slides_Ch5_W

denote the standard normal cumulative distribution

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Unformatted text preview: e derived the sampling distribution of the OLS estimator under LR.6. What if LR.6 does not hold? Melissa Tartari (Yale) Econometrics 11 / 27 Asymptotic Normality: A Review I Let Φ (.) denote the standard normal cumulative distribution fnc. Melissa Tartari (Yale) Econometrics 12 / 27 Asymptotic Normality: A Review I Let Φ (.) denote the standard normal cumulative distribution fnc. Let fZn jn = 1, ...g be a sequence of RVs such that for any scalar z lim Pr ob (Zn n !∞ Melissa Tartari (Yale) Econometrics z ) = Φ (z ) 12 / 27 Asymptotic Normality: A Review I Let Φ (.) denote the standard normal cumulative distribution fnc. Let fZn jn = 1, ...g be a sequence of RVs such that for any scalar z lim Pr ob (Zn n !∞ z ) = Φ (z ) Then Zn is said to have an asymptotic standard normal distribution and we write a Zn N (0, 1) Melissa Tartari (Yale) Econometrics 12 / 27 Asymptotic Normality: A Review I Let Φ (.) denote the standard normal cumulative distribution fnc. Let fZn jn = 1, ...g be a sequence of RVs such that for any scalar z lim Pr ob (Zn n !∞ z ) = Φ (z ) Then Zn is said to have an asymptotic standard normal distribution and we write a Zn N (0, 1) a If Zn N (0, 1) we may approximate probabilities concerning Zn with standard normal probabilities. Melissa Tartari (Yale) Econometrics 12 / 27 Asymptotic Normality: (trivial) Examples Let fZn jn = 1, ...g be such that Zn N (0, 1) for all n; then it is easy a to verify that Zn N (0, 1). Indeed, the limit of a constant sequence i.e. fΦ (z ) , Φ (z ) , ...g is the constant value Φ (z ): lim Pr ob (Zn n !∞ Melissa Tartari (Yale) z ) = lim Φ (z ) = Φ (z ) n !∞ Econometrics 13 / 27 Asymptotic Normality: (trivial) Examples Let fZn jn = 1, ...g be such that Zn N (0, 1) for all n; then it is easy a to verify that Zn N (0, 1). Indeed, the limit of a constant sequence i.e. fΦ (z ) , Φ (z ) , ...g is the constant value Φ (z ): lim Pr ob (Zn n !∞ z ) = lim Φ (z ) = Φ (z ) n !∞ Let fZn jn = 1, ...g be such that Zn N (µ, 1) for all n; then it is easy a to verify that Zn is not N (0, 1). Indeed, lim Pr ob (Zn n !∞ Melissa Tartari (Yale) z ) = lim Φ (z n !∞ Econometrics µ ) = Φ (z µ ) 6 = Φ (z ) . 13 / 27 Asymptotic Normality: (trivial) Examples Let fZn jn = 1, ...g be such that Zn N (0, 1) for all n; then it is easy a to verify that Zn N (0, 1). Indeed, the limit of a constant sequence i.e. fΦ (z ) , Φ (z ) , ...g is the constant value Φ (z ): lim Pr ob (Zn n !∞ z ) = lim Φ (z ) = Φ (z ) n !∞ Let fZn jn = 1, ...g be such that Zn N (µ, 1) for all n; then it is easy a to verify that Zn is not N (0, 1). Indeed, lim Pr ob (Zn n !∞ z ) = lim Φ (z n !∞ µ ) = Φ (z µ ) 6 = Φ (z ) . Let fZn jn = 1, ...g be such that Zn N (µ, 1) for all n; then it is easy a 0 to verify that Zn Zn µ N (0, 1). 0 Indeed,limn !∞ Pr ob (Zn z ) = limn !∞ Φ (z ) = Φ (z ). Melissa Tartari (Yale) Econometrics 13 / 27 Asymptotic Normality: the Central Limit Theorem I Let fY...
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This note was uploaded on 02/13/2014 for the course ECON 350 taught by Professor Donaldbrown during the Fall '10 term at Yale.

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