# Transforming Variables in Regression

I have a very simple problem that hopefully someone could help me with or at least point me in the right direction.

I am testing to see which factors affect index returns the most and would like to find the correct way to transform variables used in the multiple regression model. The data sample covers the time period 1990 - 2013 with a quarterly frequency (96 observations).

For my dependent variable, I have calculated quarterly index returns.

For the independent variables set, I have collected quarterly data for: CPI%, 3 year rate, 10 year rate, unemployment rate, and a couple of exchange rates.

Should I calculate quarterly % changes for all of my variables(since I used quarterly returns for my dependent), leave them unchanged(and use index values instead of returns), or change them in a different way?

Thank you,

Here is a screenshot of my current data set:

• Sorry for the delay, but... welcome to quant.SE! Could you specify the sample period of your dataset, please? – Quantopik Apr 26 '15 at 23:17
• Thanks @Quantopic! The sample period is 1990-2013. I was originally using monthly data, but some data points were unavailable for a monthly frequency so I switched to quarterly. – Jonathan Apr 28 '15 at 5:35
• You're welcome @Jay! I edited both the question and my answer with information about the dataset time-length. Remember to mark one of answers below if one of them fulfills your question. – Quantopik Apr 28 '15 at 7:53

I suggest you to implement all the analysis you cited above and analyze the results choosing the best model on model performance measures, as, for instance, the model $R^2$ value, the AIC (BIC) value, etc.; this should be the proper way to develop a model.

As regards your question particularly, the literature about the topic suggest to develop the model on the basis of the change in percentages of all variables you consider developing the factor model.

In the matter of the transformation of the dependent variable, you should consider the possibility of transforming that in returns to fulfill the assumptions of the linear regression model; indeed, the logarithmic returns are distributed according to the Normal distribution if they are in the range [-0.04; +0.04] according to the Taylor approximation, and it is more likely to produce normally distributed residuals; the values outside this range should be considered as outliers.

The same for independent variable; you should transform those in change in percentage variable, if they're not yet.

Lastly, it is easier to analyze log-log regression model results than level-log or log-level model.

As last hint, I firstly suggest you to develop a model taking into account the linear regression model assumption and testing for the reliability of those hypothesis after running the model (test for homoschedasticity, autocorrelation, multicollinearity,...). Moreover, try to build a reliable dataset; using quarterly data you can have less observation and get biased results. So, if possible, use higher frequency data than yours, as, for instance, monthly, weekly, daily data. EDIT: since your dataset is pretty small (~96 observations), you should try to get more data to improve your analysis by collecting data previous to the 1990 or later of 2013.

Basic rules for what you want to follow to have a meaningfull result

1 All your variables should be in the same terms
2.All your regression variables should be stationary (weak sense stationary) - check if there are long term dependencies
3.Test for multi-collinearity / Serial correlation / Homocedasticity each variable against the predictor.