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EXAMPLE: Three-variables, 10 observations
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With Intercept
I. Data and Summary Stats
Three-variables, 10 observations
Observations : n=10
Independent Variables : k=3
With Intercept
Data Table
| obs | yi | intcpt | xi,1 | xi,2 | xi,3
| | 1. | 1 | 1 | 73 | 11 | 27
| | 2. | 80 | 1 | 152 | 170 | 90
| | 3. | 56 | 1 | 141 | 132 | 62
| | 4. | 23 | 1 | 53 | 31 | 45
| | 5. | 73 | 1 | 99 | 172 | 152
| | 6. | 74 | 1 | 133 | 108 | 158
| | 7. | 39 | 1 | 44 | 97 | 84
| | 8. | 38 | 1 | 43 | 86 | 98
| | 9. | 38 | 1 | 133 | 49 | 100
| | 10. | 46 | 1 | 50 | 114 | 126
| | | | | |
| | sum | 468 | 10 | 921 | 970 | 942
| | mean | 46.80 | 1 | 92.10 | 97 | 94.20
| | StD ≡ σ | 24.71 | 0
| 44.43
| 54.53
| 42.90
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Means and Standard Deviations
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| Mean
Var
StD
|
| Mx= Σxi/n
Varx≡σx2 = Σ(x-Mx)2 / n-1
StDx≡σx=Varx1/2
| y
| 46.80
| 610.4
| 24.71
|
| x1
| 92.10
| 1974
| 44.43
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| x2
| 97
| 2974
| 54.53
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| x3
| 94.20
| 1841
| 42.90
| | | | | | |
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 | x3
| | y | 610.4 | 668.7 | 1225 | 781.4
| | x1 | 668.7 | 1974 | 1015 | 399.1
| | x2 | 1225 | 1015 | 2974 | 1428
| | x3 | 781.4 | 399.1 | 1428 | 1841
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Correlation Matrix -- Corr(xi,xj)=Σ[(xi-Mxi)(xj-Mxj)] / (n-1)σiσj
| y | x1 | x2 | x3
| | y | 1.000 | 0.609 | 0.909 | 0.737
| | x1 | 0.609 | 1.000 | 0.419 | 0.209
| | x2 | 0.909 | 0.419 | 1.000 | 0.610
| | x3 | 0.737 | 0.209 | 0.610 | 1.000
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The basic input matrices are:
y =
(10x1)
1
80
56
23
73
74
39
38
38
46
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| |
|
|
X =
(10x4)
| 1 | 73 | 11 | 27
| | 1 | 152 | 170 | 90
| | 1 | 141 | 132 | 62
| | 1 | 53 | 31 | 45
| | 1 | 99 | 172 | 152
| | 1 | 133 | 108 | 158
| | 1 | 44 | 97 | 84
| | 1 | 43 | 86 | 98
| | 1 | 133 | 49 | 100
| | 1 | 50 | 114 | 126
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| |
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|
X' =
(4x10)
| 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1
| | 73 | 152 | 141 | 53 | 99 | 133 | 44 | 43 | 133 | 50
| | 11 | 170 | 132 | 31 | 172 | 108 | 97 | 86 | 49 | 114
| | 27 | 90 | 62 | 45 | 152 | 158 | 84 | 98 | 100 | 126
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II. Regression Calculations
yi = alpha + b1 xi,1 + b2 xi,2 + b3 xi,3 + ui
The q.c.e. basic equation in matrix form is:
y = Xb + e
where y (dependent variable) is (nx1) or (10x1)
X (independent vars) is (nxk) or (10x4)
b (betas) is (kx1) or (4x1)
e (errors) is (nx1) or (10x1)
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 (4x1)
(2) X'X Matrix (4x4)
| 10
| 921
| 970
| 942
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| 921
| 102587
| 98473
| 90350
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| 970
| 98473
| 120856
| 104224
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| 942
| 90350
| 104224
| 105302
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(3) Determinant Det(X'X)≡|X'X|
i.e. the determinant of matrix of X'X
Det(X'X) = 40582798368264
Det(X'X) = 10*102587*120856*105302 - 10*102587*104224*104224 - 10*98473*98473*105302
... + 10*98473*104224*90350 + 10*90350*98473*104224 - 10*90350*120856*90350
... - 921*921*120856*105302 + 921*921*104224*104224 + 921*98473*970*105302
... - 921*98473*104224*942 - 921*90350*970*104224 + 921*90350*120856*942
... + 970*921*98473*105302 - 970*921*104224*90350 - 970*102587*970*105302
... + 970*102587*104224*942 + 970*90350*970*90350 - 970*90350*98473*942
... - 942*921*98473*104224 + 942*921*120856*90350 + 942*102587*970*104224
... - 942*102587*120856*942 - 942*98473*970*90350 + 942*98473*98473*942
(4) Cofactors(X'X) i.e. cofactor matrix of X 'X (4x4)
38095748040474-1742822344208070420582-199245278238
-1742822344202782723496-1051886916212594812
8070420582-10518869162813520690-1954385802
-199245278238212594812-19543858023919752854
| | | | | | | | | | | | | | | |
(5) Adj(X'X) i.e. adjugate matrix of X'X, this is just the
transpose of the cofactor matrix. (4x4)
For a symmetric matrix, will be same as cofactor matrix.
38095748040474-1742822344208070420582-199245278238
-1742822344202782723496-1051886916212594812
8070420582-10518869162813520690-1954385802
-199245278238212594812-19543858023919752854
| | | | | | | | | | | | | | | |
(6) Inverse Matrix, inv(X'X)≡(X'X)-1
= adj(X'X)/|X'X| = adj(X'X)/40582798368264 (4x4)
0.9387-0.0042940.000199-0.004910
-0.0042940.000069-0.0000260.000005
0.000199-0.0000260.000069-0.000048
-0.0049100.000005-0.0000480.000097
| | | | | | | | | | | | | | | |
(7) Beta Matrix (β)
b = [X'X-1].[X'y] , this is (4x1).
Finally we can calculate b through matrix multiplication.
Betas
| alpha | -11.38
| | β 1 | 0.1637
| | β 2 | 0.2697
| | β 3 | 0.1798
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| =
| X'X-1
| 0.9387 | -0.004294 | 0.000199 | -0.004910
| | -0.004294 | 0.000069 | -0.000026 | 0.000005
| | 0.000199 | -0.000026 | 0.000069 | -0.000048
| | -0.004910 | 0.000005 | -0.000048 | 0.000097
|
| X
| X'y
|
Yhat1= + -11.38x1 + 0.1637x73 + 0.2697x11 + 0.1798x27 = 8.3958
Yhat2= + -11.38x1 + 0.1637x152 + 0.2697x170 + 0.1798x90 = 75.5397
Yhat3= + -11.38x1 + 0.1637x141 + 0.2697x132 + 0.1798x62 = 58.4556
Yhat4= + -11.38x1 + 0.1637x53 + 0.2697x31 + 0.1798x45 = 13.7514
Yhat5= + -11.38x1 + 0.1637x99 + 0.2697x172 + 0.1798x152 = 78.5498
Yhat6= + -11.38x1 + 0.1637x133 + 0.2697x108 + 0.1798x158 = 67.9360
Yhat7= + -11.38x1 + 0.1637x44 + 0.2697x97 + 0.1798x84 = 37.0900
Yhat8= + -11.38x1 + 0.1637x43 + 0.2697x86 + 0.1798x98 = 36.4772
Yhat9= + -11.38x1 + 0.1637x133 + 0.2697x49 + 0.1798x100 = 41.5950
Yhat10= + -11.38x1 + 0.1637x50 + 0.2697x114 + 0.1798x126 = 50.2095
ESS
=(8.396 - 46.80)^2
=(75.54 - 46.80)^2
=(58.46 - 46.80)^2
=(13.75 - 46.80)^2
=(78.55 - 46.80)^2
=(67.94 - 46.80)^2
=(37.09 - 46.80)^2
=(36.48 - 46.80)^2
=(41.59 - 46.80)^2
=(50.21 - 46.80)^2
=5223.25946426799
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,1-11.38x1 = 0.1637x73 + 0.2697x11 + 0.1798x27 | = 8.396 | 1 | 46.80 | 54.70 | 1475 | 2098 | e1 = 1 - 8.396 = -7.396 | 54.70
|
| 2
| Yhat2 = Σβixi,2 + -11.38x1 = 0.1637x152 + 0.2697x170 + 0.1798x90 | = 75.54 | 80 | 46.80 | 19.89 | 826.0 | 1102 | e2 = 80 - 75.54 = 4.460 | 19.89
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| 3
| Yhat3 = Σβixi,3 + -11.38x1 = 0.1637x141 + 0.2697x132 + 0.1798x62 | = 58.46 | 56 | 46.80 | 6.030 | 135.9 | 84.64 | e3 = 56 - 58.46 = -2.456 | 6.030
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| 4
| Yhat4 = Σβixi,4 + -11.38x1 = 0.1637x53 + 0.2697x31 + 0.1798x45 | = 13.75 | 23 | 46.80 | 85.54 | 1092 | 566.4 | e4 = 23 - 13.75 = 9.249 | 85.54
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| 5
| Yhat5 = Σβixi,5 + -11.38x1 = 0.1637x99 + 0.2697x172 + 0.1798x152 | = 78.55 | 73 | 46.80 | 30.80 | 1008 | 686.4 | e5 = 73 - 78.55 = -5.550 | 30.80
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| 6
| Yhat6 = Σβixi,6 + -11.38x1 = 0.1637x133 + 0.2697x108 + 0.1798x158 | = 67.94 | 74 | 46.80 | 36.77 | 446.7 | 739.8 | e6 = 74 - 67.94 = 6.064 | 36.77
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| 7
| Yhat7 = Σβixi,7 + -11.38x1 = 0.1637x44 + 0.2697x97 + 0.1798x84 | = 37.09 | 39 | 46.80 | 3.648 | 94.28 | 60.84 | e7 = 39 - 37.09 = 1.910 | 3.648
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| 8
| Yhat8 = Σβixi,8 + -11.38x1 = 0.1637x43 + 0.2697x86 + 0.1798x98 | = 36.48 | 38 | 46.80 | 2.319 | 106.6 | 77.44 | e8 = 38 - 36.48 = 1.523 | 2.319
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| 9
| Yhat9 = Σβixi,9 + -11.38x1 = 0.1637x133 + 0.2697x49 + 0.1798x100 | = 41.59 | 38 | 46.80 | 12.92 | 27.09 | 77.44 | e9 = 38 - 41.59 = -3.595 | 12.92
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| 10
| Yhat10 = Σβixi,10 + -11.38x1 = 0.1637x50 + 0.2697x114 + 0.1798x126 | = 50.21 | 46 | 46.80 | 17.72 | 11.62 | 0.6400 | e10 = 46 - 50.21 = -4.210 | 17.72 | | | | | | RSS =
Σ(yobs - yhatobs)2 | ESS =
Σ(yhatobs - My)2 | TSS =
Σ(yobs - My)2 | | e'e=Σeobs2 | | sum-> | | | | | 270.3 | 5223 | 5494 | | 270.3
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(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 = 45.0567559553338
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| Coef value
| StD(β)
| tStat(β)
|
| alpha =
| 0.9387 * 468
+ -0.004294 * 49121
+ 0.000199 * 56421
+ -0.004910 * 51118
= -11.38
| (45.06 * 0.9387)1/2
= 6.504
| -11.38 / 6.504
= -1.750
|
| β1 =
| -0.004294 * 468
+ 0.000069 * 49121
+ -0.000026 * 56421
+ 0.000005 * 51118
= 0.1637
| (45.06 * 0.000069)1/2
= 0.05558
| 0.1637 / 0.05558
= 2.946
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| β2 =
| 0.000199 * 468
+ -0.000026 * 49121
+ 0.000069 * 56421
+ -0.000048 * 51118
= 0.2697
| (45.06 * 0.000069)1/2
= 0.05589
| 0.2697 / 0.05589
= 4.825
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| β3 =
| -0.004910 * 468
+ 0.000005 * 49121
+ -0.000048 * 56421
+ 0.000097 * 51118
= 0.1798
| (45.06 * 0.000097)1/2
= 0.06597
| 0.1798 / 0.06597
= 2.726
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(12) Table of Outputs:| yobs = | alpha | + | β1 | X obs,1 + | β2 | X obs,2 + | β3 | X obs,3 + | eobs | | -11.38 | | 0.1637 | | 0.2697 | | 0.1798 | | | | (-1.750) | | (2.946) | | (4.825) | | (2.726) | | <- tstats |
r2 = 0.950790 | adj r2 = 0.926185 |
(13) RSS = Sum{y - y_hat }^2 = 270.340535732003
TSS = Sum{y - y_avg }^2 = 5493.6
ESS(a)= Sum{y_hat - y_avg }^2 = 5223.25946426799
we use the ESSb (below) cuz smthn wrng w ESS when no intercept.
ESS(b)= TSS-RSS 5223.259464268
note: TSS = ESS + RSS
(14) r2 = ESS/TSS = 0.950789912674384
(15) adjusted r2 = ESS/TSS = 0.926184869011576
(16) F-stat = [ESS/(k-1)] / [RSS/(n-k)] = 38.6420737839032
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 = 2.97657903346925
________________________________________________________
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.
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