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Find the constrained least-squares estimator for a multiple regression model
Consider the multiple regression model
$Y=X\mathrm{\beta <!--\; \beta \; -->}+\mathrm{ϵ<!--\; ϵ\; -->}$
with the restriction that $\underset{l}{\overset{n}{\sum <!--\; \sum \; -->}}{b}_i=1$
I want to find the least squares estimator of $\mathrm{\beta <!--\; \beta \; -->}$, so I need to solve the following optimization problem
$min(Y-<!--\; -\; -->X\mathrm{\beta <!--\; \beta \; -->}{)}^t(Y-<!--\; -\; -->X\mathrm{\beta <!--\; \beta \; -->})$
$s.t.\underset{l}{\overset{n}{\sum <!--\; \sum \; -->}}{b}_i=1$
Let's set
$L=(Y-<!--\; -\; -->X\mathrm{\beta <!--\; \beta \; -->}{)}^t(Y-<!--\; -\; -->X\mathrm{\beta <!--\; \beta \; -->})-<!--\; -\; -->\mathrm{\lambda <!--\; \lambda \; -->}(U^t\mathrm{\beta <!--\; \beta \; -->}-<!--\; -\; -->1)=Y^{t}Y+{\mathrm{\beta <!--\; \beta \; -->}}^t{X}^{t}X\mathrm{\beta <!--\; \beta \; -->}+-<!--\; -\; -->2{\mathrm{\beta <!--\; \beta \; -->}}^t{X}^{t}Y-<!--\; -\; -->\mathrm{\lambda <!--\; \lambda \; -->}(U^t\mathrm{\beta <!--\; \beta \; -->}-<!--\; -\; -->1)$
where U is a dummy vector of ones (and therefore $U^T\mathrm{\beta <!--\; \beta \; -->}=\underset{l}{\overset{n}{\sum <!--\; \sum \; -->}}{b}_i$).
Take derivatives
$\frac{d}{d\mathrm{\beta <!--\; \beta \; -->}}=2X^{t}X\mathrm{\beta <!--\; \beta \; -->}-<!--\; -\; -->2X^{t}Y-<!--\; -\; -->\mathrm{\lambda <!--\; \lambda \; -->}{U}^t=0$
$\frac{d}{d\mathrm{\lambda <!--\; \lambda \; -->}}=U^t\mathrm{\beta <!--\; \beta \; -->}-<!--\; -\; -->1=0$
So from the first equation we can get an expression for $\mathrm{\beta <!--\; \beta \; -->}$, but what should I do with the $\mathrm{\lambda <!--\; \lambda \; -->}$? The second equation doesn't seem to be useful to get rid of it.

Contribution of each variable in multiple linear regression
What will be the best measure of the contribution of a variable in multiple linear regression? I was thinking of using the coefficient ratio as a marker of a variable's contribution.
For example:
If the equation is
$Y_{predicted}=a_1{X}_1+a_2{X}_2+a_3$
Then $X_1$'s contribution can be written down as:
$\frac{a_1}{a_1+a_2+a_3}$
Is there some other method possible to write down the contribution. Since in this case if any coefficient is negative, there is a possibility that a variable's contribution exceeds $100\mathrm{\%<!--\; \%\; -->}$

Multiple regression problems (restricted regression, dummy variables)
Q1.
Model 1: $Y=X_1{\mathrm{\beta <!--\; \beta \; -->}}_1+\mathrm{\epsilon <!--\; \epsilon \; -->}$
Model 2: $Y=X_1{\mathrm{\beta <!--\; \beta \; -->}}_1+X_2{\mathrm{\beta <!--\; \beta \; -->}}_2+\mathrm{\epsilon <!--\; \epsilon \; -->}$
(a) Suppose that Model 1 is true. If we estimates OLS estrimator $b_1$ for ${\mathrm{\beta <!--\; \beta \; -->}}_1$ in Model 2, what will happen to the size and power properties of the test?
(b) Suppose that Model 2 is true. If we estimates OLS estrimator $b_1$ for ${\mathrm{\beta <!--\; \beta \; -->}}_1$ in Model 1, what will happen to the size and power properties of the test?
-> Here is my guess.
(a) $b_1$ is unbiased, inefficient estimator. (I calculated it using formula for "inclusion of irrelevant variable" and $b_1=(X_1^{\prime }{M}_2{X}_1{)}^{-<!--\; -\; -->1}{X}_1^{\prime }{M}_{2}Y$ where $M_2$ is symmetric and idempotent matrix) Inefficient means that it has larger variance thus size increases and power increases too.
(b) $b_1$ is biased, efficient estimator. (I use formular for "exclusion of relevant variable" and $b_1=(X_1^{\prime }{X}_1{)}^{-<!--\; -\; -->1}{X}_1^{\prime }Y$) Um... I stuck here. What should I say using that information?
Q2.
Let $Q$ and $P$ be the quantity and price. Relation between them is different across reions of east, west, south and north, and as well, for different 4 seasons. Construct a model.
-> Actually, I don't know well about dummy variables. So any please solve this problem to help me.

Start with PCA and multiple regression or Start with multiple regression and PCA
I would like to know something easy but very important.
Imagine I have a database with 0 NA, a perfect database who has been clean. And I have to do a PCA on this database. This datebase got a lot of individuals and variables ( 95 individuals and 10 variables)
I have to do a multiple regression and a PCA.
I must start per my multiple regression and eventually delete somme individuals who has been a Cook's distance > at the limit. And after I do my PCA on " new data base"
OR I must start per my PCA on my complete database, and after I do my multiple regression.
In conclusion, I must do :
- PCA
- multiple Regression
or
-multiple Regression
-PCA
Ty for helping me !

Why is it valid to use squared Euclidean distances in high dimensions in multiple regression?
Euclidean distance is not linear in high dimensions. However, in multiple regression the idea is to minimize square distances from data points to a hyperplane.
Other data analysis techniques have been considered problematic for their reliance on Euclidean distances (nearest neighbors), and dimensionality reduction techniques have been proposed.
Why is this not a problem in multiple regression?