Is there any tool to test the tendency toward/away from stationarity?












2














I have time series data, like the following:



Time       Value
1 20
3 40
8 -10
9 30
...


I'm wondering if there is any test if this time series is toward 0 (like "decrescendo" pattern in the traceplot), or away from 0 (like "crescendo" pattern in the traceplot).



Any recommendation for the tool to visualize/summarize this pattern will be appreciated.










share|cite|improve this question





























    2














    I have time series data, like the following:



    Time       Value
    1 20
    3 40
    8 -10
    9 30
    ...


    I'm wondering if there is any test if this time series is toward 0 (like "decrescendo" pattern in the traceplot), or away from 0 (like "crescendo" pattern in the traceplot).



    Any recommendation for the tool to visualize/summarize this pattern will be appreciated.










    share|cite|improve this question



























      2












      2








      2







      I have time series data, like the following:



      Time       Value
      1 20
      3 40
      8 -10
      9 30
      ...


      I'm wondering if there is any test if this time series is toward 0 (like "decrescendo" pattern in the traceplot), or away from 0 (like "crescendo" pattern in the traceplot).



      Any recommendation for the tool to visualize/summarize this pattern will be appreciated.










      share|cite|improve this question















      I have time series data, like the following:



      Time       Value
      1 20
      3 40
      8 -10
      9 30
      ...


      I'm wondering if there is any test if this time series is toward 0 (like "decrescendo" pattern in the traceplot), or away from 0 (like "crescendo" pattern in the traceplot).



      Any recommendation for the tool to visualize/summarize this pattern will be appreciated.







      time-series






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited 1 hour ago









      Alexis

      15.7k34595




      15.7k34595










      asked 3 hours ago









      moreblue

      35411




      35411






















          2 Answers
          2






          active

          oldest

          votes


















          2














          There are many such tests.



          Generally, they fall into two categories:





          1. Tests for unit root (I.e. tests of stationarity)



            $H_{0}:$ Time series is/are stationary.



            $H_{A}:$ Time series has/have unit root (i.e. is non-stationary).



            Example: The Hadri-Lagrange multiplier test.




          2. Tests for stationarity (I.e. tests of unit root)



            $H_{0}:$ Time series has/have unit root.



            $H_{A}:$ Time series is/are stationary.



            Example: The Im-Pesaran-Shim test.




          There are many versions of these tests, for example, for making such inferences about multiple time series (e.g., a time series for each country, city, etc.). Which tests are appropriate for you depends on your study design (e.g., single or multiple time series, balanced measurements, missing observations, etc.).



          I think a wise way to approach making use of such tests is to combine inference from an appropriate version of a test for stationarity, coupled with an appropriate version of a test for unit root:




          • Reject test for stationarity, and fail to reject test for unit root: conclude data are stationary.


          • Fail to reject test for stationarity, and reject test for unit root: conclude data have unit root (are non-stationary).


          • Fail to reject test for stationarity, and fail to reject test for unit root: conclude data are under-powered to make any inference one way or the other.


          • Reject test for stationarity, and reject test for unit root: think hard about your data! For example with tests like those I mentioned above it may be the case that some time series have unit root, and some time series are stationary.



          For a single time series, the augmented Dickey-Fuller test for stationarity, and the complementary Kwiatkowski–Phillips–Schmidt–Shin test for unit root may be appropriate tools, and these tests are commonly implemented in statistical software (e.g., R, Stata, etc.).



          Further, explicit modeling of autoregressive process will allow you to make inferences not only about stationarity vs. unit root, but the middle-ground of autoregressive (memoried) processes, which will be stationary in the long run (possibly very long run), but still present difficulties for valid inference over shorter time scales. For example, using OLS regression or some other estimator, one could model the first difference of a time series ($Delta y_{t} = y_{t} - y_{t-1}$) using a single lag:



          $$Delta y_{t} = alpha + rho y_{t-1} + zeta_{t}$$



          A value of $rho$ which is undifferentiable from $0$ (equivalence tests can help with this inference) is evidence of stationarity (trend stationarity if $alpha ne 0$) over the time scale of your data. A value of $|rho| approx 1$ is evidence of unit root. A value of $0 < rho < 1$ is an autoregressive process which is likely to demand an appropriate time series model (see de Boef and Keele) the closer $rho$ is to 1 (i.e. the longer its memory gets). (There can even be $|rho|>1$, implying a runaway process... in my discipline, I don't see these much.) Of course, models with more lags are also possible.



          References



          De Boef, S. (2001). Modeling equilibrium relationships: Error correction models with strongly autoregressive data. Political Analysis, 9(1):78–94.



          De Boef, S. and Keele, L. (2008). Taking time seriously. American Journal of Political Science, 52(1):184–200.



          Dickey, D. A. and Fuller, W. A. (1979). Distribution of the estimators for autore gressive time series with a unit root. Journal of the American Statistical Association, 74(366):427–431.



          Hadri, K. (2000). Testing for stationarity in heterogeneous panel data. The Econometrics Journal, 3(2):148–161.



          Im, K. S., Pesaran, M. H., and Shin, Y. (2003). Testing for unit roots in heterogeneous panels. Journal of Econometrics, 115:53–74.



          Kwiatkowski, D., Phillips, P. C., Schmidt, P., and Shin, Y. (1992). Testing the null hypothesis of stationarity against the alternative of a unit root: how sure are we that economic time series have a unit root? Journal of Econometrics, 54(1-3):159–178.






          share|cite|improve this answer































            1














            You can plot the time-series and inspect it visually. A stationary time-series will have constant variance and have an expected value of zero. plot(Value~Time)
            You could look at the ACF and PACF plots. They should die down if the series is stationary.
            There is also the Augmented Dickey-Fuller Test for being stationary. In R use ?adf.test for more information.






            share|cite|improve this answer





















            • "A stationary time-series will have constant variance and have an expected value of zero." This is true, but it may take a long time—possibly a much longer time than the period studied—for long-memoried processes which to prove stationary in statistical inference.
              – Alexis
              1 hour ago











            Your Answer





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            2 Answers
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            active

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            2 Answers
            2






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes









            2














            There are many such tests.



            Generally, they fall into two categories:





            1. Tests for unit root (I.e. tests of stationarity)



              $H_{0}:$ Time series is/are stationary.



              $H_{A}:$ Time series has/have unit root (i.e. is non-stationary).



              Example: The Hadri-Lagrange multiplier test.




            2. Tests for stationarity (I.e. tests of unit root)



              $H_{0}:$ Time series has/have unit root.



              $H_{A}:$ Time series is/are stationary.



              Example: The Im-Pesaran-Shim test.




            There are many versions of these tests, for example, for making such inferences about multiple time series (e.g., a time series for each country, city, etc.). Which tests are appropriate for you depends on your study design (e.g., single or multiple time series, balanced measurements, missing observations, etc.).



            I think a wise way to approach making use of such tests is to combine inference from an appropriate version of a test for stationarity, coupled with an appropriate version of a test for unit root:




            • Reject test for stationarity, and fail to reject test for unit root: conclude data are stationary.


            • Fail to reject test for stationarity, and reject test for unit root: conclude data have unit root (are non-stationary).


            • Fail to reject test for stationarity, and fail to reject test for unit root: conclude data are under-powered to make any inference one way or the other.


            • Reject test for stationarity, and reject test for unit root: think hard about your data! For example with tests like those I mentioned above it may be the case that some time series have unit root, and some time series are stationary.



            For a single time series, the augmented Dickey-Fuller test for stationarity, and the complementary Kwiatkowski–Phillips–Schmidt–Shin test for unit root may be appropriate tools, and these tests are commonly implemented in statistical software (e.g., R, Stata, etc.).



            Further, explicit modeling of autoregressive process will allow you to make inferences not only about stationarity vs. unit root, but the middle-ground of autoregressive (memoried) processes, which will be stationary in the long run (possibly very long run), but still present difficulties for valid inference over shorter time scales. For example, using OLS regression or some other estimator, one could model the first difference of a time series ($Delta y_{t} = y_{t} - y_{t-1}$) using a single lag:



            $$Delta y_{t} = alpha + rho y_{t-1} + zeta_{t}$$



            A value of $rho$ which is undifferentiable from $0$ (equivalence tests can help with this inference) is evidence of stationarity (trend stationarity if $alpha ne 0$) over the time scale of your data. A value of $|rho| approx 1$ is evidence of unit root. A value of $0 < rho < 1$ is an autoregressive process which is likely to demand an appropriate time series model (see de Boef and Keele) the closer $rho$ is to 1 (i.e. the longer its memory gets). (There can even be $|rho|>1$, implying a runaway process... in my discipline, I don't see these much.) Of course, models with more lags are also possible.



            References



            De Boef, S. (2001). Modeling equilibrium relationships: Error correction models with strongly autoregressive data. Political Analysis, 9(1):78–94.



            De Boef, S. and Keele, L. (2008). Taking time seriously. American Journal of Political Science, 52(1):184–200.



            Dickey, D. A. and Fuller, W. A. (1979). Distribution of the estimators for autore gressive time series with a unit root. Journal of the American Statistical Association, 74(366):427–431.



            Hadri, K. (2000). Testing for stationarity in heterogeneous panel data. The Econometrics Journal, 3(2):148–161.



            Im, K. S., Pesaran, M. H., and Shin, Y. (2003). Testing for unit roots in heterogeneous panels. Journal of Econometrics, 115:53–74.



            Kwiatkowski, D., Phillips, P. C., Schmidt, P., and Shin, Y. (1992). Testing the null hypothesis of stationarity against the alternative of a unit root: how sure are we that economic time series have a unit root? Journal of Econometrics, 54(1-3):159–178.






            share|cite|improve this answer




























              2














              There are many such tests.



              Generally, they fall into two categories:





              1. Tests for unit root (I.e. tests of stationarity)



                $H_{0}:$ Time series is/are stationary.



                $H_{A}:$ Time series has/have unit root (i.e. is non-stationary).



                Example: The Hadri-Lagrange multiplier test.




              2. Tests for stationarity (I.e. tests of unit root)



                $H_{0}:$ Time series has/have unit root.



                $H_{A}:$ Time series is/are stationary.



                Example: The Im-Pesaran-Shim test.




              There are many versions of these tests, for example, for making such inferences about multiple time series (e.g., a time series for each country, city, etc.). Which tests are appropriate for you depends on your study design (e.g., single or multiple time series, balanced measurements, missing observations, etc.).



              I think a wise way to approach making use of such tests is to combine inference from an appropriate version of a test for stationarity, coupled with an appropriate version of a test for unit root:




              • Reject test for stationarity, and fail to reject test for unit root: conclude data are stationary.


              • Fail to reject test for stationarity, and reject test for unit root: conclude data have unit root (are non-stationary).


              • Fail to reject test for stationarity, and fail to reject test for unit root: conclude data are under-powered to make any inference one way or the other.


              • Reject test for stationarity, and reject test for unit root: think hard about your data! For example with tests like those I mentioned above it may be the case that some time series have unit root, and some time series are stationary.



              For a single time series, the augmented Dickey-Fuller test for stationarity, and the complementary Kwiatkowski–Phillips–Schmidt–Shin test for unit root may be appropriate tools, and these tests are commonly implemented in statistical software (e.g., R, Stata, etc.).



              Further, explicit modeling of autoregressive process will allow you to make inferences not only about stationarity vs. unit root, but the middle-ground of autoregressive (memoried) processes, which will be stationary in the long run (possibly very long run), but still present difficulties for valid inference over shorter time scales. For example, using OLS regression or some other estimator, one could model the first difference of a time series ($Delta y_{t} = y_{t} - y_{t-1}$) using a single lag:



              $$Delta y_{t} = alpha + rho y_{t-1} + zeta_{t}$$



              A value of $rho$ which is undifferentiable from $0$ (equivalence tests can help with this inference) is evidence of stationarity (trend stationarity if $alpha ne 0$) over the time scale of your data. A value of $|rho| approx 1$ is evidence of unit root. A value of $0 < rho < 1$ is an autoregressive process which is likely to demand an appropriate time series model (see de Boef and Keele) the closer $rho$ is to 1 (i.e. the longer its memory gets). (There can even be $|rho|>1$, implying a runaway process... in my discipline, I don't see these much.) Of course, models with more lags are also possible.



              References



              De Boef, S. (2001). Modeling equilibrium relationships: Error correction models with strongly autoregressive data. Political Analysis, 9(1):78–94.



              De Boef, S. and Keele, L. (2008). Taking time seriously. American Journal of Political Science, 52(1):184–200.



              Dickey, D. A. and Fuller, W. A. (1979). Distribution of the estimators for autore gressive time series with a unit root. Journal of the American Statistical Association, 74(366):427–431.



              Hadri, K. (2000). Testing for stationarity in heterogeneous panel data. The Econometrics Journal, 3(2):148–161.



              Im, K. S., Pesaran, M. H., and Shin, Y. (2003). Testing for unit roots in heterogeneous panels. Journal of Econometrics, 115:53–74.



              Kwiatkowski, D., Phillips, P. C., Schmidt, P., and Shin, Y. (1992). Testing the null hypothesis of stationarity against the alternative of a unit root: how sure are we that economic time series have a unit root? Journal of Econometrics, 54(1-3):159–178.






              share|cite|improve this answer


























                2












                2








                2






                There are many such tests.



                Generally, they fall into two categories:





                1. Tests for unit root (I.e. tests of stationarity)



                  $H_{0}:$ Time series is/are stationary.



                  $H_{A}:$ Time series has/have unit root (i.e. is non-stationary).



                  Example: The Hadri-Lagrange multiplier test.




                2. Tests for stationarity (I.e. tests of unit root)



                  $H_{0}:$ Time series has/have unit root.



                  $H_{A}:$ Time series is/are stationary.



                  Example: The Im-Pesaran-Shim test.




                There are many versions of these tests, for example, for making such inferences about multiple time series (e.g., a time series for each country, city, etc.). Which tests are appropriate for you depends on your study design (e.g., single or multiple time series, balanced measurements, missing observations, etc.).



                I think a wise way to approach making use of such tests is to combine inference from an appropriate version of a test for stationarity, coupled with an appropriate version of a test for unit root:




                • Reject test for stationarity, and fail to reject test for unit root: conclude data are stationary.


                • Fail to reject test for stationarity, and reject test for unit root: conclude data have unit root (are non-stationary).


                • Fail to reject test for stationarity, and fail to reject test for unit root: conclude data are under-powered to make any inference one way or the other.


                • Reject test for stationarity, and reject test for unit root: think hard about your data! For example with tests like those I mentioned above it may be the case that some time series have unit root, and some time series are stationary.



                For a single time series, the augmented Dickey-Fuller test for stationarity, and the complementary Kwiatkowski–Phillips–Schmidt–Shin test for unit root may be appropriate tools, and these tests are commonly implemented in statistical software (e.g., R, Stata, etc.).



                Further, explicit modeling of autoregressive process will allow you to make inferences not only about stationarity vs. unit root, but the middle-ground of autoregressive (memoried) processes, which will be stationary in the long run (possibly very long run), but still present difficulties for valid inference over shorter time scales. For example, using OLS regression or some other estimator, one could model the first difference of a time series ($Delta y_{t} = y_{t} - y_{t-1}$) using a single lag:



                $$Delta y_{t} = alpha + rho y_{t-1} + zeta_{t}$$



                A value of $rho$ which is undifferentiable from $0$ (equivalence tests can help with this inference) is evidence of stationarity (trend stationarity if $alpha ne 0$) over the time scale of your data. A value of $|rho| approx 1$ is evidence of unit root. A value of $0 < rho < 1$ is an autoregressive process which is likely to demand an appropriate time series model (see de Boef and Keele) the closer $rho$ is to 1 (i.e. the longer its memory gets). (There can even be $|rho|>1$, implying a runaway process... in my discipline, I don't see these much.) Of course, models with more lags are also possible.



                References



                De Boef, S. (2001). Modeling equilibrium relationships: Error correction models with strongly autoregressive data. Political Analysis, 9(1):78–94.



                De Boef, S. and Keele, L. (2008). Taking time seriously. American Journal of Political Science, 52(1):184–200.



                Dickey, D. A. and Fuller, W. A. (1979). Distribution of the estimators for autore gressive time series with a unit root. Journal of the American Statistical Association, 74(366):427–431.



                Hadri, K. (2000). Testing for stationarity in heterogeneous panel data. The Econometrics Journal, 3(2):148–161.



                Im, K. S., Pesaran, M. H., and Shin, Y. (2003). Testing for unit roots in heterogeneous panels. Journal of Econometrics, 115:53–74.



                Kwiatkowski, D., Phillips, P. C., Schmidt, P., and Shin, Y. (1992). Testing the null hypothesis of stationarity against the alternative of a unit root: how sure are we that economic time series have a unit root? Journal of Econometrics, 54(1-3):159–178.






                share|cite|improve this answer














                There are many such tests.



                Generally, they fall into two categories:





                1. Tests for unit root (I.e. tests of stationarity)



                  $H_{0}:$ Time series is/are stationary.



                  $H_{A}:$ Time series has/have unit root (i.e. is non-stationary).



                  Example: The Hadri-Lagrange multiplier test.




                2. Tests for stationarity (I.e. tests of unit root)



                  $H_{0}:$ Time series has/have unit root.



                  $H_{A}:$ Time series is/are stationary.



                  Example: The Im-Pesaran-Shim test.




                There are many versions of these tests, for example, for making such inferences about multiple time series (e.g., a time series for each country, city, etc.). Which tests are appropriate for you depends on your study design (e.g., single or multiple time series, balanced measurements, missing observations, etc.).



                I think a wise way to approach making use of such tests is to combine inference from an appropriate version of a test for stationarity, coupled with an appropriate version of a test for unit root:




                • Reject test for stationarity, and fail to reject test for unit root: conclude data are stationary.


                • Fail to reject test for stationarity, and reject test for unit root: conclude data have unit root (are non-stationary).


                • Fail to reject test for stationarity, and fail to reject test for unit root: conclude data are under-powered to make any inference one way or the other.


                • Reject test for stationarity, and reject test for unit root: think hard about your data! For example with tests like those I mentioned above it may be the case that some time series have unit root, and some time series are stationary.



                For a single time series, the augmented Dickey-Fuller test for stationarity, and the complementary Kwiatkowski–Phillips–Schmidt–Shin test for unit root may be appropriate tools, and these tests are commonly implemented in statistical software (e.g., R, Stata, etc.).



                Further, explicit modeling of autoregressive process will allow you to make inferences not only about stationarity vs. unit root, but the middle-ground of autoregressive (memoried) processes, which will be stationary in the long run (possibly very long run), but still present difficulties for valid inference over shorter time scales. For example, using OLS regression or some other estimator, one could model the first difference of a time series ($Delta y_{t} = y_{t} - y_{t-1}$) using a single lag:



                $$Delta y_{t} = alpha + rho y_{t-1} + zeta_{t}$$



                A value of $rho$ which is undifferentiable from $0$ (equivalence tests can help with this inference) is evidence of stationarity (trend stationarity if $alpha ne 0$) over the time scale of your data. A value of $|rho| approx 1$ is evidence of unit root. A value of $0 < rho < 1$ is an autoregressive process which is likely to demand an appropriate time series model (see de Boef and Keele) the closer $rho$ is to 1 (i.e. the longer its memory gets). (There can even be $|rho|>1$, implying a runaway process... in my discipline, I don't see these much.) Of course, models with more lags are also possible.



                References



                De Boef, S. (2001). Modeling equilibrium relationships: Error correction models with strongly autoregressive data. Political Analysis, 9(1):78–94.



                De Boef, S. and Keele, L. (2008). Taking time seriously. American Journal of Political Science, 52(1):184–200.



                Dickey, D. A. and Fuller, W. A. (1979). Distribution of the estimators for autore gressive time series with a unit root. Journal of the American Statistical Association, 74(366):427–431.



                Hadri, K. (2000). Testing for stationarity in heterogeneous panel data. The Econometrics Journal, 3(2):148–161.



                Im, K. S., Pesaran, M. H., and Shin, Y. (2003). Testing for unit roots in heterogeneous panels. Journal of Econometrics, 115:53–74.



                Kwiatkowski, D., Phillips, P. C., Schmidt, P., and Shin, Y. (1992). Testing the null hypothesis of stationarity against the alternative of a unit root: how sure are we that economic time series have a unit root? Journal of Econometrics, 54(1-3):159–178.







                share|cite|improve this answer














                share|cite|improve this answer



                share|cite|improve this answer








                edited 38 mins ago

























                answered 1 hour ago









                Alexis

                15.7k34595




                15.7k34595

























                    1














                    You can plot the time-series and inspect it visually. A stationary time-series will have constant variance and have an expected value of zero. plot(Value~Time)
                    You could look at the ACF and PACF plots. They should die down if the series is stationary.
                    There is also the Augmented Dickey-Fuller Test for being stationary. In R use ?adf.test for more information.






                    share|cite|improve this answer





















                    • "A stationary time-series will have constant variance and have an expected value of zero." This is true, but it may take a long time—possibly a much longer time than the period studied—for long-memoried processes which to prove stationary in statistical inference.
                      – Alexis
                      1 hour ago
















                    1














                    You can plot the time-series and inspect it visually. A stationary time-series will have constant variance and have an expected value of zero. plot(Value~Time)
                    You could look at the ACF and PACF plots. They should die down if the series is stationary.
                    There is also the Augmented Dickey-Fuller Test for being stationary. In R use ?adf.test for more information.






                    share|cite|improve this answer





















                    • "A stationary time-series will have constant variance and have an expected value of zero." This is true, but it may take a long time—possibly a much longer time than the period studied—for long-memoried processes which to prove stationary in statistical inference.
                      – Alexis
                      1 hour ago














                    1












                    1








                    1






                    You can plot the time-series and inspect it visually. A stationary time-series will have constant variance and have an expected value of zero. plot(Value~Time)
                    You could look at the ACF and PACF plots. They should die down if the series is stationary.
                    There is also the Augmented Dickey-Fuller Test for being stationary. In R use ?adf.test for more information.






                    share|cite|improve this answer












                    You can plot the time-series and inspect it visually. A stationary time-series will have constant variance and have an expected value of zero. plot(Value~Time)
                    You could look at the ACF and PACF plots. They should die down if the series is stationary.
                    There is also the Augmented Dickey-Fuller Test for being stationary. In R use ?adf.test for more information.







                    share|cite|improve this answer












                    share|cite|improve this answer



                    share|cite|improve this answer










                    answered 1 hour ago









                    Greg F.

                    263




                    263












                    • "A stationary time-series will have constant variance and have an expected value of zero." This is true, but it may take a long time—possibly a much longer time than the period studied—for long-memoried processes which to prove stationary in statistical inference.
                      – Alexis
                      1 hour ago


















                    • "A stationary time-series will have constant variance and have an expected value of zero." This is true, but it may take a long time—possibly a much longer time than the period studied—for long-memoried processes which to prove stationary in statistical inference.
                      – Alexis
                      1 hour ago
















                    "A stationary time-series will have constant variance and have an expected value of zero." This is true, but it may take a long time—possibly a much longer time than the period studied—for long-memoried processes which to prove stationary in statistical inference.
                    – Alexis
                    1 hour ago




                    "A stationary time-series will have constant variance and have an expected value of zero." This is true, but it may take a long time—possibly a much longer time than the period studied—for long-memoried processes which to prove stationary in statistical inference.
                    – Alexis
                    1 hour ago


















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