Amherst College Department of Economics Economics 360 Fall 2012

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1 Amherst College Department of Economics Economics 360 Fall 2012 Monday, December 3: Omitted Variables and the Instrumental Variable Estimation Procedure Chapter 20 Outline Revisit Omitted Explanatory Variable Bias o Review of Our Previous Explanation of Omitted Explanatory Variable Bias o Omitted Explanatory Variable Bias and the Explanatory Variable/Error Term Independence Premise The Ordinary Least Squares Estimation Procedure, Omitted Explanatory Variable Bias, and Consistency Instrumental Variable Estimation Procedure: A Two Regression Estimation Procedure o Mechanics o The Two Good Instrument Conditions Omitted Explanatory Variables Example: 2008 Presidential Election o Might the Ordinary Least Squares (OLS) Estimation Procedure Suffer from a Serious Econometric Problem? Instrument Variable (IV) Application: 2008 Presidential Election o The Mechanics o Good Instrument Conditions Revisited Justifying the Instrumental Variable (IV) Estimation Procedure Revisit the Omitted Variable Phenomenon Review: Omitting an explanatory variable from a regression will bias the OLS estimation procedure whenever two conditions are met Bias results if the omitted explanatory variable: influences the dependent variable; is correlated with an included explanatory variable When these two conditions are met, the OLS procedure to estimate the coefficient of the included explanatory variable captures two effects: Direct Effect: The effect that the included explanatory variable actually has on the dependent variable; Proxy Effect: The effect that the omitted explanatory variable has on the dependent variable because the included explanatory variable is acting as a proxy for the omitted explanatory variable Recall the goal of multiple regression analysis: Goal of Multiple Regression Analysis: Multiple regression analysis attempts to sort out the individual effect that each explanatory variable has on the dependent variable Consequently, we want the coefficient estimate of the included variable to capture onlhe direct effect and not the proxy effect Unfortunately, this does not occur when the omitted variance influences the dependent variable and when it is also correlated with an included variable To illustrate this, we considered a model with two explanatory variables, x1 and x2: + β x2 For purposes of illustration, assume that the coefficients are positive and that the explanatory variables are positively correlated: β x1 > 0 and β x2 > 0 and are positively correlated

2 2 Effect of omitting the explanatory variable x2: Intuitive Approach up Typically, β x1 > 0 Positive correlation β x2 > 0 Effect Effect When the explanatory variable x2 is omitted from a regression, the OLS estimation procedure for the value of x1 s coefficient is Effect of omitting the explanatory variable x2: Error Term Approach + β x2 + (β x2 ) + ε t where ε t = β x2 When the explanatory variable x2 is omitted from the regression, are the included explanatory variable and the new error term, ε t, correlated? and are positively correlated up Typically, up ε t = β x2 ε t up β x2 > 0 correlated OLS estimation procedure for the value of x1 s coefficient is Review: Unbiased and Consistent Estimation Procedures 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 Informally, 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 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 infinithe 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 infinithe variance approaches 0: As Sample Size : Variance[Est] 0

3 3 Question: Might the ordinary least squares (OLS) estimation procedure be consistent? Econometrics Lab 201: Ordinary Least Squares, Omitted Variables, and Consistency Estimation Corr Actual Sample Actual Mean of Magnitude Variance of Procedure X1&X2 Coef2 Size Coef Coef1 Ests of Bias Coef1 Ests OLS OLS OLS Summary: When an explanatory variable is omitted that affects the dependent variable and is correlated with an included variable the ordinary least squares (OLS) estimation procedure for the value of the included variable s coefficient is:

4 4 Instrumental Variable (IV) Estimation Procedure + β x x t + ε t where = Dependent variable x t = Explanatory variable When x t ε t = Error term are correlated t = 1, 2,, T T = Sample size x t is the explanatory variable Problem Explanatory Variable: x t is the problem explanatory variable: The explanatory variable, x t, 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 Addressing the Problem Explanatory Variable Using Instrumental Variables Choose an Instrument: A good instrument,, must meet two conditions Correlated with the problem explanatory variable, x t Uncorrelated with the error term, ε t Instrumental Variables (IV) Regression 1: Use the instrument,, to estimate the problem explanatory variable, x t Dependent variable: Problem explanatory variable, x t Explanatory variable: Instrument, Estimate of the problem explanatory variable: Estx t = a Const + a z 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, x t, with its surrogate, Estx t, the estimate of the problem explanatory variable provided bhe instrument,, form IV Regression 1 Dependent variable: Original dependent variable, Explanatory variable: Estimate of the problem explanatory variable based on the results from IV Regression 1, Estx t Justifying the Good Instrument Conditions Instrument/ Problem Explanatory Variable Correlation: The instrument,, must be correlated with the problem explanatory variable, x t Focus on IV Regression 1 Estx t = a Const + a z The estimate, Estx t, will be a good surrogate only if it is a good predictor of the problem explanatory variable, x t This will occur only if the instrument,, is correlated with the problem explanatory variable, x t

5 5 Instrument/Error Term Independence: The instrument,, must be independent of the error term, ε t Focus on IV Regression 2: + β x x t + ε t + β x Estx t + ε t Replace problem with surrogate where Estx t = a Const + a z 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, Estx t, must be of the error term, ε t To avoid violating the explanatory variable/error term independence premise Estx t must be + β x Estx t + ε t Estx t = a Const + a z must be Omitted Explanatory Variables Example: 2008 Presidential Election 2008 Presidential Election Data: Cross section data of election, population, and economic statistics from the 50 states and the District of Columbia in 2008 AdvDeg t Percent adults who have advanced degrees in state t Coll t Percent adults who graduated from college in state t HS t Percent adults who graduated from high school in state t Population density of state t (persons per square mile) RealGdpGrowth t GDP growth rate for state t in 2008 (percent) UnemTrend t Change in the unemployment rate for state t in 2008 (percent) VoteDemPartyTwo t Percent of the vote received in 2008 received bhe Democratic party in state t based on the two major parties (percent) Model: VoteDemPartyTwo t + β PopDen + β Lib Population Density Theory: States with high population densities have large urban areas which are more likelo vote for the Democratic candidate, Obama; hence, β PopDen > 0 Liberalness Theory: Since the Democratic party is more liberal than the Republican party, a high liberalness value would increase the vote of the Democratic candidate, Obama; hence, β Lib > 0 The variable reflects the liberalness of the electorate in state t If the electorate is by nature liberal in state t, the would be high; on the other hand, if the electorate is conservative the would be low

6 6 Unfortunately, we do not have any data to quantifhe liberalness of a state; according, Liberal must be omitted from the regression Ordinary Least Squares (OLS) Dependent Variable: VoteDemPartyTwo Explanatory Variable(s): Estimate SE t-statistic Prob PopDen Const Number of Observations 51 Estimated Equation: VoteDemPartyTwo = PopDen Interpretation of Estimates: b EstPopDen = 005: A 1 person increase in a state s population density increases the state s Democratic vote by 005 percentage points; that is, a 100 person increase in a state s population density increases the state s Democratic vote by 5 percentage points Critical Result: The EstPopDen coefficient estimate equals 005 The positive sign of the coefficient estimate suggests that increases in a state with a higher population density will have a greater Democratic vote This evidence supports the theory Might the Ordinary Least Squares (OLS) Estimation Procedure Suffer from a Serious Econometric Problem? VoteDemPartyTwo t + β PopDen + β Lib + β PopDen + (β Lib ) + β PopDen + ε t where ε t = β Lib Question: Is there good reason to believe that and are correlated? Included variable up and positively correlated Typically, omitted variable up ε t = β PopDen ε t up β Lib > 0 positively correlated PopDen s coefficient will be biased Summary: is a problem explanatory variable

7 7 Instrument Variable (IV) Application: 2008 Presidential Election Choose an Instrument: In this example, we shall use the percent of high school graduates, Coll t, as our instrument In doing so, we believe that it satisfies the two good instrument conditions; that is, we believe that the percent of high school graduates, Coll t, is positively correlated with the problem explanatory variable, and uncorrelated with the error term, ε t = β Lib Consequently, we believe that the instrument, Coll t, is uncorrelated with the omitted variable, Instrumental Variables (IV) Regression 1 Dependent variable: Problem explanatory variable, PopDen Explanatory variable: Instrument, Coll Ordinary Least Squares (OLS) Dependent Variable: PopDen Explanatory Variable(s): Estimate SE t-statistic Prob Coll Const Number of Observations 51 Estimated Equation: EstPopDen = 3, Coll Instrumental Variables (IV) Regression 2: Dependent variable: Original dependent variable, VoteDemPartyTwo Explanatory variable: Estimate of the problem explanatory variable based on the results from IV Regression 1, EstPopDen Ordinary Least Squares (OLS) Dependent Variable: VoteDemPartyTwo Explanatory Variable(s): Estimate SE t-statistic Prob EstPopDen Const Number of Observations 51 Estimated Equation: VoteDemPartyTwo = EstPopDen Interpretation of Estimates: b EstPopDen = 01: A 1 person increase in a state s population density increases the state s Democratic vote by 01 percentage points; that is, a 100 person increase in a state s population density increases the state s Democratic vote by 1 percentage points Critical Result: The EstPopDen coefficient estimate equals 01 The positive sign of the coefficient estimate suggests that increases in a state with a higher population density will have a greater Democratic vote This evidence supports the theory

8 8 Good Instrument Conditions Revisited Instrument/ Problem Explanatory Variable Correlation: The instrument, Coll t, must be correlated with the problem explanatory variable, We are using the instrument to create a surrogate for the problem explanatory variable in IV Regression 1: Est = 3, Coll t The estimate, Est, will be a good surrogate only if the instrument, Coll t, is correlated with the problem explanatory variable, ; that is, only if the estimate is a good predictor of the problem explanatory variable Correlation Matrix Coll PopDen Coll PopDen The correlation coefficient for Coll t and equals 60 Furthermore, the IV Regression 1 results suggest that the instrument, Coll t, will be a good predictor of the problem explanatory variable, The coefficient estimate is significant at the 1 percent level So, it is reasonable to judge that the instrument meets the first condition Instrument/Error Term Independence: The instrument, Coll t, and the error term, ε t, must be independent Otherwise, the explanatory variable/error term independence premise would be violated in IV Regression 2 Recall the model that IV Regression 2 estimates: VoteDemPartyTwo t + β PopDen Est + ε t Question: Are Est independent? Est = 3, Coll t ε t = β Lib Answer: Only if Coll t and are independent The explanatory variable/error term independence premise will be satisfied only if the surrogate, Est, and the error term, ε t, are independent Est is a linear function of Coll t is a linear function of : Est = 3, Coll t and ε t = β Lib The explanatory variable/error term independence premise will be satisfied only if the instrument, Coll t, and the omitted variable,, are independent Question: Is there reason to believe that the instrument, Coll t, and the new error term,, are correlated? NB: This can be the Achilles heel of the instrumental variable (IV) estimation procedure, however Finding a good instrument can be verricky

9 9 Justifying the Instrumental Variable (IV) Estimation Procedure Econometrics Lab 202: Instrumental Variables - Omitted Variables: Good Instrument The model upon which this simulation is based on the following model: + β x2 Defaults: The Only X1 button is selected Hence, second explanatory variable, x2, is omitted The model becomes: + (β x2 ) + ε t where ε t = β x2 The actual coefficient for the omitted explanatory variable,, equals 4 In the CorrX1&X2 list 60 is selected Hence, the included and omitted explanatory variables are correlated In the CorrX1&Z list 50 is selected Hence, the included explanatory variable and the instrument are correlated In the CorrX2&Z list 00 is selected Hence, the instrument and the omitted variable are uncorrelated; consequently, the instrument and the error term are independent

10 10 Questions: In view of the defaults: Will the ordinary least squares estimation procedure for the value of the included explanatory variable s coefficient be biased? Is the instrument a good instrument? Estimation Corr Corr Corr Sample Actual Mean of Magnitude Variance of Procedure X1&Z X2&Z X1&X2 Size Coef1 Coef1 Ests of Bias Coef1 Ests IV IV IV Bad News: Good News: Next, we explore the importance of the two good instrument conditions: Instrument/ Problem Explanatory Variable Correlation: Suppose that the instrument is more highly correlated with the problem explanatory variable: more highly correlated with Estx t is a predictor for x t Estx t is a surrogate for x t Instrument variables (IV) estimation procedure is Estimation Corr Corr Corr Sample Actual Mean of Magnitude Variance of Procedure X1&Z X2&Z X1&X2 Size Coef1 Coef1 Ests of Bias Coef1 Ests IV IV Instrument/Error Term Independence: Suppose that the instrument is correlated with the error term: Estimation Corr Corr Corr Sample Actual Mean of Magnitude Variance of Procedure X1&Z X2&Z X1&X2 Size Coef1 Coef1 Ests of Bias Coef1 Ests IV IV IV Bad news; Instrument variables (IV) estimation procedure is Bad news: Instrument variables (IV) estimation procedure is The Achilles heel of the instrumental variables (IV) estimation procedure: