Autocorrelation¶
Durbin-Watson Test¶
Positive¶
- \(H_0: \rho = 0\)
- \(H_1: \rho > 0\) (not negative)
\[ \begin{aligned} D &= \dfrac{ \sum_{i=p}^n (u_i - u_{i-p})^2 }{ \sum_{i=1}^n (u_i)^2 } \\ & \approx 2(1-\rho) \end{aligned} \]
where \(p=\) lag being tested
- If \(D>D_h\), cannot reject null hypothesis
- If \(D<D_l\), reject null hypothesis
-
If \(D_l < D<D_h\), inconclusive
-
\(D \ge 2 \implies\) no autocorrelation
- \(D \to 0 \implies\) perfect autocorrelation
Negative¶
\(D' = 4-D\)
Runs Test¶
Run: any sequence on the same side of 0
Usually one-tailed to test for +ve correlation
- +ve correlation: bounces less frequently
- -ve correlation: bounces very frequently; (not very common in data)
\[ \begin{aligned} \bar R &= \dfrac{2 n_+ n_-}{n} + 1 \\ s^2_R &= \dfrac{2 n_+ n_- (2 n_+ n_- - n)}{n^2 (n-1)} \\ \implies Z_R &= \dfrac{R-\bar R}{s_R} \sim N(0, 1) \end{aligned} \]
where
- \(R=\) number of runs in data
- \(n_+=\) number of +ve residuals
- \(n_-=\) number of -ve residuals
- \(n=\) total number of residuals