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:
- The observed \(y\), fitted \(\hat{y}_{m^*}\) values from a model with \(m^*\) breaks (estimated or pre-specified depending on function called) and fitted \(\hat{y}_0\) from a no break model for a direct comparison
- The estimated break dates with labels in chronological order and confidence interval depends on
CIargument (which is either 0.90 or 0.95) for respective dates. - The confidence interval (also depends on argument
CI) for fitted values \(\hat{y}_{m^*}\)6
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$fitted.values #fitted values of model
bigT = length(y)
#compute the null model
fixb = solve((t(zreg) %*% zreg)) %*% t(zreg) %*% y
fity_fix = zreg%*%fixb #fitted values of null model
#plots the model
tx = seq(1,bigT,1)
range_y = max(y)-min(y);
plot(tx,y,type='l',col="black", xlab='time',ylab="y",
ylim=c(min(y)-range_y/10,max(y)+range_y/10),lty=1)
#plot fitted values series for break model
lines(tx, fity,type='l', col="blue",lty=2)
#plot fitted values series for null model
lines(tx, fity_fix,type='l', col="dark red",lty=2)
#plot estimated dates + CIs
for (i in 1:m){
abline(v=date[i,1],lty=2)
if (rate_mBIC$CI[i,1] < 0){rate_mBIC$CI[i,1] = 0}
if(rate_mBIC$CI[i,2]>bigT){ rate_mBIC$CI[i,2]=bigT}
segments(rate_mBIC$CI[i,1],min(y)*(12+i/m)/10,rate_mBIC$CI[i,2],min(y)*(12+i/m)/10,lty=1,col='red')
}
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:
data(nkpc)
#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
supF_ygap
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
seqF_ygap
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
nkpc_lbs$sbtests
a) SupF tests against a fixed number of breaks
1 break 2 breaks 3 breaks 4 breaks 5 breaks
Sup F 30.592 69.630 37.894 32.765 30.607
10% CV 14.810 13.560 12.360 11.430 10.610
5% CV 16.760 14.720 13.300 12.250 11.290
2.5% CV 18.620 15.880 14.220 12.960 11.940
1% CV 20.750 17.240 15.300 13.930 12.780
b) UDmax tests against an unknown number of breaks
UDMax 10% CV 5% CV 2.5% CV 1% CV
1 69.630 15.230 17.000 18.750 20.750
nkpc_lbs$seqtests
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 24.540To 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