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

AI Thread Summary
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
Messages
4
Reaction score
0
$$\text{Consider the following decomposition of the time series }{Y}_{t}\text{ where }{Y}_{t}={m}_{t}+{\varepsilon}_{t},\text{ where }{\varepsilon}_{t}\text{ is a sequence of i.i.d }\left(0,{\sigma}^{2}\right)\text{ process. Compute the mean and variance of the process }{\nabla}_{2}{Y}_{t}\text{ when }:{m}_{t}=a+bt.$$
 
Physics news on Phys.org
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.
 

Thread 'Value of intuitionistic logic'

I'm taking a look at intuitionistic propositional logic (IPL). Basically it exclude Double Negation Elimination (DNE) from the set of axiom schemas replacing it with Ex falso quodlibet: ⊥ → p for any proposition p (including both atomic and composite propositions). In IPL, for instance, the Law of Excluded Middle (LEM) p ∨ ¬p is no longer a theorem. My question: aside from the logic formal perspective, is IPL supposed to model/address some specific "kind of world" ? Thanks.
View full post »

Thread 'Formal derivation of statement from Peano Arithmetic system'

I was reading a Bachelor thesis on Peano Arithmetic (PA). PA has the following axioms (not including the induction schema): $$\begin{align} & (A1) ~~~~ \forall x \neg (x + 1 = 0) \nonumber \\ & (A2) ~~~~ \forall xy (x + 1 =y + 1 \to x = y) \nonumber \\ & (A3) ~~~~ \forall x (x + 0 = x) \nonumber \\ & (A4) ~~~~ \forall xy (x + (y +1) = (x + y ) + 1) \nonumber \\ & (A5) ~~~~ \forall x (x \cdot 0 = 0) \nonumber \\ & (A6) ~~~~ \forall xy (x \cdot (y + 1) = (x \cdot y) + x) \nonumber...
View full post »

Similar threads

I Random variable vs Random Process
Replies
2
Views
2K
I Why does normal distribution turn into t distribution when variance is unknown?
Replies
5
Views
5K
MHB A math proof within a question about homogeneous Poisson process
Replies
1
Views
2K
A Need help about a demo with inverse weighted variance average
Replies
1
Views
1K
I Maximum likelihood to fit a parameter of this model
Replies
1
Views
2K
MHB Likelihood Ratio Test for Common Variance from Two Normal Distribution Samples
Replies
1
Views
3K
The average of a random process
Replies
3
Views
2K
I The CDF from the characteristic function
Replies
36
Views
4K
I How to calculate expectation and variance of kernel density estimator?
Replies
9
Views
3K
A Computing a variance in astrophysics context
Replies
1
Views
1K

Hot Threads

Recent Insights

Back
Top