Econometrics - Lecture 3
Regression Models: Interpretation and Comparison


Contents
nThe Linear Model: Interpretation
nSelection of Regressors
nSelection Criteria
nComparison of Competing Models
nSpecification of the Functional Form
nStructural Break
n
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Economic Models
nDescribe economic relationships (not only a set of observations), have an economic interpretation
nLinear regression model:
n yi = b1 + b2xi2 + … + bKxiK + ei = xi’b + εi
nVariables Y, X2, …, XK: observable
nObservations: yi, xi2, …, xiK, i = 1, …, N
nError term εi (disturbance term) contains all influences that are not included explicitly in the
model; unobservable
nAssumption (A1), i.e., E{εi | X} = 0 or E{εi | xi} = 0, gives
q E{yi | xi} = xi‘β
qthe model describes the expected value of yi given xi  (conditional expectation)
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Example: Wage Equation
nWage equation (Verbeek’s dataset “wages1”)
n wagei = β1 + β2 malei + β3 schooli + β4 experi + εi
n Answers questions like:
qExpected wage p.h. of a female with 12 years of education and 10 years of experience
nWage equation fitted to all 3294 observations
n wagei = -3.38 + 1.34*malei + 0.64*schooli + 0.12*experi
qExpected wage p.h. of a female with 12 years of education and 10 years of experience: 5.50 USD
n wagei = -3.38 + 1.34*0 + 0.64*12 + 0.12*10 = 5.50
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Regression Coefficients
nLinear regression model:
n yi = b1 + b2xi2 + … + bKxiK + ei = xi’b + εi
nCoefficient bk measures the change of Y if Xk changes by one unit
n
n for ∆xk = 1
n
nFor continuous regressors
n
n
n Marginal effect of changing Xk on Y
nCeteris paribus condition: measuring the effect of a change of Y due to a change ∆xk = 1 by
bk implies
qknowledge which other Xi, i ǂ k, are in the model
qthat all other Xi, i ǂ k, remain unchanged
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Example: Coefficients of Wage Equation
nWage equation
n wagei = β1 + β2 malei + β3 schooli + β4 experi + εi
n β3 measures the impact of one additional year at school upon a person’s wage, keeping gender and
years of experience fixed
n
n
nWage equation fitted to all 3294 observations
n wagei = -3.38 + 1.34*malei + 0.64*schooli + 0.12*experi
nOne extra year at school, e.g., at the university, results in an increase of 64 cents; a 4-year
study results in an increase of 2.56 USD of the wage p.h.
nThis is true for otherwise (gender, experience) identical people
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Regression Coefficients, cont’d
nThe marginal effect of a changing regressor may depend on other variables
nExamples
nWage equation: wagei = β1 + β2 malei + β3 agei + β4 agei2 + εi
n the impact of changing age depends on age:
n
n
nWage equation may contain β3 agei + β4 agei malei: marginal effect of age depends upon gender
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Elasticities
nElasticity: measures the relative change in the dependent variable Y due to a relative change in
Xk
nFor a linear regression, the elasticity of Y with respect to Xk is
n
n
n
nFor a loglinear model with (log xi)’ = (1, log xi2,…, log xik)
n log yi = (log xi)’ β + εi
n elasticities are the coefficients β (see slide 10)
n
n
n
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Example: Wage Elasticity
nWage equation, fitted to all 3294 observations:
n log(wagei) = 1.09 + 0.20 malei + 0.19 log(experi)
nThe coefficient of log(experi) measures the elasticity of wages with respect to experience:
n100% more years of experience result in an increase of wage by 0.19 or a 19% higher wage
n10% more years of experience result in a 1.9% higher wage
n
n
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Elasticities, continues slide 8
nThis follows – for log yi = (log xi)’ β + εi – from
n
n
n
n
n
n
nand
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Semi-Elasticities
nSemi-elasticity: measures the relative change in the dependent variable Y due to an (absolute)
one-unit-change in Xk
nLinear regression for
n log yi = xi’ β + εi
n the elasticity of Y with respect to Xk is
n
n
n
n βk measures the relative change in Y due to a change in Xk by one unit
nβk is called semi-elasticity of Y with respect to Xk
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Example: Wage Differential
nWage equation, fitted to all 3294 observations:
n log(wagei) = 1.09 + 0.20 malei + 0.19 log(experi)
nThe semi-elasticity of the wages with respect to gender, i.e., the relative wage differential
between males and females, is the coefficient of malei: 0.20 or 20%
nThe wage differential between males (malei =1) and females is obtained from wagef = exp{1.09 +
0.19 log(experi)} and wagem = wagef exp{0.20} = 1.22 wagef; the wage differential is 0.22 or 22%;
the coefficient 0.201) is a good approximation.
n____________________
n1) For small x, exp{x} = Skxk/k! ≈ 1+x
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Contents
nThe Linear Model: Interpretation
nSelection of Regressors
nSelection Criteria
nComparison of Competing Models
nSpecification of the Functional Form
nStructural Break
n
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Selection of Regressors
nSpecification errors:
nOmission of a relevant variable
nInclusion of an irrelevant variable
nQuestions:
nWhat are the consequences of a specification error?
nHow to avoid specification errors?
nHow to detect an erroneous specification?
n
n
n
n
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Example: Income and Consumption
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PCR: Private Consumption,
   real, in bn. EUROs
PYR: Household's Dispos-
   able Income, real, in bn.
   EUROs
1970:1-2003:4
Basis: 1995
Source: AWM-Database
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Income and Consumption
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PCR: Private Consumption,
   real, in bn. EUROs
PYR: Household's Dispos-
   able Income, real, in bn.
   EUROs
1970:1-2003:4
Basis: 1995
Source: AWM-Database
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Income and Consumption: Growth Rates
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PCR_D4: Private Consump-
   tion, real, yearly growth rate
PYR_D4: Household’s Dis-
   posable Income, real,
   yearly growth rate
1970:1-2003:4
Basis: 1995
Source: AWM-Database
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Consumption Function
nC: Private Consumption, real, yearly growth rate (PCR_D4)
nY: Household’s Disposable Income, real, yearly growth rate (PYR_D4)
nT: Trend (Ti = i/1000)
n
n
nConsumption function with trend Ti = i/1000:
n
n
n
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Consumption Function, cont’d
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OLS estimated consumption function: Output from GRETL
Dependent variable : PCR_D4
             coefficient   std. error   t-ratio    p-value
  -------------------------------------------------------------
  const       0,0162489    0,00187868         8,649     1,76e-014 ***
  PYR_D4 0,707963     0,0424086       16,69      4,94e-034 ***
  T -0,0682847   0,0188182      -3,629     0,0004    ***
Mean dependent var    0,024911   S.D. dependent var    0,015222
Sum squared resid      0,007726   S.E. of regression   0,007739
R- squared               0,745445   Adjusted R-squared 0,741498
F(2, 129)               188,8830   P-value (F)                 4,71e-39
Log-likelihood          455,9302   Akaike criterion       -905,8603
Schwarz criterion      -897,2119   Hannan-Quinn -902,3460
rho                     0,701126   Durbin-Watson      0,601668
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Misspecification: Two Models
nTwo models:
n yi = xi‘β + zi’γ + εi  (A)
n yi = xi‘β + vi  (B)
n with J-vector zi
n
n
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Misspecification: Omitted Regressor
nSpecified model is (B), but true model is (A)
n yi = xi‘β + zi’γ + εi  (A)
n yi = xi‘β + vi  (B)
nOLS estimates bB of β from (B) can be written with yi from (A):
n
n
nIf (A) is the true model but (B) is specified, i.e., J relevant regressors zi are omitted, bB is
biased by
n
n
nOmitted variable bias!
nNo bias if (a) γ = 0 or if (b) variables in xi and zi are orthogonal
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Misspecification: Irrelevant Regressor
nSpecified model is (A), but true model is (B):
n yi = xi‘β + zi’γ + εi  (A)
n yi = xi‘β + vi  (B)
nIf (B) is the true model but (A) is specified, i.e., the model contains irrelevant regressors zi
nThe OLS estimates bA
nare unbiased
nhave higher variances and standard errors than the OLS estimate bB obtained from fitting model (B)
n
n
n
n
n
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Consequences
nConsequences of specification errors:
nOmission of a relevant variable
nInclusion of a irrelevant variable
n
n
n
n
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Specification Search
nGeneral-to-specific modeling:
1.List all potential regressors, based on, e.g.,
qeconomic theory
qempirical research
qavailability of data
2.Specify the most general model: include all potential regressors
3.Iteratively, test which variables have to be dropped, re-estimate
4.Stop if no more variable has to be dropped
nThe procedure is known as the LSE (London School of Economics) method
n
n
n
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Specification Search, cont’d
nAlternative procedures
nSpecific-to-general modeling: start with a small model and add variables as long as they
contribute to explaining Y
nStepwise regression
nSpecification search can be subsumed under data mining
n
n
n
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Practice of Specification Search
nApplied research
nStarts with a – in terms of economic theory – plausible specification
nTests whether imposed restrictions are correct, such as
qTest for omitted regressors
qTest for autocorrelation of residuals
qTest for heteroskedasticity
nTests whether further restrictions need to be imposed
qTest for irrelevant regressors
nObstacles for good specification
nComplexity of economic theory
nLimited availability of data
n
n
n
n
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Contents
nThe Linear Model: Interpretation
nSelection of Regressors
nSelection Criteria
nComparison of Competing Models
nSpecification of the Functional Form
nStructural Break
n
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Regressor Selection Criteria
nCriteria for adding and deleting regressors
nt-statistic, F-statistic
nAdjusted R2
nInformation Criteria: penalty for increasing number of regressors
nAkaike’s Information Criterion
n
n
nAlternative criteria are
qSchwarz’s Bayesian Information Criterion (BIC)
qHannan-Quinn Information Criterion
n model  with smaller BIC (or AIC) is preferred
nThe corresponding probabilities for type I and type II errors can hardly be assessed
n
n
n
n
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Information Criteria
nThe most popular information criteria are
nAkaike’s Information Criterion
n
nSchwarz’s Bayesian Information Criterion
n
nHannan-Quinn Information Criterion
n
qDecide in favour of the model with the lowest value of the information criterion
n
n
n
n
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Information Criteria: Penalties
nAkaike
n 2/N
nSchwarz
q       log(N)/N
nHannan-Quinn
q       2 log(log(N))
n
n
n
n
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N
log(N)
AIC
BIC
HQIC
2
0,69
1,00
0,35
-0,73
3
1,10
0,67
0,37
0,19
4
1,39
0,50
0,35
0,65
6
1,79
0,33
0,30
1,17
8
2,08
0,25
0,26
1,46
10
2,30
0,20
0,23
1,67
30
3,40
0,07
0,11
2,45
50
3,91
0,04
0,08
2,73
100
4,61
0,02
0,05
3,05
200
5,30
0,01
0,03
3,33
500
6,21
0,00
0,01
3,65
1000
6,91
0,00
0,01
3,87

Wages: Which Regressors?
nAre school and exper relevant regressors in
n wagei = β1 + β2 malei + β3 schooli + β4 experi + εi
n or shall they be omitted?
nt-test: p-values are 4.62E-80 (school) and 1.59E-7 (exper)
nF-test: F = [(0.1326-0.0317)/2]/[(1-0.1326)/(3294-4)] = 191.24, with p-value 2.68E-79
nadj R2: 0.1318 for the wider model, much higher than 0.0315
nAIC: the wider model (AIC = 16690.2) is preferable; for the smaller model: AIC = 17048.5
nBIC: the wider model (BIC = 16714.6) is preferable; for the smaller model: BIC = 17060.7
nAll criteria suggest the wider model
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Wages, cont’d
nOLS estimated smaller wage equation (Table 2.1, Verbeek)
n
n
n
n
n
n
nwith AIC = 17048.46, BIC = 17060.66
n
n
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Wages, cont’d
nOLS estimated wider wage equation (Table 2.2, Verbeek)
n
n
n
n
n
n
n
n
n
n
nwith AIC = 16690.18, BIC = 16714.58
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The AIC Criterion
nVarious versions in literature
nVerbeek, also Greene:
n
n
nAkaike‘s original formula is
n
q AICA = - 2 l(b)/N + 2K/N = AICV + 1 + log(2π)
n
n with the log-likelihood function
n
n
n
nGRETL:
n

Contents
nThe Linear Model: Interpretation
nSelection of Regressors
nSelection Criteria
nComparison of Competing Models
nSpecification of the Functional Form
nStructural Break
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Nested Models: Comparison
nModel (B), yi = xi‘β + vi, see slide 21, is nested in model
n yi = xi‘β + zi’γ + εi   (A)
ni.e., (A) is extended by J additional regressors zi
nDo the J added regressors contribute to explaining Y?
nF-test (t-test when J = 1) for testing H0: all coefficients of added regressors are zero
q
q
qRB2 and RA2 are the R2 of the models without (B) and with (A) the J additional regressors,
respectively
nAdjusted R2: adj RA2 > adj RB2 equivalent to F > 1
nInformation Criteria: choose the model with the smaller value of the information criterion
n
n
n
n
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Comparison of Non-nested Models
n
nNon-nested models:
n yi = xi’β + εi (A)
n yi = zi’γ + vi  (B)
n at least one component in zi that is not in xi
nNon-nested or encompassing F-test: compares by F-tests artificially nested models
n yi = xi’β + z2i’δB + ε*i with z2i: regressors from zi not in xi
n yi = zi’γ + x2i’δA + v*i with x2i: regressors from xi not in zi
qTest validity of model A by testing H0: δB = 0
qAnalogously, test validity of model B by testing H0: δA = 0
qPossible results: A or B is valid, both models are valid, none is valid
nOther procedures: J-test, PE-test (see below)
n
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Wages: Which Model?
nWhich of the models is adequate?
n log(wagei) = 0.119 + 0.260 malei + 0.115 schooli    (A)
n adj R2 = 0.121, BIC = 5824.90,
n log(wagei) = 0.119 + 0.064 agei     (B)
n adj R2 = 0.069, BIC = 6004.60
nArtificially nested model
n log(wagei) =
n      = -0.472 + 0.243 malei + 0.088 schooli + 0.035 agei
nTest of model validity
qmodel A: t-test for age, p-value 5.79E-15; model A is not adequate
qmodel B: F-test for male and school: model B is not adequate
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J-Test: Comparison of Non-nested Models
n
nNon-nested models: (A) yi = xi’β + εi, (B) yi = zi’γ + vi with components of zi that are not in xi
nCombined model
n yi = (1 - δ) xi’β + δ zi’γ + ui
n  with 0 < δ < 1; δ indicates model adequacy
nTransformed model
n yi = xi’β* + δzi’c + ui = xi’β* + δŷiB + u*i
n with OLS estimate c for γ and predicted values ŷiB = zi’c obtained from fitting model B; β* =
(1-δ)β
nJ-test for validity of model A by testing H0: δ = 0
nLess computational effort than the encompassing F-test
q
n
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Wages: Which Model?
nWhich of the models is adequate?
n log(wagei) = 0.119 + 0.260 malei + 0.115 schooli    (A)
n adj R2 = 0.121, BIC = 5824.90,
n log(wagei) = 0.119 + 0.064 agei     (B)
n adj R2 = 0.069, BIC = 6004.60
nTest the validity of model B by means of the J-test
nExtend the model B to
n log(wagei) = -0.587 + 0.034 agei  + 0.826 ŷiA
n with values ŷiA predicted for log(wagei) from model A
nTest of model validity: t-test for coefficient of ŷiA, t = 15.96, p-value 2.65E-55
nModel B is not a valid model
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Linear vs. Loglinear Model
nChoice between linear and loglinear functional form
n yi = xi’β + εi (A)
q log yi = (log xi)’β + vi (B)
nIn terms of economic interpretation: Are effects additive or multiplicative?
nLog-transformation stabilizes variance, particularly if the dependent variable has a skewed
distribution (wages, income, production, firm size, sales,…)
nLoglinear models are easily interpretable in terms of elasticities
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PE-Test: Linear vs. Loglinear Model
nChoice between linear and loglinear functional form
nEstimate both models
n yi = xi’β + εi (A)
q log yi = (log xi)’β + vi (B)
n calculate the fitted values ŷ (from model A) and log ӱ (from B)
nTest H0: δLIN = 0 in
n yi = xi’β + δLIN (log (ŷi) – log ӱi) + ui
n not rejecting H0: δLIN = 0 favors the model A
nTest H0: δLOG = 0 in
n log yi = (log xi)’β + δLOG (ŷi – exp{log ӱi}) + ui
n not rejecting H0: δLOG = 0 favors the model B
nBoth null hypotheses are rejected: find a more adequate model
n
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Wages: Which Model?
nTest of validity of models by means of the PE-test
nThe fitted models are (with l_x for log(x))
n wagei = -2.046 + 1.406 malei + 0.608 schooli      (A)
n l_wagei = 0.119 + 0.260 malei + 0.115 l_schooli      (B)
nx_f: predicted value of x: d_log = log(wage_f) – l_wage_f, d_lin = wage_f – exp(l_wage_f)
nTest of validity of model A:
q wagei = -1.708 + 1.379 malei + 0.637 schooli – 4.731 d_logi
qwith p-value 0.013 for d_log; validity of model A in doubt
nTest of model validity, model B:
n l_wagei = -1.132 + 0.240 malei + 1.008 l_schooli + 0.171 d_lini
n with p-value 0.076 for d_lin; model B to be preferred
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The PE-Test
nChoice between linear and loglinear functional form
nThe auxiliary regressions are estimated for testing purposes
nIf the linear model is not rejected: accept the linear model
nIf the loglinear model is not rejected: accept the loglinear model
nIf both are rejected, neither model is appropriate, a more adequate model should be considered
nIn case of the Individual Wages example:
qLinear model (A): t-statistic is – 4.731, p-value 0.013: the model is rejected
qLoglinear model (B): t-statistic is 0.171, p-value 0.076 : the model is not rejected
n
n
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Contents
nThe Linear Model: Interpretation
nSelection of Regressors
nSelection Criteria
nComparison of Competing Models
nSpecification of the Functional Form
nStructural Break
n
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Non-linear Functional Forms
nModel specification
n yi = g(xi, β) + εi
n substitution of g(xi, β) for xi’β: allows for two types on non-linearity
ng(xi, β) non-linear in regressors (but linear in parameters)
qPowers of regressors, e.g., g(xi, β) = β1 + β2 agei + β3 agei2
qInteractions of regressors, e.g., g(xi, β) = β1 + β2 agei + β3 agei*malei
n OLS technique still works; t-test, F-test for specification check
ng(xi, β) non-linear in regression coefficients, e.g.,
qg(xi, β) = β1 xi1β2 xi2β3
q logarithmic transformation: log g(xi, β) = log β1 + β2log xi1+ β3log xi2
qg(xi, β) = β1 + β2 xiβ3
q non-linear least squares estimation, numerical procedures
n Various specification test procedures, e.g., RESET test, Chow test
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Individual Wages: Effect of Gender and Education
nEffect of gender may be depending of education level
nSeparate models for males and females
nInteraction terms between dummies for education level and male
nExample: Belgian Household Panel, 1994 (“bwages”, N=1472)
nFive education levels
nModel for log(wage) with education dummies
nModel with interaction terms between education dummies and gender dummy
nF-statistic for interaction terms:
n F(5, 1460) = {(0.4032-0.3976)/5}/{(1-0.4032)/(1472-12)} = 2.74
n with a p-value of 0.018
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Wages: Model with Education Dummies
nModel with education dummies: Verbeek, Table 3.11
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Wages: Model with Gender Interactions
nWage equation with interactions educ*male
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RESET Test
nTest of the linear model E{yi |xi} = xi’β against misspecification of the functional form:
nNull hypothesis: linear model is correct functional form
nTest of H0: RESET test (Regression Specification Error Test), Ramsey (1969)
nTest idea: linear model is extended by adding ŷi², ŷi³, ..., where ŷi is the fitted values from
the linear model; extension does not improve model fit under H0
qŷi² is a function of squares (and interactions) of the regressor variables; analogously for ŷi³,
...
qIf the F-test indicates that the additional regressor ŷi² contributes to explaining Y: the linear
relation is not adequate, another functional form is more appropriate
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The RESET Test Procedure
nTest of the linear model E{yi |xi} = xi’β against misspecification of the functional form:
nLinear model extended by adding ŷi², ..., ŷiQ
nF- (or t-) test to decide whether ŷi², ..., ŷiQ contribute as additional regressors to explaining
Y
nMaximal power Q of fitted values: typical choice is Q = 2 or Q = 3
n
nIn GRETL: Ordinary Least Squares… => Tests => Ramsey’s RESET, input of Q
n
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Wages: RESET Test
nThe fitted models are (with l_x for log(x))
n wagei = -2.046 + 1.406 malei + 0.608 schooli      (A)
n l_wagei = 0.119 + 0.260 malei + 0.115 l_schooli      (B)
nTest of specification of the functional form with Q = 3
nModel A: Test statistic: F(2, 3288) = 10.23, p-value = 3.723e-005
nModel B: Test statistic: F(2, 3288) = 4.52, p-value = 0.011
nFor both models the adequacy of the functional form is in doubt
n
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Contents
nThe Linear Model: Interpretation
nSelection of Regressors
nSelection Criteria
nComparison of Competing Models
nSpecification of the Functional Form
nStructural Break
n
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Structural Break: Chow Test
nIn time-series context, coefficients of a model may change due to a major policy change, e.g., the
oil price shock
nModeling a process with structural break
n E{yi |xi}= xi’β + gixi’ γ
n with dummy variable gi=0 before the break, gi=1 after the break
nRegressors xi, coefficients β before, β+γ after the break
nNull hypothesis: no structural break, γ=0
nTest procedure: fitting the extended model, F- (or t-) test of γ=0
n
n
n with Sr (Su): sum of squared residuals of the (un)restricted model
nChow test for structural break or structural change, Chow (1960)
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Chow Test: The Practice
nTest procedure is performed in the following steps
nFit the restricted model: Sr
nFit the extended model: Su
nCalculate F and the p-value from the F-distribution with K and N-2K d.f.
nNeeds knowledge of break point
n
nIn GRETL: Ordinary Least Squares… => Tests => Chow test
n input of the first observation period after the break point
n
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Your Homework
1.Use the data set “bwages” of Verbeek for the following analyses:
a)Estimate the model where the log hourly wages (lnwage) are explained by lnexper, male, and educ;
interpret the results.
b)Repeat exercise a) using dummy variables for the education levels, e.g., d1 for educ = 1, instead
of the variable educ; compare the models from exercises a) and b) by using (i) the non-nested
F-test and (ii) the J-test; interpret the results.
c)Use the PE-test to decide whether the model in a) (where log hourly wages, lnwage, are explained)
or the same model but with levels, wage, of hourly wages as explained variable is to be preferred;
interpret the result.
d)Estimate the model for log hourly wages (lnwage) with regressors exper, male, educ, and the
interaction male*exper as additional regressor; interpret the result.
2.
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Your Homework, cont’d
2.OLS is used to estimate β from yi = xi‘β + εi, but a relevant regressor zi is neglected: yi =
xi‘β + zi‘γ + εi. (a) Show that the estimate b is biased, and derive an expression for the bias;
(b) what test statistic can be used for testing H0: γ = 0?
3.The linear regression is specified as
n log yi = xi’β + εi
n Show that the elasticity of Y with respect to Xk is βkxik.
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