Durbin Watson test statistic I applied the DW test to my regression model in R and I got a DW test statistic of 1.78 and a p-value of 2.2e-16 = 0.  
Does this mean there is no autocorrelation between the residuals because the stat is close to 2 with a small p-value or does it mean although the stat is close to 2 the p-value is small and thus we reject the null hypothesis of there existing no autocorrelation?
 A: If you believe the DW test, then yes, it indicates that you have serial correlation. However, remember that language of hypothesis testing you can never accept anything, you can only fail to reject it. 
Further the DW test requires the full set of classical linear model assumptions, including normality and unbiasedness in order to have any power. Almost no real life application can reasonable assume this, and therefore you will hard a time convincing others about its validity. There are many much simpler (and more robust) tests to use instead of the DW, you should use these!
Of course the easy solution is to just to compute robust standard errors, for instance newey-west (which is easy to do in R), then you can simply ignore the problem
A: In R, the function durbinWatsonTest() from car package verifies if the residuals from a linear model are correlated or not:


*

*The null hypothesis ($\text{H}_0$) is that there is no correlation among residuals, i.e., they are independent.

*The alternative hypothesis ($\text{H}_a$) is that residuals are autocorrelated.


As the p value was near from zero it means one can reject the null. 
A: The Durbin Watson test looks to check for both positive and negative autocorrelation but for first order only. It should not be used for data that is autocorrelated beyond the 1st order. The following link shows both the hypothesis as well as inference
https://www.statisticshowto.datasciencecentral.com/durbin-watson-test-coefficient
From this website:
"The Hypotheses for the Durbin Watson test are:
H0 = no first order autocorrelation.
H1 = first order correlation exists.
The Durbin Watson test reports a test statistic, with a value from 0 to 4, where the rule of thumb is:
2 is no autocorrelation.
0 to <2 is positive autocorrelation (common in time series data).
>2 to 4 is negative autocorrelation (less common in time series data).

A rule of thumb is that test statistic values in the range of 1.5 to 2.5 are relatively normal. "
Note that to get a more precise conclusion, we should not just rely on the DW statistic, but rather look at the p-value. Software packages like SAS will give 2 p-values - one for test for positive first order autocorrelation and the second one for the test for negative first order autocorrelation (both p-values add upto 1). If both p-values are more than your selected Alpha (0.05 in most cases), then we can not reject the null hypothesis that "no first order autocorrelation exists.
If any one of the p-values is < 0.05 (or selected Alpha), then we know that the corresponding alternate hypothesis is true (with 1- Alpha certainty).
I hope that helps. 
A: dwtest tests against the alternative hypothesis instead to the null hypothesis. So if the p-value is bellow the level you say, then it means it accepts the alternative hypothesis and rejects the null hypothesis.
A: The p-value is the lower α (significance level or alpha level) for which you should reject the null hypothesis.
It's just a red line: if you're ok with α = 0.1, α = 0.05, α = 0.01 or any α > 2.2e-16, well, it doesn't matter. This p-value ensures that the null hypothesis must be rejected and you don't need to test again and again for each level.
The same thing to other tests and p-values. But you may not forget what are the null and the alternative hypothesis.
