Testing Weak Convergence Based on HAR Covariance Matrix ...

Testing Weak Convergence Based on HAR Covariance Matrix Estimators

Jianning Kongy, Peter C. B. Phillipsz, Donggyu Sulx August 4, 2017

Abstract

The weak convergence tests based on heteroskedasticity autocorrelation robust (HAR) covariance matrix estimators are considered. Asymptotic limits are derived. By means of numerical simulations, we evaluate the performance of the trend regression tests based on HAR estimators. We ...nd, however, that the improvement of alternative tests is very limited.

Keywords: Weak convergence. HAR estimator, HAC estimator.

JEL Classi...cation: C33

1 Introduction

Valid statistical inference on spurious trend regressions have been of interest. Sun (2004) pointed out that the valid statistical inference can be obtained by using the heteroskedasticity and autocorrelation consistent (HAC) standard error estimator with a bandwidth proportional to the sample size. Later the t statistics based on the HAC estimator without truncation is called `heteroskedasticity autocorrelation robust'(HAR) test statistics. (See Kiefer and Vogelsang, 2002; Phillips, 2005; Phillips, Zhang and Wang, 2012) Consider the following simple trend regression,

xt = at + zt;

(1)

where zt = zt 1 + ut: Since zt is I (1) ; the trend regression in (1) becomes spurious: Testing

H0 : a = 0 is not easy with the following conventional t-statistic.

ta =

a^

T

1

PT

t=1

z^t2

T

1

PT

t=1

t

1 1=2 ; with a^ = PPTtTt==11ttx2t :

Phillips acknowledges NSF support under Grant No. SES-1285258. yShandong University, China zYale University, USA; University of Auckland, New Zealand; Singapore Management University, Singapore;

University of Southampton, UK. xUniversity of Texas at Dallas, USA

1

Phillips, Zhang and Wang (2012, PZW hereafter) show that ta diverges under the null, and suggest to use the following t-ratio with HAR estimator.

tHa AR =

PT

t=1

t2

a^

1

h T

^

i

HAR

PT

t=1

t2

1 1=2 ;

where

^ HAR

=

1 T

XT

t=1

2 t

+

2 T

XL

`=1

XT `

t=1

1

` L+1

t t+`;

with L = bgT c for some g 2 (0; 1); and t = z^tt: The underlying intuition is rather simple. As the serial dependence of the regression the error, zt; goes to extremely, more larger lag length are needed. When the error becomes nonstationary, the in...nite lag length leads to a constant t-ratio.

Recently, Kong, Phillips and Sul (2017, KPS hereafter) propose the following simple trend

regression for testing the weak convergence.

Knt = a + t + ut;

(2)

where Knt is the Suppose that yit

sample cross = ai + t +

sectional yiot; then

variance Knt =

1oP f tnhe

n i=1

idyiiotsynn1crPatnii=c 1cyoimt p2o:nSeinntcseotfhae

panel data. underlying

data generating process is unknown, this trend regression is misspeci...ed unless the data generating

process is given in (2). KPS (2017) consider the following t-ratio .

^

t1 = r ^2

1

nT

PT

t=1

t~2

;

1

(3)

where

^

nT

is

the

estimate

of

the

slope

coe?

cient,

t~ =

t

T

1

PT

t=1

t;

and

^2

1

is

de...ned

as

^2

1

=

1 T

XT

t=1

u~2t

+

2 T

XM XT `

`=1 t=1

1

` M + 1 u~tu~t+`;

where u~t = K~nt

^

nT

t~ with

K~ nt

=

Knt

T

1

PT

t=1

Knt

;

and

M

=

bT

c: In fact, the trend regression

in (2) is misspeci...ed if the data generating process is given by

Knt = a + bt + et;

(4)

where et = Op n 1=2 usually: The trend regression considered by KPS in (2) is not spurious in the sense that the regression

error in (4) is Op n 1=2 ; but misspeci...ed. Interestingly, due to this misspeci...cation, the residual has a spurious trend. That is, u^t = K~nt ^t~ = btg + e~t ^t~; where `~'stands for the deviation from its time series mean. Unless = 0; the least squares estimator, ^; is not equal to zero, so that

the regression residual has a spurious trend, which inuences on the long run variance calculation.

Here we investigate whether or not the HAR type correction improves the testing result proposed

by KPS (2017).

2

2 Alternative Estimators

Even though the long run variance formula follows the Newey-West HAC estimator, the variance

of

^

nT

is

not

a

typical

sandwich

form.

Hence

the

use

of

other

long

run

variances

of

^

nT

becomes

of

interest. In KPS (2017), the hypothesis of interest is weak convergence. That is, HA : nT < 0:

When

>

0;

then

the

least

squares

estimate

of

^

nT

approaches

to

zero

as

n; T

!

1;

but

the

t1

statistic in (3) diverges negative in...npity if

1: Even when the decay value, ; becomes large,

the t1 statistic is still converging 3: Also more importantly, the t1 statistic is discontinuous

at = 0: When = 0; the limiting distribution of the t1 statistic becomes a standard normal.

However as deviates slightly from zero, the t1 statistic diverges.

We consider the following alternative t-ratios.

^

t2 = r ^2

2

nT

PT

t=1

t~2

;

1

(5)

Let p~t = u~tt~; and de...ne

^

tHAR

=

r

PT

t=1

t~2

nT

1

T

^

2 L

PT

t=1

t~2

;

1

(6)

^

tHAC

=

r

PT

t=1

t~2

nT

1

T

^

2 M

PT

t=1

t~2

;

1

(7)

where

^2

L

=

1 T

XT

t=1

p~2t

+

2 T

XL XT `

`=1 t=1

1

^2

M

=

1 T

XT

t=1

p~2t

+

2 T

XM XT `

`=1 t=1

1

^2

2

=

1 T

XT

t=1

u~2t

+

2 T

XL XT `

`=1 t=1

1

` L + 1 p~tp~t+` where L = bgT c for some g 2 (0; 1);

` M + 1 p~tp~t+` where M = bT c for some 2 (0; );

` L + 1 u~tu~t+`

Interestingly, the asymptotic behavior of the tHAC is somewhat di?erent from the t1 statistic, meanwhile the asymptotic properties of the tHAR becomes very distinct. The next theorem provides more details.

Theorem 1 (Asymptotic Properties)

Under regularity conditions, the t-ratio statistics have the following asymptotic behaviors

3

as n; T ! 1 :

t1

=

8

>>>>>>><

O O

O

T 1=2 T 1=2 T1

=2

=2 (ln T ) 1=2

=2

>>>>>>>:

OpT (1 p6=

)(1

)=2

if < 1=2; if = 1=2; if 1=2 < < 1= (1 + ) ; , if 1= (1 + ) < 1; if = 1;

(8)

3

if > 1:

8

>>>>><

O (1) O (ln T )

1=2

if < 1=2; if = 1=2;

t2

=

>>>>>:

Op(1) p6 3

if 1=2 < 1; ; if = 1; if > 1:

(9)

8 < O (1) > 0 if < 0

8 <

O

T (1

)=2 > 0

if

0

;

and tHAC

=:

0 O

T (1

)=2 < 0

if if

=0 >0

(10)

See the online appendix for the detailed proof of Theorem 1. Note that the result for the t1 statistic

in (8) is proved by KPS (2017). Except for a few speci...c cases, the asymptotic orders of the t-

ratios with HAR estimators can be obtained by replacing by unity. Here we provide some intuitive

results based on the following numerical simulation. We generate Knt by setting Knt = a + bt

by

assuming

et

=

0

for

all

t;

and

then

calculate

^

nT

;

and

the

t

ratios.

Figure 1 shows the numerical simulation results for the t1 and the t2 statistics. As KPS dpemon-

strates, the t1 statistic is discontinous at = 0; and as increases, the t1 value converges 3: Of

course, the t1 values decreases as T increases as long as 0 < < 1: The asymptotic behavior of the

t2pstatistic is somewhat similar to that of the t1 : The t2 is discontinuous at = 0 and converges 3 as ! 1: However, the biggest di?erence is the time varying behavior. The t2 statistic is

almost independent from the size T except for the case ofp = 1=2: Even when = 1=2; the t2 statistic changes little since the asymptotic order is O 1= ln T ; which is an extremely slowly

moving function.

Figure 2 plots the numerical simulation values of the tHAR and tHAC statistics. With a ...xed T; we can observe the fact that jtHARj jtHACj where the equality holds only when = 0: Moreover, surprisingly the value of jtHARj is not big enough to reject the null of no convergence even when

< 0: Also both t-ratios are not discontinuous at = 0 so that there exist regions where the test

becomes inconsistent. For example, As it shown in Figure 2, if 0 < < 0:1; then both tHAC and tHAR are not small enough to reject the null of no convergence even with T = 10; 000: The tHAR does not change over T meanwhile the absolute value of the tHAC becomes larger as T increases.

3 Concluding Remark

We investigate various types of long run variance estimators for testing the weak convergence. Among them, we show that the original estimator suggested by KPS is the most e?ective unless the convergence speed is very fast. When the convergence speed is very fast, the typical sandwich form with the HAC estimator provides better result.

4

References

[1] Kiefer, N.M., Vogelsang, T.J. (2002a). Heteroskedasticity-autocorrelation robust testing using bandwidth equal to sample size. Econometric Theory 18:1350-1366.

[2] Kong, J., P.C.B. Phillips, and D. Sul (2017). Weak convergence: Theory and Applications, Mimeo, University of Texas at Dallas.

[3] Sun, Y. (2004). A convergent t-statistic in spurious regression. Econometric Theory 20:943-962.

[4] Phillips, P.C.B. (2005). HAC estimation by automated regression. Econometric Theory 21:116? 142.

[5] Phillips, P.C.B., Y. Zhang and X. Wang (2012). Limit theory of three t statistics in spurious regressions. Mimeo, Singapore Management University.

t-ratio

25

20

t2 with T=10000

15

t1 with T=10000

t2 with T=1000

10

t1 with T=1000

5

0

-5

-10

-15

-20

-25 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 lambda

Figure 1: Numerical Simulations for t1 and t2 (g = = 0:5)

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