Estimating and testing for multiple structural changes with mbreaks package

Linh Nguyen, Pierre Perron, Yohei Yamamoto

This vignette is intended for users who use mbreaks to estimate and test for linear regression models in the presence of multiple structural changes. The package offers a set of comprehensive tools which deal with both pure and partial structural change models. In particular, it provides the Sup F tests for 0 versus a known number of structural changes and the double maximum (UD max) tests for 0 versus an unknown number of structural changes. The sequential tests for \(l\) versus \(l+1\) structural changes are also available to determine the number of structural changes (Bai and Perron (1998), Bai and Perron (2003)). The package also includes methods of estimating the number of structural changes via information criteria (Yao (1988), Liu et al. (1997)) as well as a built-in function to visualize the fit of the estimated structural break model with the estimated number \(m^*\) of structural changes. A comprehensive call to conduct all of the procedures contained in package is provided.

Econometric framework of mbreaks package

The package mbreaks provides R users with minimum but comprehensive functions to analyze multiple structural changes in linear regression models in which the regressors and errors are non-trending. The framework is based on the econometric model of the following form:

\[ y_t = x_t^{\prime}\beta + z_t^{\prime} \delta_j + u_t; \\ t=T_{j-1}+1,...,T_j, {\quad}{\text{for}}{\quad}j=1,...,m+1\] where \(\underset{(p \times 1)}{x_t}\) is a vector of regressors with fixed coefficients (if any) and \(\underset{(q \times 1)}{z_t}\) is a vector of regressors with coefficients subject to change. The break dates are \(t=T_j\) for \(j=1,...,m\) and \(T_0=0\) and \(T_{m+1}=T\) so that \(T\) is the entire sample size. If \(p=0\), the model is called a pure structural change model, and if \(p>0\), the model is called a partial structural change model. The twofold goals of this package is to enable user to:

If the number of structural changes is known (or specified), users can use dofix() function to estimate the model with the corresponding number of changes. There are 3 classes in the package, corresponding to 2 types of diagnostic tests and one for estimation based on the selected number of structural changes. In summary:

Usage, arguments and examples

Usage of main functions in mbreaks

The previous section introduced the framework on which mbreaks package is built and summarizes the classes of procedures available to users. In this section, we will illustrate the syntax of high-level functions and cover the arguments that users might want to customize to match their model with data and empirical strategy.

The mbreaks package designs high-level functions to have identical arguments with default values recommended by the literature to save users the burden. Users can use mdl(), a comprehensive function that invokes all high-level functions explained in previous section:1

#the data set for the example is real.Rda
#carry out all testing and estimating procedures via a single call
rate = mdl('rate',data=real,eps1=0.15)
#display the results from the procedures
The number of breaks is estimated by KT 
Pure change model with 2 estimated breaks.
Minimum SSR = 455.950 

Estimated date:
        Break1  Break2
Date        47      79
95% CI (37,50) (77,82)
90% CI (40,49) (78,81)

Estimated regime-specific coefficients:
           Regime 1       Regime 2      Regime 3
 (SE) 1.355 (0.155) -1.796 (0.511) 5.643 (0.603)

No full sample regressors
a) SupF tests against a fixed number of breaks

        1 break 2 breaks 3 breaks 4 breaks 5 breaks
Sup F    57.906   43.014   33.323   24.771   18.326
10% CV    7.040    6.280    5.210    4.410    3.470
5% CV     8.580    7.220    5.960    4.990    3.910
2.5% CV  10.180    8.140    6.720    5.510    4.340
1% CV    12.290    9.360    7.600    6.190    4.910

b) UDmax tests against an unknown number of breaks

   UDMax 10% CV  5% CV 2.5% CV  1% CV
1 57.906  7.460  8.880  10.390 12.370
supF(l+1|l) tests using global optimizers under the null

         supF(1|0) supF(2|1) supF(3|2) supF(4|3) supF(5|4)
Seq supF    57.906    33.927    14.725     0.033     0.000
10% CV       7.040     8.510     9.410    10.040    10.580
5% CV        8.580    10.130    11.140    11.830    12.250
2.5% CV     10.180    11.860    12.660    13.400    13.890
1% CV       12.290    13.890    14.800    15.280    15.760
To access additional information about specific procedures
  (not included above), type stored variable name + '$' + procedure name

Users should find the syntax minimal similar to lm() in stats package. It is required to specify the name of dependent variable \(y\) followed by the two types of the regressors \(z\) and \(x\) from the data frame. Note that \(z\) automatically includes a constant term. If the model is a pure structural change model, no \(x\) is specified. If none of \(z\) and \(x\) are specified, the program assumes that this is a mean shift model (because a constant term is included in \(z\) by default). The names of regressors must match the names used in the data frame, otherwise errors will be displayed and execution halted. As we will explain in the following section, the package prepares various options to be specified by users. These are set at default value if not specified in mdl() or any high-level functions of the procedures: dotest(), dosequa(), doorder(), dorepart(), and dofix()etc.

Options for high-level functions

There are two options available to all high-level functions:

Additional specific options:

Empirical examples

The vignette replicates 2 empirical exercises: i) US real interest rate and ii) New Keynesian Phillips curve.

US real interest rate

Garcia and Perron (1996),Bai and Perron (2003) considered a mean shift model:

\[y_t = \mu_j + u_t, \quad\text{for } t = T_{j-1}+1,...,T_{j}\quad\text{and } j=1,...,m.\]

for the US real interest rate series from 1961Q1 to 1986Q3. We allow heteroskedasticity and serial correlation in the errors \(u_t\) by using the heteroskedasticity and autocorrelation consistent (HAC) long-run covariance estimate using the default setting (robust=1) with the prewhitened residuals also by the default setting (prewhit=1). Here, instead of invoking mdl(), we demonstrate the specific syntax to obtain the model with the number of structural changes \(m^*\) selected by modified BIC information 'KT'

#estimating the mean-shift model with BIC (the default option is ic=`'KT'`, which use modified BIC as criterion)
rate_mBIC = doorder('rate',data=real)
#NOTE: equivalent to rate$KT; type rate$KT to compare with new result

# visualization of estimated model with modified BIC (in the argument, we can replace rate$KT with rate_mBIC for exact same graph; recall that `$` is the operator to refer to field BIC in return list from mdl())
plot_model(rate$KT, title = 'US Exchange rate')

The plot_model() function takes any estimated structural break model of class model and makes a graph with the following contents:

To show flexibility of class model in the package, we can reproduce a similar graph using information returned from stored variable rate_BIC.

  #collect model information
  m = rate_mBIC$nbreak           #number of breaks
  y = rate_mBIC$y                #vector of dependent var
  zreg = rate_mBIC$z             #matrix of regressors with changing coefs
  date = rate_mBIC$date          #estimated date
  fity = rate_mBIC$fitted.values #fitted values of model
  bigT = length(y)
  #compute the null model
  fixb = solve((t(zreg) %*% zreg)) %*% t(zreg) %*% y
  fity_fix = zreg%*%fixb    #fitted values of null model
  #plots the model
  tx = seq(1,bigT,1)
  range_y = max(y)-min(y);
  plot(tx,y,type='l',col="black", xlab='time',ylab="y", 
  #plot fitted values series for break model
  lines(tx, fity,type='l', col="blue",lty=2)
  #plot fitted values series for null model
  lines(tx, fity_fix,type='l', col="dark red",lty=2)
  #plot estimated dates + CIs
  for (i in 1:m){
    if (rate_mBIC$CI[i,1] < 0){rate_mBIC$CI[i,1] = 0}
    if(rate_mBIC$CI[i,2]>bigT){ rate_mBIC$CI[i,2]=bigT}
  legend(0,max(y)+range_y/10,legend=c("observed y",paste(m,'break y'),"0 break y"),
        lty=c(1,2,2), col=c("black","blue","red"), ncol=1)

New Keynesian Phillips Curve

Perron and Yamamoto (2015) investigates the stability of New Keynesian Phillips curve model proposed by Galı and Gertler (1999) via linear model:

\[\pi_t = \mu + \gamma \pi_{t-1} + \kappa x_t + \beta E_t \pi_{t+1} + u_t\]

where \(\pi_t\) is inflation rate at time t, \(E_t\) is an expectation operator conditional on information available up to \(t\), and \(x_t\) is a real determinant of inflation. In this example, we will reproduce specific results of the paper with ready-to-use dataset:

#x_t is GDP gap
  z_name = c('inflag','ygap','inffut')
  #we can invoke each test separately by using dotest() and doseqtests()
  supF_ygap = dotest('inf',z_name,data=nkpc,prewhit = 0, eps1 = 0.1,m=1)
  #z regressors' names are passed in the argument as an array, which equivalent to above argument call with z_name
  seqF_ygap = doseqtests('inf',c('inflag','ygap','inffut'),data=nkpc,prewhit = 0, eps1=0.1)
  #see test results

a) SupF tests against a fixed number of breaks

        1 break
Sup F    22.220
10% CV   14.810
5% CV    16.760
2.5% CV  18.620
1% CV    20.750

b) UDmax tests against an unknown number of breaks

   UDMax 10% CV  5% CV 2.5% CV  1% CV
1 22.220 15.230 17.000  18.750 20.750

supF(l+1|l) tests using global optimizers under the null

         supF(1|0) supF(2|1) supF(3|2) supF(4|3) supF(5|4)
Seq supF    22.220    12.559    22.294    13.565    18.463
10% CV      14.810    16.700    17.840    18.510    19.130
5% CV       16.760    18.560    19.530    20.240    20.720
2.5% CV     18.620    20.300    21.180    21.860    22.400
1% CV       20.750    22.400    23.550    24.130    24.540
#x_t is labor income share 
  #or invoke all tests using mdl() 
  nkpc_lbs = mdl('inf',c('inflag','lbs','inffut'),data=nkpc,prewhit = 0, eps1=0.1, m=5)

There are no breaks selected by BIC and estimation is skipped

There are no breaks selected by LWZ and estimation is skipped

There are no breaks selected by KT and estimation is skipped

a) SupF tests against a fixed number of breaks

        1 break 2 breaks 3 breaks 4 breaks 5 breaks
Sup F    30.592   69.630   37.894   32.765   30.607
10% CV   14.810   13.560   12.360   11.430   10.610
5% CV    16.760   14.720   13.300   12.250   11.290
2.5% CV  18.620   15.880   14.220   12.960   11.940
1% CV    20.750   17.240   15.300   13.930   12.780

b) UDmax tests against an unknown number of breaks

   UDMax 10% CV  5% CV 2.5% CV  1% CV
1 69.630 15.230 17.000  18.750 20.750

supF(l+1|l) tests using global optimizers under the null

         supF(1|0) supF(2|1) supF(3|2) supF(4|3) supF(5|4)
Seq supF    30.592    11.408    11.595    12.564    26.418
10% CV      14.810    16.700    17.840    18.510    19.130
5% CV       16.760    18.560    19.530    20.240    20.720
2.5% CV     18.620    20.300    21.180    21.860    22.400
1% CV       20.750    22.400    23.550    24.130    24.540

To replicate the results, we turn off prewhit option.

The values of SupF 30.6 and F(2|1) 30.6 test statistics are equivalent to 30.6 and 11.4 and the values 22.2 and 22.2 are equivalent to 22.2 and 12.6 in table VI of Perron and Yamamoto (2015) . Given the Sup F(2|1) statistics in both regressions is smaller than the 10% critical values 14.81 and both Sup F test statistic of 0 versus 1 break is larger than the 1% critical values 20.75, we conclude there is only 1 break detected.

The estimated break dates 1:Q1991 also match 1991:Q1 in the reported table. The package exactly replicates results presented in Perron and Yamamoto (2015). Given the estimated date from the sequential approach, we could split the sample into two subsamples and conduct the two stage least squares (2SLS) in each subsample as suggested by Perron and Yamamoto (2015):

Using the matrix formula for 2SLS estimator below in IV regression, we are able to obtain close estimates for first subsample as reported in table VII (Perron and Yamamoto (2015)). \[ \hat{\beta}_{IV} = (X'P_ZX)^{-1}X'P_Zy \\ P_Z = Z(Z'Z)^{-1}Z' \] where \(X = [X_1 \;\; X_2]\) is matrix of both endogenous regressor \(X_1 = \pi_{t+1}\) and exogenous regressors \(X_2 = [1,\pi_{t-1},x_t]\) and \(Z = [Z_1 \;\; Z_2]\) is matrix of instruments including excluded instruments \(Z_1\) and included instruments \(Z_2 = [1,\pi_{t-1}]\). In total, the instruments used in first stage regression are lags of inflation, labor income share, output gap, interest spread, wage inflation and commodity price (6 instruments).

The IV estimates \(\hat{\theta}_{IV}=(\hat{\mu},\hat{\gamma},\hat{\kappa},\hat{\beta})\) for 1960:Q1-1991:Q1 subsample and 1991:Q2-1 are

\(\mu\) \(\gamma(\pi_{t-1})\) \(\kappa(x_t)\) \(\beta(E_t\pi_{t+1})\)
1960:Q1-1991:Q1 0.001 0.338 -0.002 0.632
\(SE_1\) 0.002 0.115 0.009 0.130
1991:Q2-1997:Q4 0.010 0.589 -0.024 -0.866
\(SE_2\) 0.007 0.225 0.045 0.309

The table above replicates Perron and Yamamoto (2015) table VII exactly


Bai, J., 1997. Estimating multiple breaks one at a time. Econometric Theory 13, 315–352.

Bai, J., Perron, P., 2003. Computation and analysis of multiple structural change models. Journal of Applied Econometrics 18, 1–22.

Bai, J., Perron, P., 1998. Estimating and testing linear models with multiple structural changes. Econometrica 47–78.

Galı, J., Gertler, M., 1999. Inflation dynamics: A structural econometric analysis. Journal of Monetary Economics 44, 195–222.

Garcia, R., Perron, P., 1996. An analysis of the real interest rate under regime shifts. Review of Economics and Statistics 111–125.

Kurozumi, E., Tuvaandorj, P., 2011. Model selection criteria in multivariate models with multiple structural changes. Journal of Econometrics 164, 218–238.

Liu, J., Wu, S., Zidek, J.V., 1997. On segmented multivariate regression. Statistica Sinica 497–525.

Perron, P., Yamamoto, Y., 2015. Using ols to estimate and test for structural changes in models with endogenous regressors. Journal of Applied Econometrics 30, 119–144.

Yao, Y.-C., 1988. Estimating the number of change-points via schwarz’criterion. Statistics & Probability Letters 6, 181–189.

  1. Users can call independent functions to carry out specific procedures as outlined above instead of conducting all 6 main procedures provided by package via mdl()↩︎

  2. The high level functions are mdl(), dofix(), dosequa(), dorepart(), doorder(), dotest(), doseqtests()↩︎

  3. This argument is different from eps where eps sets the convergence criterion for iterative scheme when estimating partial change model.↩︎

  4. The results of pure structural change models are not affected by these options↩︎

  5. The above options are used extensively in all high-level functions of the program to specify required assumptions on structural break model. Additional options relating to formatting output in plot_model() function will not be explained in this section. For more information type ?plot_model or ?mdl() to understand the distinctions between two types of options↩︎

  6. The option CI for plot_model() is used to specify the confidence interval around estimates of break dates and fitted values. For fitted values, it is computed as: \[ (\hat{y}_t^{m^*-},\hat{y}_t^{m^*+}) = (x_t'(\hat{\beta}-Z_{CI} \hat{s.e.}(\hat{\beta}))+z_t'(\hat{\delta}-Z_{CI}\hat{s.e.}(\hat{\delta})),x_t'(\hat{\beta}+Z_{CI} \hat{s.e.}(\hat{\beta}))+z_t'(\hat{\delta}+Z_{CI}\hat{s.e.}(\hat{\delta})))\] where \[ Z_{CI} = \begin{cases} 1.96 & CI=0.95 \\ 1.65 & CI=0.90 \end{cases} \] For break dates, the confidence interval is obtained via limiting distribution of \(\hat{T}_j\) (Bai and Perron (1998)) ↩︎