# Where plus 1 came from in variance estimation [duplicate] While $$\mathrm{E}(\tilde{\mathrm{y}})=\alpha+\beta \tilde{\mathrm{x}}$$

Subject is Regression Analysis and this formula is from the "Features of Estimation ". and y is a neutral variable.

Formula for the variance of "y".

Where this plus 1 ( the one that has red underline) came from.

• Can you explain the context for this? What are $\tilde{y}$, $\tilde{x}$, $K$, $T_x$ for instance. Can you tell us what the underlying statistical model is? – jcken Mar 1 at 12:10
• For more duplicates, search our site for regression prediction interval formulas. – whuber Mar 1 at 15:02

## 1 Answer

In linear regression we have $$y = \alpha + \beta x + \varepsilon$$ where $$\varepsilon \sim N(0, \sigma^2)$$ are iid (independent and identically distributed) error terms.

For a sufficiently large sample size, the maximum likelihood estimators for $$\alpha$$ and $$\beta$$ are jointly Normal: $$\begin{pmatrix} \hat{\alpha} \\ \hat{\beta} \end{pmatrix} \sim N \left\{ \begin{pmatrix} \alpha \\ \beta \end{pmatrix}, \begin{pmatrix} \sigma^2(1/n + \bar{x}^2/S_{xx}) &-\sigma^2 \bar{x}/S_{xx} \\-\sigma^2 \bar{x}/S_{xx} & \sigma^2/S_{xx}\end{pmatrix}\right\}$$ Where $$S_{xx} - \sum_{i=1}^n(\bar{x} - x_i)^2$$. Now if I have a new value of $$x$$ denotes $$\tilde{x}$$ we have $$\tilde{y} = \alpha + \beta\tilde{x} + \varepsilon$$. The obvious (and probably best) predictor of $$\tilde{y}$$ is $$\hat{\tilde{y}} = \hat{\alpha} + \hat{\beta}\tilde{x}$$. Now suppose we want an idea of the variability of $$\tilde{y}$$. To do this we need to consider the uncertainty in our estimated of $$\alpha$$ and $$\beta$$. So we have, using standard properties of $$Var()$$, $$Cov()$$ and the assumption that $$(\hat{\alpha}, \hat{\beta})$$ is independent of $$\varepsilon$$:

\begin{align} Var(\tilde{y}) &= Var(\hat{\alpha} + \hat{\beta}\tilde{x} + \varepsilon)\\ & = Var(\hat{\alpha} + \hat{\beta}\tilde{x}) + Var(\varepsilon) \\ & = Var(\hat{\alpha} + \hat{\alpha}\tilde{x}) + \sigma^2 \\ &=Var(\hat{\alpha}) + \tilde{x}^2Var(\hat{\beta}) + 2\tilde{x}Cov(\hat{\alpha}, \hat{\beta}) + \sigma^2\end{align}

Then extracting the $$Var$$ and $$Cov$$ terms from the variance matrix of the Normal distribution above we have

\begin{align}Var(\tilde{y}) &= \sigma^2(1/n + \bar{x}^2/S_{xx}) + \tilde{x}^2\sigma^2/S_{xx} - 2\tilde{x}\bar{x}\sigma^2/S_{xx} + \sigma^2\\ &=\sigma^2 \left\{1/n + \bar{x}^2/S_{xx} + \tilde{x}^2/S_{xx} - 2\tilde{x}\bar{x}/S_{xx} + 1 \right\} \text{*notice the +1 from factoring out \sigma^2}\\ &=\sigma^2 \left\{\frac{1}{n} + \frac{(\bar{x}-\tilde{x})^2}{S_{xx}} + 1 \right\}\end{align} Now simply rearranging the additive terms leads to $$Var(\tilde{y}) = \sigma^2 \left\{1 + \frac{1}{n} + \frac{(\bar{x}-\tilde{x})^2}{S_{xx}} \right\}$$

Now my answer has slightly different notation to yours because you haven't define $$KT_x$$, however, this should show you where $$+1$$ come from.