What is the difference between Random Walk & Unit root

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Saleh.alabdali

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Is my understanding is correct related to the difference between Random Walk & Unit root?
The image is not clear on my head I’m confused!!!
Any assitant
Q1:
Random walk is a special case of AR1, where b0=0 & b1=1.
Therefore, MRL=0/(1-1)=0 undefine.
A stock that follows a random walk has no mean reversion level, so you can’t predict which direction it is likely to trend in the future. Is that right???????.
Random walk with a drift: b0=1
If a time series is a random walk wit a drift, the next predication for it’s next value is last value.Is that right???????.
By definition, all random walks have a unit root.
when AR1 model b0=1 we call it Unit root.
Therefore, MRL=1/(1-1)=0 undefine.
Example:
A stock that follows a random walk with unit root, so the predication for it’s next value is last value?????
Q2?
The first difference for random walk is Yt=change in Xt
Using first difference make the time series coverance stationary with finite MRL=0/1-0=0 = finite????
What the difference between:
MRL=0/(1-1)=0 undefine
MRL=0/1-0=0 = finite
What make them differ???

Q3?
Is the following is right?
Unit root b0=1 it could be random walk when b1=1??
Q4?
Why stock that follows a random walk is defined as having a unit root.
It may be b1=0 not 1????
 
Let
Xt = b0 + b1×X(t-1) + εt
If b1 = 1, the model has a unit root, and the mean-reverting level is undefined.
If b1 = 1 and b0 = 0, the model is a random walk.
If b1 = 1 and b0 ≠ 0, the model is a random walk with drift.
Therefore:
  • If the model has a unit root
    • It is a random walk if b0 = 0
    • It is a random walk with drift if b0 ≠ 0
  • If the model is a random walk, it has a unit root
  • If the model is a random walk with drift, it has a unit root
 
Saleh.alabdali wrote:Q1:
Random walk is a special case of AR1, where b0=0 & b1=1.
Therefore, MRL=0/(1-1)=0 undefine.
Example:
A stock that follows a random walk has no mean reversion level, so you can’t predict which direction it is likely to trend in the future???????
Random walk with a drift: b0=1
If a time series is a random walk wit a drift, the next predication for it’s next value is last value.
By definition, all random walks have a unit root.
when AR1 model b0=1 we call it Unit root.
Therefore, MRL=1/(1-1)=0 undefine.
Example:
A stock that follows a random walk with unit root, so the predication for its next value is last value?????
Click to expand...
Correct.
Saleh.alabdali wrote:Q2?
The first difference for random walk is Yt=change in Xt
Using first difference make the time series covariance stationary with finite MRL=0/(1-0)=0 = finite????
What the difference between:
MRL=0/(1-1)=0 undefine
MRL=0/(1-0)=0 = finite
What make them differ???
Click to expand...
The first has a denominator of zero, so it is undefined.
The second has a denominator of one, so it is defined, and 0/1 = 0.
Saleh.alabdali wrote:Q3?
Is the following is right?
Unit root b0=1 it could be random walk when b1=1??
Click to expand...
A unit root is not defined as b0 = 1; a unit root is defined as b1 = 1.
Saleh.alabdali wrote:Q4?
Why stock that follows a random walk is defined as having a unit root.
It may be b1=0 not 1????
Click to expand...
Because b1 = 1.
 
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