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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.

The following results were obtained from a sample of 12 agribusiness firms on their output (Y), labor input (X1), and capital input (X2) measured in arbitrary units. Hint: use 5% level of significance.

This problem involves multiple regression analysis for 12 agribusiness firms to estimate the relationship between output, labor input, and capital input. Learn how to estimate regression coefficients, compute standard errors, and test the significance of the model using OLS. Suitable for undergraduate students in economics or statistics.

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