Generalized Method of Moment

Size: px
Start display at page:

Download "Generalized Method of Moment"

Transcription

1 Generalized Method of Moment CHUNG-MING KUAN Department of Finance & CRETA National Taiwan University June 16, 2010 C.-M. Kuan (Finance & CRETA, NTU Generalized Method of Moment June 16, / 32

2 Lecture Outline 1 Generalized Method of Moments GMM Estimation Asymptotic Properties of the GMM Estimator Different GMM Estimators Examples 2 Large Sample Tests Over-Identifying Restriction Test Hausman Test Wald Test 3 Conditional Moment Restrictions Estimation of Conditional Moment Restrictions C.-M. Kuan (Finance & CRETA, NTU Generalized Method of Moment June 16, / 32

3 Moment Conditions Given y t = x t β + e t, consider the moment function: IE(x t e t = IE[x t (y t x t β]. When x tβ is correctly specified for the linear projection, we have the moment condition: IE[x t (y t x t β o ] = 0. Its sample counterpart, 1 T T x t (y t x tβ = 0, is also the FOC for OLS estimation. C.-M. Kuan (Finance & CRETA, NTU Generalized Method of Moment June 16, / 32

4 Similarly, given y t = f (x t ; β + e t and the moment function: IE[ f (x t ; βe t ] = IE{ f (x t ; β[y t f (x t ; β]}. When f (x t ; β is correctly specified for IE(y t x t, we have the moment condition IE{x t [y t f (x t ; β o ]} = 0. Its sample counterpart is the FOC for NLS estimation: 1 T T f (x t ; β[y t f (x t ; β] = 0. When the quasi-likelihood function f (x t ; θ is correctly specified, the moment condition IE[ ln f (x t ; θ o ] = 0 holds. The sample counterpart of this moment condition is the average of the score functions and yields the QMLE θ. C.-M. Kuan (Finance & CRETA, NTU Generalized Method of Moment June 16, / 32

5 Given y t = x t β + e t and the moment function: IE(z t e t = IE[z t (y t x t β]. When the variables z t are proper instrument variables such that IE[z t (y t x t β o ] = 0. The sample counterpart of this moment condition is the FOC for IV estimation: 1 T T z t (y t x tβ = 0. Note that we can not solve for unknown parameters if the number of moment conditions is more than the number of parameters. C.-M. Kuan (Finance & CRETA, NTU Generalized Method of Moment June 16, / 32

6 GMM Estimation Consider q moment functions IE[m(z t ; θ], where θ (k 1 is the parameter vector. Suppose there exists unique θ o such that IE[m(z t ; θ o ] = 0. These conditions are exactly identified if q = k and over-identified if q > k. When the conditions are exactly identified, θ o can be estimated by solving their sample counterpart: m T (θ = 1 T T m(z t ; θ = 0. This is the method of moment, which is not applicable when the conditions are over-identified. C.-M. Kuan (Finance & CRETA, NTU Generalized Method of Moment June 16, / 32

7 Given a q q symmetric and p.d. weighting matrix W o, note that the following quadratic objective function, Q(θ; W o := IE[m(z t ; θ] W o IE[m(z t ; θ], is minimized at θ = θ o. The generalized method of moments (GMM of Hansen (1982, Econometrica suggests estimating θ o by minimizing the sample counterpart of Q(θ; W o : Q T (θ; W T = [ m T (θ ] WT [ mt (θ ], where W T is a symmetric and p.d. weighting matrix, possibly dependent on the sample, and W T converges to W o in probability. The GMM estimator is thus ˆθ T (W T = arg min θ Θ Q T (θ; W T, which clearly depends on the choice of W T. C.-M. Kuan (Finance & CRETA, NTU Generalized Method of Moment June 16, / 32

8 The FOC of GMM estimation contains k equations in k unknowns: G T (θ W T m T (θ = 0, where G T (θ = T 1 T m(z t ; θ. The GMM estimator is then solved using a nonlinear optimization algorithm. For example, in the linear regression case, m T (β = 1 T T x t (y t x t β. When W T = I k, the FOC is ( ( 1 T x T t x 1 T t x T t (y t x t β = 0. Clearly, the resulting GMM estimator is the OLS estimator. C.-M. Kuan (Finance & CRETA, NTU Generalized Method of Moment June 16, / 32

9 Consistency By invoking a suitable ULLN, Q T (θ; W T is close to Q(θ; W o uniformly in θ when T becomes large. Hence, the GMM estimator ˆθ T (W T ought to be close to θ o, the minimizer of Q(θ; W o, for sufficiently large T. This is the underlying idea of establishing GMM consistency. This approach is analogous to that for NLS and QMLE consistency. Note that consistency does not depend on the weighting matrix. For example, ˆθ T (I q is consistent for θ o, which is also the minimizer of Q(θ; I q. Consider the mean value expansion: T mt (ˆθT (W T = T m T (θ o + G T (θ T T (ˆθT (W T θ o, where θ T lies between ˆθ T (W T and θ o. C.-M. Kuan (Finance & CRETA, NTU Generalized Method of Moment June 16, / 32

10 Asymptotic Distribution Using the FOC of GMM estimation, 0 = G T (ˆθ T (W T WT T mt (ˆθ T (W T = G T (ˆθT (W T WT T mt (θ o + G T (ˆθT (W T WT G T (θ T T (ˆθT (W T θ o. When m(z t ; θ obey a ULLN, so that G T (ˆθT (W T converges to G o = IE[ m(z t ; θ o ]. Then, T (ˆθT (W T θ o = { IE[ m(zt ; θ o ] W o IE[ m(z t ; θ o ] } 1 IE[ m(z t ; θ o ] W o [ T m T (θ o ] + o IP (1 = ( G o W o G 1G o o W o T mt (θ o + o IP (1. C.-M. Kuan (Finance & CRETA, NTU Generalized Method of Moment June 16, / 32

11 The asymptotic distribution of T (ˆθT (W T θ o is thus determined by T mt (θ o = 1 T T m(z t ; θ o. When m(z t ; θ o obey a central limit theorem: 1 T T m(z t ; θ o we have T (ˆθT (W T θ o D N (0, Σ o, D N (0, Ωo, where Ω o = ( G o W o G 1G o o W o Σ o W o G ( o G 1. o W o G o When W o = Σ 1 o, it is easy to see that Ω o simplifies to ( G o Σ 1 o G 1G o o Σ 1 o G o ( G o Σ 1 o G o 1 = ( G o Σ 1 o G o 1 =: Ω o. C.-M. Kuan (Finance & CRETA, NTU Generalized Method of Moment June 16, / 32

12 Asymptotic Efficiency To compare Ω o and Ω o, note that (Ω o 1 Ω 1 o = G o Σ 1 o G o G o W o G o ( G o W o Σ o W o G 1G o o W o G o = G o Σ 1/2 o [ ( I Σ 1/2 o W o G o G o W o Σ1/2 o Σ 1/2 1G o W o G o o W ] o Σ1/2 o Σ 1/2 o G o, which is p.s.d. because the matrix in the square bracket is symmetric and idempotent. Thus, Ω o Ω o is p.s.d. This shows that, given the weighting matrix whose limit is Σ 1 o, the resulting GMM estimator is asymptotically efficient. Σ 1 o is thus known as the optimal (limiting weighting matrix. C.-M. Kuan (Finance & CRETA, NTU Generalized Method of Moment June 16, / 32

13 Two-Step Optimal GMM Estimator Hansen and Singleton (1982, Econometrica: 1 Compute a preliminary, consistent estimator ˆθ 1,T based on the pre-specified weighting matrix W 0,T : [ ˆθ 1,T := arg min mt (θ ] [ W0,T mt (θ ]. θ Θ For example, we may set W 0,T = I q. 2 Compute a consistent estimator for Σ o based on ˆθ 1,T and use its inverse as 1 the optimal weighting matrix, i.e., W T (ˆθ 1,T = Σ T. 3 The two-step optimal GMM estimator is obtained as [ ˆθ 2,T := arg min mt (θ ] [ Σ 1 T mt (θ ], θ Θ which is asymptotically efficient. C.-M. Kuan (Finance & CRETA, NTU Generalized Method of Moment June 16, / 32

14 Drawbacks of Two-Step Estimators The finite-sample performance of the two-step estimator depends on the preliminary GMM estimator ˆθ 1,T and hence the initial weighting matrix W 0,T. Note that Σ T is also determined by m(z t ; θ. Hence, the correlation between m T and Σ T results in finite-sample bias in ˆθ 2,T. Computing a consistent estimator of Σ o may not be straightforward, especially when the data are serially correlated and heterogeneously distributed. C.-M. Kuan (Finance & CRETA, NTU Generalized Method of Moment June 16, / 32

15 Iterative GMM Estimator 1 At the j th iteration, compute the j th iterative GMM estimator using the weighting matrix W T (ˆθ j 1,T : [ ˆθ j,t := arg min mt (θ ] WT (ˆθ j 1,T [ m T (θ ]. θ Θ At the first iteration, we can set the initial weighting matrix W 0,T, as in the two-step estimator. 2 Use ˆθ j,t to construct the optimal weighting matrix W T (ˆθ j,t and set j = j Iterate the procedure above till one of the following convergence criteria is satisfied: For some pre-specified ε, Q T (ˆθ j,t Q T (ˆθ j 1,T, ε or ˆθ j,t ˆθ j 1,T ε. C.-M. Kuan (Finance & CRETA, NTU Generalized Method of Moment June 16, / 32

16 Continuous Updating GMM Estimator Hansen, Heaton and Yaron (1996, JBES: Continuous updating (CU estimator is based on one-time optimization: [ ˆθ T := arg min mt (θ ] WT (θ [ m T (θ ]. θ Θ The objective function clearly changes when W T depends explicitly on θ. This does not affect the limiting distribution of the resulting estimator, however; see Pakes and Pollard (1989, Econometrica. Remark: The CU estimator is invariant when the moment conditions are re-scaled, even when the scale factor is parameter dependent; the two-stage or iterative GMM estimator is sensitive to such transformation, however. C.-M. Kuan (Finance & CRETA, NTU Generalized Method of Moment June 16, / 32

17 Independently Weighted Optimal Estimator Altonji and Segal (1996, JBES: The independently weighted optimal estimator avoids possible correlation between m T and Σ T by splitting the sample and computing m T and Σ T with different sub-samples. Split the sample into l groups, and let m Tj (θ be the sample average of m(z t, θ for t in the j th group with T j observations. 1 Also let Σ T be the optimal weighting matrix based on the observations not j in the j th group. The resulting GMM estimator is obtained by solving: ˆθ T := arg min θ Θ l j=1 A common choice of l is 2. [ mtj (θ ] Σ 1 T j [ mtj (θ ]. C.-M. Kuan (Finance & CRETA, NTU Generalized Method of Moment June 16, / 32

18 Example: Regression with Symmetric Error Given the specification y t = x t β + e t, let [ ] x m(y t, x t ; β = t (y t x t β x t (y t x. t β3 The moment condition IE[m(y t, x t ; β o ] = 0 suggests estimating β o while taking into account symmetry of the error term. The gradient vector of m is: [ ] x m(y t, x t ; β = t x t 3x t x t (y t. x t β2 If the data are independent over t, [ ] ɛ 2 t Σ o = IE x t x t ɛ 4 t x t x t ɛ 4 t x t x t ɛ 6 t x. t x t C.-M. Kuan (Finance & CRETA, NTU Generalized Method of Moment June 16, / 32

19 Let ê t = y t x ˆβ t 1,T, where ˆβ 1,t is a first-step GMM estimator based on a preliminary weighting matrix. Then, Σ o may be estimated by the sample counterpart: [ ] Σ T (ˆβ 1,T = 1 T êt 2x t x t êt 4x t x t T êt 4x t x t êt 6x. t x t Note that ˆβ 1,T here may be the OLS estimator (in fact, we only need a consistent estimator for β o. The two-step GMM estimator is computed with [ ΣT (ˆβ 1,T ] 1 the weighting matrix; that is, ˆθ 2,T := arg min θ Θ m T (θ[ ΣT (ˆβ 1,T ] 1 mt (θ. C.-M. Kuan (Finance & CRETA, NTU Generalized Method of Moment June 16, / 32

20 Example: Generalized Instrumental Variables Estimator For the specification y t = x t β + e t, consider the moment condition: IE[m t (β o ] = IE[z t (y t x t β o ] = 0, where z t contains q > k instrumental variables. The GMM estimator minimizes ( ( 1 T z T t (y t x t β 1 W T T T z t (y t x t β, and solves ( T ( T x t z t W T z t (y t x t β = (X ZW T [Z (y Xβ] = 0. where Z (T q is the matrix of instrumental variables and X (T k is the matrix of regressors. C.-M. Kuan (Finance & CRETA, NTU Generalized Method of Moment June 16, / 32

21 The GMM estimator here is also the generalized instrumental variables estimator: ˆβ(W T = (X ZW T Z X 1 X ZW T Z y. When the data are independent over t and there is no condition heteroskedasticity, ( 1 T T Σ o = var z t (y t x t β o = σ2 o IE(z T T t z t. Ignoring σ 2 o in Σ o, T 1 T IE(z t z t can be estimated by Z Z/T. The two-step GMM estimator now is: ˆβ 2,T = [X Z(Z Z 1 Z X] 1 X Z(Z Z 1 Z y, which is also known as the two-stage least squares (2SLS estimator. C.-M. Kuan (Finance & CRETA, NTU Generalized Method of Moment June 16, / 32

22 The 2SLS estimator can be expressed as ˆθ 2,T = [ X X] 1 X y, where X = Z(Z Z 1 Z X is the matrix of fitted values from the OLS regression of X on Z. Hence, ˆθ 2,T is also the OLS estimator of regressing y on X (Theil, When the errors are not conditionally homoskedastic, the 2SLS estimator remains consistent (why?, but it is no longer the most efficient. A two-step GMM estimator with a properly estimated weighting matrix [ Σ T (ˆβ 1,T ] 1 is more efficient. For example, Σ T (ˆβ 1,T = 1 T T êt 2 z t z t, where ê t = y t x t ˆβ 1,T are the residuals from the first-step GMM estimation. C.-M. Kuan (Finance & CRETA, NTU Generalized Method of Moment June 16, / 32

23 Over-Identifying Restrictions Test To test whether the model for IE[m(z t, θ 0 ] = 0 is correctly specified, it is natural to check if m T (ˆθ T is sufficiently close to zero. For example, when the moment conditions are the Euler equations in different capital asset pricing models, this amounts to checking if the pricing errors are zero. The over-identifying restrictions (OIR test of Hansen (1982, also known as the J test, is based on the value of the GMM objective function: J T := T [ m T (ˆθT (W T ] WT [ mt (ˆθT (W T ]. As the test statistic involves the GMM estimator obtained from the same objective function, the limiting distribution is χ 2 (q k. C.-M. Kuan (Finance & CRETA, NTU Generalized Method of Moment June 16, / 32

24 To derive its limiting distribution, note that W 1/2 T T mt (ˆθT (W T = W 1/2 T T mt (θ o + W 1/2 T G T (θ T T (ˆθT (W T θ o. As T (ˆθ ( T (W T θ o = G o W o G 1G o o W o T mt (θ o + o IP (1, T mt (ˆθ T (W T = P o W 1/2 W 1/2 T o T mt (θ o + o IP (1 where P o = I W 1/2 ( o G o G o W o G 1G o o W1/2 o which is symmetric and idempotent with rank q k (why?, and W 1/2 D o T mt (θ o N (0, W 1/2 o Σ o W 1/2 o. C.-M. Kuan (Finance & CRETA, NTU Generalized Method of Moment June 16, / 32

25 When W o = Σ 1, the limiting distribution simplifies: o W 1/2 D o T mt (θ o N (0, I q. Consequently, J T = T m T (θ o W 1/2 o P o P o W 1/2 D o m T (θ o χ 2 (q k. Remarks: The weighting matrix in the J statistic and the weighting matrix for the GMM estimator in m T must be the same. And the weighting matrix W T for the GMM estimator must converge to W o = Σ 1 o. That is, J test requires the optimal GMM estimator. Lee and Kuan (2010 propose an OIR test that does not requires the weighting matrix to converge to Σ o and hence avoids the optimal GMM estimation. C.-M. Kuan (Finance & CRETA, NTU Generalized Method of Moment June 16, / 32

26 Hausman Test Given two estimators ˆθ T and ˇθ T of the parameter θ o, suppose that both are consistent under the null hypothesis but only one, say ˇθ T, is consistent under the alternative. The Hausman test suggests testing the null hypothesis by comparing these two estimators. The Hausman test is particularly useful for testing model specification, which may not be expressed as parameter restrictions. For example, consider testing the null hypothesis that regressors are exogenous against the alternative of endogenous regressors. Under classical conditions, the OLS estimator ˆθ T and the 2SLS estimator ˇθ T are consistent under the null, but only the 2SLS estimator is consistent under the alternative. Thus, we can test for endogeneity by checking if ˆθ T ˇθ T is sufficiently large. C.-M. Kuan (Finance & CRETA, NTU Generalized Method of Moment June 16, / 32

27 The Hausman test reads: H T = T (ˆθT ˇθ V 1 T T (ˆθT ˇθ D T χ 2 (k, where V T is a consistent estimator for the asymptotic covariance matrix of T (ˆθT ˇθ T : V (ˆθT ˇθ T = V (ˆθT + V (ˇθT 2 cov (ˆθT, ˇθ T. This asymptotic covariance matrix is simplified when ˆθ T is also asymptotically efficient under the null. In particular, we can show V 1,2 := cov (ˆθT, ˇθ T = V(ˆθT, so that V (ˆθT ˇθ T depends only on the respective asymptotic covariance matrices of these two estimators: V (ˆθT ˇθ T = V (ˇθT V (ˆθT. C.-M. Kuan (Finance & CRETA, NTU Generalized Method of Moment June 16, / 32

28 To see this, consider the following estimator which is a linear combination of ˆθ T and ˇθ T : ˆθ T = ˆθ T + [ V (ˆθT V1,2 ] V (ˆθT ˇθ T 1 (ˇθT ˆθ T. It is easy to verify that V (ˆθ T = V (ˆθT [ V (ˆθT V1,2 ] V (ˆθT ˇθ T 1 [ V (ˆθT V1,2 ]. If V (ˆθT is not the same as V1,2, the second term on the RHS is p.d. It follows that ˆθ T is asymptotically more efficient than ˆθ T. But this contradicts the assumption that ˆθ T is asymptotically more efficient. This suggests that V (ˆθ T = V1,2. C.-M. Kuan (Finance & CRETA, NTU Generalized Method of Moment June 16, / 32

29 Wald Test Hypothesis: R(θ = 0, where R : R K R r. A mean-value expansion yields R(ˆθ T = R(θ 0 + R(θ (ˆθ T θ 0, where T (ˆθ T θ 0 T [ R(ˆθT R(θ 0 ] D N (0, Ω 0. Therefore, It follows that, under the null hypothesis, D N ( 0, [ R(θo ]Ω 0 [ R(θ o ]. W T := T R (ˆθ T ( [ R (ˆθ T ] ΩT [ R (ˆθ T ] 1 R (ˆθ T D χ 2 (r. where Ω T is a consistent estimator of Ω o. For the linear hypothesis Rθ 0 = r with R a r k matrix, W T = T (Rˆθ T r (R Ω T R 1 (Rˆθ T r D χ 2 (r. C.-M. Kuan (Finance & CRETA, NTU Generalized Method of Moment June 16, / 32

30 Estimation of Conditional Moment Restrictions Consider the conditional moment restrictions: IE[h(η t ; θ o F t ] = 0, where h is r 1 and F t is the information set up to time t. Let w t be a set of variables from F t and may contain some variables in η t. The conditional moment restrictions above imply IE[D(w t h(η t ; θ o ] = 0, where D(w t is a (r n matrix of instruments. GMM estimation of θ o is based on the sample moments: 1 T T m t (θ = 1 T T D(w t h(η t ; θ. C.-M. Kuan (Finance & CRETA, NTU Generalized Method of Moment June 16, / 32

31 The optimal GMM estimator has asymptotic covariance matrix ( G 1, o Σ 1 o G o with Σ o = IE [ D(w t h(η t ; θ o h(η t ; θ o D(w t ], (n n G o = IE [ D(w t h(η t ; θ o ]. (n k For a smaller set of sample moments obtained by taking a linear transformation: 1 T T C D(w t h(η t ; θ, where C is n p with p < n, the asymptotic covariance matrix of the resulting optimal GMM estimator is [ G o C(C Σ o C 1 C G o ] 1. It is readily seen that G o Σ 1 o which is p.s.d. matrix. G o G o C(C Σ o C 1 C G o = G [ o Σ 1/2 o In Σ 1/2 o C(C Σ 1/2 o Σ 1/2 o C 1 C Σ 1/2 o ] Σ 1/2 o G o, C.-M. Kuan (Finance & CRETA, NTU Generalized Method of Moment June 16, / 32

32 Number of moment conditions: The optimal GMM estimator based on a larger set of moment conditions is asymptotically more efficient, yet it may have a larger bias in finite samples. Optimal instruments: D (w t ; θ = [ var(h(η t ; θ F t ] 1 IE [ h(ηt ; θ F t] =: V 1 t J t, (r k which contains k instruments. In this case, Σ o = IE ( J t V 1 t V t V 1 ( t J t = IE J t V 1 t J t, G o = IE [ J t V 1 t h(η t ; θ o ] = IE ( J t V 1 t J t. The optimal GMM estimator thus has the asymptotic covariance matrix ( G o Σ 1 o G o 1 ( = IE J 1. t V 1 t J t As J t and V t are conditional expectations and depend on unknown parameters, it is not easy to estimate the optimal instruments in practice. C.-M. Kuan (Finance & CRETA, NTU Generalized Method of Moment June 16, / 32

LECTURE ON HAC COVARIANCE MATRIX ESTIMATION AND THE KVB APPROACH

LECTURE ON HAC COVARIANCE MATRIX ESTIMATION AND THE KVB APPROACH LECURE ON HAC COVARIANCE MARIX ESIMAION AND HE KVB APPROACH CHUNG-MING KUAN Institute of Economics Academia Sinica October 20, 2006 ckuan@econ.sinica.edu.tw www.sinica.edu.tw/ ckuan Outline C.-M. Kuan,

More information

Econometrics II - EXAM Outline Solutions All questions have 25pts Answer each question in separate sheets

Econometrics II - EXAM Outline Solutions All questions have 25pts Answer each question in separate sheets Econometrics II - EXAM Outline Solutions All questions hae 5pts Answer each question in separate sheets. Consider the two linear simultaneous equations G with two exogeneous ariables K, y γ + y γ + x δ

More information

Econometrics II - EXAM Answer each question in separate sheets in three hours

Econometrics II - EXAM Answer each question in separate sheets in three hours Econometrics II - EXAM Answer each question in separate sheets in three hours. Let u and u be jointly Gaussian and independent of z in all the equations. a Investigate the identification of the following

More information

Introduction to Estimation Methods for Time Series models Lecture 2

Introduction to Estimation Methods for Time Series models Lecture 2 Introduction to Estimation Methods for Time Series models Lecture 2 Fulvio Corsi SNS Pisa Fulvio Corsi Introduction to Estimation () Methods for Time Series models Lecture 2 SNS Pisa 1 / 21 Estimators:

More information

Introduction to Quantile Regression

Introduction to Quantile Regression Introduction to Quantile Regression CHUNG-MING KUAN Department of Finance National Taiwan University May 31, 2010 C.-M. Kuan (National Taiwan U.) Intro. to Quantile Regression May 31, 2010 1 / 36 Lecture

More information

Quick Review on Linear Multiple Regression

Quick Review on Linear Multiple Regression Quick Review on Linear Multiple Regression Mei-Yuan Chen Department of Finance National Chung Hsing University March 6, 2007 Introduction for Conditional Mean Modeling Suppose random variables Y, X 1,

More information

Econometrics of Panel Data

Econometrics of Panel Data Econometrics of Panel Data Jakub Mućk Meeting # 6 Jakub Mućk Econometrics of Panel Data Meeting # 6 1 / 36 Outline 1 The First-Difference (FD) estimator 2 Dynamic panel data models 3 The Anderson and Hsiao

More information

Single Equation Linear GMM

Single Equation Linear GMM Single Equation Linear GMM Eric Zivot Winter 2013 Single Equation Linear GMM Consider the linear regression model Engodeneity = z 0 δ 0 + =1 z = 1 vector of explanatory variables δ 0 = 1 vector of unknown

More information

Generalized Method of Moments (GMM) Estimation

Generalized Method of Moments (GMM) Estimation Econometrics 2 Fall 2004 Generalized Method of Moments (GMM) Estimation Heino Bohn Nielsen of29 Outline of the Lecture () Introduction. (2) Moment conditions and methods of moments (MM) estimation. Ordinary

More information

Economics 582 Random Effects Estimation

Economics 582 Random Effects Estimation Economics 582 Random Effects Estimation Eric Zivot May 29, 2013 Random Effects Model Hence, the model can be re-written as = x 0 β + + [x ] = 0 (no endogeneity) [ x ] = = + x 0 β + + [x ] = 0 [ x ] = 0

More information

Instrumental Variables

Instrumental Variables Università di Pavia 2010 Instrumental Variables Eduardo Rossi Exogeneity Exogeneity Assumption: the explanatory variables which form the columns of X are exogenous. It implies that any randomness in the

More information

The outline for Unit 3

The outline for Unit 3 The outline for Unit 3 Unit 1. Introduction: The regression model. Unit 2. Estimation principles. Unit 3: Hypothesis testing principles. 3.1 Wald test. 3.2 Lagrange Multiplier. 3.3 Likelihood Ratio Test.

More information

Advanced Econometrics

Advanced Econometrics Advanced Econometrics Dr. Andrea Beccarini Center for Quantitative Economics Winter 2013/2014 Andrea Beccarini (CQE) Econometrics Winter 2013/2014 1 / 156 General information Aims and prerequisites Objective:

More information

Econ 583 Final Exam Fall 2008

Econ 583 Final Exam Fall 2008 Econ 583 Final Exam Fall 2008 Eric Zivot December 11, 2008 Exam is due at 9:00 am in my office on Friday, December 12. 1 Maximum Likelihood Estimation and Asymptotic Theory Let X 1,...,X n be iid random

More information

Spring 2017 Econ 574 Roger Koenker. Lecture 14 GEE-GMM

Spring 2017 Econ 574 Roger Koenker. Lecture 14 GEE-GMM University of Illinois Department of Economics Spring 2017 Econ 574 Roger Koenker Lecture 14 GEE-GMM Throughout the course we have emphasized methods of estimation and inference based on the principle

More information

A Course on Advanced Econometrics

A Course on Advanced Econometrics A Course on Advanced Econometrics Yongmiao Hong The Ernest S. Liu Professor of Economics & International Studies Cornell University Course Introduction: Modern economies are full of uncertainties and risk.

More information

Finite Sample Performance of A Minimum Distance Estimator Under Weak Instruments

Finite Sample Performance of A Minimum Distance Estimator Under Weak Instruments Finite Sample Performance of A Minimum Distance Estimator Under Weak Instruments Tak Wai Chau February 20, 2014 Abstract This paper investigates the nite sample performance of a minimum distance estimator

More information

Generalized Method of Moments Estimation

Generalized Method of Moments Estimation Generalized Method of Moments Estimation Lars Peter Hansen March 0, 2007 Introduction Generalized methods of moments (GMM) refers to a class of estimators which are constructed from exploiting the sample

More information

GMM Estimation and Testing

GMM Estimation and Testing GMM Estimation and Testing Whitney Newey July 2007 Idea: Estimate parameters by setting sample moments to be close to population counterpart. Definitions: β : p 1 parameter vector, with true value β 0.

More information

Lecture 4: Heteroskedasticity

Lecture 4: Heteroskedasticity Lecture 4: Heteroskedasticity Econometric Methods Warsaw School of Economics (4) Heteroskedasticity 1 / 24 Outline 1 What is heteroskedasticity? 2 Testing for heteroskedasticity White Goldfeld-Quandt Breusch-Pagan

More information

GARCH Models Estimation and Inference

GARCH Models Estimation and Inference GARCH Models Estimation and Inference Eduardo Rossi University of Pavia December 013 Rossi GARCH Financial Econometrics - 013 1 / 1 Likelihood function The procedure most often used in estimating θ 0 in

More information

ORTHOGONALITY TESTS IN LINEAR MODELS SEUNG CHAN AHN ARIZONA STATE UNIVERSITY ABSTRACT

ORTHOGONALITY TESTS IN LINEAR MODELS SEUNG CHAN AHN ARIZONA STATE UNIVERSITY ABSTRACT ORTHOGONALITY TESTS IN LINEAR MODELS SEUNG CHAN AHN ARIZONA STATE UNIVERSITY ABSTRACT This paper considers several tests of orthogonality conditions in linear models where stochastic errors may be heteroskedastic

More information

(θ θ ), θ θ = 2 L(θ ) θ θ θ θ θ (θ )= H θθ (θ ) 1 d θ (θ )

(θ θ ), θ θ = 2 L(θ ) θ θ θ θ θ (θ )= H θθ (θ ) 1 d θ (θ ) Setting RHS to be zero, 0= (θ )+ 2 L(θ ) (θ θ ), θ θ = 2 L(θ ) 1 (θ )= H θθ (θ ) 1 d θ (θ ) O =0 θ 1 θ 3 θ 2 θ Figure 1: The Newton-Raphson Algorithm where H is the Hessian matrix, d θ is the derivative

More information

The BLP Method of Demand Curve Estimation in Industrial Organization

The BLP Method of Demand Curve Estimation in Industrial Organization The BLP Method of Demand Curve Estimation in Industrial Organization 9 March 2006 Eric Rasmusen 1 IDEAS USED 1. Instrumental variables. We use instruments to correct for the endogeneity of prices, the

More information

Multiple Equation GMM with Common Coefficients: Panel Data

Multiple Equation GMM with Common Coefficients: Panel Data Multiple Equation GMM with Common Coefficients: Panel Data Eric Zivot Winter 2013 Multi-equation GMM with common coefficients Example (panel wage equation) 69 = + 69 + + 69 + 1 80 = + 80 + + 80 + 2 Note:

More information

Econometrics of Panel Data

Econometrics of Panel Data Econometrics of Panel Data Jakub Mućk Meeting # 3 Jakub Mućk Econometrics of Panel Data Meeting # 3 1 / 21 Outline 1 Fixed or Random Hausman Test 2 Between Estimator 3 Coefficient of determination (R 2

More information

Economic modelling and forecasting

Economic modelling and forecasting Economic modelling and forecasting 2-6 February 2015 Bank of England he generalised method of moments Ole Rummel Adviser, CCBS at the Bank of England ole.rummel@bankofengland.co.uk Outline Classical estimation

More information

Specification testing in panel data models estimated by fixed effects with instrumental variables

Specification testing in panel data models estimated by fixed effects with instrumental variables Specification testing in panel data models estimated by fixed effects wh instrumental variables Carrie Falls Department of Economics Michigan State Universy Abstract I show that a handful of the regressions

More information

Introduction to Estimation Methods for Time Series models. Lecture 1

Introduction to Estimation Methods for Time Series models. Lecture 1 Introduction to Estimation Methods for Time Series models Lecture 1 Fulvio Corsi SNS Pisa Fulvio Corsi Introduction to Estimation () Methods for Time Series models Lecture 1 SNS Pisa 1 / 19 Estimation

More information

Lecture: Simultaneous Equation Model (Wooldridge s Book Chapter 16)

Lecture: Simultaneous Equation Model (Wooldridge s Book Chapter 16) Lecture: Simultaneous Equation Model (Wooldridge s Book Chapter 16) 1 2 Model Consider a system of two regressions y 1 = β 1 y 2 + u 1 (1) y 2 = β 2 y 1 + u 2 (2) This is a simultaneous equation model

More information

Asymptotic Least Squares Theory

Asymptotic Least Squares Theory Asymptotic Least Squares Theory CHUNG-MING KUAN Department of Finance & CRETA December 5, 2011 C.-M. Kuan (National Taiwan Univ.) Asymptotic Least Squares Theory December 5, 2011 1 / 85 Lecture Outline

More information

MLE and GMM. Li Zhao, SJTU. Spring, Li Zhao MLE and GMM 1 / 22

MLE and GMM. Li Zhao, SJTU. Spring, Li Zhao MLE and GMM 1 / 22 MLE and GMM Li Zhao, SJTU Spring, 2017 Li Zhao MLE and GMM 1 / 22 Outline 1 MLE 2 GMM 3 Binary Choice Models Li Zhao MLE and GMM 2 / 22 Maximum Likelihood Estimation - Introduction For a linear model y

More information

Econometric Analysis of Cross Section and Panel Data

Econometric Analysis of Cross Section and Panel Data Econometric Analysis of Cross Section and Panel Data Jeffrey M. Wooldridge / The MIT Press Cambridge, Massachusetts London, England Contents Preface Acknowledgments xvii xxiii I INTRODUCTION AND BACKGROUND

More information

GMM - Generalized method of moments

GMM - Generalized method of moments GMM - Generalized method of moments GMM Intuition: Matching moments You want to estimate properties of a data set {x t } T t=1. You assume that x t has a constant mean and variance. x t (µ 0, σ 2 ) Consider

More information

Instrumental Variables and GMM: Estimation and Testing. Steven Stillman, New Zealand Department of Labour

Instrumental Variables and GMM: Estimation and Testing. Steven Stillman, New Zealand Department of Labour Instrumental Variables and GMM: Estimation and Testing Christopher F Baum, Boston College Mark E. Schaffer, Heriot Watt University Steven Stillman, New Zealand Department of Labour March 2003 Stata Journal,

More information

Classical Least Squares Theory

Classical Least Squares Theory Classical Least Squares Theory CHUNG-MING KUAN Department of Finance & CRETA National Taiwan University October 14, 2012 C.-M. Kuan (Finance & CRETA, NTU) Classical Least Squares Theory October 14, 2012

More information

IV estimators and forbidden regressions

IV estimators and forbidden regressions Economics 8379 Spring 2016 Ben Williams IV estimators and forbidden regressions Preliminary results Consider the triangular model with first stage given by x i2 = γ 1X i1 + γ 2 Z i + ν i and second stage

More information

Maximum Likelihood (ML) Estimation

Maximum Likelihood (ML) Estimation Econometrics 2 Fall 2004 Maximum Likelihood (ML) Estimation Heino Bohn Nielsen 1of32 Outline of the Lecture (1) Introduction. (2) ML estimation defined. (3) ExampleI:Binomialtrials. (4) Example II: Linear

More information

Review of Classical Least Squares. James L. Powell Department of Economics University of California, Berkeley

Review of Classical Least Squares. James L. Powell Department of Economics University of California, Berkeley Review of Classical Least Squares James L. Powell Department of Economics University of California, Berkeley The Classical Linear Model The object of least squares regression methods is to model and estimate

More information

Econ 423 Lecture Notes: Additional Topics in Time Series 1

Econ 423 Lecture Notes: Additional Topics in Time Series 1 Econ 423 Lecture Notes: Additional Topics in Time Series 1 John C. Chao April 25, 2017 1 These notes are based in large part on Chapter 16 of Stock and Watson (2011). They are for instructional purposes

More information

Specification Test for Instrumental Variables Regression with Many Instruments

Specification Test for Instrumental Variables Regression with Many Instruments Specification Test for Instrumental Variables Regression with Many Instruments Yoonseok Lee and Ryo Okui April 009 Preliminary; comments are welcome Abstract This paper considers specification testing

More information

Economics 536 Lecture 7. Introduction to Specification Testing in Dynamic Econometric Models

Economics 536 Lecture 7. Introduction to Specification Testing in Dynamic Econometric Models University of Illinois Fall 2016 Department of Economics Roger Koenker Economics 536 Lecture 7 Introduction to Specification Testing in Dynamic Econometric Models In this lecture I want to briefly describe

More information

Birkbeck Working Papers in Economics & Finance

Birkbeck Working Papers in Economics & Finance ISSN 1745-8587 Birkbeck Working Papers in Economics & Finance Department of Economics, Mathematics and Statistics BWPEF 1809 A Note on Specification Testing in Some Structural Regression Models Walter

More information

Econometrics. Week 4. Fall Institute of Economic Studies Faculty of Social Sciences Charles University in Prague

Econometrics. Week 4. Fall Institute of Economic Studies Faculty of Social Sciences Charles University in Prague Econometrics Week 4 Institute of Economic Studies Faculty of Social Sciences Charles University in Prague Fall 2012 1 / 23 Recommended Reading For the today Serial correlation and heteroskedasticity in

More information

We begin by thinking about population relationships.

We begin by thinking about population relationships. Conditional Expectation Function (CEF) We begin by thinking about population relationships. CEF Decomposition Theorem: Given some outcome Y i and some covariates X i there is always a decomposition where

More information

Chapter 6. Panel Data. Joan Llull. Quantitative Statistical Methods II Barcelona GSE

Chapter 6. Panel Data. Joan Llull. Quantitative Statistical Methods II Barcelona GSE Chapter 6. Panel Data Joan Llull Quantitative Statistical Methods II Barcelona GSE Introduction Chapter 6. Panel Data 2 Panel data The term panel data refers to data sets with repeated observations over

More information

GARCH Models Estimation and Inference. Eduardo Rossi University of Pavia

GARCH Models Estimation and Inference. Eduardo Rossi University of Pavia GARCH Models Estimation and Inference Eduardo Rossi University of Pavia Likelihood function The procedure most often used in estimating θ 0 in ARCH models involves the maximization of a likelihood function

More information

Quasi Maximum Likelihood Theory

Quasi Maximum Likelihood Theory Quasi Maximum Likelihood heory CHUNG-MING KUAN Department of Finance & CREA June 17, 2010 C.-M. Kuan (National aiwan Univ.) Quasi Maximum Likelihood heory June 17, 2010 1 / 119 Lecture Outline 1 Kullback-Leibler

More information

1 Outline. 1. Motivation. 2. SUR model. 3. Simultaneous equations. 4. Estimation

1 Outline. 1. Motivation. 2. SUR model. 3. Simultaneous equations. 4. Estimation 1 Outline. 1. Motivation 2. SUR model 3. Simultaneous equations 4. Estimation 2 Motivation. In this chapter, we will study simultaneous systems of econometric equations. Systems of simultaneous equations

More information

Maximum Likelihood Tests and Quasi-Maximum-Likelihood

Maximum Likelihood Tests and Quasi-Maximum-Likelihood Maximum Likelihood Tests and Quasi-Maximum-Likelihood Wendelin Schnedler Department of Economics University of Heidelberg 10. Dezember 2007 Wendelin Schnedler (AWI) Maximum Likelihood Tests and Quasi-Maximum-Likelihood10.

More information

Lecture 9: Quantile Methods 2

Lecture 9: Quantile Methods 2 Lecture 9: Quantile Methods 2 1. Equivariance. 2. GMM for Quantiles. 3. Endogenous Models 4. Empirical Examples 1 1. Equivariance to Monotone Transformations. Theorem (Equivariance of Quantiles under Monotone

More information

Statistics and econometrics

Statistics and econometrics 1 / 36 Slides for the course Statistics and econometrics Part 10: Asymptotic hypothesis testing European University Institute Andrea Ichino September 8, 2014 2 / 36 Outline Why do we need large sample

More information

A Course in Applied Econometrics Lecture 18: Missing Data. Jeff Wooldridge IRP Lectures, UW Madison, August Linear model with IVs: y i x i u i,

A Course in Applied Econometrics Lecture 18: Missing Data. Jeff Wooldridge IRP Lectures, UW Madison, August Linear model with IVs: y i x i u i, A Course in Applied Econometrics Lecture 18: Missing Data Jeff Wooldridge IRP Lectures, UW Madison, August 2008 1. When Can Missing Data be Ignored? 2. Inverse Probability Weighting 3. Imputation 4. Heckman-Type

More information

Instrumental Variables

Instrumental Variables Università di Pavia 2010 Instrumental Variables Eduardo Rossi Exogeneity Exogeneity Assumption: the explanatory variables which form the columns of X are exogenous. It implies that any randomness in the

More information

Interpreting Regression Results

Interpreting Regression Results Interpreting Regression Results Carlo Favero Favero () Interpreting Regression Results 1 / 42 Interpreting Regression Results Interpreting regression results is not a simple exercise. We propose to split

More information

Ultra High Dimensional Variable Selection with Endogenous Variables

Ultra High Dimensional Variable Selection with Endogenous Variables 1 / 39 Ultra High Dimensional Variable Selection with Endogenous Variables Yuan Liao Princeton University Joint work with Jianqing Fan Job Market Talk January, 2012 2 / 39 Outline 1 Examples of Ultra High

More information

9. Model Selection. statistical models. overview of model selection. information criteria. goodness-of-fit measures

9. Model Selection. statistical models. overview of model selection. information criteria. goodness-of-fit measures FE661 - Statistical Methods for Financial Engineering 9. Model Selection Jitkomut Songsiri statistical models overview of model selection information criteria goodness-of-fit measures 9-1 Statistical models

More information

Chapter 1. GMM: Basic Concepts

Chapter 1. GMM: Basic Concepts Chapter 1. GMM: Basic Concepts Contents 1 Motivating Examples 1 1.1 Instrumental variable estimator....................... 1 1.2 Estimating parameters in monetary policy rules.............. 2 1.3 Estimating

More information

Instrumental Variables Estimation in Stata

Instrumental Variables Estimation in Stata Christopher F Baum 1 Faculty Micro Resource Center Boston College March 2007 1 Thanks to Austin Nichols for the use of his material on weak instruments and Mark Schaffer for helpful comments. The standard

More information

Suggested Solution for PS #5

Suggested Solution for PS #5 Cornell University Department of Economics Econ 62 Spring 28 TA: Jae Ho Yun Suggested Solution for S #5. (Measurement Error, IV) (a) This is a measurement error problem. y i x i + t i + " i t i t i + i

More information

DSGE Methods. Estimation of DSGE models: GMM and Indirect Inference. Willi Mutschler, M.Sc.

DSGE Methods. Estimation of DSGE models: GMM and Indirect Inference. Willi Mutschler, M.Sc. DSGE Methods Estimation of DSGE models: GMM and Indirect Inference Willi Mutschler, M.Sc. Institute of Econometrics and Economic Statistics University of Münster willi.mutschler@wiwi.uni-muenster.de Summer

More information

Chapter 11 GMM: General Formulas and Application

Chapter 11 GMM: General Formulas and Application Chapter 11 GMM: General Formulas and Application Main Content General GMM Formulas esting Moments Standard Errors of Anything by Delta Method Using GMM for Regressions Prespecified weighting Matrices and

More information

Classical Least Squares Theory

Classical Least Squares Theory Classical Least Squares Theory CHUNG-MING KUAN Department of Finance & CRETA National Taiwan University October 18, 2014 C.-M. Kuan (Finance & CRETA, NTU) Classical Least Squares Theory October 18, 2014

More information

Missing dependent variables in panel data models

Missing dependent variables in panel data models Missing dependent variables in panel data models Jason Abrevaya Abstract This paper considers estimation of a fixed-effects model in which the dependent variable may be missing. For cross-sectional units

More information

Econometrics of Panel Data

Econometrics of Panel Data Econometrics of Panel Data Jakub Mućk Meeting # 2 Jakub Mućk Econometrics of Panel Data Meeting # 2 1 / 26 Outline 1 Fixed effects model The Least Squares Dummy Variable Estimator The Fixed Effect (Within

More information

Discussion of Sensitivity and Informativeness under Local Misspecification

Discussion of Sensitivity and Informativeness under Local Misspecification Discussion of Sensitivity and Informativeness under Local Misspecification Jinyong Hahn April 4, 2019 Jinyong Hahn () Discussion of Sensitivity and Informativeness under Local Misspecification April 4,

More information

This chapter reviews properties of regression estimators and test statistics based on

This chapter reviews properties of regression estimators and test statistics based on Chapter 12 COINTEGRATING AND SPURIOUS REGRESSIONS This chapter reviews properties of regression estimators and test statistics based on the estimators when the regressors and regressant are difference

More information

IV and IV-GMM. Christopher F Baum. EC 823: Applied Econometrics. Boston College, Spring 2014

IV and IV-GMM. Christopher F Baum. EC 823: Applied Econometrics. Boston College, Spring 2014 IV and IV-GMM Christopher F Baum EC 823: Applied Econometrics Boston College, Spring 2014 Christopher F Baum (BC / DIW) IV and IV-GMM Boston College, Spring 2014 1 / 1 Instrumental variables estimators

More information

ASSET PRICING MODELS

ASSET PRICING MODELS ASSE PRICING MODELS [1] CAPM (1) Some notation: R it = (gross) return on asset i at time t. R mt = (gross) return on the market portfolio at time t. R ft = return on risk-free asset at time t. X it = R

More information

Repeated observations on the same cross-section of individual units. Important advantages relative to pure cross-section data

Repeated observations on the same cross-section of individual units. Important advantages relative to pure cross-section data Panel data Repeated observations on the same cross-section of individual units. Important advantages relative to pure cross-section data - possible to control for some unobserved heterogeneity - possible

More information

GMM and SMM. 1. Hansen, L Large Sample Properties of Generalized Method of Moments Estimators, Econometrica, 50, p

GMM and SMM. 1. Hansen, L Large Sample Properties of Generalized Method of Moments Estimators, Econometrica, 50, p GMM and SMM Some useful references: 1. Hansen, L. 1982. Large Sample Properties of Generalized Method of Moments Estimators, Econometrica, 50, p. 1029-54. 2. Lee, B.S. and B. Ingram. 1991 Simulation estimation

More information

Instrumental Variables, Simultaneous and Systems of Equations

Instrumental Variables, Simultaneous and Systems of Equations Chapter 6 Instrumental Variables, Simultaneous and Systems of Equations 61 Instrumental variables In the linear regression model y i = x iβ + ε i (61) we have been assuming that bf x i and ε i are uncorrelated

More information

The generalized method of moments

The generalized method of moments Robert M. Kunst robert.kunst@univie.ac.at University of Vienna and Institute for Advanced Studies Vienna February 2008 Based on the book Generalized Method of Moments by Alastair R. Hall (2005), Oxford

More information

Advanced Econometrics

Advanced Econometrics Based on the textbook by Verbeek: A Guide to Modern Econometrics Robert M. Kunst robert.kunst@univie.ac.at University of Vienna and Institute for Advanced Studies Vienna May 16, 2013 Outline Univariate

More information

Testing Random Effects in Two-Way Spatial Panel Data Models

Testing Random Effects in Two-Way Spatial Panel Data Models Testing Random Effects in Two-Way Spatial Panel Data Models Nicolas Debarsy May 27, 2010 Abstract This paper proposes an alternative testing procedure to the Hausman test statistic to help the applied

More information

Panel Threshold Regression Models with Endogenous Threshold Variables

Panel Threshold Regression Models with Endogenous Threshold Variables Panel Threshold Regression Models with Endogenous Threshold Variables Chien-Ho Wang National Taipei University Eric S. Lin National Tsing Hua University This Version: June 29, 2010 Abstract This paper

More information

Lecture 11 Weak IV. Econ 715

Lecture 11 Weak IV. Econ 715 Lecture 11 Weak IV Instrument exogeneity and instrument relevance are two crucial requirements in empirical analysis using GMM. It now appears that in many applications of GMM and IV regressions, instruments

More information

An Introduction to Generalized Method of Moments. Chen,Rong aronge.net

An Introduction to Generalized Method of Moments. Chen,Rong   aronge.net An Introduction to Generalized Method of Moments Chen,Rong http:// aronge.net Asset Pricing, 2012 Section 1 WHY GMM? 2 Empirical Studies 3 Econometric Estimation Strategies 4 5 Maximum Likelihood Estimation

More information

Appendix A: The time series behavior of employment growth

Appendix A: The time series behavior of employment growth Unpublished appendices from The Relationship between Firm Size and Firm Growth in the U.S. Manufacturing Sector Bronwyn H. Hall Journal of Industrial Economics 35 (June 987): 583-606. Appendix A: The time

More information

Estimating Deep Parameters: GMM and SMM

Estimating Deep Parameters: GMM and SMM Estimating Deep Parameters: GMM and SMM 1 Parameterizing a Model Calibration Choose parameters from micro or other related macro studies (e.g. coeffi cient of relative risk aversion is 2). SMM with weighting

More information

The Quasi-Maximum Likelihood Method: Theory

The Quasi-Maximum Likelihood Method: Theory Chapter 9 he Quasi-Maximum Likelihood Method: heory As discussed in preceding chapters, estimating linear and nonlinear regressions by the least squares method results in an approximation to the conditional

More information

Markov-Switching Models with Endogenous Explanatory Variables. Chang-Jin Kim 1

Markov-Switching Models with Endogenous Explanatory Variables. Chang-Jin Kim 1 Markov-Switching Models with Endogenous Explanatory Variables by Chang-Jin Kim 1 Dept. of Economics, Korea University and Dept. of Economics, University of Washington First draft: August, 2002 This version:

More information

Econometrics. Week 8. Fall Institute of Economic Studies Faculty of Social Sciences Charles University in Prague

Econometrics. Week 8. Fall Institute of Economic Studies Faculty of Social Sciences Charles University in Prague Econometrics Week 8 Institute of Economic Studies Faculty of Social Sciences Charles University in Prague Fall 2012 1 / 25 Recommended Reading For the today Instrumental Variables Estimation and Two Stage

More information

Estimation of Threshold Cointegration

Estimation of Threshold Cointegration Estimation of Myung Hwan London School of Economics December 2006 Outline Model Asymptotics Inference Conclusion 1 Model Estimation Methods Literature 2 Asymptotics Consistency Convergence Rates Asymptotic

More information

Econometrics I. Ricardo Mora

Econometrics I. Ricardo Mora 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

More information

Using all observations when forecasting under structural breaks

Using all observations when forecasting under structural breaks Using all observations when forecasting under structural breaks Stanislav Anatolyev New Economic School Victor Kitov Moscow State University December 2007 Abstract We extend the idea of the trade-off window

More information

Instrumental Variables and the Problem of Endogeneity

Instrumental Variables and the Problem of Endogeneity Instrumental Variables and the Problem of Endogeneity September 15, 2015 1 / 38 Exogeneity: Important Assumption of OLS In a standard OLS framework, y = xβ + ɛ (1) and for unbiasedness we need E[x ɛ] =

More information

Understanding Regressions with Observations Collected at High Frequency over Long Span

Understanding Regressions with Observations Collected at High Frequency over Long Span Understanding Regressions with Observations Collected at High Frequency over Long Span Yoosoon Chang Department of Economics, Indiana University Joon Y. Park Department of Economics, Indiana University

More information

ADVANCED FINANCIAL ECONOMETRICS PROF. MASSIMO GUIDOLIN

ADVANCED FINANCIAL ECONOMETRICS PROF. MASSIMO GUIDOLIN Massimo Guidolin Massimo.Guidolin@unibocconi.it Dept. of Finance ADVANCED FINANCIAL ECONOMETRICS PROF. MASSIMO GUIDOLIN a.a. 14/15 p. 1 LECTURE 3: REVIEW OF BASIC ESTIMATION METHODS: GMM AND OTHER EXTREMUM

More information

Lecture 10: Panel Data

Lecture 10: Panel Data Lecture 10: Instructor: Department of Economics Stanford University 2011 Random Effect Estimator: β R y it = x itβ + u it u it = α i + ɛ it i = 1,..., N, t = 1,..., T E (α i x i ) = E (ɛ it x i ) = 0.

More information

An Encompassing Test for Non-Nested Quantile Regression Models

An Encompassing Test for Non-Nested Quantile Regression Models An Encompassing Test for Non-Nested Quantile Regression Models Chung-Ming Kuan Department of Finance National Taiwan University Hsin-Yi Lin Department of Economics National Chengchi University Abstract

More information

On the Power of Tests for Regime Switching

On the Power of Tests for Regime Switching On the Power of Tests for Regime Switching joint work with Drew Carter and Ben Hansen Douglas G. Steigerwald UC Santa Barbara May 2015 D. Steigerwald (UCSB) Regime Switching May 2015 1 / 42 Motivating

More information

GARCH Models Estimation and Inference

GARCH Models Estimation and Inference Università di Pavia GARCH Models Estimation and Inference Eduardo Rossi Likelihood function The procedure most often used in estimating θ 0 in ARCH models involves the maximization of a likelihood function

More information

A Course in Applied Econometrics Lecture 14: Control Functions and Related Methods. Jeff Wooldridge IRP Lectures, UW Madison, August 2008

A Course in Applied Econometrics Lecture 14: Control Functions and Related Methods. Jeff Wooldridge IRP Lectures, UW Madison, August 2008 A Course in Applied Econometrics Lecture 14: Control Functions and Related Methods Jeff Wooldridge IRP Lectures, UW Madison, August 2008 1. Linear-in-Parameters Models: IV versus Control Functions 2. Correlated

More information

Nonlinear GMM. Eric Zivot. Winter, 2013

Nonlinear GMM. Eric Zivot. Winter, 2013 Nonlinear GMM Eric Zivot Winter, 2013 Nonlinear GMM estimation occurs when the GMM moment conditions g(w θ) arenonlinearfunctionsofthe model parameters θ The moment conditions g(w θ) may be nonlinear functions

More information

Econ 582 Fixed Effects Estimation of Panel Data

Econ 582 Fixed Effects Estimation of Panel Data Econ 582 Fixed Effects Estimation of Panel Data Eric Zivot May 28, 2012 Panel Data Framework = x 0 β + = 1 (individuals); =1 (time periods) y 1 = X β ( ) ( 1) + ε Main question: Is x uncorrelated with?

More information

Inference Based on the Wild Bootstrap

Inference Based on the Wild Bootstrap Inference Based on the Wild Bootstrap James G. MacKinnon Department of Economics Queen s University Kingston, Ontario, Canada K7L 3N6 email: jgm@econ.queensu.ca Ottawa September 14, 2012 The Wild Bootstrap

More information

Asymptotics for Nonlinear GMM

Asymptotics for Nonlinear GMM Asymptotics for Nonlinear GMM Eric Zivot February 13, 2013 Asymptotic Properties of Nonlinear GMM Under standard regularity conditions (to be discussed later), it can be shown that where ˆθ(Ŵ) θ 0 ³ˆθ(Ŵ)

More information

PANEL DATA RANDOM AND FIXED EFFECTS MODEL. Professor Menelaos Karanasos. December Panel Data (Institute) PANEL DATA December / 1

PANEL DATA RANDOM AND FIXED EFFECTS MODEL. Professor Menelaos Karanasos. December Panel Data (Institute) PANEL DATA December / 1 PANEL DATA RANDOM AND FIXED EFFECTS MODEL Professor Menelaos Karanasos December 2011 PANEL DATA Notation y it is the value of the dependent variable for cross-section unit i at time t where i = 1,...,

More information

Regression: Ordinary Least Squares

Regression: Ordinary Least Squares Regression: Ordinary Least Squares Mark Hendricks Autumn 2017 FINM Intro: Regression Outline Regression OLS Mathematics Linear Projection Hendricks, Autumn 2017 FINM Intro: Regression: Lecture 2/32 Regression

More information