Question

4. Consider estimation of the simple regression model: y = 30+8121+w. Show a graph of the predicted values, y, associated with the estimation, and the confidence intervals for the predicted values. Is the width of the confidence interval constant for all possible cu (values of the explanatory variable, æı)? Explain your answer with the use of a figure (or figures).

Answer 1

The predicted linear regression line using OLS method would come out to be =

y^= b0+b1*X

Hence, the graph of predicted values against x would be a linear trendline with b0 as the intercept on the y axis and b1 as the (upward/downward) slope of the trend line.

This here is an example of an upward sloping linear regression line with x axis representing x variable, y representing the predcted values. The dots represent actual data points and the red line is the fitted regression line with an intercept (b0)=5.

Linear regression and confidence limits 10 Sample observations Regression line Lower confidence limit (95%) Upper confidence limit (95%) 00 6 Y values 4 2 4 00 10 N X values

The confidence interval for predicted values in an linear regression tends to be narrow around the mean of the predictor and fat around the minimum and maximum values of the predictor. This can be seen in plots of these 4 linear regressions:

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