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Let's break down and solve the problem step by step. The dataset is based on a multiple linear regression analysis where the dependent variable \( Y \) (output) is modeled as a function of two independent variables, \( X_1 \) (labor input) and \( X_2 \) (capital input). The summary statistics are provided for the sums and cross-products.

Given:
- \( \sum Y = 753 \)
- \( \sum X_1 = 643 \)
- \( \sum X_2 = 106 \)
- \( \sum Y^2 = 48,139 \)
- \( \sum X_1^2 = 34,843 \)
- \( \sum X_2^2 = 976 \)
- \( \sum YX_1 = 40,830 \)
- \( \sum YX_2 = 6,796 \)
- \( \sum X_1X_2 = 5,779 \)

We are asked to:
1. Estimate and interpret the regression coefficients.
2. Compute the average and marginal productivity.
3. Compute the standard errors.
4. Compute the coefficient of multiple determination \( R^2 \) and adjusted \( R^2 \).
5. Test the significance of individual and overall coefficients.
6. Identify economies of scale.

### Step 1: Estimate the Regression Coefficients

The multiple linear regression model can be written as:

\[
Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \epsilon
\]

To estimate the coefficients \( \beta_0 \), \( \beta_1 \), and \( \beta_2 \), we can use the normal equations from the summary statistics:

\[
\beta_1 = \frac{\sum (Y X_1) - \left(\sum Y \right)\left(\sum X_1 \right)/n}{\sum (X_1^2) - \left( \sum X_1 \right)^2 / n}
\]

\[
\beta_2 = \frac{\sum (Y X_2) - \left(\sum Y \right)\left(\sum X_2 \right)/n}{\sum (X_2^2) - \left( \sum X_2 \right)^2 / n}
\]

Where \( n = 12 \) is the number of firms in the sample.

I will compute these now.The estimated regression equation is:

\[
Y = -35.80 + 1.24X_1 + 3.64X_2
\]

### Interpretation:
- **\(\beta_0 = -35.80\)**: This is the intercept term. It represents the expected output \( Y \) when both labor input \( X_1 \) and capital input \( X_2 \) are zero. While it may not have a practical interpretation here, it is a necessary part of the regression model.
- **\(\beta_1 = 1.24\)**: This is the coefficient for labor input \( X_1 \). For each additional unit increase in labor, the expected output \( Y \) increases by 1.24 units, holding capital input \( X_2 \) constant.
- **\(\beta_2 = 3.64\)**: This is the coefficient for capital input \( X_2 \). For each additional unit increase in capital, the expected output \( Y \) increases by 3.64 units, holding labor input \( X_1 \) constant.

Next, would you like me to proceed with calculating the standard errors, \( R^2 \), and hypothesis testing, or would you like more details on this part? 

### Relative Questions:
1. How is the \( R^2 \) value interpreted in regression analysis?
2. What do the coefficients \( \beta_1 \) and \( \beta_2 \) represent in an economic context?
3. How would you test the overall significance of this regression model?
4. What does the adjusted \( R^2 \) tell you about the model’s fit?
5. How can we compute the marginal productivity of labor and capital?

### Tip:
When interpreting regression coefficients, always keep in mind the context of the variables and the units in which they are measured.

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.

This page explores a multiple regression analysis based on data from 12 agribusiness firms, analyzing output as a function of labor and capital inputs. Learn how to estimate regression coefficients, compute standard errors, and understand the significance of R^2 and adjusted R^2 values, along with testing the overall significance of the model.

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