The problem is that your statement about the variance of the MA process isn't correct. If the AR coefficient of an AR(1) process, say, $\theta$, is equal to or greater than one, the error terms in the infinite series are multiplied by an increasing and unbounded sequence of numbers (as you go back in time), and consequently the variance of the sum is infinite. With $\theta < 1$, the terms in the infinite series are multiplied by a sequence that goes to zero fast enough so that the variance of the sum is finite. With $\theta = 1$, you have an unweighted sum of an infinite number of terms with a constant variance, so the variance of the sum is infinite.
Strict stationarity is the strongest form of stationarity. It means that the joint statistical distribution of any collection of the time series variates never depends on time. So, the mean, variance and any moment of any variate is the same whichever variate you choose. However, for day to day use strict stationarity is too strict. Hence, the following weaker definition is often used instead. Stationarity of order 2 which includes a constant mean, a constant variance and an autocovariance that does not depend on time. (second-order stationary or stationary of order 2). A weaker form of stationarity that is first-order stationary which means that the mean is a constant function of time, time-varying means to obtain one which is first-order stationary.
Using traditional stationarity tests such us PP.test (Phillips-Perron Unit Root Test), kpss test or Augmented Dickey-Fuller Tests are not adequate if you are to perform regression via other methods than ARIMA (due that in Arima the orders are fixed and that no other factors that produce non stationarity are included in the model). For non Arima cases stationarity tests in the frequency domain are more adequate.
Tests in the frequency domain : The Priestley-Subba Rao (PSR) test for nonstationarity (fractal package). Based upon examining how homogeneous a set of spectral density function (SDF) estimates are across time, across frequency, or both.
The test you refer to is a test also in the frequency domain (which tests a second order unit root test) where the wavelet looks at a quantity called βj(t) which is closely related to a wavelet-based time-varying spectrum of the time series (it is a linear transform of the evolutionary wavelet spectrum of the locally stationary wavelet processes of Nason, von Sachs and Kroisandt, 2000). So we see if βj(t) function varies over time or is constant by looking at Haar wavelet coefficients of the estimate so is stationary if all haar coefficients are zero (locits package).
There are other concerns about stationarity such us long range dependence, fractional integrated processes (ARFIMA) where the term d (differenciation) refers to long term memory processes.
The effect of higher order non stationarity, long term dependencies is that they are in effect reflected systematically in the errors of a regression, however its impact and thus validity of the regression is difficult to measure
Best Answer
Yes, weak stationarity requires both constant variance and constant mean (over time). To quote from wikipedia: A wide-sense stationary random processes only require that 1st moment (i.e. the mean) and autocovariance do not vary with respect to time.