20.11.09 phd course "Econometrics" , (C) W.G.Müller, JKU Linz 1
4. Economies of Scale
,,Wo immer etwas falsch ist, ist es
zu groß"
Leopold Kohr
(1909 -1994)
20.11.09 phd course "Econometrics" , (C) W.G.Müller, JKU Linz 2
Definition:
By how much increases a firms
product, when simultaneously all
inputs are increased by the same
factor k ?
E.g.
f(kx1, kx2) > k f(x1, x2) for k>1.
20.11.09 phd course "Econometrics" , (C) W.G.Müller, JKU Linz 3
The cost function
shows the minimum costs for which a
production level of Y can be
reached. Usually the long term
average cost curve
exhibits a minimum, which indicates
the optimal amount of production.
( ) ( )
Y
Yppc
YAC
,, 21
=
20.11.09 phd course "Econometrics" , (C) W.G.Müller, JKU Linz 4
The Cobb-Douglas production
function: (p65)
usually in the form Y=c(Ka
 Lb
).
Paul Howard Douglas
(1892 - 1976)
Johan Gustaf Knut Wicksell
(1851 - 1926)
20.11.09 phd course "Econometrics" , (C) W.G.Müller, JKU Linz 5
C-D production function for
Electricity Supply Enterprises
uxxxaY fkl
a
f
a
k
a
l0=
Marc Nerlove (1933-)
Nerlove, Marc. "Returns to Scale in
Electricity Supply." In C.Christ, ed.,
Measurement in Economics:Studies in
Mathemetical Economics and
Econometrics in Memory of Yehuda
Grunfeld. Stanford, CA: Stanford
University Press, 1963
20.11.09 phd course "Econometrics" , (C) W.G.Müller, JKU Linz 6
Implies the following cost function:
where
r = al + ak + af
measures the amount of returns to scale.
u
r
p
r
a
p
r
a
p
r
a
Y
r
ac f
f
k
k
l
l
log
1
loglogloglog
1
loglog 0 +++++=
20.11.09 phd course "Econometrics" , (C) W.G.Müller, JKU Linz 7
first least squares estimation
Dependent Variable: LOG(TC)
Method: Least Squares
Sample: 1 145
Included observations: 145
Variable Coefficient Std. Error t-Statistic Prob.
C -3.526503 1.774367 -1.987471 0.0488
LOG(Y) 0.720394 0.017466 41.24448 0.0000
LOG(PL) 0.436341 0.291048 1.499209 0.1361
LOG(PK) -0.219888 0.339429 -0.647819 0.5182
LOG(PF) 0.426517 0.100369 4.249483 0.0000
R-squared 0.925955 Mean dependent var 1.724663
Adjusted R-squared 0.923840 S.D. dependent var 1.421723
S.E. of regression 0.392356 Akaike info criterion 1.000578
Sum squared resid 21.55201 Schwarz criterion 1.103224
Log likelihood -67.54189 F-statistic 437.6863
Durbin-Watson stat 1.013062 Prob(F-statistic) 0.000000
r^ = 1/0.720394 = 1.388
20.11.09 phd course "Econometrics" , (C) W.G.Müller, JKU Linz 8
The Likelihood function (pp172)
Assumption: Gaussian errors, u ~ N(0, 2
I)
Density for variables (X1,Y1), ...,(Xn,Yn)
Likelihood-function
Log-likelihood-function
2 2 /2
2
1
( ; , , ) (2 ) exp ( )'( )
2
n
p y X y X y X    

-  
= - - - 
 
2 2 /2
2
1
( , ; , ) (2 ) exp ( )'( )
2
n
L y X y X y X    

-  
= - - - 
 
2 2
2
2
2
1
log ( , ) log(2 ) log ( )'( )
2 2 2
( )
log(2 ) log
2 2 2
n n
L y X y X
n n S
     


 

= = - - - - =
- - -

Sir Ronald Aylmer Fisher 1890-1962
20.11.09 phd course "Econometrics" , (C) W.G.Müller, JKU Linz 9
Maximum Likelihood Estimator
Derivatives:
Likelihood-equations: set derivatives zero
Maximum-Likelihood (ML)-estimator:
Note: ML-estimates for  are identical to OLS-estimates
(if noises are Gaussian)
2
2 2 4
( ) 1
2
( )
2 2
S
n S

  

  
 
= -
 

= - +



1
2
( ' ) '
1
( )'( )
X X X y
y X y X
n

  
-
=
= - -

 
Daniel Bernoulli
(1700 - 1782)
20.11.09 phd course "Econometrics" , (C) W.G.Müller, JKU Linz 10
Diagnostics based on the
likelihood (p61)
Usually functions of SSE
* Logarithmic Likelihood
* Akaike`s Informationcriterion
* Schwarz` Bayesian Informationcriterion
( )2^ ^( ) 1 log(2 ) log
2
e
n
  = - + +
^2 ( ) 2k
AIC
n n

= - +

^2 ( ) logk n
BIC
n n

= - +

Gideon E. Schwarz (1933 -2007)
Hirotugu Akaike (1927 - )
20.11.09 phd course "Econometrics" , (C) W.G.Müller, JKU Linz 11
Test against the value 1,
confidence intervals (p25)
New t-statistics from (^
i-1)/ ^(X'X)-1
ii.
Generally from a confidence interval:
[ ^
i  ^ (X'X)-1
ii t1-a/2]= [0.686, 0.755].
20.11.09 phd course "Econometrics" , (C) W.G.Müller, JKU Linz 12
The Delta-method:
Standard error for r^ = f(^
1)= 1/^
1 :
Taylorexpansion around the estimated
expected value of the desired parameter,
i.e. f()  f(^
)+ (-^
) f / and thus also
for a variance-covariance matrix Kov(f,f') 
Kov(,')', where ij= fj /j .
From the diagonal of Kôv(f,f') we yield
Vâr(r^) = (f /^
1)2
Vâr(^
1 ) = Vâr(^
1 )/ ^
1
4
and a confidence interval of [1.322, 1.454]
for r.
20.11.09 phd course "Econometrics" , (C) W.G.Müller, JKU Linz 13
The trick of Papke and Wooldridge:
One estimates the parameters untransformed and
evaluates the gradient at this value. Then the
regressors are transformed such that
xi = [xi - (^
i / ^
k)xk] for ik and xk = xk /^
k
and performs a regression on the regressand.
The desired standard error can then be simply
read of the result table in the row for xk. Since
in the present example we have ^ =
(0,1/0.72,0,0,0), one just needs to multiply
logY accordingly:
Leslie E. Papke Jeffrey M. Wooldridge
20.11.09 phd course "Econometrics" , (C) W.G.Müller, JKU Linz 14
20.11.09 phd course "Econometrics" , (C) W.G.Müller, JKU Linz 15
4.2 Constraints, the Wald-test (p31)
Linear restrictions on the
parameters, i.e. they can be
presented in the form H=h,
where H denotes a g×m
matrix with the restriction
coefficients for g constraints.
Test statistics:
TW=d'(H(X'X)-1
H')-1
/^2 ~ 2
g
Abraham Wald (1902-1950)
20.11.09 phd course "Econometrics" , (C) W.G.Müller, JKU Linz 16
W  LR  LM
20.11.09 phd course "Econometrics" , (C) W.G.Müller, JKU Linz 17
Nerloves Modell A:
( ) ( ) .log
1
/log/loglog
1
log)/log( 0 u
r
pp
r
a
pp
r
a
Y
r
apc fk
k
fl
l
f ++++=
20.11.09 phd course "Econometrics" , (C) W.G.Müller, JKU Linz 18
^
S = HS
-1
(h- HŚ^
Ś),
which in the case of only one
constraint can be fulfilled by
entering it into the equation, and
here yields 1+0.007-0.593 = 0.414.
Var(S) = HS
-1
HŚ Var(Ś) HŚ'HS
-1
'
=0.099.
20.11.09 phd course "Econometrics" , (C) W.G.Müller, JKU Linz 19
4.3 Nonlinearities, Feasible
Generalized Least Squares
20.11.09 phd course "Econometrics" , (C) W.G.Müller, JKU Linz 20
20.11.09 phd course "Econometrics" , (C) W.G.Müller, JKU Linz 21
( ) ( )21
20log( / ) log log log log / log /l k
f l f k f
a a
c p a Y Y p p p p
r r
  = + + + + +
20.11.09 phd course "Econometrics" , (C) W.G.Müller, JKU Linz 22
20.11.09 phd course "Econometrics" , (C) W.G.Müller, JKU Linz 23
A feasible GLS estimator (p88)
Hayashi, 2000 suggests the
specification Var()  0+1Y-1
.
Simple, two-stage procedure:
* V^
from the residuals as above,
* then ^
FGLS =(X'V^ -1
X)-1
X'V^ -1
y.
20.11.09 phd course "Econometrics" , (C) W.G.Müller, JKU Linz 24
Separating the data
in 5 groups of 29 observations each.
Employ dummy-variables
q1 ­ q5
Programming with Eviews.
20.11.09 phd course "Econometrics" , (C) W.G.Müller, JKU Linz 25
Dependent Variable: LOG(TC/PL)
Method: Least Squares
Date: 04/17/08 Time: 11:42
Sample: 1 145
Included observations: 145
Variable Coefficient Std. Error t-Statistic Prob.
Q1 -3.343348 1.652646 -2.023027 0.0452
Q2 -6.488974 1.825601 -3.554432 0.0005
Q3 -7.332942 2.671684 -2.744689 0.0069
Q4 -6.546049 3.025269 -2.163791 0.0324
Q5 -6.714258 2.181091 -3.078394 0.0026
Q1*LOG(Y) 0.400290 0.044370 9.021601 0.0000
Q1*LOG(PF/PL) 0.466182 0.169372 2.752424 0.0068
Q1*LOG(PK/PL) -0.081356 0.371103 -0.219227 0.8268
Q2*LOG(Y) 0.658151 0.150263 4.380006 0.0000
Q2*LOG(PF/PL) 0.528265 0.189693 2.784841 0.0062
Q2*LOG(PK/PL) 0.377936 0.357290 1.057785 0.2922
Q3*LOG(Y) 0.938279 0.313146 2.996304 0.0033
Q3*LOG(PF/PL) 0.347734 0.246322 1.411703 0.1605
Q3*LOG(PK/PL) 0.250008 0.295837 0.845087 0.3997
Q4*LOG(Y) 0.912044 0.279177 3.266899 0.0014
Q4*LOG(PF/PL) 0.399690 0.158800 2.516935 0.0131
Q4*LOG(PK/PL) 0.093352 0.426200 0.219033 0.8270
Q5*LOG(Y) 1.044390 0.135455 7.710217 0.0000
Q5*LOG(PF/PL) 0.686848 0.277439 2.475674 0.0146
Q5*LOG(PK/PL) -0.289436 0.364519 -0.794021 0.4287
R-squared 0.957322 Mean dependent var 1.052996
Adjusted R-squared 0.950835 S.D. dependent var 1.412561
S.E. of regression 0.313208 Akaike info criterion 0.643546
Sum squared resid 12.26243 Schwarz criterion 1.054130
Log likelihood -26.65706 Durbin-Watson stat 1.774708
20.11.09 phd course "Econometrics" , (C) W.G.Müller, JKU Linz 26
4.4 Multicollinearity (pp43)
Perfekt collinearity: X`X is singular.
(Dummy variable trap)
Almost collinearity: some
regressors are strongly linearly
dependent: leads to imprecise
estimation.
(but OLS remains the BLUE!)
20.11.09 phd course "Econometrics" , (C) W.G.Müller, JKU Linz 27
Effects
on the variances of the coefficients:
thus depends upon
* variance of noise
* variance of regressors
* number of observations
* collinearity of regressors
( )
2
2
1
^( )
(1 )
j T
j ij j
i
Var
R x x


=
=
- -
20.11.09 phd course "Econometrics" , (C) W.G.Müller, JKU Linz 28
Micronumerosity
= not enough data!
Statistically everything fine.
No problem for predictions!
Arthur S. Goldberger
20.11.09 phd course "Econometrics" , (C) W.G.Müller, JKU Linz 29
Indicators for multicollinearity
* high R2
and little signifikances
* high correlations among regressors
* not robust against data changes
* Condition number of the design
matrix:
CI = [max eig(X`X) / min eig(X`X)]
20.11.09 phd course "Econometrics" , (C) W.G.Müller, JKU Linz 30
Supporting regressions (p43)
Regressions of single regressors on
all the rest, leads to Rj
2
.
Klein's rule of thumb: Rj
2
should
be all less than R2
.
This leads to variance inflation
factor: VIF = (1- Rj
2
)-1
Lawrence Robert Klein (1920 - )
20.11.09 phd course "Econometrics" , (C) W.G.Müller, JKU Linz 31
20.11.09 phd course "Econometrics" , (C) W.G.Müller, JKU Linz 32
Remedies for multicollinearity
* Do nothing!
* More data!
* Formalising of relationships between regressors
(equation systems)
* Spezification of relationships between
parameters (restrictions and particular lag
structures)
* Incorporate estimates from other studies or
prior information (Bayesian estimation)
* Form principal components
* (remove variabls ­ leads to bias)
* Shrinkage-estimators
20.11.09 phd course "Econometrics" , (C) W.G.Müller, JKU Linz 33
Ridge Regression
Estimate a regularised system:
(Hoerl & Kennard, 1970)
Is equivalent to the constraint:
´ = c2
( )
1
^ T T
R X X kI X Y
=
+
Andrei N. Tikhonov
(1906 ­ 1993)
20.11.09 phd course "Econometrics" , (C) W.G.Müller, JKU Linz 34
For c2
=1.5
Dependent Variable: LOG(TC/PF)
Method: Least Squares
Date: 03/20/08 Time: 19:02
Sample: 1 145
Included observations: 145
Convergence achieved after 94 iterations
LOG(TC/PF) =-@SQRT(1.5-C(1)^2-C(2)^2-C(3)^2-C(4)^2)+C(4)* LOG(Y)
+C(1)*LOG(PK/PF)+C(2)*LOG(PL/PF)+C(3)*LOG(Y)^2/2
Coefficient Std. Error t-Statistic Prob.
C(1) -0.484111 0.137611 -3.517969 0.0006
C(2) 1.032177 0.153576 6.720935 0.0000
C(3) 0.143087 0.009481 15.09227 0.0000
C(4) -0.126783 0.048722 -2.602153 0.0103
R-squared 0.946138 Mean dependent var -1.484195
Adjusted R-squared 0.944992 S.D. dependent var 1.482087
S.E. of regression 0.347604 Akaike info criterion 0.751693
Sum squared resid 17.03682 Schwarz criterion 0.833810
Log likelihood -50.49773 Durbin-Watson stat 1.501916
20.11.09 phd course "Econometrics" , (C) W.G.Müller, JKU Linz 35
The Lasso
is equivalent to the constraint:
|i|  c
special case of least angle
regression (LARS)
removes unnecessary
regressors
Robert Tibshirani
20.11.09 phd course "Econometrics" , (C) W.G.Müller, JKU Linz 36
Lasso c=3.6
Dependent Variable: LOG(TC/PF)
Method: Least Squares
Date: 03/20/08 Time: 15:18
Sample: 1 145
Included observations: 145
Convergence achieved after 4 iterations
LOG(TC/PF) =-(3.6-@ABS(C(1))-@ABS(C(2))-@ABS(C(3))-@ABS(C(4)))
+C(1)* LOG(Y)+C(2)*LOG(PK/PF)+C(3)*LOG(PL/PF)+C(4)*LOG(Y)^2/2
Coefficient Std. Error t-Statistic Prob.
C(1) 0.100061 0.055717 1.795878 0.0747
C(2) -0.154582 0.088768 -1.741419 0.0838
C(3) 0.739026 0.085006 8.693801 0.0000
C(4) 0.108797 0.009991 10.88994 0.0000
R-squared 0.957058 Mean dependent var -1.484195
Adjusted R-squared 0.956144 S.D. dependent var 1.482087
S.E. of regression 0.310375 Akaike info criterion 0.525124
Sum squared resid 13.58286 Schwarz criterion 0.607241
Log likelihood -34.07153 Durbin-Watson stat 1.682427
20.11.09 phd course "Econometrics" , (C) W.G.Müller, JKU Linz 37
EViews Standard-Output
Dependent Variable: LOG_M/V0
Method: Least Squares
Date: 01/24/08 Time: 13:54
Sample: 1 36
Included observations: 36
Variable Coefficient Std. Error t-Statistic Prob.
1/V0 0.021083 0.193736 0.108825 0.9141
LOG_Y/V0 0.882881 0.048982 18.02473 0.0000
RN/V0 -11.21374 3.285429 -3.413172 0.0019
GY/V0 -0.212149 0.291022 -0.728979 0.4717
S/V0 0.205901 0.027388 7.517951 0.0000
W/V0 0.013782 0.005821 2.367800 0.0245
R-squared 0.996072 Mean dependent var 7.333740
Adjusted R-squared 0.995417 S.D. dependent var 1.530226
S.E. of regression 0.103593 Akaike info criterion -1.545673
Sum squared resid 0.321948 Schwarz criterion -1.281753
Log likelihood 33.82211 Durbin-Watson stat 1.503403
20.11.09 phd course "Econometrics" , (C) W.G.Müller, JKU Linz 38
Adjusted coefficient of
determination (p23)
* Adding a regressor to a model yields:
­ R2
is increased
­ Increase of R2
not necessarily means that the new regressor
is relevant!
* Adjusted coefficient of determination:
* Suitable for the comparison of models
* For large n we have (n-1)/(n-k)  1
2 1
1
n RSS
R
n k TSS
=
-
-
2 2
R R<
20.11.09 phd course "Econometrics" , (C) W.G.Müller, JKU Linz 39
The Durbin-Watson-Test (1950)
(pp110)
Null hypothesis: no autocorrelation of first order
Test statistic:
* For positive autocorrelation d lies in the interval
(0,2)
* for negative in the interval (2,4);
* if d is close to 2, there is no significant
autocorrelation in the noise;
* for values of d close to 0 or 4 the errors are
highly correlated.
2
12
12
1
^ ^( )
^2(1 )
^
n
t tt
n
tt
d
 


-=
=
=
 -


Geoffrey Stuart Watson (1921 ­1998)
James Durbin (1923 ­ )
20.11.09 phd course "Econometrics" , (C) W.G.Müller, JKU Linz 40
Durbin-Watson-Test, critical values
* the citical values for d depend upon matrix X.
* Thus D & W have lower (dL) and upper bounds (dU)
for the critical values
­ d< dL: H0 is rejected
­ d> dU: H0 is not rejected
­ dL < d < dU: no decision
Critical bounds
for  = 0.05:
n k=2 k=3 k=10
dL dU dL dU dL dU
15 1.08 1.36 0.95 1.54 0.17 3.22
20 1.20 1.41 1.10 1.54 0.42 2.70
100 1.65 1.69 1.63 1.71 1.48 1.87
20.11.09 phd course "Econometrics" , (C) W.G.Müller, JKU Linz 41
Ramsey`s (1969) RESET-Test
(p66)
RESET: Regression Equation Specification Error
For checking of the functional form of the model.
Tests H0:  = 0 für Y = X + Z + v with
If H0 is true, the functional form (linearity) of the response
is correct.
Tests:
­ t-Test, if g = 1
­ F-Test
­ asymptotical Chi-square-test
(test statistic gF)
2 3^ ^( , ,...)t t tz Y Y =
James Bernard Ramsey
20.11.09 phd course "Econometrics" , (C) W.G.Müller, JKU Linz 42
The Ten Commandments of
Applied Econometrics
* Thou shalt use common sense and economic theory.
* Thou shalt ask the right questions.
* Thou shalt know the context.
* Thou shalt inspect the data.
* Thou shalt not worship complexity.
* Thou shalt look long and hard at thy results.
* Thou shalt beware the costs of data mining.
* Thou shalt be willing to compromise.
* Thou shalt not confuse significance with substance.
* Thou shalt confess in the presence of sensitivity. Peter Kennedy