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# lmctest

Leybourne-McCabe stationarity test

## Syntax

```h = lmctest(y) h = lmctest(y,'ParameterName',ParameterValue) [h,pValue] = lmctest(...) [h,pValue,stat] = lmctest(...) [h,pValue,stat,cValue] = lmctest(...) [h,pValue,stat,cValue,reg1] = lmctest(...) [h,pValue,stat,cValue,reg1,reg2] = lmctest(...) ```

## Description

`h = lmctest(y)` assesses the null hypothesis that a univariate time series `y` is a trend stationary AR(p) process, against the alternative that it is a nonstationary ARIMA(p,1,1) process.

`h = lmctest(y,'ParameterName',ParameterValue)` accepts one or more comma-separated parameter name/value pairs. Specify `ParameterName` inside single quotes. Perform multiple tests by passing a vector value for any parameter. Multiple tests yield vector results.

```[h,pValue] = lmctest(...)``` returns p-values of the test statistics.

```[h,pValue,stat] = lmctest(...)``` returns the test statistics.

```[h,pValue,stat,cValue] = lmctest(...)``` returns critical values for the tests.

```[h,pValue,stat,cValue,reg1] = lmctest(...)``` returns a structure of regression statistics from the maximum likelihood estimation of the reduced-form model.

```[h,pValue,stat,cValue,reg1,reg2] = lmctest(...)``` returns a structure of regression statistics from the OLS estimation of the filtered data on a linear trend.

## Input Arguments

 `y` Vector of time-series data. The last element is the most recent observation. The test ignores NaN values, which indicate missing entries.

### Name-Value Pair Arguments

 `'alpha'` Scalar or vector of nominal significance levels for the tests. Set values between 0.01 and 0.1. Default: `0.05` `'Lags'` Scalar or vector of nonnegative integers indicating the number `p` of lagged values of `y` to include in the structural model (equal to the number `p` of lagged changes of `y` in the reduced-form model). For best results, give a suitable value for `'lags'`. For information on selecting `'lags'`, see Determine Appropriate Lags. Default: `0` `'trend'` Scalar or vector of Boolean values indicating whether or not to include the deterministic trend term `d*t` in the structural model (equivalent to including the drift term `d` in the reduced-form model). Determine the value of `trend` by the growth characteristics of the time series `y`. Choose `trend` with a specific testing strategy in mind. If `y` is growing, set `trend` to `true` to provide a reasonable comparison of a trend-stationary null and a unit-root process with drift. If `y` does not exhibit long-term growth characteristics, set `trend` to `false`. Default: `true` `'test'` Character vector, such as `'var1'`, or cell vector of character vectors indicating which estimate of the variance ${\sigma }_{1}^{2}$ to use in computing the test statistic. Values are `'var1'` or `'var2'`. Default: `'var2'`

## Output Arguments

 `h` Vector of Boolean decisions for the tests, with length equal to the number of tests. Values of `h` equal to `1` indicate rejection of the AR(p) null in favor of the ARIMA(p,1,1) alternative. Values of `h` equal to `0` indicate a failure to reject the AR(p) null. `pValue` Vector of p-values of the test statistics, with length equal to the number of tests. 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` Vector of test statistics, with length equal to the number of tests. For details, see Test Statistics. `cValue` Vector of critical values for the tests, with length equal to the number of tests. Values are for right-tail probabilities. `reg1` Structure of regression statistics from the maximum likelihood estimation of the reduced-form model. The structure is described in Regression Statistics Structure. `reg2` Structure of regression statistics The structure is described in Regression Statistics Structure.

## Examples

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Test the growth of the U.S. unemployment rate using the data in Schwert, 1987.

Load Schwert's macroeconomic data set.

`load Data_SchwertMacro`

Focus on the unemployment rate growth over the dates considered in Leybourne and McCabe, 1999.

```UN = DataTableMth.UN; t1 = find(datesMth == datenum([1948 01 01])); t2 = find(datesMth == datenum([1985 12 01])); dUN = diff(UN(t1:t2)); % Unemployment rate growth```

Assess the null hypothesis that the unemployment rate growth is a trend stationary, AR(1) process using the estimated variance from OLS regression.

`[h1,~,stat1,cValue] = lmctest(dUN,'lags',1,'test','var1')`
```h1 = logical 0 ```
```stat1 = 0.0992 ```
```cValue = 0.1460 ```

The warning indicates that the p-value is below 0.1. `h1` = 0 indicates that there is not enough evidence to reject that the unemployment rate growth is a trend stationary, AR(1) process.

Assess the null hypothesis that the unemployment rate growth is a trend stationary, AR(1) process using the estimated variance from the maximum likelihood of the reduced-form regression model.

`[h2,~,stat2,cValue] = lmctest(dUN,'lags',1,'test','var2')`
```h2 = logical 1 ```
```stat2 = 0.1874 ```
```cValue = 0.1460 ```

`h2` = 1 indicates that there is enough evidence to suggest that the unemployment rate growth is nonstationary.

Leybourne and McCabe, 1999 report that the original LMC statistic fails to reject stationarity, while the modified LMC statistic does reject it.

## More About

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### Model Equations

`lmctest` uses the structural model

`$\begin{array}{c}y\left(t\right)=c\left(t\right)+\delta t+{b}_{1}y\left(t-1\right)+\cdots +{b}_{p}y\left(t-p\right)+{u}_{1}\left(t\right)\\ c\left(t\right)=c\left(t-1\right)+{u}_{2}\left(t\right),\end{array}$`

where

`$\begin{array}{c}{u}_{1}\left(t\right)~\text{i}\text{.i}\text{.d}\text{.}\left(0,{\sigma }_{1}^{2}\right)\\ {u}_{2}\left(t\right)~\text{i}\text{.i}\text{.d}\text{.}\left(0,{\sigma }_{2}^{2}\right),\end{array}$`

and u1 and u2 are independent of each other.

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

(1 – L)y(t) = δ + b1(1 – L)y(t – 1) + ... + bp(1 – L)y(tp) + (1 – aL)v(t),

where L is the lag operator Ly(t) = y(t–1), and v(t) ~ i.i.d(0,σ2).

The null hypothesis is that σ2 = 0 in the structural model, which is equivalent to a = 1 in the reduced-form model. The alternative is that σ2 > 0 or a < 1. Under the null, the structural model is AR(p) with intercept c(0) 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 a two-stage method that first finds maximum likelihood estimates (MLEs) of coefficients in the reduced-form model. It then regresses the filtered data

z(t) = y(t) – b1y(t–1) – ... – bpy(tp)

on an intercept and, if `'trend'` is `true`, on a trend. It forms the `stat` test statistic using the residuals e from the first regression as follows:

`$\text{stat}=\frac{{e}^{T}Ve}{{s}^{2}{T}^{2}},$`

where V(i,j) = min(i,j), s2 is an estimate of ${\sigma }_{1}^{2}$ that depends on the value of `test` (estimate of the variance), and T is the effective sample size.

### Test Choices

You can choose between test values of `'var1'` and `'var2'`. These distinguish between the algorithm for estimating the variance ${\sigma }_{1}^{2}$.

• `'var1'` — The estimate is `(e'*e)/T`, where `e` is the residual vector from the OLS regression `reg2` and `T` is the effective sample size. This is the original Leybourne-McCabe test described in , with a rate of consistency O(`T`).

• `'var2'` — The estimate is `a*`σ2, where `a` and σ2 are MLEs from the estimation `reg1` of the reduced-form model. This is the modified Leybourne-McCabe test described in , with a rate of consistency O(`T`2).

### Regression Statistics Structure

Lagging and differencing a time series reduces the sample size. Absent any presample values, if y(t) is defined for t = 1:N, then the lagged series y(tk) is defined for t = k+1:N. Differencing reduces the time base to k+2:N. With p lagged differences, the common time base is p+2:N and the effective sample size is N – (p+1).

The maximum likelihood estimation of `reg1` regresses Y = (1–L)y(t), with num = N–1, on p lagged changes of y, so that size = N – (p+1).

The OLS estimation of `reg2` regresses Y = z(t), with num = Np, on an intercept and, if `trend` is `true`, a trend, so that size = num.

The regression statistics structures have the following form:

 `num` Length of input series with `NaN`s removed `size` Effective sample size, adjusted for lags and difference `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

## Algorithms

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  using Monte Carlo simulations. Critical values and p-values reported by `lmctest` are interpolated from the tables. Tables are identical to those for `kpsstest`.

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

 shows that the test is robust when p takes values greater than the value in the data-generating process.  also notes simulation evidence that, under the null, the marginal distribution of the MLE of bp is asymptotically normal, and so may be subject to a standard t-test for significance. Estimated standard errors, however, are unreliable in cases where the MA(1) coefficient a is near 1. As a result,  proposes another test for model order, valid under both the null and the alternative, that relies only on the MLEs of bp and a, and not on their standard errors.

## References

 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.

 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.

 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.

 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.

 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.

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