This document provides an introduction and overview of nonparametric predictive regression for modeling nonlinear and time-varying effects. It summarizes the following key points: 1) Nonparametric predictive regression is proposed to model relationships that may change over time using orthogonal series estimation and local smoothing. 2) A two-step estimation procedure is used, with orthogonal series estimation in the first step to reduce bias, followed by local smoothing in the second step. 3) Asymptotic properties of the estimators are derived, showing they are consistent and asymptotically normal under certain conditions, whether the predictors are stationary or nonstationary.

1. Nonparametric Predictive Regression
Presenter: Xintian Yu
University of North Carolina at Charlotte, USA
Nov. 02, 2016
1 / 36

2. Introduction
Nonlinear effects and time-varying effects exist widely in
financial markets.
Capital asset pricing model (CAPM)
Blume, M. (1975) suggested that beta coefficients change
over time
Fabozzi, F.J. Francis, J.C. (1978) revealed that many
stocks’ beta coefficients move randomly through time
rather than remain stable
Relationship between the electricity demand and other
variables such as the income, the real price of electricity and
temperature.
Chang, Y., Martinez-Chombo, E., (2003) found that this
relationship may change over time.
Yt = β(t) Xt + t , (1.1)
2 / 36

3. Introduction
Most variables in financial industry are nonstationary.
Granger, C.W.J. (1981) and Engle, R.F., Granger, C.W.J.
(1987) introduced cointegration models in 1990s.
Cointegrating parameters are constant.
Yt = γ(zt ) Xt + t (1.2)
Cai et al. (2009) considered (1.2)
The local linear estimation
A single value of bandwidth could not make the estimation
optimal in the sense of minimizing the asymptotic mean
square error
A two-step estimation procedure
Xiao,Z. (2009) considered (1.2)
Focused on inference procedures for both parameter
instability and the hypothesis of cointegration
3 / 36

4. Introduction
Application with models (1.1)-(1.2)
y: stock price of Morgan Stanley
x: S&P 500
z: log "difference" of 5 year daily Treasury bond yield rate
and 6 month daily Treasury bill yield rate
residual of linear regression model
y∗
: log of 5 year daily Treasury bond yield rate
x∗
: log of 6 month daily Treasury bill yield rate
4 / 36

5. 2004 2008 2012
−70−50−30
time
β0
2004 2008 2012
0.0400.0500.060
time
β1
−0.4 −0.2 0.0 0.2
−30−1001020
z
γ0
−0.4 −0.2 0.0 0.2
0.0050.0150.025 z
γ1
Figure: Top two functions are estimated from model (1.1), bottom two
functions are estimated from model (1.2)
5 / 36

6. 2004 2006 2008 2010 2012 2014 2016
−15−50510
time
Estimatedresidualfrommodel1.1
2004 2006 2008 2010 2012 2014 2016
−2001030
time
Estimatedresidualfrommodel1.2
Figure: Top picture shows the residual from model (1.1), bottom
picture shows the residual from model (1.2)
6 / 36

7. Introduction
Which model is better? Which effect is true? Do these two
effects exist in the relationship between the predictors and
response?
yi = β0(ti) + γ0(zi) + {β1(ti) + γ1(zi)}xi + εi. (1.3)
We fit model (1.3) with real data and compare the goodness of
fit with models (1.1) and (1.2) in last Chapter .
7 / 36

8. Our model
For i = 1, . . . , n,
yi = β0(ti) + γ0(zi) + {β1(ti) + γ1(zi)} xi + εi, (2.1)
where
β0(ti): 1 × 1 function, γ0(zi): 1 × 1 function
β1(ti): p × 1 functions, γ1(zi): p × 1 functions
xi: p × 1, vector, either I(0) or I(1)
ti: i/n, scalar
zi: scalar, I(0)
εi: E[εi|xi, zi] = 0, E[ε2
i |xi, zi] = δ2, a strictly α-mixing
stationary process
Assume that E[γ0(zi)] = E[γ1(zi)] = 0.
8 / 36

9. Step 1: Orthogonal series estimation
Let {pκ(·), κ = 1, 2, . . .} be a standard orthogonal basis for
smooth functions on [−1, 1] and satisfy that
1
−1 pκ(x)dx = 0,
and
1
−1
pk (x)pj(x) dx =
1 if k = j;
0 otherwise.
Pκ(t, z) = [1, p1(t), . . . , pκ(t), p1(z), . . . , pκ(z)]
Θκb = (θ0,b, θ11,b, . . . , θ1κ,b, θ21,b, . . . , θ2κ,b) for b = 0, 1, . . . , p
Pκ (t, z)Θκ0 is a series approximation to β0(t) + γ0(z),
Pκ (t, z)Θκd is a series approximation to the dth component of
β1(t) + γ1(z) for d = 1, . . . , p.
9 / 36

10. Step 1: Orthogonal series estimation
Our orthogonal series estimator of β0(t) + γ0(z) and the dth
component β1d (t) + γ1d (z) of β1(t) + γ1(z) are respectively
defined as
˜β0(t) + ˜γ0(z) = Pκ (t, z)ˆΘκ0 and ˜β1d (t) + ˜γ1d (z) = Pκ (t, z)ˆΘκd
where
(ˆΘκ0, ˆΘκd ) = arg min
Θκj
n
i=1
yi−Pκ (ti, zi)Θκ0−
p
d=1
Θκd Pκ(ti, zi)xi,d
2
(3.1)
where xi,d is the dth component of xi.
10 / 36

11. Step 1: Orthogonal series estimation
B = Θκ0, Θκ1, Θκ2, . . . , Θκp]
ˆB = ˆΘκ0, ˆΘκ1, ˆΘκ2, . . . , ˆΘκp
Ai =
Pκ(ti, zi) , Pκ(ti, zi) xi,1, Pκ(ti, zi) xi,2, . . . , Pκ(ti, zi) xi,p]
Pκ(t) = 1, p1(t), . . . , pκ(t) , Pκ(z) = p1(z), . . . , pκ(z) ,
Θκbt = (θ0,b, θ11,b, . . . , θ1κ,b) ,
Θκbz = (θ21,b, . . . , θ2κ,b) for b = 0, 1, · · · p
ˆΘκbt = (ˆθ0,b, ˆθ11,b, . . . , ˆθ1κ,b) ,
ˆΘκbz = (ˆθ21,b, . . . , ˆθ2κ,b) for b = 0, 1, · · · p
Equation 3.1 can be written as
ˆB = arg min
Θκj
n
i=1
yi − Ai B
2
, (3.2)
Solution: ˆB = n
AiA
−1 n
yiAi . 11 / 36

12. Step 1: Orthogonal series estimation
It can be easily check that Pκ(t, z) = [Pκ(t) , Pκ(z) ]
Θκb = [Θκbt , Θκbz]
ˆΘκb = [ˆΘκbt , ˆΘκbz]
˜β0(ti) = Pκ (ti)ˆΘκ0t
˜β1d (ti) = Pκ (ti)ˆΘκdt , ˜β1(ti) xi = p
d=1 Pκ (ti)ˆΘκdt xi,d
˜γ0(zi) = Pκ (zi)ˆΘκ0z
˜γ1d (zi) = Pκ (zi)ˆΘκdz, ˜γ1(zi) xi = p
d=1 Pκ (zi)ˆΘκdzxi,d
Centralize ˜γ0(z) and ˜γ1(z): ˜γ∗
b(z) = ˜γb(z) − E˜γb(z) ,
˜β∗
b(t) = ˜βb(t) + E˜γb(z).
˜γ∗
b(z) and ˜β∗
b(t) are first-step estimators.
The orthogonal series estimators are used to ensure that the
biases of first-step estimators converge to zero rapidly.
12 / 36

13. Step 2: Local smoother
Taylor’s expansion for any ti in the neighborhood of t,
βk (ti) ≈ βk (t) + βk (t)(ti − t) ≡ ak + bk (ti − t) k = 0, 1
and for any zi in the neighborhood of z,
γk (zi) ≈ γk (z) + γk (z)(zi − z) ≡ ck + dk (zi − z) k = 0, 1
In the second step, we minimize
n
i=1
[yi − ˜β∗
0(ti)−˜γ∗
0(zi)− ˜β∗
1(ti) xi −{c1+d1(zi −z)} xi]2
Kh1
(zi −z)
(3.3)
and get the minimizer ˆc1 which estimates γ1(z) denoted by
ˆγ1(z), where Kh1
(·) = 1
h1
K( ·
h1
).
13 / 36

14. Step 2: Local smoother
Similarly, we minimize
n
i=1
[yi − ˜β∗
0(ti)−˜γ∗
0(zi)−˜γ∗
1(zi) xi −{a1+b1(ti −t)} xi]2
Kh2
(ti −t),
(3.4)
n
i=1
[yi −˜γ∗
0(zi)− ˜β∗
1(ti) xi −˜γ∗
1(zi) xi −{a0+b0(ti −t)}]2
Kh0
(ti −t),
(3.5)
and
n
i=1
[yi − ˜β∗
0(ti)− ˜β∗
1(ti) xi −˜γ∗
1(zi) xi −{c0+d0(zi −z)}]2
Kh4
(zi −z),
(3.6)
get minimizer ˆa0, ˆa1 and ˆc0 which estimates β0(t), β1(t) and
γ0(z) denoted by ˆβ0(t), ˆβ1(t) and ˆγ0(z).
14 / 36

15. Step 2: Local smoother
Wih1
(z) = (1, zi −z
h1
) ⊗ xi
A∗ = n
i=1 Wih1
(z) Wih1
(z)Kh1
(zi − z)
B∗ = n
i=1[yi − ˜β∗
0(ti) − ˜γ∗
0(zi) − ˜β∗
1(ti) xi]Wih1
(z) Kh1
(zi − z)
Take the first derivative of equation 3.3
ˆc1
h1
ˆd1
= [A∗]−1B∗
Similarly ˆa0, ˆc0 and ˆa1 could be easily determined.
15 / 36

16. Asymptotics when xi is stationary
Theorem 1
Under conditions (A1) ∼ (A9),
√
nh1[ ˆγ1(z) − γ1(z) −
h2
1
2 µ2(K)γ
(2)
1 (z){1 + op(1)}]
d
→
N{0, fz(z)−1δ2S−1ν0(K)}
Theorem 2
Under conditions (A1) ∼ (A9),
√
nh2[ˆβ1(t) − β1(t) −
h2
2
2 µ2(K)β
(2)
1 (t){1 + op(1)}]
d
→
N{0, δ2S−1
0 ν0(K)}
16 / 36

17. Asymptotics when xi is stationary
Theorem 3
Under conditions (A1) ∼ (A9),
√
nh4[ ˆγ0(z) − γ0(z) −
h2
4
2 µ2(K)γ
(2)
0 (z){1 + op(1)}]
d
→
N{0, fz(z)−1δ2ν0(K)}
Theorem 4
Under conditions (A1) ∼ (A9),
√
nh0[ˆβ0(t)−β0(t)−
h2
0
2 µ2(K)β
(2)
0 (t){1+op(1)}]
d
→ N{0, δ2ν0(K)}
17 / 36

18. Asymptotics when xi is nonstationary
xi = xi−1 + δi = x0 + i
s=1 δs(i ≥ 1), where δs is an I(0)
process with mean zero and variance Ωδ.
x[nr]
√
n
=
xi
√
n
=
x0
√
n
+
1
√
n
i
s=1
δs =
x0
√
n
+
1
√
n
[nr]
s=1
δs (4.1)
where r = i/n and [x] denotes the integer part of x, see Cai et
al. (2009).
Under some regularity conditions, Donsker’s theorem leads to
x[nr]
√
n
=⇒ Wδ(r) as n −→ ∞ (4.2)
where Wδ(·) is a p-dimensional Brownian motion on [0, 1] with
covariance matrix Σδ.
18 / 36

19. Asymptotics when xi is nonstationary
For any Borel measurable and totally Lebesgue integrable
function Γ(·), one has
1
n
n
i=1
Γ(x[nr])
√
n
d
−→
1
0
Γ(Wδ(s))ds as n −→ ∞ (4.3)
where
d
−→ denotes the convergence in distribution, so that, for
= 1, 2,
1
n
n
i=1
(
x[nr]
√
n
)⊗ d
−→
1
0
Wδ(s)⊗
ds ≡ Wδ, as n −→ ∞ (4.4)
see Theorem 1.2 in Berkes, I., Horváth, L. (2006) and Cai et al.
(2009)
19 / 36

20. Asymptotics when xi is nonstationary
Theorem 5
Under conditions (B1),(B2)(i),(B3) ∼(B8)
n
√
h1[ ˆγ1(z) − γ1(z) −
h2
1
2 µ2(K)γ
(2)
1 (z){1 + op(1)}]
d
→ MN Σδ(z)
where MN(Σδ(z)) is a mixed normal distribution with mean zero
and conditional covariance matrix given by
Σδ(z) = δ2ν0(K)W−1
δ,2 /fz(z)
Here, a mixed normal distribution is defined as follows.
Conditional on the random variable that appears at the
asymptotic variance, the estimator has an asymptotic normal
distribution
20 / 36

21. Asymptotics when xi is nonstationary
Theorem 6
Under conditions (B1),(B2)(ii),(B3) ∼(B8)
n
√
h2[ˆβ1(t) − β1(t) −
h2
2
2 µ2(K)β
(2)
1 (t){1 + op(1)}]
d
→ MN Σδ(t)
where MN(Σδ(t)) is a mixed normal distribution with mean zero
and conditional covariance matrix given by
Σδ(t) = δ2ν0(K)W−1
δ,2
Theorem 7
Under conditions (B1),(B2)(iii),(B3) ∼(B8),
√
nh0[ˆβ0(t)−β0(t)−
h2
0
2 µ2(K)β
(2)
0 (t){1+op(1)}]
d
→ N{0, δ2ν0(K)}
Theorem 8
Under conditions (B1),(B2)(iv),(B3) ∼(B8),
√
nh4[ˆγ0(z) − γ0(z) −
h2
4
2 µ2(K)γ
(2)
0 (z){1 + op(1)}]
d
→
N{0, f−1
z (z)δ2ν0(K)}
21 / 36

22. Simulation
B-spline
κ = 8
Standard normal kernel
We consider the following model
yi = β0(ti)+γ0(zi)+(β1(ti)+γ1(zi))xi,1+(β2(ti)+γ2(zi))xi,2+εi
= e3ti
+ 20ti + (0.5z3
i − 1.5z2
i − 0.5zi + 8)
+(−40t2
i −20ti+1.5z2
i −7.5zi−8)xi,1+(e4ti
+2ti+3z2
i −6zi−16)xi,2+εi
(5.1)
where γ0(zi) = 0.5z3
i − 1.5z2
i − 0.5zi + 8,
γ1(zi) = 1.5z2
i − 7.5zi − 8 and γ2(zi) = 3z2
i − 6zi − 16. p = 2.
ε ∼ N(0, 0.25), zi = 0.3zi−1 + Ui and Ui ∼ Uniform (−4, 4)
Eγ0(zi) = Eγ1(zi) = Eγ2(zi) = 0.
22 / 36

23. Simulation with stationary xi
x1,1 = x1,2 = 0
xi,1 = 0.9xi−1,1 + δ1i
δ1i ∼ t(3)
xi,2 = 0.6xi−1,2 + δ2i
δ2i ∼ t(7)
y is generated from above model (5.1)
x1, x2 and y all are stationary.
10 grid points for β functions and 40 grid points for γ functions
500 simulations at each grid point
Simulation results are shown in Figure 3 for n = 100.
Simulation results are shown in Figure 4 for n = 400.
23 / 36

24. 0.0 0.2 0.4 0.6 0.8 1.0
01030
t
β0
−4 −2 0 2 4
−50−30−1010
z
γ0
0.0 0.2 0.4 0.6 0.8 1.0
−60−200
t
β1
−4 −2 0 2 4
−2002040
z
γ1
0.0 0.2 0.4 0.6 0.8 1.0
0204060
t
β2
−4 −2 0 2 4
−200204060
z
γ2
Figure: Estimated function β and γ, their mediums and 95%
pointwise confidence intervals for Model 5.1 when x is stationary and
n = 100 24 / 36

25. 0.0 0.2 0.4 0.6 0.8 1.0
01030
t
β0
−4 −2 0 2 4
−50−30−1010
z
γ0
0.0 0.2 0.4 0.6 0.8 1.0
−60−200
t
β1
−4 −2 0 2 4
−2002040
z
γ1
0.0 0.2 0.4 0.6 0.8 1.0
0204060
t
β2
−4 −2 0 2 4
−200204060
z
γ2
Figure: Estimated function β and γ, their mediums and 95%
pointwise confidence intervals for Model 5.1 when x is stationary and
n = 400 25 / 36

26. Simulation with nonstationary xi
x1,1 = x1,2 = 0
xi,1 = xi−1,1 + δ1i
δ1i ∼ t(3)
xi,2 = xi−1,2 + δ2i
δ2i ∼ t(7)
y is generated from above model (5.1)
x1, x2 and y all are nonstationary, xi and yi are cointegration.
10 grid points for β functions and 40 grid points for γ functions
500 simulations at each grid point
Simulation results are shown in Figure 5 for n = 100.
Simulation results are shown in Figure 6 for n = 400.
26 / 36

27. 0.0 0.2 0.4 0.6 0.8 1.0
01030
t
β0
−4 −2 0 2 4
−50−30−1010
z
γ0
0.0 0.2 0.4 0.6 0.8 1.0
−60−200
t
β1
−4 −2 0 2 4
−2002040
z
γ1
0.0 0.2 0.4 0.6 0.8 1.0
0204060
t
β2
−4 −2 0 2 4
−200204060
z
γ2
Figure: Estimated function β and γ, their mediums and 95%
pointwise confidence intervals for Model 5.1 when x is nonstationary
and n = 100 27 / 36

28. 0.0 0.2 0.4 0.6 0.8 1.0
01030
t
β0
−4 −2 0 2 4
−50−30−1010
z
γ0
0.0 0.2 0.4 0.6 0.8 1.0
−60−200
t
β1
−4 −2 0 2 4
−2002040
z
γ1
0.0 0.2 0.4 0.6 0.8 1.0
0204060
t
β2
−4 −2 0 2 4
−200204060
z
γ2
Figure: Estimated function β and γ, their mediums and 95%
pointwise confidence intervals for Model 5.1 when x is nonstationary
and n = 400 28 / 36

29. Real example
5 year daily Treasury bond yield rate
6 month daily Treasury bill yield rate
Stock price of Morgan Stanley and price of S&P500
All data are from the Jan. 2nd, 2003 to Dec. 31st, 2015. The
sample size is 3249.
Stock price of Morgan Stanley and price of S&P500 are
nonstationary
Log "difference" of 5 year daily Treasury bond yield rate and 6
month daily Treasury bill yield rate is stationary
yi = β0(ti−1) + γ0(zi−1) + (β1(ti−1) + γ1(zi−1))xi−1 + i. (6.1)
z: log "difference" of 5 year daily Treasury bond yield rate and 6
month daily Treasury bill yield rate
y: stock price of Morgan Stanley x: price of S&P500.
29 / 36

30. ADF Test Statistic P VALUE
Stock price of Morgan Stanley -0.5896 0.4284
Price of S&P500 1.3095 0.9522
log "difference" of Treasury yield rate -3.3093 < 0.01
Table: ADF test for stock price of Morgan Stanley, price of S&P500
and log "difference" of 5 year daily Treasury bond yield rate and 6
month daily Treasury bill yield rate
30 / 36

31. 2004 2008 2012
−100−60−200
time
β0
2004 2008 2012
0.020.060.10
time
β1
−0.4 −0.2 0.0 0.2
−20−10010
yield rate
γ0
−0.4 −0.2 0.0 0.2
−0.0100.0000.010 yield rate
γ1
Figure: Estimated functions from model 6.1
31 / 36

32. Real example
Variance of the residual model 1.1 model 1.2 model 6.1
13.380 68.504 5.859
Table: variance of the residual in each model
32 / 36

33. 2004 2006 2008 2010 2012 2014 2016
0204060
time
price
2004 2006 2008 2010 2012 2014
0.020.060.10
time
functionofcoefficient
Figure: Estimated stock price and function of coefficient from model
6.1
33 / 36

34. Real example: one step forecast
Split sample into two parts: training sample, the first 3000 data
and testing sample, the remaining 249 data.
Define one step forecast ˆyj for y at time j as following:
ˆyj = ˆβ0(tj−1) + ˆγ1(zj−1) + ( ˆβ1(tj−1) + ˆγ1(zj−1))xj−1 (6.2)
where j from 3001 to 3249 and ˆβ0, ˆγ0
ˆβ1 and ˆγ1 are estimated
by the proposed two-step estimation method using only the
data from 1 to j − 1
34 / 36

35. 2004 2006 2008 2010 2012 2014 2016
0102030405060
x
yscale
Figure: ˆy and One step forecast (6.2)
35 / 36

36. Thank you
36 / 36

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