Chapter 19: Measurement Error and the Instrumental ...

1 Chapter 19: Measurement Error and the Instrumental Variables Estimation Procedure Chapter 19 Outline Introduction to Measurement Error o What Is Measurement Error ? o Modeling Measurement Error The Ordinary Least Squares (OLS) Estimation Procedure and Dependent Variable Measurement Error The Ordinary Least Squares (OLS) Estimation Procedure and Explanatory Variable Measurement Error o Summary: Explanatory Variable Measurement Error Bias o Explanatory Variable Measurement Error : Attenuation (Dilution) Bias o Might the Ordinary Least Squares (OLS) Estimation Procedure Be Consistent? Instrumental Variable (IV) Estimation Procedure: A Two Regression Procedure o Mechanics o The Good Instrument Conditions Measurement Error Example: Annual, Permanent, and Transitory Income o Definitions and Theory o Might the Ordinary Least Squares (OLS) Estimation Procedure Suffer from a Serious Econometric Problem?

2 Instrumental Variable (IV) Approach o The Mechanics o Comparison of the Ordinary Least Squares (OLS) and the Instrumental Variables (IV) Approaches o Good Instrument Conditions Revisited Justifying the Instrumental Variable (IV) Estimation Procedure Chapter 19 Prep Questions 1. Suppose that a physics assignment requires you to measure the amount of time it takes a one pound weight to fall six feet. You conduct twenty trials in which you use a very accurate stop watch to measure how long it takes the weight to fall. a. Even though you are very careful and conscientious would you expect the stop watch to report precisely the same amount of time on each trial?

3 Explain. 2 Suppose that the following equation describes the relationship between the measured elapsed time and the actual elapsed time: Measured elapsed timeActual elapsed timetttttyMeasuredyActualvyMeasuredyActu al=+== vt is a random variable. vt represents the random influences that cause your Measurement of the elapsed time to deviate from the actual elapsed time. The random influences cause you to click the stop watch a little early or a little late. b. Recall that you are careful and conscientious in attempting to measure the elapsed time. 1) In approximately what portion of the trials would you overestimate the elapsed time; that is, in approximately what portion of the trials would you expect vt to be positive?

4 2) In approximately what portion of the trials would you underestimate the elapsed time; that is, in approximately what portion of the trials would you expect vt to be negative? 3) Approximately what would the mean (average) of vt equal? 2. Economists distinguish between permanent income and annual income. Loosely speaking, permanent income equals what a household earns per year on average; that is, permanent income can be thought of as the average of annual income over an entire lifetime. In some years, annual income is more than its permanent income, but in other years it is less. The difference between the household s annual income and permanent income is called transitory income: where Households's Annual Income Household's Permanent Income Household's Transitory Incomett ttttIncTransIncAnnIncPermIncAnnIncPermIn cTrans= === or equivalently, tttIncAnnIncPermIncTrans=+ Since permanent income equals what a household earns on average, the mean of transitory income equals 0.

5 Microeconomic theory teaches that households base their consumption decisions on their permanent income. Theory: Additional permanent income increases consumption. Consider the following model to assess the theory: Model: Theory: 0tConstIncPermttIncPermConsIncPerme =++> 3 When we attempt to gather data to access this theory, we immediately encounter a difficulty. Permanent income cannot be observed. Only annual income data are available to assess the theory. So, while we would like to specify permanent income as the explanatory variable, we have no choice.

6 We must use annual disposable income. a. Can you interpret transitory income as Measurement Error ? Hint: What is the mean (average) of transitory income? b. Now, represent transitory income, IncTranst, by ut: tttIncAnnIncPermu=+ Express the model in terms of annual income. c. What is the equation for the new Error term? d. What are the ramifications of using the ordinary least squares (OLS) estimation procedure to estimate the permanent income coefficient, IncPerm, using annual income as the explanatory variable? Introduction to Measurement Error Two types of Measurement Error can be present: Dependent variable Explanatory variable We shall argue that dependent variable Measurement Error does not lead to bias.

7 On the other hand, whenever explanatory variable measure Error exists, the explanatory variable and Error term will be correlated resulting in bias. We consider dependent variable Measurement Error first. Before doing so, however, we shall describe precisely what we mean by Measurement Error . What Is Measurement Error ? Suppose that a physics assignment requires you to measure the amount of time it takes a one pound weight to fall six feet. You conduct twenty trials in which you use a very accurate stop watch to measure how long it takes the weight to fall. Question: Will your stop watch report the same amount of time on each trial?

8 Answer: No. Sometimes reported times will be lower than other reported times. Sometimes you will be a little premature in clicking the stop watch button. Other times you will be a little late. It is humanly impossible to measure the actual elapsed time perfectly. No matter how careful you are, sometimes the measured value will be a little low and other times a little high. 4 Modeling Measurement Error We can model Measurement Error with the following equation: yMeasuredt = yActualt + vt yActualt equals the actual amount of time elapsed and yMeasuredt equals the measured amount of time. vt represents Measurement Error .

9 Sometimes vt will be positive when you are a little too slow in clicking the stop watch button; other times vt will be negative when you click the button a little too quickly. vt is a random variable; we cannot predict the numerical value of vt beforehand. What can we say about vt? We can describe its distribution. Since you are conscientious in measuring the elapsed time, the mean of vt s probability distribution equals 0: Mean[vt] = 0 Measurement Error does not systematically increase or decrease the measured value of yt. The measured value of yt will not systematically overestimate or underestimate the actual value.

10 The Ordinary Least Squares (OLS) Estimation Procedure and Dependent Variable Measurement Error We begin with the equation specifying the actual relationship between the dependent and explanatory variables: Actual Relationship: yActualt = Const + xActualxActualt + et But now suppose that as a consequence of Measurement Error , the actual value of the dependent variable, yActualt, is not observable. You have no choice but to use the measured value, yMeasuredt. Recall that the measured value equals the actual value plus the Measurement Error random variable, vt: is a random variable with mean 0 : Mean[ ] 0ttttyMeasuredyActualvv=+=tv Solving for yActualt: yActualt = yMeasuredt vt 5 Let us apply this to the actual relationship: Substituting for Rearranging termsLetting tConstxActualtttttConstxActualtttConstxA ctualtttttttConstxActuayActualxActualeyA ctualyMeasuredvxActualeyMeasuredxActuale vevyMeasured =++ =++=+++ =+=+lttxActual + t represents the Error term in the regression that you will actually be running.