Kleiber - Applied econometrics in R

Fig. 6.10. OLS-based CUSUM process (left) and process of F statistics (right) for the UK driver deaths model.

Many further tests for structural change are available in strucchange. A unified and more flexible approach is implemented in gefp() (Zeileis 2005, 2006a). In practice, this multitude of tests is often a curse rather than a blessing. Unfortunately, no test is superior to any other test for all conceivable patterns of structural change. Hence, the choice of a suitable test can be facilitated if there is some prior knowledge about which types of changes are likely to occur and which parameters are a ected by it (see Zeileis 2005, for some discussion of this).

To further illustrate the wide variety of structural change tests, we consider a second example. Lutkepohl,¨ Terasvirta,¨ and Wolters (1999) establish an error correction model (ECM) for German M1 money demand, reanalyzed by Zeileis, Leisch, Kleiber, and Hornik (2005) in a structural change setting. The data frame GermanM1 contains data from 1961(1) to 1995(4) on per capita M1, price index, per capita GNP (all in logs) and an interest rate. It can be loaded along with the model used by Lutkepohl¨ et al. (1999) via

R> data("GermanM1")

R> LTW <- dm ~ dy2 + dR + dR1 + dp + m1 + y1 + R1 + season

involving the di erenced and lagged series as well as a factor season that codes the quarter of the year. To test whether a stable model can be fitted for this ECM, particularly in view of the German monetary unification on 1990- 06-01, a recursive estimates (RE) test (the “fluctuation test” of Ploberger, Kramer,¨ and Kontrus 1989), is employed using

R> m1_re <- efp(LTW, data = GermanM1, type = "RE") R> plot(m1_re)

174 6 Time Series

RE test (recursive estimates test)

Fig. 6.11. Recursive estimates fluctuation process for German M1 model.

The boundary crossing of the RE process shown in Figure 6.11 signals again that there is a deviation from structural stability (at the default 5% level), and the clear peak conveys that this is due to an abrupt change in 1990, matching the timing of the German monetary unification.

Dating structural changes

Given that there is evidence for structural change in a certain model, a natural strategy is to find a model that incorporates the changes. When the changes are abrupt, this amounts to segmenting the original data and fitting the model on each subset. In the framework of the linear regression model, the setup is

yt = x>t β(j) + "t, t = nj−1 + 1, . . . , nj, j = 1, . . . , m + 1, (6.4)

where j = 1, . . . , m is the segment index and β(j) is the segment-specific set of regression coe cients. The indices {n1, . . . , nm} denote the set of unknown breakpoints, and by convention n0 = 0 and nm+1 = n.

Estimating the breakpoints is also called dating structural changes. For the two models considered above, visual inspection already provides information on the locations of the breakpoints. However, a more formal procedure for determining the number and location of breakpoints is desirable. Bai and Perron (1998, 2003) established a general methodology for estimating breakpoints and their associated confidence intervals in OLS regression, and their

method is implemented in the function breakpoints() by Zeileis, Kleiber, Kramer,¨ and Hornik (2003). The dating procedure of Bai and Perron (2003) employs a dynamic programming algorithm based on the Bellman principle for finding those m breakpoints that minimize the residual sum of squares (RSS) of a model with m + 1 segments, given some minimal segment size of h · n observations. Here, h is a bandwidth parameter to be chosen by the user. Similar to the choice of trimming for the F statistics-based tests, the minimal proportion of observations in each segment is typically chosen to be 10% or 15%. Given h and m, the breakpoints minimizing the RSS can be determined; however, typically the number of breakpoints m is not known in advance. One possibility is to compute the optimal breakpoints for m = 0, 1, . . . breaks and choose the model that minimizes some information criterion such as the BIC. This model selection strategy is also directly available within breakpoints().

Returning to the UKDriverDeaths series, we estimate the breakpoints for a SARIMA-type model with a minimal segment size of 10% using

R> dd_bp <- breakpoints(dd ~ dd1 + dd12, data = dd_dat, h = 0.1)

The RSS and BIC as displayed by plot(dd_bp) are shown in the left panel of Figure 6.12. Although the RSS drops clearly up to m = 3 breaks, the BIC is minimal for m = 0 breaks. This is not very satisfactory, as the structural change tests clearly showed that the model parameters are not stable. As the BIC was found to be somewhat unreliable for autoregressive models by Bai and Perron (2003), we rely on the interpretation from the visualization of the structural change tests and use the model with m = 2 breaks. Its coe cients can be extracted via

R> coef(dd_bp, breaks = 2)

(Intercept) dd1 dd12 1970(1) - 1973(10) 1.45776 0.117323 0.694480 1973(11) - 1983(1) 1.53421 0.218214 0.572330 1983(2) - 1984(12) 1.68690 0.548609 0.214166

reflecting that particularly the period after the change in seatbelt legislation in 1983(1) is di erent from the observations before. The other breakpoint is in 1973(10), again matching the timing of the oil crisis and confirming the interpretation from the structural change tests. The observed and fitted series, along with confidence intervals for the breakpoints, are shown in the right panel of Figure 6.12 as generated by

R> plot(dd)

R> lines(fitted(dd_bp, breaks = 2), col = 4)

R> lines(confint(dd_bp, breaks = 2))

Readers are asked to estimate breakpoints for the GermanM1 example in an exercise.

Table 6.3. Further packages for time series analysis.

Package Description

dse Multivariate time series modeling with state-space and vector ARMA (VARMA) models (Gilbert 2007).

FinTS R companion to Tsay (2005) with data sets, functions, and script files to work some of the examples (Graves 2008).

forecast Univariate time series forecasting, including exponential smoothing, state space, and ARIMA models. Part of the forecasting bundle (Hyndman and Khandakar 2008).

fracdi ML estimation of fractionally integrated ARMA (ARFIMA) models and semiparametric estimation of the fractional di erencing parameter (Fraley, Leisch, and Maechler 2006).

longmemo Convenience functions for long-memory models; also contains several data sets (Beran, Whitcher, and Maechler 2007).

mFilter Miscellaneous time series filters, including Baxter-King, Butterworth, and Hodrick-Prescott (Balcilar 2007).

Rmetrics Suite of some 20 packages for financial engineering and computational finance (Wuertz 2008), including GARCH modeling in the package fGarch.

tsDyn Nonlinear time series models: STAR, ESTAR, LSTAR (Di Narzo and Aznarte 2008).

vars (Structural) vector autoregressive (VAR) models (Pfa 2008).

All error terms are assumed to be mutually independent. Special cases are the local linear trend model, where γt is absent, and the local linear model, where in addition σ2 = 0.

In total, there are four parameters, σ2, σ2, σ!2 , and σ"2, some (but not all) of which may be absent (and often are in practical applications).

It should be noted that, for example, the reduced form of the local trend model is ARIMA(0,2,2), but with restrictions on the parameter set. Proponents of structural time series models argue that the implied restrictions often are meaningful in practical terms and thus lead to models that are easier to interpret than results from unrestricted ARIMA fits.

Here, we fit the basic structural model to the UKDriverDeaths data using

R> dd_struct <- StructTS(log(UKDriverDeaths))

178 6 Time Series

Time

Fig. 6.13. The basic structural model for the UK driver deaths.

The resulting components

R> plot(cbind(fitted(dd_struct), residuals(dd_struct)))

are shown in Figure 6.13. This approach also clearly brings out the drop in the number of accidents in connection with the change in legislation.

More information on structural time series models in R is provided by Ripley (2002) and Venables and Ripley (2002).

GARCH models

Many financial time series exhibit volatility clustering. Figure 6.14 provides a typical example, a series of 1974 DEM/GBP exchange rate returns for the

Fig. 6.14. DEM/GBP exchange rate returns.

period 1984-01-03 through 1991-12-31, taken from Bollerslev and Ghysels (1996). This data set has recently become a benchmark in the GARCH literature, and it is also used in Greene (2003, Chapter 11). A Ljung-Box test of the MarkPound series suggests that it is white noise (note that this is not without problems since this test assumes i.i.d. data under the null hypothesis), while a test of the squares is highly significant. Thus, a GARCH model might be appropriate.

The package tseries provides a function garch() for the fitting of GARCH(p, q) models with Gaussian innovations, defaulting to the popular GARCH(1, 1)

yt = σt t, t N (0, 1) i.i.d.,

σt2 = ! + yt2−1 + βσt2−1, ! > 0, > 0, β ≥ 0.

For the exchange rate data, we obtain

R> mp <- garch(MarkPound, grad = "numerical", trace = FALSE) R> summary(mp)

Call:

garch(x = MarkPound, grad = "numerical", trace = FALSE)

Model:

GARCH(1,1)

Diagnostic Tests:

Jarque Bera Test

data: Residuals

X-squared = 1060.01, df = 2, p-value < 2.2e-16

Box-Ljung test

data: Squared.Residuals

X-squared = 2.4776, df = 1, p-value = 0.1155

which gives ML estimates along with outer-product-of-gradients (OPG) standard errors and also reports diagnostic tests of the residuals for normality (rejected) and independence (not rejected). Numerical (rather than the default analytical) gradients are employed in the garch() call because the resulting maximized log-likelihood is slightly larger. For brevity, trace = FALSE suppresses convergence diagnostics, but we encourage readers to reenable this option in applications.

More elaborate tools for volatility modeling are available in the Rmetrics collection of packages (Wuertz 2008). We mention the function garchFit() from the package fGarch, which includes, among other features, a wider selection of innovation distributions in GARCH specifications.

6.6 Exercises

1.Decompose the UKNonDurables data. Filter these data using the HoltWinters technique.

2.Repeat Exercise 1 for the DutchSales data. Compare a decomposition via decompose() with a decomposition via stl().

3.Using the AirPassengers series,

•filter the data using Holt-Winters smoothing,

•fit the airline model, and

• fit the basic structural model. Compare the results.

4.Reread Nelson and Plosser (1982) and test a selection of the extended Nelson-Plosser data, available as NelPlo in the package tseries, for unit roots. Also fit ARIMA models to these series. (You may need the xreg argument to arima() for series that appear to be trend stationary.)

5.Compute and plot the coe cients of the implied MA(1) representations (i.e., the impulse response functions) for (the stationary part of) the models fitted in the preceding exercise.

Hint: Use ARMAtoMA().

6.Stock and Watson (2007) consider an autoregressive distributed lag model for the change in the US inflation rate using the USMacroSW data. Specifically, the model is

Plot the sequence of F statistics for a single structural break for this ADL(4, 4) model using Fstats() and test for structural changes with the supF test.

7.Apply the Bai and Perron (2003) dating algorithm to the German M1 data. Do the results correspond to the German monetary reunion?

8.Fit the basic structural model to the UKNonDurables data. Compare this with the ARIMA model fitted in Section 6.2.