Análisis del proceso de implementación del Sistema de Trabajo en Grupo Sesión

Model 1(unrestricted λ):

Ai,g =λAi,g−1 +x0i,gβg+Tj,g0 πg+τj,g+αi+εi,g−λεi,g−1 (2.9)

Model 2(λ = 0)

Ai,g =x0i,gβg+Tj,g0 πg+τj,g+αi+εi,g (2.10) Model 3(λ = 1):

Ai,g−Ai,g−1 =x0i,gβg+Tj,g0 πg+τj,g+εi,g−εi,g−1 (2.11) Model 4(λ =λ0, whereλ0 is a preset value):

Ai,g −λ0Ai,g−1 =x0i,gβg+Tj,g0 πg+τj,g+αi+εi,g−λ0εi,g−1 (2.12)

We first list the main common assumptions that are required for all ap- proaches to modelling TEs, and then we discuss the specific assumptions required for each specific approach.7

Assumption A1. Strict Exogeneity of covariates conditional on unobservables:

E[εi,g|past, present and f uture values of (x, T, S), αi, τj, sg] = 0.

This assumption implies that all covariates are strictly exogenous condi- tional on all the time invariant unobservables. This statement is about the condi- tional mean of the time-varying idiosyncratic error term εi,g. In certain cases, it

might be possible to relax this strict exogeneity to sequential exogeneity whereby the conditioning is only carried out in terms of current and past values of the covariates and not in terms of future values. Hence, when the lagged dependent variable, such as Ai,g−1, is included in the VAM, it satisfies sequential exogeneity

rather than strict exogeneity.

Also note that the variables relevant to other students are assumed not

7_{Note, we have already assumedε}

affected by the outcome of student i.

Assumption A2. Random assignment of students to teachers (or classrooms)

This assumption states that all student variables are uncorrelated with all teacher variables. It is important to highlight that just having random allocation of teachers to classes may not be enough to yield consistent estimators of parameters of interest in the above models. We need to ensure that all unobservables factors in the estimation equation are uncorrelated with all observable covariates included in the model, in order to obtain consistent estimators.

We now turn to the discussion of three general approaches that have been used for modelling and estimating TE, and we discuss the additional specific as- sumptions that are needed to obtain consistent estimators in each case.

Approach 1

Early investigations were focussed on identifying high quality teachers using ob- servable characteristics only (Hanushek (1986)). This approach assumes that a very rich data source is available to account for the full effects of teachers on achievement using a variety of teacher characteristics, i.e. we can ignoreτj,g in the

above models.

Under Assumption A1, OLS estimation of Model 3 either with a pure cross-section that has one single observation on the previous score (Ai,g−1) or with

panel data, will produce consistent estimators. We can also use OLS to estimate

Model 2 and 4. However, if we only have a single cross-section with the previ- ous grade achievement, we will need the additional assumption A3.1 due to the presence of αi.

Assumption A3.1. Expected value of student heterogeneities is equal to 0, conditional on observables:

E[αi|past, present and f uture values of x, T] = 0.

Note that we have omitted τj from the conditioning variables for this par-

ticular approach. If the model estimated contains the student unobservable factor

αi, in case it cannot be eliminated prior to the estimation, then A3.1 becomes

crucial. This is because αi might still be correlated with student covariates even

dom allocation of student to teachers as stated inA2, although conditioning inT

would not be needed.

On the other hand, A3.1 is not requiered if we have multiple observations on achievement for each student, as we can use within-group transformation to eliminateαi prior to the OLS estimation. Also note that conditioning onT would

not be needed either ifstudents and teachers are randomly assigned to each other

as A2 states.

Due to the presence of lagged achievement Ai,g−1 and εi,g−1 (see equation

2.9) we cannot estimateModel 1parameters with OLS. However, an instrumental variable (IV) estimation can be used if a suitable instrument is proposed forAi,g−1.

Recent results suggest that traditional measures of teacher quality, such as additional teacher qualifications and years of experience, do not generally make a significant contribution to improving student performance. Therefore, it would be crucial to allow for unobservable teacher characteristics in these models (Rivkin

et al. (2005);Aaronson et al. (2007);Buddin and Zamarro (2009)).

Approach 2

This approach replaces the observable and unobservable variables by a set of bi- nary indicators for teachers, the so-called “Fixed Effects” (FE) approach. The main advantage of this approach is that we do not need to worry about possible correlation between unobserved (or unaccounted for) teacher characteristics and the rest of the variables included in the model because teacher indicators will control for these latent variables.

VAMs can now be estimated by OLS (also known as the Within-Group Estimator (WG)). The same estimator for TE can also be obtained using the following equivalent procedure where we first obtain the residuals from an OLS regression of teacher dummies on all of the observed covariates and then regress the test scores on these residuals.8 _{This estimation requires multiple observations}

8_{This result is due to Frisch-Waugh Theorem which shows the following:}

Consider the model y = X1β1 +X2β2 + u. Then the OLS estimator of β1 is: βˆ1 =

(X10M2X1)−1(X 0 1M2y); where M2 = I−X2(X 0 2X2)−1X 0

2 which is the symmetric idempotent

(projection) matrix, i.e. M2M2 =M2 and, M2X1 is the residual from the regression of X1on X2. It is therefore clear that the above is also equivalent to a two-step procedure where we carry

out a regression of X1 and y, on X2 to obtain the residuals M2X1 and M2y in the first step

and then regress the second set of residuals (M2y) on the first set of residuals (M2X1) to obtain

the same estimator. Because of the property of M2, we can also obtain the same estimator by

regressingy onM2X1.In our exampleX1 is the set of “teacher dummies”.

Note that estimatingy=X2β2+vby OLS and then obtaining the residuals (M2y) in step one

and then regressing this onX1to obtain an estimate of (β1) will not give the right ˆβ1 since the

new ˜β1would be (X 0

1X1)−1(X 0

1M2y).This latter procedure is sometimes used for the estimation

of TE and is known as Average Residual (AR) estimator. For example, the first stage residuals will be regressed on a set of teacher dummies which is equivalent to taking the teacher averages of the residuals (SeeGuarino et al.(2014b) for a discussion of this and other related points).

for each teacher if we want to control for SEs too.

The OLS estimator of βg in Model 3 under A1 will be consistent. How-

ever, in order to use OLS to estimateβginModel 2and4, we will also requireA2

and A3.1 to hold. Note, OLS will give consistent estimator of the effects of co- variates but not of the TEs since the estimator of TEs suffers from the“Incidental

Parameters Problem” (Neyman and Scott (1948)).

The consistency property is considered with respect to the number of stu- dents per teacher going to infinity, which cannot happen. The OLS estimation of

Model 1, even under assumptionsA1-A3.1 will not provide consistent estima- tors due to the correlation between Ai,g−1 and εi,g−1. Nevertheless, the estimator

of TE would be unbiased.

Approach 3

Here the model specification explicitly accounts for the unobserved teacher effect

τj,gand assumes this to be randomly drawn, with either including or excluding ob-

servable teacher characteristics. This estimation methodology is typically known as the “Random Effects” (RE) approach because teacher effects are treated as ran- dom effects. Common estimators for this approach are the Feasible Generalised Least Square (FGLS) and the Maximum Likelihood Estimator (MLE). Additional assumptions are now needed to obtain consistent estimators of the above models. Due to the presence ofτj,g, we now require some assumptions regarding the

relationship between this and the rest of the variables in the equation model to obtain consistent estimators. Then, a new version of A3.1 must hold under this approach.

Assumption A3.2. Expected value of student heterogeneities is equal to 0, conditional on observables and unobservables:

E[αi|past, present and f uture values of x] = 0.

If A2 holds, we do not need to condition onT orτ inA3.2. Nevertheless, for estimators following this approach we additionally need to impose A4.

Assumption A4. Expected value of teacher effects is equal to 0, conditional on observables and unobservables:

If A2 holds, we do not need to condition on x and α in A4 .

The common FGLS and MLE estimators rely on the assumption that teacher and student unobserved heterogeneities are drawn from distributions with homoskedastic variances and independent of each other.9 _{It is also customary}

to assume these unknown distributions to be N ormal for the use of maximum likelihood methods.

As before,Models 2,3and4can be estimated using FGLS or MLE. Under assumptionsA1-A4 (andN ormalityassumption for MLE) the estimators will be consistent. If model assumptions hold, this approach will provide more efficient estimators relative to the Approach 2 (Maddala et al. (1997); Guarino et al.

(2014a)).

In summary, one can either include teacher binary indicators (the so-called FE approach) or treat the teacher unobservable characteristics as random effects (RE approach) with or without including teacher observable characteristics. The RE approach using standard techniques (FGLS or MLE) will not yield consistent estimators if any of the random unobserved components is correlated with included variables in the model.

2.3.2

Teacher effects (TE): One-Step vs Two-Steps esti-