INSTRUMENTAL VARIABLES REGRESSIONS IN ESTIMATING
THE EFFECTS OF INSTITUTIONS ON ECONOMIC
DEVELOPMENT
INES BUONO & DECIO COVIELLO
Introduction
Moving form Hall and Jones HJ(99), the economic growth literature is pointing its attention on explaining economic growth at a deeper level. Instead of explaining the difference in capital accumulation, productivity, and therefore output per worker through the production function, a strand of thought is working on the idea that productivity, through capital intensity and efficiency is driven by Institutions. This suggests to treat the explaining variable Institutions as endogenous in the system of linear symultaneous equations. The object of this notes is not to explain the theoretical nature of the Institutions as first order determinant of economic development, but to highlight the empirical problems in assessing the effects of institutions on GDP per capita levels.
1.
Basic Specification and Problems:
The main regression to explain the effects of Institutional quality on the GDP level is:
(1) lg y c
= α + βIQ c
+ u c
Where lgy c is the log of per-capita GDP level expressed in purchasing parity power (Penn-World-Tables), and the IQ c is an index for the Institutions quality that differs from author to author 1 .
The problems of basic OLS estimation are:
• Measurement Error = ⇒ downward bias,
• Endogeneity = ⇒ upward-bias,
• Heterogeneity effects.
The problems of measurement error is the following:
(2) X = e
+ ν
Where e is the true variable, while X is the one measured, and error in the model. When we run the regression we have:
ν is the independent
1
From now on: lgy c
= y and IQ c
= X
1

2 INES BUONO & DECIO COVIELLO y = α + β e
+ u y = α + β ( X − ν ) + u y = α + βX + ω
(3)
Where ω = u − βν . Now we can notice that:
(4) COV ( X, ω ) = COV ( e
+ ν, u − βν ) = − βσ 2
ν
The OLS estimator of our equation is:
(5)
=
β
OLS
=
COV ( X, y )
V AR ( X )
COV ( α, X ) + βV AR ( X ) + COV ( X, ω )
V AR ( X )
= ( β +
COV ( X, ω )
) < β
V AR ( X ) where the downward bias arises from from eq.(4).
By the different way (HJ(99),and AJR(2001)) in which the IQ index is constructed it is possible to assume that a large measurement error affects the regression equation.
The Endogeneity Problem is the following: y = f ( X )
X = f ( y ) y = α + βX + u
X = a + by + ε
So when we run OLS we have that COV ( X, u ) = 0 because the X = f ( y ) and from calculation we obtain that the β
OLS is upward biased. Since empirical evidence shows a no well-determined causality among the GDP and the Institutions Index, eq.(1) is affected by the endogeneity problem 2 .
Now let’s consider the case in which there is neither endogeneity nor measurement error problem. Is it always possible to say that b
OLS
= β
IV
? The answer is No, because we can incurr in the heterogeneity in the IV estimations. This effect can give problems in the interpretation of the estimated coefficients.
Consider:
(6) X = X
1
+ X
2 where we observe and measure X but not its different components X
1 usual model:
; X
2
, and the
(7) y = β
1
X
1
+ β
2
X
2
+ u
2
Moreover, there exist a kind of easy endogeneity caused by the way in which the IQ index is constructed. The problem arises because when a country is more developed (higher GDP) it is thought to have better Institutions

3
β
OLS
=
COV ( X, y )
V AR ( X )
= β
1
COV ( X
1
, X )
V AR ( X )
+ β
2
COV ( X
2
, X )
V AR ( X )
(8) and for the instrumental variable case:
β
IV
=
COV ( Z, y )
COV ( Z, X )
= β
1
COV ( Z, X
1
)
COV ( Z, X )
+ β
2
COV ( Z, X
2
)
COV ( Z, X )
(9)
If Z is correlated with X
1 but not with X
2 then the IV estimator results to be:
β
IV
= β
1
COV ( Z, X
1
)
COV ( Z, X )
(10)
From eq.(8) and eq.(10) it is possible to see that OLS is in any case a weighted average of the two components of the endogenous variables, while in the IV estimator not always this result holds. So if we consider X as the Institutions Quality variable, X
1 as the property rights variable and X
2 as law enforcement, and we use a set of instruments that are correlated with the Institutions only through property rights, the IV estimation does not weight the effect of rule of law on the explained variable,GDP.
2.
How to solve the OLS estimations problems?
Instrumental Variables regressions are used to find the exogenous variations of the
Institutions. We estimate the following model with two stages least squares (TSLS):
(11) X = a + bZ + ν
(12) y = α + β b
+ u
Where the eq.(6) is the first stage, and eq.(7) is the second stage. To select instruments we must follow two fundamental requirements:
(1) Instruments exogeneity (the orthogonality condition): E [ Z
(2) Instruments Relevance: E [ Z 0 X ] = 0
0 u ] = 0
It is common practice to select instruments just considering the exogeneity condition through a test of the Over-Identification Restrictions. But instruments exogeneity is only one of the two criteria necessary for instruments to be valid. In fact if the instruments are not relevant (i.e are weakly correlated with the endogenous regressor), then sampling distributions of the IV statistics are in general non-normal
(Staiger and Stock (1997) for asymptotic, and Rothenberg (1984) for finite sample), and standard IV point estimates, hypothesis tests, and confidence intervals are unreliable.

4 INES BUONO & DECIO COVIELLO
3.
How to detect if instruments are weak
Various procedures are available for detecting weak instruments in the linear IV model that are summarized as follows:
• The first-stage F − statistic that must be greater than a reported threshold depending on the number of instruments (look at table 1 in Stock-Wright and Yogo (2002)).As a rule of thumb F must be bigger than 10,
• The first-stage t − tests that as a rule of thumb must be greater than 3 .
5,
• The first stage R 2 , greater than 30% .
4.
Problems given by weak instruments
4.1.
Problems in Over-Identification Test: The test is performed to detect if instruments verify the exogeneity condition -i.e if they work on GDP only through
Institutions and not directly-.
The test consist in running the following equation: y = α + x = a + bZ + ε
β b
+ δZ
1
+ u
(13)
And test H
0
: δ = 0 3 . If we reject H
0 the instrument is not good. In fact if we run the second stage regression considering the instrument as a regressor, we can incur in the multicollinearity problem because X and Z
1 vary together. Moreover, if we run the second stage regression without taking into account Z error term that becomes correlated with the regressor X
1
, this enters in the
. If we do not reject the null, we are tempted to say that Z
1 is a good instrument. The problem arises when
Z
1 is a weak instrument, because the estimated coefficient of tests are misleading(see Staiger and Stock (1997) pg.566).
Z
1 is non-standard and
4.2.
Multicollinearity in IV estimation.
Two cases are of our interest:
(1) one endogenous regressor,
(2) one endogenous and a set of exogenous regressors,
(3) two endogenous regressors.
Following the structural model of eq.(6-7) if Z is weakly correlated with X, then is almost zero, and b is almost constant. When we run eq.(7) the variables in the matrix of regressors are almost collinear.
Considering the case of one endogenous and a set of exogenous regressors we rewrite the system as follows 4 :
(14) X = a + bZ + c Π + ν
(15) y = α + β b
+ δ Π + u
If the R 2 is high we could conclude that instruments are strongly correlated, but this may be wrong because the high R 2 can arise because of the variation given by Π
3
Notice: b is given by the regression of X on the set of instruments -in this case Z
1 first stage, and in the second stage we test instrument by instrument
4 for implicity we consider only one exogenous regressor,Π
, Z
2
- in the

5 instead of that of Z. Once again in the second stage we could have multicollinearity because Π, and X move together.
The following system reports the case of more than one endogenous regressors:
(16) y = α + β b
1
+ γ b
2
+ u
(17) y
1
= a + bZ
1
+ cZ
2
+ ν
(18) y
2
= A + BZ
1
+ CZ
2
+ ε
In this case we can have that even if the instruments are detected to be strongly correlated with the endogenous variables in the eq.(12-13) it is possible to have multicollinearity in eq.(11) if the instruments are collinear among them. Notice: in the multiple endogenous case it changes the way to detect if instruments are weak: instead of using the first-stage F-statistics,t-test and R 2 , we must use the minimum eigenvalue of the concentration parameter matrix (See Stock-Wright-Yogo (2002),
Stock-Yogo (2003) for the computed thresholds). So it is important to remark that there is a big change in dealing with two or more endogenous regressors rather than with one.
5.
Partially Robust and Fully Robust Inference Remedies
The first kind of remedy focuses on testing and performing the power of different class of estimators that are the following:
• LIML estimators,
• Fuller-k estimator,
• Bias adjusted TSL,
• Jacknife estimators.
The LIML is shown to be the best median unbiased k-class estimator to second order -Rothenberg (1983)- and actually is a used estimator in applied works -Moreira
(2003)-. Even if these estimators are easy to compute they are partially robust to weak instruments problems, in the sense that they are more reliable than TSLS when instruments are weak.
The second remedy consists in construct ad hoc confidence intervals for the estimators by performing Montecarlo simulations. Based on the methods developed by
Moreira (2001), valid tests for structural coefficients are constructed, and based on the conditional distribution of non-pivotal statistics. The conditional approach is used to find critical value functions for Wald and likelihood ratio tests yielding correct rejection probabilities no matter how weak the instruments are. Together with the Anderson-Rubin and Lagrange Multiplier, the conditional Wald and likelihood ratio tests can be used to construct confidence intervals, by inverting the conditional tests, that have correct coverage probability even when instruments may be weak and that are still informative when instruments are good.

6 INES BUONO & DECIO COVIELLO
Figure 1.
Transitions resulting from business cycle type variations
References
[K (2002)] Kleibergen, F.
, Pivotal Statistics for Testing Structural Parameters in Instrumental
Variables Regression”, Econometrica, 2002, September, No.5, Vol.70, pg.1781-1803.
[Moreira (2002)] Moreira, M.
, ”Tests with Correct Size when Instruments Can Be Arbitrarily
Weak ”, Unpublished Manuscript, University of California Berkley, Dept. of Economics, 2001, September.
[SS (1997)] Staiger, D., Stock J.H.
, ”Instrumental Variable Regression With Weak Instruments”, Econometrica, 1997, May, No.5, Vol. 65, pg.557-586.
[SWY (2002)] Stock, J.H, Wright, J.H, Yogo, M.
, ”A Survey of Weak Instruments and Weak
Identification in Generalized Method of Moments”, Journal of Business and Economic
Statistics, 2002, October, Vol.4, No.20, pg.518-529.
[SY (2001)] Stock, J.H, Yogo, M.
, ”Testing for Weak Instruments in Linear IV Regression”,
Unpublished manuscript, Harvard University, 2001, revisited April 2003
E-mail address : ines.buono@upf.edu; decio.coviello@upf.edu