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Christopher Dougherty EC220 - Introduction to econometrics (chapter 11) Slideshow: assumption c.7 Original citation: Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 11). [Teaching Resource] © 2012 The Author This version available at: Available in LSE Learning Resources Online: May 2012 This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user credits the author and licenses their new creations under the identical terms.

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1 Assumption C.7, like its counterpart Assumption B.7, is essential for both the unbiasedness and the consistency of OLS estimators. ASSUMPTION C.7 ASSUMPTIONS FOR MODEL C C.7The disturbance term is distributed independently of the regressors u t is distributed independently of X jt' for all t' (including t ) and j (1)The disturbance term in any observation is distributed independently of the values of the regressors in the same observation, and (2)The disturbance term in any observation is distributed independently of the values of the regressors in the other observations.

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2 It is helpful to divide it into two parts, as shown above. Both parts are required for unbiasedness. However only the first part is required for consistency (as a necessary, but not sufficient, condition). ASSUMPTION C.7 ASSUMPTIONS FOR MODEL C C.7The disturbance term is distributed independently of the regressors u t is distributed independently of X jt' for all t' (including t ) and j (1)The disturbance term in any observation is distributed independently of the values of the regressors in the same observation, and (2)The disturbance term in any observation is distributed independently of the values of the regressors in the other observations.

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3 For cross-sectional regressions, Part (2) is rarely an issue. Since the observations are generated randomly there is seldom any reason to suppose that the disturbance term in one observation is not independent of the values of the regressors in the other observations. ASSUMPTION C.7 ASSUMPTIONS FOR MODEL C C.7The disturbance term is distributed independently of the regressors u t is distributed independently of X jt' for all t' (including t ) and j (1)The disturbance term in any observation is distributed independently of the values of the regressors in the same observation, and (2)The disturbance term in any observation is distributed independently of the values of the regressors in the other observations.

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4 Hence unbiasedness really depended on part (1). Of course, this might fail, as we saw with measurement errors in the regressors and with simultaneous equations estimation. ASSUMPTION C.7 ASSUMPTIONS FOR MODEL C C.7The disturbance term is distributed independently of the regressors u t is distributed independently of X jt' for all t' (including t ) and j (1)The disturbance term in any observation is distributed independently of the values of the regressors in the same observation, and (2)The disturbance term in any observation is distributed independently of the values of the regressors in the other observations.

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5 With time series regression, part (2) becomes a major concern. To see why, we will review the proof of the unbiasedness of the OLS estimator of the slope coefficient in a simple regression model. ASSUMPTION C.7 ASSUMPTIONS FOR MODEL C C.7The disturbance term is distributed independently of the regressors u t is distributed independently of X jt' for all t' (including t ) and j (1)The disturbance term in any observation is distributed independently of the values of the regressors in the same observation, and (2)The disturbance term in any observation is distributed independently of the values of the regressors in the other observations.

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6 The slope coefficient may be written as shown above. ASSUMPTION C.7

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7 We substitute for Y from the true model. ASSUMPTION C.7

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8 The 1 terms in the second factor in the numerator cancel each other. Rearranging what is left, we obtain the third line. ASSUMPTION C.7

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9 The first term in the numerator, when divided by the denominator, reduces to 2. Hence as usual we have decomposed the slope coefficient into the true value and an error term. ASSUMPTION C.7

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10 The error term can be decomposed as shown. ASSUMPTION C.7

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11 u is a common factor in the second component of the error term and so can be brought out of it as shown. – ASSUMPTION C.7

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12 It can then be seen that the numerator of the second component of the error term is zero. ASSUMPTION C.7

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13 We are thus able to show that the OLS slope coefficient can be decomposed into the true value and an error term that is a weighted sum of the values of the disturbance term in the observations, with weights a i defined as shown. ASSUMPTION C.7

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14 Now we will take expectations. The expectation of the right side of the equation is the sum of the expectations of the individual terms. ASSUMPTION C.7

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15 If the u i are distributed independently of the a i, we can decompose the E(a i u i ) terms as shown. ASSUMPTION C.7

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16 Unbiasedness then follows from the assumption that the expectation of u i is zero. ASSUMPTION C.7

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17 The crucial step is the previous one, which requires u i to be distributed independently of a i. a i is a function of all of the X values in the sample, not just X i. So Part (1) of Assumption C.7, that u i is distributed independently of X i, is not enough. ASSUMPTION C.7

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18 We also need Part (2), that u i is distributed independently of X j, for all j. ASSUMPTION C.7

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19 In regressions with cross-sectional data this is usually not a problem. ASSUMPTION C.7

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20 If, for example, we are relating the logarithm of earnings to schooling using a sample of individuals, it is reasonable to suppose that the disturbance term affecting individual I will be unrelated to the schooling of any other individual. CROSS-SECTIONAL DATA: LGEARN i = 1 + 2 S i + u i LGEARN j = 1 + 2 S j + u j Reasonable to assume u j and S i independent ( i ≠ j ). The main issue is whether u i is independent of S i. ASSUMPTION C.7

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21 Assuming this, the independence of u i and a i then depends only on the independence of u i and S i. ASSUMPTION C.7 CROSS-SECTIONAL DATA: LGEARN i = 1 + 2 S i + u i LGEARN j = 1 + 2 S j + u j Reasonable to assume u j and S i independent ( i ≠ j ). The main issue is whether u i is independent of S i.

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TIME SERIES DATA: Y t = 1 + 2 Y t–1 + u t Y t+1 = 1 + 2 Y t + u t+1 The disturbance term u t is automatically correlated with the explanatory variable Y t in the next observation. 22 However with time series data it is different. Suppose, for example, that you have a model with a lagged dependent variable as a regressor. Here we have a very simple model where the only regressor is the lagged dependent variable. ASSUMPTION C.7

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TIME SERIES DATA: Y t = 1 + 2 Y t–1 + u t Y t+1 = 1 + 2 Y t + u t+1 The disturbance term u t is automatically correlated with the explanatory variable Y t in the next observation. 23 We will suppose that Part (1) of Assumption C.7 is valid and that u t is distributed independently of Y t–1. ASSUMPTION C.7

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TIME SERIES DATA: Y t = 1 + 2 Y t–1 + u t Y t+1 = 1 + 2 Y t + u t+1 The disturbance term u t is automatically correlated with the explanatory variable Y t in the next observation. 24 Even if Part (1) is valid, Part (2) must be invalid in this model. ASSUMPTION C.7

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25 TIME SERIES DATA: Y t = 1 + 2 Y t–1 + u t Y t+1 = 1 + 2 Y t + u t+1 The disturbance term u t is automatically correlated with the explanatory variable Y t in the next observation. u t is a determinant of Y t and Y t is the regressor in the next observation. Hence even if u t is uncorrelated with the explanatory variable Y t–1 in the observation for Y t, it will be correlated with the explanatory variable Y t in the observation for Y t+1. ASSUMPTION C.7

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26 As a consequence u i is not independent of a i and so we cannot write E(a i u i ) = E(a i )E(u i ). It follows that the OLS slope coefficient will in general be biased. X ASSUMPTION C.7 TIME SERIES DATA: Y t = 1 + 2 Y t–1 + u t Y t+1 = 1 + 2 Y t + u t+1 The disturbance term u t is automatically correlated with the explanatory variable Y t in the next observation.

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27 X ASSUMPTION C.7 We cannot obtain a closed-form analytical expression for the bias. However we can investigate it with Monte Carlo simulation. TIME SERIES DATA: Y t = 1 + 2 Y t–1 + u t Y t+1 = 1 + 2 Y t + u t+1 The disturbance term u t is automatically correlated with the explanatory variable Y t in the next observation.

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28 We will start with the very simple model shown at the top of the slide. Y is determined only by its lagged value, with intercept 10 and slope coefficient 0.8. Y t = Y t–1 + u t n = 20 n = 1000 Sample b 1 s.e.(b 1 ) b 2 s.e.(b 2 ) b 1 s.e.(b 1 ) b 2 s.e.(b 2 ) ASSUMPTION C.7

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29 The disturbance term u will be generated using random numbers drawn from a normal distribution with mean 0 and variance 1. The sample size is 20. Y t = Y t–1 + u t n = 20 n = 1000 Sample b 1 s.e.(b 1 ) b 2 s.e.(b 2 ) b 1 s.e.(b 1 ) b 2 s.e.(b 2 ) ASSUMPTION C.7

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30 Here are the estimates of the coefficients and their standard errors for 10 samples. We will start by looking at the distribution of the estimate of the slope coefficient. 8 of the estimates are below the true value and only 2 above. Y t = Y t–1 + u t n = 20 n = 1000 Sample b 1 s.e.(b 1 ) b 2 s.e.(b 2 ) b 1 s.e.(b 1 ) b 2 s.e.(b 2 ) ASSUMPTION C.7

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31 This suggests that the estimator is downwards biased. However it is not conclusive proof because an 8–2 split will occur quite frequently even if the estimator is unbiased. (If you are good at binomial distributions, you will be able to show that it will occur 11% of the time.) Y t = Y t–1 + u t n = 20 n = 1000 Sample b 1 s.e.(b 1 ) b 2 s.e.(b 2 ) b 1 s.e.(b 1 ) b 2 s.e.(b 2 ) ASSUMPTION C.7

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32 However the suspicion of a bias is reinforced by the fact that many of the estimates below the true value are much further from it than those above. The mean value of the estimates is Y t = Y t–1 + u t n = 20 n = 1000 Sample b 1 s.e.(b 1 ) b 2 s.e.(b 2 ) b 1 s.e.(b 1 ) b 2 s.e.(b 2 ) ASSUMPTION C.7

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33 To determine whether the estimator is biased or not, we need a greater number of samples. Y t = Y t–1 + u t n = 20 n = 1000 Sample b 1 s.e.(b 1 ) b 2 s.e.(b 2 ) b 1 s.e.(b 1 ) b 2 s.e.(b 2 ) ASSUMPTION C.7

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34 The chart shows the distribution with 1 million samples. This settles the issue. The estimator is biased downwards. Y t = Y t–1 + u t 0.8 mean = (n = 20) ASSUMPTION C.7

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35 There is a further puzzle. If the disturbance terms are drawn randomly from a normal distribution, as was the case in this simulation, and the regression model assumptions are valid, the regression coefficients should also have normal distributions. Y t = Y t–1 + u t 0.8 mean = (n = 20) ASSUMPTION C.7

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36 However the distribution is not normal. It is negatively skewed. Y t = Y t–1 + u t 0.8 mean = (n = 20) ASSUMPTION C.7

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37 Nevertheless the estimator may be consistent, provided that certain conditions are satisfied. Y t = Y t–1 + u t mean = (n = 20) 0.8 ASSUMPTION C.7

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38 When we increase the sample size from 20 to 100, the bias is much smaller. (X has been assigned the values 1, …, 100. The distribution here and for all the following diagrams is for 1 million samples.) Y t = Y t–1 + u t mean = (n = 20) mean = (n = 100) 0.8 ASSUMPTION C.7

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39 If we increase the sample size to 1,000, the bias almost vanishes. (X has been assigned the values 1, …, 1,000.) Y t = Y t–1 + u t mean = (n = 20) mean = (n = 100) mean = (n = 1000) 0.8 ASSUMPTION C.7

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40 Here is a slightly more realistic model with an explanatory variable X t as well as the lagged dependent variable. Y t = X t + 0.8Y t–1 + u t 0.8 mean = (n = 20) mean = (n = 20) ASSUMPTION C.7

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41 The estimate of the coefficient of Y t–1 is again biased downwards, more severely than before (black curve). The coefficient of X t is biased upwards (red curve). Y t = X t + 0.8Y t–1 + u t 0.8 mean = (n = 20) mean = (n = 20) ASSUMPTION C.7

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42 If we increase the sample size to 100, the coefficients are much less biased. Y t = X t + 0.8Y t–1 + u t 0.8 mean = (n = 100) mean = (n = 100) mean = (n = 20) mean = (n = 20) ASSUMPTION C.7

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43 If we increase the sample size to 1,000, the bias almost disappears, as in the previous example. Y t = X t + 0.8Y t–1 + u t 0.8 mean = (n = 1000) mean = (n = 1000) ASSUMPTION C.7

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44 In both of these examples the OLS estimators were consistent, despite being biased for finite samples. We will explain this for the first example. The slope coefficient can be decomposed as shown in the usual way. ASSUMPTION C.7

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45 We will show that the plim of the error term is 0. As it stands, neither the numerator nor the denominator possess limits, so we cannot invoke the plim quotient rule. ASSUMPTION C.7

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46 We divide the numerator and the denominator by n. ASSUMPTION C.7

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47 Now we can invoke the plim quotient rule, because it can be shown that the plim of the numerator is the covariance of Y t–1 and u t and the plim of the denominator is the variance of Y t–1. ASSUMPTION C.7

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48 If Part (1) of Assumption C.7 is valid, the covariance between u t and Y t–1 is zero. In this model it is reasonable to suppose that Part(1) is valid because Y t–1 is determined before u t is generated. ASSUMPTION C.7

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49 Thus the plim of the slope coefficient is equal to the true value and the slope coefficient is consistent. ASSUMPTION C.7

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50 You will often see models with lagged dependent variables in the applied literature. Usually the problem discussed in this slideshow is ignored. This is acceptable if the sample size is large enough, but if the sample is small, there is a risk of serious bias. ASSUMPTION C.7

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51 It is obvious that Part (2) of Assumption C.7 is invalid in models with lagged dependent variables. However it is often invalid in more general models. Consider the two-equation model shown above. ASSUMPTION C.7

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52 It may be reasonable to suppose that u t is distributed independently of X t–1 because it is generated randomly at time t, by which time X t–1 has already been determined. Then Part (1) of Assumption C.7 is valid for the first equation. The same goes for the second equation. ASSUMPTION C.7

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53 However u t is a determinant of Y t, and hence of X t+1. This means that u t is correlated with the X regressor in the first equation in the observations for Y t+2, Y t+4,... etc. Again Part (2) of Assumption C.7 is violated and the OLS estimators will be biased. ASSUMPTION C.7 X t+1 ← Y t ← u t

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54 Since interactions and lags are common in economic models using time series data, the problem of biased coefficients should be taken as the working hypothesis, the rule rather than the exception. ASSUMPTION C.7 X t+1 ← Y t ← u t

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55 Fortunately Part (2) of Assumption C.7 is not required for consistency. Part (1) is a necessary condition. If it is violated, the regression coefficients will be inconsistent. ASSUMPTION C.7 X t+1 ← Y t ← u t

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56 However, Part (1) is not a sufficient condition for consistency because it is possible that the regression estimators may not tend to finite limits as the sample size becomes large. This is a relatively technical issue that will be discussed in Chapter 13. ASSUMPTION C.7 X t+1 ← Y t ← u t

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Copyright Christopher Dougherty These slideshows may be downloaded by anyone, anywhere for personal use. Subject to respect for copyright and, where appropriate, attribution, they may be used as a resource for teaching an econometrics course. There is no need to refer to the author. The content of this slideshow comes from Section 11.5 of C. Dougherty, Introduction to Econometrics, fourth edition 2011, Oxford University Press. Additional (free) resources for both students and instructors may be downloaded from the OUP Online Resource Centre Individuals studying econometrics on their own and who feel that they might benefit from participation in a formal course should consider the London School of Economics summer school course EC212 Introduction to Econometrics or the University of London International Programmes distance learning course 20 Elements of Econometrics

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