formula stata uses to calculate stdp and stdf

[Rob Shaw]

Hi

I'm trying to explain to someone the difference between the se of a prediction and the se of a forecast, for example illustrated by the stata commands

twoway (lfitci lny x, stdp)
twoway (lfitci lny x, stdpf)

Could anyone tell me the formula that is used by the stdp and stdf options in stata for twoway, preduction etc please?

thanks
Rob

[Carlo Lazzaro]

Rob may be interested in taking a look at -regression postestimation- entry in Stata 13.1 .pdf manual, especially pages 1898-1899.

[Rob Shaw]

Carlo

Thanks. I've found the formula reference you cited. However, when I calculate it I don't get the same results as Stata.

I have 11 observations in my data - the independent x is just a number 1..11. the RMSE of the regression is 0.049. So s² = 0.049^2

(X'X)-1

in excel is

=MINVERSE(MMULT(TRANSPOSE($A$2:$A$12),$A$2:$A$12))

The other parts of h, x' and x is the same as x² as I only have one independent variable

For my first observation, x=1, gives a value of 0.0022 from sqrt(s² * h) but stata gives 0.028

Can anyone point to what I'm doing wrong please?

Thanks
Rob

[Steve Samuels]

I can't interpret your Excel code (Stata matrix code would have been better) , but I see two possible errors:

1) For simple linear regression, the formula for \(h_i\) is not \(x_i^2\) or a simple function of it. Rather it is:

\[
h_i = \frac{(1 + x_i)(X'X)^{-1}(1+x_i)'}{n\sum_i(x_i - \overline{x})^2}
\]

where the design matrix \(X\) is

\[
\begin{aligned}
X = &
\begin{pmatrix}
1 & x_1 \\
\vdots & \vdots \\
1 & x_n
\end{pmatrix}
\end{aligned}
\]

2) The formula for stdf is not \( \sqrt{s^2 h_i} = s\sqrt{h_i}\), but rather \(s\sqrt(1+ h_i)\).

[Steve Samuels]

Correction:


\[
h_i = (1 + x_i)'(X'X)^{-1}(1+x_i)
\]

[Steve Samuels]

Here's a "hand" calculation that uses Stata's matrix language.

sysuse auto, clear
keep in 1/20

gen one= 1
mkmat one turn, matrix(X) // X matrix

mkmat gear, matrix(y)  // y
matrix H =  X*invsym(X'*X)*X'  //Hat (projection) matrix
local hat1 =  el("H",1,1)  // first diagonal element


matrix sse =(y-H*y)'*(y-H*y)

local rsd = sqrt((el("sse",1,1))/18) //residual SD

/* Compute forecast from regress */
regress gear turn
predict stdf, stdf

list stdf in 1
    +----------+
     |     stdf |
     |----------|
  1. | .3457681 |
     +----------+

. di `rsd'*sqrt(1+`hat1')  // hand calculation

.3457681

[Andrew Xavier]

Dear Steve Samuels,

How do I calculate stdf in the context of mixed models for the fitted predictions?