I really think this is a good question and deserves an answer. The link provided is written by a psychologist who is claiming that some home-brew method is a better way of doing time series analysis than Box-Jenkins. I hope that my attempt at an answer will encourage others, who are more knowledgeable about time series, to contribute.
From his introduction, it looks like Darlington is championing the approach of just fitting an AR model by least-squares. That is, if you want to fit the model
$$z_t = \alpha_1 z_{t-1} + \cdots + \alpha_k z_{t-k} + \varepsilon_t$$
to the time series $z_t$, you can just regress the series $z_t$ on the series with lag $1$, lag $2$, and so on up to lag $k$, using an ordinary multiple regression. This is certainly allowed; in R, it's even an option in the ar function. I tested it out, and it tends to give similar answers to the default method for fitting an AR model in R.
He also advocates regressing $z_t$ on things like $t$ or powers of $t$ to find trends. Again, this is absolutely fine. Lots of time series books discuss this, for example Shumway-Stoffer and Cowpertwait-Metcalfe. Typically, a time series analysis might proceed along the following lines: you find a trend, remove it, then fit a model to the residuals.
But it seems like he is also advocating over-fitting and then using the reduction in the mean-squared error between the fitted series and the data as evidence that his method is better. For example:
I feel correlograms are now obsolescent. Their primary purpose was to
allow workers to guess which models will fit the data best, but the
speed of modern computers (at least in regression if not in
time-series model-fitting) allows a worker to simply fit several
models and see exactly how each one fits as measured by mean squared
error. [The issue of capitalization on chance is not relevant to this
choice, since the two methods are equally susceptible to this problem.]
This is not a good idea because the test of a model is supposed to be how well it can forecast, not how well it fits the existing data. In his three examples, he uses "adjusted root mean-squared error" as his criterion for the quality of the fit. Of course, over-fitting a model is going to make an in-sample estimate of error smaller, so his claim that his models are "better" because they have smaller RMSE is wrong.
In a nutshell, since he is using the wrong criterion for assessing how good a model is, he reaches the wrong conclusions about regression vs. ARIMA. I'd wager that, if he had tested the predictive ability of the models instead, ARIMA would have come out on top. Perhaps someone can try it if they have access to the books he mentions here.
[Supplemental: for more on the regression idea, you might want to check out older time series books which were written before ARIMA became the most popular. For example, Kendall, Time-Series, 1973, Chapter 11 has a whole chapter on this method and comparisons to ARIMA.]
You can do anything you want, especially if it's a term paper or something of that nature.
To obtain useful results you can't use nonstationary data with OLS and time series. There are other more advanced methods where nonstationarity is a non issue. With OLS you have to difference real GDP and indices, and also apply log transform in many cases.
UPDATE: when using non stationary variables in OLS you run into the potentially fatal issue of spurious regression, there's a ton of literature on this subject. Basically, your regression results will turn out garbage in most cases. You may see very significant coefficients, but the significance is artificial, and disappears when you run a proper regression.
There's even more subtle phenomenon called "cointegration", but since you're working on undergrad paper, I would not worry about it. As a matter of fact, if your major is not statistics or econometrics, I would imagine your instructor will not penalize you for improper use of regressions.
Clarification: you can use non-stationary data with OLS if the series are cointegrated. However, when doing so you better show that the series are cointegrated indeed, then adjust the parameter covariance matrix accordingly if you need inference. The parameters themselves would be fine. As I mentioned in original answer this is advanced concepts that are usually outside undegrad courses.
Best Answer
I solved my issue: If you train a CART tree only with the time series data (univariate) and validate the model with the time series test part (also univariate), you will get a pretty low error rate.
The problem is, that you need independent variables (and not the time series target value itself) for further forecasting as an input. Otherwise you try to forecast data on data you don't have. That doesn't work.