0.09957
## alternative hypothesis: stationary

adf.test(diff(con_per))

## 
##  Augmented Dickey-Fuller Test
## 
## data:  diff(con_per)
## Dickey-Fuller = -2.2277, Lag order = 2, p-value = 0.4856
## alternative hypothesis: stationary

kpss.test(diff(con_per))

## Warning in kpss.test(diff(con_per)): p-value greater than printed p-value

## 
##  KPSS Test for Level Stationarity
## 
## data:  diff(con_per)
## KPSS Level = 0.15019, Truncation lag parameter = 1, p-value = 0.1

kpss.test(diff(gdp_per))

## Warning in kpss.test(diff(gdp_per)): p-value greater than printed p-value

## 
##  KPSS Test for Level Stationarity
## 
## data:  diff(gdp_per)
## KPSS Level = 0.07965, Truncation lag parameter = 1, p-value = 0.1

adf.test(diff(GDP))

## 
##  Augmented Dickey-Fuller Test
## 
## data:  diff(GDP)
## Dickey-Fuller = -3.2528, Lag order = 2, p-value = 0.09822
## alternative hypothesis: stationary

adf.test(diff(Cons))

## 
##  Augmented Dickey-Fuller Test
## 
## data:  diff(Cons)
## Dickey-Fuller = -2.2357, Lag order = 2, p-value = 0.4826
## alternative hypothesis: stationary

Zdá se, ze obe promenné jsou jiz stacionární. U \(\Delta gdp_per\) muzeme H0 zamítnout pro \(\alpha<0.01\) a u \(\Deltacpi\) pro \(\alpha<0.05\) . Tedy, m2 a cpi jsou procesy integrovany rádu 1, I(1).

Ekonomická teorie, nám ríká, ze by mel existovat slouhodobý vztah mezi spotrebou a duchodem. Pojdme to tedy otestovat.

dl_vztah=lm(con_per~gdp_per)
summary(dl_vztah)

## 
## Call:
## lm(formula = con_per ~ gdp_per)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -746.75 -350.89  -14.82  329.58 1003.94 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -2.243e+03  4.106e+02  -5.463 1.29e-05 ***
## gdp_per      8.787e-01  1.007e-02  87.222  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 515.6 on 24 degrees of freedom
## Multiple R-squared:  0.9969, Adjusted R-squared:  0.9967 
## F-statistic:  7608 on 1 and 24 DF,  p-value: < 2.2e-16

Provedeme residuálné test kointegrace. Pokud totiz jsou obe promenné kointegrované, tak vznikla residua musí být I(0), neboli nesmi obsahovat jednotkový koren, neboli musí být stacionární.

residua_con=dl_vztah$residuals
adf.test(residua_con)

## 
##  Augmented Dickey-Fuller Test
## 
## data:  residua_con
## Dickey-Fuller = -2.9263, Lag order = 2, p-value = 0.2195
## alternative hypothesis: stationary

kpss.test(residua_con)

## Warning in kpss.test(residua_con): p-value greater than printed p-value

## 
##  KPSS Test for Level Stationarity
## 
## data:  residua_con
## KPSS Level = 0.13282, Truncation lag parameter = 1, p-value = 0.1

ECM

resid_1=c(residua_con[2:length(residua_con)])

dcon=diff(con_per)
dgdp=diff(gdp_per)

data=data.frame(dcon,dgdp,resid_1)

ecm=lm(diff(con_per)~diff(gdp_per)+resid_1, data = data)
summary(ecm)

## 
## Call:
## lm(formula = diff(con_per) ~ diff(gdp_per) + resid_1, data = data)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -504.09 -135.26  -53.16  132.41  505.61 
## 
## Coefficients:
##                Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   2.995e+02  1.152e+02   2.600   0.0163 *  
## diff(gdp_per) 6.135e-01  8.135e-02   7.542 1.55e-07 ***
## resid_1       5.878e-04  1.191e-01   0.005   0.9961    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 246.3 on 22 degrees of freedom
## Multiple R-squared:  0.7912, Adjusted R-squared:  0.7723 
## F-statistic: 41.69 on 2 and 22 DF,  p-value: 3.281e-08

Kointegrace

Lukas Fryd

6 December 2017

Priklad 1

ECM