Instrumental Variables
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1 Università di Pavia 2010 Instrumental Variables Eduardo Rossi
2 Exogeneity Exogeneity Assumption: the explanatory variables which form the columns of X are exogenous. It implies that any randomness in the DGP that generated X is independent of the error terms ε in the DGP for y. This independence implies that E(ε X) = 0 the mean of the entire vector ε, that is, of every one of the ε t, is zero conditional on the entire matrix X. This assumption is acceptable for cross-section data but not for time series data. Eduardo Rossi c - Econometrics 10 2
3 Exogeneity Time series data: each observation might correspond to a year, quarter, month, or day. Even if we are willing to assume that ε t is in no way related to current and past values of the regressors, it must be related to future values if current values of the dependent variable affect future values of some of the regressors. In the context of time-series data, the exogeneity assumption is a very strong one that we may often not feel comfortable in making. Eduardo Rossi c - Econometrics 10 3
4 Innovation The assumption: E[ε t x t ] = 0 is substantially weaker than Exogeneity assumption, because this rules out the possibility that the mean of ε t may depend on the values of the regressors for any observation, while E[ε t x t ] = 0 rules out the possibility that it may depend on their values for the current observation. From the point of view of the error terms, it says that they are innovations. An innovation is a r.v. of which the mean is 0 conditional on the information in the x t, and so knowledge of the values taken by the latter is of no use in predicting the mean of the innovation. Eduardo Rossi c - Econometrics 10 4
5 Predeterminedness From the point of view of the explanatory variables x t, Predeterminedness assumption says that they are predetermined with respect to the error terms. Eduardo Rossi c - Econometrics 10 5
6 Endogeneity There is a problem of endogeneity in the linear model y t = x tβ + ε t if β is the parameter of interest and E (x t ε t ) 0. Eduardo Rossi c - Econometrics 10 6
7 Simultaneous equations Demand-supply system: q d,t = α 1 p t + α 2 x t + ε d,t q s,t = β 1 p t + ε s,t q s,t = q d,t = q t x t income exogenous. Prices and quantities are measured in logs and in deviation form their sample means. Interest centers on estimating the demand elasticity: E[ε d,t x t ] = 0 E[ε s,t x t ] = 0 E[ε 2 d,t x t ] = σ 2 s E[ε 2 s,t x t ] = σd 2 E[ε s,t x t ] = E[ε d,t x t ] = 0 E[ε s,t ε d,t x t ] = 0 Eduardo Rossi c - Econometrics 10 7
8 Simultaneous equations Reduced form of the model: p t = α 2 x t β 1 α 1 + ε d,t ε s,t β 1 α 1 = π 1 x t + v 1t q t = β 1α 2 x t β 1 α 1 + β 1ε d,t α 1 ε s,t β 1 α 1 = π 2 x t + v 2t σ2 d Cov[p t, ε d,t ] = β 1 α 1 Neither the demand not the supply equation satisfies the assumptions of the classical regression model. The price elasticity of demand (α 1 ) cannot be consistently estimated by OLS regression of q on x and p. Because the endogenous variables are all correlated with the disturbances, the OLS estimators of the parameters of equations with endogenous variables on the RHS are inconsistent. Eduardo Rossi c - Econometrics 10 8
9 measurement error Measurement error in the regressors. Suppose that (y t,x t) are joint random variables E (y t x t ) = x t β is linear. But x t is not observed. Instead we observe x t = x t + u t u t (K 1) measurement error, independent of y t and x t. Eduardo Rossi c - Econometrics 10 9
10 measurement error where y t = x t β + ε t = (x t u t ) β + ε t = x tβ + v t v t = ε t u tβ. The problem is that E (x t v t ) = E [(x t + u t ) (ε t u tβ)] = E (u t u t) β 0 If β 0 and E (u t u t) 0. It follows that if β is the OLS estimator β = β + [ ] 1 (x t x t) x t v t t t Eduardo Rossi c - Econometrics 10 10
11 measurement error then [ p lim β 1 = β + p lim N ] 1 1 (x t x t) N t x t v t t p lim β = β (E (x t x t)) 1 E (u t u t)β β. Eduardo Rossi c - Econometrics 10 11
12 Instrumental Variables Estimator Let the equation of interest be y t = x tβ + ε t (1) where x t (K 1), E (x t ε t ) 0 so that there is a problem of endogeneity. Eq. (1) is called structural equation. In matrix notation, this can be written as y = Xβ + ε any solution to the problem of endogeneity requires additional information which we call instruments. Eduardo Rossi c - Econometrics 10 12
13 Instrumental Variables Estimator Definition The vector z t, (L 1), is an instrumental variable for eq.(1) if E (z t ε t ) = 0. In a typical set-up, some regressors in x t will be uncorrelated with ε t (for example, at least the intercept). Thus we make the partition x t = x 1t x 2t K 1 K 2 where E (x 1t ε t ) = 0 yet E (x 2t ε t ) 0. We call x 1t exogenous and x 2t endogenous. x 1t is an instrument for eq.(1). Eduardo Rossi c - Econometrics 10 13
14 Instrumental Variables Estimator So we have the partition z t = x 1t z 2t K 1 L 2 Three possible cases: Just-identified if L = K Over-identified if L > K Under-identified if L < K. Eduardo Rossi c - Econometrics 10 14
15 Reduced Form The reduced form relationship between the variables or regressors x t and the instruments z t is found by linear projection. Let Γ = E (z t z t) 1 E (z t x t) be the (L K) matrix of coefficients from a projection of x t on z t, and define u t = x t z tγ as the projection error. Then the reduced form linear relationship between x t and z t is x t = Γ z t + u t (2) Eduardo Rossi c - Econometrics 10 15
16 Reduced Form In matrix notation X = ZΓ + u (3) u (N K). By construction E (z t u t) = 0 So (3) is a projection and can be estimated by OLS X = Γ = Z Γ + û (Z Z) 1 Z X Substituting X = ZΓ + u in y = Xβ + ε y = (ZΓ + u) β + ε = Zλ + v (4) where λ = Γβ, v = uβ + ε. Eduardo Rossi c - Econometrics 10 16
17 Reduced Form Observe that E (z t v t ) = E (z t u t) β + E (z t ε t ) = 0 Thus (4) is a projection equation and may be estimated by OLS. This is y = Z λ + v λ = (Z Z) 1 Z y The equation (4) is the reduced form for y. The system y = Zλ + v X = ZΓ + u OLS yields the reduced-form estimates ( λ, Γ). Eduardo Rossi c - Econometrics 10 17
18 Identification The structural parameter β relates to (λ,γ) through λ = Γβ. The parameter is identified, meaning that it can be recovered from the reduced form, if rank (Γ) = K (5) Assume that (5) holds. If L = K, then β = Γ 1 λ. If (5) is not satisfied, then β cannot be recovered from (λ,γ). Note that a necessary (although not sufficient) condition for (5) is L K. Eduardo Rossi c - Econometrics 10 18
19 Identification Since X and Z have the common variables X 1, we can rewrite some of the expressions X = [ X 1 ] X 2 Z = [ X 1 ] Z 2 we can partition Γ as Γ = Γ 11 Γ 12 Γ 21 Γ 22 = I K 1 Γ 12 0 Γ 22 L 1 K 2 L 2 K 2 Eduardo Rossi c - Econometrics 10 19
20 Identification ZΓ = = [ [ ] X 1 Z 2 I K 1 Γ 12 0 Γ 22 ] X 1 X 1 Γ 12 + Z 2 Γ 22 β is identified if rank (Γ) = K, which is true if and only if rank (Γ 22 ) = K 2 (by the upper-diagonal structure of Γ). The key to identification of the model rests on the (L 2 K 2 ) matrix Γ 22. Eduardo Rossi c - Econometrics 10 20
21 Instrumental Variable Estimation Suppose that the model is just identified (L = K). Then β = Γ 1 λ. This suggests the Indirect Least Squares estimator: β IV = = Γ 1 λ [ ] (Z Z) 1 1 [ ] Z X (Z Z) 1 Z y = (Z X) 1 (Z Z)(Z Z) 1 (Z y) = (Z X) 1 (Z y) β IV is the instrumental variables estimator of β. For identification by any given sample, the condition is just that Z X should be nonsingular. Eduardo Rossi c - Econometrics 10 21
22 Consistency of IV Estimator Since ) p ( λ, Γ (λ,γ) and Γ is invertible, β is consistent. A more direct way to see consistency is to substitute the equation y = Xβ + ε into the equation for β to obtain β IV = (Z X) 1 (Z (Xβ + ε)) Given: ( Z ) X Q ZX = p lim N = β + (Z X) 1 Z ε the IV estimator is consistent iff: ( N 1 Z X ) 1 ( N 1 Z ε ) p 0 deterministic with rank K (Asymptotic identification) or p lim N Z ε = 0 asymptotically uncorrelated. Eduardo Rossi c - Econometrics 10 22
23 The asymptotic distribution of IV estimator ( Z ) X Q ZX = p lim deterministic with rankk (Asymptotic identification) N ( Z ) Z Q ZZ = p lim positive definite N N ( βiv β) = [ ] N (Z X) 1 Z ε Under the hypothesis that E (εε X,Z) = σ 2 I n, The Central Limit Theorem gives us the limit distribution ) L ( N ( βiv β N 0, σ 2 Q 1 ZX Q ZZQ 1 ) XZ The estimate of σ 2 σ 2 = 1 N ( ) ( ) y X β y X β Eduardo Rossi c - Econometrics 10 23
24 Efficiency If the model is correctly specified the asymptotic covariance matrix: [ V ar p lim N( βiv β)] = σ 2 Q 1 ZX Q ZZQ 1 XZ N ( Z = σ [p 2 )] 1 [ ( X Z )] [ ( Z X )] 1 Z lim p lim p lim N N N { [ ( X = σ 2 )] [ ( Z Z )] 1 [ ( Z Z )] } 1 X p lim p lim p lim N N N { (X = σ 2 )( Z Z ) 1 ( Z Z ) } 1 X p lim N N N ( ) 1 1 = σ 2 p lim N X P Z X If we have some choice over what instruments to use in the matrix Z, it makes sense to choose them so as to minimize the above asymptotic covariance matrix. Eduardo Rossi c - Econometrics 10 24
25 Optimal instruments Optimal instruments: instruments that minimize the asymptotic covariance matrix. In order to determine the optimal instruments, we must know the data generating process. Quite generally, we can suppose that the explanatory variables satisfy the relation: where X = X + V x t = E[x t Ω t ] E[v t Ω t ] = 0 where Ω t is the set of uncorrelated variables with ε t. It turns out that the matrix X provides the optimal instruments. Eduardo Rossi c - Econometrics 10 25
26 Hausman Specification Test It might not be obvious that the regressors are correlated with the disturbances or that the regressors are measured with error. If not, there would be some benefit to using the OLS rather than the IV estimator. We can compare the asymptotic variance-covariance matrices under the assumption that both are consistent, that is p lim 1 N X ǫ = 0 Eduardo Rossi c - Econometrics 10 26
27 Hausman Specification Test The difference between the asymptotic covariance matrices of the two estimators is ( AV ar( β IV ) AV ar( β OLS ) = σ2 X N p lim Z(Z Z) 1 Z ) 1 X N ( σ2 X ) 1 N p lim X N = σ2 N p limn [ (X Z(Z Z) 1 Z X) 1 (X X) 1] Eduardo Rossi c - Econometrics 10 27
28 Hausman Specification Test Compare the inverses (X Z(Z Z) 1 Z X) = X P Z X = X (I N M Z )X = X X X M Z X since M Z is p.s.d. it follows that X M Z X is also so. Since (X Z(Z Z) 1 Z X) is smaller, in the matrix sense, than X X then its inverse is larger. If the OLS is consistent is the preferred estimator (asymptotically more efficient). Eduardo Rossi c - Econometrics 10 28
29 Hausman Specification Test We want to test the null hypothesis that the error terms are uncorrelated with all the regressors against the alternative that they are correlated with some of the regressors, although not with the instruments. The null hypothesis is p lim 1 N X ǫ = 0 under the null hypothesis both estimators are consistent. Under the alternative only β IV is consistent. The suggestion is to examine d = β IV β OLS Eduardo Rossi c - Econometrics 10 29
30 Hausman Specification Test Under the null hypothesis p limd = 0 We might test this hypothesis using a Wald statistic H = d [AV ar( β IV β OLS )] 1 d Eduardo Rossi c - Econometrics 10 30
31 Hausman Specification Test AV ar( β IV β OLS ) = AV ar( β IV ) + AV ar( β OLS ) 2ACov( β IV, β OLS ) We do not have an expression for the covariance term. Hausman shows that Cov[ β OLS, β OLS β IV ] = 0 Cov(X, X Y ) = E[(X E(X))(X Y E(X) + E(Y ))] then = E[(X E(X)) 2 ] E[(X E(X))(Y E(Y ))] = V ar[x] Cov(X, Y ) Cov[ β OLS, β OLS β IV ] = V ar[ β OLS ] Cov[ β IV, β OLS ] = 0 V ar[ β OLS ] = Cov[ β IV, β OLS ] Eduardo Rossi c - Econometrics 10 31
32 Hausman Specification Test AV ar( β IV β OLS ) = AV ar( β IV ) AV ar( β OLS ) Inserting this useful result into the Wald statistics and reverting to empirical estimates H = ( β IV β OLS ) [AV ar( β IV ) AV ar( β OLS )] 1 ( β IV β OLS ) Under the null hypothesis we are using two consistent estimators of σ 2. If we use s 2 as the common estimator, then the test statistic where H = ( β IV β OLS ) [( X X) 1 (X X) 1 ] 1 ( β IV β OLS ) s 2 X = (Z Z) 1 Z X Eduardo Rossi c - Econometrics 10 32
33 Hausman Specification Test Unless X and Z have no variables in common, the rank of the matrix in the statistic is less than K, and the ordinary inverse will not even exist. In most cases at least some of the variables in X appear in Z (at least the constant term!). Let X = [X 1,X 2 ] where X 2 are the suspected endogenous variables. X 1 is included in Z. A subset of K 1 variables is uncorrelated with the disturbances. The quadratic form in the statistic is used to test only K 2 = K K 1 hypotheses. In this case H is quadratic form of rank K 2. Eduardo Rossi c - Econometrics 10 33
34 Hausman Specification Test Since Z(Z Z) 1 Z is an idempotent matrix X = P Z X X X = X P Z P Z X = X P Z X = X X. Using this result and expanding d d = ( X X) 1 X y (X X) 1 X y = ( X X) 1 [ X y ( X X)(X X) 1 X y] = ( X X) 1 X [y X(X X) 1 X y] = ( X X) 1 X ǫ K 1 columns of X are in X. Suppose that these are the first K 1 columns. Thus the first K 1 rows of X ǫ are the same as the first K 1 rows of X ǫ which are 0. Eduardo Rossi c - Econometrics 10 34
35 Hausman Specification Test Thus we can write d = ( X X) 1 0 X 2ǫ = ( X X) 1 0 q the test statistic becomes H = d [( X X) 1 (X X) 1 ] 1 d s 2 = [ ] 0 q ( X X) 1 W( X 1 X) 0 q where W = s 2 [( X X) 1 (X X) 1 ] Eduardo Rossi c - Econometrics 10 35
36 Hausman Specification Test An (n m) matrix, denoted by A is the generalized inverse of the (m n) A matrix if it satisfies AA A = A 1. A is not unique in general. 2. AA and A A are idempotent. 3. I m AA and I n A A 4. rank(a) = n A A = I n 5. rank(a) = m AA = I m Eduardo Rossi c - Econometrics 10 36
37 Hausman Specification Test The Moore-Penrose Inverse. The (n m) matrix A + is the Moore-Penrose (generalized) inverse of the (m n) A if it satisfies the following four conditions: 1. AA + A = A, 2. A + AA + = A + 3. (AA + ) H = AA + 4. (A + A) H = A + A Eduardo Rossi c - Econometrics 10 37
38 Hausman Specification Test Properties 1. A (m n): A + exists and is unique. 2. A (m n): (a) rank(a) = m AA + = I m (b) rank(a) = m A + = A H (AA H ) 1 (c) rank(a) = n A + A = I n (d) rank(a) = n A + = (A H A) 1 A H 3. A nonsingular (m m) A + = A 1. Eduardo Rossi c - Econometrics 10 38
39 Hausman Specification Test Denoting P = ( X X) 1 W( X 1 X) [ H = 0 q ]P 0 q = q P 22 q P 22 is the lower K 2 K 2 submatrix of P. Eduardo Rossi c - Econometrics 10 39
40 Hausman Specification Test An alternative approach, provided by Durbin (1954) and Wu (1973), circumvents the computation of the generalized inverse. As before, we want to test that p limd = 0. The idea of the DWH test is to check whether the difference β IV β OLS is significantly different from zero in the available sample. By construction the OLS residuals are orthogonal to the columns of X, in particular those in X 1. Eduardo Rossi c - Econometrics 10 40
41 Hausman Specification Test We know that d = ( X X) 1 X ǫ testing whether β IV β OLS is significantly different from zero is equivalent to testing whether the vector X ǫ is significantly different from zero. X ǫ = X P Z ǫ since X 1 Z X 1 ǫ = X 1M X y = 0 The test is thus concerned only with the K 2 elements of X 2P Z ǫ (or X 2P Z M X y) which will not in general be identically zero, but should not differ from it significantly under H 0. Eduardo Rossi c - Econometrics 10 41
42 Hausman Specification Test An F test statistic can be computed to test the joint significance of the elements of γ in the augmented regression y = Xβ + X 2 γ + ǫ This is equivalent to the Hausman test for this model. The OLS estimates of δ are, by the FWL Theorem, the same as those from the FWL regression of M X y on M X P Z X 2 δ = (X 2P Z M X P Z X 2 ) 1 X 2P Z M X y Since the inverted matrix is positive definite, we see that testing whether δ = 0 is equivalent to testing whether X 2P Z M X y = 0 as desired. Eduardo Rossi c - Econometrics 10 42
43 Returns to schooling On average people with more education have higher wages. Two possible, and alternative explanations: Greater earning capacity means that the individual can spend more on education. Higher education causes higher wages. If the former is true OLS estimates simply reflects differences in unobserved characteristics of working individuals and an increase in a person s schooling due an exogenous shock will have no effect on this person s wage. Eduardo Rossi c - Econometrics 10 43
44 Returns to schooling Card (1999) The causal effect of education on earnings, Handbook of Labour Economics, vol.iiia. Human capital earnings function: w t = β 1 + β 2 S t + β 3 E t + βe 2 t + ε t w t : log of individual earnings S t : years of schooling E t : years of experience. In absence of information on E t, potential experience is measured as age t S t 6, assuming people start school at the age of 6. Eduardo Rossi c - Econometrics 10 44
45 Returns to schooling Additional variables to control for gender and racial dummies. It sometime argued that return to education vary across individuals: where w t = z tβ + γ t S t + u t = z tβ + γs t + ε t ε t = u t + (γ t γ)s t z t includes all observable variables (except S t ), including the experience variables and a constant. Assumption: E[ε t z t ] = 0 Eduardo Rossi c - Econometrics 10 45
46 Returns to schooling The coefficient γ has the interpretation of the average return to (an additional year of) schooling E[γ t ] = γ, and is the parameter of interest. Reduced form for S t : where S t = z tπ + v t E[v t z t ] = 0 this is the best linear approximation. OLS of β and γ are consistent if: E[ε t S t ] = E[ε t v t ] = 0. This means that there are no unobservable characteristics that both affect a person s choice of schooling and his (later) earnings. Eduardo Rossi c - Econometrics 10 46
47 Returns to schooling Reasons why schooling is correlated with ε t : 1. Ability bias. Suppose that some individuals have unobserved characteristics (ability) that enable them to get higher earnings. If these individuals also have above average schooling levels, this implies a positive correlation between ε t and v t and an OLS estimator that is upward biased. 2. Measurement error in the schooling measure. This induces a negative correlation between ε t and v t, and an OLS estimator that is downward biased. Eduardo Rossi c - Econometrics 10 47
48 Returns to schooling No instruments available for schooling as all potential candidates are included in the wage equation. The number of moment condition in E [ε t z t ] = E [(w t z tβ γs t )z t ] = 0 is one short to identify β and γ. If there is a variable in z t (e.g. z t2 ) that affects schooling but not wages, this variable can be excluded from the wage equation so as to reduce the number of unknown parameters by one, thereby making the model exactly identified. Eduardo Rossi c - Econometrics 10 48
49 Data set Data taken from the National Longitudinal Survey of Young Men (NLSYM) concerning the United States. The analysis focusses on 1976 but uses some variables that date back to earlier years. The following variables available (many are dummy variables) individuals. The average years of schooling is somewhat more than 13 years. Average experience in 1976, with age between 24 and 34, 8.86 years, the average hourly wage is $5.77. Eduardo Rossi c - Econometrics 10 49
50 Data set smsa66 1 if lived in smsa in 1966 smsa76 1 if lived in smsa in 1976 nearc2 grew up near 2-yr college nearc4 grew up near 4-yr college nearc4a grew up near 4-year public college nearc4b grew up near 4-year private college ed76 education in 1976 ed66 education in 1966 age76 age in 1976 daded dads education (imputed avg if missing) nodaded 1 if dads education imputed momed mothers education Eduardo Rossi c - Econometrics 10 50
51 Data set nomomed 1 if moms education imputed momdad14 1 if lived with mom and dad at age 14 sinmom14 1 if single mom at age 14 step14 1 if step parent at age 14 south66 1 if lived in south in 1966 south76 1 if lived in south in 1976 lwage76 log wage in 1976 (outliers trimmed) famed mom-dad education class (1-9) black 1 if black wage76 wage in 1976 (raw, cents per hour) enroll76 1 if enrolled in 1976 kww the kww score Eduardo Rossi c - Econometrics 10 51
52 Data set iqscore a normed IQ score mar76 marital status in 1976 (1 if married) libcrd14 1 if library card in home at age 14 exp76 experience in 1976 exp762 exp76 squared Eduardo Rossi c - Econometrics 10 52
53 Individual wage model OLS regression of an individual s log hourly wage upon years of schooling (ED76), experience (EXP76) and experience squared (EXP762) and three dummy variables indicating whether the individual was black, lived in a metropolitan area (SMSA76) an lived in the south (SOUTH76). LWAGE76 = EXP EXP762 (0.0676) (0.0066) (0.0003) BLACK SMSA76 (0.0176) (0.0156) SOUTH ED76 (0.0151) (0.0035) T = 3010 R 2 = F(6.3003) = ˆσ = (standard errors in parenthesis) β ED76 = implies estimated average returns to schooling of approximately 7.4% per year. Eduardo Rossi c - Econometrics 10 53
54 Individual wage model If schooling is endogenous then (EXP76) and experience squared (EXP762) are also endogenous. Three endogenous regressors, three instruments: Age (AGE76) Age squared (AGE76squared) Proximity to college (NEARC4) Reduced form: ÊD76 = AGE BLACK SMSA76 (4.2984) (0.3014) (0.1154) (0.1093) SOUTH AGE76squared NEARC4 (0.1024) (0.0052) (0.1070) T = 3010 R 2 = F(6, 3003) = 67, 294 ˆσ = Students who lived near a college in 1966 have on average 0.35 years more schooling. Eduardo Rossi c - Econometrics 10 54
55 Individual wage model LWAGE76 = BLACK SMSA76 (0.6085) (0.0774) (0.0497) SOUTH ED76 (0.0288) (0.0514) EXP EXP762 (0.0260) (0,0013) T = 3010 R 2 = F(6.3003) = 148, 06 ˆσ = (6) The estimated returns to schooling are over 13% with a relatively large standard error. The difference with OLS estimate can be due to sampling error. The downward bias of OLS could be due to measurement error or to the possibility that the true returns to schooling vary across individuals, negatively related to schooling. Eduardo Rossi c - Econometrics 10 55
56 Weak Instruments The properties of IV estimator can be very poor, and the estimator severely biased, if the instruments exhibit only weak correlation with the endogenous regressors. In this situation the normal distribution provides a very approximation to the true distribution of the IV estimator, even if N is large. Suppose the model is: ỹ t = β x t + ǫ t where ỹ t = y t y, x t = x t x, the IV estimator for β is: β IV = 1/N N i=1 z tỹ t 1/N N i=1 z t x t if the instrument is valid the estimator is consistent and converges to β = Cov(y t,z t ) Cov(x t,z t ). Eduardo Rossi c - Econometrics 10 56
57 Weak Instruments If the instrument is not correlated with the regressor, the denominator Cov(x t, z t ) = 0. In this case, the IV estimator is inconsistent and the asymptotic distribution of β IV deviates substantially from a normal distribution. The instrument is weak if there is some correlation between z t and x t, but not enough to make the asymptotic normal distribution a good approximation in finite samples. To figure out if there are Weak Instruments examine the reduced form regression and evaluate the explanatory power of the additional instruments that are not included in the equation of interest. Eduardo Rossi c - Econometrics 10 57
58 Weak Instruments Consider the LRM with one endogenous regressor where y t = x t1β 1 + β 2 x t2 + ε t E[x t1 ε t ] = 0 and where additional instruments z t2 (for x t2 ) satisfy E[z t2 ε t ] = 0 The appropriate reduced form is x t2 = x t1γ 1 + z t2γ 2 + v t if γ 2 = 0 the instruments z t2 are irrelevant and the IV estimator is inconsistent. The F test statistic for γ 2 = 0 is a measure for the information content contained in the instruments. Eduardo Rossi c - Econometrics 10 58
59 Weak Instruments Staiger and Stock (1997) provide a theoretical analysis of the properties of the IV estimator and provide some guidelines about how large the F-statistic should be for the IV estimator to have good properties: Don t worry if the F statistic exceeds 10! If the instruments are insignificant in the reduced form, don t put much confidence in the IV results. Eduardo Rossi c - Econometrics 10 59
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