哈佛计量经济学讲义

Harvard University S p r i n g T e r m Department of Economics Economics 2 140A

C. Sampling Theory

1.In developing sampling theory for univariate regression we require a few propositions from matrix theory. We introduce these propositions without proof and employ them to develop results on normal sampling theory. A matrix, say P, is orthogonal if

c-2 we say that A is a matrix

=(I

=I

is symmetric, idempotent, then there exists an orthogonal matrix P such that:

= (x

and

?i

A given value of this quadratic form, say with center at the origin as in Diagram Cl. The shaded area included in the circle represents the values of that

satisfy the inequality:

The next example we consider is the quadratic form:

=

.

c-4

Cl. Geometric representation of the quadratic form +

given value of this quadratic form,say

in the variables

and

and

We next consider an arbitrary positive definite quadratic form:=

where:=

all

a 2l where D is a diagonal matrix:

D =0

C2. Geometric representation of the quadratic form +

c-7

P’P

=

The distance from the origin for any pair of co-ordinates of the vector

=

and

and

=

C-S

Diagram C3. Geometric representation of the quadratic form .

c-9

where:

al2

A =

a22

To provide a geometric representation of this quadratic form we first consider the positive definite

quadratic form:

,

where z is a translation of x:

=

corresponds to an ellipse in the variables that is not necessarily rectangular with center at ( as represented in Diagram C4.In spaces of three and higher dimensions we refer to the locus of such points as an ellipsoid.

The shaded area included in the ellipse represented the values of

that satisfy the inequality:

I=1

c-10

C4. Geometric representation of the quadratic form Q =(x(x

c-11

3.We continue by considering the geometric representation of a linear form, say L , where:

= , a

a

= 0 .

The locus of points 0 is a straight line through

the origin as in Diagram

Geometric representation of the linear form a

Next, we consider the geometric representation of the linear form:

= 0

This is the set of vectors

as in Diagram C6.

Diagram C6. Geometric representation of the linear form = 0. In three-dimensional space the locus of such points as

, we can write:

0,

where:

=

0 implies a is non-singular.We conclude that a non-singular

transformation takes one hyperplane into another.

A hyperplane divides all vectors in the space into three parts, corresponding to values of the linear form:

= = 0 ,

respectively. We say that two vectors and

=

and on the same side of the hyperplane are transformed into two vectors

that are also on the same side of the hyperplane, since:

= = ,

and:

0 . Similarly, if we subject the vectors to a translation so that:

Z=x-p,

and:

= = a’z =

=

= x’x ,

where

.

supporting hyperplane to the corresponding circle is given by the linear form:

L(x) = = 1 ,

so that:

included in the circle satisfies the inequality:

a

c o s

cos

cos

where

0,

and that:

-a )

a), we can construct another supporting hyperplane by

considering the linear form:

+ a) = 0 .

This hyperplane can be re-written in the form:

+ a) = 0 ,

so that it is parallel to the original hyperplane: for an arbitrary positive definite

quadratic form:

C-16 a non-singular matrix T such that:

TAT’ = I .

Given this transformation, we can rewrite the quadratic form so that:

1 = =

.

The supporting hyperplanes corresponding to any vector b such that:

+ b) + b) = 0

To express these forms in terms of the original vectors

+ b)=

= ,

where:

,

+ b] = = , where:

and

b) are transformed into vectors on the same

side of the transformed hyperplane similarly, vectors on the same side of the hyperplane

L (z the same side of the transformed hyperplane

that satisfies the inequality:

We conclude that the transformation and translation results in a pair of parallel supporting hyperplanes to the ellipsoid , namely, (x The geometric

representation of these supporting hyperplanes is given in Diagram

0 there are parallel supporting hyperplanes to the ellipsoid corresponding to any given value, say

=

.

To show this we first consider the supporting hyperplanes:

=

,

and:

b’b = 1,

so that: