Final answer:
To estimate β1 in the linear panel data model, plot a scatter plot, calculate the least-squares line, and extract the slope as the estimator of β1. Verify the significance of the relationship through the correlation coefficient and check for outliers.
Step-by-step explanation:
To estimate β1 in the linear panel data model Yit = Xitβ1 + αi + λit + uit, where t denotes time, i denotes the individual, and β1 represents the coefficient of the independent variable Xit, we must account for the unobserved individual-specific time trend αi + λit.
Firstly, plot the data points by placing observations of Yit and Xit on a scatter plot. This visual representation helps identify the nature of their relationship. Subsequently, calculate the least-squares line using regression analysis, which minimizes the sum of the squares of the residuals, where the residual for each observation is the difference between the observed value of Yit and the predicted value ϯit.
The regression line can be calculated with the formula ϯit = a + bXit. Here, a is the y-intercept and may not hold practical significance as it conceptualizes the scenario when Xit is zero; which may not be meaningful depending on the context. The slope b is the estimate of β1. After obtaining the least-squares line, analyze the residuals and identify if any numerical identification of outliers is required. Outliers can heavily influence the estimated coefficients.
To verify the significance of the relationship between Yit and Xit, calculate the correlation coefficient. If it's significantly away from zero, it suggests that a linear relationship exists. This analysis is important to confirm the appropriateness of a linear model.
After refining the model to handle outliers and ensuring the significance of the relationship, the estimate of β1 can be derived from the slope of the least-squares line. Comparisons of the original and estimated coefficients, alongside diagnostic checks on residuals, help assess the model fit and the relevancy of β1's estimation.