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I'm not sure where my confusion is stemming from, but it seems that equivalent tests (Durbin Watson, and a simple significance test) for serial correlation in the errors (of lag 1) sometimes yield different results. Consider the series,

$$Y=\beta_0+ \beta_1x_t + \epsilon_t$$

As I understand it, the Durbin watson test, is performed as follows (with hypothesis spelled out)

$$\epsilon_t= \phi \epsilon_{t-1}+v_t, \hspace{0.2cm} v_t \sim \text{iid}$$ $$H_0: \phi=0 \hspace{0.2cm} \textrm{(no serial correlation)},\hspace{0.2cm} H_1: \phi \neq 0 \hspace{0.2cm} \textrm{(serial correlation)}$$ $$DW=\frac{\sum_{t=2}^T(\hat{\epsilon}_t-\hat{\epsilon}_{t-1})^2}{\sum_{t=2}^T\hat{\epsilon}_t^2}$$

Moreover, it seems to me that the Durbin Watson test test is equivalent to testing the hypothesis that $\phi=0$, which is done by first estimating $\phi$ by OLS in the following model,

$$\hat{\epsilon_t}= \phi \hat{\epsilon_{t-1}}$$

and subsequently testing whether $\phi =0$. However, I've seen these tests yield different conclusions (i.e. failure to reject with DW and a very significant coefficient for the autoregressive model of predicted residuals). What gives?

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