Chapter 2 Teacher Effects and Value-Added Modelling:
2.2 Value Added Models
Our research interest is focused on “Teacher Effectiveness” and the corresponding TE measures which are usually estimated within a Value Added Model (VAM) framework. We therefore start with the most commonly estimated VAMs and discuss how these can be derived by imposing some structural restrictions on the General Achievement Function. The estimation of TEs will be discussed in the next Section.
The theoretical model that forms the basis for the Value Added Models (VAMs) is the one initially proposed by Hanushek (1971), and it relates educa- tional outputs (achievements), such as Exam Scores, to a set of inputs, current and historical, provided by family and school characteristics. The model also includes a component of individual ability, though it is ignored for estimation pur- poses. Boardman and Murnane (1979) extended the general achievement model to include unobserved characteristics and discussed the estimation and interpre- tation of the effects of inputs in the presence of different types of data such as a simple cross section or a panel. Other empirical papers following this model are:
Todd and Wolpin (2003); Hanushek and Rivkin (2010); Harris and Sass (2014).
These more recent papers also base their VAMs on similar production functions, assuming that the current student achievement is a function not only of current educational inputs but also of all past inputs. This lead to the notion of a “Cu- mulative Production Function” (CPF).
The CPF specifies the achievement Ai,g for child i, who is in grade (class
or year) g, as:
where the vector Xi(g) includes all current and past individual and family edu-
cational inputs. Similarly, the vector Ei(g) includes the entire history of teacher
and school inputs. The input factor αi is assumed to be a time-invariant student
specific unobservable variable (e.g. student genetic endowment), and ε∗i,g can be interpreted as a measurement error or a component of luck (e.g. an individual’s bad day or a disruptive environment at the moment of an exam).
Assuming a linear separable function for Ai,g, we transform equation (2.1)
into our General Achievement Function (GAF).
Ai,g =
g X
r=1
[x0i,rβg,r+ei,r0 γg,r] +αi+εi,g (2.2)
The vector xi,r contains the individual and family educational inputs, and
ei,r consist of teacher and school inputs in grade r. The coefficients βg,r and γg,r
are vectors representing the impact of each observed variable, in all current and past grades, on current academic performance. The observable and unobservable child/family specific components fromXi(g) have been separated out and we have
assumed that there is only one unobserved child specific component αi and no
family-specific unobservable component.
We follow the literature and assume that the effect ofαi is grade invariant.
The term εi,g is assumed to be an idiosyncratic iid (independent and identically
distributed) error in the approximation. It is important to highlight the assump- tion that current inputs not only have an effect on current achievement, but also continue to have an impact on future achievements too, ceteris paribus.
Although equation (2.2) does not explicitly show it, the school educational input vector ei,r includes observable as well as unobservable variables. For ex-
ample, these unobservables which are present in two levels, may include: (i) at the school level characteristics, the ability of the principal, and the availability of resources; (ii) at class-level variables, teacher ability, and peer characteristics in the classroom.
The estimation of equation (2.2) requires a very rich data source, and so far, no one has been able to access this type of data with all historic records, and even if it were possible, estimation would be computationally challenging. Researchers have had to impose further restrictions on the parameters to make it estimable. The important issue that needs to be addressed is whether data limitations and the restrictions imposed prior to estimation will deliver an estimator that is consistent for the parameters of interest. A systematic discussion of these issues is presented in the next subsections.
2.2.1
Restricted VAMs
We take the GAF from equation (2.2) as a starting point to first discuss the set of restrictions generally imposed to obtain a manageable Value Added Model for estimation purposes. Then we discuss specific restrictions on VAMs that have been imposed in the literature due to data limitations.
The first restriction imposed is that although there are grade-specific effects of inputs, the persistence of effects on subsequent pupil achievements decays over time. In addition, it is also assumed that all input factors have the same rate of persistence parameterλ across periods, and it follows a geometric distribution. This particular form of decaying has the considerable advantage of enabling us to rewrite the initial GAF from equation (2.2) with two main components, all current inputs and a function of the previous years’ achievements.
Under the above assumptions, equation (2.2) can now be written as
Ai,g =x0i,gβg+e0i,gγg+αi+ g−1 X
r=1
λg−r[x0i,rβr+e0i,rγr] +εi,g (2.3)
where 0≤λ ≤1 andβg, =λg−rβr.
Equation (2.3) says, for example, that the effect of last year’s inputs xi,g−1
onAi,g isλβg−1 and the effect ofxi,g−2 onAi,g isλ2βg−2. This assumption has two
important implications: (i) the rate of persistence is the same for all educational input factors, and (ii) the rate of geometric decay is constant across grades: hence, the effect of an input factor in, for example, two previous periods will have the same effect λ2βg−2 on current grade g achievement, independent of whether the
current grade is 3, 4 or 5.
Thus, considering equation (2.3) for period g-1 multiplied by λto subtract both sides of equation (2.3), we get2
Ai,g−λAi,g−1 =x0i,gβg +ei,g0 γg+ (1−λ)αi+εi,g−λεi,g−1
and we rewrite the above equation as:3 Model 1:
Ai,g =λAi,g−1+x0i,gβg+e0i,gγg+αi+εi,g−λεi,g−1 (2.4)
Equation (2.4) now forms the basis for various VA estimations. This equa- tion represent the most general case of VAMs, which can be interpreted as indi-
2This trick is used in the Partial Adjustment Models to reduce the number of parameters,
and the demonstration of the transformation is shown in Appendix 2.2.
3WLOG we can write (1−λ)α
i=αi.However, as we shall see later, this term will disappear
cating that: pupil achievement in a given gradeg depends on current and previous observed and unobserved input factors (including teacher effects).
Before we turn to the estimation of TE, it should be noted that the estima- tion of equation (2.4) is complicated due to the presence of Ai,g−1 and αi, which
are correlated with εi,g−1. Hence, it requires much more stringent assumptions to
obtain consistent estimators, as we will discuss later in the VAM specifications and literature review (e.g. Aaronson et al.(2007);Kane and Staiger (2008);Rothstein
(2010); Buddin (2011)).
We next examine what additional restrictions researchers have imposed on equation (2.4) to enable them to address this issue. Of course, if a valid instrument forAi,g−1 can be found, estimation of this equation becomes easier. In that sense,
we present some of the restricted models commonly used as VAMs in the literature.
Model 2: Non-Historical with total decay of input factor effects (λ = 0)
Ai,g =x0i,gβg+e0i,gγg +αi +εi,g (2.5) Model 3: No decay with full persistence of input factor effects (λ= 1)
Ai,g−Ai,g−1 =x0i,gβg+e0i,gγg+εi,g−εi,g−1 (2.6) Model 4: Geometric decay with a fixed known persistence parameter λ0,
(0< λ0 <1)
Ai,g−λ0Ai,g−1 =x0i,gβg+e0i,gγg+αi+εi,g−λ0εi,g−1 (2.7)
In Model 4, λ is set equal to a specific parameter value to avoid the correlation problems mentioned above (e.g. λ0 is taken from estimates reported
in the literature).
However, further additional assumptions for the above models are necessary in order to generate consistent estimators depending on the estimation approach taken. In the following section we explore in detail the estimation methodology of TE and the required assumption for each estimation approach.