MHB Mean and variance of difference operators on a time series process

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The discussion centers on calculating the mean and variance of the second difference operator applied to a time series process defined as Yt = mt + εt, where εt is an i.i.d. process. The user initially computes the second difference, resulting in εt - 2εt-1 + εt-2, leading to a mean of 0 and a variance of 4σ². However, the expected answer indicates a mean of 2b and a variance of 2σ², prompting confusion about the discrepancy. The user questions whether their approach or definitions were incorrect, particularly regarding the implications of the term 2b. Clarification on the calculations and definitions used in the original answer is sought to resolve the misunderstanding.
deba123
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$$\text{Consider the following decomposition of the time series }{Y}_{t}\text{ where }{Y}_{t}={m}_{t}+{\varepsilon}_{t},\text{ where }{\varepsilon}_{t}\text{ is a sequence of i.i.d }\left(0,{\sigma}^{2}\right)\text{ process. Compute the mean and variance of the process }{\nabla}_{2}{Y}_{t}\text{ when }:{m}_{t}=a+bt.$$
 
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Welcome to MHB! :D

Can you show what you have tried so far so our helpers have some idea where you are stuck and can then offer better help?
 
MarkFL said:
Welcome to MHB! :D

Can you show what you have tried so far so our helpers have some idea where you are stuck and can then offer better help?

$$O.K\: so\: here's\: what\: i\: did:
{\nabla}_{2}{Y}_{t}=\nabla(\nabla{Y}_{t})=\nabla(({Y}_{t})-({Y}_{t-1}))={Y}_{t}-2{Y}_{t-1}+{Y}_{t-2}.
Which\: after\: expanding\: comes\: out:\:
{\varepsilon}_{t}-2{\varepsilon}_{t-1}+{\varepsilon}_{t-2}.
So\: mean\: is\: 0 \:and\: variance\: is\: 4{\sigma}^{2}.$$
$$But\:the\:answer\: is :mean=2b\:and\: variance=2{\sigma}^{2}.$$ I don't get how the answer came about something like that. Please help.
 
Did the answer detail the computations?
 
girdav said:
Did the answer detail the computations?

No it didn't. But the answer could be wrong. I just want to know where I was wrong (or right) and may be if I was using the wrong definition. Thanks for your help.
 
The term $2b$ for the variance could be explained if we had considered $m_t=a+bt^2$, but I fail to see how the $2\sigma^2$ terms come from.
 

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