12 Serial Correlation and Heteroskedasticity in Time Series Regressions
Also covered using Python and Stata
library(wooldridge)
library(lmtest)
library(stargazer)
library(car)
library(dynlm)
library(prais)
library(sandwich)
rm(list = ls())12.1 Example 12.1. Testing for AR(1) Serial Correlation in the Phillips Curve
ts_phillips <- ts(phillips)
uhat1 <- resid(dynlm(inf ~ unem + 1, data=ts_phillips))
phill_inf1 <- dynlm(uhat1 ~ L(uhat1,1))
ts_phillips2 <-ts(subset(phillips, phillips$year<1997))
uhat2 <- resid(dynlm(inf ~ unem + 1, data=ts_phillips2))
phill_inf2 <- dynlm(uhat2 ~ L(uhat2, 1))
uhat3<- resid(dynlm(cinf ~ unem + 1, data=ts_phillips2))
phill_cinf <- dynlm(uhat3 ~ L(uhat3, 1))
stargazer(phill_inf1, phill_inf2, phill_cinf, keep.stat=c("n","rsq", "adj.rsq"), no.space=TRUE, type="text")##
## ==========================================
## Dependent variable:
## -----------------------------
## uhat1 uhat2 uhat3
## (1) (2) (3)
## ------------------------------------------
## L(uhat1, 1) 0.572***
## (0.108)
## L(uhat2, 1) 0.573***
## (0.116)
## L(uhat3, 1) -0.036
## (0.124)
## Constant -0.112 -0.113 0.194
## (0.318) (0.359) (0.300)
## ------------------------------------------
## Observations 55 48 47
## R2 0.345 0.346 0.002
## Adjusted R2 0.333 0.332 -0.020
## ==========================================
## Note: *p<0.1; **p<0.05; ***p<0.0112.2 Example 12.2. Testing for AR(1) Serial Correlation in the Minimum Wage Equation
tsprminwge <- ts(prminwge)
minwg_reg <- dynlm(lprepop ~ lmincov + lprgnp + lusgnp + t, data=tsprminwge)
uhat <- resid(minwg_reg)
AR1c <- dynlm(uhat ~ lmincov + lprgnp + lusgnp + t + L(uhat, 1) , data=tsprminwge)
AR1s <- dynlm(uhat ~ L(uhat, 1))
stargazer(AR1c, AR1s, column.labels=c("AR1c", "AR1s"), no.space=TRUE, type="text")##
## ===========================================================
## Dependent variable:
## ---------------------------------------
## uhat
## AR1c AR1s
## (1) (2)
## -----------------------------------------------------------
## lmincov 0.038
## (0.035)
## lprgnp -0.078
## (0.071)
## lusgnp 0.204
## (0.195)
## t -0.003
## (0.004)
## L(uhat, 1) 0.481*** 0.417**
## (0.166) (0.159)
## Constant -0.851 -0.001
## (1.093) (0.004)
## -----------------------------------------------------------
## Observations 37 37
## R2 0.242 0.165
## Adjusted R2 0.120 0.141
## Residual Std. Error 0.028 (df = 31) 0.027 (df = 35)
## F Statistic 1.983 (df = 5; 31) 6.895** (df = 1; 35)
## ===========================================================
## Note: *p<0.1; **p<0.05; ***p<0.0112.3 Example 12.3. Testing for AR(3) Serial Correlation
tsbarium<-ts(barium)
barium_reg <- dynlm(lchnimp ~ lchempi + lgas + lrtwex + befile6 + affile6 + afdec6, data=tsbarium)
u <- resid(barium_reg)
AR3 <- dynlm(u ~ lchempi + lgas + lrtwex + befile6 + affile6 + afdec6 + L(u, 1) + L(u, 2) + L(u, 3) + 1, data = tsbarium)
summary(AR3)##
## Time series regression with "ts" data:
## Start = 4, End = 131
##
## Call:
## dynlm(formula = u ~ lchempi + lgas + lrtwex + befile6 + affile6 +
## afdec6 + L(u, 1) + L(u, 2) + L(u, 3) + 1, data = tsbarium)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.89072 -0.32250 0.05873 0.36376 1.19650
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -14.36915 20.65567 -0.696 0.4880
## lchempi -0.14316 0.47203 -0.303 0.7622
## lgas 0.62331 0.88597 0.704 0.4831
## lrtwex 0.17867 0.39103 0.457 0.6486
## befile6 -0.08592 0.25101 -0.342 0.7327
## affile6 -0.12212 0.25470 -0.479 0.6325
## afdec6 -0.06683 0.27437 -0.244 0.8080
## L(u, 1) 0.22149 0.09166 2.417 0.0172 *
## L(u, 2) 0.13404 0.09216 1.454 0.1485
## L(u, 3) 0.12554 0.09112 1.378 0.1709
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.5704 on 118 degrees of freedom
## Multiple R-squared: 0.1159, Adjusted R-squared: 0.04848
## F-statistic: 1.719 on 9 and 118 DF, p-value: 0.09202## Linear hypothesis test
##
## Hypothesis:
## L(u,0
## L(u, 2) = 0
## L(u, 3) = 0
##
## Model 1: restricted model
## Model 2: u ~ lchempi + lgas + lrtwex + befile6 + affile6 + afdec6 + L(u,
## 1) + L(u, 2) + L(u, 3) + 1
##
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 121 43.394
## 2 118 38.394 3 5.0005 5.1229 0.00229 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 112.4 Example 12.4. Prais-Winsten Estimation in the Event Study
ols <- dynlm(lchnimp ~ lchempi + lgas + lrtwex + befile6 + affile6 + afdec6, data=tsbarium)
stargazer(ols, column.labels=c("OLS"), no.space=TRUE, single.row = TRUE, type="text")##
## ===============================================
## Dependent variable:
## ---------------------------
## lchnimp
## OLS
## -----------------------------------------------
## lchempi 3.117*** (0.479)
## lgas 0.196 (0.907)
## lrtwex 0.983** (0.400)
## befile6 0.060 (0.261)
## affile6 -0.032 (0.264)
## afdec6 -0.565* (0.286)
## Constant -17.803 (21.045)
## -----------------------------------------------
## Observations 131
## R2 0.305
## Adjusted R2 0.271
## Residual Std. Error 0.597 (df = 124)
## F Statistic 9.064*** (df = 6; 124)
## ===============================================
## Note: *p<0.1; **p<0.05; ***p<0.01pw <- prais_winsten(lchnimp ~ lchempi + lgas + lrtwex + befile6 + affile6 + afdec6, data=tsbarium, index = "t")## 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 = 0.2932##
## Call:
## prais_winsten(formula = lchnimp ~ lchempi + lgas + lrtwex + befile6 +
## affile6 + afdec6, data = tsbarium, index = "t")
##
## 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): 2.08712.5 Example 12.5. Static Phillips Curve
phillips2 <- ts(subset(phillips, phillips$year<1997))
ols <- lm(inf ~ unem, data=phillips2)
stargazer(ols, column.labels=c("OLS"), no.space=TRUE, single.row = TRUE, type="text")##
## ===============================================
## Dependent variable:
## ---------------------------
## inf
## OLS
## -----------------------------------------------
## unem 0.468 (0.289)
## Constant 1.424 (1.719)
## -----------------------------------------------
## Observations 49
## R2 0.053
## Adjusted R2 0.033
## Residual Std. Error 3.131 (df = 47)
## F Statistic 2.616 (df = 1; 47)
## ===============================================
## Note: *p<0.1; **p<0.05; ***p<0.01## Iteration 0: rho = 0
## Iteration 1: rho = 0.5727
## Iteration 2: rho = 0.7307
## Iteration 3: rho = 0.7719
## Iteration 4: rho = 0.7792
## Iteration 5: rho = 0.7803
## Iteration 6: rho = 0.7805
## Iteration 7: rho = 0.7805
## Iteration 8: rho = 0.7805
## Iteration 9: rho = 0.7805##
## Call:
## prais_winsten(formula = inf ~ unem, data = phillips2, index = "year")
##
## Residuals:
## Min 1Q Median 3Q Max
## -5.5470 -2.6764 -0.3579 1.4108 10.2853
##
## AR(1) coefficient rho after 9 iterations: 0.7805
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 8.2959 2.2314 3.718 0.000535 ***
## unem -0.7157 0.3135 -2.283 0.026988 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.267 on 47 degrees of freedom
## Multiple R-squared: 0.1358, Adjusted R-squared: 0.1174
## F-statistic: 7.386 on 1 and 47 DF, p-value: 0.009174
##
## Durbin-Watson statistic (original): 0.8027
## Durbin-Watson statistic (transformed): 1.9112.6 Example 12.6. Differencing the Interest Rate Equation
tsintdef <- ts(intdef)
reg1 <- dynlm(i3 ~ inf + def, data=tsintdef)
u <- resid(reg1)
AR1u <- dynlm(u ~ L(u, 1), data=tsintdef)
reg2 <-dynlm(ci3 ~ cinf + cdef, data=tsintdef)
e <- resid(reg2)
AR1e <- dynlm(e ~ L(e, 1), data=tsintdef)
stargazer(reg1, reg2, AR1u, AR1e, column.labels=c("reg1", "reg2", "AR1u", "AR1e"), no.space=TRUE, keep.stat=c("n","rsq", "adj.rsq"), type="text")##
## ==============================================
## Dependent variable:
## ---------------------------------
## i3 ci3 u e
## reg1 reg2 AR1u AR1e
## (1) (2) (3) (4)
## ----------------------------------------------
## inf 0.606***
## (0.082)
## def 0.513***
## (0.118)
## cinf 0.149
## (0.092)
## cdef -0.181
## (0.148)
## L(u, 1) 0.623***
## (0.110)
## L(e, 1) 0.072
## (0.134)
## Constant 1.733*** 0.042 0.015 -0.041
## (0.432) (0.171) (0.190) (0.166)
## ----------------------------------------------
## Observations 56 55 55 54
## R2 0.602 0.176 0.377 0.005
## Adjusted R2 0.587 0.145 0.366 -0.014
## ==============================================
## Note: *p<0.1; **p<0.05; ***p<0.0112.7 Example 12.7. The Puerto Rican Minimum Wage
OLS <- lm(lprepop ~ lmincov + lprgnp + lusgnp + t, data=prminwge)
Newey <- coeftest(OLS, vcov.=NeweyWest(OLS, lag=2, prewhite=FALSE, adjust=TRUE, verbose=TRUE))##
## Lag truncation parameter chosen: 2stargazer(OLS, Newey, column.labels=c("OLS", "Newey"), no.space=TRUE, keep.stat=c("n","rsq", "adj.rsq"), type="text")##
## =========================================
## Dependent variable:
## ----------------------------
## lprepop
## OLS coefficient
## test
## OLS Newey
## (1) (2)
## -----------------------------------------
## lmincov -0.212*** -0.212***
## (0.040) (0.046)
## lprgnp 0.285*** 0.285***
## (0.080) (0.100)
## lusgnp 0.486** 0.486*
## (0.222) (0.279)
## t -0.027*** -0.027***
## (0.005) (0.006)
## Constant -6.663*** -6.663***
## (1.258) (1.536)
## -----------------------------------------
## Observations 38
## R2 0.889
## Adjusted R2 0.876
## =========================================
## Note: *p<0.1; **p<0.05; ***p<0.01## 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 = 0.6151##
## Call:
## prais_winsten(formula = lprepop ~ lmincov + lprgnp + lusgnp +
## t, data = tsprminwge, index = "year")
##
## 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 **
## lprgnp 0.251383 0.116462 2.158 0.03826 *
## lusgnp 0.255711 0.231750 1.103 0.27784
## 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): 1.73612.8 Example 12.8. Heteroskedasticity and the Efficient Markets Hypothesis
tsnyse <-ts(nyse)
u2 <- resid(dynlm(return ~ return_1, data=tsnyse))**2
summary(dynlm(u2 ~ return_1 + 1, data = tsnyse))##
## Time series regression with "ts" data:
## Start = 3, End = 691
##
## Call:
## dynlm(formula = u2 ~ return_1 + 1, data = tsnyse)
##
## 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-0812.9 Example 12.9. ARCH in Stock Returns
tsnyse <-ts(nyse)
u <- resid(dynlm(return ~ return_1, data=tsnyse))
u2 <- u*u
u2_reg <- dynlm(u2 ~ L(u2, 1) + 1, data = tsnyse)
u_reg <- dynlm(u ~ L(u, 1) + 1)
stargazer(u2_reg, u_reg, no.space=TRUE, type="text")##
## ===========================================================
## Dependent variable:
## ----------------------------
## u2 u
## (1) (2)
## -----------------------------------------------------------
## L(u2, 1) 0.337***
## (0.036)
## L(u, 1) 0.001
## (0.038)
## Constant 2.947*** -0.001
## (0.440) (0.081)
## -----------------------------------------------------------
## Observations 688 688
## R2 0.114 0.00000
## Adjusted R2 0.112 -0.001
## Residual Std. Error (df = 686) 10.759 2.112
## F Statistic (df = 1; 686) 87.923*** 0.001
## ===========================================================
## Note: *p<0.1; **p<0.05; ***p<0.01