Did I calculate the expected value and the variance correctly? I have the following assignment to solve but I'm not sure if I solved it correctly.

Questions


  Let the stochastic process $(Y_t)_t$ be defined by $Y_t = \mu + Y_{t-1} + \varepsilon _t$ with $(\varepsilon _t)_t\sim \mathrm{WN}(0,1)$.
  
  a) Calculate the expected value and the variance of $(\Delta Y_t)_t$.
  
  b) Prove that $(\Delta Y_t)_t\sim \mathrm{MA}(1)$ and calculate the autocovariance function of $(\Delta Y_t)_t$.


My solutions

a) \begin{eqnarray}
Y_{t+1} &amp;=&amp; \mu + Y_t + \varepsilon_{t+1}\\[1ex]
\implies \Delta Y_t = Y_{t+1}-Y_t &amp;=&amp; \mu +\varepsilon_{t+1}\\[1ex]
\mathrm E(\Delta Y_t ) &amp;=&amp; \mu + \mathrm E(\varepsilon_{t+1}) = \mu \hspace{6cm} \\[1ex]
\mathrm {Var}(\Delta Y_t) &amp;=&amp; \mathrm{Var}(\varepsilon _{t+1}) = 1
\end{eqnarray}

b) \begin{eqnarray}
(\Delta Y_t)_t &amp;=&amp; \mu + \varepsilon_{t+1} + 0\cdot \varepsilon_t \\[1ex]
\implies \mathrm{ACV} &amp;=&amp; \mathrm{Cov}(\Delta Y_t, \Delta Y_{t-h})\\
 &amp;=&amp; \mathrm{Cov}(\varepsilon_{t+1}, \varepsilon_{t-h+1}) = \left\{\begin{array}{ll}
1 &amp; h=0 \\
0 &amp; \text{otherwise}
\end{array}\right.
\end{eqnarray}

What do you think?
He who formulated the exercise is wrong, and this is why I am posting a full answer to a homework question.
This is a classic example where manipulating recursive relations can lead to different representations that may appear "different" and with different properties.  

$$\Delta Y_t \equiv Y_{t}-Y_{t-1} =  \mu + Y_{t-1} + \varepsilon_{t} - \mu - Y_{t-2} - \varepsilon_{t-1}$$

$$\implies \Delta Y_t = \Delta Y_{t-1} - \Delta \varepsilon_{t} \tag{1}$$

At the same time 

$$Y_t = \mu + Y_{t-1} + \varepsilon _t \implies \Delta Y_t = \mu +\varepsilon _t \tag{2}$$

The right hand side of Eqs. $(1)$ and $(2)$ represent the same process. 

In any case, neither of these are $MA(1)$.

Moving forward, which one to choose?  

A soft adoption of Occam's razor indicates that eq. $(2)$ is the simplest. A bit more specifically, we note that the manipulation resulting in $(1)$ hasn't "saved" us from the existence of a unit root and non-stationarity in a clear way. 

Both therefore suggest to adopt $(2)$, that says that  $\{\Delta Y_t\}_t$ is the sum of constant and a $WN$ process.