2009 Granger Lecture Granger Center for Time Series The University of Nottingham Instrumental Variables Regression, GMM, and Weak Instruments in Time Series James H. Stock Department of Economics Harvard University June 11, 2009 1
NBER-NSF Time Series Conference, SMU, Sept. 2005 2
Outline 1) IV regression and GMM 2) Problems posed by weak instruments 3) Detection of weak instruments 4) Some solutions to weak instruments 5) Current research issues & literature 3
1) IV Regression and GMM Philip Wright (1928) needed to estimate the supply equation for butter: ) = β 0 + β 1 ln( ln( ) + u t butter butter Q P t t Or: y t = Y t β + u t , (generic notation) β 1 = price elasticity of supply of butter OLS is inconsistent because ln( ) is butter P t endogenous: price and quantity are determined simultaneously by supply and demand… Philip Wright (1861-1934), MA Harvard, Econ, 1887 Lecturer, Harvard, 1913-1917 4
Figure 4, p. 296, Wright (1926), Appendix B 5
Derivation of the IV estimator in P. Wright (1928, p. 314) Now multiply each term in this equation by A (the corresponding deviation in the price of a substitute) and we shall have: eA × P = A × O – A × S 1 . Suppose this multiplication to be performed for every pair of price-output deviations and the results added, then: ∑ ∑ × − × A O A S ∑ ∑ ∑ × × − × 1 = or e = . e A P A O A S ∑ 1 × A P But A was a factor which did not affect supply conditions; hence it is uncorrelated ∑ × A O ∑ × with S 1 ; hence = 0; and hence e = . A S ∑ 1 × A P In modern notation : for an exogenous instrument, i.e. Z t s.t. EZ t u t = 0, Z ′ ( y – Y β ) = Z ′ y – Z ′ Y β = z ′ u , but E z ′ u = 0, so E Z ′ y – E Z ′ Y β = 0, ′ Z y ˆ TSLS β which suggests: = ′ Z Y 6
Classical IV regression model & notation y t = Y t β + u t , m = dim( Y t ) Equation of interest: k exogenous instruments Z t : E ( u t Z t ) = 0, k = dim( Z t ) Y t = Π′ Z t + v t , corr( u t , v t ) = ρ (vector) Auxiliary equations: Sampling assumption: ( y t , Y t , Z t ) are i.i.d. ( for now ) y = Y β + u (“second stage”) Equations in matrix form: Y = Z Π + v (“first stage”) Comments: • We assume throughout the instrument is exogenous ( E ( u t Z t ) = 0) • Included exogenous regressors have been omitted without loss of generality • Auxiliary equation is just the projection of Y on Z 7
Generalized Method of Moments GMM notation and estimator: h ( Y t ; θ ); θ 0 = true value GMM “error” term ( G equations): h ( Y t ; θ ) = y t – θ ′ Y t Note : In the linear model, × × 1 1 G k φ t ( θ ) = θ ⊗ Errors times k instruments: ( , ) h Y Z 0 t t × × 1 1 G k E φ t ( θ ) = E [ Moment conditions - k instruments: θ ⊗ ] = 0 ( , ) h Y Z 0 t t ′ ⎡ ⎤ ⎡ ⎤ T T ∑ ∑ S T ( θ ) = − − φ θ φ θ GMM objective function: 1/2 1/2 ( ) ( ) T W T ⎢ ⎥ ⎢ ⎥ t T t ⎣ ⎦ ⎣ ⎦ = = 1 1 t t ˆ θ minimizes S T ( θ ) GMM estimator: W T = Ω –1 , Ω = 2 π Efficient (infeasible) GMM: ( ) (0) S φ θ t ˆ ( θ ) –1 (each θ ) CUE (Hansen, Heaton, Yaron 1996): W T = Ω 8
Weak instruments: four examples Example #1 (time-series IV): Estimating the elasticity of intertemporal substitution, linearized Euler equation e.g. Campbell (2003), Handbook of Economics of Finance Δ c t +1 = consumption growth, t to t +1 r i , t +1 = return on i th asset, t to t +1 Log-linearized Euler equation moment condition: E t ( Δ c t +1 – τ i – ψ r i , t +1 ) = 0 where ψ = elasticity of intertemporal substitution (EIS) 1/ ψ = coeff. of relative risk aversion under power utility Resulting IV estimating equation: E [( Δ c t +1 – τ i – ψ r i , t +1 ) Z t ] = 0 (or use Z t –1 because of temporal aggregation) 9
EIS estimating equations: Δ c t +1 = τ i + ψ r i , t +1 + u i , t +1 (a) r i , t +1 = μ i + (1/ ψ ) Δ c t +1 + η i , t +1 or (b) Under homoskedasticity, standard estimation is by the TSLS estimator in (a) or by the inverse of the TSLS estimator in (b). Findings in literature (e.g. Campbell (2003), US data): • regression (a): 95% TSLS CI for ψ is (-.14, .28) • regression (b): 95% TSLS CI for 1/ ψ is (-.73, 2.14) What is going on? 10
Example #2 (cross-section IV) : Angrist-Kreuger (1991), What are the returns to education? Example #3 (linear GMM): Hybrid New Keynesian Phillips Curve e.g. Gali and Gertler (1999), where x t = labor share; see survey by Kleibergen and Mavroeidis (2008). Example #4 (nonlinear GMM): Estimating the elasticity of intertemporal substitution, nonlinear Euler equation Hansen, Heaton, Yaron (1996), Stock & Wright (2000), Neely, Roy, & Whiteman (2001) 11
Working definition of weak identification θ is weakly identified if the distributions of GMM or IV estimators and test statistics are not well approximated by their standard asymptotic normal or chi-squared limits because of limited information in the data. • Departures from standard asymptotics are what matters in practice • The source of the failures is limited information, not (for example) heavy tailed distributions, near-unit roots, unmodeled breaks, etc. • The focus is on large T . • Throughout, we assume instrument exogeneity 12
Why do weak instruments cause problems? IV regression with one Y and a single irrelevant instrument ′ ′ ′ β ( ) Z y Z Y + u Z u ˆ TSLS = β + β = = ′ ′ ′ Z Y Z Y Z Y If Z is irrelevant (as in Bound et. al. (1995)), then Y = Z Π + v = v , so 1 T ∑ Z u ′ ⎛ ⎞ ⎡ σ σ ⎤ ⎛ ⎞ 2 t t Z u z T d z ˆ TSLS – β = β = → = 1 z , where ⎟ ~ u 0, t ⎜ u uv ⎟ u ⎢ ⎥ ⎜ N ′ 1 σ σ 2 T ⎝ ⎠ ∑ z ⎣ ⎦ Z v ⎝ ⎠ v v uv v Z v t t T = 1 t Comments : ˆ • β isn’t consistent (nor should it be!) TSLS • Distribution of ˆ TSLS β is Cauchy-like (ratio of correlated normals) (Choi & Phillips (1992)) 13
• The distribution of ˆ TSLS β is a mixture of normals with nonzero mean : write z u = δ z v + η , η ⊥ z , where δ = σ uv / σ . Then 2 v σ ) 2 δ + η η , and η | z v ~ N (0, z z = δ + η z = u v 2 z z z z v v v v v ˆ TSLS – β 0 is the mixture of normals, β so the asymptotic distribution of σ 2 d ˆ TSLS ∫ – ( β 0 + δ ) η β → (1 irrelevant instrument) (0, ) ( ) N f z dz 2 z v v z v v • heavy tails (mixture is based on inverse chi-squared) ˆ • center of distribution of is β 0 + δ . But β TSLS ′ ′ / / σ Z u v u T T p ˆ OLS σ = δ , so plim ( ˆ OLS – β 0 = ) = β 0 + δ , β = → β uv ′ ′ 2 / / Z Z T v v T v so σ 2 d ˆ TSLS – plim ( ˆ OLS ∫ η β β → ) (1 irrelevant instrument) (0, ) ( ) N f z dz 2 z v v z v v 14
The unidentified and strong-instrument distributions are two ends of a spectrum . Distribution of the TSLS t -statistic (Nelson-Startz (1990a,b)): Dark line = irrelevant instruments; dashed light line = strong instruments; intermediate cases: weak instruments. The key parameter is: μ 2 = Π ′ Z ′ Z Π / σ ( concentration parameter ) 2 v = k × (numerator) noncentrality parameter of first-stage F statistic 15
Weak instrument asymptotics bridges this spectrum. Adopt nesting that makes the concentration parameter tend to a constant as the sample size increases by setting Π = C / T (weak instrument asymptotics) • This is the Pitman drift for obtaining the local power function of the first-stage F . • This nesting holds E μ 2 constant as T → ∞ . d • Under this nesting, F → noncentral χ / k with noncentrality 2 k parameter E μ 2 / k (so F = O p (1)) • Letting the parameter depend on the sample size is a common ways to obtain good approximations – e.g. local to unit roots (Bobkoski 1983, Cavanagh 1985, Chan and Wei 1987, and Phillips 1987) 16
Weak IV asymptotics for TSLS estimator, 1 included end ogenous vble: ˆ TSLS – β 0 = ( Y ′ P Z u )/( Y ′ P Z Y ) β Now − ′ ′ 1 ′ Π + Π + ⎛ ⎞⎛ ⎛ ⎞ ( ) ⎞ ( ) Z v Z Z Z Z Z v Y ′ P Y = ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ Z ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ T T T ′ − − 1/2 1/2 ′ ′ ′ ′ ′ ′ Π Π ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ Z v Z Z v Z Z Z Z Z Z Z + + = ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ T T T T T T ⎡ ⎤ ⎡ ⎤ ′ ′ − − ′ 1/2 ′ ′ 1/2 ′ 1/2 ′ 1/2 ′ ⎥ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ Z Z v Z Z Z Z Z Z Z Z v ′ ⎢ ⎥ ⎢ + + = ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ C C ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎢ ⎥ ⎢ ⎥ T T T T T T ⎣ ⎦ ⎣ ⎦ d → ( λ + z ′ ( λ + z ), v ) v where ⎛ ⎞ ⎡ ⎤ σ σ ⎛ ⎞ 2 z λ = , Q ZZ = EZ t Z t ′ , and ′ 1/2 ⎟ ~ ⎥ u 0, ⎜ u uv ⎟ ⎢ C Q ⎜ N σ σ ZZ 2 ⎝ ⎠ z ⎣ ⎦ ⎝ ⎠ v uv v 17
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