(a) To estimate the regression model (Y_i = \alpha + \beta X_i + u_i), you can use the ordinary least squares (OLS) method.
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The OLS estimates for the parameters (\alpha) and (\beta) can be calculated as follows:
[ \beta = \frac{\sum_{i=1}^{n}(X_i – \bar{X})(Y_i – \bar{Y})}{\sum_{i=1}^{n}(X_i – \bar{X})^2} ]
[ \alpha = \bar{Y} – \beta \bar{X} ]
where (\bar{X}) is the mean of X and (\bar{Y}) is the mean of Y.
(b) The coefficient of determination ((R^2)) is a measure of the proportion of the variation in the dependent variable ((Y)) that is explained by the independent variable ((X)). It is calculated as:
[ R^2 = \frac{\text{Explained Variation}}{\text{Total Variation}} = \frac{\sum_{i=1}^{n}(\hat{Y}i – \bar{Y})^2}{\sum{i=1}^{n}(Y_i – \bar{Y})^2} ]
where (\hat{Y}_i) is the predicted value of (Y_i) from the regression model.
You can find the percentage variation explained by multiplying (R^2) by 100.