Well this is my first post ever, so I decided to start with something light.And for the past couple of days I have been studying Basic Econometrics by Gujarati again just for fun ( yes for fun 🙂 )

For those of you who have done regression analysis at some point of time, the issue of comparing models must have crept up. And like a knight in shining armor, enter…..drum rolls…..R-Squared. For those unaware R-Squared measures how much of the variation in your dependent variable is explained by the independent variable. Higher the R-Squared value, better the fit of the model. But what if you wanted to compare the following two models:

Now any standard textbook will tell you you can’t compare two models unless the following conditions hold:

- The Dependent variable is the same.
- The Sample size is the same.

So now we have a problem. Luckily there is a way around. As given in Gujarati, we follow these steps:

- Estimate Model 1 to get:

2. Estimate Model 2 to get:

and the R-Squared :

3. Now take the log of the predicted values of model 1 i.e.

Notice the difference. This is key. We are taking log of the **predicted values of Model 1**.

4. Regress the log of actual values against the log of the predicted values of model 1 to get

Predicted model is:

with R-Squared :

Now this can be compared with .

This is where Gujarati signs off. When I saw this the question that occurred to me was why? Why should this work? Whats the logic behind it? After a bit of soul searching and having lunch I tried to do this:

**Note:** Hat over a value denotes the predicted values while a bar denotes the mean value.

For Model 1 R-Squared is :

For Model 2 R-Squared is :

Now the real thing. Consider the R-Squared for Model 3:

This is why we can compare model 2 and 3. The dependent variable is the same for both models. The dependent variable for model 3 comes from model 1 which establishes the connection. The chain of reasoning is as follows.

- Model 2 explains the variations in
- Model 1 explains the variations in
- We use these explained variations of model 1 as the independent variable in model 3 to explain the variations in
- Thus, in effect we are using model 1 to explain model 3.
- Hence, when we compare the two R-Squared values we are comparing model 1 with model 2.

That ends the reasoning and this blog post. Hope this helps someone. Feel free to comment and/or criticize as necessary.

Thanks for your time and patience. Ciao!!!

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Aman, it is simply great! First of all let me congratulate you for doing a great job.

The topics picked up is good. Keep it up.

And, now think about…. What if in the second model it is X and not log X?

All the best…

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