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Outline
Basic Econometrics in Transportation PrProblemoblem ooff EEstimationstimation Point Interval Estimation Methods Statistical Inference Statistical Properties Small sample Large sample Amir Samimi Hypothesis Testing
Civil Engineering Department Sharif University of Technology
Primary Sources: Basic Econometrics, by Gujarati and lecture notes of Professor Greene
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Estimation Point Estimation
The popu l ati on h as ch ar acterist ics: For sim pli city, assum e th at the re is o nly o ne u nknow n Mean, variance parameter (θ) in the PDF. Median Develop a function of the sample values such that: Percentiles
is a statistic, or an estimator and is a random variable. Get a slice (random sample) of the population. A particular numerical value of the estimator is known as an Hldbihlfi?How would you obtain the value of its parameters? estimate. This is known as the problem of estimation: is known as a point estimator because it provides a single Point estimation (point) estimate of θ. Interval estimation 4/22 5/22
Example Interval Estimation
LLetet An inintervalterval betweebetweenn two vavalueslues tthathat maymay includeinclude tthehe ttruerue θ. The key concept is the of probability distribution of an estimator.
is the sample mean, an estimator of the true mean value (μ). If in a specific case = 50, this provides an estimate of μ.
Will the interval contain the true value? The interval is a “Confidence Interval” The degree of certainty is the degree of confidence or level of significance.
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Example Estimation Methods
Suppose t hat t he d ist ribut io n o f he ig ht o f me n in a popu lat io n Least Squar es (L S) is normally distributed with mean = μ inches and σ = 2.5 A considerable time will be devoted to illustrate the LS method. inches. A sample of 100 men drawn randomly from this population had an average height of 67 inches. Establish a 95% Maximum Likelihood confidence interval for the mean height (= μ) in the population This method is of a broad application and will also be discussed as a whole. partially in the last lectures.
Method of moments (MOM) Is becoming more popular, but not at the introductory level. 8/22 9/22
Statistical Properties Small-Sample Properties
ThThee statstatisticsistics fallfall intointo two categocategories:ries: UUnbiasednessnbiasedness ˆ is an unbiased estimator of θ if E( ˆ) Small-sample, or finite-sample, properties Bias ˆ E( ˆ) Distribution of two estimators of θ: Large-sample, or asymptotic, properties.
Underlying both these sets of properties is the notion that an estimator has a probability distribution.
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Small-Sample Properties Small-Sample Properties
MinimuuVaacem Variance Linearity An estimator ˆ is said to be a linear estimator of θ if it is a linear function of the sample observations.
Example:
Best Linear Unbiased Estimator (BLUE). 12/22 13/22
Small-Sample Properties Large-Sample Properties
MinimMinimumum MeanMean-SquaSquarere-ErrErroror OOftenften aann estestimatorimator does nnotot satsatisfyisfy oonene oorr mmoreore ooff tthehe desdesirableirable . statistical properties in small samples. . But as the sample size increases indefinitely, the estimator possesses several desirable statistical properties.
Asymptotic Unbiasedness:
Sample variance of a random variable
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Large-Sample Properties Large-Sample Properties
Cons iste n cy Cons iste n cy ˆ is said to be a consistent estimator if it approaches the true value θ as the sample size gets larger and larger. A sufficient condition for consistency is that the bias and variance both tend to Unbiasedness and consistency are conceptually very much different. zero (or MSE tends to zero) as the sample size increases indefinitely.
If ˆ is a consistent estimator of θ and h(.) is a continuous function:
Note thhihat this property d oes not h hldfhold true of the expectati on operator E;
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Large-Sample Properties Hypothesis Testing
AAsymptoticsymptotic EffiEfficiencyciency AAssumessume X wwithith a knownknown PDF f(x; θ). The variance of the asymptotic distribution of ˆ is called its We obtain the point estimator ˆ from a random sample. asymptotic variance. Since the true θ is rarely known, we raise the question: If ˆ is consistent and its asymptotic variance is smaller than the Is the estimator ˆ is “compatible” with some hypothesized value of θ, say θ*? asymptotic variance of all other consistent estimators of θ, it is called asymptotically efficient. To test the null hypothesis, we use the samp le information to obtain what is known as the test statistic. Asymptotic Normality Use the confidence interval or test of significance approach to test the null ˆ An estimator is said to be asymptotically normally distributed if hypothesis. its sampling distribution tends to approach the normal distribution as the sample size n increases indefinitely.
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Example Confidence Interval Approach
From the p rev ious e xa mp le, w hic h was co nce rned w it h t he height (X) of men in a population:
Could the sample with a mean of 67, the test statistic, have come from the population with the mean value of 69? 20/22 21/22
Test of Significance Approach Errors
State tthehe nullnull hypothesishypothesis H0 aandnd tthehe aalternativelternative hypothesishypothesis H1
H0:μ = 69 and H1:μ ≠ 69 Select the test statistic X Determine the probability distribution of the test statistic . Choose t he leve l o f s ign ificance ( α) Type I error is likely to be more serious in practice . Using the probability distribution of the test statistic, establish The probability of a type I error is designated as α and is called a 100(1 − α)% confidence interval. Check if the value of the the level of significance. parameter lies in the acceptance region. The probability of not committing a type II error is called the power of the test.
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Homework 1
Stati sti cal Inf er en ce (Casell a an d B er ger , 2 001 )
1. Chapter 3, Problem 23 [20 points] 2. Chapter 4, Problem 1 [20 points] 3. Chapter 4, Problem 7 [15 points] 4. Chapter 5, Problem 18 (a,b, and d) [45 points]