I am currently studying the quantitative methods section and can’t seem to understand whether a linear trend model has to be covariance stationary, even if the DW test shows a result of 2 (fail reject the null hypothesis of no serial correlation).
In other words, if the times series x(t) = b0 + b1.t shows an upward-sloping line without showing serial correlation, is it OK to use, or do we still need to make it AR(1)?
DW test is a test for serial correlation in the regression residuals, which would violate one of the regression assumptions (that the residuals are independent).
Stationarity of the regressors is a different, but very important property. Basically, one of the assumptions of linear regression is that the model you’re estimating is linear, with residuals independent, identically distributed normal random variables with mean 0 and constant variance. Note that (most of the time) its impossible for a linear combination of nonstationary things to be stationary so having your regressors be nonstationary would imply that your residuals couldn’t possibly satisfy the above, and therefore the model can’t be linear. So this is a modeling error that’s distinct from serial correlation.
Now notice I said “most of the time” above. I can’t remember if this is covered in Level 2 or not, but there’s the notion of nonstationary time series being cointegrated. Being cointegrated basically means that there is a linear combination that IS actually stationary and so regression is still valid. So to test the validity of the model, you’d start by testing the variables for being stationary and if they fail then test cointegration. If that fails too, then you need to change your model (people commonly do so by taking differences).
To this exact question, a linear trend model is definitely not stationary. Easiest way to see that is that stationary processes need to have a mean that is constant in time, which would not be the case with your process.
Usually, however, when a process has a trend, people talk about “stationary with trend” which means if you detrend the series, what’s left is stationary. In that case, assuming the only other terms in your series are normally distributed epsilons, then your series would be stationary with trend. Series that are stationary with trend are fine to use in regression.