What is the mean and variance of an MA(4) process with time drift? For a project I need to simulate a relatively smooth time series with an upward drift.
This is what I came up with:
$$y_t = \mu + \alpha t+ \sum_{j=0}^{q=4} \theta_{j} e_{t-j}$$ where
$\theta_j = 0.2^{j}$, $\mu = 1$, $\alpha=0.01$ and I seeded the errors to be $e_t \sim N(0, 1)$
How would one estimate the mean and variance for this? I believe it is weakly stationary but not sure how to estimate it at a particular time point.
Here is a graph of one realization with T=100:


Moving-average models are always weakly stationary because they are finite linear combinations of a white noise sequence for which the first two moments are time invariant. For example, consider the $MA(1)$ model
$$r_{t}=c_{0}+a_{t}-\theta_{1}a_{t-1} \quad \text{or} \quad r_{t}=c_{0}+(1-\theta_{1}B)a_{t}$$
Taking expectation of the model, we have $$\mathbb{E}(r_{t})=c_{0}$$
which is time invariant of MA(1). Taking the variance we have :
$$Var(r_{t})=\sigma_{a}^2+\theta_{1}^2\sigma_{a}^2 = (1+\theta_{1}^2)\sigma_{a}^2.$$
where we use the fact that $a_t$ and $a_{t−1}$ are uncorrelated. Again, $Var(r_t)$ is time invariant. Your question and the previous statement applies to general $MA(q)$ models, and we obtain two general properties. First, the constant term of an $MA$ model is the mean of the series [i.e., $\mathbb{E}(r_{t})=c_{0}$]. And second in general, the variance of an $MA(q)$ model is:
$$Var(r_{t})=(1+\theta_{1}^2+\theta_{2}^2 \dots +\theta_{q}^2)\sigma_{a}^2.$$
Edit
This $ΜΑ(4)$ model is from Hamilton's book "Time Series Analysis" page 50.I have added the $a$ term in the example.Hope that answers your question.
# MA(4) :Y_t= e_t - 0.6e_t-1,_ 1 + 0.3e_t- 2 - 0.5e_t-3 + 0.5e_t-4
n_samples = 100
θ1 = -0.6
θ2 = 0.3
θ3 = -0.5
θ4 = 0.5
μ = 5
α = 0.01
y = ε = np.random.normal(0,1,size=n_samples)
for t in range(n_samples):
    y[t] = μ +α*t +ε[t] + θ1*ε[t-1] + θ2*ε[t-2] + θ3*ε[t-3] + θ4*ε[t-4]
from statsmodels.graphics import tsaplots
fig = tsaplots.plot_acf(y, lags=12)

