Pairs Trading Signals and Positioning

I am currently working on a research project for a pairs trading strategy and would like to know the correct positions to take when a signal has been triggered.
Say we are using this equation to generate signals:
\begin{equation} z = y - \beta x \end{equation} $\mu_z$ = 0, $\sigma_z$ = 0.5, and $\beta$ = 1, for simplicity.
And our signals to open are: (z >= $\mu_z$+$\sigma_z$ & z <= $\mu_z$-$\sigma_z$)
Then a signal will be triggered when :
1) y = 10, x = 9.5
2) y = 10.5, x = 10
3) y = 10, x = 10.5
4) y = 9.5, x = 10

Should the positions taken in these scenarios be different? Should we short 1 share y and go long $\beta$ shares of x for 1 and 2? Should we go long 1 share y and short $\beta$ shares of x for 3 and 4?

Or should we actually be calculating pct change in y and $\beta$x and then short which ever had the greatest pct change and go long the other?

Answer

It should be consistent with the way you calculate $\beta:$ if you use stock returns to compute it, then you should be using returns to compute the spread and your signal, and aim to be cash neutral: N $\it{dollars}$ long of x and $\beta\times N$ $\it{dollars}$ short of y. If you regress the prices of x and y to obtain $\beta,$ then pretty much, what you wrote - N $\it{shares}$ of x and $\beta\times N$ $\it{shares}$ of y.