Valuation 4: Econometrics • Why econometrics? • What are the tasks? • Specification and estimation • Hypotheses testing • Example study
Dec 19, 2015
Valuation 4: Econometrics
• Why econometrics? • What are the tasks?• Specification and estimation• Hypotheses testing• Example study
Last week we looked at
• What is so special about environmental goods?• Theory of consumer demand for market goods• Welfare effects of a price change: Equivalent
variation versus compensating variation• Consumer demand for environmental goods• Welfare effects of a quantity change:
Equivalent surplus versus compensating surplus
• Theory and practise
Why econometrics?
• Analysis– To test the validity of economic theories
• Policy making– To test the outcome of different
government economic policy moves
• Forecasting or prediction– To predict the value of other variables
What are the tasks?
• Specification– From an economic model
to an econometric model
• Estimation
• Testing hypotheses• Predictions
y=f x i i1y= + +uix
1iy = + ix
Specification – the function
• Include all relevant exogenous variables• Functional form: linear relationship?• Estimates parameters for and are
constant for all observations
Specification – disturbance (2)
• Variance is constant– Homoscedasticity vs.
heteroscedasticity 2
ivar u =
Specification – disturbance (3)
• disturbances are not autocorrelated
• disturbances are normally distributed
Data and variables• Data
– Cross-section– Time-series– Panel data
• Variables– Continuous– Discrete including dummy variables– Proxy variables
Functional forms
Function Implicit Price
– Linear
– Quadratic
– Semi-log
– Logarithm
– Inverse
i i1lny= + +uix
i i1y= + (1/ )+uix
i i1lny= + ln +uix
i i1y= + +uix 1
21/ ix
11 1
ixie y
1i
i
yx
2i i1 2y= + + +ui ix x 1 22 ix
Functional forms - Diagnostics
• RESET test
• R2 is of limited use• Box-Cox test
i i1y= + +uix
1iy = + ix
2 3 4
i i1 1 2 3i i iy= + + y + y + y +u*ix
0 1 2 3H : 0
Example using the SOEP data• The German Socio-Economic Panel Study (SOEP) offers
micro data for research in the social and economic sciences • The SOEP is a wide-ranging representative longitudinal
study of Germany‘s private households in Germany and provides information on all household members
• Some of the many topics include household composition, occupational biographies, employment, earnings, health and satisfaction indicators
• The Panel was started in 1984; in 2005, there were nearly 12,000 households, and more than 21,000 persons sampled
• We use data on the level of a household for the year 1997 and perform an OLS regression with one explanatory variable
• We try to explain differences in square meter by differences in household income
1 1iy = + with , 0ix
i i1 1y= + +u with , 0ix
Example results
. use "C:\data\kdd\data1.dta", clear(SOEP'97 (Kohler/Kreuter)) . regress sqm hhinc Source | SS df MS Number of obs = 3126-------------+------------------------------ F( 1, 3124) = 694.26 Model | 986537.128 1 986537.128 Prob > F = 0.0000 Residual | 4439145.82 3124 1420.98138 R-squared = 0.1818-------------+------------------------------ Adj R-squared = 0.1816 Total | 5425682.95 3125 1736.21854 Root MSE = 37.696 ------------------------------------------------------------------------------ sqm | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- hhinc | .0165935 .0006298 26.35 0.000 .0153588 .0178283 _cons | 55.76675 1.38561 40.25 0.000 53.04995 58.48355------------------------------------------------------------------------------
Results: The estimated coefficients
• How do square meters occupied change with higher income?
• What is the estimated size given a certain income?• Are the results significant?• What does the confidence interval tell us• How does the estimated size for a household
compare to the observed size?
------------------------------------------------------------------------------ sqm | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- hhinc | .0165935 .0006298 26.35 0.000 .0153588 .0178283 _cons | 55.76675 1.38561 40.25 0.000 53.04995 58.48355------------------------------------------------------------------------------
Estimates and observed values
010
020
030
040
0
0 5000 10000 15000Haushaltseinkommen 97
Wohnungsgroesse in qm sqmhat
Results: Analysis of variance
Sum of squares• The model is able to explain only little of the TSS
(MSS=TSS-RSS)• The higher MSS and the smaller the RSS the „better“ is
our modelDegrees of freedom• We have 3125 total degrees of freedom (n-1) of which 1
is consumed by the model, leaving 3124 for the residualMean square error• Defined as the residual sum of squares divided by the
corresponding degrees of freedom
Source | SS df MS -------------+------------------------------ Model | 986537.13 1 986537.128 Residual | 4439145.82 3124 1420.981-------------+------------------------------ Total | 5425682.95 3125 1736.219
Results: Model fit
The F-statistic • Tests that all coefficients except the intercept are zero• In our example it has 1 numerator and 3124
denominator degrees of freedomThe R-squared• MSS/TSS=1-RSS/TSSThe adjusted R-squared• Takes changes in k and n
into accountThe root mean square error• Root MSE=
Number of obs = 3126F( 1, 3124) = 694.26Prob > F = 0.0000R-squared = 0.1818Adj R-squared = 0.1816Root MSE = 37.696
RSSn K
Diagnostics0
100
200
300
Dis
tanc
e a
bove
med
ian
0 20 40 60 80Distance below median
Wohnungsgroesse in qm
2 ivar u =
iE u =0 -10
00
100
200
300
Re
sid
uals
50 100 150 200 250 300Fitted values
Homoskedasticity:
Expected value:
Diagnostics - 2
. hettest
Breusch-Pagan / Cook-Weisberg test for heteroskedasticity
Ho: Constant variance
Variables: fitted values of sqm
chi2(1) = 119.04
Prob > chi2 = 0.0000
Multiple regression. regress sqm hhinc hhsize east owner
Source | SS df MS Number of obs = 3125
-------------+------------------------------ F( 4, 3120) = 442.09
Model | 1962110.21 4 490527.553 Prob > F = 0.0000
Residual | 3461836.42 3120 1109.56295 R-squared = 0.3617
-------------+------------------------------ Adj R-squared = 0.3609
Total | 5423946.63 3124 1736.21851 Root MSE = 33.31
------------------------------------------------------------------------------
sqm | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
hhinc | .0108534 .0006002 18.08 0.000 .0096766 .0120301
hhsize | 3.044151 .4817334 6.32 0.000 2.099605 3.988698
east | -9.290054 1.321768 -7.03 0.000 -11.88168 -6.69843
owner | 35.63969 1.290836 27.61 0.000 33.10872 38.17067
_cons | 48.69397 1.612865 30.19 0.000 45.53158 51.85635
------------------------------------------------------------------------------