correlation between two random walk time series, that happen to be trended

timeseries.md

TimeSeries

Andrew Barr
July 6, 2015

create totally random time series

ts1 <- cumsum(rnorm(1000))
ts2 <- cumsum(rnorm(1000))
par(mfrow=c(1,2))
plot(ts1, type="l")
plot(ts2, type="l")

plot values of ts1 by ts2 and to regression

par(mfrow=c(1,1))
plot(ts2~ts1, pch=16)

summary(lm(ts2~ts1))

## 
## Call:
## lm(formula = ts2 ~ ts1)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -19.4187  -5.5092  -0.8079   5.5828  18.7287 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 14.18006    0.43241   32.79   <2e-16 ***
## ts1         -0.91694    0.01183  -77.54   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 7.449 on 998 degrees of freedom
## Multiple R-squared:  0.8576,	Adjusted R-squared:  0.8575 
## F-statistic:  6012 on 1 and 998 DF,  p-value: < 2.2e-16

difference data

plot(diff(ts2)~diff(ts1), pch=16)

summary(lm(diff(ts2)~diff(ts1)))

## 
## Call:
## lm(formula = diff(ts2) ~ diff(ts1))
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -2.9254 -0.7226  0.0105  0.6817  3.3244 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)  
## (Intercept)  0.074338   0.032598   2.280   0.0228 *
## diff(ts1)   -0.006398   0.031551  -0.203   0.8393  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.028 on 997 degrees of freedom
## Multiple R-squared:  4.124e-05,	Adjusted R-squared:  -0.0009617 
## F-statistic: 0.04112 on 1 and 997 DF,  p-value: 0.8393