Threshold Autoregressions and NonLinear Autoregressions

Transcription

1 Threshold Autoregressions and NonLinear Autoregressions Original Presentation: Central Bank of Chile October 29-31, 2013 Bruce Hansen (University of Wisconsin) Threshold Regression 1 / 47

2 Threshold Models A type of nonlinear time series models Strong nonlinearity Allows for switching effects Most typically univariate (for simplicity) Bruce Hansen (University of Wisconsin) Threshold Regression 2 / 47

3 Threshold Models Threshold Variable q t q t = 100(log(GDP t ) log(gdp t 4 )) = annual growth Threshold γ Split regression Coeffi cients switch if q t γ or q t > γ If growth has been above or below the threshold y t+1 = β 1 x t1 (q t γ) + β 2 x t1 (q t > γ) + e t+1 { β = 1 x t + e t q t γ β 2 x t + e t q t > γ Bruce Hansen (University of Wisconsin) Threshold Regression 3 / 47

4 Partial Threshold Model y t+1 = β 0 z t + β 1 x t1 (q t γ) + β 2 x t1 (q t > γ) + e t+1 Coeffi cients on z t do not switch More parsimonious Bruce Hansen (University of Wisconsin) Threshold Regression 4 / 47

5 Estimation y t+1 = β 0 z t + β 1 x t1 (q t γ) + β 2 x t1 (q t > γ) + e t+1 Least Squares ( β 0, β 1, β 2, γ) minimize sum-of-squared errors Equation is non-linear, so NLLS, not OLS Simple to compute by concentration method Given γ, model is linear in β Regressors are z t, x t 1 (q t γ) and x t 1 (q t > γ) Estimate by least-squares Save residuals, sum of squared errors Repeat for all thresholds γ. Find value which minimizes SSE Bruce Hansen (University of Wisconsin) Threshold Regression 5 / 47

6 Estimation Details For a grid on γ (can use sample values of q t ) Define dummy variables d 1t (γ) = 1 (q t γ) and d 2t (γ) = 1 (q t > γ) Define interaction variables x 1t (γ) = x t d 1t (γ) and x 2t (γ) = x t d 2t (γ) Regress y t+1 on z t, x 1t (γ), x 2t (γ) Sum of squared errors y t+1 = β 0 z t + β 1 x 1t (γ) + β 2 x 2t (γ) + ê t+1 (γ) S(γ) = n ê t+1 (γ) 2 t=1 Write this explicity as a function of γ as the estimates, residuals and SSE vary with γ Find ˆγ which minimizes S(γ) Useful to view plot of S(γ) against γ Given ˆγ, repeat above steps to find estimates ( β 0, β 1, β 2 ) Forecasts made from fitted model Bruce Hansen (University of Wisconsin) Threshold Regression 6 / 47

7 Example: GDP Forecasting Equation q t = 100(log(GDP t ) log(gdp t 4 )) = annual growth Threshold estimate: ˆγ = 0.18 Splits regression depend if past year s growth is above or below 0.18% 0% Bruce Hansen (University of Wisconsin) Threshold Regression 7 / 47

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9 Multi-Step Forecasts Nonlinear models (including threshold models) do not have simple iteration method for multi-step forecasts Option 1: Specify direct threshold model Option 2: Iterate one-step threshold model by simulation: Bruce Hansen (University of Wisconsin) Threshold Regression 9 / 47

10 Multi-Step Simulation Method Take fitted model y t+1 = β 0 z t + β 1 x t1 (q t ˆγ) + β 2 x t1 (q t > ˆγ) + ê t+1 Draw iid errors ê n+1,..., ê n+h from the residuals {ê 1,..., ê n } Create yn+1 (b), y n+2 (b),..., y n+h (b) forward by simulation b indexes the simulation run Repeat B times (a large number) {yn+h (b) : b = 1,..., B} constitute an iid sample from the forecast distribution for y n+h Point forecast f n+h = 1 B B b=1 y n+h (b) Interval forecast: α and 1 α quantiles of y n+h (b) Bruce Hansen (University of Wisconsin) Threshold Regression 10 / 47

11 Testing for a Threshold Null hypothesis: No threshold (linearity) Null Model: No threshold Alternative: Single Threshold y t+1 = β 0 z t + β 1 x t + ê t+1 S 0 = n êt+1 2 t=1 y t+1 = β 0 z t + β 1 x 1t(γ) + β 2 x 2t(γ) + ê t+1 (γ) S 1 (γ) = n ê t+1 (γ) 2 t=1 S 1 = S 1 ( ˆγ) = min S 1 (γ) γ Bruce Hansen (University of Wisconsin) Threshold Regression 11 / 47

12 NonStandard Testing Test is non-standard. Similar to Structural Change Tests Examine all tests for each fixed threshold critical values & p-values obtained by simulation or bootstrap Plot sequence of tests; Reject if time plot exceeds critical value Bruce Hansen (University of Wisconsin) Threshold Regression 12 / 47

13 Example: GDP Forecasting Equation q t = 100(log(GDP t ) log(gdp t 4 )) = annual growth Bootstrap p-value for threshold effect: 10.6% Bruce Hansen (University of Wisconsin) Threshold Regression 13 / 47

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15 Inference in Threshold Models Threshold estimate has nonstandard distribution Confidence intervals by inverting statistic constructed from sum-of-squares ( ) S1 (γ) S 1 LR(γ) = n Theory: [Hansen, 2000] LR(γ) d ξ P(ξ x) = ( 1 e x /2) 2 Critical values: S 1 P(ξ c) c Bruce Hansen (University of Wisconsin) Threshold Regression 15 / 47

16 Confidence Intervals for Threshold All γ such that LR(γ) c where c is critical value Easy to see in graph of LR(γ) against γ Bruce Hansen (University of Wisconsin) Threshold Regression 16 / 47

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18 Threshold Estimates Estimate: ˆγ = 0.18 Confidence Interval = [ 1.0, 2.2] Bruce Hansen (University of Wisconsin) Threshold Regression 18 / 47

19 Inference on Slope Parameters Conventional As if threshold is known Bruce Hansen (University of Wisconsin) Threshold Regression 19 / 47

20 Threshold Model Estimates q t = 100(log(GDP t ) log(gdp t 4 )) q t 0.18 q t > 0.18 Intercept (4.6) (1.11) log(gdp t ) 0.36 (0.21) 0.16 (0.08) log(gdp t 1 ) (0.21) 0.20 (0.09) Spread t 1.3 (0.8) 0.71 (0.20) Default Spread t (1.26) -2.3 (0.9) Housing Starts t 2.5 (10.6) 4.1 (2.3) Building Permits t 7.8 (10.5) -2.2 (2.0) Bruce Hansen (University of Wisconsin) Threshold Regression 20 / 47

21 NonParametric/NonLinear Autoregression Bruce Hansen (University of Wisconsin) Threshold Regression 21 / 47

22 NonParametric/NonLinear Time Series Regression Model y t+1 = g (x t ) + e t+1 E (e t+1 x t ) = 0 x t = (y t 1, y t 2,..., y t p ) or any other variables g (x t ) is arbitrary non-linear function Bruce Hansen (University of Wisconsin) Threshold Regression 22 / 47

23 Additively Separable Model x t = (x 1t,..., x pt ) g (x t ) = g 1 (x 1t ) + g 2 (x 2t ) + + g p (x pt ) Then y t+1 = g 1 (x 1t ) + g 2 (x 2t ) + + g p (x pt ) + e t+1 Greatly reduces degree of nonlinearity Bruce Hansen (University of Wisconsin) Threshold Regression 23 / 47

24 Partially Linear Model Partition x t = (x 1t, x 2t ) g (x t ) = g 1 (x 1t ) + β x 2t x 1t main variables of importance For example, if primary dependence through first lag y t+1 = g 1 (y t ) + β 1 y t β p y t p + e t+1 Bruce Hansen (University of Wisconsin) Threshold Regression 24 / 47

25 Sieve Models Approximate g(x) by a sequence g m (x), m = 1, 2,..., of increasing complexity Linear sieves g m (x) = Z m (x) β m where Z m (x) = (z 1m (x),..., z Km (x)) are nonlinear functions of x. Series : Z m (x) = (z 1 (x),..., z K (x)) Sieves : Z m (x) = (z 1m (x),..., z Km (x)) Bruce Hansen (University of Wisconsin) Threshold Regression 25 / 47

26 Polynomial (power series) z j (x) = x j gm(x) = p β j x j j=1 Stone-Weierstrass Theorem: Any continuous function g(x) can be arbitrarily well approximated on a compact set by a polynomial of suffi ciently high order For any ε > 0 there exists coeffi cients p and β j such that X Runge s phenomenon: sup g m (x) g(x) ε x X Polynomials can be poor at interpolation (can be erratic) Bruce Hansen (University of Wisconsin) Threshold Regression 26 / 47

27 Splines Piecewise smooth polynomials Join points are called knots Linear spline with one knot at τ β 00 + β 01 (x τ) g m (x) = β 10 + β 11 (x τ) x < τ x τ To enforce continuity, β 00 = β 10, g m (x) = β 0 + β 1 (x τ) + β 2 (x τ) 1 (x τ) or equivalently g m (x) = β 0 + β 1 x + β 2 (x τ) 1 (x τ) Bruce Hansen (University of Wisconsin) Threshold Regression 27 / 47

28 Quadratic Spline with One Knot β 00 + β 01 (x τ) + β 02 (x τ) 2 g m (x) = β 10 + β 11 (x τ) + β 12 (x τ) 2 x < τ x τ Continuous if β 00 = β 10 Continuous first derivative if β 01 = β 11 Imposing these constraints g m (x) = β 0 + β 1 x + β 2 x 2 + β 3 (x τ) 2 1 (x τ). Bruce Hansen (University of Wisconsin) Threshold Regression 28 / 47

29 Cubic Spline with One Knot g m (x) = β 0 + β 1 x + β 2 x 2 + β 3 x 3 + β 4 (x τ) 3 1 (x τ) Bruce Hansen (University of Wisconsin) Threshold Regression 29 / 47

30 General Case Knots at τ 1 < τ 2 < < τ N g m (x) = β 0 + p j=1 β j x j + N k=1 β p+k (x τ k ) p 1 (x τ k ) Bruce Hansen (University of Wisconsin) Threshold Regression 30 / 47

31 Uniform Approximation Stone-Weierstrass Theorem: Any continuous function g(x) can be arbitrarily well approximated on a compact set by a polynomial of suffi ciently high order For any ε > 0 there exists coeffi cients p and β j such that X Strengthened Form: sup g m (x) g(x) ε x X if the s th derivative of g(x) is continuous then the uniform approximation error satisfies sup g m (x) g(x) = O ( K α ) m x X where K m is the number of terms in g m (x) This holds for polynomials and splines Runge s phenomenon: Polynomials can be poor at interpolation (can be erratic) Bruce Hansen (University of Wisconsin) Threshold Regression 31 / 47

32 Illustration g(x) = x 1/4 (1 x) 1/2 Polynomials of order K = 3, K = 4, and K = 6 Cubic splines are quite similar Bruce Hansen (University of Wisconsin) Threshold Regression 32 / 47

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34 Runge s Phenomenon Bruce Hansen (University of Wisconsin) Threshold Regression 34 / 47

35 Placement of Knots If support of x is [0, 1], typical to set τ j = j/(n + 1) If support of x is [a, b], can set τ j = a + (b a)/(n + 1) Alternatively, can set τ j to equal the j/(n + 1) quantile of the distribution of x Bruce Hansen (University of Wisconsin) Threshold Regression 35 / 47

36 Estimation Fix number and location of knots Estimate coeffi cients by least-squares Quadratic spline y = β 0 + β 1 x + β 2 x 2 + N k=1 β 2+k (x τ k ) 2 1 (x τ k ) + e Linear model in x, x 2, (x τ 1 ) 2 1 (x τ 1 ),..., (x τ N ) 2 1 (x τ N ) Bruce Hansen (University of Wisconsin) Threshold Regression 36 / 47

37 Selection of Number of Knots Model selection Pick N to minimize AIC Or a similar criterion known as Cross-Validation (CV) Bruce Hansen (University of Wisconsin) Threshold Regression 37 / 47

38 Example: GDP Growth y t =GDP Growth x t =Housing Starts Partially Linear Model y t+1 = g(x t ) + β 1 y t 1 + β 2 y t 2 + e t+1 Polynomial Cubic Spline Bruce Hansen (University of Wisconsin) Threshold Regression 38 / 47

39 Model Selection Polynomial in Housing Starts p CV Cubic Spline in Housing Starts N CV Best fitting regression is quartic polynomial (p = 4) Cubic spline with 1 knot is close Bruce Hansen (University of Wisconsin) Threshold Regression 39 / 47

40 Polynomial=solid line Cubic Spline=dashed line Bruce Hansen (University of Wisconsin) Threshold Regression 40 / 47

41 Estimated Cubic Spline Knot=1.5 β s( β) Intercept 29 (8) y t 0.18 (0.08) y t (0.08) HS t 86 (26) HSt 2 79 (23) HSt 3 22 (6) (HS t 1.5) 2 1 (HS t > 1.5) 43 (13) Bruce Hansen (University of Wisconsin) Threshold Regression 41 / 47

42 New Example: Long and Short Rates Bi-variate model of Long (10-year) and short (3-month) bond rates Key variable: Spread: Long-Short R t =Long Rate r t =Short Rate Z t = R r r t =Spread Model R t+1 = α 0 + α p1 (L) R t + β p1 (L) r t + g 1 (Z t ) + e 1t r t+1 = γ 0 + γ p2 (L) R t + δ p2 (L) r t + g 2 (Z t ) + e 2t Bruce Hansen (University of Wisconsin) Threshold Regression 42 / 47

43 Model Selection Separately for each equation Long Rate and Short Rate Select over number of lags Number of spline terms for nonlinearity in Spread Bruce Hansen (University of Wisconsin) Threshold Regression 43 / 47

44 CV Selection: Long Rate Equation p = 0 p = 1 p = 2 p = 3 p = 4 p = 5 p = 6 Linear Quadratic Cubic Knot Knots Knots Selected Model: p = 6, Cubic spline with 1 knot at 1.53 Bruce Hansen (University of Wisconsin) Threshold Regression 44 / 47

45 CV Selection: Short Rate Equation p = 0 p = 1 p = 2 p = 3 p = 4 p = 5 p = 6 Linear Quadratic Cubic Knot Knots Knots Selected Model: p = 1, Cubic spline with 1 knot at 1.53 Bruce Hansen (University of Wisconsin) Threshold Regression 45 / 47

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47 Forecasting For h > 1, need to use forecast simulation Simulate R n+1, r n+1 forward using iid draws from residuals Create time paths Take means to estimate point forecasts Take quantiles to construct forecast intervals Bruce Hansen (University of Wisconsin) Threshold Regression 47 / 47