Monday, November 26: Explanatory Variable Explanatory Premise, Bias, and Large Sample Properties

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1 Amherst College Department of Economics Economics 360 Fall 2012 Monday, November 26: Explanatory Variable Explanatory Premise, Bias, and Large Sample Properties Chapter 18 Outline Review o Regression Model o Standard Ordinary Least Squares (OLS) Premises o Estimation Procedures Embedded within the Ordinary Least Squares (OLS) Estimation Procedure Taking Stock and a Preview: The Ordinary Least Squares (OLS) Estimation Procedure A Closer Look at the Explanatory Variable/Error Term Independence Premise Explanatory Variable/Error Term Correlation and Bias o Geometric Motivation o Confirming Our Logic Estimation Procedures: Large and Small Sample Properties o Unbiased and Consistent Estimation Procedure o Unbiased but Not Consistent Estimation Procedure o Biased but Consistent Estimation Procedure The Ordinary Least Squares (OLS) Estimation Procedure, and Consistency Instrumental Variable (IV) Estimation Procedure: A Two Regression Procedure o Mechanics o The Two Good Instrument Conditions Regression Model y t = β Const + β x + e t y t = Dependent variable = Explanatory variable e t = Error term t = 1, 2,, T T = Sample size The error term is a random variable that represents random influences: Mean[e t ] = 0 Standard Ordinary Least Squares (OLS) Premises Error Term Equal Variance Premise: The variance of the error term s probability distribution for each observation is the same; all the variances equal Var[e]: Var[e 1 ] = Var[e 2 ] = = Var[e T ] = Var[e] Error Term/Error Term Independence Premise: The error terms are independent: Cov[e i, e j ] = 0. Knowing the value of the error term from one observation does not help you predict the value of the error term for any other observation. Explanatory Variable/Error Term Independence Premise: The explanatory variables, the s, and the error terms, the e t s, are not correlated. Knowing the value of an observation s explanatory variable does not help you predict the value of that observation s error term.

2 2 Ordinary Least Squares (OLS) Estimation Procedure Several estimation procedures are embedded with the ordinary least squares estimation procedure. For our purposes, the three most important are a procedure to estimate the Σ T t=1 (yt y )( x ) value of the coefficient, β x : b x = T Σ t=1 (xt x ) 2 variance of the error term s probability distribution, Var[e]: EstVar[e] = SSR Degrees of Freedom EstVar[e] variance of the coefficient estimate s probability distribution, Var[b x ]: EstVar[b x ] = T Σ t=1 (xt x ) 2 When the standard ordinary least squares (OLS) regression premises are met: Each estimation procedure is unbiased; that is, each estimation procedure does not systematically underestimate or overestimate the actual value. The ordinary least squares (OLS) estimation procedure for the coefficient value is the best linear unbiased estimation procedure (BLUE). Crucial Point: When the ordinary least squares (OLS) estimation procedure performs its calculations, it implicitly assumes that the standard ordinary least squares (OLS) regression premises are satisfied. Taking Stock and a Preview: The Ordinary Least Squares (OLS) Estimation Procedure OLS Bias Question: Is the explanatory variable/error term independence premise satisfied or violated? Is the OLS estimation procedure for the value of the coefficient unbiased or biased? OLS Reliability Question: Are the error term equal variance and error term/error term independence premises satisfied or violated? Can the OLS calculations for the standard error be trusted? Satisfied: Explanatory Variable and Error Term Independent Satisfied ã é Violated Violated: Explanatory Variable and Error Term Correlated Is the OLS estimation procedure for the value of the coefficient BLUE? Preview: When the explanatory variable/error term independence premise is violated and the ordinary least squares (OLS) estimation procedure is biased, other estimation procedures can be used to mitigate although not completely remedy the bias problem.

3 3 Explanatory Variable/Error Term Independence Premise: The explanatory variables, the s, and the error terms, the e t s, are not correlated. Question: What happens when this premise is violated? Claim: When the explanatory variables and the error terms are correlated the ordinary least squares estimation procedure for the coefficient value is biased. Question: What does explanatory variable/error term independence and correlation looks like? Econometrics Lab 18.1: Explanatory Variable/Error Term Independence and Correlation Explanatory Variable/Error Term Independence: CorrX&E = 0. Be certain that the checkbox is cleared and click Start; after many, many repetitions, click Stop. e t The explanatory variable and error term appear to be. 0 The mean of the error term is approximately regardless of the value of x. Low x s Medium x s High x s Mean: Variance: Mean: Variance: Mean: Variance:

4 4 Explanatory Variable/Error Term Positive Correlation: CorrX&E =.6. Be certain that the checkbox is cleared and click e t Start; after many, many repetitions, click Stop. The explanatory variable and error term appear to be. When the value of the explanatory variable is 0 low the error term is typically. When the value of the explanatory variable is high the error term is typically. Low x s Medium x s High x s Mean: Variance: Mean: Variance: Mean: Variance: Explanatory Variable/Error Term Negative Correlation: CorrX&E =.6. e Be certain that the checkbox is cleared and click t Start; after many, many repetitions, click Stop. The explanatory variable and error terms appear to be. When the value of the explanatory variable is low the error term is typically. When the value of the explanatory variable is high the error term is typically. 0 Low x s Medium x s High x s Mean: Variance: Mean: Variance: Mean: Variance:

5 5 Consequences of Explanatory Variable/Error Term Correlation e t Explanatory variable and error term positively correlated e t Explanatory variable and error term negatively correlated y t y t Actual equation line Actual equation line Explanatory variable and error term are positively correlated Estimated equation is steeply sloped than actual equation OLS estimation procedure for the coefficient value biased Explanatory variable and error term are negatively correlated Estimated equation is steeply sloped than actual equation OLS estimation procedure for the coefficient value biased Econometrics Lab 18.2: Ordinary Least Squares (OLS) and Explanatory Variable/Error Term Correlation Confirming Our Suspicions Estimation Corr Sample Actual Mean of Magnitude Variance of Procedure X&E Size Coef Coef Ests of Bias Coef Ests OLS OLS OLS A new list now appears, CorrXE, which specifies the correlation between the explanatory variable and the error term. Initially, 0 is specified indicating that the explanatory variable and error term are independent (uncorrelated). Click Start and then after many, many repetitions, click Stop. Next, investigate what occurs when the explanatory variable and error term are positively or negatively correlated by changing the value of CorrXE. CorrX&E

6 6 Estimation Procedures: Unbiased versus Biased and Consistent versus Inconsistent Unbiased: Small Sample Property. The estimation procedure does not systematically underestimate or overestimate the actual value. Formally, the mean of the estimate s probability distribution equals the actual value: Mean[Est] = Actual Value. When the estimate s probability distribution is symmetric, the chances that the estimate is greater than the actual value equal the chances that it is less. NB: Unbiasedness is called a small sample property because it does not depend on the sample size. Unbiasedness depends only on the mean of the estimate s probability distribution. Consistent: Large Sample Properties. Unlike unbiasedness, both the mean and variance of the estimate s probability distribution are important for consistency: Mean of the estimate s probability distribution: Either o The estimation procedure is unbiased: Mean[Est] = Actual Value or o The estimation procedure is bias, but the magnitude of the bias diminishes as the sample size becomes larger; more, formally, as the sample size approaches infinity the mean approaches the actual value: As Sample Size : Mean[Est] Actual Value Variance of the estimate s probability distribution: The variance diminishes as the sample size becomes larger; more formally, as the sample size approaches infinity the variance approaches 0: As Sample Size : Variance[Est] 0 All Estimation Procedures Consistent Unbiased To get a better sense of the two different properties of estimation procedures we shall consider three estimation procedures: Unbiased and Consistent Unbiased but Not Consistent Biased but Consistent

7 7 Categorizing Estimation Procedures Small Sample Property: Does Mean[Est] equal the Actual Value? Yes No Unbiased Biased Does Mean[Est] Actual Value as the sample size? Yes No Large Sample Does Var[Est] 0 Does Var[Est] 0 Biased and Property: as the sample size? as the sample size? Not Consistent Yes No Yes No Unbiased and Consistent Unbiased but Not Consistent Biased but Consistent Small Sample Small Sample Small Sample Actual Value Est Actual Value Est Actual Value Est Econometrics Lab 18.3: Ordinary Least Squares (OLS) Estimation Procedure Illustrating an Estimation Procedure that Is Unbiased and Consistent Estimation Corr Sample Actual Mean of Magnitude Variance of Procedure X&E Size Coef Coef Ests of Bias Coef Ests OLS OLS OLS Econometrics Lab 18.4: Any Two Estimation Procedure Illustrating an Estimation Procedure that Is Unbiased and Consistent Estimation Sample Actual Mean of Magnitude Variance of Procedure Size Coef Coef Ests of Bias Coef Ests Any Two Any Two Any Two

8 8 Illustrating a Consistent but Biased Estimation an Procedure: Revisit Our Friend Clint Random Sample Procedure: Write the name of each individual in the population on a 3 5 card Perform the following procedure 16 times: o Thoroughly shuffle the cards. o Randomly draw one card. o Ask that individual if he/she is voting for Clint and record the answer. o Replace the card. Calculate the fraction of the sample supporting Clint. Nonrandom Sample Procedure: Leave Clint s dorm room and ask the first 16 people you run into if he/she is voting for Clint. Calculate the fraction of the sample supporting Clint. Econometrics Lab 18.5: Biased but Consistent Estimation Procedure Questions: Compared to the general student population: ActFrac Sample Size Are the students who live near Clint more likely to be Clint s friend? Are the students who live near Clint more likely to vote for him? Actual Population Fraction Since your EstFrac starting point is Clint s dorm Numerical value Mean room, is it of the estimated likely that you Var fraction in this will poll repetition students who are more supportive of Clint than the general student population? Would you be biasing your poll in Clint s favor? Non-random Sample Start Repetition: Checking Our Logic: Simulation After Many, Many Repetitions Sampling Population Sample Mean (Average) Magnitude Variance Technique Fraction Size of Estimates of Bias of Estimates Random Random Random Nonrandom Nonrandom Nonrandom Is the nonrandom procedure unbiased? Is it consistent? Stop Pause Sample Size Is Clint s estimation procedure unbiased? Mean (average) of the numerical values of the estimated fraction from all repetitions Variance of the numerical values of the estimated fraction from all repetitions

9 9 The Explanatory Variable/Error Term Premise, the Ordinary Least Squares (OLS) Estimation Procedure, and Consistency Review: We have already shown that the ordinary least squares (OLS) estimation procedure is when explanatory variable/error term correlation is present. Estimation Corr Sample Actual Mean of Magnitude Variance of Procedure X&E Size Coef Coef Ests of Bias Coef Ests OLS OLS OLS Econometrics Lab 18.6: Ordinary Least Squares (OLS) Estimation Procedure and Consistency Question: But might the ordinary least squares (OLS) estimation procedure consistent when explanatory variable/error term correlation is present? Estimation Corr Sample Actual Mean of Magnitude Variance of Procedure X&E Size Coef Coef Ests of Bias Coef Ests OLS OLS OLS There is nothing but news. In the presence of explanatory variable/error term correlation, the ordinary least squares (OLS) estimation procedure is: Question: Where do we go from here? Instrumental Variable (IV) Estimation Procedure y t = β Const + β x + ε t where y t = Dependent variable é ã = Explanatory variable When and ε t ε t = Error term are correlated t = 1, 2,, T T = Sample size is the explanatory variable Problem Explanatory Variable: is the problem explanatory variable: The explanatory variable,, is with the error term, ε t. Consequently, the explanatory variable/error term independence premise is. The ordinary least squares (OLS) estimation procedure for the coefficient value is.

10 10 Addressing the Problem Explanatory Variable Using Instrumental Variables Choose an Instrument: A good instrument, z t, must meet two conditions. Good Instrument Condition 1: Correlated with the problem explanatory variable,.. Good Instrument Condition 2: Uncorrelated with the error term, ε t. Instrumental Variables (IV) Regression 1: Use the instrument, z t, to estimate the problem explanatory variable,. Dependent variable: Problem explanatory variable,. Explanatory variable: Instrument, z t. Estimate of the problem explanatory variable: Est = a Const + a z z t where a Const and a z are the estimates of the constant and coefficient in this regression, IV Regression 1. Instrumental Variables (IV) Regression 2: In the original model, replace the problem explanatory variable,, with its surrogate, Est, the estimate of the problem explanatory variable provided by the instrument, z t, form IV Regression 1. Dependent variable: Original dependent variable, y t. Explanatory variable: Estimate of the problem explanatory variable based on the results from IV Regression 1, Est. The Good Instrument Conditions Good Instrument Condition 1: The instrument, z t, must be correlated with the problem explanatory variable,. Focus on IV Regression 1 Est = a Const + a z z t The estimate, Est, will be a good surrogate only if it is a good predictor of the problem explanatory variable,. This will occur only if the instrument, z t, is correlated with the problem explanatory variable,. Good Instrument Condition 2: The instrument, z t, must be independent of the error term, ε t. Focus on IV Regression 2: y t = β Const + β x + ε t = β Const + β x Est + ε t Replace problem with surrogate where Est = a Const + a z z t from IV Regression 1 To avoid violating the explanatory variable/error term independence premise in IV Regression 2, the surrogate for the problem explanatory variable, Est, must be of the error term, ε t. To avoid violating the explanatory variable/error term independence premise Est and ε t must be ã é y t = β Const + β x Est + ε t Estx t = a Const + a z z t é z t and ε t must be

11 11 Justifying the Instrumental Variable (IV) Estimation Procedure Econometrics Lab 18.7: Instrumental Variables (IV) Estimation Procedure and Consistency IV is selected indicating that the instrumental variable (IV) estimation procedure we just described will be used to estimate the value of the explanatory variable s coefficient. The Corr X&E list the value.30 is specified. The correlation coefficient for the explanatory variable and error term equals.30. Hence, the explanatory variable/error term independence premise is violated. Two new correlation lists appear in this simulation: Corr X&Z and Corr Z&E. The two new lists reflect the two conditions required for a good instrument. The Corr X&Z list specifies the correlation coefficient for the explanatory variable and the instrument. To be a good instrument the explanatory variable and the instrument must be correlated. The default value is.50. The Corr Z&E specifies the correlation coefficient for the instrument and error term. To be a good instrument the instrument and error term must be independent. The default value is.00; that is, the instrument and error term are independent. The default values meet the conditions necessary for a good instrument. Econometrics Lab 18.7: Instrumental Variables (IV) Estimation Procedure and Consistency Estimation Correlation Coefficients Sample Actual Mean of Magnitude Variance of Procedure X&Z Z&E X&E Size Coef Coef Ests of Bias Coef Ests IV IV IV Question: is the IV estimation procedure: Unbiased? Consistent?

12 12 Next, we explore the importance of the two good instrument conditions: Good Instrument Condition 1: Instrument/ Problem Explanatory Variable Correlation: Suppose that the instrument is more highly correlated with the problem explanatory variable: z t more highly correlated with Est is a predictor for Est is a surrogate for Instrument variables (IV) estimation procedure is. Estimation Correlation Coefficients Sample Actual Mean of Magnitude Variance of Procedure X&Z Z&E X&E Size Coef Coef Ests of Bias Coef Ests IV Good Instrument Condition 2: Instrument/Error Term Independence: Suppose that the instrument is correlated with the error term: Estimation Correlation Coefficients Sample Actual Mean of Magnitude Variance of Procedure X&Z Z&E X&E Size Coef Coef Ests of Bias Coef Ests IV IV IV Question: is the IV estimation procedure: Unbiased? Consistent?