Chapter12: Serial Correlation and Heteroskedasticity in Time Series Regressions
library(lmtest)
library(sandwich)
library(tseries)
library(wooldridge)Example 12.1
data("phillips")Static Phillips curve
lm.12.1.1 <- lm(inf ~ unem, data = phillips)
res.static <- lm.12.1.1$res
res.static_1 <- c(NA, lm.12.1.1$res[-(length(res.static))])
lm.12.1.1.test <- lm(res.static ~ res.static_1)
summary(lm.12.1.1.test)##
## Call:
## lm(formula = res.static ~ res.static_1)
##
## Residuals:
## Min 1Q Median 3Q Max
## -8.047 -1.104 -0.248 1.028 6.684
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.1118 0.3180 -0.352 0.727
## res.static_1 0.5725 0.1084 5.283 2.43e-06 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.358 on 53 degrees of freedom
## (1 observation deleted due to missingness)
## Multiple R-squared: 0.345, Adjusted R-squared: 0.3326
## F-statistic: 27.91 on 1 and 53 DF, p-value: 2.43e-06Augmented Phillips curve
lm.12.1.2 <- lm(I(inf - inf_1) ~ unem, data = phillips)
res.aug <- lm.12.1.2$res
res.aug_1 <- c(NA, lm.12.1.2$res[-length(res.aug)])
lm.12.1.2.test <- lm(res.aug ~ res.aug_1)
summary(lm.12.1.2.test)##
## Call:
## lm(formula = res.aug ~ res.aug_1)
##
## Residuals:
## Min 1Q Median 3Q Max
## -7.2635 -1.0828 -0.1021 0.7254 5.3916
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.1674 0.2655 0.631 0.531
## res.aug_1 -0.0327 0.1164 -0.281 0.780
##
## Residual standard error: 1.951 on 52 degrees of freedom
## (1 observation deleted due to missingness)
## Multiple R-squared: 0.001516, Adjusted R-squared: -0.01769
## F-statistic: 0.07896 on 1 and 52 DF, p-value: 0.7798Heteroskedasticity robust t statistic
coeftest(lm.12.1.1.test, vcov = vcovHC(lm.12.1.1.test, type = "HC0"))##
## t test of coefficients:
##
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.11181 0.30962 -0.3611 0.7194474
## res.static_1 0.57247 0.14225 4.0243 0.0001826 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1coeftest(lm.12.1.2.test, vcov = vcovHC(lm.12.1.2.test, type = "HC0"))##
## t test of coefficients:
##
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.167446 0.262215 0.6386 0.5259
## res.aug_1 -0.032697 0.193249 -0.1692 0.8663Durbin-Watson Test
# from package lmtest
dwtest(inf ~ unem, data = phillips)##
## Durbin-Watson test
##
## data: inf ~ unem
## DW = 0.80148, p-value = 1.486e-07
## alternative hypothesis: true autocorrelation is greater than 0dwtest(I(inf - inf_1) ~ unem, data = phillips)##
## Durbin-Watson test
##
## data: I(inf - inf_1) ~ unem
## DW = 1.771, p-value = 0.1673
## alternative hypothesis: true autocorrelation is greater than 0Example 12.2
data("prminwge")
lm.12.2 <- lm(lprepop ~ lmincov + lusgnp + lprgnp + t, data = prminwge)
lm.12.2.res <- lm.12.2$res
lm.12.2.res_1 <- c(NA, lm.12.2$res[-length(lm.12.2.res)])
lm.12.2.test <- lm(lm.12.2.res ~ lmincov + lusgnp + lprgnp + t + lm.12.2.res_1, data = prminwge)
summary(lm.12.2.test)##
## Call:
## lm(formula = lm.12.2.res ~ lmincov + lusgnp + lprgnp + t + lm.12.2.res_1,
## data = prminwge)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.041317 -0.018004 -0.004599 0.012378 0.067226
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.850772 1.092691 -0.779 0.44212
## lmincov 0.037500 0.035212 1.065 0.29511
## lusgnp 0.203932 0.195159 1.045 0.30412
## lprgnp -0.078466 0.070524 -1.113 0.27443
## t -0.003466 0.004074 -0.851 0.40134
## lm.12.2.res_1 0.480509 0.166444 2.887 0.00703 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.02755 on 31 degrees of freedom
## (1 observation deleted due to missingness)
## Multiple R-squared: 0.2424, Adjusted R-squared: 0.1202
## F-statistic: 1.983 on 5 and 31 DF, p-value: 0.1089For comparison the test with strict exogenous regressors
summary(lm(lm.12.2.res ~ lm.12.2.res_1))##
## Call:
## lm(formula = lm.12.2.res ~ lm.12.2.res_1)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.043815 -0.024235 -0.002278 0.015155 0.063480
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.0008953 0.0044883 -0.199 0.8431
## lm.12.2.res_1 0.4173219 0.1589351 2.626 0.0127 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.02723 on 35 degrees of freedom
## (1 observation deleted due to missingness)
## Multiple R-squared: 0.1646, Adjusted R-squared: 0.1407
## F-statistic: 6.895 on 1 and 35 DF, p-value: 0.01274Example 12.3
data("barium")
lm.12.3 <- lm(lchnimp ~ lchempi + lgas + lrtwex + befile6 + affile6 + afdec6, data = barium)
lm.12.3.res <- lm.12.3$res
lm.12.3.res_1 <- c(NA,lm.12.3$res[1:(length(lm.12.3.res)-1)])
lm.12.3.res_2 <- c(NA,NA,lm.12.3$res[1:(length(lm.12.3.res)-2)])
lm.12.3.res_3 <- c(NA,NA,NA,lm.12.3$res[1:(length(lm.12.3.res)-3)])
data <- cbind(barium, lm.12.3.res)
data <- cbind(data, lm.12.3.res_1)
data <- cbind(data, lm.12.3.res_2)
data <- cbind(data, lm.12.3.res_3)
lm.12.3.test <- lm(lm.12.3.res ~ lchempi + lgas + lrtwex + befile6 + affile6 + afdec6 +
lm.12.3.res_1 + lm.12.3.res_2 + lm.12.3.res_3, data = data)
lm.12.3.test.res <- lm(lm.12.3.res ~ lchempi + lgas + lrtwex + befile6 + affile6 + afdec6,
data = data[-(1:3),]) # Drop first 3 observations to make test statistics comparable
anova(lm.12.3.test, lm.12.3.test.res)## Analysis of Variance Table
##
## Model 1: lm.12.3.res ~ lchempi + lgas + lrtwex + befile6 + affile6 + afdec6 +
## lm.12.3.res_1 + lm.12.3.res_2 + lm.12.3.res_3
## Model 2: lm.12.3.res ~ lchempi + lgas + lrtwex + befile6 + affile6 + afdec6
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 118 38.394
## 2 121 43.394 -3 -5.0005 5.1229 0.00229 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Seasonal autocorrelation
data <- cbind(data, lm.12.3.res_12 = c(rep(NA, 12), lm.12.3$res[1:(length(lm.12.3.res) - 12)]))
summary(lm(lm.12.3.res ~ lm.12.3.res_12, data = data))##
## Call:
## lm(formula = lm.12.3.res ~ lm.12.3.res_12, data = data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.96217 -0.33069 0.03732 0.36389 1.46367
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.03789 0.05047 0.751 0.454
## lm.12.3.res_12 -0.18744 0.08425 -2.225 0.028 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.5506 on 117 degrees of freedom
## (12 observations deleted due to missingness)
## Multiple R-squared: 0.04059, Adjusted R-squared: 0.03239
## F-statistic: 4.95 on 1 and 117 DF, p-value: 0.02801Include regressors
summary(lm(lm.12.3.res ~ lchempi + lgas + lrtwex + befile6 + affile6 + afdec6 +
lm.12.3.res_12, data = data))##
## Call:
## lm(formula = lm.12.3.res ~ lchempi + lgas + lrtwex + befile6 +
## affile6 + afdec6 + lm.12.3.res_12, data = data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.0259 -0.3467 0.0158 0.3770 1.3676
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -28.57195 22.64780 -1.262 0.2097
## lchempi -0.92803 0.54647 -1.698 0.0923 .
## lgas 1.50251 1.01497 1.480 0.1416
## lrtwex -0.27439 0.38127 -0.720 0.4732
## befile6 0.04437 0.24581 0.180 0.8571
## affile6 0.06253 0.24600 0.254 0.7998
## afdec6 0.17231 0.27038 0.637 0.5253
## lm.12.3.res_12 -0.17002 0.08670 -1.961 0.0524 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.5541 on 111 degrees of freedom
## (12 observations deleted due to missingness)
## Multiple R-squared: 0.07811, Adjusted R-squared: 0.01997
## F-statistic: 1.344 on 7 and 111 DF, p-value: 0.2367Example 12.4
data("barium")
# Download my prais package
library(prais)
pw_1 <- prais_winsten(lchnimp ~ lchempi + lgas + lrtwex + befile6 +
affile6 + afdec6, data = barium)## Iteration 0: rho = 0
## Iteration 1: rho = 0.2708
## Iteration 2: rho = 0.291
## Iteration 3: rho = 0.293
## Iteration 4: rho = 0.2932
## Iteration 5: rho = 0.2932
## Iteration 6: rho = 0.2932
## Iteration 7: rho = 0.2932summary(pw_1)##
## Call:
## prais_winsten(formula = lchnimp ~ lchempi + lgas + lrtwex + befile6 +
## affile6 + afdec6, data = barium)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.99386 -0.32219 0.03747 0.40226 1.50281
##
## AR(1) coefficient rho after 7 Iterations: 0.2932
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -37.07770 22.77830 -1.628 0.1061
## lchempi 2.94095 0.63284 4.647 8.46e-06 ***
## lgas 1.04638 0.97734 1.071 0.2864
## lrtwex 1.13279 0.50666 2.236 0.0272 *
## befile6 -0.01648 0.31938 -0.052 0.9589
## affile6 -0.03316 0.32181 -0.103 0.9181
## afdec6 -0.57681 0.34199 -1.687 0.0942 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.5733 on 124 degrees of freedom
## Multiple R-squared: 0.2021, Adjusted R-squared: 0.1635
## F-statistic: 5.235 on 6 and 124 DF, p-value: 7.764e-05
##
## Durbin-Watson statistic (original): 1.458
## Durbin-Watson statistic (transformed): 2.087Example 12.5
data("phillips")First column
lm.12.5 <- lm(inf ~ unem, data = phillips)
summary(lm.12.5)##
## Call:
## lm(formula = inf ~ unem, data = phillips)
##
## Residuals:
## Min 1Q Median 3Q Max
## -5.2176 -1.7812 -0.6659 1.1473 8.8795
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.0536 1.5480 0.681 0.4990
## unem 0.5024 0.2656 1.892 0.0639 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.972 on 54 degrees of freedom
## Multiple R-squared: 0.06215, Adjusted R-squared: 0.04479
## F-statistic: 3.579 on 1 and 54 DF, p-value: 0.06389Second column
pw_2 <- prais_winsten(inf ~ unem, data=phillips)## Iteration 0: rho = 0
## Iteration 1: rho = 0.5721
## Iteration 2: rho = 0.735
## Iteration 3: rho = 0.7792
## Iteration 4: rho = 0.7871
## Iteration 5: rho = 0.7883
## Iteration 6: rho = 0.7885
## Iteration 7: rho = 0.7885
## Iteration 8: rho = 0.7885
## Iteration 9: rho = 0.7885summary(pw_2)##
## Call:
## prais_winsten(formula = inf ~ unem, data = phillips)
##
## Residuals:
## Min 1Q Median 3Q Max
## -5.258 -2.447 -1.073 1.463 10.570
##
## AR(1) coefficient rho after 9 Iterations: 0.7885
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 7.9994 2.0483 3.905 0.000264 ***
## unem -0.7140 0.2898 -2.464 0.016965 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.138 on 54 degrees of freedom
## Multiple R-squared: 0.1345, Adjusted R-squared: 0.1185
## F-statistic: 8.393 on 1 and 54 DF, p-value: 0.00543
##
## Durbin-Watson statistic (original): 0.8015
## Durbin-Watson statistic (transformed): 1.914Example 12.6
data("intdef")
summary(lm.12.6.1 <- lm(i3 ~ inf + def, data = intdef))##
## Call:
## lm(formula = i3 ~ inf + def, data = intdef)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.9948 -1.1694 0.1959 0.9602 4.7224
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.73327 0.43197 4.012 0.00019 ***
## inf 0.60587 0.08213 7.376 1.12e-09 ***
## def 0.51306 0.11838 4.334 6.57e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.843 on 53 degrees of freedom
## Multiple R-squared: 0.6021, Adjusted R-squared: 0.5871
## F-statistic: 40.09 on 2 and 53 DF, p-value: 2.483e-11res <- lm.12.6.1$residuals
res_1 <- c(NA, res[-length(res)])
summary(lm(res ~ -1 + res_1))##
## Call:
## lm(formula = res ~ -1 + res_1)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.8010 -0.6616 0.0179 0.8721 4.5901
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## res_1 0.6229 0.1087 5.73 4.61e-07 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.397 on 54 degrees of freedom
## (1 observation deleted due to missingness)
## Multiple R-squared: 0.3781, Adjusted R-squared: 0.3666
## F-statistic: 32.83 on 1 and 54 DF, p-value: 4.612e-07lm.12.6.2 <- lm(I(diff(i3)) ~ I(diff(inf)) + I(diff(def)), data = intdef)
summary(lm.12.6.2)##
## Call:
## lm(formula = I(diff(i3)) ~ I(diff(inf)) + I(diff(def)), data = intdef)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.53316 -0.96248 -0.08658 0.75884 2.93846
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.04177 0.17139 0.244 0.808
## I(diff(inf)) 0.14949 0.09216 1.622 0.111
## I(diff(def)) -0.18132 0.14768 -1.228 0.225
##
## Residual standard error: 1.265 on 52 degrees of freedom
## Multiple R-squared: 0.1763, Adjusted R-squared: 0.1446
## F-statistic: 5.566 on 2 and 52 DF, p-value: 0.006451Correlation between \(i3_t\) and \(i3_{t-1}\)
cor(intdef$i3, c(NA, intdef$i3[-length(intdef$i3)]), use = "pairwise.complete.obs")## [1] 0.8845033Regression of \(e_t\) on \(e_{t-1}\)
res <- lm.12.6.2$residuals
res_1 <- res_1 <- c(NA, res[-length(res)])
summary(lm(res ~ -1 + res_1))##
## Call:
## lm(formula = res ~ -1 + res_1)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.7439 -0.9833 -0.1421 0.7773 2.8453
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## res_1 0.07172 0.13305 0.539 0.592
##
## Residual standard error: 1.212 on 53 degrees of freedom
## (1 observation deleted due to missingness)
## Multiple R-squared: 0.005454, Adjusted R-squared: -0.01331
## F-statistic: 0.2906 on 1 and 53 DF, p-value: 0.5921Example 12.7
data("prminwge")
lm.12.7.1 <- lm(lprepop ~ lmincov + lusgnp + lprgnp + t, data = prminwge)
summary(lm.12.7.1)##
## Call:
## lm(formula = lprepop ~ lmincov + lusgnp + lprgnp + t, data = prminwge)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.054679 -0.023653 -0.004039 0.018638 0.076947
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -6.663432 1.257831 -5.298 7.67e-06 ***
## lmincov -0.212261 0.040152 -5.286 7.92e-06 ***
## lusgnp 0.486046 0.221983 2.190 0.0357 *
## lprgnp 0.285239 0.080492 3.544 0.0012 **
## t -0.026663 0.004627 -5.763 1.94e-06 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.03277 on 33 degrees of freedom
## Multiple R-squared: 0.8892, Adjusted R-squared: 0.8758
## F-statistic: 66.23 on 4 and 33 DF, p-value: 2.677e-15Still looking for the SC/heteroskedasticiy robust se.
pw_12_7_1 <- prais_winsten(lprepop ~ lmincov + lusgnp + lprgnp + t, data = prminwge)## Iteration 0: rho = 0
## Iteration 1: rho = 0.4197
## Iteration 2: rho = 0.5325
## Iteration 3: rho = 0.5796
## Iteration 4: rho = 0.5999
## Iteration 5: rho = 0.6086
## Iteration 6: rho = 0.6123
## Iteration 7: rho = 0.6139
## Iteration 8: rho = 0.6146
## Iteration 9: rho = 0.6149
## Iteration 10: rho = 0.615
## Iteration 11: rho = 0.6151
## Iteration 12: rho = 0.6151
## Iteration 13: rho = 0.6151
## Iteration 14: rho = 0.6151
## Iteration 15: rho = 0.6151
## Iteration 16: rho = 0.6151summary(pw_12_7_1)##
## Call:
## prais_winsten(formula = lprepop ~ lmincov + lusgnp + lprgnp +
## t, data = prminwge)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.079574 -0.028121 -0.004816 0.005918 0.075978
##
## AR(1) coefficient rho after 16 Iterations: 0.6151
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -4.652853 1.376470 -3.380 0.00188 **
## lmincov -0.147711 0.045842 -3.222 0.00286 **
## lusgnp 0.255711 0.231750 1.103 0.27784
## lprgnp 0.251383 0.116462 2.158 0.03826 *
## t -0.020502 0.005856 -3.501 0.00135 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.02852 on 33 degrees of freedom
## Multiple R-squared: 0.7509, Adjusted R-squared: 0.7207
## F-statistic: 24.87 on 4 and 33 DF, p-value: 1.47e-09
##
## Durbin-Watson statistic (original): 1.014
## Durbin-Watson statistic (transformed): 1.736Example 12.8
data("nyse")
lm.12.8 <- lm(return ~ return_1, data = nyse)
summary(lm.12.8)##
## Call:
## lm(formula = return ~ return_1, data = nyse)
##
## Residuals:
## Min 1Q Median 3Q Max
## -15.261 -1.302 0.098 1.316 8.065
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.17963 0.08074 2.225 0.0264 *
## return_1 0.05890 0.03802 1.549 0.1218
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.11 on 687 degrees of freedom
## (2 observations deleted due to missingness)
## Multiple R-squared: 0.003481, Adjusted R-squared: 0.00203
## F-statistic: 2.399 on 1 and 687 DF, p-value: 0.1218res <- lm.12.8$residuals^2
lm.12.8.bp <- lm(res ~ return_1, data = na.omit(nyse))
summary(lm.12.8.bp)##
## Call:
## lm(formula = res ~ return_1, data = na.omit(nyse))
##
## Residuals:
## Min 1Q Median 3Q Max
## -9.689 -3.929 -2.021 0.960 223.730
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 4.6565 0.4277 10.888 < 2e-16 ***
## return_1 -1.1041 0.2014 -5.482 5.9e-08 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 11.18 on 687 degrees of freedom
## Multiple R-squared: 0.04191, Adjusted R-squared: 0.04052
## F-statistic: 30.05 on 1 and 687 DF, p-value: 5.905e-08Example 12.9
data("nyse")
lm.12.9 <- lm(return ~ return_1, data = nyse)
summary(lm.12.9)##
## Call:
## lm(formula = return ~ return_1, data = nyse)
##
## Residuals:
## Min 1Q Median 3Q Max
## -15.261 -1.302 0.098 1.316 8.065
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.17963 0.08074 2.225 0.0264 *
## return_1 0.05890 0.03802 1.549 0.1218
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.11 on 687 degrees of freedom
## (2 observations deleted due to missingness)
## Multiple R-squared: 0.003481, Adjusted R-squared: 0.00203
## F-statistic: 2.399 on 1 and 687 DF, p-value: 0.1218u <- lm.12.8$residuals
u_1 <- c(NA, u[-length(u)])
usq <- lm.12.8$residuals^2
usq_1 <- c(NA, usq[-length(usq)])
lm.12.9.arch <- lm(usq ~ usq_1)
summary(lm.12.9.arch)##
## Call:
## lm(formula = usq ~ usq_1)
##
## Residuals:
## Min 1Q Median 3Q Max
## -23.337 -3.292 -2.157 0.556 223.981
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.94743 0.44023 6.695 4.49e-11 ***
## usq_1 0.33706 0.03595 9.377 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 10.76 on 686 degrees of freedom
## (1 observation deleted due to missingness)
## Multiple R-squared: 0.1136, Adjusted R-squared: 0.1123
## F-statistic: 87.92 on 1 and 686 DF, p-value: < 2.2e-16lm.12.9.3 <- lm(u ~ u_1)
summary(lm.12.9.3)##
## Call:
## lm(formula = u ~ u_1)
##
## Residuals:
## Min 1Q Median 3Q Max
## -15.2538 -1.3024 0.0912 1.3217 8.0613
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.001171 0.080508 -0.015 0.988
## u_1 0.001405 0.038177 0.037 0.971
##
## Residual standard error: 2.112 on 686 degrees of freedom
## (1 observation deleted due to missingness)
## Multiple R-squared: 1.974e-06, Adjusted R-squared: -0.001456
## F-statistic: 0.001354 on 1 and 686 DF, p-value: 0.9707