Structural Modelling - Weebly

What is structural modelling? Constructing structural models Education and Earnings Evaluating Structural Modelling

Structural Modelling

Selma Telalagic

University of Oxford

February 5, 2014


Structure of these lectures

• Two lectures and two classes• Two parts: understanding structural modelling & structuralmodelling vs. atheoretic approaches

• Classes: we will estimate a structural model based on Belziland Hansen (2002) and discuss two structural articles(Cherchye et al 2012, Attanasio et al 2011)

• Class 1: Simulations for model; Discussion of Cherchye et al• Class 2: Estimates of model; Discussion of Attanasio et al


Reading• *Attanasio, O. P., C. Meghir and A. Santiago. "EducationChoices in Mexico: Using a Structural Model and aRandomised Experiment to Evaluate PROGRESA." Review ofEconomic Studies 79 (2011): pp. 37-66.

• Browning, M. Notes on Identification, Estimation andStructural Modelling (2012).

• Belzil, C. and J. Hansen. "Unobserved Ability and the Returnto Schooling." Econometrica 70.5 (2002).

• *Cherchye, L., B. De Rock and F. Vermeulen. "Married withChildren: A Collective Labour Supply Model with DetailedTime Use and Intrahousehold Expenditure Information."American Economic Review 102.7 (2012): pp. 3377-3405.

• *Keane, M. P. "Structural vs. Atheoretic Approaches toEconometrics." Journal of Econometrics 156 (2010): pp.3-20.


Reading cont.

• *Reiss, P. C. and F. A. Wolak. "Structural EconometricModelling: Rationales and Examples from IndustrialOrganisation." in J.J. Heckman & E.E. Leamer (eds.).Handbook of Econometrics. Vol. 6B (2007).

• Rosenzweig, M. R. and K. Wolpin. "Natural "NaturalExperiments" in Economics." Journal of Economic Literature38 (2000): pp. 827-874.

• * denotes compulsory reading. Reiss and Wolak: only up toand including Section 4.#

• Your homework: For class 1, please read and answer questionson Cherchye et al.

• For class 2, please read and answer questions on Attanasio etal.

• See appendices for questions.


Part I: An Introduction to Structural Modelling• Structural models combine economic theory with statisticalmodels.

• They involve going to the data with a presupposed model inmind, which describes how the researcher thinks thehouseholds or firms behave.

• Structural models set out relationships between variables• Researcher believes that a variable y (e.g. wages) is related toa variable x (e.g. years of schooling)

• Set out a known function f (·), such that y = f (x ,Ψ,Φ),where Ψ is a set of known unobservables (e.g. ability) and Φis a set of unknown parameters

• To bring this model to the data, assumptions need to be madeabout the distribution of unobservables

• Economic assumptions (model) + Statistical assumptions(distribution) →Empirical model.


What is nonstructural modelling?

• Does not necessarily depend on formal economic or statisticalmodels

• Often these papers have no explicit model, only intuition ’inwords’as to why a variable is on the right-hand side orleft-hand side of a regression

• e.g. OLS or natural experiments• Researchers are interested in the joint distribution of y and x :g(x , y)

• Most common estimate is Best Linear Predictor BLP(y |x),which is obtained through Ordinary Least Squares (OLS)

• But what does the BLP really tell us? More on that in Part 2of these lectures.


Example 1: Education and Wages

• We can compare structural and nonstructural modelling in asimple example

• What is the effect of education on wages?• What is the problem with estimating the following equation:

•wi = α+ βei + ui

• The error term ui is problematic: it contains ability


Example 1 cont.

• How can we deal with this problem?

1. Instrumental variables (e.g. distance to nearest school)

2. Finding a natural experiment (a change in education policye.g. length of schooling?)

3. Running an experiment (e.g. give vouchers for free schooluniforms to some households, as in Duflo et al 2012)

4. Trying to control for the omitted variable (IQ test?)

• Can we trust these solutions?• What if some endogeneity remains? Are we sure the naturalexperiment or exogenous variable is truly exogenous?Problems with proxies - see Nicolas’lectures


Example 1 cont.

• We are estimating a linear, homogenous effect. i.e. having 2years of schooling vs. 1 year has same impact on earnings ashaving 7 vs. 8 (i.e. completing or not completing primaryschool)

• Average effect biased towards most common schoolingattainment in the sample

• How can we deal with these criticism?• One way is to estimate a full structural model• Specify explicitly how education, earnings and ability arerelated e.g. individuals maximise utility from earnings bychoosing an education level, depending on their ability

• We will do this later in the lecture


How are structural models constructed?• Two main sources of structure: economic theory andstatistical theory

• Start with deterministic economic model e.g. individualmaximising utility by choosing how much education to pursue

• A deterministic model predicts a unique level of education, asa function of individual’s characteristics

• BUT model won’t fit data perfectly → add statisticalassumptions i.e. error term

• Can we just assume i.i.d. error term? (easy)• Suppose our error term is ability

• If it is i.i.d. - then no endogeneity at all. OLS is unbiased!• But we know ability is not i.i.d.• So statistical assumption must match reality and will helpinterpret results


Four steps of structural modelling

1. Defining an economic model of the problem at hand

2. Incorporating unobservables into the economic model

3. Estimation

4. Checking how accurate the model is in describing the data(validation)

"The second step should receive significantly moreattention than it typically does. This is because theassumptions made about the unobservables will impactthe consistency of OLS parameter estimates." - Reiss andWolak (2007), p.4288.


Step 1: Defining the economic model

• What kind of assumptions might we make?• a) Agents’objectives (e.g. utility maximisation, profitmaximisation)

• b) Agents’constraints (e.g. budget constraints, productionfunction) e.g. time constraint

• c) Heterogeneity: how a) and b) vary across agents e.g. someindividuals find education ’easier’

• d) How (if at all) agents interact (e.g. Nash bargaining)


Step 2: Unobservables

• Incorporating unobservables is a crucial part of structuralmodelling

• We can incorporate them in three ways:

• a) Uncertainty (either researchers’or agents’)• b) Mistakes in optimisation by agents• c) Measurement error


Uncertainty

• Whose uncertainty is it? The agent’s, or the researcher’s?• Example 2.• Data: firms’output (Q), total costs (TC ), input prices(pKi , pLi )

• No data on labour or capital• Goal: estimate parameters of production function

Qi = AiLαi K

βi

• How? Need relationship between output and total cost thatincludes parameters

• Assume: price of product set by regulator (firm can’tinfluence)


Example 2 cont.• Uncertainty type 1: researcher. Firm knows its productivity(Ai ) but researcher doesn’t

• This is called "unobserved heterogeneity" (agents vary insome parameter that researcher doesn’t see; e.g. ability!)

• Assume: Ai are i.i.d. random variables• Assume: regulated price = minimum average cost• Notice that we are making a lot of assumptions.. this is aregular part of structural modelling

• Firm maximises its profit:

π(Ki , Li ) = pri AiLαi K

βi − pKiKi − pLiLi ,

• Total cost function

TCi = C0pγKip

1−γLi Qδ

i A−δi ,


Example 2 cont.

• Take natural logs:

lnTCi = lnC0 + γ ln pKi + (1− γ) ln pLi + δ lnQi − δ lnAi

• This looks like a regression equation BUT it is deterministic• We need an error term• Recall that Ai is unobserved by the researcher → enters errorterm

lnTCi = C1 + γ ln pKi + (1− γ) ln pLi + δ lnQi − δ ln ui ,

• This we can estimate with OLS. Moved from unobservables toestimation equation by making assumptions.

• OLS unbiased because we assumed Ai uncorrelated with inputprices and quantity


Example 2 cont.

• Uncertainty type 2: Both researcher and agent• Assume: both researcher and agent have same expectation ofAi

• Assume: regulator sets price such that expected profits = 0• Firm maximises expected profit

E [π(peri ,Ki , Li )] = peri E [AiL

αi K

βi ]− pKiKi − pLiLi .

• With cost function

TCi =α+ β

βpKiKi ,


Example 2 cont.

• SubstitutingQai = D0TC

α+βi p−β

Ki p−αLi Ai .

• Taking natural log

lnQai = lnD0 + (α+ β) lnTCi − β ln pKi − α ln pLi + lnAi .

• Notice: Q is now on LHS and TC is on RHS!

• This is again a deterministic equation. Assume Ai iid asbefore gives regression equation

lnQai = D1 + (α+ β) lnTCi − β ln pKi − α ln pLi + ηi ,

• What do we learn from Example 2? What we assume aboutuncertainty has a profound effect on how we estimate


Other types of error

• Optimisation errors: agents do not behave perfectly rationally• e.g. first-order conditions of utility maximisation not satisfiedexactly

• This can yield predictions similar to unobserved heterogeneity• Measurement error: researcher observes data that differs fromthe true value by some margin

• Easy way to move from deterministic model to statisticalmodel

• BUT what do we assume about measurement error? Is iti.i.d.? In that case, wouldn’t be an appropriate way ofincorporating ability in a model of earnings and wages


Step 3: Estimation

• Specify functional forms (e.g. the utility function isCobb-Douglas)

• Specifies distributional assumptions (e.g. the error term isnormally distributed)

• Choose an estimation technique (e.g. OLS/2SLS if linearprediction, or ML if probabilities of events are given)

• In the examples of these lectures, we focus on OLS/2SLS, butwe will try our hand at ML in the class


Step 4: Validation• Researcher needs to provide tests of the model• Structural modelling involves a lot of assumptions• How sensitive are the results to the assumptions? → showwhether predictions change if change underlying assumption

• How sensitive are the results to functional form? → showempirical results if e.g. we change form of utility function

• Does the model fit the data well? → significance ofcoeffi cients

• Does the model perform well in out-of-sample prediction? →split sample and focus on one subsample, or do policyprediction

• A rejected model is not necessarily a wrong or uselessmodel

• Models won’t fit the data 100% + a rejected model is a usefulpiece of information


Belzil and Hansen (2002)

• Belzil and Hansen (2002) answer the earnings-educationquestion in a structural way

• Obtain structural estimates of the local (and average) returnsto schooling in a model that does not contain certain earlierrestrictive assumptions

• Able to estimate heterogeneity in impact of schooling e.g.show convex returns: low until Grade 11, then increasing in anincreasing way thereafter

• We will not do this! Instead we will build a simple structuralmodel that is inspired by Belzil and Hansen


Our education and earnings model

• This model is attributed to Quinn (2013).• Assumption 1: Utility. Every student has identicalpreferences. A given student i receives utility v si from a yearof schooling (’school ability’)

• Homogenous return• The student i receives a wage determined by his or herschooling achievement (Si ) and the student’s idiosyncratic‘market ability’, vwi

• Student receives the following in-period utilities for schoolingand work:

USit (·) = v si ;UWit (·) = lnwit = ψ(Si ) + vwi .


Education and earnings cont.

• The object of interest for our estimation is the function ψ(S),which determines the causal returns to education

• Assumption 2: Education choices. Students may onlychoose zero, four, seven or 11 years of education.

• Assumption 3: Lifetime utility. Every student hastime-separable preferences using exponential discounting withdiscount factor β. That is, if the instantaneous utility fromperiod t is ut , the student’s utility from time t = 0 to t = Tis:

U =T

∑t=0

βtut .


Education and earnings cont.

• Assumption 4: Leaving school. Once a student leavesschool, the student cannot return.

• Assumption 5: Uncertainty. Conditional on knowing v si andvwi , everything is deterministic. That is, no student faces anyuncertainty – including, for example, about (i) whether ornot the student will get a job, or (ii) how much the studentwill subsequently earn.


Solving the model

Value functions for four education choices:

V0(0) =T

∑t=0

βt · vwi

V0(4) =3

∑t=0

βt · v si +T

∑t=4

βt · (ψ(4) + vwi )

V0(7) =6

∑t=0

βt · v si +T

∑t=7

βt · (ψ(7) + vwi )

V0(11) =10

∑t=0

βt · v si +T

∑t=11

βt · (ψ(11) + vwi ) .


Solving the model cont.Student prefers four years of education to no years of education if

V0(4) ≥ V0(0)3

∑t=0

βt · v si +T

∑t=4

βt · (ψ(4) + vwi ) ≥T

∑t=0

βt · vwi

3

∑t=0

βt · v si +T

∑t=4

βt · ψ(4) ≥3

∑t=0

βt · vwi

3

∑t=0

βt · (v si − vwi ) ≥ −T

∑t=4

βt · ψ(4)

(v si − vwi ) ·(1− β4

1− β

)≥ −

(β4 − βT+1

1− β

)· ψ(4)

v si ≥ vwi −(

β4 − βT+1

1− β4

)· ψ(4)


Solving the model cont.

• We can use the same approach to compare 4 and 7 years, 7and 11 years.. giving a series of cutoffs (see Appendix 2)

• This is the deterministic solution to the model: if we know anindividual’s utility function and market and school ability, wewill know exactly which schooling choice that individual willpredict

• What happens next? The models needs to be fit to the data→ we need error terms


Error terms

• Assumption 6: Errors. The joint distribution of v si and vwi is

bivariate normal with zero covariance:(v sivwi

)∼ N

((00

),

(σ2s 00 σ2w

)).

• i.e. market ability and schooling ability are uncorrelated (verysimplified!)


Object of interest

• What are we really interested in? The shape of the functionψ(S) i.e. the returns to schooling at different levels ofschooling

• If we assume that ψ(0) = 0, the key parameters are:ψ(4),ψ(7) and ψ(11)

• We would also like to estimate β (the discount factor) and thevariance of the unobservables: σ2s and σ2w

• Intuitively: unobserved market & schooling ability → choice ofS AND unobserved market ability & S → earnings

• We need the probability of observing all these thingsTOGETHER (not just one part of the model)


Use Bayes’Rule

Pr(A&B) = Pr(A|B) ∗ Pr(B)Pr(S&earnings) = Pr(schooling |earnings) ∗ Pr(earnings)

= Pr(schooling |market ability) ∗ Pr(market ability)L (Θ;Si , vwi ) = L (Θ;Si | vwi )× L (Θ; vwi ) .` (Θ;Si , vwi ) = ` (Θ;Si | vwi ) + ` (Θ; vwi ) .


The first bit: Distribution of market ability

• Recall: lnwit = ψ(Si ) + vwi• So

lnwit = ψ4 ·D4i + ψ7 ·D7i + ψ11 ·D11i + vwivwi = lnwit − (ψ4 ·D4i + ψ7 ·D7i + ψ11 ·D11i )

• Recall, vwi is normally distributed, so the log likelihood of vwiis

` (Θ; vwi ) = ln[

σ−1w · φ(lnwit − ψ4 ·D4i − ψ7 ·D7i − ψ11 ·D11i

σw

)],

• where φ is the pdf of the normal distribution


The second bit: Schooling|market ability

Ordered probit i.e. what is the probability of falling within acertain range?

`(Θ;Si | vwi ) =

ln{

Φ(vwi +γ4

σs

)−Φ

(vwi −∞

σs

)}if S = 0

ln{

Φ(vwi +γ7

σs

)−Φ

(vwi +γ4

σs

)}if S = 4

ln{

Φ(vwi +γ11

σs

)−Φ

(vwi +γ7

σs

)}if S = 7

ln{

Φ(vwi +∞

σs

)−Φ

(vwi +γ11

σs

)}if S = 11.


Summary of education & earnings model

• Using a structural model, we have tried to answer thequestion: what is the effect of education on earnings?

• Assumed utility functions for agent• Assumed unobservable ability• Assumed distribution for ability• Solved for probability of observing different levels of education(combined with ability and earnings)

• Next step: estimate the model! In Friday’s class.


Summary of structural modelling

• Structural modelling involves combining economic theory withstatistical theory to generate testable models

• We begin with a deterministic model (e.g. utilitymaximisation)

• We introduce uncertainty or unobservables• This allows us to move from the model to the data

• We estimate the model using linear or nonlinear techniques• We evaluate the model by seeing how good the fit is &out-of-sample predictive power


Part II: Structural vs. atheoretic approaches

• In the second part of this short course, we discuss structuralvs. atheoretic approaches to econometrics

• Why add structure?

1. Allows the estimation of parameters that cannot be obtainedfrom nonexperimental data e.g. marginal cost

2. Evaluation of proposed policies (e.g. Cherchye et al 2012)

3. Compare two competing models, by comparing their predictivepower.

4. Assumptions are made explicit

5. Results can be used by other researchers.


Disadvantages of structural modelling

1. Models can be diffi cult to solve. Solutions often requiresimplifying more than we would like to.

2. Sensitivity to assumptions.

3. Diffi cult for non-economists to understand.


What is the atheoretic approach to econometrics?

• ’Causal reduced form’/naturalexperiments/quasi-experiments/atheoretic

• Education and wages: need exogenous variation in educationto estimate causal effect

• Controlled experiment, treatment group and control group• BUT unethical for education• Look for natural experiments: but how many naturalexperiments are truly random?

• Rosenzweig and Wolpin (2000): only five true naturalexperiments: twins, human cloning, birth date, gender and theweather


Problems with CRF: Nonlinearities• Even if we manage to find natural variation: how do weinterpret it?

• Usually estimated with OLS and 2SLS, which assume linearrelationships

• Nonlinearities are a problem (e.g. Belzil and Hansen’s resulton non-convexity of returns to education)

• What happens if we estimate a nonlinear relationship with alinear equation?

• OLS yields an estimate of BLP (y |x). the BLP will notnecessarily measure the causal effect of a one-unit change in xon y

• BLP and E [y |x ] are not equivalent.• BLP (y |x) may not even have the same sign as ∂E [y |x ]

∂x !• Unless we have an economic model that predicts a linearrelationship between x and y , diffi cult to interpret thecoeffi cient


Problems with CRF: Heterogeneity• Illustrate problem of heterogeneity using Angrist (1990)• To estimate the effect of being a war veteran on subsequentearnings, Angrist used the draft lottery in the U.S.

• Men with certain lottery numbers were ’draft eligible’forserving in Vietnam War

• Assumption: draft eligibility is correlated with having served inthe war but uncorrelated with individual characteristics suchas ability

• Draft eligibility can be used as an IV to estimate the effect ofserving in the war on earnings

• Estimates serving in the war caused an approximate 15%decrease in the earnings of veterans

• But what does this estimate really mean?• Is it the expected effect on earnings for someone randomlychosen to participate in war? Probably not. Could beparticularly negative effect for small part of population.


Problems with CRF: Interpretation requires theory

• Continuing with the example of Angrist: what is the cause ofthe negative effect?

• For the result to be helpful for policy, we need a mechanism• e.g. it may be that veterans had fewer years of schooling(then schooling must go into the regression - otherwise it is inerror term and veteran status no longer exogenous)

• e.g. being drafted may have caused individuals to reduce theirinvestment in their human capital

• Even an ideal instrument can be endogenous, dependingon economic assumptions

• How do we choose between these competing theories withouta model?


Advantages of CRF over structural modelling

• Requires significantly less theory than structural modelling →Takes less time to do

• Easy for non-economists to understand• Non-parametric: not sensitive to functional form assumptions.


Identification vs. Estimation

• There are two steps to empirical analysis: identification andestimation

• Identification asks whether we can estimate a parameter, if wehad population data available (intuitively: Even if we had allthe data we could dream up, would we be able to estimatethis coeffi cient?)

• Estimation involves estimating an identified parameter, usingour sample data

• Estimation is all the econometrics you have studied so far inthis course

• Identification is part of the theory bit of empirical work• I will use an example to illustrate identification in the contextof structural modelling


Child cognitive ability• Suppose we are interested in the effect of maternal contacttime on child cognition

• Idea: If mothers increase daycare time and reduce contacttime, this may negatively influence child cognition

• How to measure child cognition? Test scores• How to measure mothers ’quality’time? Diffi cult. What isquality time?

• Can we just regress child cognition on daycare time?

IQtesti = α+ βdaycarei + ui

• NO. Mothers who put children in daycare probablyDIFFERENT from mothers who don’t

• i.e. ’treatment’and ’control’groups are fundamentallydifferent

• What about ’controlling’for mothers’characteristics? Can wereally capture all unobserved heterogeneity?


Child cognitive ability

• Ideally, we’d like an instrumental variable for daycarei• What about using welfare rules?• In the U.S., different states implemented rules at differenttimes, which aimed to encourage mothers into work

• Mothers’work ↑ daycare time ↑• Identifying assumption for IV: welfare rules are uncorrelatedwith unobservable heterogeneity in child/mothercharacteristics

• Seems plausible!• "As far as instruments go, I think this is about as good as itgets." - Keane (2010), p. 7.


Child cognitive ability: The model• We are interested in the impact of mothers’’quality time’with children on child cognitive ability

• Let’s build a structural model (that incorporates our IV)• First, specify a production function for child cognitive ability:

ln (Ait ) = α0 + α1Tit + α2Cit + α3 ln Git +ωi

• Ait is child i’s cognitive ability t periods after birth, Tit iscumulative maternal quality time, Cit is daycare time, Git isgoods inputs and ωi is the child’s innate ability

• Two problems:• Git , Tit and ωi are not observed (→re-express in terms ofobservables)

• Daycare time, which is observable, may be endogenous(→instrument with welfare rules)


Unobservables in terms of observables

• Decompose innate ability:

ωi = β0 + β1Ei + ωi

• Decompose daycare time:

Cit = π0 + π1Ei + π2ωi + π3cc + π4Rit + εcit

• Decompose maternal quality time and goods inputs:

Tit = (φ0 + φ1Ei + φ2ωi ) t + φ3Cit + φ4Hit+φ5 ln Iit (W ,H;R) + εTit

ln Git = γ0 + γ1Ei + γ2ωi + γ3Cit + γ4Hit+γ5 ln Iit (W ,H;R) + γ6t + εgit


The ’hybrid’production function

lnAit = (α0 + α3γ0 + β0) + (α2 + α1φ3 + α3γ3) Cit+ (α1φ5 + α3γ5) ln Iit + (α1φ4 + α3γ4) Hit+ {α1 (φ0 + φ1Ei ) + α3γ6} t + (β1 + α3γ1)Ei

+{(1+ α3γ2) ωi + α1φ2ωi t + α1ε

Tit + α3ε

git

}• Where is Rit?• Identifying assumption for instrument: Rit uncorrelated with

ωi , εTit , ε

git (child ability; mother’s taste for investing in child)

• We are being explicit about what is suffi cient for ourinstrument to be valid.


Identification• We are interested in the effect of daycare on a child’scognitive ability

• Estimation yields α2 + α1φ3 + α3γ3 = λ• This is the total effect• The parameter λ is "just identified" = if we had populationdata, the data could be used to estimate the parameter(uniquely)

• In contrast, the individual parameters (α2, α1, etc) are notidentified or "under-identified." = Even with ideal data, wewould not be able to estimate these parameters

• e.g. what is effect of daycare relative to maternal time? Wedon’t know.

• Suppose (hypothetically) we had another equationlnAit = a+ ...+ bCit .

• Two predicted coeffi cients in the regression: b and λ →"over-identified"


Identification cont.

• Can we infer whether mother’s quality time is better for childcognition than daycare time?

• Estimation yields α2 + α1φ3 + α3γ3 = λ

• Mother’s quality time is captured by α1φ3• We know that φ3 < 0

• Suppose α3γ3 = 0 (goods have no impact)

• Then even if α1 > α2, the total effect may be positive(i.e.daycare looks like it is better), because φ3 is not ’negativeenough’(mothers substitute only weakly)

• Why? Mothers may vary their quality time as daycare timevaries. Even an ’ideal instrument’would not resolve this. Onlyif we assume mothers don’t adapt.. which is unlikely


Identification cont.• Consider a simpler example (a different assumption)• Mothers’time and daycare time are substituted one-for-one:Tit = T − Cit

• This simplifes a lot of the parameters:φo = T , φ1 = φ2 = φ4 = φ5 = 0, φ3 = −1

• The coeffi cient on Cit is now α2 − α1 + α3γ3• Effect of daycare time relative to maternal time + effect ofgoods inputs

• Assume: effect of goods inputs=0• Then structural model identifies parameter of interest• Now, the parameter ’effect of daycare time relative tomaternal time’is just identified (before it wasunder/unidentified!)

• What is identified depends on a priori economic assumptions


Implicit vs Explicit Assumptions

• A further criticism of structural modelling is that it has "toomany assumptions" (Keane 2010, p. 11)

• But it’s not about the number of assumptions• It’s about whether assumptions are implicit or made explicit• Go back to child cognition example• Bernal (2008): dynamic model of childcare and employmentchoices for women

• Result: a mother that works full-time and uses one year ofchild care reduces her child’s test scores by approximately1.8%.


Bernal (2008)

• Bernal makes two assumptions that have generatedcontroversy:

1. Mothers know their child’s innate ability, ωit .

2. Mothers know the cognitive ability production function.

• Critics argue that by using simpler methods such as fixedeffects to remove unobservable ability, these assumptionswould not need to be made

• Is this correct?• What kinds of assumptions do we need to make with fixedeffects?


Fixed effects & child cognition• Fixed effects estimators remove unobservables by assumingthat they are time invariant

• e.g. subtract lagged observation or the mean of allobservations

• Key assumption (strict exogeneity): conditional on theunobservable, there is no relationship between previous valuesof x and today’s value of x

• Necessary for consistency of fixed effect estimator• BUT what if mothers are unsure of their child’s ability andlearn about it based on test scores?

• A particularly low test score in the previous period may inducehigher levels of goods inputs and mother’s time in thefollowing period

• Even conditioning on the child’s ability, there is a relationshipover time in T and G


Fixed effects cont.

• If child test scores today affect time input tomorrow, the strictexogeneity assumption is invalidated

• Fixed effects is inconsistent• What is the solution? Assume: mothers know the child’sability perfectly (they don’t learn) and they know the exactfrom of the production function

• But these are Bernal’s assumptions!• Bernal’s assumptions are not restrictive. They are justexplicit.


Three econometric models

• We have talked about identication• Let’s talk some more about estimation• This topic is not specific to structural modelling, but relatedto understanding different approaches to econometricmodelling

• Modelling categories: Structural vs. nonstructural; parametricvs. semi-parametric vs. nonparametric

• Methods: OLS vs. ML vs...• e.g. What is a semi-parametric structural model estimated byOLS?

• Begin with relationship between wages and schooling:

W = f (S) + u


Parametric models

• A parametric model involves specifying a functional form forf (S) and a distribution for the error term

• The simplest functional form is a linear one:

W = α+ βS + u,

u ∼ N(0, σ2)

• This equation could be interpreted as a structural model, if wethink f (S) should be linear

• Parameters of interest: α, β and σ2

• Estimation procedure: OLS


Parametric models cont.

• What if we had assumed a nonlinear form for f (S)? e.g.f (S) = Sγ

• Use maximum likelihood

• Note: maximum likelihood + additive normal error →nonlinear least squares.

• Error terms don’t have to be additive

W = α+ βSu

u ∼ N(µ, σ2)

• Use maximum likelihood or general method of moments

• Parametric models involve specific functional forms• Belzil and Hansen: model parametric when we assumed shapeof utility function and joint normal of abilities


Semi-parametric models

• Inbetween parametric & nonparametric models (obviously)• For example:

W = α+ βSγ + u,

E (u|S) = 0

• One assumption: functional form of f (S)

• Left one unspecified: the distribution of the error term• No distribution → no maximum likelihood

• Use other methods e.g. GMM


Semi-parametric OLS

• Suppose you know the value of γ: e.g. γ = 2:

W = α+ βS2 + u,

E (u|S) = 0

• Estimation equation is linear in parameters• Necessary condition for use of OLS• With these assumptions, we can use OLS• So OLS can be used for semi-parametric analysis


Non-parametric models

• In nonparametric estimation, we make no assumptionsregarding the function f (S)

• Instead, we estimate it!• Simplest: histogram (=distribution of observations)

• More complicated methods exist e.g. ’kernel densities’• Why not always use non-parametric models?• Diffi cult to have more than one explanatory variable• Ineffi cient if parametric assumption is correct


Example

• What is a semi-parametric structural model estimated byOLS?

• Example 1!• Structural model of firm’s profit maximisation• Semi-parametric: assume functional form, but do not assumedistribution for errors (only that i.i.d.)

• Estimated by OLS (since Ai are i.i.d.)• NB. If we had said that the Ai are distributed normally, thiswould have been a parametric structural model estimated byOLS.


Summary & practical advice

• Structural modelling involves combining economic modelswith statistical models to generate empirical models

• We build a deterministic economic model and addunobservables/error terms

• If the predicted equations are linear, we estimate with OLS (orIV, if endogenous); If nonlinear, use ML

• Structural modelling is time intensive BUT it’s notall-or-nothing

• The most important thing to learn from these lectures is:empirics can’t work independently of theory

• Writing down models can help you think about how tointerpret your coeffi cients, even if you don’t estimate themodels directly

• Theory is necessary for empirical work!

Documents

Structural Modelling - Weebly