econometircs2

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lucia831124
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econometircs2
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2011-04-18 05:44:05
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econometrics2
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  1. Two relationships
    • Deterministic
    • Stochastic: y=f(x)+e
    • => e=disturbance, random error
  2. Why is there randomness?
    • 1. Measurement error
    • 2. We cannot observe all independent variables
  3. The Classical Multiple Linear Regression Model
    • y=x*beta+eps
    • y=E[y|x]+eps
  4. CMLRM assumptions (5+1)
    • 1. Linearity: dep.=linear fn. (indep. & disturbance term)
    • 2. X (n X k) has full rank = k. i.e. n>=k
    • 3. Exogeneity of the regressors = E[epsi|x]=0 <=> Cov[eps,x]=0
    • 4. Spherical Disturbances: E[eps*eps'|x]=sigma2*In
    • 4.1. Var[epsi|x]=sigma2: homoskedasticity
    • 4.2. Cov[epsi,epsj|x]=0: nonautocorrelation
    • 5. Indep. vari.s are not stochastic (fixed in repeated sample)
    • (6. Normality: eps|x ~ N(0,sigma2)
  5. How to estimate b? y=Xb+e
    • 1. Method of Moments
    • 2. Maximum Likelihood
    • 3. Least Squares
  6. Method of Moments
    population mean = sample mean
  7. Maximum Likelihood
    • based on Normality
    • Maximize the log-likelihood fn.
    • b_ml = b_mm
  8. Least Squares
    • min. e'e
    • b_ls=b_ml=b_mm (in the CMLR)=inv(x'x)x'y
  9. CLRM: residual maker matrix
    e=[I-x(x'x)-1x']y
  10. CLRM: properties of residual maker matrix M
    • 1.My=e
    • 2.Mx=0
    • 3.Me=e
  11. CLRM: projection matrix
    • x*inv(x'x)x'
    • =>x*inv(x'x)x'y = xb = y_hat = y-e
  12. CLRM: properties of projection matrix P
    • 1.Py=y_hat
    • 2.Px=x
    • 3.Pe=0
  13. Simple vs. Multiple regression
    • simple: y=b0+b1x+e
    • multiple: y=b0+b1x1+...+bkxk+e
  14. Patitioned Regression
    • y=x1b1+x2b2+e
    • b1=inv(x1'x1)x1'(y-x2b2)
    • b2=inv(x2'x2)x2'(y-x1b1)
    • If x1'x2=0 (independent, orthogonal) then b1=inv(x1'x1)x1'y & b2=inv(x2'x2)x2'y
  15. Frisch-Waugh-Lovell Thm
    • In the linear LS regression of y or 2 sets of variables, x1 and x2, subvector b2 is a set of coefficients obtained when residuals from a regression of y on x1 alone (M1y) are regressed on a set of residuals from a regression of each colomn of x2 on x1 (M1x2)
    • : b2=inv((M1x2)'(M1x2))(M1x2)'(M1y)
  16. Coro1 of Frisch-Waugh-Lovell Thm
    Slopes in a multiple regression with a constant term are obtained by regressing deviation of y from its mean on deviation of x from their mean
  17. CLRM: Goodness of fit
    • SST = SSR + SSE
    • <=> Total sum of squares = regression sum of squares + error sum of squares
    • As SSR is higher, the model is better
  18. Coefficient of Determination
    • R2=SSR/SST
    • =1-SSE/SST
  19. 2 Problems of Coefficient of Determination
    • 1. More regressors => higher R2
    • 2. w/o constant => R2>1 or <0 possible
  20. Fixing more vari = higher coeffi. of determination
    Adjusted R2=1-[(SSE/n-k)/(SST/n-1)]
  21. b_ols (Small Sample Properties)
    • Unbiased
    • Efficient = BLUE by Gauss-Markov thm.
  22. Gauss-Markov thm
    In the CLRM with regressor matrix X, the LS estimator b is Best Linear Unbiased Estimator or the minimum variance (efficient) linear unbiased estimator of beta.. regardless of whether X is deterministic or stochastic
  23. s2_ols
    • Unbiased
    • => est. Var(b|x)=s2(x'x)-1
  24. b_ols (large sample property)
    • Consistent
    • Asymptotic efficiency (b_ols=b_ml; by Cramer Rao Lower Bound)
    • Asysmptotic dist (asy. normally dist)
    • => plim(x'x/n)=Q, then sqr(n)(b-beta) converges in distribution N(0,sigma2Q-1)
  25. s2_ols (large sample property)
    • Consistent
    • => Est.Asy.Var(b)=s2(x'x)-1
  26. OLS_Hypothesis testing: Z & t dist
    • (bk-betak)/sqr(sigma2(x'x)-1kk)~Z(0,1)
    • => same/sqr(s2(x'x)-1kk)~t(n-k)
  27. OLS: t-test interval
    Pr[-t(a/2)<=statistic<=t(a/2)]=1-a
  28. Type I error vs. Type II error
    • Type I error: incorrectly reject true H0
    • Type II error: incorrectly fail to reject (accept) false H0
    • (type I) a: level of significance
    • 1-a: confidence coefficient
    • (type II) 1-b: power of the test
  29. 2 Potential Problems of OLS
    • 1. Multicollinearity
    • 2. Missing observations
  30. How to handle Multicollinearity
    • 1. nothing if bi is significant
    • 2. Get more data
    • 3. Drop one of collinear vari.s
    • 4. Group collinear vari.s together
  31. How to handel missing obs.
    • 1. yn, xn: no problem
    • 2. ynt, xn: filling in for y is not a good idea
    • 3. yn, xnt
    • >> zero-order method: replace with x_bar
    • >> modified zero-order method: 2nd col. of x=0 if complete / x=1 if missing
    • >> another way: reg. x on y and x_hat replaced
  32. Type I error vs. Type II error
    • Type I error: incorrectly reject true H0
    • Type II error: incorrectly fail to reject false H0
  33. Inference & Test: Rb=q
    Wald test ~ Chi [J]

    cf. (n-k)s^2/sigma^2~Chi[n-k]
  34. If H0: b_k=beta_k (J=1)
    • F test[1,n-k] = t-test^2[n-k]
    • therefore, r.v.~F[1,n-k], then sqrt(r.v.)~t[n-k]
  35. Test unrestriced vs. restricted models
    F[J,n-k]
  36. If H0: all beta_k=0
    [R^2/(k-1)]/[(1-R^2)/(n-k)]~F[k-1,n-k]
  37. Large sample test (2)
    • 1. Asymptotic t-test: asymptotically, t->std. normal dist (Z(0,1))
    • 2. Asymptotic F-test: Asymptotically J*F~chi(J)
  38. Test non-linear restrictions
    Asymptotically, wald~chi(J)
  39. Measures of Accuracy of Prediction
    • 1. Root mean squared error
    • 2. Mean absolute error
    • 3. Theil U-statistic
  40. Regarding Accuracy of Prediction: compare y_hat & y_i.. however, what if we don't know y_i?
    Divide smaple into two groups, and use a group A to predict a group B, and compare the them as y_hat & y_i
  41. Binary variables
    • Dummies
    • 1. binary case
    • 2. several categories
    • 3. several groupings
    • 4. threshold effects
    • 5. interaction terms >> intercept dummies & interaction dummies (e.g. b1*x1+b2*x1*D)
  42. Structural Changes (coefficient)
    • compare two groups' parameters
    • stat~F(# of restrictions, d.f.)
    • e.g. s x's are different ~ F(s, n-k-s)
  43. Structural changes (variance)
    W=(b1-b2)'[Var(b1)+Var(b2)]^-1 (b1-b2)~Chi(J)
  44. Omit relevant vari.
    coefficient: Biased, but more efficient
  45. Include irrelevant vari.
    Coefficient: Unbiased, but less efficient
  46. Model building
    • 1. simple>>general
    • 2. general>>simple (recommended) since omission is worse than including irrelevant variables (<=> Kennedy's book)
  47. Model selection criteria (4)
    • 1. adj. R^2
    • 2. Akaike Info. criterion
    • 3. Bayesian (Schwarz) info. criterion
    • 4. Prediction criterion
  48. Choosing b/w nonnested models
    • 1. encompassing model
    • H0: y=xb+e
    • H1: y=zr+e
    • y=xb_bar+zr_bar+(x,z)d+e
    • F-test: b(or z)_bar=0 >> reject H0 or H1

    • 2. J-test
    • y=(1-lambda)xb+lambdazr+e
    • regress y on z, get r_hat, and regress y on x & zr_har >> get lambda_hat & test lambda=0
  49. When? Generalized Least Squares
    • 1. Heteroskedasticity
    • 2. Autocorrelation
    • >> violate the assumption of spherical disturbances of OLS
  50. b_ols in GLS cases: small sample property
    • 1. ubiased
    • 2. efficiency is not guaranteed
  51. b_ola in GLS cases: Asymtotic property
    • 1. consistent
    • 2. asy'ly normally dist.
    • 3. aymptotic efficiency (NO!)
  52. b_GLS (Sigma known), E(eps eps'|x)=sigma^2*Sigma
    • inv(Sigma)=pp'
    • then x*>>px, y*>>py, eps*>>peps
    • b_gls=inv(x*'x*)x*'y*
  53. Small sample property of b_gls (Sigma known)
    • 1. unbiased
    • 2. efficient (the same with OLS case, thus BLUE)
  54. sigma^2_gls
    • unbised
    • consistent
  55. b_gls: Asymptotic properties
    • 1. consistent
    • 2. asy'ly nomally dist.
    • 3. asy'ly efficient
  56. Sigma completely unknow
    • GLS impossible
    • 1. do OLS >> unbiaed estimator
    • 2. Est. Asy. var(b) >> White's Heteroskedasticity consistent estimator
  57. Sigma partially known: Feasible GLS >> procedure
    • 1. Reg. OLS
    • 2. Reg ei^2 = az+ui >> get a_hat >> Sigma_hat = Sigma(a_hat)
    • 3. b_FLS=inv[x' inv(Sigma_hat x)]x' inv(Sigma_hat) y
  58. Sigma partially known: MLE
    in the log-likelihood fn. inv(Sigma) = Matrix of inv(fn.(a))
  59. Sigma partially known >> FGLS, MLE, or GMM
  60. 4 tests for Heteroskedasticity
    • 1. eyeball test
    • 2. White's general test~Chi(p-1): all sigma^2 are same
    • 2. Goldfeld-Quandt test~F(n1-k,n2-k): two groups' sigma^2 are same
    • 4. Brewsch-Pagan (Godfrey LM test): LM stat.~Chi(p)
  61. Common reasons of Endogeneity (violate exogeneity=Cov(eps,xi) not 0)
    • measurement error
    • lagged dep. vari.
    • simultaneity
    • omitted vari.
  62. b_ols using in endogeneity case
    • 1. biased
    • 2. inconsistent
  63. small/large sample properties: b_iv=inv(z'x)z'y (instrumental variables) when L=K
    • 1. biased
    • 2. var-cov(estimator) is larger than that of OLS >> based on MSE criterion, OLS can be preferred
    • 3. consistent
    • 4. Asy'ly normally dist.
    • 5. Est. Asy. Var (b_iv) is also consistent
  64. properties of b_iv (L>K): regress z on x >> x_hat >> replace x with x_hat
    • 1. biased
    • 2. consistent
    • 3. Asy'ly normally dist.
    • 4. Asy. Var(b_iv)-Asy. Var(b_ols) >0

    • b_iv: biased & consistent, but less efficient
    • b_ols: biased & inconsistent
  65. Hausman test (general)
    • H0: plim(theta_hat-theta_tilde)=0
    • >> stat.=(theta_hat-theta_telde)'inv(V_H/n)(theta_hat-theta_telde)~Chi(# of theta's =parm.s)
    • where V_H=V(theta_hat)+V(theta_tilde)-2Cov(theta_hat, theta_tilde)

    • If theta_hat is efficient under H0, then Cov(.)=V(theta_hat)
    • Then H=(theta_hat-theta_telde)'inv[(V(theta_tilde)-V(theta_hat))/n](theta_hat-theta_telde)~Chi(# of theta's =parm.s)
  66. Hausman test (IV case)
    • H0: plim x'eps/n=0
    • H1: not 0 >> only iv is consistent
  67. Endogeneity test (2)
    • Hausman test
    • Wu test
  68. IV in GLS case
    • b_iv
    • biased
    • consistent
    • asy'ly normally dist. & asy. Var.(b_iv): Sigma apprears!
  69. Weak instrument problem
    z is correlated with x weakly
  70. Resulats of weak instrument (2)
    • 1. Var(b_iv) goes up
    • 2. in large samples, it'd be less consistent than b_ols
  71. 3 test of weak instrument
    • 1. R^2 measures
    • 2. Godfrey test
    • 3. F-statistic measures
  72. Alternatives to IV
    • 1. limited info. ML
    • y=xb; and x1=zr+u >> likelihood
    • 2. split sample IV
    • (y1,x1,z1) (y2,x2,z2)
    • get r_hat from a group 1, regress z1 on x1 >> predict x2_hat=z2*r_hat
    • reduce biasedness
  73. Test z'eps=0
    • 1. L=K.. we cannot test
    • 2. L>K
    • a. Sargan test
    • b. C-test

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