Econometrics: Weak Instruments
One criteria for an instrument to be valid is that it needs to be exogenous – in the traditional simultaneous equations approach this occurs if they are excluded from the equation of interest. But there is a second criteria. – instrumental relevance. Recent work suggests that many applications of instrumental variables (IV) regressions suffer from “weak instruments” or “weak identification”. That is instruments are only weakly correlated with the included endogenous variables.
The linear IV regression model with a single endogenous regressor and no included exogenous variables is:
(1) y = Yβ + u
And
(2) Y = ZΠ + v
Where y and Y are Tx1 vectors of observations on endogenous variables, Z is a TxK matrix of instruments and u and v are Tx1 error vectors. The errors are assumed to be iid (idenpendent and identical districted) N(0, Σ) where the elements of Σare σ2u, σ2uvand σ2v . Let ρ = σ2uv/ (σuσv) – the correlation between the error terms. Equation (1) is the structural equation, β the scalar parameter and (2) relates the endogenous regressor to the instruments.
The Concentration parameter
The concentration parameter μ2 is a measure of the strength of the instruments:
(3) μ2= Π′ZZΠ/σ2v
μ2can be thought of in terms of the F statistic for testing the hypothesis Π=0 (i.e. the instruments are of no value). For large values of μ2/K,F-1 can be thought of as an estimator of μ2/K.
We can also link the bias in the 2SLS estimator to μ2via the following formula:
A practical approach to detecting weak instruments is to define a set of instruments to be weak if μ2/K is small enough that inferences based on conventional normal distributions are misleading. For hypothesis testing one could define instruments to be strong if μ2/K is large enough that a 5% hypothesis test rejects no more than [say] 15% of the time.
The F statistic is useful for making inferences about μ2/K. However, simply using this to test the hypothesis of non-identification (Π=0) is an inadequate test for weak instruments. Instead Stock and Yogo propose using F to test the hypothesis that μ2/K is less than or equal to the weak instrument threshold. Table 1 reports threshold values
Table 1: Test for 5% test>15%
Threshold μ2/KThreshold F statistic
11.828.96
24.6211.59
36.3612.83
59.2015.09
1015.5520.88
1521.6926.80
Source Stock and Yogo
This can be extended to the case of multiple endogenous variables.
EXAMPLE
In an influential article Angrist and Krueger !991) proposed using the quarter of birth as an instrument to circumvent bias in estimating the returns to education. This is exogenous and should be uncorrelated with ability (the difficulty in measuring the impact of education is that ability os an omitted variable with which education is correlated, hence the coefficient on education ina wage equation will overestimate the impact of education per se). With a large sample of in excess of 300,000 people the instruments they use are however weak and their results misleading illustrating that this is not jts a small sample problem.