To assume that other variable do not change in order to allow for an evaluation of partial variation in a dependent variable due to variation in the only independent while other variables do not change

https://stats.stackexchange.com/questions/180155/what-does-it-mean-intuitively-to-hold-other-variables-constant-in-regression

https://blog.minitab.com/blog/adventures-in-statistics-2/a-tribute-to-regression-analysis

Okay, what you ask goes by the name of "sensitivity analysis” Here are some issues when using multiple regression: 1. Multicollinearity Look at correlations among the predictor variables, any large pairwise correlation will affect your Equation. May need Ridge regression and/or PCA- based regression. To decide which action, calculate correlation matrix among your x-variables (sat C) and then take INVERSE of this matrix say C-1. Look at the diagonal elements, if any larger than ‘10’ then your equation may not be correct 2. calculate principal components & see amount of variance accounted for by 1st & 2nd PCAs, if relatively larger than others then do as in (1) above 3. Sensitivity analysis Let us assume you have 4 x-variables predicting ‘1’ y-variable. a. Fix x-variables 2, 3,&4 at their Mean values and allow x-variable ‘1’ vary over its range by certain steps. Plot & analyse thus obtained “y-values vs x(1) values”. The regression equation now is your net effect of x(1) relative to other variables equal to their average values… b. Repeat above for X(2) fixing X(1), X(3) & X(4) at their mean c. Repeat as in (a) for X(3) fixing X(1), X(2) & X(4) at their mean d. Repeat as in (a) for X(4) fixing X(1), X(2) & X(3) at their mean *** regression slope from a, b, c & d is your sensitivity coefficient …

To assume that other variable do not change in order to allow for an evaluation of partial variation in a dependent variable due to variation in the only independent while other variables do not change

Akinyemi has given a good answer to the question

In your hypothetical case, 0.5 is the increase of wage for an additional year of education for both rural and urban employees. A rural employee earns 0,3 more than an urban one with the same education. This example is too simple to be good for explaining the purpose of "holding other variables constant". For that you should take a more complicated equation or - better - a model with several equations.

Mousumi's links do a good job of explaining this. Mewa's comments are correct, but the issue of multicollinearity doesn't really address this question, since it is a violation of the Gauss-Markov conditions upon which regression analysis is based. It has been many years since I have dealt with these issues so I am somewhat rusty, but I think a more intuitive approach may be helpful.

Your sample result of W= C + .5 *Education + .3 *Rural can easily be viewed in two dimensional space as 2 parallel lines (equations). For Urban workers (i.e. the Rural variable = 0) the predictive equation is W = C +.5* Education, while for Rural workers the predictive equation is W = (C +.3) + .5*Education. In both cases, an additional year of education raises the wage by .5. For a binary dummy (like Rural), this is straightforward. If you had a dummy with more values this is extended fairly easily. However, even for a continuous variable, while you potentially have an infinitude of equations, what you are saying is for each admissible value of the variable (or other variables if there are more) there is a unique relationship between W and Education with the coefficient showing the effect of an additional year of education with the values of the other variables held constant, in other words it is basically a partial derivative of W wrt Education.

This begs the question of how does it statistically hold this variable (or there variables) constant. The answer is to compare it to a regression of W on Education where you are not statistically holding Rural constant. The result generally would be a different constant and more importantly a different coefficient on Education. This would especially be true if for example Urban people had more education(or less) on average. The simple regression would be combining the effects of Education and Rural as one effect, rather than separating them into 2 separate effects.

Hope this is helpful.

I do not know how you determine the .5 and .3 coefficients but if you want to keep them constant, here is a trick you can use: First, introduce one more variable, say Y, that you think affects your dependent variable W. Then substract from the W variable the .5*Educ.+ .3* and run the regression (W- .5 Educ, - .3 Rural) against Y. As you can see you have held The .5 and .3 coefficients constant. therefore W = C+ a*Y + .e Educ. + .3Rural. Now, systematically change the coefficients of Educ. and Rural (that is (the .5 and .3) and compare the estimated correlations coefficients . Those that maximize the correlation coefficient will be the Maximum Likelihood Estimates. Obviously this mean you will have to run several regressions . You may start by changing one variable at a time and after you determine the correlation coefficient of say E then start working with the R variable and experiment by changing by a bit both variables. Good luck.

mathematically, the 0.5 which is beta in the regression is obtained as a derivative of Y on the X (here, education) so it's the net effect of 1unit increase of X(education) independently of the effect of any other explicative variable. Accordingly, 0.5 is the effect of an increase in the education of a person in rural or urban areas. Please see any reference books in econometrics GREEN, GUJARATI, Tsay ...

If multicollinearity exists between variables, the 0.5 is no longer the net effect of education since it will be confused with the effects of the correlated variables which are no longer held constant.

It mean you only change one variable while other variable do not change (remain unchanged). This assumption will let you to observe the effect of the only variable changes on the dependent variable of the model(regression).

Take the partial derivative of the wage with respect to education. In common parlance, this is the change in the wage when education changes by 1. This is a hypothesized linear relationship so going from education of 1 to 2 is the same as from 12 to 13. Now with respect to holding a variable constant, it simply doesn't matter what the rural area variable is. There is no interaction between these explanatory variables in the equation you have written out.

Obviously, Santiago's question provoked some misunderstandings. This may be due to the supplement "in a regression", which could be understood as holding variables constant during the estimation procedure. But this would not make sense, and Santiago's hypothetical example shows that he means the interpretation of a result of a regression. Therefore, one can assume that statistical problems (like collinearity) have already been solved (i.e. not part of the question). In this simple example, one can derive the effects of changes of the independent variables directly from the equation, i.e. there is no need to hold one of the two variables constant. It is clear, that every year more of education brings a .5 higher wage for both rural and urban employees, and rural employees earn 0.3 more than their urban colleagues with the same education.

But if you have a more complicated model y=f(x,y) with x and y vectors, partly lagged and non-linear, it is not that simple. To calculate the effect of a change of x(i), often a baseline solution is calculated. As a next step, one calculates y with the same x, with the exception that x(i) is changed. The difference of both solutions is the effect of the change of x(i).