Regression Examples

Example : alpha=1 alpha=0 One-Variable Regression Example From J. Johnston (1984) p19. With Intercept No Intercept Two-Variable Regression Example From J. Johnston (1984) p178. With Intercept No Intercept Three-Independent Variables Regression Example With Intercept No Intercept Three-variables, 20 observations With Intercept No Intercept Three-variables, 10 observations With Intercept No Intercept

         



EXAMPLE:   Two-Variable Regression Example
     |     
No Intercept


From J. Johnston (1984) p178.


I. Data and Summary Stats 
Two-Variable Regression Example
Observations :  n=5
Independent Variables :  k=2
No Intercept


Data Table 

 obs
yi
xi,1
xi,2
1. 
3
3
5
2. 
1
1
4
3. 
8
5
6
4. 
3
2
4
5. 
5
4
6

sum 
20
15
25
mean 
4
3
5
StD ≡ σ 
2.646
1.581
1





Means and Standard Deviations 



 Mean 
 Var 
 StD 


 Mx= Σxi/n   
 Varx≡σx2 = Σ(x-Mx)2 / n-1 
 StDx≡σx=Varx1/2

y
4
7.000
2.646

x1
3
 2.500
 1.581

x2
5
 1
 1



Covariance Matrix  -- Cov(xi,xj)=Σ[(xi-Mxi)(xj-Mxj)] / n-1 
     NOTE: be careful of MS Excel's COVAR() function,
     which divides by n instead of n-1.

  y
  x1
  x2
y
7
4
2.250
x1
4
2.500
1.500
x2
2.250
1.500
1



Correlation Matrix  -- Corr(xi,xj)=Σ[(xi-Mxi)(xj-Mxj)] / (n-1)σiσj

  y
  x1
  x2
y
 1.000
 0.956
 0.850
x1
 0.956
 1.000
 0.949
x2
 0.850
 0.949
 1.000



The basic input matrices are: 



 
  y =  
 (5x1)   

3
1
8
3
5


   


  X = 
 (5x2)

3
5
1
4
5
6
2
4
4
6



   

 

   X' = 
 (2x5)   


3
1
5
2
4
5
4
6
4
6









II. Regression Calculations

yi =  b1 xi,1 +  b2 xi,2 +  ui

The q.c.e. basic equation in matrix form is: 
  y = Xb + e
  where y (dependent variable) is (nx1) or  (5x1)
        X (independent vars) is (nxk) or  (5x2)
        b (betas) is (kx1) or  (2x1)
        e (errors) is (nx1) or  (5x1)
Minimizing sum or squared errors using calculus results in the OLS eqn:
  b=(X'X)-1.X'y
To minimize the sum of squared errors of
a k dimensional line that describes the relationship 
between the k independent variables and y we
find the set of slopes (betas) that minimizes
Σi=1 to nei2
Re-written in linear algebra we seek to min e'e
Rearranging the regression model equation, we get e = y - Xb
So e'e = (y-Xb)'(y-Xb) = y'y - 2b'X'y + b'X'Xb   (see Judge et al (1985) p14 )
Differentiating by b we get 0 = - 2X'y + X'Xb -> 2X'Xb=2X'y
Rearranging, dividing both sides by 2 -> b = X'X-1X'y
So to obtain the elements of the (kx1) vector b we need the elements 
of the (kxk) matrix X'X-1 and of the (kx1) matrix X'y. 
Caclulating X'y is easy (see (1) below) but X'X-1 requires 
first calculation of X'X then finding cofactors -- see (4) -- and 
the deteminant - see (3) - in order to invert.



(1) X'y Matrix  (2x1)


76
109



(2) X'X Matrix  (2x2)



55	
81	

81	
129	


(3) Determinant   Det(X'X)≡|X'X|  
    i.e. the determinant of matrix of X'X 

Det(X'X) = 534
Det(X'X) =    55*129
         - 81*81
        

(4) Cofactors(X'X) i.e. cofactor matrix of X 'X  (2x2)

129
-81
-81
55


(5) Adj(X'X) i.e. adjugate matrix of X'X, this is just the 
    transpose of the cofactor matrix.  (2x2)
    For a symmetric matrix, will be same as cofactor matrix.

129
-81
-81
55



(6) Inverse Matrix, inv(X'X)≡(X'X)-1
    = adj(X'X)/|X'X| = adj(X'X)/534   (2x2)

0.2416
-0.1517
-0.1517
0.1030


(7) Beta Matrix (β)
    b = [X'X-1].[X'y] , this is  (2x1).
    Finally we can calculate b through matrix multiplication.

 Betas 

 β 1 
 1.826 
 β 2 
 -0.3015 

   =    
X'X-1 

 0.2416 
 -0.1517 
 -0.1517 
 0.1030 

   X   
 X'y 
 76 
 109 



Yhat1= + 1.826x3 + -0.3015x5 = 3.9700
Yhat2= + 1.826x1 + -0.3015x4 = 0.6199
Yhat3= + 1.826x5 + -0.3015x6 = 7.3202
Yhat4= + 1.826x2 + -0.3015x4 = 2.4457
Yhat5= + 1.826x4 + -0.3015x6 = 5.4944


ESS
=(3.970 - 4)^2
=(0.6199 - 4)^2
=(7.320 - 4)^2
=(2.446 - 4)^2
=(5.494 - 4)^2
=27.0992509363296

REPORT 


 obs 
 calculation of yhatobs
yhatobs = Σβixi,obs
 yhatobs
a
 yobs
 (data)
 Meany
 (yobs - yhatobs)2
 (yhatobs - My)2
 (yobs - My)2
a
eobs=yobs-yhatobs
eobs2

1
Yhat1 = Σβixi,1 = 1.826x3 + -0.3015x5
 = 3.970
3
4
0.9410
0.000898
1
 e1 = 3 - 3.970 = -0.9700
0.9410

2
Yhat2 = Σβixi,2 = 1.826x1 + -0.3015x4
 = 0.6199
1
4
0.1445
11.43
9
 e2 = 1 - 0.6199 = 0.3801
0.1445

3
Yhat3 = Σβixi,3 = 1.826x5 + -0.3015x6
 = 7.320
8
4
0.4621
11.02
16
 e3 = 8 - 7.320 = 0.6798
0.4621

4
Yhat4 = Σβixi,4 = 1.826x2 + -0.3015x4
 = 2.446
3
4
0.3073
2.416
1
 e4 = 3 - 2.446 = 0.5543
0.3073

5
Yhat5 = Σβixi,5 = 1.826x4 + -0.3015x6
 = 5.494
5
4
0.2444
2.233
1
 e5 = 5 - 5.494 = -0.4944
0.2444
RSS = 
Σ(yobs - yhatobs)2
ESS = 
Σ(yhatobs - My)2
TSS = 
Σ(yobs - My)2
e'e=Σeobs2
sum->
2.099
27.10
28
2.099




(11) Betas and their t-Stats
     from the covar matrix of b=σ2(X'X)-1
     the var(βi) = σ2vii where vii is the ith diag element of X'X-1
     where σ2 = e'e / n-k  (k=num of ind vars plus 1 for the intercept if present).
     and where vii is the ith diag element of X'X-1
     Std(βi) = sqr root of Var(βi) 
     TStat(βi) = βi / Std(βi)
     Estimate of σ2 = 0.699750312109863



 Coef value 
 StD(β)
 tStat(β)

β1 = 
   0.2416 * 76
 + -0.1517 * 109
 = 1.826
(0.6998 * 0.2416)1/2 
 = 0.4111
1.826 / 0.4111
 = 4.441

β2 = 
   -0.1517 * 76
 + 0.1030 * 109
 = -0.3015
(0.6998 * 0.1030)1/2 
 = 0.2685
-0.3015 / 0.2685
 = -1.123


(12) Table of Outputs:
 yobs =
 β1
 X obs,1 + 
 β2
 X obs,2 + 
 eobs
 1.826
 
 -0.3015
 
(4.441)
(-1.123)
 <- tstats 
 r2 = 0.925027    |    adj r2 = 0.850054


(13)  RSS   = Sum{y     - y_hat }^2  = 2.09925093632959
      TSS   = Sum{y     - y_avg }^2  = 28
      ESS(a)= Sum{y_hat - y_avg }^2  = 27.0992509363296
           we use the ESSb (below) cuz smthn wrng w ESS when no intercept.
      ESS(b)= TSS-RSS 25.9007490636704
      note:  TSS = ESS + RSS 
(14)  r2 = ESS/TSS = 0.925026752273943
(15)  adjusted r2 = ESS/TSS = 0.850053504547887
(16)  F-stat = [ESS/(k-1)] / [RSS/(n-k)] = 18.507136485281
               see Johnston(1984) p186
               F measures the joint significance of 
               all explanatory variables.
      Alternatively:  F-stat = r2/(k-1) / (1-r2)/(n-k)
(17)  Durbin-Watson Statistic (DW or d) measures autocorrelation.
      DW = 1.18890207242953
 ________________________________________________________

Note, RSS, ESS and TSS stand for ... 
      Residual Sum of Squares (RSS),
      Explained Sum of Squares (ESS), and 
      Total Sum of Squares (TSS).
      However ESS is sometimes referred to as the Regression Sum of Squares.
      and     RSS is sometimes referred to as the Sum of Squares Rresidual.
Note, an alternative way of calculating TSS, ESS is...
      TSS = y'Ay   
      ESS = bv'Xv'Ay  where bv' Xv' are b' & X' wo intercept row col
      RSS = TSS-ESS 

Bibliography 

J. Johnston (1984) Econometric Methods, 3rd ed.
Judge et al (1985) The Theory and Practice of Econometrics 2rd ed, Wiley, New York.
Donald F. Morrison (1990) Multivariate Statistical Methods, 3rd edition, McGraw Hill, New York. 
A. H. Studenmund (1997) Using Econometrics: A Practical Guide, 3rd edition.  Addison-Wesley, Reading.