Wed

07

Nov

2012

When it comes to causality tests, the typical Granger-causality test can be problematic. Testing for Granger-causality
using *F*-statistics when one or both time series are non-stationary can lead to spurious causality (He & Maekawa,
1999).

Professor Giles gives an excellent example of how the TY method can be implemented.

More formal explanations can be found in the original TY (1995) paper or for example here.

In this post, I will show how Professor Giles' example can be implemented in R.

The procedure is based on the following steps:

1. Test for integration (structural breaks need to be taken into account). Determine max order of integration (*m*). If none of the series in
integrated, the usual Granger-causality test can be done.

2. Set up a VAR-model in the levels (do not difference the data).

3. Determine lag length. Let the lag length be p. The VAR model is thus VAR(p).

4. Carry out tests for misspecification, especially for residual serial correlation.

5. Add the maximum order of integration to the number of lags. This is the augmented VAR-model, VAR(p+m).

6. Carry out a Wald test for the ** first p variables only** with p degrees of freedom.

You may want to do a test of cointegration. If series are cointegrated, there must be a causality. However, Toda and Yamamoto (1995) noted that one advantage of the TY-method is that you don't have to test for cointegration and, therefore, a pretest bias can be avoided.

__________

The example is about causalities between prices in Robusta and Arabica coffee. The excel-file can be downloaded here. But in order to be loaded into R, the data should be put in the csv. format. The csv. file is available here.

Update: If you want examine the data interactively, have a look here.

The script below tests for causality between these two time series. The script is annotated, but let me know if I can clarify anything or if there is room for improvement.

```
``````
library(fUnitRoots)
library(urca)
library(vars)
library(aod)
library(zoo)
library(tseries)
#Load data
cof <- read.csv("http://www.christophpfeiffer.org/app/download/6938079586/coffee_data.csv", header=T,sep=";")
names(cof)
#Adjust Date format
cof["Date"]<-paste(sub("M","-",cof$Date),"-01",sep="")
#Visualize
plot(as.Date(cof$Date),cof$Arabica,type="l",col="black",lwd=2)
lines(as.Date(cof$Date),cof$Robusta,col="blue",lty=2,lwd=1)
legend("topleft",c("Arabica","Robusta"),col=c("black","blue"),lty=c(1,2),lwd=c(2,1),bty="n")
#Possible structural break in 1970s. Therefore only values from 1976:01 onwards are regarded
cof1<-cof[193:615,]
#Visualize
plot(as.Date(cof1$Date),cof1$Arabica,type="l",col="black",lwd=2,ylim=range(cof1$Robusta))
lines(as.Date(cof1$Date),cof1$Robusta,col="blue",lty=2,lwd=1)
legend("topright",c("Arabica","Robusta"),col=c("black","blue"),lty=c(1,2),lwd=c(2,1),bty="n")
#Test for unit roots
adf.test(cof$Arabica)
adf.test(cof$Robusta)
kpss.test(cof$Arabica)
kpss.test(cof$Arabica)
adf.test(diff(cof$Arabica,1))
adf.test(diff(cof$Robusta,1))
kpss.test(diff(cof$Arabica,1))
kpss.test(diff(cof$Robusta,1))
# Since first order differencing eliminates the unit root, the maximum order of integration
# is concluded to be I(1).
#Set up VAR-Model
#select lag order // either 2 or 6
VARselect(cof1[,2:3],lag=20,type="both")
#VAR Model, lag=2
V.2<-VAR(cof1[,2:3],p=2,type="both")
serial.test(V.2)
#VAR-Model, lag=6
V.6<-VAR(cof1[,2:3],p=6,type="both")
serial.test(V.6)
#Stability analysis (thanks to Erdogan Cevher)
1/roots(V.6)[[1]] # ">1"
1/roots(V.6)[[2]] # ">1"
# Model with p=6 is less likely to be serially correlated. Thus model with p=6 is selected.
# Wald-test for the first 6 lags
# VAR model is seperately set up as a linear model; makes the wald test easier
#lag variables
arab<-zoo(cof1["Arabica"])
robu<-zoo(cof1["Robusta"])
arab.l<-lag(arab,-(0:7),na.pad=T)
robu.l<-lag(robu,-(0:7),na.pad=T)
lm1<-lm(arab~arab.l[,2:8]+robu.l[,2:8]+index(arab))
lm2<-lm(robu~arab.l[,2:8]+robu.l[,2:8]+index(arab))
#Wald-test (H0: Robusta does not Granger-cause Arabica)
vcov(lm1)
wald.test(b=coef(lm1), Sigma=vcov(lm1), Terms= c(9:14),df=6)
# Could not be rejected (X2=8.6; p=0.2)
#Wald.test (H0: Arabica does not Granger-cause Robusta)
vcov(lm2)
wald.test(b=coef(lm2), Sigma=vcov(lm2), Terms= c(2:7),df=6)
# Could be rejected at 10% (X2=12.3; p=0.056)
# It seems that Arabica Granger-causes Robusta prices, but not the other way around.
```

You can download the R-code as well as the csv. file in "Files".

Let me know if you have any suggestions.

--- C

**References**

He, Z.; Maekawa, K. (1999). On spurious Granger causality. *Economic letters*, 73(3), 307–313.

Toda H.Y.; Yamamoto T. (1995). Statistical inference in vector autoregressions with possibly integrated
processes. *Journal of Econometrics*, 66, 225–250.

Write a comment

## Comments: 53

#1Dave Giles (

Wednesday, 07 November 2012 23:42)Christoph. This is just great! Thanks for sharing this.

#2Claudio D. Shikida (

Thursday, 08 November 2012 16:10)Just change this for the tests.

It's not adf.test, but ur.df. For example:

ur.df(cof$Arabica)

ur.df(cof$Robusta)

ur.kpss(cof$Arabica)

ur.kpss(cof$Arabica)

Great job! Thanks!

#3Christoph(Thursday, 08 November 2012 17:38)Claudio, thanks for pointing out.

adf.test(), kpss.test() work es well, but we need the "tseries" package loaded. Should be fine now.

#4Cindy M.(Thursday, 06 December 2012 17:53)Thanks for sharing this! Two questions for you.

In lm1<-lm(arab~arab.l[,2:8]+robu.l[,2:8]+index(arab)) , can you please explain why you need to have the index(arab) term in the regression?

Also in:

>#Wald.test (H0: Arabica does not Granger-cause Robusta)

>vcov(lm2)

>wald.test(b=coef(lm1), Sigma=vcov(lm1), Terms= c(2:7),df=6)

Should this wald.test be using lm2 instead of lm1?

#5christophpfeiffer (

Thursday, 06 December 2012 18:08)Hi Cindy,

thanks for the catch, it is indeed lm2 and not lm1.

index(arab) caputres the trend index(robu) would of course also work and yields the same result.

-- Christoph

#6Cindy M.(Thursday, 06 December 2012 18:23)Thanks for the quick response!

#7london(Thursday, 21 March 2013 20:12)Dear Christoph,

I used the code on a data set which has 22 observations and 9 variables all variables entering into the vAR model. I determined p=k+dmax=3. this gives me 27 coefficient estimates. I have NAs in the results. Just wondering if I am doing something wrong or overfitting?

#8christophpfeiffer (

Thursday, 21 March 2013 21:52)Hi,

it seems that you have relatively few observations which makes statistical analysis difficult. Any chance obtain more data? Have you considered bootstrapping?

-- Christoph

#9london(Thursday, 21 March 2013 23:31)Dear Christoph,

I am trying to run a bootstrap regression, however, the regression is producing NAs in place of coefficients for the last 2 coefficients after every bootstrap.

would appreciate any suggestion!

#10christophpfeiffer (

Friday, 22 March 2013 17:32)If you like, you can send me your R-code and the data and I'll have a look.

christophpp@gmail.com

#11london(Friday, 22 March 2013 23:54)Dear Christoph,

Many thanks, I will email you my code and data set.

#12Aviral Kumar Tiwari(Sunday, 19 May 2013 10:08)I tried to use your R-codes but in my computer it it shows "Error: could not find function "wald.test""

when i used

> wald.test(b=coef(lm1), Sigma=vcov(lm1), Terms= c(9:14),df=6)

Please help me ..

#13christophpfeiffer (

Sunday, 19 May 2013 12:14)Hi Aviral,

make sure you have all packages installed. So you shouldn't get any error messages for the first 6 lines. You can install packages with the function

install.packages("...")

where ... is the name of the package. Specifically, for the function wald.test() you need the package "aod" to be installed. For this you need to execute the following line:

install.packages("aod")

once.

Hope this helps

Christoph

#14Aviral Kumar Tiwari(Sunday, 19 May 2013 13:55)thankssssss it worked out ...

#15Mike(Friday, 07 June 2013 00:49)Christoph,

The output for wald.test also gives the F-statistic, but your comments show the test is determined by the X2 results. What is the interpretation of the F output results in this case? Thanks!

#16christophpfeiffer (

Sunday, 16 June 2013 12:05)Mike,

sorry for the late answer, I just got back from vacation.

Toda and Yamamoto have shown that if you add an additional lag to a correctly specified VAR-model for which at least one time-series is integrated, the parameters asymptotically follow a chi-squared distribution. Assuming these two conditions are met, looking at the F-statistic for the augmented model would simply yield a meaningless answer.

#17Rick(Sunday, 14 July 2013 02:35)In your wald.test, why are you using Terms c(9:14) and c(2:7)? I would have thought it would be c(8:13) and (1:6) because the last lag is supposed to be exogenous and just there for the asymptotics?

#18Christoph(Monday, 15 July 2013 09:36)Hi Rick,

for the first test we need to assess if the coefficients of Robusta are significant for the price of Arabica. For the second test we are testing if the coefficients of Arabica are significant for the price of Robusta. These are the models:

Arabica = Intercept+A*X_1+B*X_2 + Trend # lm1

No. of terms: (1) (7) (7) (1) = 16

Robusta = Intercept+A*X_1+B*X_2 + Trend # lm2

No. of terms: (1) (7) (7) (1) = 16

With A being the coefficients for the values of Arabica, B coefficients for Robusta, X_1 and X_2 the respective lagged values. For each matrix of lagged values we have 7 terms but only want to test the first 6. Then, for the first Wald test we need to look at terms 9 to 14 and for the second model at terms 2 to 7. You can look at the coefficients of each model with coef(lm1) or coef(lm2).

#19Martin(Monday, 28 October 2013 13:21)I'm having trouble interpreting the serial.tests. What's the significance of the degrees of freedom and the p-values? (The p-value for serial.test(V.6) is .5.)

#20Bob(Wednesday, 11 December 2013 19:44)Any way to do this without the packages?

#21christophpfeiffer (

Saturday, 14 December 2013 15:14)Bob,

of course, but why would you do it without the packages? If you want to look deeper, you can just look at the functions provided by the packages.

#22christophpfeiffer (

Saturday, 14 December 2013 15:20)Martin,

sorry for the late reply. The serial test used here is a portmanteau test with the null hypothesis that residual errors are not serially correlated. A higher p-value indicates a favorable model which is why a model with 6 vs. 2 lags is chosen.

#23Yolande(Friday, 14 March 2014 20:18)I was wondering how the method and the R code would change if one of the series has unit root and the other does not.

#24Christoph(Friday, 14 March 2014 21:36)Yolande,

good question. The first step says:

1. Test for integration (structural breaks need to be taken into account). Determine max order of integration (m). If none of the series in integrated, the usual Granger-causality test can be done.

So, if one series has an order of integration of 1 and the other is not integrated (order of integration is 0), the maximum order of integration would be m=1.

Jumping to step 5:

5. Add the maximum order of integration to the number of lags. This is the augmented VAR-model, VAR(p+m).

This means we would still have to use the augmented model even if one series is not integrated at all. It's the maximum order of integration that counts. The only case in which you can use the standard Granger test for non-causality is when neither of the series are integrated.

#25Cory (

Monday, 17 March 2014 18:00)Christoph,

I was wondering how I could change the code in my blog to follow the procedure you outline above. I don't understand the idea of including an additional lag, but not testing for it.

http://r-datameister.blogspot.com/2014_02_01_archive.html

#26Yolande(Monday, 17 March 2014 18:00)Thank you for the prompt reply.

#27Yolande(Tuesday, 18 March 2014 15:06)I do have an additional question: Where in the code and algorithm the model v.7 was used after it is set

#28Pablo(Thursday, 03 April 2014 22:38)Hi,

I was wondering if you've ever used the function "causality" from the package "vars"? Are the results from this function compatible with the ones from your code?

Thanks a lot

#29Christoph(Saturday, 05 April 2014 16:17)Cory,

there are some issues with differencing the data first for integrated time series and then using the standard Granger causality test (e.g. Enders, W. Applied Econometric Times Series ).

You can think of the idea to add an additional lag and then not using it for the test as a trick to get rid of the problems involved in testing integrated time series for causality.

#30Christoph(Saturday, 05 April 2014 16:18)Yolande,

you are very welcome. :)

Good point, model v.7 is actually not necessary for the following tests.

#31Christoph(Saturday, 05 April 2014 16:20)Hi Pablo,

as far as I know, the causality function from 'vars' only uses the standard Granger causality test and would not work in this context.

#32Yolande(Monday, 07 April 2014 21:33)Hi Christoph,

This is a general question.

Why would I got -Infinity in my varselect. The lag length is 3 based on all criteria.

> VARselect(M,lag.max=20,type="both")

$selection

AIC(n) HQ(n) SC(n) FPE(n)

3 3 3 3

$criteria

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

AIC(n) 5.517709 4.919171 -Inf -Inf -Inf -Inf -Inf -Inf -Inf -Inf -Inf -Inf -Inf -Inf -Inf -Inf -Inf

HQ(n) 5.139389 4.351692 -Inf -Inf -Inf -Inf -Inf -Inf -Inf -Inf -Inf -Inf -Inf -Inf -Inf -Inf -Inf

SC(n) 5.693020 5.182138 -Inf -Inf -Inf -Inf -Inf -Inf -Inf -Inf -Inf -Inf -Inf -Inf -Inf -Inf -Inf

FPE(n) 284.562815 237.788270 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

18 19 20

AIC(n) -Inf -Inf -Inf

HQ(n) -Inf -Inf -Inf

SC(n) -Inf -Inf -Inf

FPE(n) 0 0 0

Thanks

#33christophpfeiffer (

Friday, 18 April 2014 19:31)Hi Yolande,

this probably happens because the number of lags available in your dataset is too low to calculate the criteria for lags higher than two. If this does not explain it, you can send me the dataset and I will have a look.

#34Lohit(Friday, 02 May 2014 11:50)Hi Christoph,

When I execute the code I get the following error. How to debug this. Can you please help me with it. I am new R software. Thank you.

> #Wald-test (H0: Robusta does not Granger-cause Arabica)

> vcov(lm1)

Error in if (is.finite(resvar) && resvar < (mean(f)^2 + var(f)) * 1e-30) warning("essentially perfect fit: summary may be unreliable") :

missing value where TRUE/FALSE needed

#35Erdogan CEVHER(Saturday, 17 May 2014 00:32)Hi Christoph,

Your presentation is very nice. By the way, allowing the others to ask questions and your immediate answers are very nice. I grasped the topic. I want to gift you my econometry book as soon as it is finished.

#36Erdogan CEVHER(Saturday, 17 May 2014 15:45)Hi Christoph,

First, thanks millions of times for the above R code.

In #29, you said: "there are some issues with differencing the data first for integrated time series and then using the standard Granger causality test (e.g. Enders, W. Applied Econometric Times Series )."

However, what W. Enders' say in "Applied Econometric Times Series" 3E (2010) p.321 (equivalent of above 347 in the below link) that:

"The issue of differencing is important. If the VAR can be written entirely in first differences, hypothesis tests can be performed on any equation or any set of equations using t-tests or F-tests. This follows because all of the variables are stationary. As you will see in the next chapter, it is possible to write the VAR in first differences if the variables are I(1) and are not cointegrated. If the variables in question are cointegrated, the VAR cannot be written in first differences; hence, causality tests cannot be performed using t-tests or F-tests."

Book: http://tr.scribd.com/doc/215018030/180318659-Applied-Econometric-Time-Series-3rd-Edition-Walter-Enders-PDF

That is to say, the "issues" you mentioned are restricted to both of the variables are nonstationary AND COINTEGRATED.

When only one of the variables in a system of 2 variables is non-stationary, and the variables are not cointegrated, then Enders (2010) does not mention anything about the problematic nature of classical G-causality(1969).

Hence all in all: in a system of 2 variables:

Cases of Variables Solution Source

--------------- ----------------------- ------------------------

1. Both Stationary .. Classical G-causality(1969) .. Granger (1969)

2. Only one nonstationary ..(AFAIK:) Toda-Yamamoto(1995) .. ??????

3. Both nonstationary and not cointegrated .. TY(1995) .. TY(1995)

4. Both nonstationary and cointegrated (VAR-->VECM) .. (I am not sure:)TY(1995) .. ??????

I will be glad if you clarify my "AFAIK"s, "I am not sure"s and "????"s.

Best and Warm Regards,

#37Erdogan CEVHER(Saturday, 17 May 2014 16:16)Hi Chris, again me!

Look what Enders 3E 2010 p. 397 say (for my cases of Variables, 3rd case):

"There are three consequences if the I(1) variables are not cointegrated and you estimate the VAR in levels...:

For a VAR in levels, test for Granger causality conducted on the I(1) variables do not have a standard F-distribution. If you use first differences, you can use the standard F-distribution to test for Granger causality".

Hence, as far as I understand, (in the case of non-cointegrated non-stationary I(1) variables) Enders2010 suggest classical G-causality(1969) on 1st differences. However, many books says, "in a 2-variable system in which AT LEAST ONE OF THE VARIABLES IS NONSTATIONARY, TY1995 is applied. Hence, for the above 3rd case, does it seem that Both G1969 (on 1st differences) and TY1995 (on levels) be applied?

Also, what do you think for my above 4th case?

Regards...

#38Erdogan Cevher(Saturday, 17 May 2014 17:39)As if it seems no matter (stationary/non-stationary, integrated/cointegrated) what the variables in 2-variable system, TY1995 correctly finds G-causality:

Eiji Kurozumi, Khashbaatar Dashtseren 2011: "Statistical Inference in Possibly Integrated/Cointegrated Vector Autoregressions: Application to Testing for Structural Changes":

".....Toda and Yamamoto (1995) propose to estimate a model with intentionally augmented lags, and show that the estimated parameter of interest has a limiting normal distribution IRRESPECTIVE OF WHETHER THE VARIABLES ARE (TREND) STATIONARY, INTEGRATED, OR COINTEGRATED...."

Anyway, Chris, I wanna learn what you think as well.

#39Erdogan CEVHER(Saturday, 17 May 2014 22:06)An offer to the above code: The revealing of why the relevant packages was loaded; that is to say, for which function... would be better:

library(fUnitRoots)

library(urca)

library(vars) # VARselect, serial.test are in vars

library(aod) # wald.test is in aod

library(zoo)

library(tseries) #adf.test, kpss.test are in tseries

#40Erdogan CEVHER(Sunday, 18 May 2014 00:35)Another offer to the above code: I looked at the above code and observed the following code snippet:

# Model with additional lag is set up.

V.7<-VAR(cof1[,2:3],p=7,type="both")

is upset since it is not being used. I decided the above snippet to make happy! Notice that the difference between your R code and Dave Giles' Eviews code is the absence of STABILITY ANALYSIS OF THE VAR!

Here, you can add the following lines to the code to perform VAR stability analysis, and completely equalize your R code with Giles' Eviews:

# Stability analysis of VAR(7)

1/roots(V.7)[[1]] # result became ">1"

1/roots(V.7)[[2]] # result became ">1"

# Since the above two value exceeds (in absolute value) 1, V.7 is stable.

# Fact: VAR is stable if the eigenvalues of RHO part of VAR exceeds 1

equivalently you can use:

roots(V.7)[[1]] # result became "<1"

roots(V.7)[[2]] # result became "<1"

and get the same stability of the VAR(7).

Best and Warm Regards,

Erdogan CEVHER

#41Erdogan CEVHER(Sunday, 18 May 2014 01:03)Third Offer to the above code:

Though p values of serial.test's are enough to decide, it would be nicer to show the ACFs of the residuals of the VARs as well. Hence, I offer to add the followings just after the relevant tests:

plot(serial.test(V.2))

plot(serial.test(V.6))

#42Erdogan CEVHER(Sunday, 18 May 2014 01:07)In my #39,

Instead of:

library(vars) # VARselect, serial.test are in vars

....etc......

the following is nicer:

library(vars) # for VARselect, serial.test

....etc......

#43christophpfeiffer (

Saturday, 24 May 2014 17:16)Hi Erdogan,

thanks for making my argument more precise. The issues in differencing integrated time series first and then applying the standard Granger-causality test are that the following two conditions have to be met:

1. The time-series have to be integrated by the same order.

2. The time-series must not be conintegrated.

Personally, I would simply stick to the TY-method as soon as one series is integrated.

The TY-method also works when time series are cointegrated. There is no need to test for cointegration first (see TY 1995).

Thanks for the stability analysis.

All the best and good luck for your book!

#44christophpfeiffer (

Saturday, 24 May 2014 17:39)Lohit,

make sure you have installed all the packages (you only need to due this once):

install.packages("fUnitRoots")

install.packages("...")

Let me know if this helped.

#45agapi Somwaru(Sunday, 08 June 2014 03:11)Hi Christoph,

I installed all packages but I still get the error below

>vcov(lm1)

Error in if (is.finite(resvar) && resvar < (mean(f)^2 + var(f)) * 1e-30) warning("essentially perfect fit: summary may be unreliable") :

missing value where TRUE/FALSE needed

best regards

Agapi

#46christophpfeiffer (

Sunday, 08 June 2014 12:53)Agapi,

if you just copy and paste the entire code and successfully download the data this should not happen.

A few things to look at:

1. Have you copied the entire code?

2. Do you get any previous warnings?

3. Have you installed R correctly?

Looking at the warning, it seems that there is something wrong with the data you were supposed to download ('essentially perfect fit').

#Load data

cof <- read.csv("http://www.christophpfeiffer.org/app/download/6938079586/coffee_data.csv", header=T,sep=";")

#47NicholasG(Tuesday, 10 June 2014 01:47)Hi Christopher,

Thanks for the great post. I'm testing for Granger Causality under the TY approach you outline here but when using the function wald.test with d.o.f=p the response when the function is run is a Chi-squared test with df=p+m.

If you look up the help for the package wald.test {aod}, under the arguments for df it says:

"A numeric vector giving the degrees of freedom to be used in an F test, i.e. the degrees of freedom of the residuals of the model from which b and Sigma were fitted"

Given that we want an asymptotically chi-square statistic distributed with p df under the null, does the package wald.test have the ability to specify the degrees of freedom for a Chi-squared test, or does df only work for an F test?

If this is the case do you know any other Wald test package for R that would allow you to specify the degrees of freedom specifically for the Chi-Squared statistic?

#48Alexei V(Tuesday, 10 June 2014 22:27)Hi Christoph, thanks for the post. Question - looking at these two lines:

lm1<-lm(arab~arab.l[,2:8]+robu.l[,2:8]+index(arab))

lm2<-lm(robu~arab.l[,2:8]+robu.l[,2:8]+index(arab))

In the expression for lm2, shouldn't it be index(robu)? Seems that comment #18 above suggests that also?

Thanks!

#49christophpfeiffer (

Wednesday, 11 June 2014 00:34)Hi Alexei,

it does not matter for the end result. index(arab) and index(robu) should be identical.

#50agapi(Wednesday, 11 June 2014 02:34)Hi Christopher,

One of your points were if R was correctly installed.

I had to re-installed 32bit version of R.

Many thanks!

Agapi

#51Alexei V(Wednesday, 11 June 2014 19:49)Thanks.

Would you be able to point me in the direction of how to build a calibration dataset for this kind of modeling in general? E.g., generate two dataseries with specific lags, coefficients, amounts of noise, etc?

#52christophpfeiffer (

Monday, 16 June 2014 23:25)Hello Alexei,

sure. But I need some more specifics on your dataset.

What kind of data do you have and what is your outcome?

#53christophpfeiffer (

Tuesday, 17 June 2014 23:24)Hi Nicholas,

the df parameter in the Wald test only refers to the F-distribtuion and is actually not necessary for the TY-test.

I am not aware of any other wald test function in R. But in any case have a look here:

http://www.statlect.com/Wald_test.htm

and here:

http://www.utstat.toronto.edu/~brunner/oldclass/appliedf12/lectures/2101f12WaldWith