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Is it always true that in the simple linear regression model the width of the prediction interval corresponding to a new observation x=xo increases linearly with xo?

Thanks in advance

pianca Strict
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  • Assuming $x$ is your explanatory variable, this should not be the case in general. In heteroscedastic models where the variance is increasing with $x$ then it will of course happen. Maybe you could give some more info about the motivation of your question. – Dayne Nov 14 '19 at 08:12
  • okay so in the case that variance is increasing with x. then the width of the prediction interval increases linearly with xo, but how can i explain why this is true? – pianca Strict Nov 14 '19 at 08:33
  • In simple linear regression we model the $E(y|x)$ as function of $x$. So for a given $x$ $y$ can have any value around $E(y|x)$. If the variation in $y$ around $E(y|x)$ is increasing in $x$ then of course with higher $x$ your prediction interval of would be higher. – Dayne Nov 14 '19 at 08:40
  • thank you so much, i have another small question about linear regression model if you can help me, is it true or false that the simple linear regression model through the origin can be applied to any dataset? – pianca Strict Nov 14 '19 at 08:52
  • You can forcefully fit a linear model from origin to any dataset (numerical variables) of course but whether is a wise choice or not would completely depend on data. – Dayne Nov 15 '19 at 02:00

1 Answers1

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The width of prediction intervals does not increase linearly; the uncertainty in parameters involving constant and slope induces a spreading out that increases a little more slowly than linearly as you move further away from the mean of the predictor variable; the limits of the prediction interval will asymptote to a pair of lines that cross at $(\bar{x},\bar{y})$.

There's an intuitive illustration of the way the confidence interval behaves here: Shape of confidence interval for predicted values in linear regression

The prediction interval has similar behavior but includes an additional term that has a constant spread.

Glen_b
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