Question

A low F-statistic means thay all of the coefficients in the regression model could actually be 0?

shadow price in a minimization problem cannot be positive?

Answer 1

Hi,

The F-test of overall significance indicates whether your linear regression model provides a better fit to the data than a model that contains no independent variables.

The F-test for overall significance has the following two hypotheses:

The null hypothesis states that the model with no independent variables fits the data as well as your model.

The alternative hypothesis says that your model fits the data better than the intercept-only model.

Compare the p-value for the F-test to your significance level. If the p-value is less than the significance level, your sample data provide sufficient evidence to conclude that your regression model fits the data better than the model with no independent variables.
Generally speaking, if none of your independent variables are statistically significant, the overall F-test is also not statistically significant.

So, yes you are partially right.
A low F statistic doesn't mean that all independent statistics would actually be zero but what it means is that, since they are approximately equals 0 in our hypothesis testing meaning that they don't play any statistically significant role in our regression model.

A shadow price value is associated with each constraint of the model. It is the instantaneous change in the objective value of the optimal solution obtained by changing the right hand side constraint by one unit.

A reduced cost value is associated with each variable of the model. It is the amount by which an objective function parameter would have to improve before it would be possible for a corresponding variable to assume a positive value in the optimal solution.

Yes, here it's true that shadow price in a minimization problem cannot be positive because shadow price of a constraint in a minimization problem directly means that constraint is of <= type i.e. it can take negative values or zero. So it can't be positive.

I hope I answered your question well.
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