The relationship between Y and X is given by where ui

Question

The relationship between Y and X is given by $y_i = \alpha + \beta x_i + u_i$ where u_i follows classical assumptions. Consider the following set of data and answer the questions.

| Y | 11 | 12 | 13 | 14 | 15 | |---|----|----|----|----|----| | X | 18 | 16 | 19 | 22 | 20 |

a) Estimate parameters of the model from the following data by using OLS method.

b) What is the estimate of error variance in the above case?

c) Find the value of R2 for the above data.

31 Aug 2025

Answer :

Word Count : 618

We are asked to estimate the parameters of a simple linear regression model $Y = \beta_0 + \beta_1 X + u$, where $u$ satisfies the classical assumptions, using the given data. Let’s solve it step by step. --- Step 1: Calculate the OLS estimates of $\beta_0$ and $\beta_1$ The formulas for simple linear regression are: $$ \hat{\beta}_1 = \frac{\sum (X_i - \bar{X})(Y_i - \bar{Y})}{\sum (X_i - \bar{X})^2} $$ $$ \hat{\beta}_0 = \bar{Y} - \hat{\beta}_1 \bar{X} $$ Step 1a: Compute the means of X and Y $$ \bar{X} = \frac{18 + 16 + 19 + 22 + 20}{5} = \frac{95}{5} = 19 $$ $$ ____ __________ _____ _______ _______ __________ ____ ______ _____ ____ _____.
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X	18	16	19	22	20

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