Yes, if I understand the question correctly. So you have three cars: A, B, and C. For each car you have three independent variables: V1, V2, V3. So you have three equations:

1) Speed = V1 + V2 + V3 for car A

2) Speed = V1 + V2 + V3 for car B

3) Speed = V1 + V2 + V3 for car C

Approach 1) make a model Speed = V1 + V2 + V3 + Car type + V1*Car type + V2*Car Type + V3* Car Type (you should include the other interaction terms if you have enough data to support the model). If the independent variable by car type interaction terms are significant, then there is an effect of car type on the variable.

Approach 2) If car type has no effect on the beta, then you can mix all the data and it will have no effect on your outcome. So Take all the data for car A and car B, and dump it into a large pot. Withdraw (with replacement=bootstrap, without replacement=randomization test) the data to get two groups with the same number of observations as in your original groups. Calculate the difference in the betas, and repeat this process some 10,000 times, and sort the data. Calculate the difference in beta for the observed data, and count the number of times you found a results as large or larger than the observed results in the randomized data. You can work this approach for any number of comparisons: so Car A versus Car B versus Car C ... versus Car W. Doing the multiple comparison approach is better if you have a large number of car types unless you are always comparing a treatment to a "standard type" in which case pairwise will work fine.

Sir,

Thank you for the reply. I have six classes of cars. So, I have to add interaction variable for each of the independent variable and run the regression for every model? And if the t stats of interaction variables are significant, we can conclude that the class of car has influence on the variable? If so, can we use the same beta values for comparison? Problem is that I want to compare the magnitudes of betas rather than examine if they are JUST different from one to the other.

Also, can we use the same procedure for categorical independent variables?

Car is a categorical variable. It equals A for the first car type, B for the second, and so forth. Approach 1 is then a single model that covers the entire analysis. Some programs have a short cut notation to include all of the interaction terms.

If the t-stat is significant then you can conclude that the effect of variable V1 on speed changes depending on what type of car is being examined. It is also phrased as there is a significant interaction between car type and speed. So yes. However, in the simplest approach you will not be able to identify which car or cars are significantly different, only that car type has an effect. Some programs will allow you to take slices and visualize the interaction term. If you want to keep it simple you could just do this pairwise: in model 1, CarType=A and B; in model 2 CarType=A and C; ...; in model 15 CarType E and F.

Be not too proud of your beta values. Your beta values may be real, or they may be an artifact of your experiment. Maybe in your application you have measured all the important variables -- but can you prove that? If you have missed a variable for whatever reason, then your beta values may be entirely wrong (see http://optimalprediction.com/files/pdf/V1A13.pdf). It is also possible that your independent variables are not independent of one another. So if Variable 2 is in the model, Variable 1 will change its value. However, if it is important to keep the beta values for each model separate, then use approach 2.

Both approaches will work with categorical independent variables. However, this adds another source of error in your betas. For example: at low levels of nitrogen, nitrogen fertilization improves plant growth. At very high levels of nitrogen the fertilizer can become toxic and reduce growth. Now I will make up some numbers: So I apply nitrogen at 30 kg/ha and 400 kg/ha. It happens that peak growth for this plant is are 180 kg/ha. By great misfortune, we picked two points that are equal: one from promoting growth and the other from toxicity retarding growth. We now conclude that these plants are unresponsive to nitrogen. I could rearrange this example to get any answer I want because the statistical design is interacting with the plant response to nitrogen fertilization. In your case, I could make a categorical variable "spoiler" where a car with a spoiler in back gets a 1 otherwise zero. Or it might be 1 for manufacturer 1, and a 2 if the other manufacturer. So the shape/style/surface of the spoiler will have an effect and you are making a proxy variable that combines all these effects. This is condensing a large number of variables into a single value ... and there is a great deal that is hidden/missing/unknown. The more of these categorical variables that are in your model, the more likely your betas are not giving you what you are hoping for. This doesn't mean that there is a problem, just that there is more opportunity for such problems. There are always things that we don't measure, and that is just part of science.

Srijith -

You said that you want "...to conclude unique effects of each independent variable on different vehicle classes?"

The interaction effects can get very messy and are very important as Timothy indicated, but to answer the part of your question I quoted above, looking at each independent variable, one at a time, I think you can obtain valuable information by plotting one independent variable per scatterplot, with a regression line plotted for each dependent variable against that same one regressor, if you have continuous data. Then you will have simple regression lines that may be directly compared to each other under each independent variable. If regression is through the origin, then you could compare each slope directly to another on the same scatterplot, and consider the standard error of each slope as well. The link attached takes you to an example that would be for one such independent variable, with two dependent variables compared. This is for continuous data, and scatterplots can be very enlightening for such data.

Scatterplots may help you explore your data further, perhaps by looking at linear combinations of regressors as a single independent variable, in other sets of regression lines on scatterplots.

This may give you the beginnings of a clearer picture of your data before you look at the more complicated aspects of multiple regression. Also, you stated that you were doing multiple linear regression, but scatterplots can help you to consider whether or not some relationships may be nonlinear. However, you should have good theoretical reasons behind your conclusions as data have measurement errors and randomly occurring data that may be anomalous, and though scatterplots can give you many good leads, you have to beware of occasionally spurious results.

I hope this was relevant to your question.

Cheers - Jim 

PS - It sounds like, for your data, and unlike the paper at the attached link as I recall, you would need to consider covariance in each case. There are examples you can find through an Internet search. 

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Thank you sir for the valuable suggestions. I am confident that there are several factors whose effects are different for one group to the other. I just want to present it for a conference paper. So I am looking for some robust methods with some strong statistical support. Will try out the scatter plots. I guess I already did it sometimes ago. However, my data is quite scattered and hence number of covariates required to explain the variance is also high

Final question. I built regression model combining all the groups along with interaction variables as follows.

Let, A, B and C be the dummy variables for the groups. X1 is the independent variable and Y is the dependent variable. The output model of the regression in R is as follows.

Y ~ b0 + b1*X1 + b2*B + b3*C + b4*B*X1 + b5*C*X1.

Question: What does each of the parameter estimates represent?

Just addressing the intercept term, I think it tempting to think of it as the overall result of unknown covariates. However, perhaps especially with multiple regression, I think it just as much or more likely influenced by any model misspecification. If your estimated intercept term is small with regard to its estimated standard error, then that may be the best situation, because if there is no theoretical reason to have an intercept, you could check your model without it. If there is no theoretical reason for an intercept, yet you have an estimated one that is at least several times the magnitude of its standard error, then you may have substantial model specification, missing regressors, or both. Data quality could also be a problem. If there is a good theoretical reason for an intercept term, and you have one whose standard error is relatively low, that might be a good sign. 

Thank you so much sir. This helped a lot.