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adftest

Augmented Dickey-Fuller test

Description

example

h = adftest(y) returns the rejection decision h from conducting an augmented Dickey-Fuller test for a unit root in a univariate time series y.

example

[h,pValue,stat,cValue] = adftest(y) also returns the p-value pValue, test statistic stat, and critical value cValue of the test.

example

StatTbl = adftest(Tbl) returns the table StatTbl containing variables for the test results, statistics, and settings from conducting an augmented Dickey-Fuller test for a unit root in the last variable of the input table or timetable Tbl. To select a different variable in Tbl to test, use the DataVariable name-value argument.

example

[___] = adftest(___,Name=Value) uses additional options specified by one or more name-value arguments, using any input-argument combination in the previous syntaxes. adftest returns the output-argument combination for the corresponding input arguments.

Some options control the number of tests to conduct. The following conditions apply when adftest conducts multiple tests:

  • adftest treats each test as separate from all other tests.

  • If you specify y, all outputs are vectors.

  • If you specify Tbl, each row of StatTbl contains the results of the corresponding test.

For example, adftest(Tbl,DataVariable="GDP",Alpha=0.025,Lags=[0 1]) conducts two tests, at a level of significance of 0.025, for the presence of a unit root in the variable GDP of the table Tbl. The first test includes 0 lagged difference terms in the AR model, and the second test includes 1 lagged difference term in the AR model.

example

[___,reg] = adftest(___) additionally returns a structure of regression statistics for the hypothesis test reg.

Examples

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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, 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
DataTable.LogGDP = log(DataTable.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(DataTable,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 t-statistics and corresponding 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.

Input Arguments

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Univariate time series data, specified as a numeric vector. Each element of y represents an observation.

Data Types: double

Time series data, specified as a table or timetable. Each row of Tbl is an observation.

Specify a single series (variable) to test by using the DataVariable argument. The selected variable must be numeric.

Note

adftest removes missing observations, represented by NaN values, from the input series.

Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Before R2021a, use commas to separate each name and value, and enclose Name in quotes.

Example: adftest(Tbl,DataVariable="GDP",Alpha=0.025,Lags=[0 1]) conducts two separate tests at a level of significance of 0.025 on the variable GGDP of the input table Tbl. The first test includes 0 lagged difference terms in the AR model, and the second test includes 1 lagged difference term in the AR model.

Number p of lagged difference terms to include in the AR model, specified as a nonnegative integer or vector of nonnegative integers.

adftest conducts a separate test for each element in Lags.

Example: Lags=[0 1] includes no lags in the AR model for the first test, and then includes Δyt – 1 in the AR model for the second test.

Data Types: double

Model variant, specified as a model variant name, or a string vector or cell vector of model names. This table contains the supported model variant names.

Model Variant NameDescription
"AR"

Autoregressive model variant, which specifies a test of the null model

yt=yt1+β1Δyt1+β2Δyt2++βpΔytp+εt

against the alternative model

yt=ϕyt1+β1Δyt1+β2Δyt2++βpΔytp+εt,

with AR(1) coefficient ϕ<1.

"ARD"

Autoregressive model with drift variant, which specifies a test of the null model

yt=yt1+β1Δyt1+β2Δyt2++βpΔytp+εt

against the alternative model

yt=c+ϕyt1+β1Δyt1+β2Δyt2++βpΔytp+εt,

with drift coefficient c and AR(1) coefficient ϕ<1.

"TS"

Trend-stationary model variant, which specifies a test of the null model

yt=c+yt1+β1Δyt1+β2Δyt2++βpΔytp+εt

against the alternative model

yt=c+δt+ϕyt1+β1Δyt1+β2Δyt2++βpΔytp+εt,

with drift coefficient c, deterministic trend coefficient δ, and AR(1) coefficient ϕ<1.

adftest conducts a separate test for each model variant name in Model.

Example: Model=["AR" "ARD"] uses the stationary AR model as the alternative hypothesis for the first test, and then uses the stationary AR model with drift as the alternative hypothesis for the second test.

Data Types: char | cell | string

Test statistic, specified as a test name, or a string vector or cell vector of test names. This table contains the supported test names.

Test NameDescription
"t1"

Standard t statistic

t1=ϕ^1SE(ϕ^),

computed using the OLS estimate of the AR(1) coefficient ϕ^ and its standard error SE(ϕ^), in the alternative model.

The test assesses the significance of the restriction, ϕ1=0.

"t2"

Lag-adjusted, unstudentized t statistic

t2=T(ϕ^1)1β^1β^p,

computed using the OLS estimates of the AR(1) coefficient and stationary coefficients in the alternative model. T is the effective sample size, adjusted for lags and missing values.

The test assesses the significance of the restriction, ϕ1=0.

"F"

F statistic for assessing the significance of a joint restriction on the alternative model.

  • For model variant (Model argument) "ARD", the restrictions areϕ1=0 and c = 0.

  • For model variant "TS", the restrictions areϕ1=0 and δ = 0.

An F statistic is invalid for model variant "AR".

adftest conducts a separate test for each test name in Test.

Example: Test="F" computes the F test statistic for all tests.

Data Types: char | cell | string

Nominal significance level for the hypothesis test, specified as a numeric scalar between 0.001 and 0.999 or a numeric vector of such values.

adftest conducts a separate test for each value in Alpha.

Example: Alpha=[0.01 0.05] uses a level of significance of 0.01 for the first test, and then uses a level of significance of 0.05 for the second test.

Data Types: double

Variable in Tbl to test, specified as a string scalar or character vector containing a variable name in Tbl.Properties.VariableNames, or an integer or logical vector representing the index of a name. The selected variable must be numeric.

Example: DataVariable="GDP"

Example: DataVariable=[false true false false] or DataVariable=2 tests the second table variable.

Data Types: double | logical | char | string

Note

  • When adftest conducts multiple tests, the function applies all single settings (scalars or character vectors) to each test.

  • All vector-valued specifications that control the number of tests must have equal length.

  • If you specify the vector y and any value is a row vector, all outputs are row vectors.

  • A lagged and differenced time series has a reduced sample size. Absent presample values, if the test series yt is defined for t = 1,…,T, the lagged series yt– k is defined for t = k+1,…,T. The first difference applied to the lagged series yt– k further reduces the time base to k+2,…,T. With p lagged differences, the common time base is p+2,…,T and the effective sample size is T–(p+1).

Output Arguments

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Test rejection decisions, returned as a logical scalar or vector with length equal to the number of tests. adftest returns h when you supply the input y.

  • Values of 1 indicate rejection of the unit-root null model in favor of the alternative model.

  • Values of 0 indicate failure to reject the unit-root null model.

Test statistic p-values, returned as a numeric scalar or vector with length equal to the number of tests. adftest returns pValue when you supply the input y.

  • The p-value of test statistic (Test) "t1" or "t2" is a left-tail probability.

  • The p-value of test statistic "F" is a right-tail probability.

When test statistics are outside tabulated critical values, adftest returns maximum (0.999) or minimum (0.001) p-values.

Test statistics, returned as a numeric scalar or vector with length equal to the number of tests. adftest returns stat when you supply the input y.

adftest computes test statistics using ordinary least squares (OLS) estimates of the coefficients in the alternative model.

Critical values, returned as a numeric scalar or vector with length equal to the number of tests. adftest returns cValue when you supply the input y.

  • The critical value of test statistic (Test) "t1" or "t2" is for a left-tail probability.

  • The critical value of test statistic "F" is for a right-tail probability.

Test summary, returned as a table with variables for the outputs h, pValue, stat, and cValue, and with a row for each test. adftest returns StatTbl when you supply the input Tbl.

StatTbl contains variables for the test settings specified by Lags, Alpha, Model, and Test.

Regression statistics from the OLS estimation of coefficients in the alternative model, returned as a structure array with number of records equal to the number of tests.

Each element of reg has the fields in this table. You can access a field using dot notation, for example, reg(1).coeff contains the coefficient estimates of the first test.

FieldDescription
numLength of input series with NaNs removed
sizeEffective sample size T, num adjusted for lags
namesRegression coefficient names
coeffEstimated coefficient values
seEstimated coefficient standard errors
CovEstimated coefficient covariance matrix
tStatst statistics of coefficients and p-values
FStatF statistic and p-value
yMuMean of the lag-adjusted input series
ySigmaStandard deviation of the lag-adjusted input series
yHatFitted values of the lag-adjusted input series
resRegression residuals
DWStatDurbin-Watson statistic
SSRRegression sum of squares
SSEError sum of squares
SSTTotal sum of squares
MSEMean square error
RMSEStandard error of the regression
RSqR2 statistic
aRSqAdjusted R2 statistic
LLLoglikelihood of data under Gaussian innovations
AICAkaike information criterion
BICBayesian (Schwarz) information criterion
HQCHannan-Quinn information criterion

More About

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Augmented Dickey-Fuller Test for Unit Root

The augmented Dickey-Fuller test for a unit root assesses the null hypothesis of a unit root in the time series yt, where

yt=c+δt+ϕyt1+β1Δyt1++βpΔytp+εt,

and

  • Δ is the differencing operator such that Δyt=ytyt1.

  • p is the number of lagged difference terms (see Lags).

  • c is the drift coefficient (see Model).

  • δ is the deterministic trend coefficient (see Model).

  • εt is a mean zero innovation process.

The null hypothesis of a unit root is

H0:ϕ=1.

Under the alternative hypothesis, ϕ<1.

Variants of the model allow for different growth characteristics (see Model). The model with δ = 0 has no trend component, and the model with c = 0 and δ = 0 has no drift or trend.

A test that fails to reject the null hypothesis, fails to reject the possibility of a unit root.

Tips

  • To draw valid inferences from the test, determine a suitable value for Lags.

    One method is to begin with a maximum lag, such as the one recommended in [7], and then test down by assessing the significance of β^p, the coefficient of the largest lagged change in yt. The usual t statistic is appropriate, as returned in the reg output structure.

    Another method is to combine a measure of fit, such as the SSR, with information criteria, such as AIC, BIC, and HQC. These statistics are also returned in the reg output structure. For more details, see [6].

  • With a specific testing strategy in mind, determine the value of Model by the growth characteristics of yt. If you include too many regressors (see Lags), the test loses power; if you include too few regressors, the test is biased towards favoring the null model [4]. In general, if a series grows, the "TS" model (see Model) provides a reasonable trend-stationary alternative to a unit-root process with drift. If a series is does not grow, the "AR" and "ARD" models provide reasonable stationary alternatives to a unit-root process without drift. The "ARD" alternative model has a mean of c/(1 – a); the "AR" alternative model has mean 0.

Algorithms

Dickey-Fuller statistics follow nonstandard distributions under the null hypothesis (even asymptotically). adftest uses tabulated critical values, generated by Monte Carlo simulations, for a range of sample sizes and significance levels of the null model with Gaussian innovations and five million replications per sample size. adftest interpolates critical values cValue and p-values pValue from the tables. Tables for tests of Test types "t1" and "t2" are identical to those for pptest. For small samples, tabulated values are valid only for Gaussian innovations. For large samples, values are also valid for non-Gaussian innovations.

References

[1] Davidson, R., and J. G. MacKinnon. Econometric Theory and Methods. Oxford, UK: Oxford University Press, 2004.

[2] Dickey, D. A., and W. A. Fuller. "Distribution of the Estimators for Autoregressive Time Series with a Unit Root." Journal of the American Statistical Association. Vol. 74, 1979, pp. 427–431.

[3] Dickey, D. A., and W. A. Fuller. "Likelihood Ratio Statistics for Autoregressive Time Series with a Unit Root." Econometrica. Vol. 49, 1981, pp. 1057–1072.

[4] Elder, J., and P. E. Kennedy. "Testing for Unit Roots: What Should Students Be Taught?" Journal of Economic Education. Vol. 32, 2001, pp. 137–146.

[5] Hamilton, James D. Time Series Analysis. Princeton, NJ: Princeton University Press, 1994.

[6] Ng, S., and P. Perron. "Unit Root Tests in ARMA Models with Data-Dependent Methods for the Selection of the Truncation Lag." Journal of the American Statistical Association. Vol. 90, 1995, pp. 268–281.

[7] Schwert, W. "Tests for Unit Roots: A Monte Carlo Investigation." Journal of Business and Economic Statistics. Vol. 7, 1989, pp. 147–159.

Version History

Introduced in R2009b