econometircs2

• Deterministic
• Stochastic: y=f(x)+e
• => e=disturbance, random error

• 1. Measurement error
• 2. We cannot observe all independent variables

• y=x*beta+eps
• y=E[y|x]+eps

• 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[eps_i|x]=0 <=> Cov[eps,x]=0
• 4. Spherical Disturbances: E[eps*eps'|x]=sigma²*I_n
• 4.1. Var[eps_i|x]=sigma2: homoskedasticity
• 4.2. Cov[eps_i,eps_j|x]=0: nonautocorrelation
• 5. Indep. vari.s are not stochastic (fixed in repeated sample)
• (6. Normality: eps|x ~ N(0,sigma²)

• 1. Method of Moments
• 2. Maximum Likelihood
• 3. Least Squares

population mean = sample mean

• based on Normality
• Maximize the log-likelihood fn.
• b_ml = b_mm

• min. e'e
• b_ls=b_ml=b_mm (in the CMLR)=inv(x'x)x'y

e=[I-x(x'x)^-1x']y

• 1.My=e
• 2.Mx=0
• 3.Me=e

• x*inv(x'x)x'
• =>x*inv(x'x)x'y = xb = y_hat = y-e

• 1.Py=y_hat
• 2.Px=x
• 3.Pe=0

• simple: y=b₀+b₁x+e
• multiple: y=b₀+b₁x₁+...+b_kx_k+e

• 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

• 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)

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

• SST = SSR + SSE
• <=> Total sum of squares = regression sum of squares + error sum of squares
• As SSR is higher, the model is better

• R²=SSR/SST
• =1-SSE/SST

• 1. More regressors => higher R²
• 2. w/o constant => R²>1 or <0 possible

Adjusted R2=1-[(SSE/n-k)/(SST/n-1)]

• Unbiased
• Efficient = BLUE by 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

• Unbiased
• => est. Var(b|x)=s²(x'x)^-1

• 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)

• Consistent
• => Est.Asy.Var(b)=s²(x'x)^-1

• (b_k-beta_k)/sqr(sigma²(x'x)^-1_kk)~Z(0,1)
• => same/sqr(s²(x'x)^-1_kk)~t(n-k)

Pr[-t_(a/2)<=statistic<=t_(a/2)]=1-a

• 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

• 1. Multicollinearity
• 2. Missing observations

• 1. nothing if bi is significant
• 2. Get more data
• 3. Drop one of collinear vari.s
• 4. Group collinear vari.s together

• 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

• Type I error: incorrectly reject true H0
• Type II error: incorrectly fail to reject false H0

Wald test ~ Chi [J]

cf. (n-k)s^2/sigma^2~Chi[n-k]

• F test[1,n-k] = t-test^2[n-k]
• therefore, r.v.~F[1,n-k], then sqrt(r.v.)~t[n-k]

F[J,n-k]

[R^2/(k-1)]/[(1-R^2)/(n-k)]~F[k-1,n-k]

• 1. Asymptotic t-test: asymptotically, t->std. normal dist (Z(0,1))
• 2. Asymptotic F-test: Asymptotically J*F~chi(J)

Asymptotically, wald~chi(J)

• 1. Root mean squared error
• 2. Mean absolute error
• 3. Theil U-statistic

Divide smaple into two groups, and use a group A to predict a group B, and compare the them as y_hat & y_i

• 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)

• compare two groups' parameters
• stat~F(# of restrictions, d.f.)
• e.g. s x's are different ~ F(s, n-k-s)

W=(b1-b2)'[Var(b1)+Var(b2)]^-1 (b1-b2)~Chi(J)

coefficient: Biased, but more efficient

Coefficient: Unbiased, but less efficient

• 1. simple>>general
• 2. general>>simple (recommended) since omission is worse than including irrelevant variables (<=> Kennedy's book)

• 1. adj. R^2
• 2. Akaike Info. criterion
• 3. Bayesian (Schwarz) info. criterion
• 4. Prediction criterion

• 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

• 1. Heteroskedasticity
• 2. Autocorrelation
• >> violate the assumption of spherical disturbances of OLS

• 1. ubiased
• 2. efficiency is not guaranteed

• 1. consistent
• 2. asy'ly normally dist.
• 3. aymptotic efficiency (NO!)

• inv(Sigma)=pp'
• then x*>>px, y*>>py, eps*>>peps
• b_gls=inv(x*'x*)x*'y*

• 1. unbiased
• 2. efficient (the same with OLS case, thus BLUE)

• unbised
• consistent

• 1. consistent
• 2. asy'ly nomally dist.
• 3. asy'ly efficient

• GLS impossible
• 1. do OLS >> unbiaed estimator
• 2. Est. Asy. var(b) >> White's Heteroskedasticity consistent estimator

• 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

in the log-likelihood fn. inv(Sigma) = Matrix of inv(fn.(a))

• 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)

• measurement error
• lagged dep. vari.
• simultaneity
• omitted vari.

• 1. biased
• 2. inconsistent

• 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

• 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

• 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)

• H0: plim x'eps/n=0
• H1: not 0 >> only iv is consistent

• Hausman test
• Wu test

• b_iv
• biased
• consistent
• asy'ly normally dist. & asy. Var.(b_iv): Sigma apprears!

z is correlated with x weakly

• 1. Var(b_iv) goes up
• 2. in large samples, it'd be less consistent than b_ols

• 1. R^2 measures
• 2. Godfrey test
• 3. F-statistic measures

• 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

• 1. L=K.. we cannot test
• 2. L>K
• a. Sargan test
• b. C-test