adftest

Augmented Dickey-Fuller test

Syntax

h = adftest(y)

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

StatTbl = adftest(Tbl)

[___] = adftest(___,Name=Value)

[___,reg] = adftest(___)

Description

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

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

StatTbl = adftest(Tbl) returns a table 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. To select a different variable to test, use the DataVariable name-value argument.

[___] = adftest(___,Name=Value) specifies options using one or more name-value arguments in addition to any of the input argument combinations in 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.

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

Examples

Conduct Dickey-Fuller Test Without Augmentation on Vector of Data

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.

Return Test p-Value and Decision Statistics

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

Conduct Dickey-Fuller Test Without Augmentation on Table Variable

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.

Conduct Augmented Dickey-Fuller Test Against Trend-Stationary Alternative

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 = 1×3 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.

Choose Number of Lags for Test by Inspecting OLS Statistics

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.

Input Arguments

y — Univariate time series data

numeric vector

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

Data Types: double

Tbl — Time series data

table | timetable

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.

Lags — Number of lagged difference terms

0 (default) | nonnegative integer | vector of nonnegative integers

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 Δy_{t – 1} in the AR model for the second test.

Data Types: double

Model — Model variant

"AR" (default) | "ARD" | "TS" | character vector | string vector | cell vector of character vectors

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 Name	Description
"AR"	Autoregressive model variant, which specifies a test of the null model $$y_t=y_{t-1}+{\beta }_1\Delta y_{t-1}+{\beta }_2\Delta y_{t-2}+\dots +{\beta }_p\Delta y_{t-p}+{\epsilon }_t$$ against the alternative model $$y_t=\varphi y_{t-1}+{\beta }_1\Delta y_{t-1}+{\beta }_2\Delta y_{t-2}+\dots +{\beta }_p\Delta y_{t-p}+{\epsilon }_t,$$ with AR(1) coefficient $$\varphi <1.$$
"ARD"	Autoregressive model with drift variant, which specifies a test of the null model $$y_t=y_{t-1}+{\beta }_1\Delta y_{t-1}+{\beta }_2\Delta y_{t-2}+\dots +{\beta }_p\Delta y_{t-p}+{\epsilon }_t$$ against the alternative model $$y_t=c+\varphi y_{t-1}+{\beta }_1\Delta y_{t-1}+{\beta }_2\Delta y_{t-2}+\dots +{\beta }_p\Delta y_{t-p}+{\epsilon }_t,$$ with drift coefficient c and AR(1) coefficient $$\varphi <1.$$
"TS"	Trend-stationary model variant, which specifies a test of the null model $$y_t=c+y_{t-1}+{\beta }_1\Delta y_{t-1}+{\beta }_2\Delta y_{t-2}+\dots +{\beta }_p\Delta y_{t-p}+{\epsilon }_t$$ against the alternative model $$y_t=c+\delta t+\varphi y_{t-1}+{\beta }_1\Delta y_{t-1}+{\beta }_2\Delta y_{t-2}+\dots +{\beta }_p\Delta y_{t-p}+{\epsilon }_t,$$ with drift coefficient c, deterministic trend coefficient δ, and AR(1) coefficient $$\varphi <1.$$

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 — Test statistic

"t1" (default) | "t2" | "F" | character vector | string vector | cell vector of character vectors

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 Name	Description
"t1"	Standard t statistic $$t_1=\frac{\widehat{\varphi }-1}{\text{SE}\left(\widehat{\varphi }\right)},$$ computed using the OLS estimate of the AR(1) coefficient $$\widehat{\varphi }$$ and its standard error SE($$\widehat{\varphi }$$), in the alternative model. The test assesses the significance of the restriction, $$\varphi -1=0.$$
"t2"	Lag-adjusted, unstudentized t statistic $$t_2=\frac{T(\widehat{\varphi }-1)}{1-{\widehat{\beta }}_1-\dots -{\widehat{\beta }}_p},$$ 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, $$\varphi -1=0.$$
"F"	F statistic for assessing the significance of a joint restriction on the alternative model. For model variant (Model argument) "ARD", the restrictions are$$\varphi -1=0$$ and c = 0. For model variant "TS", the restrictions are$$\varphi -1=0$$ 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

Alpha — Nominal significance level

0.05 (default) | numeric scalar | numeric vector

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

DataVariable — Variable in Tbl to test

last variable (default) | string scalar | character vector | integer | logical vector

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 y_t is defined for t = 1,…,T, the lagged series y_{t– k} is defined for t = k+1,…,T. The first difference applied to the lagged series y_{t– 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

h — Test rejection decisions

logical scalar | logical vector

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.

pValue — Test statistic p-values

numeric scalar | numeric vector

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.

stat — Test statistics

numeric scalar | numeric vector

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.

cValue — Critical values

numeric scalar | numeric vector

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.

StatTbl — Test summary

table

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.

reg — Regression statistics

structure array

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.

Field	Description
num	Length of input series with NaNs removed
size	Effective sample size T, num adjusted for lags
names	Regression coefficient names
coeff	Estimated coefficient values
se	Estimated coefficient standard errors
Cov	Estimated coefficient covariance matrix
tStats	t statistics of coefficients and p-values
FStat	F statistic and p-value
yMu	Mean of the lag-adjusted input series
ySigma	Standard deviation of the lag-adjusted input series
yHat	Fitted values of the lag-adjusted input series
res	Regression residuals
DWStat	Durbin-Watson statistic
SSR	Regression sum of squares
SSE	Error sum of squares
SST	Total sum of squares
MSE	Mean square error
RMSE	Standard error of the regression
RSq	R2 statistic
aRSq	Adjusted R2 statistic
LL	Loglikelihood of data under Gaussian innovations
AIC	Akaike information criterion
BIC	Bayesian (Schwarz) information criterion
HQC	Hannan-Quinn information criterion

More About

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 y_t, where

$$y_t=c+\delta t+\varphi y_{t-1}+{\beta }_1\Delta y_{t-1}+\dots +{\beta }_p\Delta y_{t-p}+{\epsilon }_t,$$

and

• Δ is the differencing operator such that $$\Delta y_t=y_t-y_{t-1}.$$

• 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).

• ε_t is a mean zero innovation process.

The null hypothesis of a unit root is

$$H_0:\varphi =1.$$

Under the alternative hypothesis, $$\varphi <1.$$

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 $${\widehat{\beta }}_p$$, the coefficient of the largest lagged change in y_t. 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 y_t. 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