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Outline Topics Parametric distributionsRandom sampling and sampling distribution of Y ¯ Law of large numbers and central limit theorem Applied Statistics for Economics3. Parametric Probability Distributions, Random Sampling, and the Law of Large Numbers SFC - juliohuato@gmail.com Spring 2012 SFC - juliohuato@gmail.com Applied Statistics for Economics 3. Parametric Probability Dis
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Outline Topics Parametric distributions Random sampling and sampling distribution of Y ¯ Law of large numbers and central limit theoremTopicsParametric distributions ¯Random sampling and sampling distribution of YLaw of large numbers and central limit theorem SFC - juliohuato@gmail.com Applied Statistics for Economics 3. Parametric Probability Dis
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Outline Topics Parametric distributions Random sampling and sampling distribution of Y ¯ Law of large numbers and central limit theoremTopics The topics for this chapter are: 1. The normal, chi-square, F , and t distributions 2. Random sampling and the distribution of the sample average 3. Large-sample approximations and laws of large numbers SFC - juliohuato@gmail.com Applied Statistics for Economics 3. Parametric Probability Dis
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Outline Topics Parametric distributions Random sampling and sampling distribution of Y ¯ Law of large numbers and central limit theoremParametric distributions The most widely used distributions in econometrics are the following: 1. Normal N(µ, σ 2 ) 2. Chi-squared χ2 m 3. Student tm 4. F distribution Fm,n SFC - juliohuato@gmail.com Applied Statistics for Economics 3. Parametric Probability Dis
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Outline Topics Parametric distributions Random sampling and sampling distribution of Y ¯ Law of large numbers and central limit theoremNormal distribution The normal distribution has the bell shape probability density. The normal density with mean µ and variance σ 2 is symmetric around its mean. It has approximately 68% of its probability mass between µ − σ and µ + σ; 95% between µ − 2σ and µ + 2σ; and 99.7% between µ − 3σ and µ + 3σ. The normal with mean µ and variance σ 2 is denoted as N(µ, σ 2 ). The standard normal distribution is the normal distribution with mean µ = 0 and variance σ 2 = 1. It’s denoted as N(0, 1). SFC - juliohuato@gmail.com Applied Statistics for Economics 3. Parametric Probability Dis
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Outline Topics Parametric distributions Random sampling and sampling distribution of Y ¯ Law of large numbers and central limit theoremNormal distribution Random variables with a standard normal distribution are denoted as Z . The standard normal cumulative distribution function is denoted by Φ: Pr(Z ≤ c) = Φ(c), where c is a constant. The textbook tables give you the values of the standard normal cumulative function. So does Excel. If you have a normally distributed r.v. Y and want to ﬁnd speciﬁc probabilities using the tables, standardize it ﬁrst: (Y − µ) Z= σ SFC - juliohuato@gmail.com Applied Statistics for Economics 3. Parametric Probability Dis
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Outline Topics Parametric distributions Random sampling and sampling distribution of Y ¯ Law of large numbers and central limit theoremNormal distribution Let Y ∼ N(µ, σ 2 ). Then Z = (Y − µ)/σ. Let c1 and c2 be two numbers such that c1 < c2 and let d1 = (c1 − µ)/σ and d2 = (c2 − µ)/σ. Then: Pr(Y ≤ c2 ) = Pr(Z ≤ d2 ) = Φ(d2 ) Pr(Y ≥ c1 ) = Pr(Z ≥ d1 ) = 1 − Φ(d1 ) Pr(c1 ≤ Y ≤ c2 ) = Pr(d1 ≤ Z ≤ d2 ) = Φ(d2 ) − Φ(d1 ) SFC - juliohuato@gmail.com Applied Statistics for Economics 3. Parametric Probability Dis
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Outline Topics Parametric distributions Random sampling and sampling distribution of Y ¯ Law of large numbers and central limit theoremMultivariate normal distribution The normal distribution generalized to many r.v.’s is called the multivariate normal. For two, X and Y , it’s called the bivariate normal. If X and Y have a bivariate normal distribution with covariance σXY , while a and b are constants, then aX + bY ∼ N(aµX + bµY , a2 σX + a2 σX + 2abσXY ). 2 2 Similarly, if n r.v.’s have a multivariate normal distribution, then: 1. any linear combination of these variables is normally distributed, 2. the marginal distribution of each of the variables is normal, and 3. the r.v.’s are independent if, also, their covariances are zero.1 1 We said before that if two r.v.’s are independent, then their covariance is zero. We also said the converse is not necessarily true. In the special case of a joint normal distribution, the converse is true. SFC - juliohuato@gmail.com Applied Statistics for Economics 3. Parametric Probability Dis
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Outline Topics Parametric distributions Random sampling and sampling distribution of Y ¯ Law of large numbers and central limit theoremChi-squared The chi-squared distribution is the distribution of the sum of m squared independent standard normal r.v.’s. This distribution depends on m (the ‘degrees of freedom’ of the distribution). Let Z1 , Z2 , Z3 be three independent standard normal r.v.’s Then 2 2 2 Z1 + Z2 + Z3 has a chi-squared distribution with 3 degrees of freedom. Formally and in general: (Z1 + · · · + Zm ) ∼ χ2 2 2 m SFC - juliohuato@gmail.com Applied Statistics for Economics 3. Parametric Probability Dis
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Outline Topics Parametric distributions Random sampling and sampling distribution of Y ¯ Law of large numbers and central limit theoremStudent t distribution The Student t distribution with m degrees of freedom is deﬁned as the distribution of the ratio of a standard normal variable, divided by the square root of an independently distributed chi-squared r.v. with m degrees of freedom divided by m. Let Z be a standard normal r.v., W a r.v. with a chi-squared distribution with m degrees of freedom, and Z and W are independently distributed. Then Z / W /m ∼ tm The t density function has a bell shape, similar to the normal. But when m is small (20 or less) the tails are fatter. With m > 30, the t is approximated well by the standard normal, and t∞ converges to the standard normal. SFC - juliohuato@gmail.com Applied Statistics for Economics 3. Parametric Probability Dis
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Outline Topics Parametric distributions Random sampling and sampling distribution of Y ¯ Law of large numbers and central limit theoremThe F distribution The F distribution with (m, n) d.f. is deﬁned as the distribution of the ratio of a chi-squared r.v. with m d.f., divided by m, to an independently distributed chi-squared r.v. with n d.f., divided by n. Let W be a chi-squared r.v. with m d.f., V a chi-squared r.v. with n d.f., where W and V are independently distributed. Then W /m ∼ Fm,n V /n When the d.f. of the denominator (n) increase indeﬁnitely, then the r.v. V approximates the mean of an inﬁnite number of chi-squared r.v.’s. And the mean of an inﬁnite number of chi-squared r.v.’s is 1, because the mean of a standard normal r.v. is 1. In other words, the Fm,∞ distribution of W /m converges to the χ2 distribution of W /m. V /n m SFC - juliohuato@gmail.com Applied Statistics for Economics 3. Parametric Probability Dis
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Outline Topics Parametric distributions Random sampling and sampling distribution of Y ¯ Law of large numbers and central limit theoremRandom sampling Virtually all the statistical and econometric procedures we’ll use involve averages of a sample of data. That’s why we need to characterize the distribution of sample averages. Random sampling is randomly drawing a sample from a larger population. The average of a sample is, therefore, a r.v. – because it depends on the particular sample used. Since it is a random variable, the average sample has a probability distribution (the sampling distribution). But before we talk about the average of a random sample, let’s say more about random sampling in general. SFC - juliohuato@gmail.com Applied Statistics for Economics 3. Parametric Probability Dis
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Outline Topics Parametric distributions Random sampling and sampling distribution of Y ¯ Law of large numbers and central limit theoremRandom sampling To say it diﬀerently, random sampling is the selection at random of n objects from a population such that each member of the population is equally likely to be included in the sample. Example: Suppose you record the length of your commute to school and the weather on a sample of days picked randomly. The population from which you draw your sample is all your commuting days. If you draw your sample randomly, each day of commute will have an equal chance to be picked. Since the choice of days is random, learning about the weather on a given sampled day won’t tell you anything about the length of commute on any other sample day. That is, the value of the commuting time on each sample day is an independently distributed r.v. Let the observations in the sample be Y1 , . . . , Yn . Because the days are picked randomly, the value of the r.v. on day i, Yi is itself random. If you pick diﬀerent days, you get diﬀerent values of Y . Because of random sampling, you can treat Yi as a r.v.: before it is sampled, Yi can have many possible values; after sampled, YApplied Statistics for Economics 3. Parametric Probability Dis SFC - juliohuato@gmail.com i has a speciﬁc value.
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Outline Topics Parametric distributions Random sampling and sampling distribution of Y ¯ Law of large numbers and central limit theoremi.i.d. Since Y1 , . . . , Yn are drawn randomly from the same population (e.g., commuting days), the marginal distribution of Yi is the same for each i = 1, . . . , n. And this marginal distribution is the marginal distribution of the population variable Y being sampled. When Yi has the same marginal distribution for i = 1, . . . , n, then Y1 , . . . , Yn are said to be identically distributed. And when Y1 , . . . , Yn are drawn from the same distribution and are independently distributed, they are said to be i.i.d. (independently and identically distributed). Formally: In a simple random sample, n objects are drawn at random from a population and each object is equally likely to be drawn. The value of the r.v. Y for the ith randomly drawn object is Yi . Since each object is equally likely to be drawn and the distribution of Yi is the same for all i, the r.v.’s Y1 , . . . , Yn are i.i.d.; that is the distribution of Yi is the same for all i = 1, . . . , n and Y1 is distributed independently of Y2 , . . . , Yn , etc. SFC - juliohuato@gmail.com Applied Statistics for Economics 3. Parametric Probability Dis
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Outline Topics Parametric distributions Random sampling and sampling distribution of Y ¯ Law of large numbers and central limit theoremSampling distribution of the sample average ¯ The sample average, Y , of the n observations Y1 , Y2 , . . . , Yn is: n ¯ 1 1 Y = (Y1 + Y2 + · · · + Yn ) = Yi n n i=1 By drawing a random sample, we ensure that the sample average is a r.v. Since the sample is random, each Yi is random. Since the n observations are random, their average is random. If we had drawn a diﬀerent sample, the Y ’s would have been diﬀerent and their average would have been ¯ diﬀerent. From sample to sample, the value of Y changes. ¯ Since Y is a r.v., it has a probability distribution. It is called the sampling distribution of Y : the probability of the possible values of Y ¯ that could be computed for diﬀerent possible samples Y1 , Y2 , . . . , Yn . The sample average and their sampling distributions play a key role in statistics. SFC - juliohuato@gmail.com Applied Statistics for Economics 3. Parametric Probability Dis
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Outline Topics Parametric distributions Random sampling and sampling distribution of Y ¯ Law of large numbers and central limit theorem ¯Mean of Y 2 Let the observations Y1 , Y2 , . . . , Yn be i.i.d. and µY and σY be the mean and variance of Yi . (All Yi have the same mean and variance since the observations are i.i.d. draws.) If n = 2, then mean of Y1 + Y2 is E (Y1 + Y2 ) = µY + µY = 2µY . Therefore, the mean of the sample average is E [ 1 (Y1 + Y2 )] = ( 1 )2µY = µY . In general, 2 2 n ¯ 1 E (Y ) = E (Yi ) = µY n i=1 Question: What’s the variance of (aX + bY )? SFC - juliohuato@gmail.com Applied Statistics for Economics 3. Parametric Probability Dis
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Outline Topics Parametric distributions Random sampling and sampling distribution of Y ¯ Law of large numbers and central limit theorem ¯Variance of Y We learned before that var(aX + bY ) = a2 σX + 2abσXY + b 2 σY . 2 2 2 With two i.i.d. draws (n = 2), var(Y1 + Y2 ) = 2σY . And var(Y ¯ ) = 1 σ2 . 2 Y Why does the covariance term drops out? For general n, since Y1 , Y2 , . . . , Yn are i.i.d. (Yi = Yj ) for i = j, so the cov(Y1 , Y2 ) = 0, n ¯ 1 var(Y ) = var Yi n i=1 2 2 σY σY = ¯ n The standard deviation: ¯ σY s.d.(Y ) = √ n SFC - juliohuato@gmail.com Applied Statistics for Economics 3. Parametric Probability Dis
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Outline Topics Parametric distributions Random sampling and sampling distribution of Y ¯ Law of large numbers and central limit theorem ¯Mean, variance, and s.d. of Y Just to summarize these results: ¯ E (Y ) = µY 2 ¯ σY var(Y ) = n σ ¯ ) = √Y s.d.(Y n Note: These results hold regardless of the distribution of Y . But if 2 ¯ Y1 , . . . , Yn are i.i.d. draws from Y ∼ N(µY , σY ), then E (Y ) = µY and var(Y ¯ ) = σ 2 /n. In other words, Y ∼ N(µY , σ 2 /n). ¯ Y Y Random sampling ensures that the observations are i.i.d. draws from the population r.v. SFC - juliohuato@gmail.com Applied Statistics for Economics 3. Parametric Probability Dis
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Outline Topics Parametric distributions Random sampling and sampling distribution of Y ¯ Law of large numbers and central limit theoremLaw of large numbers Sampling distributions are key in developing statistical and econometric procedures. That’s why it is important to understand, mathematically, the sampling distribution of Y . ¯ There are two approaches to characterizing the sampling distribution of ¯ Y : (1) the ‘exact’ approach and (2) the ‘approximate’ approach. The exact approach requires the mathematical derivation of a formula for the sampling distribution that holds for any value of n. The result is ¯ called the exact or ﬁnite-sample distribution of Y . As we learned, if Y ¯ is a normal r.v. and Y1 , . . . , Yn are i.i.d., then the exact distribution of Y 2 is normal with mean µY and variance σY /n. SFC - juliohuato@gmail.com Applied Statistics for Economics 3. Parametric Probability Dis
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Outline Topics Parametric distributions Random sampling and sampling distribution of Y ¯ Law of large numbers and central limit theoremLaw of large numbers What if Y is not a normal r.v.? Then, the derivation of the exact ¯ probability distribution of Y is very hard. That’s why we use the approximate or large-sample approach. The resulting sampling distribution is often called an asymptotic distribution (asymptotic means that the approximation becomes exact in the limit when n is very large). The beauty of this is that the approximations can be very accurate once the sample size goes over, say, n = 30. If we use really large samples (thousands or tens of thousands of observations), then we can comfortably rely on asymptotic distributions since they become adequate approximations to the exact sampling distributions. SFC - juliohuato@gmail.com Applied Statistics for Economics 3. Parametric Probability Dis
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Outline Topics Parametric distributions Random sampling and sampling distribution of Y ¯ Law of large numbers and central limit theoremLaw of large numbers In deriving asymptotic sampling distributions, we will invoke two strong mathematical facts: (1) the law of large numbers and (2) the central limit theorem. The law of large numbers says that if the observations in a sample Yi , i, . . . , n are i.i.d. with E (Yi ) = µY and if large outliers are unlikely (in 2 other words, if the variance of Yi is ﬁnite: var(Yi ) = σY < ∞), then Y ¯ converges in probability to µY . ¯ The sample average Y converges in probability to (or “is consistent ¯ for”) µY if the probability that Y is “close” to µY becomes arbitrarily close to one as n increases. (Usually, when statisticians say that a given sample average is consistent, they mean that the sample average converges in probability to the population average. In other words, they say that the higher n is, the closer the sample average gets to the population average. This concept is key in estimating the population average from a sample.) SFC - juliohuato@gmail.com Applied Statistics for Economics 3. Parametric Probability Dis
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Outline Topics Parametric distributions Random sampling and sampling distribution of Y ¯ Law of large numbers and central limit theoremCentral limit theorem If the observations in a sample Y1 , . . . , Yn are i.i.d. with E (Yi ) = µY and 2 2 var(Yi ) = σY , where 0 < σY < ∞, and regardless of the distribution of Yi , then as n increases indeﬁnitely (n → ∞) the distribution of Y ¯ becomes arbitrarily well approximated by a normal distribution with mean ¯ 2 2 E (Y ) = µY and variance σY = σY /n. ¯ ¯ 2 In other words, the distribution of (Y − µY )/σY (where σY = σY /n) ¯ ¯ 2 becomes arbitrarily well approximated by the standard normal distribution. SFC - juliohuato@gmail.com Applied Statistics for Economics 3. Parametric Probability Dis
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Outline Topics Parametric distributions Random sampling and sampling distribution of Y ¯ Law of large numbers and central limit theoremCentral limit theorem How large should n be for this approximation to normality to be good? It ¯ depends on the distribution of Yi . If Yi is normal, then Y is normal for any n (even if small). If Yi has a distribution very far from normal, then the approximation requires that n ≥ 30. For sure, when n ≥ 100, the ¯ distribution of Y should look pretty normal. ¯ Since the distribution of Y approaches the normal as n grows large, then ¯ Y is said to be asymptotically normally distributed. We’re ready for statistics! SFC - juliohuato@gmail.com Applied Statistics for Economics 3. Parametric Probability Dis
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