Dear Dr. Anusuya Nallathambi ,

Your question is not very clear. Please be clear in your hypothesis and objective statement.

With the available information, I feel you can look into schild plot analysis, since your problem statement is not clear, I am not 100% sure, just check it.

Also, I Have attached reference files for comparing the slopes of two or more regression plots, It may be useful to you please check.

All the best

Regards

Vishwanatha

I agree with Vishwanath. We need more information on what is the problem you are trying to investigate.

If I understood it right, your problem is similar to this: you have a time series for an object, let's say "Candy production" per year, and another time series with another object, let's say "total wage paid" per year. Both of them has constant growth rates, let's say 4% a year for the Candy production, and 2% for the total wage paid. Since them both has constant growth rates, the correlation will be high (if truly constant, it will be 1).

What you want to compare is how they differ from each other, right?

If the growth rate of them both are truly constant (without any oscillation what so ever) them, the only difference between them will be the angle of the curve that describe each growth rate. There will be no other difference, since the growth has no oscillation (therefore, there will be no distance between observed and expected values, and that means you will not have any distribution of errors, since there aren't any). That is why correlation is 1.

So, to compare the curves, all you have to do is to compare the growth rates. In this scenario I described, the only difference of the two curves will be the 2 p.p there exist between 2% and 4%. You can also built a regression with one of the variables as dependent, and the other as independent. It will give you the coefficient that describe the angular difference between them. But if the problem is something like what I just described, using econometric methods would just be an over analysis of something very simple that observational statistic will give the precise answer.

If this is not the case, please, try to explain better what the problem is.

First of all, you should look at the stability of the time series. for this you can look Unit Root Test result. Then, if these series is stationary, you can study Regression analysis, if they are not stationary, You should select cointegration test. Detailed information can be found in our manuscript.

Vishwanath shastry G.L. I apologize for the confusing question, but Apoena Canuto Cosenza has captured my problem accurately. As Mr.Cosenza stated, I want to calculate an estimate that shows the difference between the slopes (i.e. 2% and 4%) and I want to calculate this estimate for different sets of variables. For example the "candy production" Vs "wage rise" of factory1, of factory2, factory3 etc and determine which factory has the greatest difference in slope between these variables.

As Mr.Cosenza stated I am trying out the regression method, for now with dummy data to better understand its behavior.

However I have a question, can the regression method still give valid answers if one of the variables have spurious noise contained in it ?

For initial understanding of the concept I tried to imagine two variables with constant growth rate. But in real world data, the one of the variables may be very noisy, even though it shown a distinct growth rate when plotted on a paper. In this case can I still use the regression model to test the difference in slope ?

Thank you very much for taking the time to answer this unclear query. Also. thank you, Mr.Shastry for the reference paper.

Anusuya Nallathambi , there is no problem is the variable is noisy. But it maybe a problem if it is too spurious. Let me explain it better:

If one of the variables have high random oscillation, but a clear growth tendency, your regression may end up with (a) a high standard error (but possibly acceptable); (b) a low Standard R squared (i.e., low explanatory power), that will have to be noted on the final result, but will not invalidate your regressions results. Actually, a low R² is something to be described as a difference between 2 variables, so it is a relevant find.

But, if the oscillations are not random, but actually a result of conjunctural factors (let's say there was a national strike of Candy Workers, causing the production to fall), the regression may end up being spurious. That can be verified by the Durbin-Watson of the errors. In that case, your coefficients will be not precise.

One way to solve this is to change the regression parameters. You can either change the model (from a liner model to a quadratic or cubic, for instance), if it makes sense; or to use a dummy (or angular) variable to compensate for the conjunctural factor (dummy "Strike" = 1 at months that the Strike was happening. The regression for production will be: Candy = constant + b(wage) + c(strike)).

How to detect similar problem before running the regression? Simple: build a X-Y dispersion Graph with both variables, and try to identify what kind of picture it shows you. Is it a linear relation? or a exponential one? Is there conjunctures? etc.

If you want, I can send you my "step by step" that I use in my classes.

Apoena Canuto Cosenza thank you for the clear explanation. I understand. Judging from the real world data I have, it feels like adding another explanatory dummy variable feels like worth a try. Also, I would find it helpful if you could send the materials you use in your classes since I don't have experience in statistical analysis. Thank you very much for taking the time.

Anusuya Nallathambi ,

I made a translation from my original step-by-step sheet. The original sheet is based on a class I gave for economic history grad-students.

It is at my profile. You can check it here: Method Ordinary Least-Square: Step-By-Step sheet

Any doubts, or comments, just send me a message and I will answer it.

I am assuming that, by "slope" you mean "rate of change". If this is constant then there's no random component so staistical testing is not appropriate - just measure and compare. If non-constant then many possibilities. For example, measure rate of change at each sample point in both series and test for difference in mean between the two rate of change samples. (If your time series is short then this won't be 100% reliable because it's assumption of normality and independence is probably not satisfied.)