Least Squares Algebra

Report
Econometrics I
Professor William Greene
Stern School of Business
Department of Economics
3-1/27
Part 3: Least Squares Algebra
Econometrics I
Part 3 – Least Squares Algebra
3-2/27
Part 3: Least Squares Algebra
Vocabulary



3-3/27
Some terms to be used in the discussion.

Population characteristics and entities vs.
sample quantities and analogs

Residuals and disturbances

Population regression line and sample
regression
Objective: Learn about the conditional mean
function. Estimate  and 2
First step: Mechanics of fitting a line
(hyperplane) to a set of data
Part 3: Least Squares Algebra
Fitting Criteria




The set of points in the sample
Fitting criteria - what are they:

LAD

Least squares

and so on
Why least squares?
A fundamental result:
Sample moments are “good” estimators of
their population counterparts
We will spend the next few weeks using this
principle and applying it to least squares
computation.
3-4/27
Part 3: Least Squares Algebra
An Analogy Principle for Estimating 
In the population
E[y | X ]
=
E[y - X |X] =
E[xi i]
=
Σi E[xi i]
=
E[Σi xi i]
=
E[X(y - X) ] =
Continuing
Summing,
Exchange Σi and E[]
X so
0
0
Σi 0 = 0
E[ X ] = 0
0
Choose b, the estimator of  to mimic this population
result: i.e., mimic the population mean with the sample
mean
Find b such that
1
N
X e = 0 
1
X ( y - X b )
N
As we will see, the solution is the least squares coefficient
vector.
3-5/27
Part 3: Least Squares Algebra
Population and Sample Moments
We showed that E[i|xi] = 0 and Cov[xi,i] = 0.
If so, and if E[y|X] = X, then
 = (Var[xi])-1 Cov[xi,yi].
This will provide a population analog to the
statistics we compute with the data.
3-6/27
Part 3: Least Squares Algebra
U.S. Gasoline Market, 1960-1995
3-7/27
Part 3: Least Squares Algebra
Least Squares

Example will be, Gi on
xi = [1, PGi , Yi]

Fitting criterion: Fitted equation will be
yi = b1xi1 + b2xi2 + ... + bKxiK.

Criterion is based on residuals:
ei = yi - b1xi1 + b2xi2 + ... + bKxiK
Make ei as small as possible.
Form a criterion and minimize it.
3-8/27
Part 3: Least Squares Algebra
Fitting Criteria

Sum of residuals: 

Sum of squares: 

Sum of absolute values of residuals:  i 1 e i

Absolute value of sum of residuals  i 1 e i

We focus on  i 1 e now and  i 1 e i later
3-9/27
N
i 1
N
ei
2
e
i 1 i
N
N
N
2
i
N
Part 3: Least Squares Algebra
Least Squares Algebra
2
e
 i 1 i  e e = (y - X b )'(y - X b )
N
A d ig re ssio n o n m u ltiv a ria te ca lcu lu s.
M a trix a n d v e cto r d e riv a tiv e s.
D e riv a tiv e o f a sca la r w ith re sp e ct to a v e cto r
D e riv a tiv e o f a co lu m n v e cto r w rt a ro w v e cto r
O th e r d e riv a tiv e s
3-10/27
Part 3: Least Squares Algebra
Least Squares Normal Equations
  i 1 e
N
b
2
i
2
  i 1 ( y i - x i b )
N

b
 (y - X b )'(y - X b )
b
 (1x1) /  (kx1)
  2 X '(y - X b )
= 0
(-2)(N xK )'(N x1)
= (-2)(K xN )(N x1) = K x1
N ote: D erivative of 1x1 w rt K x1 is a K x1 vector.
S olution:  2 X '(y - X b ) = 0  X 'y = X 'X b
3-11/27
Part 3: Least Squares Algebra
Least Squares Solution
-1
A ssum ing it exists: b = ( X 'X ) X 'y
N ote the analogy:
 =  V ar( x ) 
1
 1

b= 
X 'X 
N

 C ov( x ,y) 
1
 1

X 'y 

N

S uggests som ething desirable about least squares
3-12/27
Part 3: Least Squares Algebra
Second Order Conditions
N ecessary C ondition: First derivatives = 0
 (y - X b )'(y - X b )
b
  2 X '(y - X b )
S ufficient C ondition: S econd derivative s ...
 (y - X b )'(y - X b )
2
b b 
=
=
  (y - X b )'(y - X b ) 



b


b 
 K  1 colum n vector
 1  K row vector
= 2 X 'X
3-13/27
Part 3: Least Squares Algebra
Does b Minimize e’e?
  iN1 x i21
 N
2
 i 1 x i 2 x i1
 e'e

 2 X 'X = 2

b b '
...
 N
  i 1 x iK x i 1
 i 1 x i1 x i 2
...
 i 1 x i 2
...
...
...
 i 1 x iK x i 2
...
N
N
2
N
N
 i 1 x i 1 x iK 

N
 i 1 x i 2 x iK 

...

N
2
 i 1 x iK 
If there w ere a single b, w e w ould require this to be
p o sitive, w hich it w ould be; 2 x'x = 2  i 1 x i  0.
N
2
T he m atrix counterpart of a positive num ber is a
p ositive d efin ite m a trix .
3-14/27
Part 3: Least Squares Algebra
Sample Moments - Algebra
  iN1 x i21
 N
 i 1 x i 2 x i1
X 'X = 

...
 N
  i 1 x iK x i 1
=  i 1
N
 x i1 
 
x
 i2   x
i1
 ... 
 
 x ik 
 i 1 x i1 x i 2
...
 i 1 x i 2
...
...
...
 i 1 x iK x i 2
...
N
N
2
N
xi 2
...
x iK
N
 i 1 x i1 x iK 

N
 i 1 x i 2 x iK 
N
=  i 1

...

N
2
 i 1 x iK 
 x i21

 x i 2 x i1
 ...

 x iK x i 1
x i1 x i 2
...
xi 2
2
...
...
...
x iK x i 2
...
x i 1 x iK 

x i 2 x iK 
... 

2
x iK 

N
=  i 1 x i x i
3-15/27
Part 3: Least Squares Algebra
Positive Definite Matrix
M a trix C is p o sitiv e d e fin ite if a 'C a is > 0
fo r a n y a.
G e n e ra lly h a rd to c h e c k . R e q u ire s a lo o k a t
c h a ra c te ristic ro o ts (la te r in th e c o u rse ).
Fo r so m e m a tric e s, it is e a sy to v e r ify . X 'X i s
o n e o f th e se .
a 'X 'X a = ( a 'X ')( X a ) = ( X a ) '( X a ) = v 'v =

K
v 0
k=1 k
2
C o u ld v = 0 ? v = 0 m e a n s X a = 0 . Is th is p o ssib le ?
-1
C o n c lu sio n : b = ( X 'X ) X 'y d o e s in d e e d m in im ize e 'e .
3-16/27
Part 3: Least Squares Algebra
Algebraic Results - 1
In th e p o p u la tio n : E [ X '  ] = 0
In th e sa m p le :
1
N
3-17/27

N
i 1
x ie i  0
Part 3: Least Squares Algebra
Residuals vs. Disturbances
D istu rb a n c e s (p o p u la tio n ) y i  x i   i
y = E [ y| X ] + ε
P a rtitio n in g y :
= c o n d itio n a l m e a n + d is tu rb a n c e
R e sid u a ls (sa m p le )
y i  x ib  e i
P a rtitio n in g y :
y = Xb + e
=
p ro je c tio n + re s id u a l
( N o te : P ro je c tio n in to th e c o lu m n s p a c e
o f X , i.e ., th e
se t o f lin e a r c o m b in a tio n s o f th e c o lu m n s o f X - X b is o n e o f th e se . )
3-18/27
Part 3: Least Squares Algebra
Algebraic Results - 2







3-19/27
A “residual maker” M = (I - X(X’X)-1X’)
e = y - Xb= y - X(X’X)-1X’y = My
My = The residuals that result when y is regressed on X
MX = 0
(This result is fundamental!)
How do we interpret this result in terms of residuals?
When a column of X is regressed on all of X, we get a
perfect fit and zero residuals.
(Therefore) My = MXb + Me = Me = e
(You should be able to prove this.
y = Py + My, P = X(X’X)-1X’ = (I - M).
PM = MP = 0.
Py is the projection of y into the column space of X.
Part 3: Least Squares Algebra
The M Matrix
M = I- X(X’X)-1X’ is an nxn matrix
 M is symmetric – M = M’
 M is idempotent – M*M = M
(just multiply it out)
 M is singular; M-1 does not exist.
(We will prove this later as a side result
in another derivation.)

3-20/27
Part 3: Least Squares Algebra
Results when X Contains a Constant Term



X = [1,x2,…,xK]
The first column of X is a column of ones
Since X’e = 0, x1’e = 0 – the residuals sum to
zero. y  X b + e
D e fin e i  [1,1, ...,1] '  a co lu m n o f n o n e s
i'y =

N
i= 1
y i  ny
i'y  i'X b + i'e = i'X b
im p lie s (a fte r d iv id in g b y N )
y  x b (th e re g re ssio n lin e p a sse s th ro u g h th e m e a n s)
T h e se d o n o t a p p ly if th e m o d e l h a s n o co n sta n t te rm .
3-21/27
Part 3: Least Squares Algebra
Least Squares Algebra
3-22/27
Part 3: Least Squares Algebra
Least Squares
3-23/27
Part 3: Least Squares Algebra
Residuals
3-24/27
Part 3: Least Squares Algebra
Least Squares Residuals
3-25/27
Part 3: Least Squares Algebra
Least Squares Algebra-3
X
I
e
X
X X
X
M
M is NxN potentially huge
3-26/27
Part 3: Least Squares Algebra
Least Squares Algebra-4
MX =
3-27/27
Part 3: Least Squares Algebra

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