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96

"The data are those used in an illustrative example by Gary Smith, "An Example of Ridge Regression Difficulties," Canadian Journal of Statistics, Vol. 8, No. 2, 1980, pp. 217-225.

Let us define

= real consumption per capita

Y = real per capita current income

Yp = real per capita permanent income Yj = real per capita transitory income

Y = Yj + Yp and Yp and Yy are uncorrelated Suppose that we formulate the consumption function as

= aY+ fiYp ¥u (7.5)

This equation can alternatively be written as

= , + (a + fi)Yp + (7.6)

= (a + ) - fiYr + (7.7)

All these equations are equivalent. However, the corelations between the explanatory variables will be different depending on 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. In equation (7.6), since Y 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 a and (3 or (a + (3) are estimable. Consider, for instance, the following data.*

var(C) = 7.3 cov(C, Y) = 8.3 cov(T, Yp) = 9.0 var(F) = 10.0 cov(C, Yp) = 8.0 cov(y, Yt) = 1.0

\ar{Yp) = 9.0 cov(C, Yj) = 0.3 cov{Yp, Yj) = 0 by definition

var(Tr) = LO

For these data the estimation of equation (7.5) gives (all variables measured as deviations from their means)

= 0.30F + 0.59 Fp &l = 0.1

(0 32) (0 33)

(The 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).



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 a and p are no more precise in equation (7.6) or (7.7).

Let us consider the estimation of equation (7.6). We get

= . + 0.89

(0 32) (0.11)

The estimate at (a + P) is thus 0.89 and the standard error is 0.11. Thus a + P is indeed more precisely estimated than either a or p. As for a, it is not precisely estimated even though the explanatory varWbles in this equation are uncorrelated. The reason is that the variance of is very low [see formula (7.1)].

We can summarize the conclusions from this illustrative example as follows:

1. It is difficult to define multicoUinearity in terms of the correlations between the explanatory variables because the explanatory variables can be redefined in a number of different ways and these can give drastically different measures of intercorrelations. In some cases, these redefinitions may not make sense, but in the example above involving measured income, permanent income, and transitory income, these redefinitions make sense.

2. Just because the explanatory variables are uncorrelated it does not mean that we have no problems with inference. Note that the estimate of a and its standard error are the same in equation (7.5) (with the correlation among the explanatory variables equal to 0.95) and in equation (7.6) (with the explanatory variables uncorrelated).

3. Often, though the individual parameters are not precisely estimable, some linear combinations of the parameters are. For instance, in our example, a + p is estimable with good precision. Sometimes, these linear combinations do not make economic sense. But at other times they do.

We will present yet another example to illustrate some problems in judging whether multicoUinearity is serious or not and also to illustrate the fact that even if individual parameters are not estimable with precision, some linear functions of the parameters are.

In Table 7.1 we present data on C, Y, and L for the period from the first quarter of 1952 to the second quarter of 1961. is consumption expenditures, 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:

= -7.160 + 0.95213 /-2 = 0.9933 (7.8)

(-1.93) (73.25)

= -10.627 + 0.68166 + 0.37252Z, JR = 0.9953 (7.9)

(-3.25) . (9.60) (3.%)

The data are from Z. Griliches et al., "Notes on Estimated Aggregate Quarterly Consumption Functions," Econometnca, July 1962.



7.4 PROBLEMS WITH MEASURING MULTICOLLINEARITY

Table 7.1 Data on Consumption, Income, and Liquid Assets

Year,

Year,

Quarter

Quarter

1952

220.0

238.1

182.7

1957

268.9

291.1

218.2

222.7

240.9

183.0

270.4

294.6

218.5

223.8

245.8

184.4

273.4

296.1

219.8

230.2

248.8

187.0

272.1

29.3

219.5

1953

234.0

253.3

189.4

1958

268.9

291.3

220.5

236.2

256.1

192.2

270.9

292.6

222.7

236.0

255.9

193.8

274.4

299.9

225.0

234.1

255.9

194.8

278.7

302.1

229.4

1954

233.4

254.4

197.3

1959

283.8

305.9

232.2

236.4

254.8

197.0

289.7

312.5

235.2

239.0

257.0

200.3

290.8

311.3

237.2

243.2

260.9

204.2

292.8

313.2

237.7

1955

248.7

263.0

207.6

1960

295.4

315.4

238.0

253.7

271.5

209.4

299.5

320.3

238.4

259.9

276.5

211.1

298.6

321.0

240.1

261.8

281.4

213.2

299.6

320.1

243.3

1956

263.2

282.0

214.1

1961

297.0

318.4

246.1

263.7

286.2

216.5

301.6

324.8

250.0

263.4

287.7

217.3

266.9

291.0

217.3

9.307

(1.80)

+ 0.76207

(37.20)

r LY

= 0.9758

(7.10)

(Figures in parentheses are r-ratios, not standard errors.)

Equation (7.10) shows that L and are very highly correlated. In fact, substituting the value of L in terms of from (7.10) into equation (7.9) and simplifying, we get equation (7.8) correct to four decimal places! However, looking at the r-ratios in equation (7.9) we might conclude that multicollinearity is not a problem.

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:

= -6.980 + 0.95145F

(-1.74) (67 04)

= -13.391

(-3 71)

+ 0.63258 + 0.45065L

(8.32) (4 24)

= 0.9925

= 0.9951

L = 14.255 + 0.70758

(2 69) (37 80)

= 0.9768

(7.11)

(7.12)

(7-. 13)

Comparing (7.11) with (7.8) and (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



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