One thing that is really confusing me is in the text it states that:
“… the FX rate is a random walk, as we now suspect. If so, the first differenced series will be covariance stationary.”
Elsewhere is also says that:
“… modelling the first-differenced time series with an AR(1) model does not help us predict the future, as b0=0 and b1=1.”
Does this mean that if you first difference a series and b0=0 and b1=0 then the original time series is a random walk, but if b0=0 and b1 is not equal to 0 it isn’t? AAAGH!
FX spot rate are not random walks, they are approximately random walks. Indeed, they have mean revert levels which they tend to achieve in the long-run (20-30 years).
Not always a first-difference is enough, maybe it is needed 2 or more.
If you first difference a random walk, b0 and b1 will be statistically equal to zero because they can not find a constant pattern (a constant difference) that can be used as an average, so this reveals that the original time serie was a random walk.
If you get that b1 is statistically significant, then the time serie was not a random walk. A graphic of the serie is a very important help.
Could someone please help me out by posting a graph displaying why the first difference (b1) does not differ from zero when there is random walk? I still cannot wrap my head around this issue.
Is first-differencing mean regress dependent yt on independent xt−xt−1, I don’t understand.
Suppose the original series is
4
8
10
12
20
Then the first difference series is
4
4
2
2
8
Now you try to regress (predict) the first difference series through a regression model in which dependent variable = intercept + coefficient * previous term of the difference series + error term.
And you know how to arrive at previous term of the arrived difference series given the original series you can find it
Hope it helps 