Econometrics I Department of Economics Universidad Carlos III de Madrid Master in Industrial Economics and Markets
Outline Motivation 1 Motivation 2 3 4
Motivation
The Analogy Principle The () is a framework for deriving estimators estimators use assumptions about the moments of the variables to derive an objective function The assumed moments of the random variables provide population moment conditions We use the data to compute the analogous sample moment conditions estimates make the sample moment conditions as true as possible: This step is implemented by minimizing an objective function
The (MM) is a particular type of In the (MM), we have the same number of sample moment conditions as we have parameters In, we may have more sample moment conditions than we have parameters
Unconditional Mean We want to estimate µ = E [y] Population condition: E [y] µ = 0 The sample moment condition 1 N N i=1 y i =ˆµ = 0 The is obtained by solving the sample moment condition ˆµ = 1 N N y i i=1 This method dates back to Pearson (1895)
Ordinary Least Squares Ordinary least squares (OLS) is an MM estimator y i = β x i + u i, E [u i x i ] = 0 E [y i β x i x i ] = 0 E [x i (y i β x i )] = 0 E [x i (y i β x i )] = 0 are the population moment conditions The corresponding sample moment conditions: 1 N N i=1 This is the OLS estimator. [x i (y i ˆβ MM x i )] = 0
Instrumental Variables Instrumental Variables is an MM estimator y i = β x i + u i u i = y i β x i is such that E [y i β x i z i ] = 0, where x i,z i R k E [y i β x i z i ] = 0 E [z i (y i β x i )] = 0 E [z i (y i β x i )] = 0 are the population moment conditions The corresponding sample moment conditions: 1 N N i=1 This is the IV estimator. [z i (y i ˆβ MM x i )] = 0
vs. MM Motivation MM only works when the number of moment conditions equals the number of parameters to estimate If there are more moment conditions than parameters, the system of equations is algebraically over-identied and cannot be solved chooses the estimates that minimize a quadratic form of the moment conditions gets as close to solving the over-identied system as possible reduces to MM when the number of parameters equals the number of moment conditions
Denition Motivation Let θ be a k 1 vector of parameters such that E [m (w i,θ)] = 0 qx1 where m () is a q 1, q k, vector of known functions and w i is the data on person i For any c, the q sample moments are m N (c) = 1 N N i=1 m(w i,c) where A N is a qxq matrix ˆθ arg minm N (c) A N m N (c) c
and the Moments Conditions where A N is a qxq matrix ˆθ arg minm N (c) A N m N (c) c First Order Conditions: 2 m N A N m N ( ˆβ ) = 0 k where m N is a kxq matrix with the k rst derivatives of vector m N The estimator imposes in the sample k linear combinations of q moment conditions
Ordinary Least Squares Ordinary least squares (OLS) is a estimator y i = β x i + u i Dene the kx1 vector of moments as: m N (b) = 1 N N i=1 [x i (y i bx i )] ˆβ arg min b m N (b) A N m N (b), A N is kxk ( ) ( ) FOC: 2 m N A N m N ˆβ = 0 k m N ˆβ = 0 k This is the same system of equations and has the same solution as the OLS estimator.
Instrumental Variables IV is a estimator y i = β x i + u i, E [u i z i ] = 0 Dene the kx1 vector of moments as: m N (b) = 1 N N i=1 [z i (y i bx i )] ˆβ arg min b m N (b) A N m N (b), A N is kxk ( ) ( ) FOC: 2 m N A N m N ˆβ = 0 k m N ˆβ = 0 k This is the same system of equations and has the same solution as the IV estimator.
2SLS and Motivation 2SLS is a estimator for a particular A N where x i R k and z i R q, q > k y i = β x i + u i, E [u i z i ] = 0 Take A N = ( 1 N N i=1 (z i z i )) 1 The estimator dened by m N A N m N ( w i, ˆβ ) = 0 is the 2SLS estimator with instruments z i.
Some Properties of When k = q (just-identied case), m N (ˆθ MM ) = 0 ˆθ MM ˆθ Under general conditions, for a given weight matrix A N, is both consistent and asymptotically normal plim ˆθ = θ ) N (ˆθ d θ N (0,W ) where W can be consistently estimated
Consistency Motivation Under general conditions, if 1 plim A N = A positive denite 2 plim m N (c) = E [m (w,c)], and 3 AE [m (w,c)] = 0, then plim ˆθ = θ
Asymptotic Normality Under general conditions, if 1 ˆθ is consistent, 2 plim m N c = D (c), 3 D AD (D D (θ)) is non-singular, and 4 then NmN (θ) d N (0,V ) ) N (ˆθ d θ N (0,W ) where W = (D AD) 1 D AVAD(D AD) 1
The Weight Matrix A N A N only aects the eciency of the estimator Setting A N such that A = I yields consistent, but usually inecient estimates Setting A N such that A = k [AsyVar (m N (θ))] 1 for any k > 0 yields an ecient estimator Hence, in order to obtain an optimal estimator we need a consistent estimate of AsyVar (m N (θ)) This can be done in a two-step procedure
An Ecient Motivation We can take two steps to get an ecient estimator 1 Get ˆθ 1 arg min c m N (c) m N (c) to get [ ))] 1 A N = Var ˆ (m N (ˆθ 1 2 Get ˆθ 2 arg min c m N (c) A N m N (c)
2SLS and Eciency Under conditional homoskedasticity, i.e., Var(u i z i ) = σ 2 E (z i z i ), 2SLS is optimal If conditional homoskedasticity is violated, then 2SLS is not optimal and the standard formula for the asymptotic variance will not be consistent We can still use 2SLS as a consistent estimator To do the correct inference, we would need an estimate of the variance robust to heteroskedasticity Under heteroskedasticity, we can estimate a two-step optimal estimator: 1 Compute 2SLS and obtain A N = ( 1 N N i=1 û2 i (z i z i )) 1 2 Obtain the ( estimator such that m N A N m N w i, ˆβ ) = 0
The Sargan test Motivation If all q > k restrictions are right, [ ( ))] 1 Nm N (ˆθ) Var ˆ m N (ˆθ mn (ˆθ) d χ 2 q k where ˆθ is the most ecient estimator and ( )) a consistent estimator of Var m N (ˆθ. ( )) Var ˆ m N (ˆθ is the basic idea is that when all restrictions are ) right, then there will be q k linear combinations of m N (ˆθ that should be close to zero but are not exactly close to zero
Incremental Sargan test Suppose we have q > k restrictions. We can rst test use q 1 > k, with q 1 < q, restrictions and get ˆθ 1 and use the Sargan test ) [ ( ))] 1 ) S 1 Nm N (ˆθ 1 Var ˆ m N (ˆθ 1 mn (ˆθ 1 d χ 2 q 1 k We can also use all q > k, restrictions and get ˆθ and use the Sargan test (ˆθ) [ ( (ˆθ))] 1 ) S Nm N Var ˆ d m N mn (ˆθ χ 2 q k We can test the validity of the q q 1 additional restrictions S d S S 1 d χ 2 q q 1
Motivation OLS, IV, and 2SLS are shown to be particular cases of They will be ecient only under restrictive assumptions It is always possible to obtain a two-step consistent and asymptotically ecient estimator When the model is over-identied, we can test the over-identifying restrictions or a subset of them