Confidence Sets Based on Shrinkage Estimators
|
|
- Sharon Andrews
- 5 years ago
- Views:
Transcription
1 Confidence Sets Based on Shrinkage Estimators Mikkel Plagborg-Møller Harvard University June 2017
2 Shrinkage estimators in applied work ˆβ shrink = argmin β { ˆQ(β) + λc(β) } Shrinkage/penalized estimators popular in economics: Random effects. High-dimensional prediction. Smoothing jagged functions. Shiller (1973); Hodrick & Prescott (1981); Breitung & Roling (2015); Barnichon & Brownlees (2017) Estimating fixed effects. Chetty et al. (2014); Chamberlain (2016) Shrinking toward theory. Hansen (2016); Fessler & Kasy (2017) Shrinkage parameter λ often data-dependent. 2
3 Challenges of shrinkage inference How to calculate SEs for shrinkage estimators? With data-dependent shrinkage parameter λ, asy. distribution often discontinuous in true parameters. Example For finite-dim parameters, impossible to estimate CDF of ˆβ shrink uniformly consistently. Leeb & Pötscher (2005) Standard bootstrap typically doesn t work. Beran (2010) Applied researchers often just undersmooth (i.e., SE for usual point estimator). Not always valid. 3
4 This project Class of generalized ridge regression estimators: Vinod (1978) ˆβ M,W (λ) = argmin β R n { β ˆβ 2 W + λ Mβ 2}. Shrinkage parameter λ selected by unbiased risk estimate. Gaussian location model: ˆβ N n (β, Σ), known Σ. Conditional QLR test for linear hypothesis on β. Exact size. Conditional QLR confidence set by test inversion. Simulations show favorable average length/area of CSs. Uniform asymptotic validity even when data is non-gaussian. 4
5 Relationship to literature Large stats lit uses analytically convenient transformations and priors. Casella & Hwang (1982, 1984, 1987, 2012); Tseng & Brown (1997) My starting point: How to calculate SEs for given ridge estimator? Arbitrary correlation structure, arbitrary shrinkage hypothesis. CSs tied to (and always contain) meaningful point estimator. Tests/CSs have Empirical Bayes (random effects) interpretation. But I do not start from decision-theoretic first principles. Impossible to uniformly dominate expected volume of Wald ellipsoid for 1-D or 2-D problems. Stein (1962); Brown (1966); Joshi (1969) 5
6 Other related literature Shrinkage: Stein (1956); James & Stein (1961); Bock (1975); Oman (1982); Casella & Hwang (1987) Unbiased risk estimate: Mallows (1973); Stein (1973, 1981); Berger (1985); Claeskens & Hjort (2003); Hansen (2010) Asymptotics for shrinkage: Leeb & Pötscher (2005); Hansen (2016) Uniformity: Andrews, Cheng & Guggenberger (2011); McCloskey (2015) Post-regularization inference: Chernozhukov, Hansen & Spindler (2015) Conditional inference: Andrews & Mikusheva (2016) Adaptive confidence sets: Pratt (1961); Brown, Casella & Hwang (1995); Wasserman (2006); Armstrong & Kolesár (2016) 6
7 Outline 1 Shrinkage estimators and unbiased risk estimate 2 Testing 3 Confidence sets 4 Simulation study 5 Uniform asymptotic validity 6 Applications 7 Summary
8 Gaussian location model For now, consider finite-sample Gaussian location model β R n unknown. Σ symmetric p.d. and known. ˆβ N n (β, Σ). Will later consider asymptotic framework for which the Gaussian model is the right limit experiment. Plug in consistent estimator ˆΣ. 7
9 General shrinkage estimator class { ˆβ M,W (λ) = argmin β ˆβ 2 W + λ Mβ 2} = Θ M,W (λ) ˆβ, β R n Θ M,W (λ) = (I n + λw 1 M M) 1. M R m n, W R n n symmetric p.d. Example: M = Penalizes jaggedness R(n 2) n. Whittaker (1923); Shiller (1972); Hodrick & Prescott (1981); Wahba (1990) 8
10 8 6 response, basis points horizon, months y t : GZ excess bond premium. x t : high-freq. FFF shock. Controls: 2 lags of y t, x t, log(cpi), log(ip), 1yrTreas. Sample:
11 Projection shrinkage Shrinkage particularly tractable when W = I n and M = P R n n is orthogonal projection matrix: P = P = P 2. Projection shrinkage towards linear subspace span(i n P). Stein (1956); Oman (1982a,b); Bock (1985); Casella & Hwang (1987) ˆβ P (λ) = argmin { β ˆβ 2 + λ Pβ 2} β R n = λ P ˆβ + (I n P) ˆβ. Example: I n P = proj. matrix from regression onto basis functions. 10
12 5 response, basis points horizon, months y t : GZ excess bond premium. x t : high-freq. FFF shock. Controls: 2 lags of y t, x t, log(cpi), log(ip), 1yrTreas. Sample:
13 MSE risk criterion: Unbiased risk estimate R M,W (λ; β ) = E β Unbiased risk estimate (URE): ( ) ˆβ M,W (λ) β 2 W. Bias/var. Mallows (1973); Stein (1973, 1981); Berger (1985); Hansen (2010) ˆR M,W (λ) = ˆβ M,W (λ) ˆβ 2 W + 2 tr{w Θ M,W (λ)σ}. Define ˆλ M,W = argmin λ 0 ˆR M,W (λ). May equal. lim λ ˆβM,W (λ) well defined if M full rank or proj. 12
14 MSE risk criterion: Unbiased risk estimate R M,W (λ; β ) = E β Unbiased risk estimate (URE): ( ) ˆβ M,W (λ) β 2 W. Bias/var. Mallows (1973); Stein (1973, 1981); Berger (1985); Hansen (2010) ˆR M,W (λ) = ˆβ M,W (λ) ˆβ 2 W + 2 tr{w Θ M,W (λ)σ}. Define ˆλ M,W = argmin λ 0 ˆR M,W (λ). May equal. lim λ ˆβM,W (λ) well defined if M full rank or proj. Suff. cond. for unique minimum: All nonzero eig.val s of MW 1 M are equal (e.g., proj shrink). Assume a.s. unique min. for rest of talk. 12
15 1 estimated MSE, normalized ˆR P ( x 1 x ), x [0, 1) λ/(1+λ) y t : GZ excess bond premium. x t : high-freq. FFF shock. Controls: 2 lags of y t, x t, log(cpi), log(ip), 1yrTreas. Sample:
16 Optimal projection shrinkage For projection shrinkage, can minimize URE in closed form: ˆβ P (ˆλ P ) = ( 1 tr(σ ) P) P ˆβ 2 + P ˆβ + (I n P) ˆβ, James-Stein shrinkage towards linear subspace. Stein (1956); James & Stein (1961); Oman (1982a,b); Bock (1985) Σ P = PΣP. 14
17 Optimal projection shrinkage For projection shrinkage, can minimize URE in closed form: ˆβ P (ˆλ P ) = ( 1 tr(σ ) P) P ˆβ 2 + P ˆβ + (I n P) ˆβ, James-Stein shrinkage towards linear subspace. Stein (1956); James & Stein (1961); Oman (1982a,b); Bock (1985) Proposition (Hansen, 2016): If tr(σ P ) > 4ρ(Σ P ), E β Σ P = PΣP. ( ˆβ P (ˆλ P ) β 2) ( < E β ˆβ β 2) for all β. Necessary cond n: rk(p) > 4. E.g., if I n P is projection onto p basis functions, then need n > p
18 Outline 1 Shrinkage estimators and unbiased risk estimate 2 Testing 3 Confidence sets 4 Simulation study 5 Uniform asymptotic validity 6 Applications 7 Summary
19 Hypothesis testing in shrinkage applications R R r n full row rank. No UMP test exists. H 0 : Rβ = b, H 1 : Rβ b. Wald test is UMP unbiased (r = 1), UMP invariant, and admissible. If we re already using shrinkage point estimator, might want hypothesis test tied to this estimator as well. Obtain CS by inversion. My proposed test is biased+noninvariant, so may achieve higher power than usual Wald test for some DGPs. 15
20 Empirical Bayes quasi-likelihood ratio test Base hypothesis test on (negative) quasi-log-likelihood ˆL M,W (β) = β ˆβ 2 W + ˆλ M,W Mβ 2. Empirical Bayes (random effects) interpretation: β data N ( ˆβ M,W (ˆλ M,W ), (W + ˆλ M,W M M) 1). QLR test statistic of Rβ = b: min β : Rβ=b ˆL M,W (β) min ˆL M,W (β) β = R ˆβ M,W (ˆλ M,W ) b 2 (R(W +ˆλ M,W M M) 1 R ) 1 16
21 Null distribution impractical LR M,W (b) = R ˆβ M,W (ˆλ M,W ) b 2 (R(W +ˆλ M,W M M) 1 R ) 1 Assume Var(RZ MZ) nonsingular, Z N n (0, I n ). Then LR well defined even when ˆλ M,W =. Holds in many cases. If Var(RZ MZ) singular, can use ad hoc LR statistic LR M,W (b) = R ˆβ M,W (ˆλ M,W ) b 2 (RW 1 R ) 1. 17
22 Null distribution impractical LR M,W (b) = R ˆβ M,W (ˆλ M,W ) b 2 (R(W +ˆλ M,W M M) 1 R ) 1 Assume Var(RZ MZ) nonsingular, Z N n (0, I n ). Then LR well defined even when ˆλ M,W =. Holds in many cases. If Var(RZ MZ) singular, can use ad hoc LR statistic LR M,W (b) = R ˆβ M,W (ˆλ M,W ) b 2 (RW 1 R ) 1. Practical problem: Null distribution of LR statistic depends on Mβ. Solution: Condition on sufficient statistic for n r nuisance param s. 17
23 Sufficient statistic for nuisance parameters Define ζ = ΣR (RΣR ) 1 R n r and P = ζr R n n. Statistic ˆν = (I n P) ˆβ is S-ancillary wrt. Rβ : ˆβ ˆν F Rβ,Σ, ˆν F (In P)β,Σ. It would be uncontroversial to condition on ˆν in the absence of prior information linking Rβ and (I n P)β. In practice, the prior information Mβ 1 may not substantially constrain the relationship between Rβ and (I n P)β. Then conditioning wastes little information. Severini (1995) I condition on ˆν. Later: connection to Empirical Bayes HPD set. 18
24 Critical value by simulation Conditional QLR test rejects H 0 if LR M,W (b) > q 1 α,m,w (b, ˆν). Conditional critical value given ˆν = ν: q 1 α,m,w (b, ν) = quantile 1 α ( R β( λ; U) b 2 (R(W + λ(u)m M) 1 R ) 1 ), where U N r (b, RΣR ), β(λ; U) = Θ M,W (λ)(ζu + ν) for all λ 0, { } λ(u) = argmin β(λ; U) (ζu + ν) 2 W + 2 tr(w Θ M,W (λ)σ). λ 0 By design, conditional (and thus unconditional) size = 1 α. 19
25 Outline 1 Shrinkage estimators and unbiased risk estimate 2 Testing 3 Confidence sets 4 Simulation study 5 Uniform asymptotic validity 6 Applications 7 Summary
26 Confidence set by test inversion Invert CQLR test to obtain CS for b = Rβ : Ĉ M,W = { b R r : LR } M,W (b) q 1 α,m,w (b, ˆν). Do this by grid search. Simulate quantile at each point. Feasible in one or two dimensions (proj. shrinkage fast). Uniform band If M full rank or proj., can compute simple, finite upper bound on critical value. More Ĉ M,W contained in bounded ellipsoid centered at R ˆβ M,W (ˆλ M,W ). Limits grid search. 20
27 Properties of shrinkage confidence set 1 ĈM,W always contains shrinkage point estimate. 2 Generally not symmetric around point estimate. 3 Not always convex. 4 Converges a.s. to usual Wald ellipsoid as Mβ, M fixed. 5 Expected volume depends on β only through Mβ. Appears difficult to characterize expected volume. Even for projection shrinkage, conditional power of CQLR test depends on 6 parameters. 21
28 Empirical Bayes HPD set ˆL M,W (β) = β ˆβ 2 W + ˆλ M,W Mβ 2, β data N ( ˆβ M,W (ˆλ M,W ), (W + ˆλ M,W M M) 1). Empirical Bayes 1 α Highest Posterior Density set for Rβ : Ĉ EB = Doesn t control frequentist coverage. { b R r : LR } M,W (b) χ 2 r,1 α. Like shrinkage CS, but non-random critical value. 22
29 Minimum coverage discrepancy with EB HPD set Symmetric set difference: A B = (A B)\(A B). Proposition (following Andrews & Mikusheva, 2016) Let C be any similar confidence set for Rβ (like ĈM,W ): P β ( Rβ C ) = 1 α for all β R n. Then P β ( ) ( ) Rβ ĈM,W ĈEB P β Rβ C ĈEB for all β R n. Proof 23
30 Outline 1 Shrinkage estimators and unbiased risk estimate 2 Testing 3 Confidence sets 4 Simulation study 5 Uniform asymptotic validity 6 Applications 7 Summary
31 Illustration: bivariate shrinkage toward average Bivariate model, projection shrinkage toward average: Lindley (1962) ˆβ = ( ˆβ 1 ˆβ 2 ) e 1 ˆβ P (ˆλ P ) = ˆβ 1 + ˆβ 2 2 Parameter of interest: β 1. N 2 (( β 1 β 2 ), ( 1 ρ ρ 1 ( ) 2(1 ρ) + 1 ( ˆβ 1 ˆβ 2 ) 2 + )), ˆβ 1 ˆβ 2. 2 Both MSE of shrinkage estimator and expected length of shrinkage CI depend on DGP only through β 2 β 1 and ρ. 24
32 Illustration: bivariate shrinkage toward average RMSE avg. length of 90% CI = 0.0 = 0.3 =
33 Simulation study of confidence intervals β i = ˆβ N n (β, Σ), 1 i 1 n 1 if K = 0, sin 2πK(i 1) n 1 if K > 0, Σ ij = σ i σ j κ i j, σ i = σ 0 ( 1 + (i 1) ϕ 1 n 1 Consider projection shrinkage toward quadratic polynomial. Lower bound on expected length relative to Wald CI: Pratt (1961) ). (1 α)φ 1 (1 α) + (2π) 1/2 e 1 2 (Φ 1 (1 α)) 2 Φ 1 (1 α/2) for α =
34 Simulation study of confidence intervals MSE ˆβ(ˆλ) Length Ĉ n K κ σ 0 ϕ Tot 1st Mid 1st Mid MSE relative to ˆβ, average length relative to Wald. Conf. level = 90%. 1st = β 1, Mid = β 1+[n/2]. 27
35 Simulation study of 2-D confidence sets Same design, but now construct 2-D confidence set for (β 1, β 1+[n/2] ). Lower bound on expected area relative to Wald ellipse: Pratt (1961); Brown, Casella & Hwang (1995) 2 0 r Φ ( Φ 1 (1 α) r ) dr χ 2 1 α, for α =
36 Simulation study of 2-D confidence sets Area n K κ σ 0 ϕ Ĉ Ĉ adhoc Average area relative to Wald. Conf. level = 90%. 29
37 Takeaways from simulations Shrinkage CS works well when shrinkage point estimator works well. Shrinkage may be harmful when... 1 Mβ conveys little info about Rβ. 2 Mβ neither small nor large. 3 Correlations are high. 4 Variance of MLE of nuisance parameters large relative to variance of MLE of parameter of interest (e.g., small n). 30
38 Outline 1 Shrinkage estimators and unbiased risk estimate 2 Testing 3 Confidence sets 4 Simulation study 5 Uniform asymptotic validity 6 Applications 7 Summary
39 Uniform asymptotic size control CQLR test achieves uniform asymptotic size control, provided ˆβ is uniformly asy. normal, and ˆΣ is uniformly consistent for Σ. Uniform frequentist validity contrasts with other approaches. Undersmoothing: Pretend λ is small, ignore bias of shrinkage estimator as well as variability in λ. Switching rule: Use Wald SE if M ˆβ > c, otherwise use asymptotics under assumption Mβ = 0. Random effects: Treat random effects assumption as part of the DGP rather than just a prior. Size control wrt. random effects distribution. 31
40 Assumption: Preliminary estimator well-behaved Assumption Define S = {A S n + : c 1/ρ(A 1 ) ρ(a) c} for fixed c, c > 0. The distribution of the data F T for sample size T is indexed by three parameters β B R n, Σ S, and γ Γ. The estimators ( ˆβ, ˆΣ) R n S n + satisfy the following: For every sequence {β T, Σ T, γ T } T 1 B S Γ and every subsequence {k T } T 1 of {T } T 1, there exists a further subsequence { k T } T 1 such that k T ˆΣ 1/2 ( ˆβ β kt ) d N n(0, I n ), F k T (β k T,Σ k T,γ k ) T (ˆΣ Σ kt ) p 0, as T. F k T (β k T,Σ k T,γ k ) T S n + = set of symmetric positive definite n n matrices. 32
41 Shrinkage test is uniformly valid Let LR and ˆq 1 α denote CQLR test statistic and quantile obtained by plugging in T 1 ˆΣ in place of Σ. (Suppress M, W.) Proposition Let the previous assumption hold. Assume either rk(m) = m or M = P. Assume also Var(RZ MZ) is nonsingular, Z N n (0, I n ). Then ( lim inf inf Prob F T T (β,σ,γ) LR(Rβ) ˆq 1 α (Rβ, ˆν)) = 1 α. (β,σ,γ) B S Γ 33
42 Shrinkage test is uniformly valid Let LR and ˆq 1 α denote CQLR test statistic and quantile obtained by plugging in T 1 ˆΣ in place of Σ. (Suppress M, W.) Proposition Let the previous assumption hold. Assume either rk(m) = m or M = P. Assume also Var(RZ MZ) is nonsingular, Z N n (0, I n ). Then ( lim inf inf Prob F T T (β,σ,γ) LR(Rβ) ˆq 1 α (Rβ, ˆν)) = 1 α. (β,σ,γ) B S Γ Caveat: I have only written down the full proof for proj. shrinkage. I believe I have the arguments worked out for the general case. Proof idea: Consider drifting parameters β T... 1 If T Mβ T, we converge to non-shrinkage case. 2 If T Mβ T is bounded, we re in the Gaussian model in the limit. 33
43 Outline 1 Shrinkage estimators and unbiased risk estimate 2 Testing 3 Confidence sets 4 Simulation study 5 Uniform asymptotic validity 6 Applications 7 Summary
44 Treatment effect heterogeneity NSW job training experiment. Lalonde (1986); Dehejia & Wahba (1999) Outcome: earnings (absolute $) 3 years after treatment assignment. 297 treated, 425 control. Bin subjects by age decile subjects per bin. ˆβ R 10 : ATE estimate by bin. Projection shrinkage toward average ˆβ. 34
45 Treatment effect heterogeneity: confidence intervals Conf. level = 90%. Vertical axis = ATE ($), horizontal axis = age (years). 35
46 Treatment effect heterogeneity: 2-D confidence set ages ages 17- Conf. level = 90%. Axes = ATE ($). Ad hoc QLR statistic. 36
47 MIDAS forecasting Predict monthly PCE inflation using daily commodity prices, 1991:2 2017:2. MIDAS specification (lag lengths chosen by AIC): 6 25 p PCE,t = µ + γ l p PCE,t l + β j z t,j + ε t. l=1 j=1 z t,j : j-th daily observation of log Bloomberg commodity price index (BCOM) on or after 1st day of month t. ˆβ R 25 : least-squares estimator. Projection shrinkage toward straight line. Breitung & Roling (2015) 37
48 MIDAS forecasting: confidence intervals Conf. level = 90%. Vertical axis = inflation (log points), horizontal axis = lags (days). 38
49 MIDAS forecasting: 2-D confidence set Conf. level = 90%. Axes = inflation (log points). 39
50 Outline 1 Shrinkage estimators and unbiased risk estimate 2 Testing 3 Confidence sets 4 Simulation study 5 Uniform asymptotic validity 6 Applications 7 Summary
51 Summary Considered setting where generalized ridge regression point estimator is of interest: smoothing, shrinking toward average, etc. Proposed conditional QLR test based on same quasi-log-likelihood as shrinkage point estimator. Exact conditional size in Gaussian location model. Asymptotic uniform size control more generally. Shrinkage confidence set by test inversion. Contains shrinkage point estimate. Minimum coverage discrepancy w. EB HPD set among similar CSs. Computationally feasible in 1 2 dimensions. Proj. shrinkage fast. Promising simulation evidence. 40
52 Thank you
53 Non-standard asymptotics: example ˆβ N n (β, T 1 I n ) James-Stein estimator of β R n : ˆβ JS = ( 1 n 2 ) T ˆβ 2 ˆβ. If β 0: T ( ˆβ JS β ) d N n (0, I n ). If β = 0: ( T ( ˆβ JS β ) d 1 n 2 Z 2 ) Z, Z N n (0, I n ). Back 42
54 W = I n for simplicity. URE captures bias/variance tradeoff Risk decomposition: Claeskens & Hjort (2003) R M,In (λ) = tr { [I n Θ M,In (λ)] 2 β β } + tr { Θ M,In (λ) 2 Σ }. }{{}}{{} bias squared variance Unbiased estimate: β β = E( ˆβ ˆβ ) Σ. Plug in: R M,In (λ) = tr { [I n Θ M,In (λ)] 2 ( ˆβ ˆβ Σ) } + tr { Θ M,In (λ) 2 Σ } = ˆR M,In (λ) tr(σ). Back 43
55 Triangle inequality: Bound on critical value LR M,W (Rβ) R( ˆβ M,W (ˆλ M,W ) ˆβ) V (ˆλ) 1 + R( ˆβ β) V (ˆλ) 1. Let Z N n (0, W 1 ). For any β R n and A R n n symm. p.d., ( R(β ˆβ) 2 β ˆβ 2 V (ˆλ) 1 A ρ RA 1 R Var(RZ MZ) 1). Since ˆR M,W (ˆλ M,W ) ˆR M,W (0), { ˆβ M,W (ˆλ M,W ) ˆβ 2 W 2 tr MΣM (MW 1 M ) 1}. Under the null H 0 : Rβ = Rβ, R( ˆβ β) 2 (RΣR ) 1 χ 2 (r). Back 44
56 Uniform confidence band Supremum test statistic of H 0 : β i = β i, i = 1,..., n: ŜLR M,W (β) = sup i=1,...,n ˆβ i,m,w (ˆλ M,W ) β i e i (W + ˆλ M,W M M) 1. e i Simulate null critical value q 1 α,m,w (β) for any β. Simultaneous confidence band: rectangular envelope of inverted test. n C M,W = inf β i, sup β i. i=1 β : ŜLR(β) q 1 α (β) β : ŜLR(β) q 1 α (β) Computationally challenging. Can sample from band. Inoue & Kilian (2016) Back 45
57 Coverage discrepancy: proof sketch Proof is a confidence set reinterpretation of Andrews & Mikusheva (2016) result on conditional testing. =1 α ( ) { [ }}{ P β Rβ C ĈEB = E β 1(Rβ C) ] [ ] +E β 1(Rβ ĈEB) [ 2E β 1(Rβ C)1(Rβ ] ĈEB) 46
58 Coverage discrepancy: proof sketch Proof is a confidence set reinterpretation of Andrews & Mikusheva (2016) result on conditional testing. =1 α ( ) { [ }}{ P β Rβ C ĈEB = E β 1(Rβ C) ] [ ] +E β 1(Rβ ĈEB) [ 2E β 1(Rβ C)1(Rβ ] ĈEB) ( ) ( ) P β Rβ C ĈEB P β Rβ ĈM,W ĈEB [{ = 2E β 1(Rβ ĈM,W ) 1(Rβ C) } ] 1(Rβ ĈEB) [{ = 2E β 1(Rβ ĈM,W ) 1(Rβ C) } )] 1( LR M,W (Rβ ) χ 2 r,1 α 46
59 Similarity of C and completeness of the Gaussian family imply conditional similarity (like ĈM,W ): ( P β Rβ C ) ˆν = 1 α. By law of iterated expectations, [{ } ( )] 1(Rβ ĈM,W ) 1(Rβ C) 1 q 1 α,m,w (Rβ, ˆν) χ 2 r,1 α = 0. E β 47
60 Similarity of C and completeness of the Gaussian family imply conditional similarity (like ĈM,W ): ( P β Rβ C ) ˆν = 1 α. By law of iterated expectations, [{ } ( )] 1(Rβ ĈM,W ) 1(Rβ C) 1 q 1 α,m,w (Rβ, ˆν) χ 2 r,1 α = 0. E β ( ) ( ) P β Rβ C ĈEB P β Rβ ĈM,W ĈEB [ { = 2E β 1(Rβ ĈM,W ) 1(Rβ C) } { ) ( )} ] 1( LR M,W (Rβ ) χ 2 r,1 α 1 q 1 α,m,w (Rβ, ˆν) χ 2 r,1 α Variable inside the expectation is a.s. nonnegative by def n of ĈM,W. 47
61 Similarity of C and completeness of the Gaussian family imply conditional similarity (like ĈM,W ): ( P β Rβ C ) ˆν = 1 α. By law of iterated expectations, [{ } ( )] 1(Rβ ĈM,W ) 1(Rβ C) 1 q 1 α,m,w (Rβ, ˆν) χ 2 r,1 α = 0. E β ( ) ( ) P β Rβ C ĈEB P β Rβ ĈM,W ĈEB [ { = 2E β 1(Rβ ĈM,W ) 1(Rβ C) } { ) ( )} ] 1( LR M,W (Rβ ) χ 2 r,1 α 1 q 1 α,m,w (Rβ, ˆν) χ 2 r,1 α Variable inside the expectation is a.s. nonnegative by def n of ĈM,W. Crucial: EB set inverts same test stat., but non-random crit. val. Back 47
Confidence Sets Based on Shrinkage Estimators
Confidence Sets Based on Shrinkage Estimators Mikkel Plagborg-Møller April 12, 2017 Shrinkage estimators in applied work { } ˆβ shrink = argmin β ˆQ(β) + λc(β) Shrinkage/penalized estimators popular in
More informationPart III. A Decision-Theoretic Approach and Bayesian testing
Part III A Decision-Theoretic Approach and Bayesian testing 1 Chapter 10 Bayesian Inference as a Decision Problem The decision-theoretic framework starts with the following situation. We would like to
More informationThis model of the conditional expectation is linear in the parameters. A more practical and relaxed attitude towards linear regression is to say that
Linear Regression For (X, Y ) a pair of random variables with values in R p R we assume that E(Y X) = β 0 + with β R p+1. p X j β j = (1, X T )β j=1 This model of the conditional expectation is linear
More informationMachine learning, shrinkage estimation, and economic theory
Machine learning, shrinkage estimation, and economic theory Maximilian Kasy December 14, 2018 1 / 43 Introduction Recent years saw a boom of machine learning methods. Impressive advances in domains such
More informationLecture 20 May 18, Empirical Bayes Interpretation [Efron & Morris 1973]
Stats 300C: Theory of Statistics Spring 2018 Lecture 20 May 18, 2018 Prof. Emmanuel Candes Scribe: Will Fithian and E. Candes 1 Outline 1. Stein s Phenomenon 2. Empirical Bayes Interpretation of James-Stein
More informationHabilitationsvortrag: Machine learning, shrinkage estimation, and economic theory
Habilitationsvortrag: Machine learning, shrinkage estimation, and economic theory Maximilian Kasy May 25, 218 1 / 27 Introduction Recent years saw a boom of machine learning methods. Impressive advances
More informationfinite-sample optimal estimation and inference on average treatment effects under unconfoundedness
finite-sample optimal estimation and inference on average treatment effects under unconfoundedness Timothy Armstrong (Yale University) Michal Kolesár (Princeton University) September 2017 Introduction
More informationFixed Effects, Invariance, and Spatial Variation in Intergenerational Mobility
American Economic Review: Papers & Proceedings 2016, 106(5): 400 404 http://dx.doi.org/10.1257/aer.p20161082 Fixed Effects, Invariance, and Spatial Variation in Intergenerational Mobility By Gary Chamberlain*
More informationSTA 732: Inference. Notes 10. Parameter Estimation from a Decision Theoretic Angle. Other resources
STA 732: Inference Notes 10. Parameter Estimation from a Decision Theoretic Angle Other resources 1 Statistical rules, loss and risk We saw that a major focus of classical statistics is comparing various
More informationData Mining Stat 588
Data Mining Stat 588 Lecture 02: Linear Methods for Regression Department of Statistics & Biostatistics Rutgers University September 13 2011 Regression Problem Quantitative generic output variable Y. Generic
More informationLong-Run Covariability
Long-Run Covariability Ulrich K. Müller and Mark W. Watson Princeton University October 2016 Motivation Study the long-run covariability/relationship between economic variables great ratios, long-run Phillips
More information401 Review. 6. Power analysis for one/two-sample hypothesis tests and for correlation analysis.
401 Review Major topics of the course 1. Univariate analysis 2. Bivariate analysis 3. Simple linear regression 4. Linear algebra 5. Multiple regression analysis Major analysis methods 1. Graphical analysis
More informationStatistics 203: Introduction to Regression and Analysis of Variance Penalized models
Statistics 203: Introduction to Regression and Analysis of Variance Penalized models Jonathan Taylor - p. 1/15 Today s class Bias-Variance tradeoff. Penalized regression. Cross-validation. - p. 2/15 Bias-variance
More informationLECTURE 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 informationThe 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 informationModel comparison and selection
BS2 Statistical Inference, Lectures 9 and 10, Hilary Term 2008 March 2, 2008 Hypothesis testing Consider two alternative models M 1 = {f (x; θ), θ Θ 1 } and M 2 = {f (x; θ), θ Θ 2 } for a sample (X = x)
More informationStatistical Inference
Statistical Inference Liu Yang Florida State University October 27, 2016 Liu Yang, Libo Wang (Florida State University) Statistical Inference October 27, 2016 1 / 27 Outline The Bayesian Lasso Trevor Park
More informationHypothesis Testing. 1 Definitions of test statistics. CB: chapter 8; section 10.3
Hypothesis Testing CB: chapter 8; section 0.3 Hypothesis: statement about an unknown population parameter Examples: The average age of males in Sweden is 7. (statement about population mean) The lowest
More informationRegression, Ridge Regression, Lasso
Regression, Ridge Regression, Lasso Fabio G. Cozman - fgcozman@usp.br October 2, 2018 A general definition Regression studies the relationship between a response variable Y and covariates X 1,..., X n.
More informationAveraging Estimators for Regressions with a Possible Structural Break
Averaging Estimators for Regressions with a Possible Structural Break Bruce E. Hansen University of Wisconsin y www.ssc.wisc.edu/~bhansen September 2007 Preliminary Abstract This paper investigates selection
More informationUnderstanding 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 informationMachine Learning for OR & FE
Machine Learning for OR & FE Regression II: Regularization and Shrinkage Methods Martin Haugh Department of Industrial Engineering and Operations Research Columbia University Email: martin.b.haugh@gmail.com
More informationLeast Squares Model Averaging. Bruce E. Hansen University of Wisconsin. January 2006 Revised: August 2006
Least Squares Model Averaging Bruce E. Hansen University of Wisconsin January 2006 Revised: August 2006 Introduction This paper developes a model averaging estimator for linear regression. Model averaging
More informationSupplement to Quantile-Based Nonparametric Inference for First-Price Auctions
Supplement to Quantile-Based Nonparametric Inference for First-Price Auctions Vadim Marmer University of British Columbia Artyom Shneyerov CIRANO, CIREQ, and Concordia University August 30, 2010 Abstract
More informationCentral Bank of Chile October 29-31, 2013 Bruce Hansen (University of Wisconsin) Structural Breaks October 29-31, / 91. Bruce E.
Forecasting Lecture 3 Structural Breaks Central Bank of Chile October 29-31, 2013 Bruce Hansen (University of Wisconsin) Structural Breaks October 29-31, 2013 1 / 91 Bruce E. Hansen Organization Detection
More informationROBUST CONFIDENCE SETS IN THE PRESENCE OF WEAK INSTRUMENTS By Anna Mikusheva 1, MIT, Department of Economics. Abstract
ROBUST CONFIDENCE SETS IN THE PRESENCE OF WEAK INSTRUMENTS By Anna Mikusheva 1, MIT, Department of Economics Abstract This paper considers instrumental variable regression with a single endogenous variable
More informationLinear Model Selection and Regularization
Linear Model Selection and Regularization Recall the linear model Y = β 0 + β 1 X 1 + + β p X p + ɛ. In the lectures that follow, we consider some approaches for extending the linear model framework. In
More informationA more powerful subvector Anderson and Rubin test in linear instrumental variables regression. Patrik Guggenberger Pennsylvania State University
A more powerful subvector Anderson and Rubin test in linear instrumental variables regression Patrik Guggenberger Pennsylvania State University Joint work with Frank Kleibergen (University of Amsterdam)
More informationSimultaneous Confidence Bands: Theoretical Comparisons and Recommendations for Practice
Simultaneous Confidence Bands: Theoretical Comparisons and Recommendations for Practice PRELIMINARY AND INCOMPLETE José Luis Montiel Olea Columbia University montiel.olea@gmail.com Mikkel Plagborg-Møller
More informationRidge regression. Patrick Breheny. February 8. Penalized regression Ridge regression Bayesian interpretation
Patrick Breheny February 8 Patrick Breheny High-Dimensional Data Analysis (BIOS 7600) 1/27 Introduction Basic idea Standardization Large-scale testing is, of course, a big area and we could keep talking
More informationModel Selection and Geometry
Model Selection and Geometry Pascal Massart Université Paris-Sud, Orsay Leipzig, February Purpose of the talk! Concentration of measure plays a fundamental role in the theory of model selection! Model
More informationCointegrated VAR s. Eduardo Rossi University of Pavia. November Rossi Cointegrated VAR s Financial Econometrics / 56
Cointegrated VAR s Eduardo Rossi University of Pavia November 2013 Rossi Cointegrated VAR s Financial Econometrics - 2013 1 / 56 VAR y t = (y 1t,..., y nt ) is (n 1) vector. y t VAR(p): Φ(L)y t = ɛ t The
More informationoptimal inference in a class of nonparametric models
optimal inference in a class of nonparametric models Timothy Armstrong (Yale University) Michal Kolesár (Princeton University) September 2015 setup Interested in inference on linear functional Lf in regression
More informationg-priors for Linear Regression
Stat60: Bayesian Modeling and Inference Lecture Date: March 15, 010 g-priors for Linear Regression Lecturer: Michael I. Jordan Scribe: Andrew H. Chan 1 Linear regression and g-priors In the last lecture,
More informationOptimizing forecasts for inflation and interest rates by time-series model averaging
Optimizing forecasts for inflation and interest rates by time-series model averaging Presented at the ISF 2008, Nice 1 Introduction 2 The rival prediction models 3 Prediction horse race 4 Parametric bootstrap
More informationEfficient Shrinkage in Parametric Models
Efficient Shrinkage in Parametric Models Bruce E. Hansen University of Wisconsin September 2012 Revised: June 2015 Abstract This paper introduces shrinkage for general parametric models. We show how to
More informationδ -method and M-estimation
Econ 2110, fall 2016, Part IVb Asymptotic Theory: δ -method and M-estimation Maximilian Kasy Department of Economics, Harvard University 1 / 40 Example Suppose we estimate the average effect of class size
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 informationLinear Algebra Massoud Malek
CSUEB Linear Algebra Massoud Malek Inner Product and Normed Space In all that follows, the n n identity matrix is denoted by I n, the n n zero matrix by Z n, and the zero vector by θ n An inner product
More informationEconomics 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 informationFall 2017 STAT 532 Homework Peter Hoff. 1. Let P be a probability measure on a collection of sets A.
1. Let P be a probability measure on a collection of sets A. (a) For each n N, let H n be a set in A such that H n H n+1. Show that P (H n ) monotonically converges to P ( k=1 H k) as n. (b) For each n
More informationCross-Validation with Confidence
Cross-Validation with Confidence Jing Lei Department of Statistics, Carnegie Mellon University WHOA-PSI Workshop, St Louis, 2017 Quotes from Day 1 and Day 2 Good model or pure model? Occam s razor We really
More informationEcon 5150: Applied Econometrics Dynamic Demand Model Model Selection. Sung Y. Park CUHK
Econ 5150: Applied Econometrics Dynamic Demand Model Model Selection Sung Y. Park CUHK Simple dynamic models A typical simple model: y t = α 0 + α 1 y t 1 + α 2 y t 2 + x tβ 0 x t 1β 1 + u t, where y t
More informationA Very Brief Summary of Statistical Inference, and Examples
A Very Brief Summary of Statistical Inference, and Examples Trinity Term 2008 Prof. Gesine Reinert 1 Data x = x 1, x 2,..., x n, realisations of random variables X 1, X 2,..., X n with distribution (model)
More informationCross-Validation with Confidence
Cross-Validation with Confidence Jing Lei Department of Statistics, Carnegie Mellon University UMN Statistics Seminar, Mar 30, 2017 Overview Parameter est. Model selection Point est. MLE, M-est.,... Cross-validation
More informationTime Series and Forecasting Lecture 4 NonLinear Time Series
Time Series and Forecasting Lecture 4 NonLinear Time Series Bruce E. Hansen Summer School in Economics and Econometrics University of Crete July 23-27, 2012 Bruce Hansen (University of Wisconsin) Foundations
More informationIEOR 165 Lecture 7 1 Bias-Variance Tradeoff
IEOR 165 Lecture 7 Bias-Variance Tradeoff 1 Bias-Variance Tradeoff Consider the case of parametric regression with β R, and suppose we would like to analyze the error of the estimate ˆβ in comparison to
More informationMA 575 Linear Models: Cedric E. Ginestet, Boston University Non-parametric Inference, Polynomial Regression Week 9, Lecture 2
MA 575 Linear Models: Cedric E. Ginestet, Boston University Non-parametric Inference, Polynomial Regression Week 9, Lecture 2 1 Bootstrapped Bias and CIs Given a multiple regression model with mean and
More informationVALIDITY OF SUBSAMPLING AND PLUG-IN ASYMPTOTIC INFERENCE FOR PARAMETERS DEFINED BY MOMENT INEQUALITIES
Econometric Theory, 2009, Page 1 of 41. Printed in the United States of America. doi:10.1017/s0266466608090257 VALIDITY OF SUBSAMPLING AND PLUG-IN ASYMPTOTIC INFERENCE FOR PARAMETERS DEFINED BY MOMENT
More informationROBUST CONFIDENCE SETS IN THE PRESENCE OF WEAK INSTRUMENTS By Anna Mikusheva 1, MIT, Department of Economics. Abstract
ROBUST CONFIDENCE SETS IN THE PRESENCE OF WEAK INSTRUMENTS By Anna Mikusheva 1, MIT, Department of Economics Abstract This paper considers instrumental variable regression with a single endogenous variable
More informationBayesian Inference and the Parametric Bootstrap. Bradley Efron Stanford University
Bayesian Inference and the Parametric Bootstrap Bradley Efron Stanford University Importance Sampling for Bayes Posterior Distribution Newton and Raftery (1994 JRSS-B) Nonparametric Bootstrap: good choice
More informationEcon 2148, fall 2017 Gaussian process priors, reproducing kernel Hilbert spaces, and Splines
Econ 2148, fall 2017 Gaussian process priors, reproducing kernel Hilbert spaces, and Splines Maximilian Kasy Department of Economics, Harvard University 1 / 37 Agenda 6 equivalent representations of the
More informationApplied Econometrics (QEM)
Applied Econometrics (QEM) based on Prinicples of Econometrics Jakub Mućk Department of Quantitative Economics Jakub Mućk Applied Econometrics (QEM) Meeting #3 1 / 42 Outline 1 2 3 t-test P-value Linear
More informationLet us first identify some classes of hypotheses. simple versus simple. H 0 : θ = θ 0 versus H 1 : θ = θ 1. (1) one-sided
Let us first identify some classes of hypotheses. simple versus simple H 0 : θ = θ 0 versus H 1 : θ = θ 1. (1) one-sided H 0 : θ θ 0 versus H 1 : θ > θ 0. (2) two-sided; null on extremes H 0 : θ θ 1 or
More informationRegime switching models
Regime switching models Structural change and nonlinearities Matthieu Stigler Matthieu.Stigler at gmail.com April 30, 2009 Version 1.1 This document is released under the Creative Commons Attribution-Noncommercial
More informationLecture 8 Inequality Testing and Moment Inequality Models
Lecture 8 Inequality Testing and Moment Inequality Models Inequality Testing In the previous lecture, we discussed how to test the nonlinear hypothesis H 0 : h(θ 0 ) 0 when the sample information comes
More informationPrevious lecture. P-value based combination. Fixed vs random effects models. Meta vs. pooled- analysis. New random effects testing.
Previous lecture P-value based combination. Fixed vs random effects models. Meta vs. pooled- analysis. New random effects testing. Interaction Outline: Definition of interaction Additive versus multiplicative
More informationEcon 2148, spring 2019 Statistical decision theory
Econ 2148, spring 2019 Statistical decision theory Maximilian Kasy Department of Economics, Harvard University 1 / 53 Takeaways for this part of class 1. A general framework to think about what makes a
More informationAnalysis Methods for Supersaturated Design: Some Comparisons
Journal of Data Science 1(2003), 249-260 Analysis Methods for Supersaturated Design: Some Comparisons Runze Li 1 and Dennis K. J. Lin 2 The Pennsylvania State University Abstract: Supersaturated designs
More informationBayesian methods in economics and finance
1/26 Bayesian methods in economics and finance Linear regression: Bayesian model selection and sparsity priors Linear Regression 2/26 Linear regression Model for relationship between (several) independent
More informationHigh-dimensional regression with unknown variance
High-dimensional regression with unknown variance Christophe Giraud Ecole Polytechnique march 2012 Setting Gaussian regression with unknown variance: Y i = f i + ε i with ε i i.i.d. N (0, σ 2 ) f = (f
More informationConsistent high-dimensional Bayesian variable selection via penalized credible regions
Consistent high-dimensional Bayesian variable selection via penalized credible regions Howard Bondell bondell@stat.ncsu.edu Joint work with Brian Reich Howard Bondell p. 1 Outline High-Dimensional Variable
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 informationNonparametric Regression. Badr Missaoui
Badr Missaoui Outline Kernel and local polynomial regression. Penalized regression. We are given n pairs of observations (X 1, Y 1 ),...,(X n, Y n ) where Y i = r(x i ) + ε i, i = 1,..., n and r(x) = E(Y
More informationEcon 2140, spring 2018, Part IIa Statistical Decision Theory
Econ 2140, spring 2018, Part IIa Maximilian Kasy Department of Economics, Harvard University 1 / 35 Examples of decision problems Decide whether or not the hypothesis of no racial discrimination in job
More informationOPTIMAL INFERENCE IN A CLASS OF REGRESSION MODELS. Timothy B. Armstrong and Michal Kolesár. May 2016 Revised May 2017
OPTIMAL INFERENCE IN A CLASS OF REGRESSION MODELS By Timothy B. Armstrong and Michal Kolesár May 2016 Revised May 2017 COWLES FOUNDATION DISCUSSION PAPER NO. 2043R COWLES FOUNDATION FOR RESEARCH IN ECONOMICS
More informationTesting Statistical Hypotheses
E.L. Lehmann Joseph P. Romano Testing Statistical Hypotheses Third Edition 4y Springer Preface vii I Small-Sample Theory 1 1 The General Decision Problem 3 1.1 Statistical Inference and Statistical Decisions
More informationBIOS 312: Precision of Statistical Inference
and Power/Sample Size and Standard Errors BIOS 312: of Statistical Inference Chris Slaughter Department of Biostatistics, Vanderbilt University School of Medicine January 3, 2013 Outline Overview and Power/Sample
More informationSize Distortion and Modi cation of Classical Vuong Tests
Size Distortion and Modi cation of Classical Vuong Tests Xiaoxia Shi University of Wisconsin at Madison March 2011 X. Shi (UW-Mdsn) H 0 : LR = 0 IUPUI 1 / 30 Vuong Test (Vuong, 1989) Data fx i g n i=1.
More informationLecture 2: Statistical Decision Theory (Part I)
Lecture 2: Statistical Decision Theory (Part I) Hao Helen Zhang Hao Helen Zhang Lecture 2: Statistical Decision Theory (Part I) 1 / 35 Outline of This Note Part I: Statistics Decision Theory (from Statistical
More informationQuantile Regression for Panel/Longitudinal Data
Quantile Regression for Panel/Longitudinal Data Roger Koenker University of Illinois, Urbana-Champaign University of Minho 12-14 June 2017 y it 0 5 10 15 20 25 i = 1 i = 2 i = 3 0 2 4 6 8 Roger Koenker
More informationLikelihood Ratio Tests. that Certain Variance Components Are Zero. Ciprian M. Crainiceanu. Department of Statistical Science
1 Likelihood Ratio Tests that Certain Variance Components Are Zero Ciprian M. Crainiceanu Department of Statistical Science www.people.cornell.edu/pages/cmc59 Work done jointly with David Ruppert, School
More informationMultiscale Adaptive Inference on Conditional Moment Inequalities
Multiscale Adaptive Inference on Conditional Moment Inequalities Timothy B. Armstrong 1 Hock Peng Chan 2 1 Yale University 2 National University of Singapore June 2013 Conditional moment inequality models
More informationSome Curiosities Arising in Objective Bayesian Analysis
. Some Curiosities Arising in Objective Bayesian Analysis Jim Berger Duke University Statistical and Applied Mathematical Institute Yale University May 15, 2009 1 Three vignettes related to John s work
More informationLectures on Structural Change
Lectures on Structural Change Eric Zivot Department of Economics, University of Washington April5,2003 1 Overview of Testing for and Estimating Structural Change in Econometric Models 1. Day 1: Tests of
More informationWorking Paper Series. Selecting models with judgment. No 2188 / October Simone Manganelli
Working Paper Series Simone Manganelli Selecting models with judgment No 2188 / October 2018 Disclaimer: This paper should not be reported as representing the views of the European Central Bank (ECB).
More informationInference in Nonparametric Series Estimation with Data-Dependent Number of Series Terms
Inference in Nonparametric Series Estimation with Data-Dependent Number of Series Terms Byunghoon ang Department of Economics, University of Wisconsin-Madison First version December 9, 204; Revised November
More informationVector Auto-Regressive Models
Vector Auto-Regressive Models Laurent Ferrara 1 1 University of Paris Nanterre M2 Oct. 2018 Overview of the presentation 1. Vector Auto-Regressions Definition Estimation Testing 2. Impulse responses functions
More informationWhat s New in Econometrics. Lecture 13
What s New in Econometrics Lecture 13 Weak Instruments and Many Instruments Guido Imbens NBER Summer Institute, 2007 Outline 1. Introduction 2. Motivation 3. Weak Instruments 4. Many Weak) Instruments
More informationVAR Models and Applications
VAR Models and Applications Laurent Ferrara 1 1 University of Paris West M2 EIPMC Oct. 2016 Overview of the presentation 1. Vector Auto-Regressions Definition Estimation Testing 2. Impulse responses functions
More informationParameter estimation and forecasting. Cristiano Porciani AIfA, Uni-Bonn
Parameter estimation and forecasting Cristiano Porciani AIfA, Uni-Bonn Questions? C. Porciani Estimation & forecasting 2 Temperature fluctuations Variance at multipole l (angle ~180o/l) C. Porciani Estimation
More informationCarl N. Morris. University of Texas
EMPIRICAL BAYES: A FREQUENCY-BAYES COMPROMISE Carl N. Morris University of Texas Empirical Bayes research has expanded significantly since the ground-breaking paper (1956) of Herbert Robbins, and its province
More informationProgram Evaluation with High-Dimensional Data
Program Evaluation with High-Dimensional Data Alexandre Belloni Duke Victor Chernozhukov MIT Iván Fernández-Val BU Christian Hansen Booth ESWC 215 August 17, 215 Introduction Goal is to perform inference
More informationThreshold Autoregressions and NonLinear Autoregressions
Threshold Autoregressions and NonLinear Autoregressions Original Presentation: Central Bank of Chile October 29-31, 2013 Bruce Hansen (University of Wisconsin) Threshold Regression 1 / 47 Threshold Models
More informationWhy experimenters should not randomize, and what they should do instead
Why experimenters should not randomize, and what they should do instead Maximilian Kasy Department of Economics, Harvard University Maximilian Kasy (Harvard) Experimental design 1 / 42 project STAR Introduction
More informationLecture 3. Inference about multivariate normal distribution
Lecture 3. Inference about multivariate normal distribution 3.1 Point and Interval Estimation Let X 1,..., X n be i.i.d. N p (µ, Σ). We are interested in evaluation of the maximum likelihood estimates
More informationSome properties of Likelihood Ratio Tests in Linear Mixed Models
Some properties of Likelihood Ratio Tests in Linear Mixed Models Ciprian M. Crainiceanu David Ruppert Timothy J. Vogelsang September 19, 2003 Abstract We calculate the finite sample probability mass-at-zero
More informationLinear Models and Estimation by Least Squares
Linear Models and Estimation by Least Squares Jin-Lung Lin 1 Introduction Causal relation investigation lies in the heart of economics. Effect (Dependent variable) cause (Independent variable) Example:
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 informationLecture 8: Information Theory and Statistics
Lecture 8: Information Theory and Statistics Part II: Hypothesis Testing and I-Hsiang Wang Department of Electrical Engineering National Taiwan University ihwang@ntu.edu.tw December 23, 2015 1 / 50 I-Hsiang
More informationInstrumental Variables Estimation and Weak-Identification-Robust. Inference Based on a Conditional Quantile Restriction
Instrumental Variables Estimation and Weak-Identification-Robust Inference Based on a Conditional Quantile Restriction Vadim Marmer Department of Economics University of British Columbia vadim.marmer@gmail.com
More informationST5215: Advanced Statistical Theory
Department of Statistics & Applied Probability Wednesday, October 5, 2011 Lecture 13: Basic elements and notions in decision theory Basic elements X : a sample from a population P P Decision: an action
More informationEstimation under Ambiguity
Estimation under Ambiguity R. Giacomini (UCL), T. Kitagawa (UCL), H. Uhlig (Chicago) Giacomini, Kitagawa, Uhlig Ambiguity 1 / 33 Introduction Questions: How to perform posterior analysis (inference/decision)
More informationLecture 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 informationSTAT 200C: High-dimensional Statistics
STAT 200C: High-dimensional Statistics Arash A. Amini May 30, 2018 1 / 57 Table of Contents 1 Sparse linear models Basis Pursuit and restricted null space property Sufficient conditions for RNS 2 / 57
More informationSIMILAR-ON-THE-BOUNDARY TESTS FOR MOMENT INEQUALITIES EXIST, BUT HAVE POOR POWER. Donald W. K. Andrews. August 2011
SIMILAR-ON-THE-BOUNDARY TESTS FOR MOMENT INEQUALITIES EXIST, BUT HAVE POOR POWER By Donald W. K. Andrews August 2011 COWLES FOUNDATION DISCUSSION PAPER NO. 1815 COWLES FOUNDATION FOR RESEARCH IN ECONOMICS
More informationReview 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 informationBootstrapping high dimensional vector: interplay between dependence and dimensionality
Bootstrapping high dimensional vector: interplay between dependence and dimensionality Xianyang Zhang Joint work with Guang Cheng University of Missouri-Columbia LDHD: Transition Workshop, 2014 Xianyang
More informationImproved Inference for First Order Autocorrelation using Likelihood Analysis
Improved Inference for First Order Autocorrelation using Likelihood Analysis M. Rekkas Y. Sun A. Wong Abstract Testing for first-order autocorrelation in small samples using the standard asymptotic test
More informationRobust Backtesting Tests for Value-at-Risk Models
Robust Backtesting Tests for Value-at-Risk Models Jose Olmo City University London (joint work with Juan Carlos Escanciano, Indiana University) Far East and South Asia Meeting of the Econometric Society
More information