Fit the random walk model with drift to the data

I have a time series data of exchange rates.

I can apply many tests, such as variance ratio test, to see if it is a random walk or not.

However, I would like to get an estimation of a drift that the random walk can have.

Is there any idea how can I do that?

Answer 1

[Walter Roberson]

https://workspace.imperial.ac.uk/lifesciences/Public/postgraduate/courses/QB%20MSc/TestingForTheRandomWalkHypothesisWithndWithoutMeasurementError.pdf

Answer 2

[Image Analyst]

Well there are analytical formulas that you can use. Or you could do a Monte Carlo simulation. I'm attaching some random walk Monte Carlo simulations for what it's worth.

[Astrik]

Thank you for provided example. May be I do not get something very basic. I need to estimate the value of possible trend, in order to perform pseudo out of sample forecast estimation and compare its performance with other models. How can simulation help me here?

[Image Analyst]

As you know, random walks do not bounce around the starting point. They tend to eventually wander away. There is even theory that says how far away is the expected distance as a function of the number of steps taken. With a Monte Carlo experiment, you could plot out this function without even knowing exactly what the analytical function is. You just build it up empirically by running the experiment a bunch of times.

Of course it is possible your time series data could have come from a random walk, after all you know the old saying about monkeys writing Shakespeare. How likely it was that your data came from a random walk might be beyond my statistical abilities, like if you want a confidence p value or something.

Perhaps all you want to do is to fit your time series data to a polynomial or something. We're not sure.

[Walter Roberson]

The PDF I linked to has formulae for estimating the trend.