For the ARIMA (0,0,1) model, some books write the equation as $$Z_t = \mu - \theta Z_{t-1}$$ whereas some books write the equation as
$$Z_t = \mu + \theta Z_{t-1}$$
Why is there either a negative sign or positive sign before the moving average parameter in an ARIMA model?
Purely personal preference. Note that in both cases $\theta$ is any real number so the sign does not matter. It is much common though to use plus sign for the MA part and negative signs for AR part. The latter preference can be explained with the following example. Suppose process $X_t$ satisfies the ARIMA(1,0,1) equation:
$$X_t-\rho X_{t-1}=Z_t+\theta Z_{t-1}$$
Then
$$X_t=\rho X_{t-1}+Z_t+\theta Z_{t-1}$$
So all the signs on the right hand side are positive. Mathematically this does not matter, but many people find all the positive signs aesthetically pleasing.
