Predicting Levels Estimation from a regression with a Natural Log as Y variable

[Daniel Coveney]

Hello,

I am forecasting prices of bitcoin using a one-step ahead, regression method. To deal with unit root issues, I have to run a linear regression on the natural log of bitcoin prices from a training dataset, then predict 21 periods (daily data) from the training set. I was wondering if there is an easy way to back-out the level estimations of bitcoin prices from this regression? My current regression is as follows:

ln(bitcoin)_t = beta₀ + L(1/4).ln(bitcoin)_t + beta_n(X)_t + u_t

where,
beta₀ = a constant
L(1/4).ln(bitcoin) = lagged ln(bitcoin) through time periods 1-4
beta_n = a series of additional predictors included in the regression, including a linear time trend
u_t= error term

When using the predict newvar, xb command I get the natural log predictions for bitcoin, however I would like the level predictions for bitcoin using this regression. Is there a way to do this using the predict command?

If there is anymore information I could provide to facilitate a clearer answer to my question, let me know.

[Joao Santos Silva]

Dear Daniel,

That is one of the problems of estimating models where the dependent variable is in logs (search for the retransformation problem). If your data is homoskedastic, you can use a procedure described in Jeff Wooldridge's introductory textbook. If on top of that the errors are normal, you can do confidence intervals for the predictions.

Best wishes,

Joao

[Daniel Coveney]

Thank you Joao,

For the purposes of my project, I suppose I'll keep everything in log form and just compare forecasts from there. Your recommendation was very helpful in figuring out this problem.