Figures

Example 1

Spirious correlations

Example 2

• consider time series \(\{x_t\}\) and \(\{y_t\}\), following two independent random walks

\(x_t = x_t-1 + \varepsilon_x,t\) \(y_t = y_t-1 + \varepsilon_y,t\) where \(\varepsilon_x,t ~ N(0, \varsigma^2_x), \varepsilon_y,t~ N(0, \varsigma^2_y)\)

• {xt} and {yt} are thus unrelated

• it can be shown that if \(t \rightarrow \infty\) then \(\beta_1 \not\rightarrow 0\) and its \(t\)-statistics \(\rightarrow \pm \infty\)

Example 2

library(zoo)

set.seed(42)
n <- 500
x <- as.ts( cumsum( rnorm(n, mean=0, sd=1) ) )
y <- as.ts( cumsum( rnorm(n, mean=0, sd=1) ) )

Example 2

Correlation coefficient between yt and xt is \(\varrho_y,x\) = 0.6746103

Example 2

• consider linear regression model \[ y = \beta_0 + \beta_1 x + \varepsilon \]

• \(\hat \beta_1=1.5954\) is highly signifficant with \(p\)-value \(1.1643\times 10^{-67}\)

  m1 <- lm(y~x)
  summary(m1)

  ## 
  ## Call:
  ## lm(formula = y ~ x)
  ## 
  ## Residuals:
  ##     Min      1Q  Median      3Q     Max 
  ## -17.170  -7.887  -0.464   7.566  18.375 
  ## 
  ## Coefficients:
  ##             Estimate Std. Error t value Pr(>|t|)    
  ## (Intercept)  -3.2330     0.5857   -5.52  5.5e-08 ***
  ## x             1.5954     0.0782   20.39  < 2e-16 ***
  ## ---
  ## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
  ## 
  ## Residual standard error: 9.4 on 498 degrees of freedom
  ## Multiple R-squared:  0.455,  Adjusted R-squared:  0.454 
  ## F-statistic:  416 on 1 and 498 DF,  p-value: <2e-16