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ECMT2150 Model Specification
Creation date：2024-04-20 14:59:44
Belonging category：计算机computer science
Model Specification

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ECMT2150 – Lecture 6

Topics Today
Week 6
Model Specification
Specification errors
=> including irrelevant variables
=> omitting relevant variables – Omitted Variable Bias
More on multicollinearity and sampling variance
Endogeneity
Proxy Variables
Reference: Chapters 3.3b; 3.4a; Chp 9.2
Recap
• Econometrics is about estimating economic relationships, e.g.
between wages and education, etc.
• Simple and multiple linear regression – many examples
– Incorporated non-linearities: logs & polynomials
– Incorporated qualitative and categorical information
– Discussed goodness of fit
Recap
• Derived the OLS estimator – understand the steps/the
intuition. Derivation is not examinable
• Statistical Properties of OLS that hold for any sample under
given assumptions
– Expected values/unbiasedness under MLR.1 – MLR.4
– Variance formulas under MLR.1 – MLR.5
– Gauss-Markov Theorem under MLR.1 – MLR.5
– Exact sampling distributions/tests under MLR.1 – MLR.6
• Asymptotic Properties of OLS (Consistency and Asymptotic
normality)
Recap
Why?
• Importance of the Zero Conditional Mean Assumption:
E(u|x) = E(u) = 0
• Reliability of estimator and inference rest on whether the
assumptions hold
• Causality - understanding what we have to assume to obtain
causal estimates of our parameters of interest
• + we need the variance formulas and sampling distributional
assumptions to conduct inference
Recap
• Inference
– Hypothesis tests
• T-tests:
– One-sided
– Two sided (tests of statistical significance)
• p-values
• Confidence intervals
– Testing more general alternatives
• An estimate is equal to a constant
• One estimate is equal to another => Linear combination of
parameters
• Testing multiple linear restrictions
Specification Errors
Specification errors
• Until now we have assumed our population model
= 0 + 11 + … + +
has been correctly specified
• A bit unrealistic – we can never be sure of the true
population model
• Types of specification errors:
– Choice of independent variables
• Over-specification & sampling variance effects
• Omitted variables & Endogeneity
– Heteroskedasticity
– Functional Form
– Measurement Error
– Missing Data, Non-random Sampling & Outliers
Misspecification in the choice of
independent variables
Misspecification in the choice of independent variables
A. Including irrelevant variables in a regression
model (over-specifying)
– This model satisfies MLR.1-MLR.4
– x3 may be correlated with x1 and x2
– Crucially, in the population, x3 has no effect on y after we
control for x1 and x2
⇒ Inclusion of x3 has no cost in terms of bias in the estimates of
any of the parameters, because
– However, including irrelevant variables may increase the
sampling variance (more on this shortly)
= 0 in the population
Misspecification in the choice of independent variables
B. Omitting relevant variables (Wooldridge 3.3b)
⇒Omitted Variable Bias
⇒Violate MLR.4, E(u|x)=0
The simple case:
but due to our ignorance or data unavailability, we estimate
True model (contains x1 and x2)
Estimated model (x2 is omitted)
Misspecification in the choice of independent variables
B. Omitting relevant variables (Wooldridge 3.3b)
Example: Omitting ability in a wage equation
True model:
We estimate:
Omitting a relevant variable causes bias when the omitted
variable is correlated with any of the other explanatory variables
in the model
Omitted Variable Bias – the simple case
Let‘s look at this in more detail:
• If x1 and x2 are correlated, assume a linear regression
relationship between them:
• The true model is:
If y is only regressed
on x1 this will be the
estimated intercept
If y is only regressed
on x1, this will be the
estimated slope on x1
error term
And the bias =
Conclusion: All estimated coefficients will be biased.
Omitted Variable Bias – the simple case
Our example again: Omitting ability in a wage equation
Will be positive
The return to education 1 will be over-estimated because 21 > 0.
It will look as if people with many years of education earn very high
wages, but this is partly due to the effect of ability - the fact that
people with more education are also more able on average.
Omitted Variable Bias – the simple case
Summarising the direction of the bias:
Omitted Variable Bias – the simple case
What about the size of the bias?
Our example again: Omitting ability in a wage equationln = 0 + 1 + 2 +
As above, the return to education 1 will be over-estimated
because 21 > 0.
By how much?
For example, if the return to educ in the population is 8.6%
• a bias of 21= 0.1 percentage points – not so worrying
• a bias of 21= 3 percentage points – big concern
Omitted Variable Bias – the simple case
Q: When is there no omitted variable bias?
A: When the omitted factors are
i) unrelated => 1, 2 = 0, 1 = 0
ii) when they don‘t affect the outcome => 2 = 0
In our wages example:
• 1, 2 > 0 if individuals with high innate ability
tend to have higher education.
• 2 > 0 if individuals with high ability tend to have higher
productivity and wages.
 Together, we would expect that OLS overestimates 1
Omitted Variable Bias: more general cases
(Wooldridge 3.3c)
– No general statements possible about direction of bias
– If 1, 3 ≠ 0, we can analyse as per the simple
case if and only if 2is uncorrelated with 1 and 3
others
• Example: Omitting ability in a wage equation
True model (contains x1, x2,
and x3)
Estimated model (x3 is omitted)
If experience is approximately uncorrelated with educ and abil, then the
direction of the omitted variable bias can be as analyzed in the simple two
variable case.
Omitting relevant variables =>
Inconsistency
• Not only is the OLS estimator biased when we omit
relevant variables, it is also inconsistent
see section 5.1a, Wooldridge
• We can show that:
�1 = 1 + 2 ∑ 1 − ̅1 2∑( 1 − ̅1 2) + ∑ 1 − ̅1 ∑( 1 − ̅1 2)
• Then taking the plim on both sides, we have:
�1 = 1 + 2 (1, 2)(1)
Sampling Variances
Specification Errors and Multi-collinearity
• So omitting a relevant variable can cause bias.
• Including irrelevant variables does not cause bias
• BUT, you might consider leaving them out so as to not
unnecessarily inflate the sampling variance
⇒there is a trade-off to be made with the effect on the
variance of our slope parameters
Sampling Variance: Mis-specified Models
(Wooldridge 3.4b)
• The choice of whether to include a particular variable in a
regression can be made by analyzing this trade-off
between bias and variance
• It might be the case that the likely omitted variable bias in
the misspecified model 2 is compensated by a smaller
variance
True population
model
Estimated model 1
Estimated model 2
Sampling Variance: Misspecified Models
(Wooldridge 3.4b)
• Variance in the misspecified model
• Case 1:
• Case 2:
Conditional on x1 and x2,
the variance in model 2
is always smaller than
that in model 1
Conclusion: Do not include irrelevant regressors
Trade off bias and variance
Sampling Variance: Misspecified Models
Caution!! Bias will not vanish even in large samples. But the variance
of � will decrease with a large sample
Multicollinearity & sampling variance
– Recall:
– Linear relationships between explanatory variables can create
problems
– High multicollinearity can occur when R2j is ‘close’ to 1
– Ideally, we have little correlation between xj and other
independent variables – Yet this may not be the case.
– For example, examining the effect of various school expenditure
categories on school performance
• It is expected that wealthier schools will spend more on everything than less
wealthy schools
• It can be difficult to estimate the effect of any category of school expenditure on
student performance when there is little variation in one category
Multicollinearity & sampling variance
Average standardized
test score of a school Expenditure
on teachers
Expenditure on
instructional
materials
Other expenditures
The different expenditure categories will be strongly correlated: if a school has a
lot of resources it will spend a lot on everything.
It will be hard to estimate the differential effects of different expenditure
categories because all expenditures are either high or low.
For precise estimates of the differential effects, one would need information
about situations where expenditure categories change differentially.
So,... often the sampling variance of the estimated effects will be large.
Multicollinearity & sampling variance
• Because effects cannot be disentangled, it may be better to lump
all expenditure categories together
• In other cases, dropping some of the x‘s may reduce
multicollinearity (but this may lead to omitted variable bias!)
• Only the sampling variance of the variables involved in
multicollinearity will be inflated; the estimates of other effects
may be very precise
• Note that multicollinearity is not a violation of MLR.3
• Multicollinearity may be detected through variance inflation
Case category
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CMP7000A CMP-7038B Math 263 LAW3016 CS371 STAT 334 FIN0008 ACC3004 STAT 371 ​COMP1921 MTH201 ACCT2000 UN3412 CIV6000 MATHS 3021 MATE7014 ECON1 CIBC6035 LAW121G COMM 321 CSC8204R TTM2033 CSCI 2600 MAE 115 LNG302 CENV6175W1 ECON30019 CPS2015 BST969 DDES9015 AMATH242 CS 1151 N1592 ACCT600 EG503G SMM519 EC318 Ecstatistics EEE 471 MARK2051 Topology COM00037H ACCT3000 MTHM501 EG501V COM00055H BUSN9085 GGR273 MATH 2B ITD102 MA214 ECON0060 ECE3114 ECMM447 Mechanics CCT361H5S BSAN3210 WELF2001 Fin3001 RESP5001 STA 137 Linguistics 101 CSCI368 CMPSC497 LUBS3655 OMBA 5355 BUSM2562 ​BFF5902 CYBR7001 CULT2005 ECON 7720 MTRX3760 FSE598 ​LUE1001 MGSC 7100 CS3240 COSC7500 ACCT3091 INFOGR TRAN 2080 DESIGN CIVE215001 CIVE2360 CPSC 8430 FNCE2003 Stoichiometry INFS6023 Quantitative Method Math 21D ECE5179 Design ​ECON8022 Chemical DMS 213 ADM1370 BFF5902 BFF2701 AGRI10051 AMN400 ECON7322 ​FINC5090 ​MATH1052 ​SciComp Elec4622 ​FINS5516 ENG 101 MMM143 CS431CA2 BSAN2201 EBSC7070 ES9v9-10 PLIN0009 DATA 8 MATH 101 BCPM0097 ​MAT 237 BISM2201 EEE8099 Physics 11 ECE2238B Biostat 234 ​ECON30009 INF10025 CHEM252 CCMA4008 CIVE 3007 STAT0031 ELEC ENG 3108 MGMT90140 CHEM 252 STAT3016 MMM095 ​MMGT6016 Legal STAT 311 ELEC9764 COMP 370 TCS 3053 EECS 1021 FIT1051 C/C++ EE3C COMP2003J COM398 COMP51915 Simulator FIN42330 8PRO102 PGD ELEC5160 XJCO1921 COMP5123M INFS 2042 CHE1148H ECA5313 IN3329 Microsoft Modeling B15 2TT COMP3687 COMP2432 DMS2030 KAN4TSF CFKG CSE 5322 CSE 445 STATS 3DA3 CHC5223 COMP1048 EL3105 EECS 3221 CS231 ME 4434 ITP4510 ITP4501 CSCI 5708 Controller EECE 5699 CPEN 231L CPN programs CSC8016 MATU9D2 ENGN4213 2. SVM. (10 points) Consider an SVM classifier using RBF kernel, and finish the following python code to search for the optimum kernel parameter γ. You are required to calculate the RBF function and search yourself, instead of using functions in Scikit-Learn (sklearn). def parameter search(gamma list ,Xtr,ytr,Xts,yts ,...): """ Parameters gamma list: A list of RBF kernel parameter gamma to search for Xtr: Training data, shape (ntr,nrow,ncol) ytr: Training labels, shape (ntr,) Xts: Test data, shape (nts,nrow,ncol) yts: Test labels, shape (nts,) ... Add other parameters as needed Returns: gamma optimum: The gamma with minimum error on Xts, yts """ ... return gamma optimum
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