You'd better to read carefuly chapter 12 in Wooldridge "introductory Econometrics" 2nd Edn

Since beta1 changed sign, the huge difference between OLS and Cochrane-Orcutt or Prais-Winston is always an indication of problem. choice: try differenced series; work with OLS and Newy and West standard Error., And others...

Thanks a lot!can you tell me some details?

i think you should make a unit root test on Y, X and the residuals

test cointegration

Thanks a lot!can you tell me some details

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When it comes to the details, we need equations and formula! So You can try to read Wooldridge, he explains it very well, especially regarding the intuitions.