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The Durbin-Watson statistic to detect autocorrelation in the error terms ranges from 0 to 4. Currently, I am working out why it cannot exceed 4 analytically. The lower boundary case is obvious considering the actual statistic which is

$$DW = \frac{\sum\limits_{t=2}^{T}(\hat{\epsilon}_t - \hat{\epsilon}_{t-1})^2}{\sum\limits_{t=1}^{T} \hat{\epsilon}_t^2};$$

but for the upper boundary case, i.e. DW < 4, I ran in one particular difficulty. After expansion of the numerator I get

$$\sum\limits_{t=2}^{T}(\hat{\epsilon}_t - \hat{\epsilon}_{t-1})^2 = \sum\limits_{t=2}^{T} \hat{\epsilon}_t^2 - 2\hat{\epsilon}_t\hat{\epsilon}_{t-1} + \hat{\epsilon}_{t-1}^2 = \sum\limits_{t=2}^{T} \hat{\epsilon}_t^2 -2 \sum\limits_{t=2}^{T} \hat{\epsilon}_t\hat{\epsilon}_{t-1} + \sum\limits_{t=2}^{T} \hat{\epsilon}_{t-1}^2.$$

Obviously the last and first term are both less or equal than $\sum\limits_{t=1}^{T}\epsilon_t^2$ since they miss a non-negative term. Hence,

$$\sum\limits_{t=2}^{T} \hat{\epsilon}_t^2 + \sum\limits_{t=2}^{T} \hat{\epsilon}_{t-1}^2 \leq 2\sum\limits_{t=1}^{T} \hat{\epsilon}_t^2.$$

What I can't seem to figure out analytically is why necessarily

$$-2\sum\limits_{t=2}^{T}\hat{\epsilon}_t\hat{\epsilon}_{t-1} \leq 2 \sum\limits_{t=1}^{T} \hat{\epsilon}_t^2$$

must hold, i.e. why

$$0 \leq \sum\limits_{t=1}^{T} \hat{\epsilon}_t^2 + \sum\limits_{t=2}^{T} \hat{\epsilon}_t\hat{\epsilon}_{t-1},$$

considering that the error terms might be negative? Surely, there is some kind of estimation that yields the result but what is it?

Richard Hardy
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Taufi
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  • the mysterious term is, I believe the correlation of the noise process with the noise process lagged by 1 ($\epsilon_t$ has mean zero. The min value of correlation -1, so the max of -2*(mystery term) is 2 . – meh Jun 10 '17 at 15:22
  • Yes, the term itself is actually not mysterious at all, it is just the autocorrelation. I was under the impression that autocorrelation is only in the boundaries of +1 and -1 in case it is somehow normalized. Here, the residuals might take any value. – Taufi Jun 10 '17 at 15:31
  • See this answer https://stats.stackexchange.com/questions/153286/understanding-the-durbin-watson-test/153310#153310 – Christoph Hanck Jun 10 '17 at 15:42
  • Cauchy-Schwarz, of course. How could I have not seen this! Thank you, Christoph Hanck. – Taufi Jun 10 '17 at 15:46
  • The fact that correlation varies between +/-1 is the Cauchy Schwarz inequality. – meh Jun 10 '17 at 15:55

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