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Understanding Multicollinearity in Regression Analysis

Learn about detecting, consequences, and solutions for multicollinearity in regression models, with emphasis on high standard errors due to intercorrelated variables. Explore illustrative examples and potential solutions suggested in literature.

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Understanding Multicollinearity in Regression Analysis

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  1. Chapter 7 Multicollinearity

  2. What is in this Chapter? • How do we detect this problem? • What are the consequences? • What are the solutions? • An example by Gauss

  3. What is in this Chapter? • In Chapter 4 we stated that one of the assumptions in the basic regression model is that the explanatory variables are not exactly linearly related. If they are, then not all parameters are estimable • What we are concerned with in this chapter is the case where the individual parameters are not estimable with sufficient precision (because of high standard errors) • This often occurs if the explanatory variables are highly intercorrelated (although this condition is not necessary).

  4. What is in this Chapter? • This chapter is very important, because multicollinearity is one of the most misunderstood problems in multiple regression • There have been several measures for multicollinearity suggested in the literature (variance-inflation factors VIF, condition numbers CN, etc.) • This chapter argues that all these are useless and misleading • They all depend on the correlation structure of the explanatory variables only.

  5. What is in this Chapter? • It is argued here that this is only one of several factors determining high standard errors • High intercorrelations among the explanatory variables are neither necessary nor sufficient to cause the multicollinearity problem • The best indicators of the problem are the t-ratios of the individual coefficients.

  6. What is in this Chapter? • This chapter also discusses the solutions offered for the multicollinearity problem: • principal component regression • dropping of variables • However, they are ad hoc and do not help • The only solutions are to get more data or to seek prior information

  7. 7.1 Introduction • Very often the data we use in multiple regression analysis cannot give decisive (significant) answers to the questions we pose. • This is because the standard errors are very high or the t-ratios are very low. • This sort of situation occurs when the explanatory variables display little variation and/or high intercorrelations.

  8. 7.1 Introduction • The situation where the explanatory variables are highly intercorrelated is referred to as multicollinearity • When the explanatory variables are highly intercorrelated, it becomes difficult to disentangle the separate effects of each of the explanatory variables on the explained variable

  9. 7.2 Some Illustrative Examples • Thus only(β1 +2β2) would be estimable. • We cannot get estimates of β1 and β2 separately. • In this case we say that there is “perfect multicollinearity,” because x1 and x2 are perfectly correlated (with =1). • In actual practice we encounter cases where r2 is not exactly 1 but close to 1.

  10. 7.2 Some Illustrative Examples • As an illustration, consider the case where so that the normal equations are • The solution is . • Suppose that we drop an observation and the new values are

  11. 7.2 Some Illustrative Examples • Now when we solve the equations • We get

  12. 7.2 Some Illustrative Examples • Thus very small changes in the variances and covariances produce drastic changes in the estimates of regression parameters. • It is easy to see that the correlation coefficient between the two explanatory variables is given by which is very high.

  13. 7.2 Some Illustrative Examples • In practice, addition or deletion of observations would produce changes in the variances and covariances • Thus one of the consequences of high correlation between x1 and x2 is that the parameter estimates would be very sensitive to the addition or deletion of observations • This aspect of multicollinearity can be checked in practice by deleting or adding some observations and examining the sensitivity of the estimates to such perturbations

  14. 7.2 Some Illustrative Examples

  15. 7.2 Some Illustrative Examples Thus the variance of will be high if: 1. σ2 is high. 2. S11 is low. 3. is high.

  16. 7.2 Some Illustrative Examples • Even if is high, if σ2 is low and S11 high, we will not have the problem of high standard errors. • On the other hand, even if is low, the standard errors can be high if σ2 is high and S11 is low (i.e., there is not sufficient variation in x1). • What this suggests is that high value of do not tell us anything whether we have a multicollinearity problem or not. • When we have more than two explanatory variables, the simple correlations among them become all the more meaningless.

  17. 7.2 Some Illustrative Examples • As an illustration, consider the following example with 20 observations on x1, x2, and x3: x1 =(1, 1, 1, 1, 1, 0, 0, 0, 0, 0, and 10 zeros) x2 =(0, 0, 0, 0, 0, 1, 1, 1, 1, 1, and 10 zeros) x3 =(1, 1, 1, 1, 1, 1, 1, 1, 1, 1, and 10 zeros)

  18. 7.2 Some Illustrative Examples • Obviously, x3=x1+x2 and we have perfect multicollinearity. • But we can see that ,and thus the simple correlations are not high. • In the case of more than two explanatory variables, what we have to consider are multiple correlations of each of the explanatory variables with the other explanatory variables.

  19. 7.2 Some Illustrative Examples • Note that the standard error formulas corresponding to equations (7.1) and (7.2) are where σ2 and Sii are defined as before in the case of two explanatory variables and represents the squared multiple correlation coefficient between xi and the other explanatory variables.

  20. 7.3 Some Measures of Multicollinearity • It is important to be familiar with two measures that are often suggested in the discussion of multicollinearity : the variance inflation factor (VIF) and the condition number (CN). • The VIF is defined as where is the squared multiple correlation coefficient between xi and the other explanatory variables.

  21. 7.3 Some Measures of Multicollinearity • A measure that considers the correlations of the explanatory variable with the explained variable is Theil’s measure, which is defined as where R2 = squared multiple correlation from a regression of y on x1, x2,…..,xk = squared multiple correlation from a regression of y on x1, x2,…..,xk with xi omitted

  22. 7.3 Some Measures of Multicollinearity • The quantity is termed the “incremental contribution” to the squared multiple correlation by Theil. • If x1, x2,…..,xk are mutually uncorrelated, then m will be 0 because the incremental contributions all add up to . • In other cases m can be negative as well as highly positive.

  23. 7.4 Problems with Measuring Multicollinearity • Let us define C= real consumption per capita Y= real per capita current income Yp= real per capita permanent income YT= real per capita transitory income Y=YT+Yp and Yp and YT are uncorrelated

  24. 7.4 Problems with Measuring Multicollinearity • Suppose that we formulate the consumption function as • All these equation are equivalent. However, the correlations between the explanatory variables will be different depending in which of the three equations is considered. • In equation (7.5), since Y and Yp are often highly correlated, we would say that there is high multicollinearity.

  25. 7.4 Problems with Measuring Multicollinearity • In equation (7.6), since YT and Yp are uncorrelated, we would say that there is no multicollinearity. • However, the two equations are essentially the same. • What we should be talking about is the precision with which α and β or (α+β ) are estimable.

  26. 7.4 Problems with Measuring Multicollinearity Consider, for instance, the following data：

  27. 7.4 Problems with Measuring Multicollinearity • For these data the estimation of equation (7.5) gives (figures in parentheses are standard errors) • One reason for the imprecision in the estimates is that Y and Yp are highly correlated (the correlation coefficient is 0.95).

  28. 7.4 Problems with Measuring Multicollinearity • For equation (7.6) the correlation between the explanatory variables is zero and for equation (7.7) it is 0.32. • The least squares estimates of α and β are no more precise in equation (7.6) or (7.7). • Let us consider the estimation of equation (7.6). We get

  29. 7.4 Problems with Measuring Multicollinearity • The estimate at (α+β) is thus 0.89 and the standard error is 0.11. • Thus (α+β) is indeed more precisely estimated than either αorβ. • As for α, it is not precisely estimated even though the explanatory variables in this equation are uncorrelated. • The reason is that the variance of YT is very low [ see formula (7.1)]

  30. 7.4 Problems with Measuring Multicollinearity • In Table 7.1 we present data in C, Y, and L for the period from the first quarter of 1952 to the second quarter of 1961. • C is consumption expenditures, Y is disposable income, and L is liquid assets at the end of the previous quarter. • All figures are in billions of 1954 dollars. • Using the 38 observations we get the following regression equations .

  31. 7.4 Problems with Measuring Multicollinearity

  32. 7.4 Problems with Measuring Multicollinearity • Equation(7.10) shows that L and Y are very highly correlated. • In fact, substituting the value of L in terms Y from (7.10) into equation (7.9) and simplifying, we get equation (7.8) correct to four decimal place! • However, looking at the t-ratios in equation (7.9) we might conclude that multicollinearity is not a problem.

  33. 7.4 Problems with Measuring Multicollinearity • Are we justified in this conclusion? • Let us consider the stability of the coefficients with deletion of some observations. • Using only the first 36 observations we get the following results:

  34. 7.4 Problems with Measuring Multicollinearity • Comparing equation (7.11) with (7.8) and equation (7.12) with (7.9) we see that the coefficients in the latter equation show far greater changes than in the former equation. • Of course, if one applies the tests for stability discussed in Section 4.11, one might conclude that the results are not statistically significant at the 5% level. • Note that the test for stability that we use us the “predictive” test for stability.

  35. 7.4 Problems with Measuring Multicollinearity • Finally, we might consider predicting C for the first two quarters of 1961 using equations (7.11) and (7.12). • The predictions are:

  36. 7.4 Problems with Measuring Multicollinearity • Thus the prediction from the equation including L is further off from the true value than the predictions from the equations excluding L. • Thus if prediction was the sole criterion, one might as well drop the variable L.

  37. 7.4 Problems with Measuring Multicollinearity • The example above illustrates four different ways of looking at the multicollinearity problem: • 1. Correlation between the explanatory variables L and Y, which is high. • 2. Standard errors or t-ratios for the estimated coefficients • In this example the t-ratios are significant, suggesting that multicollinearity might not be serious.

  38. 7.4 Problems with Measuring Multicollinearity • 3. Stability of the estimated coefficients when some observations are deleted. • 4. Examining the predictions from the model: • If multicollinearity is a serious problem, the predictions from the model would be worse than those from a model that includes only a subset of the set of explanatory variables.

  39. 7.4 Problems with Measuring Multicollinearity • The last criterion should be applied if prediction is the object of the analysis. Otherwise, it would be advisable to consider the second and third criteria. • The first criterion is not useful, as we have so frequently emphasized.

  40. 7.6 Principal Component Regression • Another solution that is often suggested for the multicollinearity problem is the principal component regression. • Suppose that we k explanatory variables. • Then we can consider linear functions of these variables

  41. 7.6 Principal Component Regression • Suppose we choose the a’s so that the variance of z1 is maximized subject to the condition that • This is called the normalization condition. • Z1 is then said to be the first principal component. • It is the linear function of the x’s that has the highest variance.

  42. 7.6 Principal Component Regression • The process of maximizing the variance of the linear function z subject to the condition that the sum of squares of the coefficients of the x’s is equal to 1, produces k solutions. • Corresponding to these we construct k linear functions z1, z2,…..,zk. These are called the principal components of the x’s.

  43. 7.6 Principal Component Regression • They can be ordered so that var(z1)>var(z2)>…..>var(zk) • z1, the one with the highest variance, is called the first principal component • z2, with the next highest variance, is called the second principal component, and so on • Unlike the x’s, which are correlated, the z’s are orthogonal or uncorrelated.

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