Simple Linear Regression - Difference between predicting and estimating?Here is what my notes say about estimation and prediction:Estimating the conditional mean"We need to estimate the conditional mean&nbsp;β0+β1x0&nbsp;at a value&nbsp;x0, so we use&nbsp;Y0^=β0^+β1^x0&nbsp;as a natural estimator." here we getY0^~N(β0+β1x0,σ2h00)&nbsp;&nbsp;where&nbsp;&nbsp;h00=1n+(x0−x―)2(n−1)sx2with a confidence interval for&nbsp;E(Y0)=β0+β1x0&nbsp;is(b0^+b1^x0−csh00,b0^+b1^x0+csh00)where&nbsp;c=tn−2,1−α2&nbsp;Where these results are found by looking at the shape of the distribution and at&nbsp;E(Y0^)&nbsp;and&nbsp;var(Y0^)Predicting observations"We want to predict the observation&nbsp;Y0=β0+β1x0+ϵ0&nbsp;at a value&nbsp;x0"E(Y0^−Y0)=0&nbsp;&nbsp;and&nbsp;&nbsp;var(Y0^−Y0)=σ2(1+h00)Hence a prediction interval is of the form(b0^+b1^x0−csh00+1,b0^+b1^x0+csh00+1)

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Simple Linear Regression - Difference between predicting and estimating?Here is what my notes say about estimation and prediction:Estimating the conditional mean&quot;We need to estimate the conditional mean&amp;nbsp;β0+β1x0&amp;nbsp;at a value&amp;nbsp;x0, so we use&amp;nbsp;Y0^=β0^+β1^x0&amp;nbsp;as a natural estimator.&quot; here we getY0^~N(β0+β1x0,σ2h00)&amp;nbsp;&amp;nbsp;where&amp;nbsp;&amp;nbsp;h00=1n+(x0−x―)2(n−1)sx2with a confidence interval for&amp;nbsp;E(Y0)=β0+β1x0&amp;nbsp;is(b0^+b1^x0−csh00,b0^+b1^x0+csh00)where&amp;nbsp;c=tn−2,1−α2&amp;nbsp;Where these results are found by looking at the shape of the distribution and at&amp;nbsp;E(Y0^)&amp;nbsp;and&amp;nbsp;var(Y0^)Predicting observations&quot;We want to predict the observation&amp;nbsp;Y0=β0+β1x0+ϵ0&amp;nbsp;at a value&amp;nbsp;x0&quot;E(Y0^−Y0)=0&amp;nbsp;&amp;nbsp;and&amp;nbsp;&amp;nbsp;var(Y0^−Y0)=σ2(1+h00)Hence a prediction interval is of the form(b0^+b1^x0−csh00+1,b0^+b1^x0+csh00+1)

Colonninisxi · Accepted Answer

You have to distinguish between estimating E[Y∣X=x0] and {Y}(x0)For the former you have that E[Y∣X=x]^=β0^+β^1x0,such that Var(E[Y∣X=x]^)=σ2[1n+(x0−x―)2(n−1)sx2]=σ2h00,where for the latter you have thatVar(Y^(x0))=Var(E[Y∣X=x]^)+Var(ϵ0)=σ2(1+h00).Namely, for the former case you are estimating the conditional mean of Y at x0, while for the latter you are estimating (predicting) the value itself. The conditional mean &quot;smooths out&quot; the variance of the error term as E[Y∣X=x]=β0+β1x, however the prediction of a new value Y(x0) should consider also the (estimated) variance of the error term (to account for the fluctuation around the conditional mean).

Simple Linear Regression - Difference between predicting and estimating? Here is what my notes s

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