Chapter 6: Further Issues
Chapter 6 in Wooldridge (2013) deals with some more issues that might have an impact on your estimates. Among these are a change in the scale of your variables, so-called beta-factors and the interpretation of variables that appear together with their squared values, i.e. interaction terms.
Before you start, set your working directory, load the foreign library and download the data we are going to use in this chapter.
library(QuantPsyc)
library(wooldridge)The impact of scales on the estimated coefficients
Table 6.1
data("bwght")
lm.1 <- lm(bwght ~ cigs + faminc, data = bwght)
lm.2 <- lm(bwghtlbs ~ cigs + faminc, data = bwght)
lm.3 <- lm(bwght ~ packs + faminc, data = bwght)
summary(lm.1)##
## Call:
## lm(formula = bwght ~ cigs + faminc, data = bwght)
##
## Residuals:
## Min 1Q Median 3Q Max
## -96.061 -11.543 0.638 13.126 150.083
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 116.97413 1.04898 111.512 < 2e-16 ***
## cigs -0.46341 0.09158 -5.060 4.75e-07 ***
## faminc 0.09276 0.02919 3.178 0.00151 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 20.06 on 1385 degrees of freedom
## Multiple R-squared: 0.0298, Adjusted R-squared: 0.0284
## F-statistic: 21.27 on 2 and 1385 DF, p-value: 7.942e-10summary(lm.2)##
## Call:
## lm(formula = bwghtlbs ~ cigs + faminc, data = bwght)
##
## Residuals:
## Min 1Q Median 3Q Max
## -6.0038 -0.7215 0.0399 0.8204 9.3802
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 7.310883 0.065562 111.512 < 2e-16 ***
## cigs -0.028963 0.005724 -5.060 4.75e-07 ***
## faminc 0.005798 0.001824 3.178 0.00151 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.254 on 1385 degrees of freedom
## Multiple R-squared: 0.0298, Adjusted R-squared: 0.0284
## F-statistic: 21.27 on 2 and 1385 DF, p-value: 7.942e-10summary(lm.3)##
## Call:
## lm(formula = bwght ~ packs + faminc, data = bwght)
##
## Residuals:
## Min 1Q Median 3Q Max
## -96.061 -11.543 0.638 13.126 150.083
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 116.97413 1.04898 111.512 < 2e-16 ***
## packs -9.26815 1.83154 -5.060 4.75e-07 ***
## faminc 0.09276 0.02919 3.178 0.00151 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 20.06 on 1385 degrees of freedom
## Multiple R-squared: 0.0298, Adjusted R-squared: 0.0284
## F-statistic: 21.27 on 2 and 1385 DF, p-value: 7.942e-10Beta coefficients (Example 6.1)
To estimate beta coefficients I use the QuantPsyc package. Its lm.beta function is neat, however, it does not provide standard errors of the coefficients. Thus, I use the Make.Z() function to obtain z-values for the whole sample, turn it into a data frame and estimate the model as usual.
data("hprice2")
lm.1 <- lm(price ~ nox + crime + rooms + dist + stratio, data = hprice2)
lm.beta(lm.1) # Beta coefficients with no standard errors.## nox crime rooms dist stratio
## -0.3404460 -0.1432828 0.5138878 -0.2348385 -0.2702799hprice2.2 <- as.data.frame(Make.Z(hprice2)) # Generate z-values
lm.2 <- lm(price ~ nox + crime + rooms + dist + stratio, data = hprice2.2)
summary(lm.1)##
## Call:
## lm(formula = price ~ nox + crime + rooms + dist + stratio, data = hprice2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -13914 -3201 -662 2110 38064
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 20871.13 5054.60 4.129 4.27e-05 ***
## nox -2706.43 354.09 -7.643 1.09e-13 ***
## crime -153.60 32.93 -4.665 3.97e-06 ***
## rooms 6735.50 393.60 17.112 < 2e-16 ***
## dist -1026.81 188.11 -5.459 7.57e-08 ***
## stratio -1149.20 127.43 -9.018 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5586 on 500 degrees of freedom
## Multiple R-squared: 0.6357, Adjusted R-squared: 0.632
## F-statistic: 174.5 on 5 and 500 DF, p-value: < 2.2e-16summary(lm.2)##
## Call:
## lm(formula = price ~ nox + crime + rooms + dist + stratio, data = hprice2.2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.5110 -0.3476 -0.0719 0.2291 4.1334
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 4.744e-16 2.697e-02 0.000 1
## nox -3.404e-01 4.454e-02 -7.643 1.09e-13 ***
## crime -1.433e-01 3.072e-02 -4.665 3.97e-06 ***
## rooms 5.139e-01 3.003e-02 17.112 < 2e-16 ***
## dist -2.348e-01 4.302e-02 -5.459 7.57e-08 ***
## stratio -2.703e-01 2.997e-02 -9.018 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.6066 on 500 degrees of freedom
## Multiple R-squared: 0.6357, Adjusted R-squared: 0.632
## F-statistic: 174.5 on 5 and 500 DF, p-value: < 2.2e-16Joint interpretation of different transformations of an exogenous variable
If you use simple and squared values of an exogenous variable in your regression, you have to interpret them jointly. In this example I estimate the model, save the summary and calculate the value of the exogenous variable at which the joint effect is highest by extracting the coefficients into the solution of the first derivation of the model formula.
# Equation 6.12
data("wage1")
lm.1 <- lm(wage ~ exper + expersq, data = wage1)
s.1 <- summary(lm.1)
abs(s.1$coefficient[2, 1] / (2 * s.1$coefficient[3, 1]))## [1] 24.3153Example 6.2
This example is similar to the previous. Again, one exogenous variable is squared and used as an additional estimator to take account of a possible change in the parameter with an increase in the size of the variable. The curve function uses the estimated parameters to simulate such a relationship. The effect on the endogenous variable first decreases at lower values of x, i.e. the number of rooms, and starts to increase with higher values.
data("hprice2")
ldist <- log(hprice2$dist)
rooms.sq <- hprice2$rooms^2
lm.1 <- lm(lprice ~ lnox + ldist + rooms + rooms.sq + stratio, data = hprice2)
summary(lm.1)##
## Call:
## lm(formula = lprice ~ lnox + ldist + rooms + rooms.sq + stratio,
## data = hprice2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.04285 -0.12774 0.02038 0.12650 1.25272
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 13.385478 0.566473 23.630 < 2e-16 ***
## lnox -0.901682 0.114687 -7.862 2.34e-14 ***
## ldist -0.086781 0.043281 -2.005 0.04549 *
## rooms -0.545113 0.165454 -3.295 0.00106 **
## rooms.sq 0.062261 0.012805 4.862 1.56e-06 ***
## stratio -0.047590 0.005854 -8.129 3.42e-15 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.2592 on 500 degrees of freedom
## Multiple R-squared: 0.6028, Adjusted R-squared: 0.5988
## F-statistic: 151.8 on 5 and 500 DF, p-value: < 2.2e-16# Figure 6.2
curve(lm.1$coefficients[1] + lm.1$coefficients[4] * x + lm.1$coefficients[5] * x^2, xlim = c(3, 9), ylab = "Effect")
Example 6.3
This example looks at interaction terms. I use the with function just to spare the attend$-part before each of the variables. I calculate squared values and the interaction term, estimate both models, print the summary - note that you have to use the print command since R will not return this line of code otherwise - and calculate an F-test via anova. The last line gives thee marginal effect of the attendance rate on the outcome variable.
The problem with this coefficient is that there is no standard error for it. This is achieved by demeaning the second variable of the interaction term and estimating the model again.
data("attend")
lm.1 <- lm(stndfnl ~ atndrte + priGPA + ACT + I(priGPA^2) + I(ACT^2) + I(priGPA*atndrte), data = attend)
lm.2 <- lm(stndfnl ~ priGPA + ACT + I(priGPA^2) + I(ACT^2), data = attend)
summary(lm.1)##
## Call:
## lm(formula = stndfnl ~ atndrte + priGPA + ACT + I(priGPA^2) +
## I(ACT^2) + I(priGPA * atndrte), data = attend)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.1698 -0.5316 -0.0177 0.5737 2.3344
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.050293 1.360319 1.507 0.132225
## atndrte -0.006713 0.010232 -0.656 0.512005
## priGPA -1.628540 0.481003 -3.386 0.000751 ***
## ACT -0.128039 0.098492 -1.300 0.194047
## I(priGPA^2) 0.295905 0.101049 2.928 0.003523 **
## I(ACT^2) 0.004533 0.002176 2.083 0.037634 *
## I(priGPA * atndrte) 0.005586 0.004317 1.294 0.196173
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.8729 on 673 degrees of freedom
## Multiple R-squared: 0.2287, Adjusted R-squared: 0.2218
## F-statistic: 33.25 on 6 and 673 DF, p-value: < 2.2e-16anova(lm.1, lm.2)## Analysis of Variance Table
##
## Model 1: stndfnl ~ atndrte + priGPA + ACT + I(priGPA^2) + I(ACT^2) + I(priGPA *
## atndrte)
## Model 2: stndfnl ~ priGPA + ACT + I(priGPA^2) + I(ACT^2)
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 673 512.76
## 2 675 519.34 -2 -6.5814 4.319 0.01368 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1lm.1$coefficients[2] + mean(attend$priGPA) * lm.1$coefficient[7]## atndrte
## 0.007736558Typo in the text
priGPA.m.atndrte <- (attend$priGPA - mean(attend$priGPA)) * attend$atndrte
lm.1 <- lm(stndfnl ~ atndrte + priGPA + ACT + I(priGPA^2) + I(ACT^2) + priGPA.m.atndrte, data = attend)
summary(lm.1)##
## Call:
## lm(formula = stndfnl ~ atndrte + priGPA + ACT + I(priGPA^2) +
## I(ACT^2) + priGPA.m.atndrte, data = attend)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.1698 -0.5316 -0.0177 0.5737 2.3344
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.050293 1.360319 1.507 0.132225
## atndrte 0.007737 0.002633 2.938 0.003412 **
## priGPA -1.628540 0.481003 -3.386 0.000751 ***
## ACT -0.128039 0.098492 -1.300 0.194047
## I(priGPA^2) 0.295905 0.101049 2.928 0.003523 **
## I(ACT^2) 0.004533 0.002176 2.083 0.037634 *
## priGPA.m.atndrte 0.005586 0.004317 1.294 0.196173
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.8729 on 673 degrees of freedom
## Multiple R-squared: 0.2287, Adjusted R-squared: 0.2218
## F-statistic: 33.25 on 6 and 673 DF, p-value: < 2.2e-16Equations 6.23 and 6.24
See how the estimates and R-squared differ between the two models.
data("rdchem")
lm.1 <- lm(rdintens ~ lsales, data = rdchem)
lm.2 <- lm(rdintens ~ sales + salessq, data = rdchem)
summary(lm.1)##
## Call:
## lm(formula = rdintens ~ lsales, data = rdchem)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.9710 -1.4323 -0.5639 0.9874 5.7748
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.1043 1.5782 0.7 0.490
## lsales 0.3017 0.2155 1.4 0.172
##
## Residual standard error: 1.845 on 30 degrees of freedom
## Multiple R-squared: 0.06133, Adjusted R-squared: 0.03004
## F-statistic: 1.96 on 1 and 30 DF, p-value: 0.1718summary(lm.2)##
## Call:
## lm(formula = rdintens ~ sales + salessq, data = rdchem)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.1418 -1.3630 -0.2257 1.0688 5.5808
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.613e+00 4.294e-01 6.084 1.27e-06 ***
## sales 3.006e-04 1.393e-04 2.158 0.0394 *
## salessq -6.946e-09 3.726e-09 -1.864 0.0725 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.788 on 29 degrees of freedom
## Multiple R-squared: 0.1484, Adjusted R-squared: 0.08969
## F-statistic: 2.527 on 2 and 29 DF, p-value: 0.09733Example 6.4
See how the estimates and R-squared differ between the following models.
data("ceosal1")
lm.1 <- lm(salary ~ sales + roe, data = ceosal1)
lm.2 <- lm(lsalary ~ lsales + roe, data = ceosal1)
summary(lm.1)##
## Call:
## lm(formula = salary ~ sales + roe, data = ceosal1)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1501.8 -492.6 -232.0 123.3 13575.2
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 8.306e+02 2.239e+02 3.710 0.000267 ***
## sales 1.634e-02 8.874e-03 1.842 0.066973 .
## roe 1.963e+01 1.108e+01 1.772 0.077823 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1359 on 206 degrees of freedom
## Multiple R-squared: 0.02917, Adjusted R-squared: 0.01975
## F-statistic: 3.095 on 2 and 206 DF, p-value: 0.04739summary(lm.2)##
## Call:
## lm(formula = lsalary ~ lsales + roe, data = ceosal1)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.9464 -0.2888 -0.0322 0.2261 2.7830
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 4.362167 0.293878 14.843 < 2e-16 ***
## lsales 0.275087 0.033254 8.272 1.62e-14 ***
## roe 0.017872 0.003955 4.519 1.05e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.4822 on 206 degrees of freedom
## Multiple R-squared: 0.282, Adjusted R-squared: 0.275
## F-statistic: 40.45 on 2 and 206 DF, p-value: 1.519e-15summary(lm.1)$adj.r.squared## [1] 0.01974617summary(lm.2)$adj.r.squared## [1] 0.2750178Example 6.5
Compare the usual estimation of a model with a regression, where interesting values - like means - are subtracted from each variable before the estimation.
data("gpa2")
lm.1 <- lm(colgpa ~ sat + hsperc + hsize + hsizesq, data = gpa2)
summary(lm.1)##
## Call:
## lm(formula = colgpa ~ sat + hsperc + hsize + hsizesq, data = gpa2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.57543 -0.35081 0.03342 0.39945 1.81683
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.493e+00 7.534e-02 19.812 < 2e-16 ***
## sat 1.492e-03 6.521e-05 22.886 < 2e-16 ***
## hsperc -1.386e-02 5.610e-04 -24.698 < 2e-16 ***
## hsize -6.088e-02 1.650e-02 -3.690 0.000228 ***
## hsizesq 5.460e-03 2.270e-03 2.406 0.016191 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.5599 on 4132 degrees of freedom
## Multiple R-squared: 0.2781, Adjusted R-squared: 0.2774
## F-statistic: 398 on 4 and 4132 DF, p-value: < 2.2e-16summary(lm.1)$sigma## [1] 0.5598638lm.1$coefficients## (Intercept) sat hsperc hsize hsizesq
## 1.492652209 0.001492497 -0.013855782 -0.060881453 0.005460295lm.1$coefficients[1] + lm.1$coefficients[2] * 1200 + lm.1$coefficients[3] * 30 +
lm.1$coefficients[4] * 5 + lm.1$coefficients[5] * 5^2## (Intercept)
## 2.700075lm.2 <- lm(colgpa ~ I(sat - 1200) + I(hsperc - 30) + I(hsize - 5) + I(hsizesq - 25), data = gpa2) # Estimate the model
s.2 <- summary(lm.2)
s.2##
## Call:
## lm(formula = colgpa ~ I(sat - 1200) + I(hsperc - 30) + I(hsize -
## 5) + I(hsizesq - 25), data = gpa2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.57543 -0.35081 0.03342 0.39945 1.81683
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.700e+00 1.988e-02 135.833 < 2e-16 ***
## I(sat - 1200) 1.492e-03 6.521e-05 22.886 < 2e-16 ***
## I(hsperc - 30) -1.386e-02 5.610e-04 -24.698 < 2e-16 ***
## I(hsize - 5) -6.088e-02 1.650e-02 -3.690 0.000228 ***
## I(hsizesq - 25) 5.460e-03 2.270e-03 2.406 0.016191 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.5599 on 4132 degrees of freedom
## Multiple R-squared: 0.2781, Adjusted R-squared: 0.2774
## F-statistic: 398 on 4 and 4132 DF, p-value: < 2.2e-16ci.low <- s.2$coefficients[1, 1] - 1.96 * s.2$coefficients[1, 2] # Create confidence intervals
ci.high <- s.2$coefficients[1, 1] + 1.96 * s.2$coefficients[1, 2]
c(ci.low, ci.high)## [1] 2.661115 2.739036Example 6.6
data("gpa2")
lm.2 <- lm(colgpa ~ I(sat - 1200) + I(hsperc - 30) + I(hsize - 5) + I(hsizesq - 25), data = gpa2)
s.2 <- summary(lm.2)
se <- sqrt(s.2$coefficients[1, 2]^2 + s.2$sigma^2)
se## [1] 0.5602166ci.low <- s.2$coefficients[1, 1] - 1.96 * se
ci.high <- s.2$coefficients[1, 1] + 1.96 * se
c(ci.low, ci.high)## [1] 1.602051 3.798100Example from page 203
data("hprice1")
lm.1 <- lm(price ~ lotsize + sqrft + bdrms, data = hprice1)
min(lm.1$residuals)## [1] -120.0264Example 6.7
data("ceosal2")
lm.1 <- lm(lsalary ~ lsales + lmktval + ceoten, data = ceosal2)
summary(lm.1)##
## Call:
## lm(formula = lsalary ~ lsales + lmktval + ceoten, data = ceosal2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.49693 -0.29472 0.00964 0.30417 1.85286
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 4.503795 0.257234 17.509 < 2e-16 ***
## lsales 0.162854 0.039242 4.150 5.21e-05 ***
## lmktval 0.109243 0.049595 2.203 0.0289 *
## ceoten 0.011705 0.005326 2.198 0.0293 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.5048 on 173 degrees of freedom
## Multiple R-squared: 0.3182, Adjusted R-squared: 0.3063
## F-statistic: 26.91 on 3 and 173 DF, p-value: 2.474e-14a.1 <- sum(exp(lm.1$residuals)) / 177
m <- exp(lm.1$fitted.values)
summary(lm.3 <- lm(salary ~ m - 1, data = ceosal2))##
## Call:
## lm(formula = salary ~ m - 1, data = ceosal2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1256.4 -240.9 -73.7 217.5 4371.1
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## m 1.1169 0.0471 23.71 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 511.9 on 176 degrees of freedom
## Multiple R-squared: 0.7616, Adjusted R-squared: 0.7603
## F-statistic: 562.4 on 1 and 176 DF, p-value: < 2.2e-16a.2 <- summary(lm.3 <- lm(salary ~ m - 1, data = ceosal2))$coefficient[1]
lm.1$coefficients[1] + lm.1$coefficients[2] * log(5000) + lm.1$coefficients[3] * log(10000) + lm.1$coefficients[4] * 10## (Intercept)
## 7.014077exp(lm.1$coefficients[1] + lm.1$coefficients[2] * log(5000) + lm.1$coefficients[3] * log(10000) + lm.1$coefficients[4] * 10)## (Intercept)
## 1112.18a.1 * exp(lm.1$coefficients[1] + lm.1$coefficients[2] * log(5000) + lm.1$coefficients[3] * log(10000) + lm.1$coefficients[4] * 10)## (Intercept)
## 1263.059a.2 * exp(lm.1$coefficients[1] + lm.1$coefficients[2] * log(5000) + lm.1$coefficients[3] * log(10000) + lm.1$coefficients[4] * 10)## (Intercept)
## 1242.145Example 6.8
data("ceosal2")
lm.1 <- lm(lsalary ~ lsales + lmktval + ceoten, data = ceosal2)
m <- exp(lm.1$fitted.values)
cor(ceosal2$salary, m)^2## [1] 0.2430808lm.2 <- lm(salary ~ sales + mktval + ceoten, data = ceosal2)
summary(lm.2)$r.squared## [1] 0.2012744ceosal2.2 <- ceosal2
ceosal2.2$lsales <- ceosal2.2$lsales - log(5000)
ceosal2.2$lmktval <- ceosal2.2$lmktval - log(10000)
ceosal2.2$ceoten <- ceosal2.2$ceoten - 10
lm.3 <- lm(lsalary ~ lsales + lmktval + ceoten, data = ceosal2.2)
s.3 <- summary(lm.3)
s.3$sigma## [1] 0.504769s.3$coefficients[1, 2]## [1] 0.0745556se <- sqrt(s.3$coefficients[1, 2]^2 + s.3$sigma^2)
se## [1] 0.5102453ci.low <- exp(-1.96 * se) * exp(s.3$coefficients[1, 1])
ci.high <- exp(1.96 * se) * exp(s.3$coefficients[1, 1])
c(ci.low, ci.high)## [1] 409.115 3023.462