Question
The relationship between Y and X is given by where ui 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.
Answer :
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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 $$ \bar{X} = \frac{18 + 16 + 19 + 22 + 20}{5} = \frac{95}{5} = 19 $$ $$ _____ ___ ____ _________ _______ ______ ________ _______ ________.
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We are asked to estimate the parameters of a simple linear regression model $Y = \beta_0 + \beta_1 X + u$, where $u$ satisfies the classical assumptions, using the given data. Let’s solve it step by step. --- Step 1: Calculate the OLS estimates of $\beta_0$ and $\beta_1$ The formulas for simple linear regression are: $$ \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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