Both variables (dependent and independent) show autocorrelation effects. Data is time-series and stationary
When I run the regression residuals appear not to be correlated.
My Durbin-Watson statistic is greater than upper critical value, so there is an evidence that error terms are not positively correlated. Also when I plot ACF for errors it looks like there is no correlation there and Ljung-Box statistic is smaller than critical value.
Can I trust my regression output, are the t-statistics reliable?
and 
and 
. Since anomalies are present the appropriate regression needs to take into account these effects. Following are the three models ( including any necessary lag structures in the two inputs ) and the appropriate ARIMA structure obtained from an automatic transfer function run using AUTOBOX ( a piece of software I have been developing for the last 42 years )
and 
and 
. We now take the three cleansed series returned from the modelling process and estimate a minimally sufficient common model which in this case would be a comtemporary and 1 lag PDL on tweets and a contemporary PDL on wiki with an ARIMA of (1,0,0)(0,0,0). Estimating this model locally and globally provides insight as to the commonality of coefficients .
with coefficients 
. The test for commonality is easily rejected with an F value of 79 with 3,291 df. Note that the DW statistic is 2.63 from the composite analysis. The summary of coefffici
ents is presented here. The OP poster reflected that the only software he has access to is insufficient to be able to answer this thorny research question.
Best Answer
The t-statistics are reliable in the absence of autocorrelation of the errors. The fact that the residuals don't display significant autocorrelation indicates, in a not terribly rigorous way, that the autocorrelation in your dependent variable is due to the autocorrelation in your independent variable. However, it's also important to remember that the difference between statistical significance and insignificance is not itself statistically significant in many cases, e.g., a t-statistic of 1.8 vs. a t-statistic of 2.8 is a difference of 1.0, hence the lack of rigor in the statement above.
An alternative approach would be to model the data using time series analysis techniques, which, for R, are very briefly described in CRAN task view: Time Series Analysis. These techniques can get you sharper parameter estimates by explicitly modeling cross-time correlation structures, whereas, if you don't model them explicitly, you are implicitly assuming that the only such structure in the data is due to the independent variable.