This problem involves multiple regression analysis using data from a sample of 12 agribusiness firms. The goal is to estimate a regression model where output $Y$ is the dependent variable, and labor input $X_1$ and capital input $X_2$ are the independent variables.

Given:

• $\sum Y = 753$
• $\sum X_1 = 643$
• $\sum X_2 = 106$
• $\sum Y^2 = 48,139$
• $\sum X_1^2 = 34,843$
• $\sum X_2^2 = 976$
• $\sum YX_1 = 40,830$
• $\sum YX_2 = 6,796$
• $\sum X_1X_2 = 5,779$

These values allow us to:

1. Estimate the regression coefficients using the least squares method.
2. Compute the average and marginal productivity of labor and capital.
3. Calculate the standard errors, adjusted $R^2$, and perform significance tests.

Let me begin by estimating the regression coefficients for the model $Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2$.

I'll calculate that now.The estimated regression equation is:

$Y = 3.651 + 0.855 X_1 + 1.506 X_2$

Where:

• $\beta_0 = 3.651$ is the intercept.
• $\beta_1 = 0.855$ is the coefficient for labor input $X_1$.
• $\beta_2 = 1.506$ is the coefficient for capital input $X_2$.

These coefficients suggest the following:

• For every unit increase in labor input $X_1$, the output $Y$ increases by 0.855 units, holding $X_2$ constant.
• For every unit increase in capital input $X_2$, the output $Y$ increases by 1.506 units, holding $X_1$ constant.

Would you like me to continue with the remaining parts (such as calculating standard errors, adjusted $R^2$, and significance tests)?

Here are some related questions you might explore:

1. What assumptions must be checked in multiple regression analysis?
2. How is the significance of individual regression coefficients tested?
3. What does the adjusted $R^2$ tell us about the model?
4. How can multicollinearity affect the regression estimates?
5. How is the marginal productivity of an input calculated from the regression coefficients?

Tip: In multiple regression, the coefficients are interpreted while holding other variables constant, which helps in understanding partial effects of each independent variable on the dependent variable.