Lectures on Structural Change

Transcription

1 Lectures on Structural Change Eric Zivot Department of Economics, University of Washington April5, Overview of Testing for and Estimating Structural Change in Econometric Models 1. Day 1: Tests of Parameter Constancy 2. Day 2: Estimation of Models with Structural Change 3. Day 3: Time Varying Parameter Models 2 Some Preliminary Asymptotic Theory Reference: Stock, J.H. (1994) Unit Roots, Structural Breaks and Trends, in Handbook of Econometrics, Vol. IV. 3 Tests of Parameter Constancy in Linear Models 3.1 Motivation Diagnostics for model adequacy Provide information about out-of-sample forecasting accuracy Within-sample parameter constancy is a necessary condition for superexogeneity 3.2 Example Data Sets Simulated Data Consider the linear regression model y t = α + βx t + ε t,,...,t = 200 x t iid N(0, 1) ε t iid N(0,σ 2 ) 1

2 No structural change parameterization: α = 0,β = 1,σ = 0.5 Structural change cases Break in intercept: α =1fort>100 Break in slope: β =3fort>100 Break in error variance: σ =0.25 for t>100 Random walk in slope: β = β t = β t 1 +η t,η t iid N(0, 0.1) and β 0 =1. (show simulated data) US/DM Monthly Exchange rate data Let s t = log of spot exchange rate in month t f t = log of forward exchange rate in month t The forward rate unbiased hypothesis is typically investigated using the so-called differences regression s t+1 = α + β(f t s t )+ε t+1 f t s t = i US t i DM t = forward discount If the forward rate f t is an unbiased forecast of the future spot rate s t+1 then we should find α =0andβ =1 The forward discount is often modeled as an AR(1) model Statistical Issues f t s t = δ + φ(f t 1 s t 1 )+u t s t+1 is close to random walk with large variance f t s t behaves like highly persistent AR(1) with small variance f t s t appears to be unstable over time 3.3 Chow Forecast Test Reference: Chow, G.C. (1960). Tests of Equality between Sets of Coefficients in Two Linear Regressions, Econometrica, 52, Consider the linear regression model with k variables y t = x 0 tβ + u t,u t (0,σ 2 ),,...,n y = Xβ + u 2

3 Parameter constancy hypothesis Intuition H 0 : β is constant If parameters are constant then out-of-sample forecasts should be unbiased (forecast errors have mean zero) Test construction: Split sample into n 1 >kand n 2 = n n 1 observations y = µ y1 y 2 n1 n 2, X = µ X1 X 2 n1 n 2 Fit model using first n 1 observations ˆβ 1 = (X 0 1X 1 ) 1 X 0 1y 1 û 1 = y 1 X 1ˆβ 1 ˆσ 2 1 = û 0 1û 1 /(n 1 k) Use ˆβ 1 and X 2 to predict y 2 using next n 2 observations ŷ 2 = X 2ˆβ1 Compute out-of-sample prediction errors Under H 0 : β is constant and û 2 = y 2 ŷ 2 = y 2 X 2ˆβ 1 û 2 = u 2 X 2 (ˆβ 1 β) E[û 2 ] = 0 ³ var(û 2 ) = σ 2 I n2 + X 2 (X 0 1 X 1) 1 X 0 2 Further, If the errors u are Gaussian then û 2 N(0,var(û 2 )) û 0 2var(û 2 ) 1 û 2 χ 2 (n 2 ) (n 1 k)ˆσ 2 1/σ 2 χ 2 (n 1 k) This motivates the Chow forecast test statistic ³ û 0 2 I n2 + X 2 (X 0 1 X 1) 1 X 0 2 û 2 Chow FCST (n 2 )= n 2ˆσ F (n 2,n 1 k)

4 Decision: Reject H 0 at 5% level if Remarks: Chow FCST (n 2 ) >cv 0.05 Test is a general specification test for unbiased forecasts Popular with LSE methodology Implementation requires apriorisplitting of data into fit and forecast samples Application: Simulated Data Chow Forecast Test n 2 Model No SC Mean shift 9.130*** 1.329* Slope shift 9.055*** 2.067*** 1.545* Var shift RW slope 2.183*** CUSUM and CUSUMSQ Tests Reference: Brown, R.L., J. Durbin and J.M. Evans (1975). Techniques for Testing the Constancy of Regression Relationships over Time, Journal of the Royal Statistical Society, Series B, 35, Recursive least squares estimation The recursive least squares (RLS) estimates of β are based on estimating y t = β 0 tx t + ξ t,,...,n by least squares recursively for t = k +1,...,n giving n k least squares (RLS) estimates (ˆβ k+1,...,ˆβ T ). RLS estimates may be efficiently computed using the Kalman Filter If β is constant over time then ˆβ t should quickly settle down near a common value. 4

5 3.4.2 Recursive residuals Formal tests for structural stability of the regression coefficients may be computed from the standardized 1 step ahead recursive residuals Intuition: w t = v t ft = y t ˆβ 0 t 1x t ft f t = ˆσ 2 h 1+x 0 t(x 0 t X t) 1 x t i If β i changes in the next period then the forecast error will not have mean zero w t are recursive Chow Forecast t-statistics with n 2 = CUSUM statistic The CUSUM statistic of Brown, Durbin and Evans (1975) is CUSUM t = ˆσ 2 w = tx j=k+1 1 n k ŵ j ˆσ w (w t w) 2 Under the null hypothesis that β is constant, CUSUM t has mean zero and variance that is proportional to t k CUSUMSQ statistic THE CUSUMSQ statistic is CUSUMSQ t = P t j=k+1 ŵ2 j P n j=k+1 ŵ2 j Under the null that β is constant, CUSUMSQ t behaves like a χ 2 (t) andconfidence bounds can be easily derived Application: Simulated Data Remarks (insert graphs here) Ploberger and Kramer (1990) show the CUSUM test can be constructed with OLS residuals instead of recursive residuals 5

6 CUSUM Test is essentially a test to detect instability in intercept alone CUSUM Test has power only in direction of the mean regressors CUSUMSQ has power for changing variance There are tests with better power Application: Exchange Rate Regression (insert graphs here) 3.5 Nyblom s Parameter Stability Test Reference:Nyblom, J. (1989). Testing for the Constancy of Parameters Over Time, Journal of the American Statistical Association, 84 (405), Consider the linear regression model with k variables y t = x 0 tβ + ε t,,...,n The time varying parameter (TVP) alternative model assumes Thehypothesesofinterestare β = β t = β t 1 + η t,η it (0,σ 2 η i ),i=1,...,k H 0 : β is constant σ 2 η i =0foralli H 1 : σ 2 η i > 0forsomei Nyblom (1989) derives the locally best invariant test as the Lagrange multiplier test. The score assuming Gaussian errors is x tˆε t = 0 Define ˆε t = y t x 0 tˆβ ˆβ= (X 0 X) 1 X 0 y f t = x tˆε t tx S t = f t = cumulative sums j=1 V = n 1 X 0 X Note that f t = 0 j=1 6

7 Nyblom derives the LM statistic L = = 1 nˆσ 2 S t V 1 S t 1 nˆσ 2 tr " V 1 n X S t S 0 t # Under mild assumptions regarding the behavior of the regressors, the limiting distribution of L under the null is a Camer-von Mises distribution: L Z 1 0 B μ k (λ)bμ k (λ)0 dλ B μ k (λ) = W k(λ) λw k (1) W k (λ) = k dimensional Brownian motion Decision: Reject H 0 at 5% level if Remarks: L>cv 0.05 Distribution of L is non-standard and depends on k. Critical values are computed by simulation and are given in Nyblom, Hansen (1992) and Hansen (1997) Test is for constancy of all parameters Test is not informative about the date or type of structural change Test is applicable for models estimated by methods other than OLS Distribution of L is different if x t is non-stationary (unit root, deterministic trend). See Hansen (1992) Application: Simulated Data Nyblom Test Model L c No SC.332 Mean shift Slope shift var shift.351 RW slope

8 3.5.2 Application: Exchange rate regression Nyblom Test Model L c AR(1) 1.27 Diff reg Hansen s Parameter Stability Tests References 1. Hansen, B.E. (1992). Testing for Parameter Instability in Linear Models Journal of Policy Modeling, 14(4), Hansen, B.E. (1992). Tests for Parameter Instability in Regressions with I(1) Processes, Journal of Business and Economic Statistics, 10, Idea: Extension of Nyblom s LM test to individual coefficients. Under the null of constant parameters, the score vector from the linear model with Gaussian errors is x itˆε t = 0,i=1,...,k X (ˆε 2 t ˆσ 2 ) = 0 ˆε t = y t x 0 tˆβ ˆσ 2 = X n n 1 ˆε 2 t Define f it = S it = ½ xitˆε t ˆε 2 t ˆσ 2 i =1,...k i = k +1 tx f ij,i=1,...,k+1 j=1 Note that f it =0, i =1,...,k+1 i= Individual Coefficient Tests Hansen s LM test for H 0 : β i is constant, i =1,...,k 8

9 and for is L i = V i = H 0 : σ 2 is constant 1 nv i Sit, 2 i=1,...,k fit 2 Under H 0 : β i is constant or H 0 : σ 2 is constant L i Z 1 0 B μ 1 (λ)bμ 1 (λ)dλ Decision: Reject H 0 at 5% level if L i >cv 0.05 = Joint Test for All Coefficients For testing the joint hypothesis H 0 : β and σ 2 are constant define the (k +1) 1 vectors f t = (f 1t,...,f k+1,t ) 0 S t = (S 1t,...,S k+1,t ) 0 Hansen s LM statistic for testing the constancy of all parameters is à L c = 1 S 0 n tv 1 S t = 1 n! n tr X V 1 S t S 0 t V = f t ft 0 Under the null of no-structural change L c Z 1 0 B μ k+1 (λ)bμ k+1 (λ)dλ Decision: Reject H 0 at 5% level if L c >cv 0.05 Remarks 9

10 Tests are very easy to compute and are robust to heteroskedasticity Null distribution is non-standard and depends upon number of parameters tested for stability Individual tests are informative about the type of structural change Tests are not informative about the date of structural change Hansen s L 1 test for constancy of intercept is analogous to the CUSUM test Hansen s L k+1 test for constancy of variance is analogous to CUSUMSQ test Hansen s L c test for constancy of all parameters is similar to Nybolom s test Distribution of tests is different if data are nonstationary (unit root, deterministic trend) - see Hansen (1992), JBES Application: Simulated Data Hansen Tests Model α β σ 2 Joint No SC Mean shift Slope shift var shift RW slope Application: Exchange rate regression cont d Hansen Tests Model intercept slope variance Joint AR(1) Diff reg Tests for Single Structural Change Consider the linear regression model with k variables No structural change null hypothesis y t = x 0 tβ t + ε t,...,n H 0 : β t = β 10

11 Single break date alternative hypothesis H 1 : ½ βt = β, t m =breakdate β t = β + γ, t>mand γ 6= 0 k < m < n k λ = m = break fraction n Remarks: Under no break null γ = 0. Pure structural change model: all coefficients change (γ i 6= 0 for i = 1,...,k) Partial structural change model: some coefficients change (γ i 6=0forsome i) m =[λ n], [ ] =integerpart 4.1 Chow s Test with Known Break Date Assume: m or λ is known For a data interval [r,...,s]suchthats r>kdefine ˆβ r,s = OLS estimate of β ˆε r,s = OLS residual vector SSR r,s = ˆε 0 r,sˆε r,s = sum of squared residuals Chow s breakpoint test for testing H 0 vs. H 1 with m known is ³ m F n = F n (λ) = (SSR 1,n (SSR 1,m + SSR m+1,n ))/k n (SSR 1,m + SSR m+1,n ) /(n 2k) The Chow test may also be computed as the F-statistic for testing γ = 0 from the dummy variable regression Under H 0 : γ = 0 with m known Decision: Reject H 0 at 5% level if y t = x 0 tβ + D t (m)x 0 tγ + ε t D t (m) = 1 if t>m;0 otherwise F n (λ) F (k, n 2k) k F n (λ) d χ 2 (k) F n (λ) > F 0.95 (k, n k) k F n (λ) > χ (k) 11

12 4.1.1 Application: Simulated Data Chow Breakpoint Test F 200 (0.5) F 200 (0.25) F 200 (0.75) No SC Mean shift 377*** 13.03*** 11.21*** Slope shift 374*** 10.97*** 17.57*** Var shift RW slope 80.14*** 4.058** Quandt s LR Test with Unknown Break Date References: 1. Quandt, R.E. (1960). Tests of Hypotheses that a Linear System Obeys Two Separate Regimes, Journal of the American Statistical Association, 55, Davies, R.A. (1977). Hypothesis Testing When a Nuisance Parameter is Present only Under the Alternative, Biometrika, 64, Kim, H.-J., and D. Siegmund (1989). The Likelihood Ratio Test for a Change-Point in Simple Linear Regression, Biometrika, 76, 3, Andrews, D.W.K. (1993). Tests for Parameter Instability and Structural Change with Unknown Change Point, Econometrica, 59, Hansen, B.E. (1997). Approximate Asymptotic P Values for Structural- Change Tests, Journal of Business and Economic Statistics, 15, Assume: m or λ is unknown. Quandt considered the LR statistic for testing H 0 : γ = 0 vs. H 1 : γ 6= 0 when m is unknown. This turns out to be the maximal F n (λ) statisticovera rangeofbreakdatesm 0,...,m 1 : Remarks ³ m QLR = max F n m [m 0,m 1] n λ i = m i n = max λ [λ 0,λ 1] F n(λ) = trimming parameters, i=0, 1 QLR is also know as Andrews sup F statistic Trimming parameters λ 0 and λ 1 must be set Cannot have λ 0 =1andλ 1 = 1 because breaks are hard to identify near beginning and end of sample Information about location of break can be used to specify λ 0 and λ 1 12

13 Andrews recommends λ 0 =0.15 and λ 1 =0.85 if there is no knowledge of break date Implicitly, the break data m and break fraction λ are estimated using ³ m ˆm = argmax F n m n ˆλ = ˆm/n Under the null, m defined under the alternative is not identified. This is an example of the Davies problem. Davies (1977) showed that if estimated parameters are unidentified under the null, standard χ 2 inference does not obtain. Under H 0 : γ = 0, Kim and Siegmund (1989) showed B μ k k QLR sup (λ)0 B μ k (λ) λ [λ 0,λ 1 ] λ(1 λ) B μ k (λ) = W k(λ) λw k (1) = Brownian Bridge Decision: Reject H 0 at 5% level if k QLR > cv 0.05 Remarks Distribution of QLR is non-standard and depends on the number of variables k and the trimming parameters λ 0 and λ 1 Critical values for various values of λ 0 and λ 1 computed by simulation are given in Andrews (1993), and are larger than χ 2 (k) criticalvalues. For λ 0 =0.15 and λ 1 =0.85 5% critical values k χ 2 (k) QLR k QLR P-values can be computed using techniques from Hansen (1997) Graphical plot of F n (λ) statistics is informative to locate the break date Application: Simulated data QLR or sup-f Test QLR bm λ b No SC Mean shift 377*** Slope shift 374*** Var shift RW slope 113*** (insert graphs here) 13

14 4.2.2 Application: Exchange rate data QLR or sup-f Test QLR bm λ b AR(1) 12.13*** 1989: Diff reg : Optimal Tests with Unknown Break Date References: 1. Andrews, D.W.K. and W. Ploberger (1994). Optimal Tests When a Nuisance Parameter Is Present Only Under the Alternative, Econometrica, 62, Andrews and Ploberger (1994) derive tests for structural change with an unknown break date with optimal power. These tests turn out to be weighted averages of the Chow breakpoint statistics F n ( m n ) used to compute the QLR statistic: Ã 1 Xm 2 µ µ! 1 t ExpF n = ln exp m 2 m k F n n t=m 1 1 Xm 2 µ t AveF n = k F n m 2 m 1 +1 n t=m 1 k = number of regressors being tested Remarks Asymptotic null distributions are non-standard and depend on k, λ 0 and λ 1 Critical values are given in Andrews and Ploberger; P-values can be computed using techniques of Hansen (1997) Tests can have higher power than QLR statistic Tests are not informative about location of break date 4.4 Empirical Application Reference: Stock and Watson (199?), Journal of Business and Economic Statistics. 14