Today we are going to discuss autocorrelation problem which is a violation of one of the Gauss-Markov assumptions. Outline of the lecture is presented below.

Outline

Durbin Watson Test for AC

Breusch Godfrey Test for AC

White HC Robust Standard Errors

Newey West HAC Robust Standard Errors

EGLS

Cochrane Orcutt Method

Prais Winsten Method

Durbin Watson Test for AC

To investigate auto-correlation we will focus on US macro data and try to obtain a consumption function. To be able to do this we will deal with a time series model. Since autocorrelation is much more common in time series data we have to resort to a time series model. In our model we will use USMacroG dataset from AER package.

install.packages("dynlm", repos = "https://cran.r-project.org/")

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library(dynlm)

## Loading required package: zoo

## 
## Attaching package: 'zoo'

## The following objects are masked from 'package:base':
## 
##     as.Date, as.Date.numeric

library(AER)

## Loading required package: car

## Loading required package: carData

## Loading required package: lmtest

## Loading required package: sandwich

## Loading required package: survival

data("USMacroG")
consump1 <- dynlm(consumption ~ dpi + L(dpi), data = USMacroG)

So far we have constructed a DL(1) model for the investigation of autocorrelation. Lagging component is obtained by using L() command. If we had specified as L(,x), x would be the order of lag. To conduct durbin watson test we can use two packages: “lmtest” and “car”. In car package command for Durbin-Watson test is durbinWatsonTest() and in car package function for conducting Durbin Watson test is dwtest().

install.packages("car", repos = "https://cran.r-project.org/")

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library(car)
durbinWatsonTest(consump1)

##  lag Autocorrelation D-W Statistic p-value
##    1       0.9244708     0.0866355       0
##  Alternative hypothesis: rho != 0

durbinWatsonTest(consump1, max.lag = 4)

##  lag Autocorrelation D-W Statistic p-value
##    1       0.9244708     0.0866355       0
##    2       0.8634632     0.1342431       0
##    3       0.7947730     0.2123351       0
##    4       0.7183643     0.2914617       0
##  Alternative hypothesis: rho[lag] != 0

install.packages("lmtest", repos = "https://cran.r-project.org/")

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library(lmtest)
dwtest(consump1)

## 
##  Durbin-Watson test
## 
## data:  consump1
## DW = 0.086636, p-value < 2.2e-16
## alternative hypothesis: true autocorrelation is greater than 0

For the arguments of the function you can consult to help segment of the R Studio.

Breusch Godfrey Test for AC

Durbin Watson test fails to capture AC when a lagged term of the dependent variable included. In other words, BG test is applicable in AR(p), ARDL(p,q), MA(q), and DL(q) models. BG test is implemented as bgtest() function of lmtest package.

library(lmtest)
bgtest(consump1)

## 
##  Breusch-Godfrey test for serial correlation of order up to 1
## 
## data:  consump1
## LM test = 192.97, df = 1, p-value < 2.2e-16

To see the difference between DW and BG tests we can construct an ARDL(2,1) model.

consump2 <- dynlm(consumption ~ dpi + L(dpi) + L(consumption, 2)+ L(consumption), data = USMacroG)
durbinWatsonTest(consump2, max.lag = 4)

##  lag Autocorrelation D-W Statistic p-value
##    1     -0.10692838      2.184723   0.190
##    2      0.21865835      1.500568   0.000
##    3      0.16513471      1.594958   0.002
##    4     -0.05008718      1.987786   0.976
##  Alternative hypothesis: rho[lag] != 0

bgtest(consump2)

## 
##  Breusch-Godfrey test for serial correlation of order up to 1
## 
## data:  consump2
## LM test = 11.215, df = 1, p-value = 0.0008116

White HC Robust Standard Errors

For investigation of the robust standard errors one should be familiar with the sandwich package. vcovHC(x, …) function provides us White’s HC consistent standard errors. Usage of the function is provided below.

install.packages("sandwich", repos = "https://cran.r-project.org/")

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library(sandwich)
coeftest(consump1, vcov = vcovHC(consump1, type = "HC1"))

## 
## t test of coefficients:
## 
##               Estimate Std. Error t value  Pr(>|t|)    
## (Intercept) -81.079588  15.527398 -5.2217 4.429e-07 ***
## dpi           0.891168   0.215252  4.1401 5.116e-05 ***
## L(dpi)        0.030913   0.215612  0.1434    0.8861    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

summary(consump1)

## 
## Time series regression with "ts" data:
## Start = 1950(2), End = 2000(4)
## 
## Call:
## dynlm(formula = consumption ~ dpi + L(dpi), data = USMacroG)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -190.02  -56.68    1.58   49.91  323.94 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -81.07959   14.50814  -5.589 7.43e-08 ***
## dpi           0.89117    0.20625   4.321 2.45e-05 ***
## L(dpi)        0.03091    0.20754   0.149    0.882    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 87.58 on 200 degrees of freedom
## Multiple R-squared:  0.9964, Adjusted R-squared:  0.9964 
## F-statistic: 2.785e+04 on 2 and 200 DF,  p-value: < 2.2e-16

Let us delve deeper into arguments of vcovHC function from the URL provided below. https://cran.r-project.org/web/packages/sandwich/vignettes/sandwich.pdf

Newey West HAC Robust Standard Errors

Method is pretty similar with the those of White standard errors but a minor alteration is needed. Instead of using vcovHC function while specifying the method for variance-covariance matrix, we are going to use NeweyWest() function.

coeftest(consump1, vcov = NeweyWest(consump1))

## 
## t test of coefficients:
## 
##               Estimate Std. Error t value Pr(>|t|)  
## (Intercept) -81.079588 100.825730 -0.8042  0.42226  
## dpi           0.891168   0.423003  2.1068  0.03638 *
## L(dpi)        0.030913   0.398930  0.0775  0.93831  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

coeftest(consump1)

## 
## t test of coefficients:
## 
##               Estimate Std. Error t value  Pr(>|t|)    
## (Intercept) -81.079588  14.508136 -5.5886 7.430e-08 ***
## dpi           0.891168   0.206252  4.3208 2.448e-05 ***
## L(dpi)        0.030913   0.207542  0.1490    0.8817    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Cochrane Orcutt Method

Execution of Cochrane Orcutt iterative EGLS method requires orcutt package. Reference manual of the orcutt package can be found at the URL provided below. https://cran.r-project.org/web/packages/orcutt/orcutt.pdf

Let us construct a new model by following the footsteps of Mr. Eruygur. Our new model again focuses on consumption function and obtained from USConsump1993 dataset.

install.packages("orcutt", repos = "https://cran.r-project.org/")

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library(orcutt)
library(AER)
data("USConsump1993")
consump93 <- lm(expenditure ~ income, data = USConsump1993) 
summary(consump93)

## 
## Call:
## lm(formula = expenditure ~ income, data = USConsump1993)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -294.52  -67.02    4.64   90.02  325.84 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -65.795821  90.990824  -0.723    0.474    
## income        0.915623   0.008648 105.874   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 153.6 on 42 degrees of freedom
## Multiple R-squared:  0.9963, Adjusted R-squared:  0.9962 
## F-statistic: 1.121e+04 on 1 and 42 DF,  p-value: < 2.2e-16

cochrane.orcutt(consump93)

## Cochrane-orcutt estimation for first order autocorrelation 
##  
## Call:
## lm(formula = expenditure ~ income, data = USConsump1993)
## 
##  number of interaction: 4
##  rho 0.782893
## 
## Durbin-Watson statistic 
## (original):    0.46078 , p-value: 3.274e-11
## (transformed): 1.95670 , p-value: 3.832e-01
##  
##  coefficients: 
## (Intercept)      income 
## -170.317934    0.926381

Prais Winsten Method

This method requires prais package and it is quiet similar to Cochrane-Orcutt estimation method. Following code showa its application.

install.packages("prais", repos = "https://cran.r-project.org/")

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library(prais)
library(AER)
data("USConsump1993")
prais_winsten(consump93, data = USConsump1993)

## Iteration 0: rho = 0
## Iteration 1: rho = 0.7921
## Iteration 2: rho = 0.8021
## Iteration 3: rho = 0.8037
## Iteration 4: rho = 0.804
## Iteration 5: rho = 0.804
## Iteration 6: rho = 0.804
## Iteration 7: rho = 0.804
## Iteration 8: rho = 0.804

## 
## Call:
## prais_winsten(formula = consump93, data = USConsump1993)
## 
## Coefficients:
## (Intercept)       income  
##    -15.1436       0.9142  
## 
## AR(1) coefficient rho: 0.804

Autocorrelation: Tests and Remedies

Arda Yalcin

17/11/2019

Outline

Durbin Watson Test for AC

Breusch Godfrey Test for AC

White HC Robust Standard Errors

Newey West HAC Robust Standard Errors

EGLS

Cochrane Orcutt Method

Prais Winsten Method

Durbin Watson Test for AC

Breusch Godfrey Test for AC

White HC Robust Standard Errors

Newey West HAC Robust Standard Errors

Cochrane Orcutt Method

Prais Winsten Method