A General Dependence Test and Applications*

David Johnson**

Robert McClelland***

Division of Price and Index Number Research

Bureau of Labor Statistics

2 Massachusetts Ave., NE, Room 3105

Washington, D.C. 20212

**(202) 606-6579 Ext 599; JohnsoDS@oplc.psb.bls.gov

***(202) 606-6579 Ext 607; McClelland_R@bls.gov

Keywords: Misspecification tests, U-Statistics, nonlinear dynamics, chaos

 * The authors wish to thank Rochelle Antoniewicz, Ralph Bradley, Alan Dorfman, Ted

Jaditz, Bruce Mizrach, Chera Sayers, participants at the 1993 Eastern Economic

Association meetings and the 1993 American Statistical Association meetings, and the

editor and two referees of this jounral for helpful comments. The views expressed are

those of the authors and do not necessarily represent an official position of the Bureau

of Labor Statistics or the views of other BLS staff members.
Abstract:

We describe a test, based on the correlation integral, for the

independence of a variable and a vector that can be used to detect model

misspecification in serially dependent data. In Monte Carlo simulations

this test performs nearly as well or better than the BDS test in univariate

time series and complements the BDS test in distributed lag models.

Finally, we apply our test to detect misspecification in models of U.S.

unemployment data.
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Recent work on nonlinear dynamics has focused on several concepts,

including the use of the correlation integral in nonparametric tests of

misspecification for time series models.1 By detecting serial dependence in a time

series they allow researchers to check for dependence without specifying an

alternative model. In this paper we use the correlation integral to describe a test

for the independence of a variable and a vector that can be used to detect model

misspecification in both independently and identically distributed (iid) and

serially dependent data.

To clarify how our test fits into the family of nonparametric specification

tests, we can distinguish members of the family by their null hypotheses. For

example, Wooldridge (1992) and Bradley and McClelland (1994) propose linearity

tests in iid data that are consistent against almost all alternatives, the former

using a sieve estimator and the latter a kernel estimator. In addition, Lee, White

and Granger (1993) propose a linearity test for use on serially dependent data

that is based upon the test of Bierens (1990).

Although requiring iid data, the Bierens test has a more general null

hypothesis than linearity: it is consistent against almost all misspecifications of

the first conditional moment of a nonlinear least squares model.2 Similarly,

Bradley and McClelland (1993) and Lewbel (1993) describe conditional moment

tests for use with iid data without requiring linearity and de Jong (1992) extends

the Bierens approach to time series data.

While the above tests work on individual moments of a distribution, other

tests have even broader null hypotheses by checking assumptions about the

whole distribution of the residual or the dependent variable. This is useful for

 1 For example, see Brock, Dechert and Schienkman [1987], Baek and Brock [1988] and [1992], Brock,

Hsieh and LeBaron [1991].

2 For a more detailed discussion of the Bierens test and some potential problems, see Bradley and

McClelland (1993).
Page 2 Revision

procedures that require strong distributional assumptions, such as the adaptive

estimation method of Manksi (1984) or some quasi-maximum likelihood methods,

or more generally when the researcher wishes to test distributional assumptions

or a priori knowledge other than simple conditional moment restrictions.

Examples of this set of tests include the many types of Pearson χ2 independence

tests, such as the tests for iid data in Andrews (1988). Tests for serial

independence of a variable or vector, such as Robinson (1991) and Skaug and

Tjøstheim (1993) also fall into this class.

Because these tests are nonparametric, assumptions about the

distributional form of the variable are not necessary. Both the Robinson test and

the test of Skaug and Tjøstheim are consistent against all alternatives when a

single lag is chosen. However, the consistency properties against dependence in

the more useful case of higher lags is unknown. It is also unclear if either of

these tests is appropriate for use upon residuals from an estimated model.

Finally, it should be noted that the Skaug and Tjøstheim test does not have a

standard normal or χ2 distribution, which can make comparisons with other

tests more difficult.3

Another example of a time series independence test is the BDS test (see

Brock, Hsieh and LeBaron (1991)). This test checks for serial dependence of a

random variable by estimating the correlation integral of the series, which

measures spatial correlation in phase space with the frequency with which pairs

of observations are close. The BDS test is asymptotically normally distributed

under the null hypothesis of independence and De Lima (1996) shows that

correlation integral-based tests make relatively weak assumptions about the

 3 The Skaug and Tjøstheim test has the distribution of the test of Blum, et.al. (1961), which is defined

only by its characteristic function. Although the Robinson test does have a standard distribution,

Robinson notes that a parameter affecting the power of the test may also distort its size.
Page 3 Revision

necessary moment conditions. By using a nuisance parameter theorem, Brock,

Hsieh and LeBaron also show that the test can be used as a specification test by

checking for the dependence of the residuals from a regression. However, it has

been pointed out that consistent estimation of ordinary least squares coefficients

does not require that residuals are iid, only that they are a martingale difference

sequence (MDS).

In this paper, we use the correlation integral to test for the dependence of

the residual upon the set of explanatory variables. By checking the dependence

of the residuals upon the regressors our test resembles the test of Andrews, using

the correlation integral instead of the standard χ2 test. Unlike the tests of

Andrews (1988), Bierens (1990), Wooldridge (1992), and Bradley and McClelland

(1993), our test can be used with serially dependent data. As in the BDS test,

our test is normally distributed and does not require bootstrapping of the

distribution. It also does not use a pseudo-random number generator, as in the

tests of Bierens (1990), de Jong (1992), Lee, White and Granger (1993), and

Bradley and McClelland (1994).

Because we use correlation integrals, our test shares the moment condition

properties of the BDS test. Unlike the BDS test however, our test explicitly allows

for serial dependence in the residuals, so that our test is insensitive to

dependence among residuals that do not affect the regressors. We more directly

test for correct specification since we check for dependence of the estimated

residual upon the regressors, rather than the dependence of the residuals upon

the regressors through the past residuals. This means that in univariate time

series the GD test may have greater finite sample power against functional form

misspecification. The BDS test, however, should have better power against

dependence in the residuals upon their own histories (e.g., autoregressive

conditional heteroskedasticity).
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In the case where regressors do not include lagged values of the dependent

variable, the BDS and GD tests are essentially complementary. While the BDS

test checks for time dependence in the estimated residuals (which can come from

time dependence in the true residuals), the GD test checks for contemporaneous

dependence between the regressors and the estimated residuals. A rejection of

the null hypothesis by the BDS test but not by the GD test indicates time

dependence in the actual residuals but a correct functional form, while acceptance of the null by the BDS test but rejection by the GD test indicates functional

form misspecification but no dependence in the residuals.

To compare the size and power properties of these two tests we use Monte

Carlo simulations under the null and several alternative hypotheses. We show

that for some univariate time series models the GD test performs nearly as well or

better than the BDS test. We also use Monte Carlo simulations to illustrate the

complementarity of the BDS and GD tests in a distributed lag model.

As an example we use both tests to examine various models of U.S.

unemployment data. While we find evidence of nonlinearities and asymmetries,

none of the existing models appears to adequately fit the data. However, using

the null of a four-regime threshold autoregressive model, the GD test rejects the

null while the BDS fails to reject. This illustrates the complementarity of the

tests and indicates that dependence in the residuals is likely to be less of a

concern than the issue of proper functional form specification in each regime.

The remainder of the paper is organized as follows. The next section

describes our test and compares and contrasts it with the BDS test. Section two

presents the results of Monte Carlo simulations. Section three applies our test to

detect misspecifications in models of unemployment rates and the final section

concludes.
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I The GD Test for Independence

Our test can be applied to general nonlinear models of the form

yt = G(xt,β) + et, (1.)

where yt is an observed random variable produced by the data generating process

G(·,·) with a K-dimensional vector of observed explanatory variables xt and

parameter vector β, and an unobserved disturbance et distributed independently

of xt. We assume that xt and et satisfy the following two conditions: (1) {(xt,et)} is

an absolutely regular stationary sequence with mixing coefficients4 α(t) such that,

for some λ < 1

2 and δ > 0, α(t)δ/(2+δ)

 = O(t(-2+λ)) and (2) {(xt,et)} has a smooth joint

distribution with a bounded density. Note that assumption (1) excludes unitroot processes but allows Martingdale difference sequences in {xt} and {et}. Under

the assumption of independence, the regressors xt hold no information about the

disturbance et, so that the conditional distribution of et given xt is equal to the

marginal distribution of et.

Equation (1.) can be estimated with

yt = G(xt,βT) + êt, t = 1,...,T (2.)

where êt is the estimated residual and βT is a consistent estimator of β. Using the

results of Brock, Hsieh and LeBaron (1991) and de Lima (1996), we can test the

independence of xt and et by examining the estimated residuals êt. Combining

equations (1.) and (2.), the estimated residuals can be described as follows:

êt = G(xt,β) - G(xt,βT) + et. (3.)

 

4 Denker and Keller (1983)define as α(t) = supj

 E[sup{|P(A | j

1) - P(A)| : A ∈ ∞j+t}] such that ji

 is the

σ-algebra generated by {(xt,ut) : 1 ≤ i ≤ t ≤ j ≤ ∞}. A sequence is absolutely regular if the mixing

coefficients converge to zero.
Page 6 Revision

If βT is a consistent estimator of β and we have correctly specified the data

generating process, then in the limit êt is equal to et and hence independent of xt.

A test of the independence of êt and xt is then a test of the correct specification.

In general, et is independent of xt if the joint probability of some event is

equal to the product of the marginal probabilities of that event. For our test, we

note that if xt and et are independent then

Pr[||(xt,et) - (xs,es)|| < ε] = Pr[||xt - xs|| < ε]⋅Pr[|et - es| < ε] for all ε, (4.)

where ||⋅|| is the sup norm.5 This means that the probability that the pairs, (xt,et)

and (xs,es), are close is equal to the product of the probability that xt and xs are

close and the probability that et and es are close.

These probabilities can be given in terms of correlation integrals, i.e.,

C((x,e),ε) ≡ Pr[||(xt,et) - (xs,es)|| < ε] = EI((xt,et) - (xs,es),ε), where I(⋅) is an indicator

function given by:

I((xn,en) - (xm,em),ε) = 



 1 if ||(xn,en) - (xm,em)|| < ε

0 otherwise . (5.)

Hence, independence implies that C((x,e),ε) = C(x,ε)⋅C(e,ε).

To estimate these correlation integrals we use the theory of U-statistics. An

estimator of C((x,e),ε) is CT((x,e),ε), which is a U-statistic of the following form:

CT((x,e),ε) = 2

T(T-1) ∑

1≤t<s≤T

 I(et - es,ε) ⋅











 ∏

k=1

K

 I(xk

t - x

k

s,ε) (6.)

The correlation integral estimator for the regressors, CT(x,ε) , is given by:

CT(x,ε) = 2

T(T-1) ∑

1≤t<s≤T

 









 ∏

k=1

K

 I(xk

t - x

k

s,ε) (7.)

The correlation integral estimator for the disturbances, CT(e,ε), is defined

similarly.

 

5 For the sup norm, ||(xt

,ut) - (xs,us)|| < ε if and only if |ut - us| < ε and |xk

t - x

k

s| < ε for k: 1 ≤ k ≤ K.
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Using the results of Brock, Hsieh and LeBaron (1991) and Denker and

Keller (1986), it can be shown that if our assumptions described above hold then

for a fixed ε

T[CT((x,e),ε) - CT(x,ε)CT(e,ε)] ⇒ (0,σ

2

C), (8.)

where

σ

2

C = 4[K0(x,ε)K0(e,ε) - K0(x,ε)C(e,ε)2 - K0(e,ε)C(x,ε)2 + C(e,ε)2C(x,ε)2 +

2 ∑

j ≥ 1

(Kj(x,ε)Kj(e,ε) - Kj(x,ε)C(e,ε)2 - Kj(e,ε)C(x,ε)2 + C(e,ε)2C(x,ε)2)], (9.)

K0(x,ε) = K0(xi,ε) = E[E[I(xt - xs,ε)|xt = xi]2] , (10.)

Kj(x,ε) = Kj(xi,ε) = E[E[I(xt - xs,ε)|xt = xi]E[I(xt - xs,ε)|xs = xj+i]] , (11.)

and

C(x,ε) = E[I(xt - xs,ε)] (similarly for K(e,ε) and C(e,ε)).6

Hence, the GD test statistic is given by:

T WT(x,e,ε)

sC

, (19.)

where WT(x,e,ε) = CT(x,e,ε) - CT(x,ε) ⋅ CT(e,ε) and s2

C is calculated by substituting

the sample analogs, CT(⋅,ε), K0T(⋅,ε) and KjT(⋅,ε), for C(⋅,ε), K0(⋅,ε) and Kj(⋅,ε) in

equation (9.).

Equation (8.) states that the distribution of our statistic converges to a

normal distribution when using the disturbances, et. To apply our tests to

residuals êt

, we must show that under the appropriate conditions the GD test

evaluated at the residuals converge to the statistics evaluated at the

disturbances, i.e., that

T 







 WT(x,ê,ε) 

s(ê) - WT(x,e,ε)s(e) →p 0 (20.)

 

6 This follows from proposition 1 in Johnson and McClelland [1996], where the variance σ

2

C is

calculated using the results of Denker and Keller [1986] to allow for the vector-pair, {(xt

,ut)}, to follow

the mixing condition. A formal proof is available from the authors upon request.
Page 8 Revision

This type of convergence is shown in nuisance parameter theorems such as

those in Baek and Brock (1992) and de Lima (1996). De Lima (1996) shows a

nuisance parameter theorem for a family of BDS tests applied to univariate time

series models. A nuisance parameter theorem for the GD test would be a simple

application of de Lima’s results if our regressors were the K lags of a univariate

time series, that is, (x1

t ,...,x

K

t ) = (yt-1,...,yt-K) and the disturbances were iid. In this

case, we can apply de Lima’s results by requiring the series { yt } to satisfy the

conditions in his theorems.

Nuisance Parameter Theorem: Suppose that the data generating process is

given by yt = G(xt,β) + et, and the following conditions are satisfied:

i) The regressors and disturbances are contemporaneously independent and

the vector {xt,et} is a strong mixing processes (absolutely continuous

processes are strong mixing) with mixing coefficients that satisfy the

summability condition ∑α(t) 1/2 < ∞. (i.e., the processes are strong mixing

processes of order two). G(⋅) is a measurable function of xt.

ii) For all d there is some constant C1 such that

E sup

{ : - < d} β ββ T T

 | I(êt - ês,ε) - I(et - es,ε) | < C1d.

iii) T(βT-β) = O(1).

iv) The series {(xt,et)} has a joint distribution, F(⋅,⋅) that is continuously

differentiable with a bounded density.

Then the following convergences hold:

T(WT(x, ê,ε) - WT(x,e,ε)) →p 0 and (s2

C(ê) - s2C(e)) →p 0.

The definition of W(⋅,⋅,⋅) and the independence of xt, and et imply that both

of these convergences are satisfied if T(CT(ê ,ε) - CT(e,ε)) →p 0. We can now apply
Page 9 Revision

Theorem 2.1 in de Lima (1996) to show this convergence.7 Here, (i) takes the

place of assumption (A) in de Lima (1996). For a univariate time series, our

assumption (i) is weaker than assumption (A). De Lima assumes that the series

{yt} is strong mixing of order two, which implies that {xt}, where xt is the k lags of

yt (i.e., (x1

t ,...,x

K

t ) = (yt-1,...,yt-K)), is strong mixing of order two. De Lima’s additional

assumption that the disturbances are iid along with the fact that {xt

} is a mixing

process implies that {xt,et} is also strong mixing of order two. De Lima uses

assumption (A) to show that the residuals and the kernal are strong mixing

processes of order two.

In addition, assumption (ii) is assumption (B) in de Lima (which given our

indicator kernel also implies de Lima’s assumption (C)), (iii) is assumption (D) and

(iv) is assumption (E).

While there is a nuisance parameter theorem for the BDS test, it differs

from the GD test because it examines the independence among the residuals.

Using the 2-histories of the residuals, e2 ≡ {(et

, et+1)}, if the series is iid then the

correlation integral for 2-histories is equal to the product of two one-history

correlation integrals, i.e.,

C(e2,ε) = C(e,ε)⋅C(e,ε) (21.)

so that the BDS statistic is given by:

T 

CT(e2,ε) - CT(e1,ε)2

sB

, (22.)

where s2

B is the sample variance given in Brock, Hsieh and LeBaron (1991) and

the correlation integral for the 2-histories is:

CT(ê2,ε) = 2

(T-1)(T-2) ∑

1≤i<j≤Τ−1

 I((êi,êi+1)-(êj,êj+1),ε). (23.)

 7 A formal proof is available from the authors upon request.
Page 10 Revision

We can compare the BDS and GD test in a univariate time series model in

which the lagged value is the only regressor, i.e.,

yt = G(yt-1,b) + et. (24.)

Using (3), misspecification implies that

êt = G(yt-1,b) - G(G(yt-2,βT) + êt-1,βT) + et. (25.)

 The BDS test will detect dependence of êt on êt-1 because êt-1 affects êt through

G(⋅,⋅). Alternatively, the GD test examines the dependence between êt and yt-1

directly.

II Monte Carlo Simulations

In this section, we describe the relative properties of the GD test and the

BDS test in univariate and distributed lag time series models. All tests use 5,000

iterations, errors with a standard normal distribution and a window width, ε, set

to one standard deviation. We apply the BDS test to the two-histories of the

residuals. When calculating the GD test we follow Hiemstra and Jones (1994)

and include a weight, w(j), on the crossproduct terms in (9.) such that

w(j) = 1 - j/(trunc(T.25)+1). (26.)

This weight determines the rate at which the terms in the summation in (9.)

increase with the sample size.8

In figure one we show the results of simulations when the null hypothesis

is true.9 In this case we correctly specified a AR model in which the lags were

selected using the Bayesian information criteria. The horizontal axis depicts

critical points for a normal distribution, while the vertical axis indexes the actual

 8 The second component of (9) becomes:

 2 ∑

j = 1

trunc(T.25)

 w(j)(Kj(x,ε)Kj(u,ε) - Kj(x,ε)C(u,ε)2 - Kj(u,ε)C(x,ε)2 + C(u,ε)2C(x,ε)2)].

9 The authors would like to thank the editor for suggesting this graphical method of presenting test

statistic simulation results. For more information, see Davidson and Mackinnon (1994).
Page 11 Revision

percent of simulations that exceed that critical point. The curve of a perfectlysized statistic will follow a line formed of points where the actual percent of

simulations that exceeds the critical point equals the critical points , while the

curve of a statistic that rejects too often will be above that line..

As figure one shows, the BDS test and the GD test for regressors formed

with one and four lags from the series reject too frequently with 50 observation

data sets, with the rates for the BDS test everywhere above the GD test lines.10

Because of the use of the sup norm in the correlation integral, the size of the GD

test changes with the number of lags. However, this does not occur if we

aggregate the regressors, such as by using the conditional expected value of the

dependent variable in place of the regressors. Figure two shows results of

simulations with 250 observations in each data set. As expected, both the GD

test and the BDS test are converging to the proper size.

Although it is impossible to draw any global conclusions, it is also useful to

examine the power of the GD test and the BDS test against the models in table

two. Models one, two, and three are inspired by those in Lee, White and Granger

(1993). Model four is an AR(1) model with ARCH disturbances. This model

allows us to compare the BDS and GD tests when the disturbances are an MDS

process. Finally, model five is from Rothman (1992b). In each case, the model is

misspecified by an AR model with the number of lags chosen by the Bayesian

information criteria. The horizontal axis in the figures now lists the rejection rates

generated by the simulations described in figures one and two. More powerful

tests will be reflected in curves that most closely approximate an inverted "L".

Figures three through seven show the results from these simulations. In

figures three and four it appears that the GD test and the BDS test are

comparable against nonlinear AR and nonlinear MA alternatives. In figure five

 

10 The lines for two and three lags lie everywhere between the lines for the one and four lags. 
Page 12 Revision

the GD test seems to outperform the BDS test, to the point where the GD test

rejects on data sets with 50 observations at approximately the same rate as the

BDS test with 250 observations. Figure six shows that, for data sets with 250

observations, the BDS test detects the dependence in the residuals, while the GD

test shows almost no power. Finally, figure seven illustrates that the GD test

appears to strongly outperform the BDS test against sign AR models with 250

observations and is approximately equal in power (which is to say no power) with

50 observations.11

To demonstrate how the BDS test and the GD test complement each other,

table two presents a distributed lag model with two variables that are

independent of each other but time dependent. The true model is given by these

two variables minus their product and a disturbance term that is either iid

(model six) or autocorrelated (model seven). In the estimation of model six the

product term is omitted, while in model seven the autocorrelation in the error

term is not modeled. As figures eight and nine show, when the cross-product

term is omitted from estimation only the GD test detects the misspecification,

while failure to model the error term properly is detected only by the BDS test.

III An Application to Unemployment Rates

In this section we use the BDS test and the GD test to examine various

models of the U.S. unemployment rate. This series provides a good application

because of the current interest in applying models of nonlinearities and

asymmetries to macroeconomic data. For example, Brock and Sayers (1988),

Frank and Stengos (1988), Ham and Sayers (1990) and Rothman (1992a)

 11 The results for models 1, 2 and 3 can be compared to the results of the neural net test given in Lee,

Granger and White (1993). At the 5% critical value they report better power against model 3, while

the power of the GD exceeds that of the neural net test in model 1 and the tests have the same power

against model 2.
Page 13 Revision

demonstrate this asymmetry by examining nonlinearities in unemployment rates,

while Potter (1991), Sichel (1991), Beaudry and Koop (1993) and Pesaran and

Potter (1994) use GNP to demonstrate these business cycle asymmetries.

To examine these models, we use aggregate seasonally adjusted monthly

unemployment rates from January 1948 to December 1993. Since the published

seasonally adjusted unemployment rate series is only reported at one decimal

point, we calculate the unemployment rate from the seasonally adjusted levels of

the unemployed and labor force. This allows us to retain another significant

digit.

Figure 10 shows the unemployment rate data in which the shaded areas

are the periods of recession (as calculated by the National Bureau of Economic

Research and published in USDC (1994)). This figure illustrates the observation

of Keynes (1936) that expansions (where the rate is falling) are longer and slower

than contractions (the shaded areas). The figure also illustrates the insight of

Sichel (1991) of a third recovery phase in which the rate falls faster than in the

expansion phase. Kydland and Prescott (1990) report that Mitchell suggested in

the 1920's that there are four phases: prosperity (expansion), crisis (recession),

depression (contraction), revival (recovery). 

While many suggest the presence of asymmetries in the unemployment

data, a simple AR model fits the data fairly well (see also Brock and Sayers

(1988)). However, the coefficient on the first lag is close to one, suggesting the

presence of a unit root. Applying the test of Dickey and Fuller (1981), we fail to

reject the null hypothesis of a unit root. Hence, for the remainder of the analysis

the data is transformed by taking the log first differences.

A reasonable next step is to estimate a linear model on the transformed

data, as in Ham and Sayers (1990) and Rothman (1992a). The Akaike

Information Criteria (AIC) suggests an AR(10) fits the data fairly well, yet the BDS
Page 14 Revision

and GD tests both reject the null of independence. Similar to Ham and Sayers

and Rothman, this suggests that there may be nonlinearities unaccounted for by

the AR model. 

To model these nonlinearities, many researchers use threshold

autoregressive models since this class of models can be viewed as piecewise

linear approximations to general nonlinear models. In general, a self-exciting

autoregressive (SETAR) model with K regimes, SETAR(K,p1,...,pK), is given by:

yt = α0

(k) + ∑

i=1

p(k)

αi

(k)yt-i + σ(k)ut , for τk-1 < yt-d ≤ τk, k = 1,...,K. (27.)

In (25.), d represents the delay parameter, τk represent the thresholds that define

the regimes (-∞ = τ0<τ1<...<τK<τK+1=+∞) and ut is an iid disturbance term. This

specification allows the order of the AR(pk) process and the variance of the

disturbance term to vary across regimes.

To estimate the parameters we use the procedures of Tong (1983) and

conduct a grid search to choose the threshold value, delay parameter and orders

of the regimes by minimizing the normalized (the sum of the AIC for each regime)

AIC.12 We allow the delay parameter to vary from one to three and the switching

parameter to vary from the 15th to the 85th percentile of the distribution of the

series. Finally, we standardize the residuals from each regime before applying

our tests.

First, we follow Ham and Sayers (1990) and fit a SETAR model with two

regimes to the data. As expected, the regimes are determined by whether the

unemployment rate is increasing or decreasing. The first regime consists of the

expansionary months in which the rate was decreasing in the previous month

(xt-1 ≤ -.0145), while the second regime represents the contractionary periods. In

 12 Selecting the parameters by minimizing the mean squared error does not significantly alter the

results.
Page 15 Revision

fact, the expansionary regime is longer and slower with an order of 11 and

smaller coefficients on the first two lagged terms. Although there is an

improvement in the AIC, both the BDS and GD tests still reject the null

hypothesis, even though the residuals have been standardized.

These rejections of both an AR and SETAR model contrast with those of

Hansen (1994), who provides a test which has a null of an AR(p) and an

alternative of a SETAR(2,p1,p2). Using a model similar to Potter's model of U.S.

GNP growth rates, Hansen shows that the hypothesis of a single regime (AR(p)

model) cannot be rejected. Pesaran and Potter (1994), however, find that when

the alternative is a more general SETAR model, the null hypothesis of linearity

can be rejected.

To attempt to capture this remaining dependence, we estimate three

models that are generalizations of the two-regime SETAR model: an exponential

autoregressive (EAR) model (as in Rothman (1992a)), a threshold model similar to

that in Beaudry and Koop (1993) and a four-regime SETAR model.

The advantage of the EAR model is that it allows for smooth transitions

between regimes. Basically, the EAR(p) model is an AR(p) model with additional

terms that depend on the delay parameter and is defined by:

yt

 = ∑

i=1

p

φiyt-i + ut , (28.)

where

φi

 = ai + πiexp(-γy2

t-1). (29.)

 Table 3 shows that both the BDS and GD tests indicate misspecification.

Next we estimate a threshold model similar to that in Beaudry and Koop

(1993). Their model is an AR(p) model with an additional depth-of-recession

variable. While they apply their model to GNP, we apply their model to

unemployment data by measuring the depth of a recession as the difference
Page 16 Revision

between the current unemployment rate and the previous twelve month low. Our

application of their model is given by:13

yt = α0 + ∑

i=1

p

αiyt-i + CDRt-1 + ut, (30.)

where

CDRt-1 = yt-1 - min{yt-12-1,...,yt-1}. (31.)

Again, table 3 shows that the BDS and GD tests suggest misspecification.

Finally, because Pesaran and Potter (1994) find that periods of high

expansions and contraction are significantly different than mild expansions and

contractions, we estimate a four regime SETAR model. The AIC suggests that

this model fits better than the previous models and the switching parameters of

-.016, 0, and .038 capture the regimes suggested by Mitchell. The first regime

consists of fairly substantial expansions (recoveries), the second consists of mild

expansions, the third consists of mild contractions and the fourth consists of

recessions. Unlike the previous models, the BDS test fails to reject the null of

independence. The GD test, however, rejects the null. This suggests that the

problem is not dependence among the residuals, but dependence between the

residuals and the regressors. We can show that the acceptance of the four

regime model by the BDS is obtained in part because the residuals in each

regime are standardized before the test is applied. When the BDS test is applied

to the non-standardized residuals, it rejects the null of independence with a value

of 3.32.

We can also use the GD test to examine each regime separately. As table 3

show, the GD test for the first three regimes is about the same as for the whole

series and is about 1.93 for the last regime. These values coupled with the BDS

 13 Pesaran and Potter (1994) show how this model can be viewed as a generalized SETAR model.
Page 17 Revision

test's failure to reject suggests that for this model the specification of the

functional form within each regime is more of a concern than dependence in the

residual.

IV Conclusion

In this paper we have presented a nonparametric test for the independence

of a variable from a vector and described how this test can be used as a

specification test in both cross-sectional and time series models. Monte Carlo

simulations suggest that the test has power in several models in iid and serially

dependent data. Finally, we have used the GD test in conjunction with the BDS

test to detect misspecification in models of U.S. unemployment rates.
Page 18 Revision

Figures

Figure 1: Size of Tests with 50 Observations

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 0.05 0.1 0.15 0.2 0.25

GD 1 lag

GD 4 lags

BDS

45o Line

Figure 2: Size of Tests with 250 Observations

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.05 0.1 0.15 0.2 0.25

GD 1 lag

GD 4 lags

BDS

45o Line
Page 19 Revision

Figure 3: Power of Tests Against Model 1 (Nonlinear AR)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.05 0.1 0.15 0.2 0.25

BDS (50 OBS)

BDS (250 obs)

GD (50 obs)

 GD (250 obs)

45o Line

Figure 4: Power of Tests Against Model 2 (Nonlinear MA)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.05 0.1 0.15 0.2 0.25

BDS (50 obs)

BDS (250 obs)

 GD (50 obs)

 GD (250 obs)

45o Line
Page 20 Revision

Figure 5: Power of Tests Against Model 3 (Sign AR)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.05 0.1 0.15 0.2 0.25

BDS (50 obs)

BDS (250 obs)

 GD (50 obs)

 GD (250 obs)

45o Line

Figure 6: Power of Tests Against Model 4 (AR/ARCH)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.05 0.1 0.15 0.2 0.25

 BDS (50)

BDS (250)

 GD (50)

 GD (250)

45o Line
Page 21 Revision

Figure 7: Power of Tests Against Model 5 (SETAR)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.05 0.1 0.15 0.2 0.25

BDS (50 obs)

BDS (250 obs)

 GD (50 obs)

 GD (250 obs)

45o Line

Figure 8: Power of Tests Against Distributed Lag with Misspecified Functional

Form

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.05 0.1 0.15 0.2 0.25

BDS (50 obs)

BDS (250 obs)

 GD (50 obs)

 GD (250 obs)

45o Line
Page 22 Revision

Figure 9: Power of Tests Against Distributed Lag with Misspecified Residual

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.05 0.1 0.15 0.2 0.25

BDS (50 obs)

BDS (250 obs)

 GD (50 obs)

 GD (250 obs)

45o Line
Page 23 Revision

Figure 10: Monthly Unemployment Rate 1948-1993 (seasonally adjusted)

0

2

4

6

8

10

12

1948194919501951195219531954195519561957195819591960196119621963196419651966196719681969197019711972197319741975197619771978197919801981198219831984198519861987198819891990199119921993

Period of Recession

Unemployment rate
Page 24 Revision

Tables

Table 1: Models Used in Time Series Misspecification Testing

Number True Model Name of Model

Model 1 yt = .7yt-1/(yt-2+2) + ut Nonlinear AR

Model 2 yt = .8ut-2ut-1 + ut Nonlinear MA

Model 3

yt = 



-1 + ut if yt-1< 0

1 + ut otherwise

Sign AR

Model 4 yt = .9yt-1+ ut, ut ~ N(0, (1+ .25ut-1)2) AR/ARCH

Model 5

yt

 = 



0.62 +1.25yt-1 -0.43yt-2 +u1t if yt-2≤3.25

2.25 +1.52yt-1 -1.24yt-2 +u2t otherwise

u1t ~ N(0,.0381); u2t ~ N(0,.0626)

SETAR

In each case, u is drawn from a standard Normal distribution

Table 2: Models used in Misspecification of Distributed Lag Models

Number True Model Model Misspecification

Model 6 yt = 2xt + 1.5zt - xtzt + et

xt = .5xt-1 - .2xt-2 + u1,t

zt = .5zt-1 + u2,t

et = u3,t

xtzt omitted

Model 7 yt = 2xt + 1.5zt - xtzt + et

xt = .5xt-1 - .2xt-2 + u1,t

zt = .5zt-1 + u2,t

et = .7et-1 + u4,t

et normalized to have zero mean

and a standard deviation of 1

et-1 omitted

 u is drawn from a standard Normal distribution
Page 25 Revision

Table 3: Statistics on Alternative Models of Monthly Unemployment Rates

Model AIC σ2 BDS Test GD Test

AR(10) -6.486 .00147 4.692 5.403

SETAR(2;11,7) -6.566 .00133 5.079 5.357

 Regime I: xt-1≤ -.0145

 Regime II: xt-1> -.0145

-6.775

-6.459

.00101

.00150

2.048

5.817

EAR -6.568 .00134 4.748 4.254

Beaudry/Koop -6.495 .00144 4.193 5.389

SETAR(4;10,9,2,3) -6.624 .00129 1.069 2.489

 Regime I: xt-1≤ -.0156 -6.723 .00107 2.664

 Regime II: -.0156< xt-1≤0 -7.029 .00076 2.365

 Regime III: 0<xt-1≤ .0384 -6.556 .00138 2.595

 Regime IV: .0384 ≤ xt-1 -5.904 .00249 1.929
Page 26 Revision

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