slides_Ch4_W[1]

# Bh the natural idea is to replace 2 with an estimator

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Unformatted text preview: estimator because sd βj jx is a function of the unknown σ2 . Melissa Tartari (Yale) Econometrics 12 / 41 The Sampling Distribution When sigma is not Known When σ2 is not known, ˆ βj ˆ sd [ β jx] j i h ˆ is not an estimator because sd βj jx is a function of the unknown σ2 . bh The natural idea is to replace σ2 with an estimator σ2 with good i i h ˆ ˆ jx with se β jx . properties and, accordingly, replace sd βj j Melissa Tartari (Yale) Econometrics 12 / 41 The Sampling Distribution When sigma is not Known When σ2 is not known, ˆ βj ˆ sd [ β jx] j i h ˆ is not an estimator because sd βj jx is a function of the unknown σ2 . bh The natural idea is to replace σ2 with an estimator σ2 with good i i h ˆ ˆ jx with se β jx . properties and, accordingly, replace sd βj j We introduced such an estimator in Chapter 3, see equation (3.58). Melissa Tartari (Yale) Econometrics 12 / 41 The Sampling Distribution When sigma is not Known When σ2 is not known, ˆ βj ˆ sd [ β jx] j i h ˆ is not an estimator because sd βj jx is a function of the unknown σ2 . bh The natural idea is to replace σ2 with an estimator σ2 with good i i h ˆ ˆ jx with se β jx . properties and, accordingly, replace sd βj j We introduced such an estimator in Chapter 3, see equation (3.58). We will next derive its samplying distribution: this is necessary for deriving the sampling distribution of the (feasible) estimator Melissa Tartari (Yale) Econometrics ˆ βj . ˆ se [ βj jx] 12 / 41 The Sampling Distribution When sigma is not Known Theorem 4.2: Under CLM assumptions, ˆ βj βj h i ˆ se βj jx Melissa Tartari (Yale) Econometrics tn K1 (4.3) 13 / 41 The Sampling Distribution When sigma is not Known Theorem 4.2: Under CLM assumptions, ˆ βj βj h i ˆ se βj jx tn K1 (4.3) QUESTION: what is the di¤erence between (4.1 bis) and (4.3) that motivates the di¤erent result (normal versus t-student)? Melissa Tartari (Yale) Econometrics 13 / 41 The Sampling Distribution When sigma is not Known Theorem 4.2: Under CLM assumptions, ˆ βj βj h i ˆ se βj jx tn K1 (4.3) QUESTION: what is the di¤erence between (4.1 bis) and (4.3) that motivates the di¤erent result (normal versus t-student)? h i ˆ ANSWER: in (4.1 bis) at the denominator we had sd βj jx while h i ˆ now in (4.3) we have se βj jx ; WHY? because the unknown σ h i ˆ ˆ appearing in sd βj jx has been substituted with the RV σ. Melissa Tartari (Yale) Econometrics 13 / 41 The Sampling Distribution (sigma not Known): “Proof” From Theorem 4.1, under LR.1-LR.6 and conditional on x, ˆ βj βj h i jx ˆ sd βj jx Melissa Tartari (Yale) Econometrics N (0, 1) (4.1 bis) 14 / 41 The Sampling Distribution (sigma not Known): “Proof” From Theorem 4.1, under LR.1-LR.6 and conditional on x, ˆ βj βj h i jx ˆ sd βj jx now write N (0, 1) (4.1 bis) h i ˆ ˆ ˆ βj βj β j β j sd β j j x z zj h i=r j h i= h i =q wj 2 β jx ˆ] se [ j ˆ ˆ ˆ se βj jx sd βj jx se βj jx nK 2 β jx sd [ ˆ j ] 1 where we let zj ˆ βj βj h i ˆ sd βj jx M elissa Tartari (Yale) and wj Econometrics (n K 1) dˆ Var βj jx ˆ Var βj jx 14 / 41 The Sampling Distribution (sigma not Known): “Proof” cont. From previous slide: r ˆ βj βj i = zj / h n ˆ se βj jx Melissa Tartari (Yale) Econometrics wj K 1 15 / 41 The Sampling Distribution (sigma not Known): “Proof” cont. From previous slide: We know that zj jx Melissa Tartari (Yale) r ˆ βj βj i = zj / h n ˆ se βj jx wj K 1 N (0, 1). Econometrics 15 / 41 The Sampling Distribution (sigma not Known): “Proof” cont. From previous slide: r ˆ βj βj i = zj / h n ˆ se βj jx We know that zj jx N (0, 1). We can prove that wj jx χ2n ( Melissa Tartari (Yale) wj K 1 K 1) : Econometrics 15 / 41 The Sampling Distribution (sigma not Known): “Proof” cont. From previous slide: r ˆ βj βj i = zj / h n ˆ se βj jx We know that zj jx N (0, 1). We can prove that wj jx χ2n ( wj K 1 K 1) : dˆ the intuition for this last result is that Var βj jx is ˆ σ2 = ∑n=1 ui2 /n K 1 i.e. to the (rescaled) sum of squares of ˆ i independent and identically distributed normal random variables. Melissa Tartari (Yale) Econometrics 15 / 41 The Sampling Distribution (sigma not Known): “Proof” cont. From previous slide: r ˆ βj βj i = zj / h n ˆ se βj jx We know that zj jx N (0, 1). We can prove that wj jx χ2n ( wj K 1 K 1) : dˆ the intuition for this last result is that Var βj jx is ˆ σ2 = ∑n=1 ui2 /n K 1 i.e. to the (rescaled) sum of squares of ˆ i independent and identically distributed normal random variables. We can show that zj jx and wj jx are independent RVs. Melissa Tartari (Yale) Econometrics 15 / 41 The Sampling Distribution (sigma not Known): “Proof” cont. From previous slide: r ˆ βj βj i = zj / h n ˆ se βj jx We know that zj jx N (0, 1). We can prove that wj jx χ2n ( wj K 1 K 1) : dˆ the intuition for this last result is that Var βj jx is ˆ σ2 = ∑n=1 ui2 /n K 1 i.e. to the (rescaled) sum of squares of ˆ i independent and identically distributed nor...
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