Below is the question from my statistics class and my answer about multiple regression. Are my IVs in the correct order? how good is the MR equation? The professor used parenthesis $($with e$).$ Should I include the coefficients or not? Class $1$ Question:

Choose a MR model that you would like to assess and indicate how you would enhance the traditional MR to include expanded error. Your answer can be simple with the traditional MR model and then the enhanced MR model with expanded error. Indicate how the dependent variable and independent variables relationships might change if the expanded errors are allowed to correlate. The traditional formula is: fy $=$ x$1+$ x$2+$ x$3+$ x$4+$ x$5+$ e$.$ When we allow for expanded error we have fy $=$ x$1($with e$)+$ x$2($with e$)+$ x$3($with e$)+$ x$4($with e$)+$ x$5($with e$)$

Please include the one dependent variable and a minimum of three and no more than five independent variables in your answer.

Class $1$ Answer:

A Model to assess an Operations Research System Analyst $($ORSA$)$ Productivity.

The dependent variable is productivity score $($y$).$

The independent variables are:

$$ x$1=$ years of experience

$$ x$2=$ training $($hours per year$)$

$$ x$3=$ number of completed projects $($per year$)$

$$ x$4=$ position $($lead$,$ supporting role$)$

When expanded errors are allowed, the model becomes:

fy $=$ a$1$x$1+$ a$2$x$2+$ a$3$x$3+$a$4$x$4+$ e$1($x$1)+$e$2($x$2)+$ e$3($x$3)+$ e$4($x$4)$

where a$1$ through a$4$ are the coefficients and e$1($x$1)$ through e$4($x$4)$ are the expanded error terms.

The coefficients a$1$through a$5$ will capture the main effects of the independent variables, while the expanded error terms account for correlated errors.

If we allow the errors $($e$)$ to correlate the greatest correlated errors might occur between:

$1.$ e$1($years of experience$)$ and e$2($hours of training$)$: If more experienced ORSAs are receiving less training $($e$1$ and e$3$ are negatively correlated$),$ this might indicate that more experienced ORSAs benefit less from training $($e$.$g$.,$ training may have less impact on more experienced ORSAs, potentially reducing productivity$).$

$2.$ e$4($position$)$ and e$2($hours of training$)$: Those in lead positions might receive or require different training levels compared to those in supporting roles, leading to a potential correlation here as well. Misaligned training for leads might reduce their productivity.

$3.$ e$1($years of experience$)$ and e$3($number of completed projects$)$: More experienced ORSAs might complete more projects, leading to a positive correlation between these errors, suggesting that experience contributes directly to project completion, which in turn affects productivity.