Christopher Dougherty EC220 - Introduction to econometrics (chapter 8) Slideshow: Friedman’s critique of OLS estimation of the consumption function Original

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: http://learningresources.lse.ac.uk/134/

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. http://creativecommons.org/licenses/by-sa/3.0/

http://learningresources.lse.ac.uk/

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, CP, is proportional to permanent income, YP.

True model Q = 1 + 2Z + v CP = 2YP


Permanent consumption and income are subjective notions of sustainable, medium-term, consumption and income, respectively. They cannot be measured directly.


2


Actual (measured) income, Y, has two components: permanent income, YP, and transitory income, YT (a random, short-run component). Likewise actual (measured) consumption, C, has two components: permanent consumption, CP, and transitory consumption, CT.


Measurement X = Z + w Y = YP + YT errors Y = Q + r C = CP + CT

3


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 CT and YT.



4


From the first three equations we can derive an equation relating actual consumption to actual income.



True model, in Y = 1 + 2X + v C = 2Y + CT – 2YT

measured + r – 2w = 2Y + uvariables = 1 + 2X + u

5





Assumptions v, w, and r distributed YT and CT distributed

independently of independently of

each other and Z each other and YP and Q and CP


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






22

2

22wZ

w

22

2

22TP

T

YY

Y

OLS2 plim b

Even with these assumptions, OLS would tend to underestimate the propensity to consume, the bias in large samples being as shown.

7






22

2

22wZ

w

22

2

22TP

T

YY

Y

OLS2 plim b

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

True model CP = 2YP CP = 0.9YP

We will illustrate the analysis with a Monte Carlo experiment. We choose 0.9 for the propensity to consume out of permanent income.


9


Data for the YP = 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.


10



Measurement Y = YP + YT YT = 400N(0,1)errors C = CP + CT CT = 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.


11




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.


000,160000,325000,160

9.09.0

22

2

22TP

T

YY

Y

61.029.09.0

OLS2b

12




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, b2

OLS will tend to 0.61 in large samples.


000,160000,325000,160

9.09.0

61.029.09.0 22

2

22TP

T

YY

Y

OLS2b

13

Here is the nonstochastic part of the relationship.


0

500

1000

1500

2000

2500

3000

3500

4000

0 500 1000 1500 2000 2500 3000 3500 4000 YP

CP

14

The part of most interest to us will be enlarged.


0

500

1000

1500

2000

2500

3000

3500

4000

0 500 1000 1500 2000 2500 3000 3500 4000 YP

CP

15

Next we need to add the measurement error to the model, the transitory income.


1200

1700

2200

2700

3200

3700

4200

1200 1700 2200 2700 3200 3700 4200 YP

CP

16

Consumption is unaffected by transitory income, so the observations are shifted in the Y dimension only.


1200

1700

2200

2700

3200

3700

4200

1200 1700 2200 2700 3200 3700 4200 YP,Y

CP

17


The regression line underestimates the true slope.

1200

1700

2200

2700

3200

3700

4200

1200 1700 2200 2700 3200 3700 4200 YP,Y

CP

18

We repeat the experiment with another set of random numbers to generate the transitory component of income in the 20 observations.


1200

1700

2200

2700

3200

3700

4200

1200 1700 2200 2700 3200 3700 4200 YP,Y

CP

19

Again, the regression line underestimates the true slope.


1200

1700

2200

2700

3200

3700

4200

1200 1700 2200 2700 3200 3700 4200 YP,Y

CP

20

We will repeat the experiment again. A fresh set of random numbers is generated for the transitory component of income.


1200

1700

2200

2700

3200

3700

4200

1200 1700 2200 2700 3200 3700 4200

CP

YP,Y

21

Once again the slope is underestimated.


1200

1700

2200

2700

3200

3700

4200

1200 1700 2200 2700 3200 3700 4200 YP,Y

CP

22

Here is a table showing 10 sets of results of the experiment. The true values of 1 and 2 are 0 and 0.9.


Sample b1 s.e.(b1) b2 s.e.(b2)

1 1,001 251 0.56 0.082 755 357 0.62 0.113 756 376 0.68 0.134 668 290 0.66 0.095 675 179 0.64 0.066 982 289 0.57 0.107 918 229 0.56 0.078 625 504 0.66 0.169 918 181 0.58 0.06

10 679 243 0.65 0.08

23



1 1,001 251 0.56 0.082 755 357 0.62 0.113 756 376 0.68 0.134 668 290 0.66 0.095 675 179 0.64 0.066 982 289 0.57 0.107 918 229 0.56 0.078 625 504 0.66 0.169 918 181 0.58 0.06

10 679 243 0.65 0.08

There is a strong downward bias in b2, and its values do seem to be distributed around the limiting value of 0.61.

24



1 1,001 251 0.56 0.082 755 357 0.62 0.113 756 376 0.68 0.134 668 290 0.66 0.095 675 179 0.64 0.066 982 289 0.57 0.107 918 229 0.56 0.078 625 504 0.66 0.169 918 181 0.58 0.06

10 679 243 0.65 0.08

Since 1 is zero, b1 should be randomly positive and negative. b1 is clearly upwards biased as a consequence of the slope being underestimated.

25



1 1,001 251 0.56 0.082 755 357 0.62 0.113 756 376 0.68 0.134 668 290 0.66 0.095 675 179 0.64 0.066 982 289 0.57 0.107 918 229 0.56 0.078 625 504 0.66 0.169 918 181 0.58 0.06

10 679 243 0.65 0.08

The standard errors are invalidated by the measurement error bias and we should not attempt to perform any tests.

26


This chart shows the distribution of the estimates of the slope coefficient when the Monte Carlo simulation was repeated 1,000,000 times.

0

1

2

3

4

5

6

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3

61.0 plim OLS2 b

20n63.0mean

27


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,

0

1

2

3

4

5

6

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3

61.0 plim OLS2 b

20n63.0mean

28


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.

0

1

2

3

4

5

6

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3

61.0 plim OLS2 b

20n63.0mean

29

Copyright Christopher Dougherty 2011.

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 http://www.oup.com/uk/orc/bin/9780199567089/.

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 courseEC212 Introduction to Econometrics http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspxor the University of London International Programmes distance learning course20 Elements of Econometricswww.londoninternational.ac.uk/lse.

11.07.24

Documents

Christopher Dougherty EC220 - Introduction to econometrics (chapter 8) Slideshow: Friedman’s critique of OLS estimation of the consumption function Original