Introduction to Econometrics: Review of Statistics - Estimation and Hypothesis Testing - P, Study notes of Introduction to Econometrics

Download Introduction to Econometrics: Review of Statistics - Estimation and Hypothesis Testing - P and more Study notes Introduction to Econometrics in PDF only on Docsity! Introduction to Econometrics: Review of Statistics Introduction to Econometrics: Review of Statistics ECO 4305 Dr. Peter M. Summers Texas Tech University September 11, 2008 Introduction to Econometrics: Review of Statistics Estimation Estimating the population mean Recall that our goal is to learn about a population (eg, mean earnings of college grads) by examining a sample. We just saw that the sample average, Ȳ , is a consistent estimator of the population mean, µY ; that’s one argument for using Ȳ to learn about µY . Others are: I Ȳ is unbiased: E (Ȳ ) = µY I Ȳ is efficient: If Ỹ is any other linear, unbiased estimator of µY (eg, a weighted average), then var(Ỹ ) ≥ var(Ȳ ). Ȳ is BLUE. I Ȳ makes best use of the information in the sample I Ȳ is the least squares estimator: the estimator that minimizes the sum of squared errors made by predicting the value of Y . Introduction to Econometrics: Review of Statistics Hypothesis testing Confidence intervals Remember that if n is big, then the Central Limit Theorem says that the standardized sample average has a standard Normal distribution: Ȳ − µY σy −→p N(0, 1) Also recall what a standard Normal looks like (fig 2.5, 3.1): Introduction to Econometrics: Review of Statistics Hypothesis testing The Normal distribution (density) Copyright © 2003 by Pearson Education, Inc. 2-15 Introduction to Econometrics: Review of Statistics Hypothesis testing Constructing a (95%) confidence interval I If we knew σȲ = σY / √ n, we could calculate the region holding 95% of the probability for the standard Normal. I We don’t know σY , but can estimate it: I sample variance: s2Y = 1 n−1 ∑n i=1 ( Yi − Ȳ )2 I gives estimate of σ2Y I standard error: SE (Ȳ ) = sY / √ n I Then a 95% confidence interval for Ȳ is Ȳ ± 1.96× SE (Ȳ ) Introduction to Econometrics: Review of Statistics Hypothesis testing Confidence intervals and hypothesis tests I Our null hypothesis is that there’s no difference in real earnings between the two groups: H0 : µD = 0 I Construct a 95% confidence interval for Dahe : Dahe ± 1.96× SE (Dahe) I If this confidence interval doesn’t include 0, we reject the null hypothesis (H0) at the 5% level I “Dahe is statistically significantly different from 0 at the 5% level” I “There is a significant difference in real earnings between...” Introduction to Econometrics: Review of Statistics Hypothesis testing Confidence intervals and hypothesis tests I Average real earnings of high school grads... CIhs = ahehs ± 1.96× shs/ √ nhs = $13.81± 1.96× 6.73/ √ 4346 = $13.81± 0.20 I ... and college grads CIbs = $20.31± 1.96× 9.55/ √ 3640 = $20.31± 0.31 Introduction to Econometrics: Review of Statistics Hypothesis testing Confidence intervals and hypothesis tests I Now we want a 95% CI for the difference (Dahe): CI = Dahe ± 1.96× SE (Dahe) I Average of the difference is the difference of the averages Dahe = ahebs − ahehs I Standard error of difference (eq’n (3.19), p. 84) SE (Dahe) = √ s2bs nbs + s2hs nhs Introduction to Econometrics: Review of Statistics Hypothesis testing Hypothesis tests and p-values I Rather than (or in addition to) computing ap confidence interval, we can also compute a p-value. For example, suppose we reject H0 at the 5% level. Can we also reject it at 4%? 2%? 1%? I Suppose that H0 is true (there really is no difference between earnings of high school and college graduates). Testing this, we got a t-statistic of 34.49. What’s the probability of getting a statistic this large (in absolute value) by random chance if the null is true? Introduction to Econometrics: Review of Statistics Hypothesis testing Hypothesis tests and p-values I Let tact be the value of t that we got. The 2-sided probability of getting a value that large is p − value = Pr(t ≤ −tact) + Pr(t ≥ tact) = 2Φ(−|tact |) NB: typo, p. 74, eqn. 3.6 I Tests H0 : µ = µ0 against H1 : µ 6= µ0 I The one-sided p-value is p − value = Pr(t ≤ −tact) or Pr(t ≥ tact), depending on the “direction” I Tests H0 : µ = µ0 against H1 : µ > µ0 (or <) Introduction to Econometrics: Review of Statistics Hypothesis testing Example: problem 3.15 Sept. 2004: n = 755, 405 prefer GWB, 350 prefer JFK Oct. 2004: n = 756, 378 prefer GWB, 378 prefer JFK a 95% CI for GWB, Sept poll: p̂ = 405/755 = 0.536 SE (p̂) = √ p̂(1− p̂)/n = 0.0181 95%CI = p̂ ± 1.96SE (p̂) = 0.536± 0.036