1. Understanding Nonstationarity and Recognizing it When You See it
D. A. Dickey
North Carolina State University

2. Nonstationary Forecast

3. ”Trend Stationary” Forecast
Nonstationary Forecast

4. DYt = m (1- r) + (r-1)Yt-1 + et DYt = (r-1)(Yt-1 - m) + et
Autoregressive Model
AR(1)
Yt - m = r (Yt-1-m) + et
Yt = m (1- r) + rYt-1 + et
DYt = m (1- r) + (r-1)Yt et
DYt = (r-1)(Yt-1 - m) + et
where DYt is Yt -Yt-1
AR(p)
Yt - m = a1(Yt-1-m) + a2(Yt-2-m) ap(Yt-1-m) + et

5. AR(1) Stationary  |r| < 1
OLS Regression Estimators – Stationary case
Mann and Wald (1940’s) : For |r| < 1
More exciting algebra coming up ……

6. AR(1) Stationary  |r| < 1
OLS Regression Estimators – Stationary case
Same limit if sample mean replaced by m
(2) AR(p)  Multivariate Normal Limits 

7. |r| < 1
Yt-m = r(Yt-1-m) + et = r(r(Yt-2-m) + et-1) + et = ... = et + ret-1 +r2et-2 + … +rk-1 et-k+1+ rk (Yt-k-m)
Yt=m (converges for |r| < 1)
Var{Yt } = s2/(1-r2)
r = 1
But if r=1, then Yt = Yt-1 + et, a random walk.
Yt = Y0 + et + et-1 + et-2 + … + e1
Var{Yt - Y0} = ts2
E{Yt} = E{Y0}

8. Forecast of Yt+L converges to m (exponentially fast)
AR(1) |r| < 1
E{Yt} = m
Var{Yt } is constant
Forecast of Yt+L converges to m (exponentially fast)
Forecast error variance is bounded
AR(1) r = 1
Yt = Yt-1 + et
E{Yt} = E{Y0}
Var{Yt} grows without bound
Forecast not mean reverting

9. E = MC2
r = ?

10. Nonstationary (r=1) cases:
Case 1: m known (=0)
Regression Estimators (Yt on Yt-1 noint ) /n n
/n2

11. r=1  Nonstationary
Recall stationary results:
Note: all results independent of s 2

12. Where are my clothes?
H0:r=1 H1:|r|<1 ?

13. DF Distribution ??
Numerator:
e e e3 … en
e e e1e e1e3 … e1en
e e e2e3 … e2en
e e32 … e3en
: :
en en2 :
Y1e2
Y2e3 …
Yn-1en

14. Denominator For n Observations:
(eigenvalues are reciprocals of each other)

15. Results:
eTAne =
n-2 eTAne =
Graph of
gi,502 and limit :

16. Histograms for n=50:
-1.96
-8.1

17. Theory 1: Donsker’s Theorem (pg. 68, 137 Billingsley)
{et} an iid(0,s2)
sequence 
(n=100)
Sn =
e1+e2+ …+en
X(t,n) = S[nt]/(n1/2s)=Sn normalized

18. Theory 1: Donsker’s Theorem (pg. 137 Billingsley)
Donsker: X(t,n) converges in law to W(z), a “Wiener Process”
plots of X(t,n) versus z= t/n for n=20, 100, 2000
20 realizations of
X(t,100) vs. z=t/n

19. Theory 2: Continuous mapping theorem
(Billingsley pg. 72)
h( ) a continuous functional => h( X(t,n) ) h(W(t))
For our estimators,
and
so……
Distribution is …. ???????

20. Nice proof Grandpa!
As you see, I’m very excited.

21. Extension 1: Add a mean (intercept)
New quadratic forms.
New distributions
Estimator independent of Y0

22. Extension 2: Add linear trend
Regress Yt
on 1, t, Yt-1 annihilates Y0 , bt
New quadratic forms.
New distributions

23. The 6 Distributions coefficient n(rj-1) t test t f(t) = 0 mean trend
-8.1
-14.1
-21.8
t test t
- 1.96
-1.95
-2.93
-3.50
f(t) = mean trend

24. t percentiles, n=50 t percentiles, limit pr<t 0.01 0.025 0.05 0.10
0.50
0.90
0.95
0.975
0.99
f(t)
---
-2.62
-2.25
-1.95
-1.61
-0.49
0.91
1.31
1.66
2.08 1
-3.59
-3.32
-2.93
-2.60
-1.55
-0.41
-0.04
0.28
0.66
(1,t)
-4.16
-3.80
-3.50
-3.18
-2.16
-1.19
-0.87
-0.58
-0.24
t percentiles, limit
pr<t
0.01
0.025
0.05
0.10
0.50
0.90
0.95
0.975
0.99
f(t)
---
-2.58
-2.23
-1.95
-1.62
-0.51
0.89
1.28
1.62
2.01 1
-3.42
-3.12
-2.86
-2.57
-1.57
-0.44
-0.08
0.23
0.60
(1,t)
-3.96
-3.67
-3.41
-3.13
-2.18
-1.25
-0.94
-0.66
-0.32

25. Higher Order Models stationary: “characteristic eqn.”
roots 0.5, 0.8 ( < 1)
note: (1-.5)(1-.8) = -0.1
nonstationary

26. Higher Order Models- General AR(2)
roots: (m - a )( m - b ) = m2 - ( a + b )m + ab
AR(2): ( Yt - m ) = ( a + b ) ( Yt-1 - m ) - ab ( Yt-2 - m ) + et
(0 if unit root)
nonstationary
t test same as AR(1).
Coefficient requires
modification
t test  N(0,1) !!

27. Tests Regress: These coefficients  normal! |   | on (1, t) Yt-1
|   |
on (1, t)
Yt-1
( “ADF” test )
r-1
( t )
augmenting affects limit distn.
“ does not affect “ “

28. Silver example:
Nonstationary Forecast
Stationary Forecast

29. Is AR(2) sufficient ? test vs. AR(5).
proc reg; model D = Y1 D1-D4; test D2=0, D3=0, D4=0;
Source df Coeff t Pr>|t|
Intercept Yt
Yt-1-Yt
Yt-2-Yt
Yt-3-Yt
Yt-4-Yt
F413 = 1152 / 871 = Pr>F = X

30. Fit AR(2) and do unit root test
Method 1: OLS output and tabled critical value (-2.86)
proc reg; model D = Y1 D1;
Source df Coeff. t Pr>|t|
Intercept X
Yt X
Yt-1-Yt 
Method 2: OLS output and tabled critical values
proc arima; identify var=silver stationarity = (dickey=(1));
Augmented Dickey-Fuller Unit Root Tests
Type Lags t Prob<t
Zero Mean
Single Mean 
Trend

31. ?
First part ACF IACF PACF

32. Full data ACF IACF PACF

33. Amazon.com Stock ln(Closing Price)
Levels
Differences

34. Levels Differences Type Lags Tau Pr < Tau Zero Mean 2 1.85 0.9849
Augmented Dickey-Fuller Unit Root Tests
Type Lags Tau Pr < Tau
Zero Mean
Single Mean
Trend
Differences
Augmented Dickey-Fuller Unit Root Tests
Type Lags Tau Pr<Tau
Zero Mean <.0001
Single Mean <.0001
Trend <.0001

35. Are differences white noise (p=q=0) ?
Autocorrelation Check for White Noise
To Chi Pr >
Lag Square DF ChiSq Autocorrelations 

36. Amazon.com Stock Volume
Levels
Differences

37.  Augmented Dickey-Fuller Unit Root Tests Type Lags Tau Pr < Tau
Zero Mean
Single Mean
Trend <.0001
Maximum Likelihood Estimation
Approx
Parameter Estimate t Value Pr > |t| Lag Variable
MU < volume
MA1, < volume
AR1, < volume
AR1, < volume
NUM < date
To Chi Pr >
Lag Square DF ChiSq Autocorrelations 

38. Amazon.com Spread = ln(High/Low)
Levels
Differences

39.  Augmented Dickey-Fuller Unit Root Tests Type Lags Tau Pr<Tau
Zero Mean
Single Mean <.0001
Trend <.0001
Maximum Likelihood Estimation
Approx
Parm Estimate t Value Pr>|t| Lag Variable
MU spread
MA1, < spread
AR1, < spread
AR1, < spread
NUM date
To Chi Pr >
Lag Square DF ChiSq Autocorrelations 

40. S.E. Said: Use AR(k) model even if MA terms in true model.
N. Fountis: Vector Process with One Unit Root
D. Lee: Double Unit Root Effect
M. Chang: Overdifference Checks
G. Gonzalez-Farias: Exact MLE
K. Shin: Multivariate Exact MLE
T. Lee: Seasonal Exact MLE
Y. Akdi, B. Evans – Periodograms of Unit Root Processes

41. H. Kim: Panel Data tests
S. Huang: Nonlinear AR processes
S. Huh: Intervals: Order Statistics
S. Kim: Intervals: Level Adjustment & Robustness
J. Zhang: Long Period Seasonal.
Q. Zhang: Comparing Seasonal Cointegration Methods.