Monitoring Structural Change in Dynamic Econometric Models
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1 Monitoring Structural Change in Dynamic Econometric Models Achim Zeileis Friedrich Leisch Christian Kleiber Kurt Hornik
2 Contents Model frame Generalized fluctuation tests OLS-based processes Rescaling of estimates-based processes Boundaries Applications German M1 money demand U.S. labor productivity Software
3 Model frame Consider the linear regression model in a monitoring situation y i = x i β i + u i (i = 1,..., n,...). Technical assumptions: lim sup n 1 n ni=1 x i 2+δ <, for some δ > 0. 1 n ni=1 x i x i p Q; Q finite, regular, nonstochastic. {u i } is a homoskedastic martingale difference sequence.
4 Model frame Basic assumption: The regression relationship is stable (β i = β 0 ) during the history period i = 1,..., n. Null hypothesis: H 0 : β i = β 0 (i > n). Alternative: H 1 : β i β 0 for some i > n.
5 Generalized fluctuation tests The generalized fluctuation test framework includes formal significance tests but its philosophy is basically that of data analysis as expounded by Tukey. Essentially, the techniques are designed to bring out departures from constancy in a graphic way instead of parametrizing particular types of departure in advance and then developing formal significance tests intended to have high power against these particular alternatives. (Brown, Durbin, Evans, 1975)
6 Generalized fluctuation tests empirical fluctuation processes reflect fluctuation in residuals coefficient estimates theoretical limiting process is known choose boundaries which are crossed by the limiting process only with a known probability α. if the empirical fluctuation process crosses the theoretical boundaries the fluctuation is improbably large reject the null hypothesis.
7 Generalized fluctuation tests Chu, Stinchcombe, White (1996) Extension of fluctuation tests to the monitoring situation: processes based on recursive estimates and recursive residuals. Leisch, Hornik, Kuan (2000) Generalized framework for estimates-based tests for monitoring. Contains the test of Chu et al., and considered in particular moving estimates.
8 Generalized fluctuation tests Processes based on estimates: ˆβ (i) = ( X (i) X (i) ) 1 X(i) y (i) Recursive estimates (RE) process: Y n (t) = i ˆσ n Q 1 2(n) where i = k + t(n k) and t 0. (ˆβ (i) ˆβ (n)),
9 Generalized fluctuation tests Processes based on estimates: ˆβ (i) = ( X (i) X (i) ) 1 X(i) y (i) Recursive estimates (RE) process: Y n (t) = i ˆσ n Q 1 2(n) where i = k + t(n k) and t 0. (ˆβ (i) ˆβ (n)), Moving estimates (ME) process: where t h. Z n (t h) = nh ˆσ n Q 1 2(n) (ˆβ ( nt nh, nh ) ˆβ (n)),
10 Generalized fluctuation tests Limiting processes: bridge. (increments of a) k-dimensional Brownian Boundaries: RE: b(t) = ME: c(t) = λ t(t 1) log + t [ ( )] λ 2 t + log t 1 Check for crossings in the monitoring period 1 < t < T. Significance level is determined by λ.
![ME: c(t) = λ t(t 1) log + t [ ( )] λ 2 t + log t 1 Check for](/docs-images/41/6105378/images/page_10.jpg "crossings in the monitoring period 1 < t < T.")
11 OLS-based processes Processes based on OLS residuals: û i = y i x i ˆβ (n) OLS-based CUSUM process: W 0 n (t) = 1 ˆσ n nt i=1 û i (t 0).
12 OLS-based processes Processes based on OLS residuals: û i = y i x i ˆβ (n) OLS-based CUSUM process: W 0 n (t) = 1 ˆσ n nt i=1 û i (t 0). OLS-based MOSUM process: M 0 n(t h) = 1 ˆσ n ηt û i i= ηt nh +1 (t h).
13 OLS-based processes Limiting processes: bridge. (increments of a) 1-dimensional Brownian the same boundaries can be used. Advantages: easy to interpret, easy to compute.
14 Rescaling Kuan & Chen (1994): Empirical size of (historical) estimates-based tests can be seriously distorted in dynamic models if the whole inverse sample covariance matrix estimate Q (n) = 1/n X (n) X (n) is used to scale the fluctuation process. Improvement: use Q (i) instead. In a monitoring situation rescaling cannot improve the size of the RE test but it does so for the ME test!
15 Rescaling Example: AR(1) process with ϱ = 0.9 but without a shift: rescaled not rescaled empirical fluctuation process empirical fluctuation process Time Time
16 German M1 money demand Lütkepohl, Teräsvirta, Wolters (1999) investigate the linearity and stability of German M1 money demand: stable regression relation for the time before the monetary unification on but a clear structural instability afterwards. Data: seasonally unadjusted quarterly data, 1961(1) to 1995(4) Error Correction Model (in logs) with variables: M1 (real, per capita) m t, price index p t, GNP (real, per capita) y t and long-run interest rate R t : m t = 0.30 y t R t 1.00 R t p t 0.12m t y t R t Q Q2 0.11Q3 + û t,
17 German M1 money demand Historical residual-based tests...do not discover shift: Standard CUSUM test OLS based CUSUM test empirical fluctuation process empirical fluctuation process Time Time The shift has an estimated angle of
18 German M1 money demand Historical estimates-based tests discover shift ex post: RE test ME test empirical fluctuation process empirical fluctuation process Time Time
19 German M1 money demand Monitoring discovers shift online: Monitoring with OLS based CUSUM test empirical fluctuation process Time
20 German M1 money demand Monitoring discovers shift online: Monitoring with ME test empirical fluctuation process Time
21 Software All methods implemented in the R system for statistical computing and graphics in the contributed package strucchange. Both are available under the GPL (General Public Licence) from the Comprehensive R Archive Network (CRAN):
22 Software Further functionality: Historical tests: Generalized fluctuation tests (Kuan & Hornik): CUSUM, MOSUM, RE, ME, Nyblom-Hansen, F tests (Andrews & Ploberger): supf, avef, expf. Dating breaks (Bai & Perron): breakpoint estimation, confidence intervals.
23 Software Documented in: A. Zeileis, F. Leisch, K. Hornik, C. Kleiber (2002), strucchange: An R Package for Testing for Structural Change in Linear Regression Models, Journal of Statistical Software, 7(2), A. Zeileis, C. Kleiber, W. Krämer, K. Hornik (2003), Testing and Dating Structural Changes in Practice, Computational Statistics & Data Analysis, forthcoming.
24 Software R> LTW.model <- dm ~ dy2 + dr + dr1 + dp + m1 + y1 + R1 + season R> re <- efp(ltw.model, type = "RE", data = GermanM1) R> plot(re)
25 Software R> LTW.model <- dm ~ dy2 + dr + dr1 + dp + m1 + y1 + R1 + season R> re <- efp(ltw.model, type = "RE", data = GermanM1) R> plot(re) Fluctuation test (recursive estimates test) empirical fluctuation process Time
26 Software R> sctest(re) Fluctuation test (recursive estimates test) data: re FL = , p-value =
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