Monday, June 1, 2015

The Toda–Yamamoto Approach to Granger Causality Test

(These paragraphs are from my Masters Thesis)

The dynamic Granger causality can be captured from the vector error correction model derived from the long-run cointegrating relationship Granger (1988). The Granger causality proposed by Granger (1969) has probable shortcomings of specification bias and spurious regression. Engel and Granger (1987) have defined X and Y as being cointegrated if the linear combination of X and Y is stationary but each variable is not stationary. Engel and Granger (1987) pointed out that while these two variables are non-stationary and cointegrated, the standard Granger -causal inference will be invalid.

To mitigate these problems, Toda and Yamamoto (1995) and Dolado and Lutkepohl (1996) (TYDL henceforth) based on augmented VAR modeling, introduced a modified Wald test statistic (MWALD). This procedure has been found to be superior to ordinary Granger - causality tests since it does not require pre-testing for the cointegrating properties of the system and thus avoids the potential bias associated with unit roots and cointegration tests as it can be applied regardless of whether a series is I(0), I(1) or I(2), non-cointegrated or cointegrated of an arbitrary order.

TYDL approach, first involves finding the maximum order of integration dmax of the series that are to be incorporated in the model. For this conventional ADF unit root test is applied in each series and the maximal order of integration is identified. Say, with ADF unit root test, three variable are found to be I(0), I(1) and I(1) respectively, then the maximal order of integration is 1. TYDL approach, secondly, specifies a well behaved kth optimal lag order vector autoregressive model in levels (not in the difference series).

The number of optimal lags is usually determined by a selection criterion such as the Akaike Information criterion (AIC), Bayesian information criterion (BIC), or Schwarz Info Criterion (SIC) or the democracy of these criterion which ever makes the VAR well behaved in term of AR unit root graph, VAR residual serial correlation LM-stat, VAR residual normality tests. TYDL approach, thirdly, intentionally over-fits the underlying model is with additional dmax order of integration.The dmax is the maximal order of integration of the series in the model.

A stylized working procedure is given in this figure.

Lets suppose, we have 3 variables, Log Real GDP (LRGDP), log of money supply (LM2) and inflation (INF), they are I(1), I(1) and I(0) respectively. The given as: (k + dmax ) VAR for LRGDP, LM2 and INF is given as:

where, LRGDP, LM2 and INF are Y1t, Y2t and Y3t respectively.

Finally, to draw valid casual inferences, TYDL procedure utilizes a modified Wald test statistic (MWALD) restricting the parameters of k-th optimal lag order of the vector autoregressive. The MWALD statistic has an asymptotic chi-square distribution when VAR (k + dmax ) is estimated.

To test the causality running from LRGDP to INF

To test the causality running from LM2 to INF

To test the causality running jointly from LRGDP and LM2 to INF
To test the causality running from LM2 to LRGDP
To test the causality running from INF to LRGDP

To test the causality running jointly from INF and LM2 to LRGDP

Here is the Video Post:


Dolado, J.J., Lütkepohl, H., 1996. Making wald tests work for cointegrated VAR systems. Econ. Rev. 15, 369–386.

Engle, R.F. and C.W.J. Granger (1987), “Co-integration and error-correction: representation, estimation, and testing”. Econometrica, 55: 251-276.

Granger, C.W.J., 1969. Investigating causal relation by econometric and cross-sectional method. Econometrica 37, 424–438.

Granger, C.W.J. (1988), “Some recent developments in the concept of causality”, Journal of Econometrics, 39: 199-211.

Toda, H. Y., & Yamamoto, T. (1995). Econometrics, 66, 225–250.


  1. Hi sir , I wonder if I can use this test with 6 variables simultaneously

    1. Could you please cite any reference that TY can be used in ">2"-variable systems?

  2. can u please guide....can we test posittive or negative casuality through toda model..i mean how can we explain that by increasing inflation, trade openness is increasing or decreasing .we can say that there is bidirectional cauality but what sign it show? variables are causing each other but in which directin? and can we find shor n long run causality with this test..
    if yes then kindly explain the method i really need it urgently thank u

    1. Sir i need your reply urgently
      kindly reply

    2. This will be shown in your long and short run dynamic analysis. This test is to attest to the two test.. ..Granger causality is a spectral analysis.. .it shows no sign.. But the sign may be reveal maybe through var or vecm alongside the cointegrating equation

  3. Hi, sir if I have k+dmax=2 (where lag length =1 and the variables are integrated at I(1)), can I still go ahead with the TY non-granger causality test?

  4. hi i am very happy about the work that you are doing very nice. Please I would like to know if you can help me on my thesis to charaterize économic cycles and forecasting a the output gap?

    1. Hi Tmac,

      To identify the potential GDP and outputGAP you can run

      Hodrick-Prescott filter
      Baxter-King filter
      Christiano-Fitzgerald filter
      Butterworth filter
      Trigonometric regression filter

      Once you find the out put gap (using each filter), you can develop ARIMA for each of output gap. Make your ARIMA best one!

      Then With ARIMA you can forecast the possible output gap then do model averaging.

      To make your work more validated you can perform training and testing exercise (as I done in ARIMA tutorials). Also perform forecaste accurarcy test.

      Cheers. I hope this will solve your problem!

      Good day.

  5. Hi. I want to test causality between two rates , saving and growth, thank you to respond on my two questions :
    1- most of the two series (I do it for many countries) are I(0) or I(1) than I conclude that dmax=1, the length k=1 (or 2 for other countries) for the first case should I put in Eviews lag from 1 to 1 or 1 to 2 in VAR lag ?
    2- when I used the causality test I found any result, the two series are cointegrated (one is I(0) the other I(1) i dont know if the priblelen is with lags or the data ... thank you

  6. Hi Shishir,
    This article is very helpful. I have a query though. What if you have 3 variables, which are I(1), but not cointegrated. Would we still have to follow the Toda-Yamamoto approach to test for causality, or can we use the normal Granger causality Wald test results?

    1. Hi, you can do simple ols or the VAR model to test the causality.

    2. Thanks Shishir for your reply. I saw a blog by Dave Giles, which said irrespective nature of cointegration (i.e. whether variables are cointegrated or not), one should use the Toda-Yamamoto approach to test for causality. Any thoughts?

  7. Hi Shishir,

    Your video is very informative for applying TY test.

    I was actually looking for the Dynamic Granger Causality analysis, which is based on the time-varying property of the variables. While going through the paper titled "Time-varying analysis of CO2 emissions, energy consumption, and
    economic growth nexus: Statistical experience in next 11 countries" by Shahbaz et al., I encountered this new technique. I have been searching for any concrete material on this technique, and I haven't found anything yet.

    It will be extremely helpful for me, if you provide me with some material (hands-on or STATA code is preferable) on this topic.

    Awaiting your reply.

    1. Thank you for the post. I havenot yet started to work on them! But If I come across those materials I will share with you!

  8. I really want to use ARDL bound test, but it seems like all variables are I(1) (I tested for unit root in my series with ADF, PP, KPSS and ZA tests). is ARDL bound test viable in my case (all variables are I(1)).
    or i should use the traditional Johansen test?
    can i use both?

  9. Also can i use an ARDL model (and its cointegration and long run form) to capture the short run and long run relations and make inferences about them.
    or i strictly have to use a VECM in presence of cointegration.

  10. i have 4 variables.

    ZA test reveals that the series are I(1) allowing breaks in the intercept and both (intercept and trend). however, when allowing breaks in trend only, the test's results show that 3 of my 4 variables are I(2).
    Should I conclude that my variables are I(1)?
    Whats the best way to test for cointegration in this case? johansen or ARDL Bound test?

    Thanks in advance!

    1. ZA1992 test is very old. NarayanPopp2010 and NarayanLiu2015 ara both plausible in H0,H1 hypothesis, and also allows 2 structural breaks in time. ZA1992's H0,H1 hypothesis are not conformal to the daily economic life. A time series may break, but it is unlikely that it changes its nature: sto to det or vice versa.

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  12. hi sir, may i know how to add dummy(structural break) in Eview system?

  13. Hello. If series are I(1), I(1), I(1) and I(2). Can we use Toda & Yamamoto test in this situation?

  14. Dear Shishir,
    Congratulation for your Blog, it's very informative! Let me ask you. When I'm developing a VARX, my exogenous variables must be stationary too? Because my multipliers are very sensitive to this point.
    I'm waiting for more videos.
    Thank you.