adftest

Test a time series for a unit root using the default autoregressive model without augmented difference terms. Input the time series data as a numeric vector.

Load the Canadian inflation rate data and extract the CPI-based inflation rate INF_C.

load Data_Canada
y = DataTable.INF_C;

Test the time series for a unit root.

h = adftest(y)

h = logical
   0

The result h = 0 indicates that this test fails to reject the null hypothesis of a unit root against the autoregressive alternative.

Load Canadian inflation rate data and extract the CPI-based inflation rate INF_C.

load Data_Canada
y = DataTable.INF_C;

Test the time series for a unit root. Return the test decision, $\mathit{p}$-value, test statistic, and critical value.

[h,pValue,stat,cValue] = adftest(y)

h = logical
   0

pValue = 
0.3255

stat = 
-0.8769

cValue = 
-1.9476

Test a time series, which is one variable in a table, for a unit root using the default autoregressive model without augmented difference terms.

Load Canadian inflation rate data, which contains yearly measurements on five time series variables in the table DataTable.

load Data_Canada

Test the long-term bond rate series INT_L, the last variable in the table, for a unit root.

StatTbl = adftest(DataTable)

StatTbl=1×8 table
                h      pValue     stat      cValue     Lags    Alpha    Model      Test 
              _____    ______    _______    _______    ____    _____    ______    ______

    Test 1    false    0.7358    0.24601    -1.9476     0      0.05     {'AR'}    {'T1'}

adftest returns test results and settings in the table StatTbl, where variables correspond to test results (h, pValue, stat, and cValue) and settings (Lags, Alpha, Model, and Test), and rows correspond to individual tests (in this case, adftest conducts one test).

By default, adftest tests the last variable in the table. To select a variable from an input table to test, set the DataVariable option.

Test a time series for a unit root against a trend-stationary alternative augmented with lagged difference terms.

Load a GDP data set. Compute the log of the series.

load Data_GDP;
Y = log(Data);

Test for a unit root against a trend-stationary alternative, augmenting the model with 0, 1, and 2 lagged difference terms.

h = adftest(Y,Model="TS",Lags=0:2)

h = 1x3 logical array

   0   0   0

adftest treats the three lag choices as three separate tests, and returns a vector with rejection decisions for each test. The values h = 0 indicate that all three tests fail to reject the null hypothesis of a unit root against the trend-stationary alternative.

Test a time series for a unit root against trend-stationary alternatives augmented with different numbers of lagged difference terms. Look at the regression statistics corresponding to each of the alternative models to choose how many lagged difference terms to include in the augmented model.

Load a US macroeconomic data set Data_USEconModel.mat. Compute the log of the GDP and include the result as a new variable called LogGDP in the data set.

load Data_USEconModel
DataTimeTable.LogGDP = log(DataTimeTable.GDP);

Test for a unit root in the logged GDP series using three different choices for the number of lagged difference terms. Return the regression statistics for each alternative model.

[StatTbl,reg] = adftest(DataTimeTable,DataVariable="LogGDP",Model="TS",Lags=0:2);
StatTbl

StatTbl=3×8 table
                h      pValue        stat       cValue     Lags    Alpha    Model      Test 
              _____    _______    __________    _______    ____    _____    ______    ______

    Test 1    false      0.999        1.0247    -3.4302     0      0.05     {'TS'}    {'T1'}
    Test 2    false    0.99565    -0.0020747    -3.4303     1      0.05     {'TS'}    {'T1'}
    Test 3    false    0.99214      -0.21274    -3.4304     2      0.05     {'TS'}    {'T1'}

adftest treats each of the three lag choices as separate tests, and returns results and settings for each test along the rows of the table StatTbl. reg is a 3-by-1 structure array containing regression statistics corresponding to each of the three alternative models.

Display the names of the coefficients, their $\mathit{t}$-statistics and corresponding $\mathit{p}$-values, and the BIC resulting from the regression of the three alternative models.

model1 = array2table([reg(1).tStats.t reg(1).tStats.pVal], ...
    RowNames=reg(1).names,VariableNames=["tStat" "pValue"])

model1=3×2 table
          tStat        pValue   
         ________    ___________

    c    -0.43299         0.6654
    d     -1.2195        0.22383
    a      167.66    8.5908e-255

model2 = array2table([reg(2).tStats.t reg(2).tStats.pVal], ...
    RowNames=reg(2).names,VariableNames=["tStat" "pValue"])

model2=4×2 table
           tStat        pValue   
          ________    ___________

    c      0.35537        0.72262
    d     -0.13077        0.89607
    a       185.44    1.0349e-263
    b1      8.1646     1.7553e-14

model3 = array2table([reg(3).tStats.t reg(3).tStats.pVal], ...
    RowNames=reg(3).names,VariableNames=["tStat" "pValue"])

model3=5×2 table
           tStat        pValue   
          ________    ___________

    c      0.52121         0.6027
    d     0.089107        0.92907
    a       184.56    1.7276e-261
    b1      6.4983     4.6217e-10
    b2      1.8871       0.060353

The first model has no added difference terms, the second model has one difference term (b1), and the third model has two difference terms (b1 and b2). These results indicate that the coefficient of the first difference term is significantly different from zero in both the second and third models, but the coefficient of the second term in the third model is not at a 0.05 significance level. This result suggests augmenting the model with one lagged difference term is adequate.

Compare the BIC for each of the three alternatives.

reg.BIC

ans = 
-1.5114e+03

ans = 
-1.5589e+03

ans = 
-1.5496e+03

Of the three alternative models, the model augmented with one lagged difference term is the best because it yields the lowest BIC.