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 b_{0}=0 and b_{1}=1.”

Does this mean that if you first difference a series and b_{0}=0 and b_{1}=0 then the original time series is a random walk, but if b_{0}=0 and b_{1} 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 b_{1} 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 (b_{1}) 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 y_{t} on independent x_{t}−x_{t−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