This note provides explanations for an unexpected result, namely, the estimated parameter of the
correlation coefficient of the trend shock and cycle shock in the state–space model is almost always
(positive or negative) unity, even when the true variance of the trend shock is zero. It is shown that the
set of the true parameter values lies on the restriction that requires the variance–covariance matrix of
the errors to be nonsingular, therefore, almost always the likelihood function has its (constrained) global
maximum on the boundary where the correlation coefficient implies perfect correlation

Wada, Tatsuma

Munich Personal RePEc Archive

MPRA
Munich Personal RePEc Archive
On the Correlations of Trend-Cycle
Errors
Tatsuma Wada
19. December 2011
Online at http://mpra.ub.uni-muenchen.de/41754/
MPRA Paper No. 41754, posted 7. October 2012 23:27 UTC
On the Correlations of Trend-Cycle Errors∗
Tatsuma Wada†
Wayne State University
This version: December 19, 2011
Abstract
This note provides explanations for an unexpected result, namely, the estimated parameter
of the correlation coeﬃcient of the trend shock and cycle shock in the state-space model is almost
always (positive or negative) unity, even when the true variance of the trend shock is zero. It is
shown that the set of the true parameter values lies on the restriction that requires the variance-
covariance matrix of the errors to be nonsingular, therefore, almost always the likelihood function
has its (constrained) global maximum on the boundary where the correlation coeﬃcient implies
perfect correlation.
JEL Classification Number: C13, C22.
Keywords: Trend-Cycle Decomposition, Unit-root, Maximum likelihood.
1 Introduction
When the trend-cycle decomposition of economic time series data is implemented through a state-
space (or unobserved components, UC) model, the correlation of a shock (error) to the trend and
a shock (error) to the cycle is often assumed to be zero. This is due to the fact that the correlation
coeﬃcient is generally unidentified, as Watson (1986) demonstrates. A recent influential paper
by Morley et al. (2003), however, shows that the correlation of errors can be identified only if
identification conditions are satisfied; and that whether or not allowing such a correlation is a key
to understanding the substantial diﬀerences between “business cycles” estimated by the Beveridge-
Nelson decomposition and by the UC model.
Among others, a study by Perron and Wada (2009) finds that US GDP follows a stationary
process, when the trend function is allowed to have sudden changes. Although there is no shocks to
trend,1 the maximum likelihood estimator implies perfect correlation. In addition, their simulation
shows that the estimated correlation parameter is almost always (positive or negative) unity. This
unexpected result is worth exploring. Since, in practice, researchers may estimate the correlation of
the errors by simply assuming the data are non-stationary without applying a variety of unit root
∗I am grateful to an anonymous referee for useful comments. All remaining errors are mine.
†Department of Economics, Wayne State University, 656 W. Kirby St., Detroit, MI, 48202 (tat-
suma.wada@wayne.edu).
1Except the shock that causes the trend to change suddenly.
1
tests; hence, knowing properties of the estimator when the true parameter is not identified, serves
the purpose of diagnosing model misspecification. In line with the idea of detecting misspecification
via correlated UCmodels, Morley et al. (2011) propose a likelihood ratio test of stationarity utilizing
the fact that the correlation is observed only when trend errors exist.
This note demonstrates that such a perfect correlation, when the true data generating process
(DGP) is stationary, is artificially created due to the restriction requiring that the variance-
covariance matrix of the errors be positive-semi-definite. Since the true parameters lie on the
boundary of this restriction, where the correlation coeﬃcient is positive or negative unity, or one
of the variances is zero, the estimated correlation coeﬃcient is almost always 1 or -1, as Table 1
displays,2 rather than the undefined correlation with zero variance. The rest of this note is orga-
nized as follows. Section 2 explains our model. The likelihood function and its properties for a
simple model under the restriction are analyzed in Section 3. An extension to the AR(p) error in
the cyclical component is presented in Section 4. Section 5 concludes.
2 Model
Our model is given by:
 =   +  (1)
  =  −1 + 
() = 
where  is observable variable;   and  are the trend and cyclical components, respectively; ()
is the lag-polynomial; the shock to the trend  and the shock to the cycle  are drawn from the
bivariate Normal distribution:
⎡
⎣ 

⎤
⎦ ∼  
⎛
⎝
⎡
⎣ 0
0
⎤
⎦ 
⎡
⎣ 2 
 2
⎤
⎦
⎞
⎠ 
The correlation coeﬃcient is defined as  =  ().
3 A Simple White Noise Case
3.1 The Likelihood Function
Let the data DGP be a white noise process:
 = 
where  is a zero-mean, normally distributed iid process with a variance of 2. Consider the
state-space model:
 =  + 
 = −1 + 
2As Table1 reveals, it is still possible to obtain an estimate that is neither 1 nor -1. The relative frequency of
finding such estimates is, however, only about 10%. Note also that the results displayed in Table 1 are diﬀerent from
Perron and Wada’s (2009) simulation results, since their model includes a change in the slope of the deterministic
trend, in addition to the stationary component.
2
Assuming that  and  are Normally distributed iid processes, the vector form of the model
is (Tanaka 1996)
 = +  ∼  ¡0 2 ¡ +  0¢¢
where
 = (1      )0 ;  = (1      )0 ;  = 
2
2 ;
with the random walk generating matrix
 =
⎡
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
1 0 0 · · · 0
1 1 0 · · · 0
1 1
. . . . . .
...
...
... 1 0
1 1 · · · 1 1
⎤
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

The log-likelihood function (without constant) is
 = −1
2
log
¯¯2 ¡ +  0¢¯¯− 122 0 ¡ +  0¢−1 
In order to allow the correlation in errors  and , let us assume
 =  +
s
 − 2
  (2)
where  is a constant that represents the covariance of  and ; shocks  and  are independent:
⎡
⎣ 

⎤
⎦ ∼ 
⎛
⎝0 2
⎡
⎣  0
0 1
⎤
⎦
⎞
⎠ ;  = 
2
2 
By (2),  and  are now correlated,3 thereby allowing us to find the likelihood function (see
Online Appendix4 for details). Since the variance of  2, is non-zero, it is convenient to find the
concentrated likelihood function (with respect to 2):
 ( ) = −1
2
log ||− 
2
log 0−1
3Alternatively, we can write the error processes as

 


 =



−2
 
0 1



 


 
so that shocks  and  are contemporaneously correlated:

 


 ∼ 

0 2

  
 1



 ;  = 2 
4http://www.clas.wayne.edu/multimedia/usercontent/File/Economics/wada/tech_appendix_rho.pdf
3
where  =  +  0 +  ( +  0). Using the concentrated likelihood function, we obtain the
following three lemmas that clarify the properties of the likelihood function.
Lemma 1 The first order conditions are given by:
 ( )
 = −
1
2
©−1 ¡ + 0¢ª+ 
2
0−1 ( +  0)−1
0−1 = 0
and  ( )
 = −
1
2
©−1 0ª+ 
2
0−1 0−1
0−1 = 0
Lemma 2 At  =  = 0,
 ( )
 = −

2
µ
( + 1)
2
− 
0 0
0
¶
and for  →∞, we have
−2 ( ) ⇒
1
2
∙Z 1
0
[ ()]2  − 1
2
¸
where “⇒” represents weak convergence in distribution; and  () is the standard Wiener process
on C[0 1].
Lemma 3 At  =  = 0,
 ( )
 = −

2
µ
2− 
0 ( +  0) 
0
¶
and for  →∞, we have
−1 ( ) ⇒
Z 1
0
 ()  () = 1
2
h
 (1)2 − 1
i

3.2 The Restriction
We impose the restriction that the variance-covariance matrix of the errors is positive-semi-definite.
In our case, this restriction is  ≥ 2. Figure 1 displays such a restriction on the parameter space:
The horizontal axis represents , the covariance parameter, while the vertical axis represents ,
the ratio of the variance of the trend errors to the variance of the cycle error. As one can see,
the boundary is a parabola: along the quadratic curve, the correlation coeﬃcient is -1 (when
corresponding  is negative) or 1 (when corresponding  is positive), except for the point where
 =  = 0, at which the correlation is undefined and at which the set of the true parameter values
is located. It is obvious from Figure 1 that the true parameter values are, in fact, close by the
restriction that implies a perfect correlation.
As is well known, the gradient at the true parameter values is not necessarily zero, since the
gradient itself is composed of random variables. Because  R 10 [ ()]2  = 12 and  h (1)2i =
1,5 the gradient becomes zero only as the expected value, asymptotically. Still, the fact that the
probability of 2 (1) being less than unity is 0.683, implies −1 ( )  is often negative at the
5  1
0
[ ()]2  =  1
0
 [ ()]2  =   1
0
 = 1
2
; and  [ (1)]2 =  2 (1) = 1.
4
true parameter values. Hence, the likelihood function has its global maximum at somewhere other
than the true parameter values; many times it can be found on the boundary of the parameter
space with negative . As an example, see Figure 2: The likelihood function does not have its
global maximum at  =  = 0. It is important to keep in mind that increasing the sample size does
not prevent one from finding the artificial perfect correlation.
4 An AR(p) Case
Our framework can be extended to an AR(p) case. To do so, first we modify our model:
 = +Φ−1
where
Φ−1 =
⎡
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
1 0 0
 1 0
2  1 . . .
. . . 0
−1 2  1
⎤
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
for an AR(1) case. The log-likelihood function (concentrated with respect to 2) is then
 ( Φ) = −1
2
log ||− 
2
log 0Φ0−1Φ (3)
where  =  +  0Φ0 + Φ + Φ 0Φ0. Similar to the previous section, our strategy here is that
we concentrate the likelihood function with respect to Φ, so that (3) is a function of ( ):
 ( Φ ( )) = −1
2
log | ( )|− 
2
log 0Φ ( )0 ( )−1Φ ( )  (4)
Note that Φ ( ) is obtained by solving  ( Φ)  = 0 with respect to .
Assuming DGP to be  = Φ−1, Lemmas 1-3 are altered to:
Lemma 4 The first order conditions are given by:
 ( Φ ( ))
 = −
1
2
©−1 ¡ 0Φ0 +Φ¢ª+ 
2
0Φ0−1 ( 0Φ0 +Φ)−1Φ
0Φ0−1Φ = 0
and  ( Φ ( ))
 = −
1
2
©−1Φ 0Φ0ª+ 
2
0Φ0−1Φ 0Φ0−1Φ
0Φ0−1Φ = 0
Lemma 5 At  =  = 0 and for  →∞, we have
−2 ( Φ ( )) ⇒
(1− )2
2
∙Z 1
0
[ ()]2  − 1
2
¸
where “⇒” represents weak convergence in distribution; and  () is the standard Wiener process
on C[0 1].
5
Lemma 6 At  =  = 0 and for  →∞, we have
−1 ( Φ ( )) ⇒ (1− )
Z 1
0
 ()  () = (1− )
2
n
[ (1)]2 − 1
o

Clearly,  = 0 is the previous case. Also, it is not diﬃcult to extend the Lemmas noted above
to an AR(2) model, which is argued in Morley et al. (2003).6 For a finite sample, Figure 3 is
computed as follows: first, b is obtained as a function of ( ), i.e., by solving  ( Φ)  = 0
with respect to , using the MATLAB function “fzero.” Then, given b, the likelihood function is
computed by (4).
5 Conclusion
When data are generated by a stationary process, the correlation in the error of a stochastic trend
and errors of cycles in the state-space model is undefined because there is no stochastic trend. If
one allows for such a correlation and estimate the parameters, the correlation parameter will be
unidentified, and unexpectedly, the estimated parameters will almost always be 1 or -1. It is shown
that the following two facts explain such a result: (i) We impose the restriction that requires the
variance covariance matrix of the errors to be positive semi-definite (in other words, either the
variance of the trend error is zero or the correlation is perfect). The set of the true parameter
values lies on the boundary in the parameter space and in the neighborhood of parameters that
imply perfect correlation. (ii) The likelihood function has its (constrained) global maximum at the
true parameters only on average. Almost always its global maximum is on the boundary.
However, caution is necessary. Unlike Morley et al. (2011), whose proposed likelihood ratio
test of stationarity compares the likelihood value with the restriction of  =  = 0 to the likelihood
value without the restriction, our diagnosis principle does not fit the framework of a rigorous test
of stationarity. This is because an estimated perfect correlation does not necessarily mean that the
true DGP is stationary; it might as well be that the true DGP is non-stationary with a perfect
correlation.
References
[1] Fuller, W.A. (1996) “Introduction to Statistical Time Series” John Wiley & Sons, Inc.
[2] Morley, J.C., C.R. Nelson and E. Zivot (2003) “Why are Beveridge-Nelson and Unobserved-
Component Decompositions of GDP So Diﬀerent?” Review of Economics and Statistics 85,
235-243.
[3] Morley, J.C., I. Panovska and T.M. Sinclair (2011) “A Likelihood Ratio Test of Stationarity
Based on a Correlated Unobserved Components Model,” RPF Working Paper 2008-11, George
Washington University.
[4] Perron, P. and T. Wada (2009) “Let’s Take a Break: Trends and Cycles in US Real GDP,”
Journal of Monetary Economics 56 (6) 749-765
6 In an AR(2) case, for example, the coeﬃcients of the limit distributions in Lemmas 5 and 6 need to be replaced
by (1− 1 − 2)2 and (1− 1 − 2), respectively. See Online Appendix.
6
[5] Tanaka, K. (1996) “Time Series Analysis” John Wiley & Sons, Inc.
[6] Watson, M. (1986) “Univariate Detrending Methods with Stochastic Trend,” Journal of Mon-
etary Economics 18, 49-75.
6 Appendix
• Proof of Lemma2
 ( )

¯¯¯¯
==0
= −
2
µ
( + 1)
2
− 
0 0
0
¶
= −
2
µ
( + 1)
2
− 
0 0
0
¶

Since
−10→ 
2
−20 0 = −2
X
=1
2 ⇒ 2
Z
[ ()]2 
where
 = −1 +  with 0 = 0;
and “→ ” denotes convergence in probability. We have then
−2  ( )
¯¯¯¯
==0
⇒ 1
2
∙Z 1
0
[ ()]2  − 1
2
¸
• Proof of Lemma 3
 ( )

¯¯¯¯
==0
= −1
2
 ¡ +  0¢+ 
2
0 ( +  0) 
0
= − + 
2
0 ( +  0) 
0
= − +  
0
0
Note further that
−10→ 
2
−10 = −1
X
=1
 = −1
X
=1
−1 + −1
X
=1
2
⇒ 1
2
2
n
[ (1)]2 − 1
o
+ 2
7
Hence,
−1  ( )
¯¯¯¯
==0
⇒ −1 + 1
2
n
[ (1)]2 − 1
o
+ 1
=
1
2
n
[ (1)]2 − 1
o

• Proof of Lemma 4 (Sketch)
The derivations are similar to those given in Lemma 1. Note that the Envelope theorem:
 ( Φ ( ))
Φ ( )
Φ ( )
 = 0;
 ( Φ ( ))
Φ ( )
Φ ( )
 = 0
is used to obtain the results.
• Proof of Lemma 5
At  =  = 0,  =  and  = 0 holds. Then, we can prove that the estimator for 
is a consistent estimator. To show this, noting that Φ = (1− )  + −1, the first order
condition for ,
0 =
 ( Φ)
 = −
1
2

µ
−1
¶
−
2
−2
³
0 Φ 0−1Φ
´
− 
³−1Φ0Φ0−1´
0Φ0−1Φ 
is simplified at  =  = 0 to
0 = 0Φ
0
Φ = 0 ¡−1 − ¢0Φ
and hence,7 b = P=2 −1P
=2 2−1
→ 
Since the least square estimator is consistent under the DGP process considered here (see
Fuller 1996, for example), our estimator for the autoregressive parameters is consistent.
7The first order condition is now
0 −1 − 0Φ =




0 0
−1 0
. . .
. . .
0 −1 0




1
2
...





0



1 0
− 1
. . .
. . .
0 − 1




1
2
...





= −

=2
( − −1) −1
8
Next, we note that
Φ 0Φ0 = b2 + b³1− b´ ¡ +  0¢+ ³1− b´2 0;
 ¡Φ 0Φ0¢ = b2 + b³1− b´ 2 + ³1− b´2  ( + 1)
2

Since b →  and the fact that DGP is  = Φ−1, the same argument as Lemma 2 leads to
the limit distribution.
• Proof of Lemma 6
The limit distribution is obtained since
Φ + 0Φ0 =
³
1− b´ ¡ +  0¢+ 2b;  ¡Φ +  0Φ0¢ = 2
Table 1: Frequency Distributions of the Estimated Correlation 
Data Generating Process
Estimated  AR(0) AR(1) AR(2)
  −099 710 654 711
−099 ≤  ≤ 099 85 116 102
099   205 230 187
||  099 915 (915%) 884 (884%) 898 (898%)
Notes: 1) The estimated frequencies are are computed from the sample size  = 500 with 1,000 replications. 2) For
the AR(1) model, DGP is  = 09 −1 + , where  ∼ i i d (0 1) and 0 = 0. 3) For the AR(2) model, DGP is
 = 128 −1 − 038 −2 + , where  ∼ i i d (0 1) and 0 = −1 = 0 . For more details, see Online Appendix.
9
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
S
D
e
l
t
a
Palameter Space
True parameter values
Positeve−semi−definite restriction
Figure 1: Parameter space and the restriction
−0.04
−0.03
−0.02
−0.01
0
0.01
0.02
0.03
0.04
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
x 10−3
−10000
−9000
−8000
−7000
−6000
−5000
−4000
−3000
−2000
−1000
0
S
Likelihood Surface: AR0
Delta
Global Max
True Parameter Values
Figure 2: Typical Likelihood Surface: AR(0) with T=500.
−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
0.2
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
−10000
−9000
−8000
−7000
−6000
−5000
−4000
−3000
−2000
−1000
0
S
Likelihood Surface: AR1
Delta
True Parameter Values
Global Max
Figure 3: Typical Likelihood Surface: AR(1) with T=500.


On the Correlations of Trend-Cycle Errors

https://mpra.ub.uni-muenchen.de/41754/1/MPRA_paper_41754.pdf

On the Correlations of Trend-Cycle Errors

Abstract

Similar works

Full text

Available Versions

Munich Personal RePEc Archive