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### 494lec7bw

Course: STAT 494, Fall 2009
School: Washington
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Word Count: 1450

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494 1 POLS Lecture 7: Review of the Linear Regression Model (Part I) Kevin Quinn University of Washington 2 Outline Least Squares as a Fit Criterion Sampling Properties of the OLS estimator Importance of Assumptions The Least Squares Estimator as a Maximum Likelihood Estimator 3 Example: Relationship Between Household Income and Household Expenditures At each of 10 household income levels we take a...

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494 1 POLS Lecture 7: Review of the Linear Regression Model (Part I) Kevin Quinn University of Washington 2 Outline Least Squares as a Fit Criterion Sampling Properties of the OLS estimator Importance of Assumptions The Least Squares Estimator as a Maximum Likelihood Estimator 3 Example: Relationship Between Household Income and Household Expenditures At each of 10 household income levels we take a random sample of size 100 from the population of US households, for a total sample size of 1000 We record the level of expenditures for each household Plotting these data we might see: Expenditures (Thousands of Dollars) 0 10 20 30 40 50 60 2 4 Income 6 8 10 4 5 It appears that the relationship between income and expenditures is essentially linear As a result, it seems reasonable to express this relationship as: yi = 1 + xi2 + i, i = 1, . . . , 1000 where yi denotes the expenditure level in household i, xi denotes the income level of household i, and i denotes a random disturbance Expenditures (Thousands of Dollars) 0 10 20 30 40 50 60 2 4 Income 6 8 10 6 7 Questions: What are "good" estimators of 1 and 2? How can we express our uncertainty about the underlying true values of 1 and 2? What assumptions need to be maintained and how might we know if they don't hold? 8 Least Squares as a Fit Criterion Consider the following linear model in scalar form: yi = 1 + xi22 + + xik k + i i = 1, . . . , n We can write this in matrix form as: y = X + 9 Or in other words: y1 1 x12 y2 1 x22 . = . . . . . yn 1 xn2 y is n 1 X is n k is k 1 and is n 1 x1k x2k ... . . xnk 1 2 . . k 1 2 + . . n 10 The Ordinary Least Squares (OLS) Estimator One possible estimator for is to choose the coefficient vector that minimizes the sum of squared residuals 11 Advantages: Has a closed form solution Can be calculated quickly This estimator has very desirable sampling properties within the class of linear unbiased estimators as long as some additional assumptions hold As we'll see in a second, this estimator is the maximum likelihood estimator if a few additional assumptions are maintained 12 Disadvantages: Not robust to outliers in either y or (X, y) space Better estimators exist outside the class of linear unbiased estimators 13 Notation: the true value of the coefficient vector ~ an arbitrary value of the coefficient vector ^ OLS the OLS estimator of We can write the sum of squared residuals as: n n ~ S() = i=1 ~i2 = i=1 ~ (yi - xi)2 ^ The OLS estimator ( OLS) is defined to be the value of ~ ~ that minimizes S() 14 ~ We can write S() in matrix form as: ~ ~ ~ S() = ~ ~ = (y - X) (y - X) ~ ~ ~ ~ = y y - X y - y X + X X ~ ~ ~ = y y - 2 X y + X X The necessary condition for a minimum is that the ~ ~ gradient of S() with respect to be equal to 0 15 Calculating the gradient and setting it equal to 0 we have: ~ S() ~ = -2X y + 2X X = 0 ~ ^ This implies that OLS must satisfy the so-called normal equations: ^ X X OLS = X y 16 Assuming that (X X)-1 exists (which it will as long as X has full rank) we can solve the normal equations and ^ write our estimator OLS as: ^ OLS = (X X)-1X y 17 Example: Regression on a Constant Consider the following model: yi = 1 + i, i = 1, , n 18 The data are: y= 2 1 2 3 2 x= 1 1 1 1 1 19 Note that xx = n xi = 5 i=1 -1 (x x) xy = n 1 = 5 yi = 10 i=1 So ^1 = (x x)-1x y = 1 10 = 2 5 Which, as we would expect, is the mean of y 20 Example: A Bivariate Relationship Consider the following model: yi = 1 + xi22 + i 21 The data are: y= 1.5 1.0 2.5 1.0 4.0 3.0 4.5 4.0 X= 1 1 1 1 1 1 1 1 1 1 2 2 3 3 4 4 22 Note that XX = n n i=1 xi2 -1 n i=1 xi2 n x2 i=1 i2 = 8 20 20 60 (X X) Xy = So = 0.75 -0.25 -0.25 0.10 n i=1 yi n i=1 xi2yi = 21.5 64.5 0.0 1.075 ^ = (X X)-1X y = y 1.0 1.0 1.5 2.0 2.5 3.0 3.5 4.0 x 1.5 2.0 2.5 3.0 3.5 4.0 4.5 23 y 1.0 1.0 1.5 2.0 2.5 3.0 3.5 4.0 x 1.5 2.0 2.5 3.0 3.5 4.0 4.5 24 25 Sampling Properties Frequentist inference focuses on the performance of an estimator over repeated samples In other words, if we were to draw a large number of ^ random samples from some population and calculate ^ for each sample, what would the distribution of look like over these repeated samples? 26 In order to draw some conclusions about the repeated sampling properties of the OLS estimator we need to rely on a number of assumptions 27 The (Functional Assumptions 1. Form) y = X + E[ ] = 0 E[ ] = 2I 2. (Zero Mean Disturbances) or equivalently E[y] = X 3. (Variance-Covariance of Disturbances) 4. (Relationship between X and ) 5. (Nature of X) column rank E[ |X] = 0 X is a nonstochastic matrix with full 28 6. (Normality of Disturbances) N (0, 2I) 29 If these assumptions hold what can we say about the ^ repeated sampling properties of OLS? 30 ^ The mean of OLS ^ OLS] = E[(X X)-1X y] E[ = E[(X X) X (X + )] = E[(X X)-1X X + (X X)-1X ] = E[(I + (X X)-1X ] = E[ + (X X)-1X ] = + (X X)-1X E[ ] = + (X X)-1X 0 = -1 31 ^ OLS is unbiased On average the OLS estimator yields the true value of Note, we didn't need to assume normal disturbances to prove unbiasedness Unbiasedness doesn't say anything about the variability ^ of OLS 32 ^ Variance-covariance matrix of OLS ^ ^ ^ ^ ^ Var( OLS) = E[( OLS - E[ OLS])( OLS - E[ OLS]) ] ^ ^ = E[( OLS - )( OLS - ) ] = E[( + (X X)-1X - )( + (X X)-1X - ) ] = E[(X X)-1X = (X X)-1X E[ X(X X)-1] ]X(X X)-1 = (X X)-1X ( 2I)X(X X)-1 = 2(X X)-1X X(X X)-1 = 2(X X)-1 33 Once again, we didn't need to assume normality of the disturbances to derive the variance covariance matrix of ^ OLS However, we did assume we know 2 In practice, we'll estimate 2 as ^^ = ^ n-k 2 34 If we are willing to assume the disturbances are normally distributed then we can make even stronger statements ^ about the sampling distribution of OLS In particular (assuming normal disturbances): ^ OLS| 2 N (, 2(X X)-1) 35 Other properties of the OLS estimator (assuming the assumptions hold) ^ OLS is consistent ^ OLS is the best (minimum mean squared error) linear unbiased estimator ^ Assuming normality, OLS is the best (minimum mean squared error) unbiased estimator 36 What Happens if Assumptions Don't Hold? Functional form assumption (y = X + ) doesn't hold If functional form isn't correct all bets are off Omitted variable bias 37 Assumption of zero mean disturbances (E[ ] = 0) doesn't hold As long as a constant term is included this is relatively benign 38 Variance-Covariance of Disturbances is not equal to 2I Suppose E[ ] = 2 ^ As long as E[ |X] = 0 OLS is still unbiased ^ Var( OLS) is now ^ Var( OLS) = 2(X X)-1(X X)(X X)-1 so standard errors will not be correct 2 may be a biased estimator of 2 ^ ^ OLS still oftentimes consistent although this depends on X and 39 E[ |X] = 0 ^ OLS is no longer unbiased ^ OLS is no longer consistent 40 X is stochastic If X is stochastic but perfectly measured then all of ^ the results above still go through ( OLS) is unbiased, consistent, and best linear unbiased However, if X is measured imperfectly, i.e. there is measuremen...

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Washington - STAT - 494
1POLS Lecture 8: Review of the Linear Regression Model (Part II)Kevin Quinn University of Washington2Outline Residual Diagnostics Leverage and Influence Example3Residual Diagnostics In deriving the sampling properties of the OLS esti
Washington - STAT - 494
1POLS Lecture 8: Review of the Linear Regression Model (Part II)Kevin Quinn University of Washington2Outline Residual Diagnostics Leverage and Influence Example3Residual Diagnostics In deriving the sampling properties of the OLS esti
Washington - STAT - 494
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Washington - STAT - 494
Homework Assignment 1 CSSS/POLS 494 Advanced Quantitative Political MethodologyProfessor: Kevin Quinn, Political Science and CSSS Spring Quarter 2002 Due before class on April 16Problem 1Consider two random variables X and Y . X can take values e
Washington - STAT - 494
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Washington - STAT - 494
Homework Assignment 2 CSSS/POLS 494 Advanced Quantitative Political MethodologyProfessor: Kevin Quinn, Political Science and CSSS Spring Quarter 2002 Due before class on April 23Problem 1Calculate the rst derivatives of the following functions wi
Washington - STAT - 494
Homework Assignment 3 CSSS/POLS 494 Advanced Quantitative Political MethodologyProfessor: Kevin Quinn, Political Science and CSSS Spring Quarter 2002 Due before class on April 30Problem 1Consider the following dataset: y = (0, 0, 0, 1, 0) that is
Washington - STAT - 494
Homework Assignment 4 CSSS/POLS 494 Advanced Quantitative Political MethodologyProfessor: Kevin Quinn, Political Science and CSSS Spring Quarter 2002 Due before class on May 14Problem 1Let 3 1 4 1 2 -3 2 0 9 10 1 0.5 e= f = 4 A= B= C= D= 5 0 1
Washington - STAT - 494
%!PS-Adobe-2.0 %Creator: dvips(k) 5.86e Copyright 2001 Radical Eye Software %Title: 494hw4.dvi %Pages: 1 %PageOrder: Ascend %BoundingBox: 0 0 596 842 %EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -o 494hw4.ps -O 0.0in,0.5i
Washington - STAT - 494
Homework Assignment 5 CSSS/POLS 494 Advanced Quantitative Political MethodologyProfessor: Kevin Quinn, Political Science and CSSS Spring Quarter 2002 Due before class on May 21Problem 1Part a) Using the data in the file hw5.dat available on the c
Washington - STAT - 494
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