lmctest

Leybourne-McCabe stationarity test

Syntax

h = lmctest(y)

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

StatTbl = lmctest(Tbl)

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

[___,reg1,reg2] = lmctest(___)

Description

h = lmctest(y) returns the rejection decision from conducting the Leybourne-McCabe stationarity test for assessing whether an input univariate time series is stationary.

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

StatTbl = lmctest(Tbl) returns a table containing variables for the test results, statistics, and settings from conducting the Leybourne-McCabe stationarity test on the last variable of an input table or timetable. To select a different variable to test, use the DataVariable name-value argument.

[___] = lmctest(___,Name=Value) specifies options using one or more name-value arguments in addition to any of the input argument combinations in previous syntaxes. lmctest returns the output argument combination for the corresponding input arguments.

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

• lmctest 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, lmctest(Tbl,DataVariable="GDP",Alpha=0.025,Lags=[0 1]) conducts two tests, at a level of significance of 0.025, on the variable GDP of the table Tbl. The first test includes 0 lagged terms in the structural model, and the second test includes 1 lagged term in the structural model.

[___,reg1,reg2] = lmctest(___) additionally returns structures of regression statistics, which are required to form the test statistic.

• reg1 – Maximum likelihood estimation of the reduced-form model

• reg2 – Deterministic local level model of filtered response data, with Gaussian noise and an optional linear trend

Examples

Conduct Leybourne-McCabe Stationarity Test on Vector of Data

Test a time series for stationarity using the default options of lmctest. Input the time series data as a numeric vector.

Load Schwert's macroeconomic data set. Extract the monthly unemployment rate UN.

load Data_SchwertMacro
un = DataTableMth.UN;

Represent the series as a growth rate by applying the first difference.

unr = diff(un);

The first difference operation causes unr to have one less observation than un. The timebase of unr starts at observation 2.

Plot the unemployment growth rate.

dts = datetime(datesMth,ConvertFrom="datenum");
plot(dts(2:end),unr)
title("Unemployment Growth Rate")

Assess the null hypothesis of the Leybourne-McCabe stationarity test that the unemployment growth rate series is a trend-stationary AR(0) model. Use default options.

h = lmctest(unr)

h = logical
   0

h = 0 indicates that, at a 5% level of significance, the test fails to reject the null hypothesis that the unemployment growth rate series is a trend-stationary AR(0) model.

Return Test p-Value and Decision Statistics

Load Schwert's macroeconomic data set Data_SchwertMacro.mat. Extract the monthly unemployment rate UN, and apply the first difference to the series.

load Data_SchwertMacro
unr = diff(DataTableMth.UN);

Assess the null hypothesis that the series is a trend-stationary AR(0) process. Return the test decision, $\mathit{p}$-value, test statistic, and critical value.

[h,pValue,stats,cValue] = lmctest(unr)

h = logical
   0

pValue = 
0.1000

stats = 
0.0978

cValue = 
0.1460

pValue = 0.1000 is the maximum tabulated value; its actual value can be larger than 0.1000.

Conduct Leybourne-McCabe Stationarity Test on Table Variable

Test whether a time series, which is one variable in a table, for stationarity using the default options.

Load Schwert's macroeconomic data set. Convert the table of monthly series to a timetable.

load Data_SchwertMacro
dates = datetime(datesMth,ConvertFrom="datenum");
TT = table2timetable(DataTableMth,RowTimes=dates);

Apply the first difference to all monthly series.

DTT = varfun(@diff,TT(:,2:end));
DTT.Properties.VariableNames{end}

ans = 
'diff_SIG'

Assess the null hypothesis of the Leybourne-McCabe stationarity test that the rate of the volatility of returns to Standard & Poor's composite index series is a trend-stationary AR(0) model.

StatTbl = lmctest(DTT)

StatTbl=1×8 table
                h      pValue      stat       cValue    Lags    Alpha    Trend      Test  
              _____    ______    _________    ______    ____    _____    _____    ________

    Test 1    false     0.1      0.0027953    0.146      0      0.05     true     {'var2'}

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

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

Assess Whether Series Is Trend Stationary and AR(p)

Test the growth of the US unemployment rate using the data in [5].

Load Schwert's macroeconomic data set. Convert the table of monthly series to a timetable. Apply the first difference to all variables in the timetable.

load Data_SchwertMacro
dates = datetime(datesMth,ConvertFrom="datenum");
TT = table2timetable(DataTableMth,RowTimes=dates);
TT.Dates = [];
DTT = varfun(@diff,TT);

Define the time range of the sample considered in [4]. Extract a subtable with the corresponding dates.

trLM = timerange("1948-01-01","1985-12-01","closed");
DTT4 = DTT(trLM,:);

Assess the null hypothesis that the unemployment rate growth is a trend-stationary, AR(1) process. Conduct the same test twice. For the first test, specify the test statistic that uses the estimated variance from OLS regression (Test = "var1"). For the second test, specify the test statistic that uses estimated variance from the maximum likelihood of the reduced-form regression model (Test = "var2").

StatTbl = lmctest(DTT4,DataVariable="diff_UN", ...
    Lags=1,Test=["var1" "var2"])

StatTbl=2×8 table
                h       pValue       stat      cValue    Lags    Alpha    Trend      Test  
              _____    ________    ________    ______    ____    _____    _____    ________

    Test 1    false         0.1    0.099166    0.146      1      0.05     true     {'var1'}
    Test 2    true     0.020721     0.18741    0.146      1      0.05     true     {'var2'}

Row Test 1 of StatTbl contains the results of the first test StatTbl.h(1) = 0 indicates that, at a 5% level of significance, there is not enough evidence to reject that the unemployment growth rate is a trend-stationary, AR(1) process. Row Test 2 of StatTbl contains the results of the second test StatTbl.h(2) = 1 indicates that, at a 5% level of significance, there is enough evidence to reject that the unemployment growth rate is a trend-stationary, AR(1) process, which implies that the unemployment rate growth is nonstationary.

Leybourne and McCabe report that the original LMC statistic fails to reject stationarity, while the modified LMC statistic does reject it [4].

Inspect Regression Statistics

Load Schwert's macroeconomic data set. Convert the table of monthly series to a timetable. Apply the first difference to all variables in the timetable.

load Data_SchwertMacro
dates = datetime(datesMth,ConvertFrom="datenum");
TT = table2timetable(DataTableMth,RowTimes=dates);
TT.Dates = [];
DTT = varfun(@diff,TT);

Assess the null hypothesis that the unemployment rate growth is a trend-stationary, AR(1) process. Return the regression statistics from the maximum likelihood estimation of reduced-form model and from the OLS estimation of deterministic local level model of filtered response data.

[StatTbl,reg1,reg2] = lmctest(DTT,DataVariable="diff_UN",Lags=1);

Display the coefficients of both regressions.

rownames = ["c_0"; "delta"; "b_1"];
varnames = ["Coefficient"; "SE"; "PValue"];
coeff = reg1.coeff;
se = reg1.se;
pval = reg1.tStats.pVal;
table(coeff,se,pval,RowNames=rownames,VariableNames=varnames)

ans=3×3 table
             Coefficient       SE          PValue  
             ___________    _________    __________

    c_0      -0.00051942    0.0041996       0.90162
    delta       -0.25044     0.051797    1.8304e-06
    b_1          0.63665     0.046451    5.6215e-36

rownames = ["Intercept"; "Trend"];
coeff = reg2.coeff;
se = reg2.se;
pval = reg2.tStats.pVal;
table(coeff,se,pval,RowNames=rownames,VariableNames=varnames)

ans=2×3 table
                 Coefficient        SE        PValue 
                 ___________    __________    _______

    Intercept       0.013904      0.024521    0.57099
    Trend        -2.2372e-05    9.3396e-05    0.81079

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

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

Lags — Number p of lagged response terms

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

Number p of lagged values of y_t to include in the structural model, specified as a nonnegative integer or vector of nonnegative integers. p is equal to the number of lagged differences of y_t in the reduced-form model.

lmctest conducts a separate test for each element in Lags.

Lags > 0 decreases the sample size (see Algorithms).

Tip

To draw valid inferences from a Leybourne-McCabe stationarity test, you must determine a suitable value for the Lags argument. The test is robust when p is greater than its true value in the data-generating process, and simulation evidence shows that, under the null hypothesis, the marginal distribution of the MLE of b_p is asymptotically normal [3]. Therefore, you can use a standard t-test to determine whether b_p is significant. However, estimated coefficient standard errors are unreliable when the MA(1) coefficient a is near 1. For a model-order test, valid under both the null and alternative hypotheses, that relies only on the MLEs of b_p and a, and not on their standard errors, see [4].

Example: Lags=[0 1] includes no lags in the structural model for the first test, and then includes y_{t - 1} in the structural model for the second test.

Data Types: double

Trend — Flag for including deterministic trend term δt

true (default) | false | logical vector

Flag for including deterministic trend δt in the structural model, specified as a logical scalar or vector. Trend=true is equivalent to including the drift term δ in the reduced-form model.

lmctest conducts a separate test for each element in Trend.

Tip

With a specific testing strategy in mind, determine the value of the Trend argument by the growth characteristics of the input time series.

• If the input series grows, include a trend term by setting Trend to true (default). This setting provides a reasonable comparison of a trend stationary null and a unit root process with drift.

• If a series does not exhibit long-term growth characteristics, exclude a trend term by setting Trend to false.

Example: Trend=false excludes δt from the structural model for all tests.

Data Types: logical

Test — Variance estimate

"var1" (default) | "var2" | character vector | string vector | cell vector of character vectors

Variance estimate $${\widehat{\sigma }}_1^2$$ to use for the test statistic, specified as an estimate name, or a string vector or cell vector of estimate names, in this table.

Test Name	Description
"var1"	$${\widehat{\sigma }}_1^2=\frac{e_2\prime e_2}{T},$$ where e2 is the residual vector from the regression of the deterministic local level model of the filtered responses.
"var2"	$${\widehat{\sigma }}_1^2=a{\sigma }^2$$, where a and σ2 are MLEs from the reduced-form model regression.

For more details, see Test Statistics.

lmctest conducts a separate test for each estimate name in Test.

Example: Test=["var1" "var2"] conducts two tests. The first test uses the variance estimate described by "var1" to compute the test statistic, and the second test uses the variance estimate "var2".

Data Types: char | cell | string

Alpha — Nominal significance level

0.05 (default) | numeric scalar | numeric vector

Nominal significance level for the hypothesis test, specified a numeric scalar between 0.01 and 0.10 or a numeric vector of such values.

lmctest 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

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. lmctest returns h when you supply the input y.

• Values of 1 indicate rejection of the AR(p) null hypothesis in favor of the ARIMA(p,1,1) alternative.

• Values of 0 indicate failure to reject the AR(p) null hypothesis.

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. lmctest returns pValue when you supply the input y.

The p-values are right-tail probabilities.

When test statistics are outside tabulated critical values, lmctest returns maximum (0.10) or minimum (0.01) 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. lmctest returns stat when you supply the input y.

For details, see Test Statistics.

cValue — Critical values

numeric scalar | numeric vector

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

Critical values are for right-tail probabilities.

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. lmctest returns StatTbl when you supply the input Tbl.

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

reg1 — Regression statistics from maximum likelihood estimation of reduced-form model

structure array

Regression statistics from the maximum likelihood estimation of the reduced-form model, returned as a structure array with the number of records equal to the number of tests.

Each element of reg1 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.

num	Length of input series with NaNs removed, T – 1
size	Effective sample size, adjusted for lags and difference, T – (p + 1)
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

reg2 — Regression statistics from estimation of deterministic local level model of filtered response data

structure array

Regression statistics from the estimation of the deterministic local level model of the filtered response data, returned as a structure array with the number of records equal to the number of tests.

reg2 has the same fields as reg1, but with the differences described in the following table.

num	Length of input series with NaNs removed, T – p
size	Effective sample size, adjusted for lags and difference, T – p

More About

Leybourne-McCabe Stationarity Test

The Leybourne-McCabe stationarity test assesses the null hypothesis that a response series y_t is a trend-stationary, degree p autoregressive model (AR(p)) against the alternative hypothesis that y_t is nonstationary.

Structural Model for Hypotheses

The structural model for the response series is

$$\begin{array}{c}{y}_t=c_t+\delta t+b_1{y}_{t-1}+\cdots +b_p{y}_{t-1}+u_{1,t}\\ c_t=c_{t-1}+u_{2,t},\end{array}$$

where

• $$\begin{array}{c}{u}_{1,t}~\text{iid}\left(0,{\sigma }_1^2\right)\\ u_{2,t}~\text{iid}\left(0,{\sigma }_2^2\right),\end{array}$$

• u_1,t and u_2,t are independent.

• The number of lags p is the value of the Lags option.

• The trend term δt is present when Trend=true.

• T is the sample size without missing observations.

The model is second-order equivalent in moments to the reduced-form ARIMA(p,1,1) model

$$\Delta y_t=\delta +b_1\Delta y_{t-1}+\mathrm{...}+b_p\Delta y_{t-p}+\left(1-aL\right)v_t,$$

where L is the lag operator Ly_t = y_t–1 and v_t ~ iid(0,σ²).

The null hypothesis is that σ₂² = 0 in the structural model, which is equivalent to a = 1 in the reduced-form model. The alternative is that σ₂² > 0 or a < 1. Under the null hypothesis, the structural model is AR(p) with intercept c₀ and trend δt; the reduced-form model is an over-differenced ARIMA(p,1,1) representation of the same process.

Test Statistics

lmctest computes test statistics using this two-stage method:

1. Perform OLS regression of the reduced-form model to obtain maximum likelihood estimates (MLEs) of the coefficients (results are stored in the output reg1). Specifically, regress Y = Δy_t on p lagged differences of y. lmctest stores the regression results in reg1.

2. Regress filtered data z_t onto x_t where

   • $$z_t=y_t-b_1{y}_{t-1}-\mathrm{...}-b_p{y}_{t-p}.$$

   • t = p+1,…,T

   • x_t is one of the following:

     • 1, for an intercept-only model when Trend=false

     • [1 t], for a model with an intercept and deterministic tome trend when Trend=true

   lmctest stores the regression results in reg2.

The test statistic s^* (output stat) is

$$s^{\ast }=\frac{e_1\prime V{e}_1}{s^2{T}^2},$$

wherea

• e₁ is a vector of the residuals from the reduced-form model regression.

• V(i,j) = min(i,j)

• s² is an estimate of $${\sigma }_1^2$$ that depends on the value of the variance-estimation algorithm option Test:

  • "var1" — The estimate is

    $${\widehat{\sigma }}_1^2=\frac{e_2\prime e_2}{T},$$

    where e₂ is the residual vector from the regression z_t onto x_t (output reg2). This selection is the original Leybourne-McCabe test, as described in [3], and it has a rate of consistency O(T).

  • "var2" (default) — The estimate is aσ², where a and σ² are MLEs from the reduced-form model regression (output reg1). This selection is the modified Leybourne-McCabe test, as described in [4], and it has a rate of consistency O(T²).

Tips

• The alternative hypothesis that σ²₂ > 0 implies 0 < a < 1. As a result, an alternative model with a = 0 and a random walk, reduced-form model with iid errors is not possible. The class of I(1) alternatives represented by such a model is appropriate for economic series with significant MA(1) components [3]. To test for a random walk, use vratiotest.

Algorithms

• The value of the Lags option lags the response in the structural model, and the reduced-form model operates on the first difference of the response. In general, when a time series is lagged or differenced, the sample size is reduced. Without a presample, if y_t is defined for t = 1,…,T, the lagged series y_t–k is defined for t = k+1,…,T. When y_t–k is differenced, the time base reduces to k+2,…,T. p lagged differences reduce the common time base to p+2,…,T and the effective sample size is T – (p+1).

• Test statistics follow nonstandard distributions under the null, even asymptotically. Asymptotic critical values for a standard set of significance levels between 0.01 and 0.1, for models with and without a trend, have been tabulated in [2] using Monte Carlo simulations. Critical values cValue and p-values pValue reported by lmctest are interpolated from the tables. The tabulated tables are identical to those for kpsstest.

• Bootstrapped critical values, used by tests with a unit root null (such as adftest and pptest), are not possible for lmctest [1]. As a result, size distortions for small samples may be significant, especially for highly persistent processes.

References

[1] Caner, M., and L. Kilian. "Size Distortions of Tests of the Null Hypothesis of Stationarity: Evidence and Implications for the PPP Debate." Journal of International Money and Finance. Vol. 20, 2001, pp. 639–657.

[2] Kwiatkowski, D., P. C. B. Phillips, P. Schmidt, and Y. Shin. “Testing the Null Hypothesis of Stationarity against the Alternative of a Unit Root.” Journal of Econometrics. Vol. 54, 1992, pp. 159–178.

[3] Leybourne, S. J., and B. P. M. McCabe. "A Consistent Test for a Unit Root." Journal of Business and Economic Statistics. Vol. 12, 1994, pp. 157–166.

[4] Leybourne, S. J., and B. P. M. McCabe. "Modified Stationarity Tests with Data-Dependent Model-Selection Rules." Journal of Business and Economic Statistics. Vol. 17, 1999, pp. 264–270.

[5] Schwert, G. W. "Effects of Model Specification on Tests for Unit Roots in Macroeconomic Data." Journal of Monetary Economics. Vol. 20, 1987, pp. 73–103.

Version History

Introduced in R2010a