Is a high $R^2$ and F statistic evidence of autocorrelation?

Is a high $R^2$ and $F$ statistic between numeric data and its corresponding time evidence that the variable is serially autocorrelated? For time series analysis, is it necessary to perform a test like the Durbin-Watson to determine if there is first-order autocorrelation? Does $R^2$ and $F$ statistics give any indication?