Chapter 11 GMM: General Formulas and Application
|
|
- Arron Goodwin
- 5 years ago
- Views:
Transcription
1 Chapter 11 GMM: General Formulas and Application
2 Main Content General GMM Formulas esting Moments Standard Errors of Anything by Delta Method Using GMM for Regressions Prespecified weighting Matrices and Moment Conditions Estimating on One Group of Moments,esting on Another Estimating the Spectral Density Matrix
3 11.1 General GMM Formulas GMM procedures can be used to implement a host of estimation and testing exercises. o estimate the parameter, you just have to remember (or look up) a few very general formulas, and then map them into your case. Express a model as E[ f( xt, b)] Everything is a vector: f can represent a vector of L sample moments, x can be M data series, b t can be N parameters.
4 Definition of the GMM Estimate We estimate parameters to set some linear combination of sample means of f to zero b: set ag ( b ) = 0 where a t is a matrix that defines which linear combination of g ( b) will be set to zero. If you estimate b by min g ( b) Wg( b),the firstorder condition are his is mean a b 1 g( b) = f( xt, b) g = W b t = 1 g Wg b ( b ) = 0
5 Standard Error of Estimate Hansen (1982), heorem 3.1 tells us that the asymptotic distribution of the GMM estimate is 1 1 ( b b) N[0,( ad) asa ( ad) ] where f g ( b) d E[ ( xt, b)] = b b S E[ f( xt, b), f( xt j, b)] j= a= plim a In practical terms,this means to use var( b) = ( ad) asa ( ad)
6 Distribution of the moments Hansen s Lemma 4.1 gives the sampling distribution of the moments : g ( b) g b N I d ad a S I d ad a 1 1 ( ) [0,( ( ) ) ( ( ) )] 1 he I dad ( ) a terms account for the fact that in each sample some linear combinationsof g are set to zero. hen S is singular.
7 2 χ est A sum of squared standard normals is 2 distributed χ, so we have g b I dad asi dad a g b ( )[( ( ) ) ( ( ) )] ( ) 2 is distributed χ which has degrees of freedom given by number of nonzero linear combinations of g, the number of moments less the number of estimated parameters
8 It does, but with a hitch: he variancecovariance matrix is singular, so you have to pseudo-invert it. For example, you can perform an eigenvalue decomposition = QΛQ' and then invert only the non-zero eigenvalues.
9 Efficient Estimates Hansen shows that one particular choice is statistically optimal, a = ds 1 his choice is the first order condition to 1 min g( b) S g( b) that we studied in the last { b} Chapter. With this weighting matrix, the standard error of b reduces to b b N ds d 1 1 ( ) [0,( ) ]
10 With the optimal weights S 1 the variance of the moments simplifies to Proof: cov( g ) = ( S d( ds d) d ) 1 1 var( g ( b)) = ( I d( ad) a) S( I d( ad) a) a= ds 1 1 I d ad a S I d ad a 1 1 ( ( ) ) ( ( ) )) = I d ds d ds S I d ds d ds ( ( ) ) ( ( ) ) = S dds dd I S dds d d ( ( ) )( ( ) ) = S d( ds d) d d( ds d) d + d( ds d) d = S d( ds d) d 1 1
11 Using this matrix in a test, there is an equivalent and simpler way to construct this test 1 g b S g b moments parameters 2 ( ) ( ) χ (# # )
12 Alternatively, note that S 1 is a pseudo-inverse of the second stage cov( g ) Proof: A pseudo inverse times cov( g ) should result in an idempotent matrix of the same rank as S cov( g ) = S ( S d( ds d) d ) = I S d( ds d) d hen, check that the result is idempotent = ( I S d( ds d) d)( I S d( ds d) d) I S d( ds d) d) cov( ) his derivation not only verifies that J has the same distribution as g cov( g) g, but that they are numerically the same in every sample. g
13 Model Comparisons You often want to compare one model to another. If one model can be expressed as a special or restricted case of the other or unrestricted model we can perform a statistical comparison that looks very much like a likelihood ratio test. 2 J ( restricted) J ( unrestricted) χ (# restriction) If the restricted model is really true, it should nor rise much. 2 his is a χ difference test due to Newey and West(1987), who call it the D-test
14 11.2 est Moments How to test one or a group of pricing error. (1)Use the formula for var( g ) 2 (2)A χ difference test We can use the sampling distribution of g, to evaluate the significance of individual pricing errors, to construct a t-test(for a 2 single moment) or a χ test(for groups of moments)
15 Alternatively, you can use the difference approach. Start with a general model that includes all the moments, and form an estimate of the spectral density matrix S. Set to zero the moments you want to test, and denote g s ( b) the vector of moments, including the zeros(s for smaller) g ( b ) S g ( b ) g ( b ) S g ( b ) χ s s s s moments) (#eliminated If moments we want to test truly are zero,the criterion should not be that much lower 2 χ
16 11.3 Standard Errors of Anything by Delta Method we want to estimate a quantity that is a nonlinear function of sample means b= φ[ E( x )] = φ( u) In this case, we have For example, a correlation coefficient can be written as a function of sample means as hus, take corr( x, y ) = t t 1 dφ dφ var( b) = [ ] cov( xt, x t j )[ ] du du t Exy ( ) Ex ( ) Ey ( ) t t t t ( t ) ( t) ( t ) ( t) Ex E x Ey E y u = E x E x E y E y E x y 2 2 [ ( t), ( t ), ( t), ( t ), ( t t)]
17 11.4 Using GMM for Regression Mapping any statistical procedure into GMM makes it easy to develop an asymptotic distribution that corrects for statistical problems such as non-normality, serial correlation and conditional heteroskedasticity. For example, I map OLS regressions into GMM. When errors do not obey the OLS assumptions, OLS is consistent, and often more robust than GLS, but its standard errors need to be corrected.
18 OLS picks parameters β to minimize the variance of the residual: β We find from the first order condition, which states that the residual is orthogonal to the right hand variable: g ( β) = E[ x ( y x β)] = 0 t t t It is exactly identified. We set the sample moments exactly to zero and there is no weighting matrix (a = I). We can solve for the estimate analytically, β = 2 min E[( yt xt) ] { β } β 1 [ E( xx t t )] E( xy t t) his is the familiar OLS formula. But its standard error need to be corrected.
19 We can use GMM to obtain the standard errors 1 1 through ( b b) N[0,( ds d) ], so that d = E( x x ) t t f( xt, β) = xt( yt xtβ) = xtεt 1 var( β) = E( xx ) [ E( ε xx ε )] E( xx) 1 1 t t t t t j t j t t j=
20 Serially Uncorrelated, Homoskedastic Errors Formally,the OLS assumptions are E( ε x, x, ε, ε ) = 0 t t t 1 t 1 t E( εt xt, xt 1, εt 1, εt 2 ) = cons tant = σe he first assumption means that only the j=0 term enter the sum 2 E( εtxx t t jε t j) = E( εt xx t t ) j= he second assumption means that 2 2 E( εt xx t t ) = E( εt ) E( xx t t) Hence the standard errors reduces to our old form var( β) = σ ( XX ) ε
21 Heteroskedastic Errors If we delete the condition homoskedasticity assumption 2 2 ( εt t, t 1, εt 1, εt 2 ) = tan = σ he standard errors are E x x cons t ε 1 = 1 2 var( β) Exx ( t t ) E( εt xx t t) Exx ( t t ) hese are known as heteroskedasticity consistent standard errors or white standard errors after White (1980)
22 Hansen-Hodrick Errors When the regression notation is yt+ k= β kxt+ εt+ k under the null that one-period returns are unforecastable, we still see correlation in the e t due to the overlapping data. Unforecastable returns imply E( εtεt j) = 0 for j K Under this condition, the standard errors are 1 var( β ) ( ) [ ( ε ε )] ( ) k 1 1 k = Exx t t E txx t t j t j Exx t t j= k+ 1
23 11.5 Prespecified Weighting Matrices and Moment Conditions In the last chapter, our final estimates were based on the efficinet S 1 weighting matrix. A prespecified weighting matrix lets you specify which moments or linear combination of moments GMM will value in the minimization. So you can also go one step further and impose which linear combinations a of moment conditions will be set to zero in estimation rather than use the choice resulting from a minimization.
24 1 2 For example, if g,, but = [ g, g] W = I g / b= [1,10] so that the second moment is 10 times more sensitive to the parameter value than the first moment, then GMM with fixed weighting matrix set 1 g 2 1* g + 10* = 0 If we want GMM to pay equal attention to the two moment, we can fix the a matrix directly. Using a prespecified weighting matrix is not the same thing as ignoring correlation of the error u t in the distribution theory.
25 How to Use Prespecified Weighting Matrices If we use weighting matrix W, the first-order conditions to min g ( bwg ) ( b) are { b} g ( b) Wg ( b ) = d Wg ( b ) = 0 b So the variance-covariance matrix of the estimated coefficients is 1 var( b) = ( d Wd) d WSWd( d Wd) 1 1 he variance-covariance matrix of the moments 1 g = I d dwd dw S I Wd dwd d 1 1 var( ) ( ( ) ) ( ( ) ) 2 he above equation can be the basis of χ test for the overidentifying restrictions. g
26 If we interpret () i 1 to be a generalized inverse, then 1 2 g var( g ) g χ (# moment # parameters) If var( g ) is singular, you can inverting only the nonzero eigenvalues.
27 Motivations for Prespecified Weighting Matrices Robustness, as with OLS vs. GLS When errors are autocorrelated or heteroskedastic and we correctly model the error covariance matrix and the regression is perfectly specified, the GLS procedure can improve efficiency. If the error covariance matrix is incorrectly, the GLS estimates can be much worse than OLS. In these cases, it is often a good idea to use OLS estimates. But we need to correct the standard error of the OLS estimates
28 For GMM, first-stage or other fixed weighting matrix estimates may give up something in asymptotic efficiency, standard errors and model fit tests. hey are still consistent and more robust to statistical and economic problems.but we use the S matrix in computing standard error. When the parameter estimates have a great different between the first stage and the second stage, we should decide what cause this. It is truly due to efficiency gain or a model misspecification.
29 Near-Singular S he spectral density matrix is often nearly singular, since asset returns are highly correlated with each other. As a result, second stage GMM tries to minimize differences and differences of differences of asset returns in order to extract statistically orthogonal components with lowest variance. his feature leads GMM to place a lot of weight on poorly estimated, economically uninteresting, or otherwise non-robust aspects of the data.
30 For example, suppose that S is given by So We can write where S 1 ρ S = ρ 1 hen, the GMM criterion is is equivalent to gc ρ 1 ρ ρ 1 = 2 CC = S 1 1 ρ 2 2 C = 1 ρ 1 ρ 0 1 min( )( Cg) 1 min g S g
31 Cg gives the linear combination of moments that efficient GMM is trying to minimize. As ρ 1,for the matrix C, the (2,2) element stay at 1, but the (1,1) and (1,2) elements get very large. If ρ = 0.95 then C = 0 1 his mean that GMM pay a little attention to the second moment, and play three times as much weight on the difference between first and second moment. hrough the decomposition of S, we can see what moments GMM is prizing.
32 GMM wants to focus on well-measured moments. In asset pricing applications, the errors are close to uncorrelated over time, so GMM is looking for e portfolios with small values of var( mt + 1Rt + 1). hose will be assets with small return variance. hen, GMM will pay most attention to correctly pricing the sample minimum-variance portfolio. his cause that sample minimum-variance portfolio may have little to do with the true minimum-variance portfolio. Like any portfolio on the sample frontier, its composition largely reflects luck.
33
34 Economically Interesting Moment he optimal weighting matrix makes GMM pay close attention to linear combinations of moments with small sampling error in both estimation and evaluation. We want to force the estimation and evaluation to pay attention to economically interesting moments instead.
35 Level Playing Field he S matrix changes as the model and as its parameters change. As the S matrix changes, which assets the GMM estimate tries hard to price well changes as well. For example we take a model m t and create a new model by simply adding noise, unrelated to asset returns (in sample), m t = mt + εt then the moment e condition g = E ( mr t t ) is unchanged. However, the 2 e e spectral density matrix S = E(( mt +εt) Rt Rt ) rise dramatically. his can reduce the J, leading to a false sense of improvement.
36 Level Playing Field If the sample contains a nearly riskfree portfolio of the test assets, or a portfolio with apprently small variances of m t+1 R e t+1, then J test will focus on the pricing of this portfolio and will lead to a false rejection, since there is an eigenvalue of S that is too small. Some stylized facts, such as the RMSE, pricing errors, are as interesting as the statistical tests.
37 Some Prespecified Weighting Matrices When the second-moment matrix of payoffs in place of S (Hansen and 1 W = E( xx ) Jagannathan (1997)). he minimum distance (second moment) between a candidate discount factor y and the space of true discount factors is the same as the minimum 1 value of the GMM criterion with W = E( xx ) as weighting matrix. Why is this true?
38
39 Proof :he distance between y and the nearest valid m is the same as the distance between is the same as the distance between proj( y X ) and. From the OLS formula, x 1 proj( y X ) E( yx ) E( xx ) x is the portfolio of that price x hen, the distance between y and is x = 1 x = pe( xx) x y x = proj( y X ) x x = ( ) ( ) ( ) x 1 1 E yx E xx x p E xx x = ( ( ) ) ( ) 1 E yx p E xx x = = 1 [ E( xy ) p] E( xx ) [ E( xy ) p] 1 gexx ( ) g
40 Identity Matrix Using the identity matrix weights has a particular advantage with large systems in which S is nearly singular. It avoids most of the problems associated with inverting a near-singular S matrix.
41 Comparing the Second-Moment and Identity Matrices he second moment matrix and the optimal weighting matrix S give an objective that is invariant to the initial choice of assets or portfolios. 1 [ E( yax) Ap] E( Axx A ) [ E( yax) Ap] 1 = [ E( yx) p] E( xx ) [ E( yx) p] It is not true of the identity or other fixed matrices. he results depend on the initial choice of portfolios. he second-moment matrix is often even more nearly singular than the spectral density matrix. It is no help on overcoming the near singularity of S.
42 Estimating on One Group of Moment, esting on Another We can use one set of moment for estimate and another for testing We can also using one set of asset returns and then see how the model does out of sample on another set of asset We can do all this very simply by using an appropriate weighting matrix or a prespecified moment matrix, for example a = [ I N,0 N+ M] a
43 11.7 Estimating the Spectral Density Matrix he optimal weighting matrix S depend on population moments, and depend on the parameters b. S = E( uu ), u ( m ( b) x p ) j= t t j t t t 1 here are a lot of parameters. How do we estimate this matrix in practice?
44 Use a sensible first-stage W, or transform the data In the first-stage b estimates, we should use a sensible weighting matrix. Sometimes, some moments will have different unit than other moment. For example the moment formed by * d Rt + 1 pt and the moment formed by R *1 t + 1. It is also useful to start with moments that are not horrendously correlated with each other.
45 a For example, you might consider and a b rather than and R. R R R a R b W = 0 1 =
46 Remove means Under the null, Eu ( t ) = 0 whether we estimate the covariance matrix by removing mean. 1 1 [( ut u)( ut u) ], u = ut t= 1 t= 1 Hansen and Singleton (1982) advocate removing the means in sample. But this mothed also make that estimate S matrices are often nearly singular. Since E( uu ) = cov( uu, ) + EuEu ( ) ( ) EuEu ( ) ( ) is a singular matrix
47 Downweight highter-order correlations When we estimate S, we want to construct consistent estimates that are automatically positive definite in every sample. For example the Newey and West estimate, it is k k j 1 Sˆ = ( ) ( uu t t j ) j= k k t= 1 he Newey-West estimator is base on the variance of kth sums. So it is positive definite k var( ut j) = ke( uu t t) + ( k 1)[ E( uu t t 1) + E( ut 1u t)] + j= 1 k k j + [ Euu ( t t k) + Eu ( t ku t)] = k Euu ( t t j) k j= k
48 What value of k, or how wide a window if of another shape, should you use? he rate at which k should increase with sample size, but not as quickly as the sample size increases.
49 Consider parametric structures for autocorrelation and heteroskedasticity GMM is not inherently tied to nonparametric covariance matrix estimates. We can impose a parametric structure on the S matrix. For example, if we model a scalar u as an 2 AR(1) with parameter σ and ρ u then 2 j ( ) 21+ ρ S = E uu t t j = σu ρ = σu j= 1 ρ So we only need to estimate two parameter j=
50 Use the null limit correlations In the asset pricing setup, the null hypothesis specifies that E ( ) ( 1) 0 t ut+ 1 Et mt+ 1Rt+ 1 as well as In this situation, you can get = = Eu ( t + 1) = 0 ˆ 1 S = uu t t. = However, the null might not be correct, if the null is not correct, you have a inconsistent estimate. If the null is not correct,estimating extra lags that should be zero under the null only loses a little bit of power. t 1
51 Monte Carlo evidence suggest that adding null hypothesis can help with the power of test statistics. Small-sample performance of the nonparametric estimators with many lags is not very good. We can test the autocorrelated of u t to decide whether the model is right. If there is a lot of correlation, this is an indication that something is wrong with the estimate. ˆ 1 S = uu t t. = t 1
52 Size problems; consider a factor or other parametric cross-sectional structure when the number of moments is more than around 1/10 the number of data points, S estimates tend to become unstable and near-singular. It might be better to estimate an S imposing a factor structure on all the primitive assets. One might also use a highly structured estimate of S as weighting matrix, while using a less constrained estimate for the standard errors.
53 Alternatives to the two-stage procedure: iteration and one-step. Iterate: we can use this formula bˆ = min g ( b) S ( b) g ( b) { b} Where b 1 is a first-stage estimate, held fixed in the minimization over b, then use ˆb 2 2 to find Sb ( ˆ 2), find ˆ 1 b3 = min[ g( b) S( b2) g( b) { b} and so on. We can find this estimate serial converge to one value.
54 his procedure is also likely to produce estimates that do not depend on the initial weighting matrix. Pick b and S simultaneously. When search for b, the S also change. hen the object become into 1 min[ g( b) S ( b) g( b)] { b} he first-order conditions are 1 g 1 S ( b) 2*( ) S ( b) g( b) + g( b) g( b) = 0 b b
55 In the iteration method, each step involves a numerical search over g( b) Sg( b), may be much quicker to minimize once over g( b. ) S( b) g( b) On the other hand, the latter is not a locally quadratic form, so the search may run into greater numerical difficulties.
56 he End hank you!
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 informationGMM, HAC estimators, & Standard Errors for Business Cycle Statistics
GMM, HAC estimators, & Standard Errors for Business Cycle Statistics Wouter J. Den Haan London School of Economics c Wouter J. Den Haan Overview Generic GMM problem Estimation Heteroskedastic and Autocorrelation
More informationEconomic 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 informationEstimating 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 informationAn 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 informationu t = T u t = ū = 5.4 t=1 The sample mean is itself is a random variable. What if we sample again,
17 GMM Notes Note: These notes are slightly amplified versions of the blackboards for the GMM lectures in my Coursera Asset Pricing class. They are not meant as substitutes for the readings, and are incomplete
More informationGeneralized 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 informationAn estimate of the long-run covariance matrix, Ω, is necessary to calculate asymptotic
Chapter 6 ESTIMATION OF THE LONG-RUN COVARIANCE MATRIX An estimate of the long-run covariance matrix, Ω, is necessary to calculate asymptotic standard errors for the OLS and linear IV estimators presented
More informationHeteroskedasticity and Autocorrelation Consistent Standard Errors
NBER Summer Institute Minicourse What s New in Econometrics: ime Series Lecture 9 July 6, 008 Heteroskedasticity and Autocorrelation Consistent Standard Errors Lecture 9, July, 008 Outline. What are HAC
More informationASSET 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 informationProblem Set #6: OLS. Economics 835: Econometrics. Fall 2012
Problem Set #6: OLS Economics 835: Econometrics Fall 202 A preliminary result Suppose we have a random sample of size n on the scalar random variables (x, y) with finite means, variances, and covariance.
More informationEconometrics 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 informationGMM - 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 informationFinal Exam. Economics 835: Econometrics. Fall 2010
Final Exam Economics 835: Econometrics Fall 2010 Please answer the question I ask - no more and no less - and remember that the correct answer is often short and simple. 1 Some short questions a) For each
More informationSTAT 100C: Linear models
STAT 100C: Linear models Arash A. Amini June 9, 2018 1 / 56 Table of Contents Multiple linear regression Linear model setup Estimation of β Geometric interpretation Estimation of σ 2 Hat matrix Gram matrix
More informationChapter 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 informationVolatility. Gerald P. Dwyer. February Clemson University
Volatility Gerald P. Dwyer Clemson University February 2016 Outline 1 Volatility Characteristics of Time Series Heteroskedasticity Simpler Estimation Strategies Exponentially Weighted Moving Average Use
More informationEconometrics Summary Algebraic and Statistical Preliminaries
Econometrics Summary Algebraic and Statistical Preliminaries Elasticity: The point elasticity of Y with respect to L is given by α = ( Y/ L)/(Y/L). The arc elasticity is given by ( Y/ L)/(Y/L), when L
More informationLinear Regression with Time Series Data
u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f e c o n o m i c s Econometrics II Linear Regression with Time Series Data Morten Nyboe Tabor u n i v e r s i t y o f c o p e n h a g
More informationProf. Dr. Roland Füss Lecture Series in Applied Econometrics Summer Term Introduction to Time Series Analysis
Introduction to Time Series Analysis 1 Contents: I. Basics of Time Series Analysis... 4 I.1 Stationarity... 5 I.2 Autocorrelation Function... 9 I.3 Partial Autocorrelation Function (PACF)... 14 I.4 Transformation
More informationHeteroskedasticity and Autocorrelation
Lesson 7 Heteroskedasticity and Autocorrelation Pilar González and Susan Orbe Dpt. Applied Economics III (Econometrics and Statistics) Pilar González and Susan Orbe OCW 2014 Lesson 7. Heteroskedasticity
More informationLinear Regression with Time Series Data
u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f e c o n o m i c s Econometrics II Linear Regression with Time Series Data Morten Nyboe Tabor u n i v e r s i t y o f c o p e n h a g
More informationGeneralized Method of Moment
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, 2010 1 / 32 Lecture
More informationGeneralized 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 informationNext is material on matrix rank. Please see the handout
B90.330 / C.005 NOTES for Wednesday 0.APR.7 Suppose that the model is β + ε, but ε does not have the desired variance matrix. Say that ε is normal, but Var(ε) σ W. The form of W is W w 0 0 0 0 0 0 w 0
More informationDepartment of Economics, UCSD UC San Diego
Department of Economics, UCSD UC San Diego itle: Spurious Regressions with Stationary Series Author: Granger, Clive W.J., University of California, San Diego Hyung, Namwon, University of Seoul Jeon, Yongil,
More informationMultiple 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 informationInverse of a Square Matrix. For an N N square matrix A, the inverse of A, 1
Inverse of a Square Matrix For an N N square matrix A, the inverse of A, 1 A, exists if and only if A is of full rank, i.e., if and only if no column of A is a linear combination 1 of the others. A is
More informationLECTURE 11. Introduction to Econometrics. Autocorrelation
LECTURE 11 Introduction to Econometrics Autocorrelation November 29, 2016 1 / 24 ON PREVIOUS LECTURES We discussed the specification of a regression equation Specification consists of choosing: 1. correct
More informationLecture: 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 informationTesting Overidentifying Restrictions with Many Instruments and Heteroskedasticity
Testing Overidentifying Restrictions with Many Instruments and Heteroskedasticity John C. Chao, Department of Economics, University of Maryland, chao@econ.umd.edu. Jerry A. Hausman, Department of Economics,
More informationDrawing Inferences from Statistics Based on Multiyear Asset Returns
Drawing Inferences from Statistics Based on Multiyear Asset Returns Matthew Richardson ames H. Stock FE 1989 1 Motivation Fama and French (1988, Poterba and Summer (1988 document significant negative correlations
More informationChapter 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 informationEconometrics 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 informationDSGE 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 informationEconometrics Honor s Exam Review Session. Spring 2012 Eunice Han
Econometrics Honor s Exam Review Session Spring 2012 Eunice Han Topics 1. OLS The Assumptions Omitted Variable Bias Conditional Mean Independence Hypothesis Testing and Confidence Intervals Homoskedasticity
More information1 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 informationStatistics 910, #5 1. Regression Methods
Statistics 910, #5 1 Overview Regression Methods 1. Idea: effects of dependence 2. Examples of estimation (in R) 3. Review of regression 4. Comparisons and relative efficiencies Idea Decomposition Well-known
More informationDSGE-Models. Limited Information Estimation General Method of Moments and Indirect Inference
DSGE-Models General Method of Moments and Indirect Inference Dr. Andrea Beccarini Willi Mutschler, M.Sc. Institute of Econometrics and Economic Statistics University of Münster willi.mutschler@uni-muenster.de
More informationLecture 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 informationMULTIPLE REGRESSION AND ISSUES IN REGRESSION ANALYSIS
MULTIPLE REGRESSION AND ISSUES IN REGRESSION ANALYSIS Page 1 MSR = Mean Regression Sum of Squares MSE = Mean Squared Error RSS = Regression Sum of Squares SSE = Sum of Squared Errors/Residuals α = Level
More information11. Further Issues in Using OLS with TS Data
11. Further Issues in Using OLS with TS Data With TS, including lags of the dependent variable often allow us to fit much better the variation in y Exact distribution theory is rarely available in TS applications,
More informationPanel Data Models. Chapter 5. Financial Econometrics. Michael Hauser WS17/18 1 / 63
1 / 63 Panel Data Models Chapter 5 Financial Econometrics Michael Hauser WS17/18 2 / 63 Content Data structures: Times series, cross sectional, panel data, pooled data Static linear panel data models:
More informationECONOMETRICS HONOR S EXAM REVIEW SESSION
ECONOMETRICS HONOR S EXAM REVIEW SESSION Eunice Han ehan@fas.harvard.edu March 26 th, 2013 Harvard University Information 2 Exam: April 3 rd 3-6pm @ Emerson 105 Bring a calculator and extra pens. Notes
More informationCh.10 Autocorrelated Disturbances (June 15, 2016)
Ch10 Autocorrelated Disturbances (June 15, 2016) In a time-series linear regression model setting, Y t = x tβ + u t, t = 1, 2,, T, (10-1) a common problem is autocorrelation, or serial correlation of the
More informationMotivation Non-linear Rational Expectations The Permanent Income Hypothesis The Log of Gravity Non-linear IV Estimation Summary.
Econometrics I Department of Economics Universidad Carlos III de Madrid Master in Industrial Economics and Markets Outline Motivation 1 Motivation 2 3 4 5 Motivation Hansen's contributions GMM was developed
More informationR = µ + Bf Arbitrage Pricing Model, APM
4.2 Arbitrage Pricing Model, APM Empirical evidence indicates that the CAPM beta does not completely explain the cross section of expected asset returns. This suggests that additional factors may be required.
More informationA Non-Parametric Approach of Heteroskedasticity Robust Estimation of Vector-Autoregressive (VAR) Models
Journal of Finance and Investment Analysis, vol.1, no.1, 2012, 55-67 ISSN: 2241-0988 (print version), 2241-0996 (online) International Scientific Press, 2012 A Non-Parametric Approach of Heteroskedasticity
More informationA 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 informationInterpreting 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 informationPanel Data. March 2, () Applied Economoetrics: Topic 6 March 2, / 43
Panel Data March 2, 212 () Applied Economoetrics: Topic March 2, 212 1 / 43 Overview Many economic applications involve panel data. Panel data has both cross-sectional and time series aspects. Regression
More informationUnit roots in vector time series. Scalar autoregression True model: y t 1 y t1 2 y t2 p y tp t Estimated model: y t c y t1 1 y t1 2 y t2
Unit roots in vector time series A. Vector autoregressions with unit roots Scalar autoregression True model: y t y t y t p y tp t Estimated model: y t c y t y t y t p y tp t Results: T j j is asymptotically
More informationHeteroskedasticity. We now consider the implications of relaxing the assumption that the conditional
Heteroskedasticity We now consider the implications of relaxing the assumption that the conditional variance V (u i x i ) = σ 2 is common to all observations i = 1,..., In many applications, we may suspect
More informationThe Linear Regression Model
The Linear Regression Model Carlo Favero Favero () The Linear Regression Model 1 / 67 OLS To illustrate how estimation can be performed to derive conditional expectations, consider the following general
More informationQuick 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 informationThe 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 informationECON 366: ECONOMETRICS II. SPRING TERM 2005: LAB EXERCISE #10 Nonspherical Errors Continued. Brief Suggested Solutions
DEPARTMENT OF ECONOMICS UNIVERSITY OF VICTORIA ECON 366: ECONOMETRICS II SPRING TERM 2005: LAB EXERCISE #10 Nonspherical Errors Continued Brief Suggested Solutions 1. In Lab 8 we considered the following
More informationEC408 Topics in Applied Econometrics. B Fingleton, Dept of Economics, Strathclyde University
EC408 Topics in Applied Econometrics B Fingleton, Dept of Economics, Strathclyde University Applied Econometrics What is spurious regression? How do we check for stochastic trends? Cointegration and Error
More informationMultivariate Time Series: VAR(p) Processes and Models
Multivariate Time Series: VAR(p) Processes and Models A VAR(p) model, for p > 0 is X t = φ 0 + Φ 1 X t 1 + + Φ p X t p + A t, where X t, φ 0, and X t i are k-vectors, Φ 1,..., Φ p are k k matrices, with
More informationLinear Regression with Time Series Data
Econometrics 2 Linear Regression with Time Series Data Heino Bohn Nielsen 1of21 Outline (1) The linear regression model, identification and estimation. (2) Assumptions and results: (a) Consistency. (b)
More informationWe 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 informationRepeated 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 informationAsymptotic distribution of GMM Estimator
Asymptotic distribution of GMM Estimator Eduardo Rossi University of Pavia Econometria finanziaria 2010 Rossi (2010) GMM 2010 1 / 45 Outline 1 Asymptotic Normality of the GMM Estimator 2 Long Run Covariance
More informationIntermediate Econometrics
Intermediate Econometrics Heteroskedasticity Text: Wooldridge, 8 July 17, 2011 Heteroskedasticity Assumption of homoskedasticity, Var(u i x i1,..., x ik ) = E(u 2 i x i1,..., x ik ) = σ 2. That is, the
More information1 Introduction to Generalized Least Squares
ECONOMICS 7344, Spring 2017 Bent E. Sørensen April 12, 2017 1 Introduction to Generalized Least Squares Consider the model Y = Xβ + ɛ, where the N K matrix of regressors X is fixed, independent of the
More informationLinear models. Linear models are computationally convenient and remain widely used in. applied econometric research
Linear models Linear models are computationally convenient and remain widely used in applied econometric research Our main focus in these lectures will be on single equation linear models of the form y
More informationEcon 836 Final Exam. 2 w N 2 u N 2. 2 v N
1) [4 points] Let Econ 836 Final Exam Y Xβ+ ε, X w+ u, w N w~ N(, σi ), u N u~ N(, σi ), ε N ε~ Nu ( γσ, I ), where X is a just one column. Let denote the OLS estimator, and define residuals e as e Y X.
More informationTime Series 2. Robert Almgren. Sept. 21, 2009
Time Series 2 Robert Almgren Sept. 21, 2009 This week we will talk about linear time series models: AR, MA, ARMA, ARIMA, etc. First we will talk about theory and after we will talk about fitting the models
More informationMultivariate Time Series Analysis and Its Applications [Tsay (2005), chapter 8]
1 Multivariate Time Series Analysis and Its Applications [Tsay (2005), chapter 8] Insights: Price movements in one market can spread easily and instantly to another market [economic globalization and internet
More informationLeast Squares Estimation-Finite-Sample Properties
Least Squares Estimation-Finite-Sample Properties Ping Yu School of Economics and Finance The University of Hong Kong Ping Yu (HKU) Finite-Sample 1 / 29 Terminology and Assumptions 1 Terminology and Assumptions
More informationThe Hansen Singleton analysis
The Hansen Singleton analysis November 15, 2018 1 Estimation of macroeconomic rational expectations model. The Hansen Singleton (1982) paper. We start by looking at the application of GMM that first showed
More informationGeneralized Method of Moments: I. Chapter 9, R. Davidson and J.G. MacKinnon, Econometric Theory and Methods, 2004, Oxford.
Generalized Method of Moments: I References Chapter 9, R. Davidson and J.G. MacKinnon, Econometric heory and Methods, 2004, Oxford. Chapter 5, B. E. Hansen, Econometrics, 2006. http://www.ssc.wisc.edu/~bhansen/notes/notes.htm
More informationEcon 423 Lecture Notes
Econ 423 Lecture Notes (hese notes are modified versions of lecture notes provided by Stock and Watson, 2007. hey are for instructional purposes only and are not to be distributed outside of the classroom.)
More informationSpecification 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 informationLinear Model Under General Variance
Linear Model Under General Variance We have a sample of T random variables y 1, y 2,, y T, satisfying the linear model Y = X β + e, where Y = (y 1,, y T )' is a (T 1) vector of random variables, X = (T
More information1 The Multiple Regression Model: Freeing Up the Classical Assumptions
1 The Multiple Regression Model: Freeing Up the Classical Assumptions Some or all of classical assumptions were crucial for many of the derivations of the previous chapters. Derivation of the OLS estimator
More informationNonlinear 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 informationHeteroscedasticity and Autocorrelation
Heteroscedasticity and Autocorrelation Carlo Favero Favero () Heteroscedasticity and Autocorrelation 1 / 17 Heteroscedasticity, Autocorrelation, and the GLS estimator Let us reconsider the single equation
More informationLECTURE 13: TIME SERIES I
1 LECTURE 13: TIME SERIES I AUTOCORRELATION: Consider y = X + u where y is T 1, X is T K, is K 1 and u is T 1. We are using T and not N for sample size to emphasize that this is a time series. The natural
More information13. Time Series Analysis: Asymptotics Weakly Dependent and Random Walk Process. Strict Exogeneity
Outline: Further Issues in Using OLS with Time Series Data 13. Time Series Analysis: Asymptotics Weakly Dependent and Random Walk Process I. Stationary and Weakly Dependent Time Series III. Highly Persistent
More informationGMM estimation is an alternative to the likelihood principle and it has been
GENERALIZEDMEHODOF MOMENS ESIMAION Econometrics 2 Heino Bohn Nielsen November 22, 2005 GMM estimation is an alternative to the likelihood principle and it has been widely used the last 20 years. his note
More informationLecture 4: Testing Stuff
Lecture 4: esting Stuff. esting Hypotheses usually has three steps a. First specify a Null Hypothesis, usually denoted, which describes a model of H 0 interest. Usually, we express H 0 as a restricted
More informationIris Wang.
Chapter 10: Multicollinearity Iris Wang iris.wang@kau.se Econometric problems Multicollinearity What does it mean? A high degree of correlation amongst the explanatory variables What are its consequences?
More informationLinear Regression with 1 Regressor. Introduction to Econometrics Spring 2012 Ken Simons
Linear Regression with 1 Regressor Introduction to Econometrics Spring 2012 Ken Simons Linear Regression with 1 Regressor 1. The regression equation 2. Estimating the equation 3. Assumptions required for
More informationEconometrics I KS. Module 2: Multivariate Linear Regression. Alexander Ahammer. This version: April 16, 2018
Econometrics I KS Module 2: Multivariate Linear Regression Alexander Ahammer Department of Economics Johannes Kepler University of Linz This version: April 16, 2018 Alexander Ahammer (JKU) Module 2: Multivariate
More informationMFE Financial Econometrics 2018 Final Exam Model Solutions
MFE Financial Econometrics 2018 Final Exam Model Solutions Tuesday 12 th March, 2019 1. If (X, ε) N (0, I 2 ) what is the distribution of Y = µ + β X + ε? Y N ( µ, β 2 + 1 ) 2. What is the Cramer-Rao lower
More informationFinancial Econometrics
Financial Econometrics Multivariate Time Series Analysis: VAR Gerald P. Dwyer Trinity College, Dublin January 2013 GPD (TCD) VAR 01/13 1 / 25 Structural equations Suppose have simultaneous system for supply
More informationEcon 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 informationSingle 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 informationThis 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 informationSpring 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 informationThe Statistical Property of Ordinary Least Squares
The Statistical Property of Ordinary Least Squares The linear equation, on which we apply the OLS is y t = X t β + u t Then, as we have derived, the OLS estimator is ˆβ = [ X T X] 1 X T y Then, substituting
More information1. How can you tell if there is serial correlation? 2. AR to model serial correlation. 3. Ignoring serial correlation. 4. GLS. 5. Projects.
1. How can you tell if there is serial correlation? 2. AR to model serial correlation. 3. Ignoring serial correlation. 4. GLS. 5. Projects. 1) Identifying serial correlation. Plot Y t versus Y t 1. See
More informationFinQuiz Notes
Reading 10 Multiple Regression and Issues in Regression Analysis 2. MULTIPLE LINEAR REGRESSION Multiple linear regression is a method used to model the linear relationship between a dependent variable
More informationECON3327: Financial Econometrics, Spring 2016
ECON3327: Financial Econometrics, Spring 2016 Wooldridge, Introductory Econometrics (5th ed, 2012) Chapter 11: OLS with time series data Stationary and weakly dependent time series The notion of a stationary
More informationEcon 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 informationFollow links for Class Use and other Permissions. For more information send to:
COPYRIGH NOICE: Kenneth. Singleton: Empirical Dynamic Asset Pricing is published by Princeton University Press and copyrighted, 00, by Princeton University Press. All rights reserved. No part of this book
More informationSpatial Regression. 11. Spatial Two Stage Least Squares. Luc Anselin. Copyright 2017 by Luc Anselin, All Rights Reserved
Spatial Regression 11. Spatial Two Stage Least Squares Luc Anselin http://spatial.uchicago.edu 1 endogeneity and instruments spatial 2SLS best and optimal estimators HAC standard errors 2 Endogeneity and
More informationSingle Equation Linear GMM with Serially Correlated Moment Conditions
Single Equation Linear GMM with Serially Correlated Moment Conditions Eric Zivot November 2, 2011 Univariate Time Series Let {y t } be an ergodic-stationary time series with E[y t ]=μ and var(y t )
More informationTESTING FOR CO-INTEGRATION
Bo Sjö 2010-12-05 TESTING FOR CO-INTEGRATION To be used in combination with Sjö (2008) Testing for Unit Roots and Cointegration A Guide. Instructions: Use the Johansen method to test for Purchasing Power
More information