data <- barium
head(data,5)## chnimp bchlimp befile6 affile6 afdec6 befile12 affile12 afdec12 chempi
## 1 220.4620 9578.376 0 0 0 0 0 0 100.1
## 2 94.7980 11219.480 0 0 0 0 0 0 100.9
## 3 219.3575 9719.900 0 0 0 0 0 0 101.1
## 4 317.4215 12920.950 0 0 0 0 0 0 102.5
## 5 114.6390 9790.446 0 0 0 0 0 0 104.1
## gas rtwex spr sum fall lchnimp lgas lrtwex lchempi t feb mar
## 1 7830000128 86.74 0 0 0 5.395725 22.78123 4.462915 4.606170 1 1 0
## 2 8819999744 85.63 1 0 0 4.551748 22.90029 4.450036 4.614130 2 0 1
## 3 8449999872 85.42 1 0 0 5.390703 22.85743 4.447580 4.616110 3 0 0
## 4 9240000512 87.29 1 0 0 5.760231 22.94681 4.469236 4.629863 4 0 0
## 5 9150000128 86.60 0 1 0 4.741788 22.93702 4.461300 4.645352 5 0 0
## apr may jun jul aug sep oct nov dec percchn
## 1 0 0 0 0 0 0 0 0 0 2.3016636
## 2 0 0 0 0 0 0 0 0 0 0.8449411
## 3 1 0 0 0 0 0 0 0 0 2.2567875
## 4 0 1 0 0 0 0 0 0 0 2.4566422
## 5 0 0 1 0 0 0 0 0 0 1.1709272When I add the time trend to equation 10.22, there is only one significant variable - time trend variable.
model_1 <- lm(log(chnimp)~ log(chempi)+log(gas)+log(rtwex)+befile6+affile6+afdec6+t, data= data)
summary(model_1)##
## Call:
## lm(formula = log(chnimp) ~ log(chempi) + log(gas) + log(rtwex) +
## befile6 + affile6 + afdec6 + t, data = data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.94317 -0.31168 0.03172 0.36366 1.21218
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -2.367308 20.782159 -0.114 0.90949
## log(chempi) -0.686233 1.239711 -0.554 0.58090
## log(gas) 0.465669 0.876178 0.531 0.59605
## log(rtwex) 0.078222 0.472440 0.166 0.86877
## befile6 0.090470 0.251289 0.360 0.71945
## affile6 0.097006 0.257313 0.377 0.70683
## afdec6 -0.351502 0.282542 -1.244 0.21584
## t 0.012706 0.003844 3.305 0.00124 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.5748 on 123 degrees of freedom
## Multiple R-squared: 0.3616, Adjusted R-squared: 0.3252
## F-statistic: 9.951 on 7 and 123 DF, p-value: 8.358e-10We have the Fstatistic = 10.99 > 7.158; Hence, the join effect of other variables are significant.
model_2 <- lm(log(chnimp)~ log(chempi)+log(gas)+log(rtwex)+befile6+affile6+afdec6, data= data)
summary(model_2)##
## Call:
## lm(formula = log(chnimp) ~ log(chempi) + log(gas) + log(rtwex) +
## befile6 + affile6 + afdec6, data = data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.03356 -0.39080 0.03048 0.40248 1.51720
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -17.80277 21.04537 -0.846 0.3992
## log(chempi) 3.11719 0.47920 6.505 1.72e-09 ***
## log(gas) 0.19634 0.90662 0.217 0.8289
## log(rtwex) 0.98302 0.40015 2.457 0.0154 *
## befile6 0.05957 0.26097 0.228 0.8198
## affile6 -0.03241 0.26430 -0.123 0.9026
## afdec6 -0.56524 0.28584 -1.978 0.0502 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.5974 on 124 degrees of freedom
## Multiple R-squared: 0.3049, Adjusted R-squared: 0.2712
## F-statistic: 9.064 on 6 and 124 DF, p-value: 3.255e-08F= ((0.362 - 0.305)/1)/((1-0.362)/123)
F## [1] 10.98903critical_value = qf(0.95,1,123, TRUE)
critical_value## [1] 7.157589We have the Fstatistic = 0.8481 < 2.528; Hence, the join effect of monthly dummy variables are insignificant.
model_3 <- lm(log(chnimp)~ log(chempi)+log(gas)+log(rtwex)+befile6+affile6+afdec6+feb+mar+apr+may+jun+jul+aug+sep+oct+nov+dec, data= data)
summary(model_3)##
## Call:
## lm(formula = log(chnimp) ~ log(chempi) + log(gas) + log(rtwex) +
## befile6 + affile6 + afdec6 + feb + mar + apr + may + jun +
## jul + aug + sep + oct + nov + dec, data = data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.98535 -0.36207 0.07366 0.41786 1.37734
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 16.779215 32.428645 0.517 0.6059
## log(chempi) 3.265062 0.492930 6.624 1.24e-09 ***
## log(gas) -1.278140 1.389008 -0.920 0.3594
## log(rtwex) 0.663045 0.471304 1.407 0.1622
## befile6 0.139703 0.266808 0.524 0.6016
## affile6 0.012632 0.278687 0.045 0.9639
## afdec6 -0.521300 0.301950 -1.726 0.0870 .
## feb -0.417711 0.304444 -1.372 0.1728
## mar 0.059052 0.264731 0.223 0.8239
## apr -0.451483 0.268386 -1.682 0.0953 .
## may 0.033309 0.269242 0.124 0.9018
## jun -0.206332 0.269252 -0.766 0.4451
## jul 0.003837 0.278767 0.014 0.9890
## aug -0.157064 0.277993 -0.565 0.5732
## sep -0.134161 0.267656 -0.501 0.6172
## oct 0.051693 0.266851 0.194 0.8467
## nov -0.246260 0.262827 -0.937 0.3508
## dec 0.132838 0.271423 0.489 0.6255
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.6012 on 113 degrees of freedom
## Multiple R-squared: 0.3583, Adjusted R-squared: 0.2618
## F-statistic: 3.712 on 17 and 113 DF, p-value: 1.282e-05F= ((0.358-0.305)/11)/((1-0.358)/113)
F## [1] 0.84806critical_value = qf(0.95,11,113, TRUE)
critical_value## [1] 2.041383