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Christopher Dougherty EC220 - Introduction to econometrics (chapter 8) Slideshow: Friedman’s critique of OLS estimation of the consumption function Original citation: Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 8). [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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FRIEDMAN’S CRITIQUE OF OLS ESTIMATION OF THE CONSUMPTION FUNCTION 1 Milton Friedman’s Permanent Income Hypothesis provides a classic example of measurement error theory. The basic idea is that permanent consumption, C P, is proportional to permanent income, Y P. True model Q= 1 + 2 Z + vC P = 2 Y P

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FRIEDMAN’S CRITIQUE OF OLS ESTIMATION OF THE CONSUMPTION FUNCTION Permanent consumption and income are subjective notions of sustainable, medium-term, consumption and income, respectively. They cannot be measured directly. True model Q= 1 + 2 Z + vC P = 2 Y P 2

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FRIEDMAN’S CRITIQUE OF OLS ESTIMATION OF THE CONSUMPTION FUNCTION Actual (measured) income, Y, has two components: permanent income, Y P, and transitory income, Y T (a random, short-run component). Likewise actual (measured) consumption, C, has two components: permanent consumption, C P, and transitory consumption, C T. True model Q= 1 + 2 Z + vC P = 2 Y P Measurement X= Z + wY= Y P + Y T errors Y= Q + rC= C P + C T 3

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FRIEDMAN’S CRITIQUE OF OLS ESTIMATION OF THE CONSUMPTION FUNCTION According to the Permanent Income Hypothesis, an OLS regression of actual consumption on actual income will be subject to measurement error in both the dependent and the explanatory variables, the measurement errors being the transitory components C T and Y T. True model Q= 1 + 2 Z + vC P = 2 Y P Measurement X= Z + wY= Y P + Y T errors Y= Q + rC= C P + C T 4

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FRIEDMAN’S CRITIQUE OF OLS ESTIMATION OF THE CONSUMPTION FUNCTION From the first three equations we can derive an equation relating actual consumption to actual income. True model Q= 1 + 2 Z + vC P = 2 Y P Measurement X= Z + wY= Y P + Y T errors Y= Q + rC= C P + C T True model, in Y= 1 + 2 X + vC= 2 Y + C T – 2 Y T measured + r – 2 w= 2 Y + u variables = 1 + 2 X + u 5

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True model Q= 1 + 2 Z + vC P = 2 Y P Measurement X= Z + wY= Y P + Y T errors Y= Q + rC= C P + C T True model, in Y= 1 + 2 X + vC= 2 Y + C T – 2 Y T measured + r – 2 w= 2 Y + u variables = 1 + 2 X + u Assumptions v, w, and r distributed Y T and C T distributedindependently of each other and Z each other and Y P and Q and C P FRIEDMAN’S CRITIQUE OF OLS ESTIMATION OF THE CONSUMPTION FUNCTION To simplify the analysis, we will assume that the transitory components of consumption and income are independent of their permanent components and of each other. 6

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True model Q= 1 + 2 Z + vC P = 2 Y P Measurement X= Z + wY= Y P + Y T errors Y= Q + rC= C P + C T True model, in Y= 1 + 2 X + vC= 2 Y + C T – 2 Y T measured + r – 2 w= 2 Y + u variables = 1 + 2 X + u FRIEDMAN’S CRITIQUE OF OLS ESTIMATION OF THE CONSUMPTION FUNCTION Even with these assumptions, OLS would tend to underestimate the propensity to consume, the bias in large samples being as shown. 7

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True model Q= 1 + 2 Z + vC P = 2 Y P Measurement X= Z + wY= Y P + Y T errors Y= Q + rC= C P + C T True model, in Y= 1 + 2 X + vC= 2 Y + C T – 2 Y T measured + r – 2 w= 2 Y + u variables = 1 + 2 X + u FRIEDMAN’S CRITIQUE OF OLS ESTIMATION OF THE CONSUMPTION FUNCTION If the assumptions concerning the covariances of the transitory components are relaxed, the analysis has to be modified. See, for example, the discussion of the Liviatan study in the text. 8

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True model C P = 2 Y P C P = 0.9Y P We will illustrate the analysis with a Monte Carlo experiment. We choose 0.9 for the propensity to consume out of permanent income. FRIEDMAN’S CRITIQUE OF OLS ESTIMATION OF THE CONSUMPTION FUNCTION 9

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True model C P = 2 Y P C P = 0.9Y P Data for the Y P = 2,000, 2,100, regressor(s)..., 3,900 The sample size is 20 and the data for permanent income are 2,000, 2,100, increasing in steps of 100 to 3,900. FRIEDMAN’S CRITIQUE OF OLS ESTIMATION OF THE CONSUMPTION FUNCTION 10

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True model C P = 2 Y P C P = 0.9Y P Data for the Y P = 2,000, 2,100, regressor(s)..., 3,900 Measurement Y= Y P + Y T Y T = 400N(0,1) errors C= C P + C T C T = 0 The measurement error in income (the transitory component) will be generated from a normal distribution with mean 0 and unit variance, scaled up by a factor of 400. FRIEDMAN’S CRITIQUE OF OLS ESTIMATION OF THE CONSUMPTION FUNCTION 11

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True model C P = 2 Y P C P = 0.9Y P Data for the Y P = 2,000, 2,100, regressor(s)..., 3,900 Measurement Y= Y P + Y T Y T = 400N(0,1) errors C= C P + C T C T = 0 plim Why the scaling factor of 400? The large sample bias depends, in part, on the variance of the measurement error, and if a scaling factor had not been used, the bias would have been imperceptible. FRIEDMAN’S CRITIQUE OF OLS ESTIMATION OF THE CONSUMPTION FUNCTION 12

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True model C P = 2 Y P C P = 0.9Y P Data for the Y P = 2,000, 2,100, regressor(s)..., 3,900 Measurement Y= Y P + Y T Y T = 400N(0,1) errors C= C P + C T C T = 0 plim The scaling factor of 400 was chosen, after a bit of trial and error, because it produced a visible bias. Given our choice of parameters and data, b 2 OLS will tend to 0.61 in large samples. FRIEDMAN’S CRITIQUE OF OLS ESTIMATION OF THE CONSUMPTION FUNCTION 13

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Here is the nonstochastic part of the relationship. FRIEDMAN’S CRITIQUE OF OLS ESTIMATION OF THE CONSUMPTION FUNCTION YPYP CPCP 14

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The part of most interest to us will be enlarged. FRIEDMAN’S CRITIQUE OF OLS ESTIMATION OF THE CONSUMPTION FUNCTION YPYP CPCP 15

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Next we need to add the measurement error to the model, the transitory income. FRIEDMAN’S CRITIQUE OF OLS ESTIMATION OF THE CONSUMPTION FUNCTION YPYP CPCP 16

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Consumption is unaffected by transitory income, so the observations are shifted in the Y dimension only. FRIEDMAN’S CRITIQUE OF OLS ESTIMATION OF THE CONSUMPTION FUNCTION YP,YYP,Y CPCP 17

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FRIEDMAN’S CRITIQUE OF OLS ESTIMATION OF THE CONSUMPTION FUNCTION The regression line underestimates the true slope. YP,YYP,Y CPCP 18

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We repeat the experiment with another set of random numbers to generate the transitory component of income in the 20 observations. FRIEDMAN’S CRITIQUE OF OLS ESTIMATION OF THE CONSUMPTION FUNCTION YP,YYP,Y CPCP 19

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Again, the regression line underestimates the true slope. FRIEDMAN’S CRITIQUE OF OLS ESTIMATION OF THE CONSUMPTION FUNCTION YP,YYP,Y CPCP 20

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We will repeat the experiment again. A fresh set of random numbers is generated for the transitory component of income. FRIEDMAN’S CRITIQUE OF OLS ESTIMATION OF THE CONSUMPTION FUNCTION CPCP YP,YYP,Y 21

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Once again the slope is underestimated. FRIEDMAN’S CRITIQUE OF OLS ESTIMATION OF THE CONSUMPTION FUNCTION YP,YYP,Y CPCP 22

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Here is a table showing 10 sets of results of the experiment. The true values of 1 and 2 are 0 and 0.9. FRIEDMAN’S CRITIQUE OF OLS ESTIMATION OF THE CONSUMPTION FUNCTION Sample b 1 s.e.(b 1 ) b 2 s.e.(b 2 ) 11,

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FRIEDMAN’S CRITIQUE OF OLS ESTIMATION OF THE CONSUMPTION FUNCTION Sample b 1 s.e.(b 1 ) b 2 s.e.(b 2 ) 11, There is a strong downward bias in b 2, and its values do seem to be distributed around the limiting value of

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FRIEDMAN’S CRITIQUE OF OLS ESTIMATION OF THE CONSUMPTION FUNCTION Sample b 1 s.e.(b 1 ) b 2 s.e.(b 2 ) 11, Since 1 is zero, b 1 should be randomly positive and negative. b 1 is clearly upwards biased as a consequence of the slope being underestimated. 25

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FRIEDMAN’S CRITIQUE OF OLS ESTIMATION OF THE CONSUMPTION FUNCTION Sample b 1 s.e.(b 1 ) b 2 s.e.(b 2 ) 11, The standard errors are invalidated by the measurement error bias and we should not attempt to perform any tests. 26

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FRIEDMAN’S CRITIQUE OF OLS ESTIMATION OF THE CONSUMPTION FUNCTION This chart shows the distribution of the estimates of the slope coefficient when the Monte Carlo simulation was repeated 1,000,000 times. mean 27

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FRIEDMAN’S CRITIQUE OF OLS ESTIMATION OF THE CONSUMPTION FUNCTION It can be seen that the probability limit for the slope coefficient, 0.61, does provide a good guide to the central tendency of the distribution of the slope coefficient, even though the sample size, 20, in each simulation is quite small, mean 28

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FRIEDMAN’S CRITIQUE OF OLS ESTIMATION OF THE CONSUMPTION FUNCTION The mean value of the slope coefficient in the Monte Carlo experiment was 0.63, so it appears that the probability limit overstates the finite sample bias, at least for n = 20, but not by much. mean 29

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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 8.4 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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