The method of instrumental variables (IV) and the generalized method of moments (GMM) has become a central technique in health economics as a method to help to disentangle the complex question of causality. However the application of these techniques require data on a sufficient number of instrumental variables which are both independent and relevant. We argue that in general such instruments cannot exist. This is a reason for the widespread finding of weak instruments.

George S. Tavlas

P. A. V. B. Swamy

Stephen Hall

English

Research Papers in Economics

 
 
 
 
DEPARTMENT OF ECONOMICS 
   
 
 
 
 
 
 
 
 
 The Nonexistence of Instrumental Variables  
 
 
 
Stephen Hall, University of Lecester, UK 
George S. Tavlas, Bank of Greece, Economics Research Department, Greece 
P. A. V. B. Swamy, Federal Reserve Board (Retired), USA 
 
 
 
 
 
 
 
 
Working Paper No. 11/24 
March 2011 
 1 
 
The Nonexistence of Instrumental Variables  
Stephen G. Hall
b c, , George S Tavlas
b, and P.A.V.B. Swamy
a                                                                                               
aBoard of Governors of the Federal Reserve System (retired), Washington, DC, USA                                
b Bank of Greece, 21, El, Venizelos Ave, 102 50 Athens, Greece. Tel.++30210 320 2370; Fax: 
++30210 320 2432. Email: gtavlas@bankofgreece.gr  
cDept of Economics, University of Leicester 
 
Abstract 
     The method of instrumental variables (IV) and the generalized method of moments (GMM) 
has become a central technique in health economics as a method to help to disentangle the 
complex question of causality. However the application of these techniques require data on a 
sufficient number of instrumental variables which are both independent and relevant. We argue 
that in general such instruments cannot exist. This is a reason for the widespread finding of weak 
instruments..   
JEL classification: C32, C51 
   
1. Introduction    
     Researchers are becoming increasingly aware that there are often serious problems with the 
use of instrumental variable based techniques (both instrumental variable (IV) estimation and 
versions of generalized methods of moments (GMM) which use instrumental variables). 
Auld(2006) points out that while many researchers in health economics resort to instrumental 
variable techniques to identify causal effects there are serious difficulties with finding valid 
instruments in many contexts. A valid instrument must be uncorrelated with the errors in an 
equation (independent) and correlated with the variable which it is being used to replace 
(relevant). Increasingly, researchers are finding that when a set of instruments are independent of 2 
 
the error they often have little relevance, this is the well known problem of ‘weak instruments’. 
Recently Basu, Heckman, Navarro-Lozano and Urzua(2007), following on from earlier work by 
Imbens and Angrist (1994) have shown that there can also be serious problems with the use of 
instrumental variables where treatment effects are heterogeneous across patients. In this case 
researchers aim to achieve an average treatment effect (ATE) but IV does not provide a 
consistent estimate of this and instead only estimates a local average treatment effect (LATE), 
see Jones’s (2009) survey for many examples of IV applied in the Health Economics context and 
for a thorough discussion of ways to estimate a causal relationship. This argument is similar, 
although less general, to the criticism  of IV made below. Here we show that any type of model 
misspecification may be represented by a model without errors but with coefficients which vary 
across each individual or each time period (we refer to time here but by simply replacing t with i 
the analysis goes through exactly with respect to cross section data and with only a minor 
complication panels may also be included in the proofs below). Once this has been done then the 
problems identified by Basu et al become relevant to all uses of IV. 
 
We show that, in general, valid instrumental variables cannot, in principle exist. We do this by 
first developing a general representation of how misspecification arises in an econometric model, 
in section 2, and then showing that this representation implies that valid instrumental variables 
cannot exist. In section 3 we then illustrate this argument in the context of a very simple model 
involving measurement error. Section 4 concludes. 
2. A General Representation of Misspecification 
 Economic theory suggests relationships between variables but it does not usually give clear 
guidance as to the correct functional form or the exact set of variables which might be relevant. 3 
 
For example, consider an economic variable, denoted by 
*
t y , and its determinants, denoted by 
*
jt x , j = 1, …,  t L . Here the total number  t L  of determinants may be time dependent. Typically, 
data on 
*
t y  and on a subset K - 1 of the  t L  determinants are available. The remaining  t L  - K + 1 
determinants are omitted from the model either because they are unobserved or for some other 
reason. Moreover, these data may contain measurement errors. Let  t y  = 
*
t y  +  0t v  and  jt x  = 
*
jt x  + 
jt v , j = 1, …, K – 1, where the variables without an asterisk are observable, the variables with an 
asterisk are unobservable, and vs are measurement errors. The theoretical relationship is 
          
* **
1 ( ,..., )
t t t t Lt y fx x =      (t = 1, …, T)                                                                                      (1) 
with unknown functional form.  
     Without misspecifying the relationship in (1), we can write  
          
1
* **
0
1
t L K
t t jt jt gt gt
j gK
y xx αα α
−
==
= ++ ∑∑                                                                                           (2) 
where the time profiles of the coefficients are determined by the correct functional form of 
model (1). These time profiles are unknown, since the correct functional form is unknown
1
The correlations between the omitted determinants and the observed determinants are implied by  
  .     
Allowing the coefficients of equation (2) to vary freely defines an infinite class of functional 
forms, which surely encompasses the correct (but unknown) functional form of (1) as a special 
case.      
          
1
**
0
1
K
gt gt jgt jt
j
xx λλ
−
=
= + ∑           (g = K, …,  t L )                                                                          (3) 
where  0gt λ  is a portion of 
*
gt x  remaining after the effects of the 
*
jt x ’s have been removed from 
                                                           
1 It is possible to represent any functional form exactly by a time varying Parameter model, we refer to this as the 
Swamy theorem . Recently, Granger(2008) has also confirmed this theorem although he attributes the proof to 
Halbert White . 4 
 
*
gt x . Since we do not have data on the  t L  - K + 1 
*
gt x  variables, we can eliminate them from 
equation (2) by substituting (3) into (2), which gives  
          
1
**
00
1
()
tt LL K
t t gt gt jt gt jgt jt
gK j gK
yx α αλ α αλ
−
= ==
= + ++ ∑ ∑∑                                                                     (4)  
     Note that equation (4) shows 
*
t y   as a function of K  –  1 included determinants and the 
remainders of the excluded variables - - i.e., what remains after subtracting the effects on the 
excluded variables of the K – 1 observable determinants. Equation (4) accounts for both the 
unknown functional form (since it is derived from equation (2)) and the full set of (time-varying) 
determinants of 
*
t y . It does not, however, account for measurement errors. In this connection, 
consider model (2) again. It is not in an estimable form. Such a form is derived below.  
     In terms of the observable variables, equation (2) can be written as  
          
1
0
1
K
t t jt jt
j
yx γγ
−
=
= + ∑                                                                                                                (5)    
We call the 
*
gt x ’s “excluded variables” because they are excluded from model (5). The  jt x ’s are 
the included explanatory variables. Model (5) coincides with model (2) if     
           0t γ  =  0t α  +  0
t L
gt gt gK αλ
= ∑  +  0t v                                                                                           (6) 
           jt γ  = ( )(1 )
t L jt
jt gt jgt gK
jt
v
x
α αλ
= +− ∑   ( j = 1, …, K-1)                                                           (7) 
These equations are derived by establishing the correspondence between equations (4) and (5).
2
     The terms on the right-hand side of equations (6) and (7) provide crucial information. 
Equation (4) shows that the 
  
0gt λ ’s in conjunction with the 
*
jt x ’s are at least sufficient to 
                                                           
2 For the derivation, see Swamy and Tavlas (2007).  5 
 
determine 
*
t y . The interpretation of the terms on the right-hand side of equation (7) and their 
implications are as follows:  
•  The term  jt α  is equal to 
** / t jt yx ∂∂   (if 
*
t y  is a continuous function of 
*
jt x ) and corresponds 
to the bias-free effect of 
*
jt x  on 
*
t y  , as can be seen from (2). 
•  The term 
t L
gt jgt gK αλ
= ∑  measures omitted-variables bias. Note that each term in this sum 
is the products of two coefficients - - the effect of the excluded variable 
*
gt x  on 
*
t y  (i.e., 
gt α ) and the effect of the included variable 
*
jt x  on the excluded variable 
*
gt x  (i.e.,  jgt λ ).  
•  The term ( )( ( / ))
t L
jt gt jgt jt jt gK vx α αλ
= +− ∑   measures measurement-errors bias.
3
•  The explanatory variables of model (5) are correlated with their own coefficients because 
the measurement-error bias component of 
  These 
biases exist whenever estimates of some theoretical variables are used as explanatory 
variables.  
jt γ  is a function of  jt x .  
 
Having derived the model in (5) which explicitly includes all these forms of misspecification it is 
now possible to show why valid instruments can not be found for this model. Under IV or GMM 
we are imposing constant parameters on (5) we can therefore re-write 5 as; 
∑ ∑ ∑
=
−
=
= − + + + + =
L
K g
K
j
L
k g jt
jt
jt
jgt gt jt t gt t t
t x
x
y
1
1
0 0 ) 1 )( (
ν
λ α α ν α α
        
(8) 
And then imposing a constant parameter model we get 
 
                                                           
3 The minus sign in the expression reflects the fact that the second parenthetical term on the right-hand side of (7) in 
one minus the ratio (/) jt jt vx .   6 
 
∑ ∑ ∑ ∑
=
−
=
=
−
=
− − + + − + + + + =
L
K g
K
j
L
k g jt j
jt
jt
jgt gt jt t gt t
K
j
jt j t
t x
x
x y
1
1
0 0 0
1
1
0 ) ) 1 )( (( ) ( β
ν
λ α α β ν α α β β
(9) 
Where the last two  terms in (9) become the error term in the model. The problem with 
instrumental variables in this context now becomes apparent; we need to find a variable which is 
correlated with xjt but which is not correlated with the error term, which itself contains xjt. Such a 
variable cannot in general exist. The only case in which a valid instrument may exist is when 
0 = jgt λ  For all j,g and t, which implies that there is no correlation between the included and 
excluded variables at any point in time, or that there are no omitted variables. And that   j jt β α =  
for all t, which implies that there is no time variation in the true parameter which, in turn, implies 
that we have the correct functional form. This amounts to saying that there is no misspecification 
in the model and, hence, there is no need for instruments in the first place. 
3. A Simple Example 
Consider a very trivial example where the only misspecification is measurement error in the 
independent variable. Assume we have a perfectly fitting relationship in the true variables 
* * X Y β =                  (10) 
where the measured value of X is given by 
v X X + = *                 (11) 
Then the model we estimate is 
ε β + = X Y
1 *                (12) 
where  ε and v  are error terms. Then there are two ways we can demonstrate the problem with IV 
applied to (12). First we may consider the issue from a TVC perspective and we may write an 
exact version of 10 as 
X Y t β = *                 (13) 7 
 
Then if we apply a fixed parameter model to this equation we get 
X X Y t ) ( *
2 2 β β β − + =             (14) 
And the last term is the error term in (12) and we can see that no valid instruments can exist for 
X since X is also in the error term. We can also see the same problem from a more conventional 
perspective. If we perform a fixed parameter regression then we can rewrite (12) as 
) * ( *
1 1 X X X Y β β β − + =             (15) 
Where again the term in brackets in (15) is the error term in (12) and we can see that the error 
term again contains the same variable that we are trying to instrument and so no valid instrument 
can exist. 
4. Conclusion 
     The instrumental variables that are correlated with the  jt x ’s of model (5) but not with the 
error terms of model (9) do not in general exist because these error terms also involve the  jt x ’s. 
These arguments make it clear why practical work with IV methods is plagued by the problem of 
weak instruments. The general difficulty with IV has become increasingly obvious in Health 
Economics as well as other areas of our discipline Basu et al (2007). We have argued here that 
the problems of IV are endemic to all models not just to certain ones such as those with 
heterogeneous coefficients. We would argue that a much better way forward in terms of practical 
estimation rests on a recognition of all the potential sources of misspecification which are 
apparent in (5) and which starts from a time varying coefficient model as outlined in Swamy, 
Hall, Tavlas and Hondroyianis(2010). 
 
 Acknowledgments 8 
 
     The views expressed in this paper are the authors’ own and are not to be interpreted as those 
of their respective past or present institutions.        
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The Nonexistence of Instrumental Variables

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