EC220 - Introduction to econometrics (chapter 3)

EC220 - Introduction to econometrics (chapter 3)
Christopher Dougherty EC220 - Introduction to econometrics (chapter 3) Slideshow: alleviation of multicollinearity Original citation: Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 3). [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.

POSSIBLE MEASURES FOR ALLEVIATING MULTICOLLINEARITY
What can you do about multicollinearity if you encounter it? We will discuss some possible measures, looking at the model with two explanatory variables. 1

Before doing this, two important points should be emphasized. First, multicollinearity does not cause the regression coefficients to be biased. 2

The problem is that they have unsatisfactorily large variances. 3

Second, the standard errors and t tests remain valid. The standard errors are larger than they would have been in the absence of multicollinearity, warning us that the regression estimates are erratic. 4

Since the problem of multicollinearity is caused by the variances of the coefficients being unsatisfactorily large, we will seek ways of reducing them. 5

(1) Reduce by including further relevant variables in the model. We will focus on the slope coefficient and look at the various components of its variance. We might be able to reduce it by bringing more variables into the model and reducing su2, the variance of the disturbance term. 6

. reg EARNINGS S EXP EXPSQ Source | SS df MS Number of obs = F( 3, 536) = Model | Prob > F = Residual | R-squared = Adj R-squared = Total | Root MSE = . reg EARNINGS S EXP EXPSQ MALE ASVABC F( 5, 534) = Model | Prob > F = Residual | R-squared = Adj R-squared = Total | Root MSE = However they account for only a small proportion of the variance in earnings and the reduction in the estimate of the variance of the disturbance term is likewise small. 11

(2) Increase the number of observations. Surveys: increase the budget, use clustering The next factor to look at is n, the number of observations. If you are working with cross-section data (individuals, households, enterprises, etc) and you are undertaking a survey, you could increase the size of the sample by negotiating a bigger budget. 16

(2) Increase the number of observations. Surveys: increase the budget, use clustering Alternatively, you could make a fixed budget go further by using a technique known as clustering. You divide the country geographically by zip code or postal area. 17

(2) Increase the number of observations. Surveys: increase the budget, use clustering You select a number of these randomly, perhaps using stratified random sampling to make sure that metropolitan, other urban, and rural areas are properly represented. 18

(2) Increase the number of observations. Surveys: increase the budget, use clustering You then confine the survey to the areas selected. This reduces the travel time and cost of the fieldworkers, allowing them to interview a greater number of respondents. 19

(2) Increase the number of observations. Surveys: increase the budget, use clustering Time series: use quarterly instead of annual data If you are working with time series data, you may be able to increase the sample by working with shorter time intervals for the data, for example quarterly or even monthly data instead of annual data. 20

(3) Increase MSD(X2). A third possible way of reducing the problem of multicollinearity might be to increase the variation in the explanatory variables. This is possible only at the design stage of a survey. 27

(3) Increase MSD(X2). For example, if you were planning a household survey with the aim of investigating how expenditure patterns vary with income, you should make sure that the sample included relatively rich and relatively poor households as well as middle-income households. 28

(4) Reduce Another possibility might be to reduce the correlation between the explanatory variables. This is possible only at the design stage of a survey and even then it is not easy. 29

(5) Combine the correlated variables. If the correlated variables are similar conceptually, it may be reasonable to combine them into some overall index. 30

(6) Drop some of the correlated variables. Dropping some of the correlated variables, if they have insignificant coefficients, may alleviate multicollinearity. 33

(6) Drop some of the correlated variables. However, this approach to multicollinearity is dangerous. It is possible that some of the variables with insignificant coefficients really do belong in the model and that the only reason their coefficients are insignificant is because there is a problem of multicollinearity. 34

(6) Drop some of the correlated variables. If that is the case, their omission may cause omitted variable bias, to be discussed in Chapter 6. 35

(7) Empirical restriction A further way of dealing with the problem of multicollinearity is to use extraneous information, if available, concerning the coefficient of one of the variables. 36

(7) Empirical restriction For example, suppose that Y in the equation above is the demand for a category of consumer expenditure, X is aggregate disposable personal income, and P is a price index for the category. 37

(7) Empirical restriction To fit a model of this type you would use time series data. If X and P are highly correlated, which is often the case with time series variables, the problem of multicollinearity might be eliminated in the following way. 38

(7) Empirical restriction Obtain data on income and expenditure on the category from a household survey and regress Y' on X'. (The ' marks are to indicate that the data are household data, not aggregate data.) 39

(7) Empirical restriction This is a simple regression because there will be relatively little variation in the price paid by the households. 40

(7) Empirical restriction Now substitute b' for b2 in the time series model. Subtract b' X from both sides, and regress Z = Y – b' X on price. This is a simple regression, so multicollinearity has been eliminated. 2 2 2 41

(7) Empirical restriction There are some problems with this technique. First, the b2 coefficients may be conceptually different in time series and cross-section contexts. 42

(7) Empirical restriction Second, since we subtract the estimated income component b' X, not the true income component b 2X, from Y when constructing Z, we have introduced an element of measurement error in the dependent variable. 2 43

(8) Theoretical restriction Last, but by no means least, is the use of a theoretical restriction, which is defined as a hypothetical relationship among the parameters of a regression model. 44

(8) Theoretical restriction It will be explained using an educational attainment model as an example. Suppose that we hypothesize that highest grade completed, S, depends on ASVABC, and highest grade completed by the respondent's mother and father, SM and SF, respectively. 45

(8) Theoretical restriction Suppose that we hypothesize that mother's and father's education are equally important. We can then impose the restriction b3 = b4. 51

(8) Theoretical restriction This allows us to rewrite the equation as shown. 52

(8) Theoretical restriction Defining SP to be the sum of SM and SF, the equation may be rewritten as shown. The problem caused by the correlation between SM and SF has been eliminated. 53

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 3.4 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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