orthog.hlp

是一个经济学管理应用软件 很难找的 但是经济学学生又必须用到

{smcl}
{* 10mar2005}{...}
{cmd:help orthog}, {cmd:help orthpoly}{right:dialogs:  {bf:{dialog orthog}}  {bf:{dialog orthpoly}}}
{hline}

{title:Title}

{p2colset 5 19 21 2}{...}
{p2col :{hi:[R] orthog} {hline 2}}Orthogonalize variables and compute
orthogonal polynomials{p_end}
{p2colreset}{...}


{title:Syntax}

{phang}
Orthogonalize variables

{p 8 15 2}
{cmd:orthog}
[{varlist}]
{ifin}
{weight}
{cmd:,}
{opth g:enerate(newvarlist)}
[{opt mat:rix(matname)}]

{phang}
Compute orthogonal polynomial

{p 8 17 2}
{cmd:orthpoly}
{varname}
{ifin}
{weight}
{cmd:,}{break}
{c -(}{opth g:enerate(newvarlist)}{c |}{opt p:oly(matname)}{c )-}
[{opt d:egree(#)}]

{p 4 6 2}
{opt orthpoly} requires that {opt generate(newvarlist)} or {opt poly(matname)}, or both, be specified.{p_end}
{p 4 6 2}
{it:varlist} may contain time-series operators; see {help tsvarlist}.{p_end}
{p 4 6 2}
{opt iweight}s, {opt fweight}s, {opt pweight}s, and {opt aweight}s are
allowed; see {help weight}.
{p_end}


{title:Description}

{pstd}
{opt orthog} orthogonalizes a set of variables, creating a new set of
orthogonal variables (all of type {opt double}), using a
modified Gram-Schmidt procedure.
Note that the order of the variables determines the orthogonalization;
hence, the "most important" variables should be listed first.

{pstd}Note that execution time is proportional to the square of the number of
variables.  With a large number (>10) of variables, {cmd:orthog} will be
fairly slow.

{pstd}
{opt orthpoly} computes orthogonal polynomials for a single variable.


{title:Options for orthog}

{dlgtab:Main}

{phang}
{opth generate(newvarlist)} is required.  {opt generate()} creates new
variables of type {opt double}.
For {opt orthog}, {it:newvarlist}
will contain the orthogonalized {varlist}.
If {it:varlist} contains d variables, then so will {it:newvarlist}.
{it:newvarlist} can be specified by giving a list of exactly d new variable
names, or it can be abbreviated using the styles {it:newvar}{cmd:1-}
{it:newvard} or {it:newvar}{opt *}.  For these two styles of abbreviation, new
variables {it:newvar}{cmd:1}, {it:newvar}{cmd:2}, ..., {it:newvard} are
generated.  See {help orthog##orthog_examples:Examples} below.

{phang}
{opth matrix(matname)} creates a (d + 1) * (d + 1) matrix 
containing the matrix R defined by X = QR, where X is the N * (d + 1)
matrix representation of {varlist} plus a column of ones and Q is the 
N * (d + 1) matrix representation of {it:newvarlist} plus a column of ones 
(d = number of variables in {it:varlist}, and N = number of observations).


{title:Options for orthpoly}

{dlgtab:Main}

{phang}
{opth generate(newvarlist)} or {opt poly()}, or both, must be specified.
{opt generate()} creates new orthogonal variables of type {opt double}.
{it:newvarlist} will contain orthogonal polynomials of degree 1, 2, ..., d
evaluated at {it:varname}, where d is as specified by {opt degree(d)}.
{it:newvarlist} can be specified by giving a list of exactly d new variable
names, or it can be abbreviated using the styles {it:newvar}{cmd:1-}
{it:newvard} or {it:newvar}{opt *}.  For these two styles of abbreviation, new
variables {it:newvar}{cmd:1}, {it:newvar}{cmd:2}, ..., {it:newvard} are
generated.  See {help orthog##orthpoly_examples:Examples} below.

{phang}
{opth poly(matname)} creates a (d + 1) * (d + 1) matrix
called {it:matname} containing the coefficients of the orthogonal polynomials.
The orthogonal polynomial of degree i {ul:<} d is

{pin2}
{it:matname}{cmd:[}i{cmd:,} d + 1{cmd:] +}
{it:matname}{cmd:[}i{cmd:, 1]*}{it:varname} {cmd:+}
{it:matname}{cmd:[}i{cmd:, 2]*}{it:varname}^2
{cmd:+}...{cmd:+}
{it:matname}{cmd:[}i{cmd:,} i{cmd:]*}{it:varname}^i

{pmore}
Note that the coefficients corresponding to the constant term are placed in
the last column of the matrix.  The last row of the matrix is all zero, except
for the last column, which corresponds to the constant term.

{phang}
{opt degree(#)} specifies the highest-degree polynomial to include.
Orthogonal polynomials of degree 1, 2, ..., d = {it:#} are computed.
The default is {bind:d = 1}.


{title:Remarks}

{pstd}
Orthogonal variables are useful for two reasons.  The first is numerical
accuracy for highly collinear variables.  Stata's {helpb regress} and other
estimation commands can face a large amount of collinearity and still produce
accurate results.  But, at some point, these commands will drop variables due
to collinearity.  If you know with certainty that the variables are not
perfectly collinear, you may want to retain all of their effects in the model.
If you use {cmd:orthog} or {cmd:orthpoly} to produce a set of orthogonal
variables, all variables will be present in the estimation results.

{pstd}
Users are more likely to find orthogonal variables useful for the second
reason:  ease of interpreting results.  {cmd:orthog} and {cmd:orthpoly} create
a set of variables such that the "effects" of all the preceding variable have
been removed from each variable.  For example, if we issue the command

{phang2}{cmd:. orthog x1 x2 x3, generate(q1 q2 q3)}

{pstd}
the effect of the constant is removed from {cmd:x1} to produce {cmd:q1}; the
constant and {cmd:x1} are removed from {cmd:x2} to produce {cmd:q2}; and
finally the constant, {cmd:x1}, and {cmd:x2} are removed from {cmd:x3} to
produce {cmd:q3}.  Hence,

{p 8 13 2}{cmd:q1} = r01 + r11*{cmd:x1}{p_end}
{p 8 13 2}{cmd:q2} = r02 + r12*{cmd:x1} + r22*{cmd:x2}{p_end}
{p 8 13 2}{cmd:q3} = r03 + r13*{cmd:x1} + r23*{cmd:x2} + r33*{cmd:x3}

{pstd}
This can be generalized and written in matrix notation as

	X = QR

{pstd}
where X is the N * (d + 1) matrix representation of {it:varlist} plus a column
of ones, and Q is the N * (d + 1) matrix representation of {it:newvarlist}
plus a column of ones (d = number of variables in {it:varlist} and N = number
of observations).  The (d + 1) * (d + 1) matrix R is a permuted upper
triangular matrix; i.e., R would be upper triangular if the constant were
first, but the constant is last, so the first row/column has been permuted
with the last/column.  Since Stata's estimation commands list the constant
term last, this allows R, obtained via the {cmd:matrix()} option, to be used
to transform estimation results.


{marker orthog_examples}{...}
{title:Examples of orthog}

{phang2}{cmd:. orthog x1 x2 x3, gen(u1 u2 u3)}{p_end}
{phang2}{cmd:. orthog x1 x2 x3, gen(u1-u3)}{p_end}
{phang2}{cmd:. orthog x1 x2 x3, gen(u*)}

{phang2}{cmd:. orthog x1-x3, gen(u*) matrix(R)}

{pstd}
The matrix R created by the {cmd:matrix()} option can be used to transform
coefficients from a regression:

{phang2}{cmd:. orthog x*, gen(u*) mat(R)}{p_end}
{phang2}{cmd:. regress y u*}{p_end}
{phang2}{cmd:. matrix b1 = e(b)*inv(R)'} {space 1} [note that the transpose of inv(R) is used]

{phang2}{cmd:. regress y x*}{p_end}
{phang2}{cmd:. matrix b2 = e(b)}

{pstd}Then b1 and b2 will be the same.

{pstd}
The matrix R can also be used to recover X (original {it:varlist}) from Q
(orthogonalized {it:newvarlist}) one variable at a time:

{phang2}{cmd:. orthog price weight mpg, gen(upr uwei umpg) mat(R)}{p_end}
{phang2}{cmd:. matrix c = R[1...,"price"]}{p_end}
{phang2}{cmd:. matrix c = c'} {space 2} [{cmd:matrix score} requires a row vector]{p_end}
{phang2}{cmd:. matrix score double samepr = c}{p_end}
{phang2}{cmd:. compare price samepr}

{pstd}
That is, the variable {opt samepr} is the same as the original {opt price}.
This procedure can be performed as a check of the numerical soundness of the
orthogonalization.


{marker orthpoly_examples}{...}
{title:Examples of orthpoly}

{phang}{cmd:. orthpoly weight, deg(4) generate(pw1 pw2 pw3 pw4)}{p_end}
{phang}{cmd:. orthpoly weight, deg(4) generate(pw1-pw4)}{p_end}
{phang}{cmd:. orthpoly weight, deg(4) generate(pw*)}{p_end}

{phang}{cmd:. orthpoly weight, deg(4) poly(P)}{p_end}


{title:Also see}

{psee}
Manual:  {bf:[R] orthog}

{psee}
Online:  {helpb matrix}, {helpb regress}
{p_end}