cointegration test ardl bounds testing - … · cointegration test – ardl bounds testing ......

Prepared by Kelly Wong 2009

Cointegration Test – ARDL Bounds Testing

The VAR(p) model can be rewritten in vector ECM form as:

∆𝑧𝑡 = 𝑎0 + 𝑎1𝑡𝑟𝑒𝑛𝑑 + 𝛑𝑧𝑡−1 + ∀𝑖∆𝑧𝑡−𝑖

𝜌−1

𝑖=1

+ 𝜀𝑡

where

∆ = 1 – L is the difference operator,

zt = f(yt, xt)

we now partition the long-run multiplier matrix 𝜋 conformably with zt = (yt, x’t)’ as

𝜋 = 𝜋𝑦𝑦 𝜋𝑦𝑥

𝜋𝑥𝑦 𝜋𝑥𝑥

Under the assumption 1, 3, and 4 (see Pesaran et al. 2001), 𝜋 has rank r and is given by

𝜋 = 𝜋𝑦𝑦 𝜋𝑦𝑥

0 𝜋𝑥𝑥

Consequently, the conditional ECM may be written as following:

∆𝑦𝑡 = 𝑎0 + 𝑎1𝑡𝑟𝑒𝑛𝑑 + 𝛑𝑦𝑦 𝑦𝑡−1 + 𝛑𝑦𝑥 .𝑥𝑥𝑡−1 + ∀𝑖∆𝑧𝑡−𝑖

𝜌−1

𝑖=1

+𝑤 ′∆𝑥𝑡 + 𝜀𝑡

If the 𝜋𝑦𝑦 ≠ 0 and 𝜋𝑦𝑥 .𝑥 = 0′ , the yt is (trend) stationary or I(0), whatever the value r.

Consequently, the differenced variable ∆𝑦𝑡 depends only on its own lagged level yt-1 in the

conditional ECM. Second, if 𝜋𝑦𝑦 = 0 , the ∆𝑦𝑡 depends only on the lagged level xt-1 in the

conditional ECM model. Therefore, in order to test for the absence of level effects in the

conditional ECM model and more crucially, the absence of a level relationship between yt and xt,

the emphasis in this approach is a test of the joint hypothesis the 𝜋𝑦𝑦 = 0 and 𝜋𝑦𝑥 .𝑥 = 0′ in the

above model.

According to Pesaran et al. (2001), there are 5 cases provided for testing the cointegrating bound

test:

Case 1: (no intercepts; no trends) a0 and a1 = 0.

Case 2: (restricted intercepts; no trends) a0 = - (𝜋𝑦𝑦 , 𝜋𝑦𝑥 .𝑥)𝜇 and a1 = 0.


Case 3: (unrestricted intercepts; no trends) 𝑎0 ≠ 0 and 𝑎1 = 0.

Case 4: (unrestricted intercepts; restricted trends) 𝑎0 ≠ 0 and a1 = - (𝜋𝑦𝑦 , 𝜋𝑦𝑥 .𝑥)𝜇

Case 5: (unrestricted intercepts; unrestricted trends) 𝑎0 ≠ 0 and 𝑎1 ≠ 0

______________________________________________________________________________

If the Wald F-statistic fall above the upper critical value – cointegrated

If the Wald F-statistic falls between the lower bound and upper bound critical value –

inconclusive

If the Wald F-statistic falls below the lower bound critical value – no cointegration

______________________________________________________________________________

Example: Based on the theory, the financial development will be affected the national poverty.

So, it is likely to investigate whether the financial development have decreases the Malaysian

poverty gap. Testing the regression model - RPCC = f(RGDPC, UN, WAGE, TAX, LL) by using

the secondary data from poverty and fd.xls file.

Rewriting the ARDL Bound Cointegration Test model:

∆𝑅𝑃𝐶𝐶𝑡 = 𝑐 + 𝛽1𝑅𝐺𝐷𝑃𝐶𝑡−1 + 𝛽2𝑈𝑁𝑡−1 + 𝛽3𝑊𝐴𝐺𝐸𝑡−1 + 𝛽4𝑇𝐴𝑋𝑡−1 + 𝛽5𝐿𝐿𝑡−1 +

𝛼1𝑖

𝑝

𝑖=1

∆𝑅𝑃𝐶𝐶𝑡−𝑖 + 𝛼2𝑖

𝑝

𝑖=0

∆𝑈𝑁𝑡−𝑖 + 𝛼3𝑖

𝑝

𝑖=0

∆𝑊𝐴𝐺𝐸𝑡−𝑖 +

𝛼4𝑖

𝑝

𝑖=0

∆𝑇𝐴𝑋𝑡−𝑖 + 𝛼5𝑖

𝑝

𝑖=0

∆𝐿𝐿𝑡−𝑖 + 𝛾𝐷𝑈𝑀

where RPCC = real Consumption Per capita

C = constant

RGDPC = real GDP Per capita

UN = unemployment

WAGE = Wages

TAX = Individual Tax

LL = Liquid Liability

p = optimum lag length uses in model


DUM = 1 for crisis and 0 for otherwise; the crisis is refer to the Malaysian oil

crisis at 1973, 1974, 1980 and 1981; commodities crisis at 1985 to 1986; and 1997/98 for

Asian financial Crisis.

Firstly, we should examine this Bound Testing model start from the higher lag length. For

example, in this model have only 38 observations and consider as small size sample time series

model. In order to avoid the overparameter problem, this example start with the maximum lag

order 2 (maximum lag) and then reduces to maximum lag 1:

Definition for the Eviews code:

d = change

-1 = lag one variable = time T – 1

c = constant term

Go to Quick – Estimate Equation –

Type in the model.


Dependent Variable: D(RPCC)

Method: Least Squares

Sample (adjusted): 1973 2004

Included observations: 32 after adjustments Variable Coefficient Std. Error t-Statistic Prob. C 13.77331 2.126913 6.475728 0.0003

RPCC(-1) 0.102209 0.124537 0.820713 0.4389

RGDPC(-1) -2.033942 0.344902 -5.897155 0.0006

UN(-1) -0.089575 0.026223 -3.415877 0.0112

WAGE(-1) -0.520555 0.080194 -6.491173 0.0003

TAX(-1) 0.563704 0.096699 5.829448 0.0006

LL(-1) -0.050592 0.056886 -0.889364 0.4033

D(RPCC(-1)) -0.618619 0.221448 -2.793518 0.0268

D(RPCC(-2)) -0.508969 0.173976 -2.925506 0.0222

D(RGDPC) 0.558362 0.143865 3.881151 0.0060

D(RGDPC(-1)) 1.912944 0.372013 5.142141 0.0013

D(RGDPC(-2)) 0.683437 0.278168 2.456918 0.0437

D(UN) -0.110758 0.045845 -2.415939 0.0464

D(UN(-1)) -0.015647 0.031613 -0.494933 0.6358

D(UN(-2)) -0.026159 0.024813 -1.054212 0.3268

D(WAGE) -0.179064 0.035233 -5.082250 0.0014

D(WAGE(-1)) 0.124112 0.048005 2.585401 0.0362

D(WAGE(-2)) -0.067168 0.022898 -2.933300 0.0219

D(TAX) 0.166026 0.044277 3.749698 0.0072

D(TAX(-1)) -0.216132 0.044101 -4.900844 0.0018

D(TAX(-2)) -0.069359 0.028300 -2.450836 0.0441

D(LL) -0.107055 0.033535 -3.192336 0.0152

D(LL(-1)) -0.103247 0.028797 -3.585338 0.0089

D(LL(-2)) -0.022240 0.021554 -1.031829 0.3365

DUM -0.008776 0.014612 -0.600579 0.5670 R-squared 0.992074 Mean dependent var 0.031780

Adjusted R-squared 0.964899 S.D. dependent var 0.056345

S.E. of regression 0.010556 Akaike info criterion -6.221481

Sum squared resid 0.000780 Schwarz criterion -5.076375

Log likelihood 124.5437 Hannan-Quinn criter. -5.841910

F-statistic 36.50675 Durbin-Watson stat 2.664106

Prob(F-statistic) 0.000030

Diagnostic Checking for Serial LM test – Click ―View‖ – ―Residual Test‖ – ―Serial Correlation

LM Test‖:

Breusch-Godfrey Serial Correlation LM Test: F-statistic 1.264529 Prob. F(2,5) 0.3594

Obs*R-squared 10.74900 Prob. Chi-Square(2) 0.0046




After the Lag 2 model, we have attempts to reduce the lag to be lag 1 models. The empirical

result is shows as following:





RPCC(-1) -0.113193 0.180452 -0.627273 0.5406

RGDPC(-1) -0.753554 0.275372 -2.736499 0.0161

UN(-1) -0.036190 0.039083 -0.925960 0.3702

WAGE(-1) -0.232102 0.066913 -3.468741 0.0038

TAX(-1) 0.238435 0.076828 3.103505 0.0078

LL(-1) -0.002917 0.065584 -0.044480 0.9652

D(RPCC(-1)) 0.047289 0.192419 0.245759 0.8094

D(RGDPC) 0.808263 0.231515 3.491185 0.0036

D(RGDPC(-1)) 0.739431 0.295086 2.505812 0.0252

D(UN) -0.025586 0.065172 -0.392589 0.7005

D(UN(-1)) 0.033920 0.042539 0.797381 0.4385

D(WAGE) -0.124827 0.057956 -2.153823 0.0492

D(WAGE(-1)) -0.000677 0.041739 -0.016213 0.9873

D(TAX) 0.074923 0.049829 1.503584 0.1549

D(TAX(-1)) -0.068402 0.044450 -1.538861 0.1461

D(LL) -0.057921 0.043217 -1.340219 0.2015

D(LL(-1)) -0.060747 0.031568 -1.924284 0.0749








Series LM Test:






After the Eviews output, we need to prepaid a table to record the AIC, SBC, and Serial LM test.

For example:

Table Exp.1. Selected Maximum Lag Base on difference Method

P AIC SBC )2(SCx )4(SCx

2 -6.2215 -5.0764 10.74*** 15.30***

1 -4.6322 -3.7706 4.4051 10.945**

Note: p is the lag order of the underlying VAR model for the conditional ECM ( ), with zero restrictions on

the coefficients of lagged changes in the independent variables. AICp = (-2l / T) + (2k / T) and SBCp = (-

2l / T) + (k * logT / T) denote Akaike’s and Schwarz’s Bayesian Information Criteria for a given lag order

p, where l is the maximized log-likelihood value of the model, k is the number of freely estimated

coefficients and T is the sample size. The AIC and SBC are often used in model selection for non-nested

alternatives—lowest values of the AIC and SBC are preferred (refer to Eviews Users Guide 4.0, pp. 279).

Xsc (1) and Xsc (4) are LM statistics for testing no residual serial correlation against orders 2 and 4. The

symbols ***, and ** denote significance at 0.01, and 0.05 levels, respectively.

The above estimated results showed that serial correlation problem exists in all lag models.

Hence, we should reconsider other methods for chosen the optimum lag length such as Hendry’s

General to specific model. Again, we start the model at maximum lag 2. After that, delete the

higher insignificant lag for the changes variables. For example: the D(LL(-2)) should be drop out

from the model first and then following by D(UN(-2)) and D(UN(-1)). At the end, the result

should be as follow:






RPCC(-1) 0.055598 0.104682 0.531110 0.6069

RGDPC(-1) -1.817123 0.270645 -6.714037 0.0001

UN(-1) -0.074733 0.021336 -3.502689 0.0057

WAGE(-1) -0.469390 0.062083 -7.560639 0.0000

TAX(-1) 0.523991 0.082461 6.354384 0.0001

LL(-1) -0.087001 0.042415 -2.051171 0.0674

D(RPCC(-1)) -0.513899 0.177106 -2.901654 0.0158

D(RPCC(-2)) -0.442000 0.151922 -2.909383 0.0156

D(RGDPC) 0.634967 0.115742 5.486054 0.0003

D(RGDPC(-1)) 1.749137 0.290044 6.030600 0.0001

D(RGDPC(-2)) 0.609775 0.226866 2.687824 0.0228

D(UN) -0.081451 0.034959 -2.329896 0.0421

D(WAGE) -0.166054 0.029206 -5.685552 0.0002

D(WAGE(-1)) 0.103437 0.039845 2.595963 0.0267

D(WAGE(-2)) -0.068902 0.020769 -3.317622 0.0078

D(TAX) 0.152515 0.039787 3.833305 0.0033

D(TAX(-1)) -0.203931 0.038001 -5.366529 0.0003

D(TAX(-2)) -0.072995 0.025175 -2.899487 0.0158

D(LL) -0.111729 0.030455 -3.668680 0.0043

D(LL(-1)) -0.075093 0.016830 -4.461815 0.0012












After the Hendry’s

method applied, the

problems of serial

correlation were

overcome.

After the Hendry’s

method applied, the

problems of serial

correlation were not

significant at 5%

significance level.


After specified the optimum lag model, then we should process to the ARDL Cointegration

Bound Testing. Firstly, we need to understand the code used in the Eviews program represented.

Second, according to Pesaran et al. (2001), if the coefficients between lag one variables are

jointly fall above the upper bound critical value, then indicates that these estimated variables

exists a long-run cointegration relationship. In order to testing this hypothesis, we need estimate

the coefficient RPCC(-1) = RGDPC(-1) = UN(-1) = WAGE(-1) = TAX(-1) = LL(-1) = 0;

rewriting in the Eviews code as c(2) = c(3) = c(4) = c(5) = c(6) = c(7) = 0. Hence, now should

click on the ―view‖ button – ―Coefficient Tests‖ – ―Wald coefficient restrictions‖ .





RPCC(-1) 0.055598 0.104682 0.531110 0.6069

RGDPC(-1) -1.817123 0.270645 -6.714037 0.0001

UN(-1) -0.074733 0.021336 -3.502689 0.0057

WAGE(-1) -0.469390 0.062083 -7.560639 0.0000

TAX(-1) 0.523991 0.082461 6.354384 0.0001

LL(-1) -0.087001 0.042415 -2.051171 0.0674

D(RPCC(-1)) -0.513899 0.177106 -2.901654 0.0158

D(RPCC(-2)) -0.442000 0.151922 -2.909383 0.0156

D(RGDPC) 0.634967 0.115742 5.486054 0.0003

D(RGDPC(-1)) 1.749137 0.290044 6.030600 0.0001

D(RGDPC(-2)) 0.609775 0.226866 2.687824 0.0228

D(UN) -0.081451 0.034959 -2.329896 0.0421

D(WAGE) -0.166054 0.029206 -5.685552 0.0002

D(WAGE(-1)) 0.103437 0.039845 2.595963 0.0267

D(WAGE(-2)) -0.068902 0.020769 -3.317622 0.0078

D(TAX) 0.152515 0.039787 3.833305 0.0033

D(TAX(-1)) -0.203931 0.038001 -5.366529 0.0003

D(TAX(-2)) -0.072995 0.025175 -2.899487 0.0158

D(LL) -0.111729 0.030455 -3.668680 0.0043

D(LL(-1)) -0.075093 0.016830 -4.461815 0.0012








C(1)

C(2)

C(3)

C(4)

C(5)

C(6)

C(7)

C(8)

c(9)

C(10)

C(11)

C(12)

C(13)

C(14)

C(15)

C(16)

C(17)

C(18)

C(19)

C(20)

C(21)

C(22)


Wald Test:

Equation: Untitled Test Statistic Value df Probability F-statistic 14.63692 (6, 10) 0.0002

Chi-square 87.82149 6 0.0000

After click on the Wald –

Coefficient Restrictions

test, the Eviews Program

will occurs a empty

windows for put in the

null hypothesis testing. In

our Example, we should

type in the code as

showed in the figure.

Compare this F-statistic with the

Narayan (2005) Critical value based

on the case you chose before, if your

sample size is relative small.


Prepare a table to record the information, such as:

Table Exp.2. F-statistics for testing the existence of levels Poverty equation

Model F-statistic

Model 1: RPCC = f (RGDPC, UN, WAGE, TAX, LL)

14.6369

Narayan (2005) k = 6, n=35

Critical Value Lower bound Upper bound

1%

5%

10%

4.016

2.864

2.387

5.797

4.324

3.671

Notes: *, **, and *** denote significant at 10%, 5%, and 1% significance level, respectively. Critical

values are cited from Narayan (2005) (Table Case III: Unrestricted intercept and no trend; pg. 1988).

In this Example, the model shows that the financial development and other determinant variables

are strongly cointegrated with poverty in Malaysia. The results showed that the F-statistic

compute by Wald Test is highly significant at 1% significance level.


Using Eviews to construct an ARDL Bound Test

1. These criteria are suggested for author to select the optimum lag in the ARDL modeling,

namely

1. Akaike Information Criterion

2. Schwarz Bayesian Criterion

3. General to specific model

For Example: Our regression model – RPCC = f(RGDPC, UN, WAGE, TAX, LL) (See the file poverty

and fd) and the selected optimum lag length is (1, 1, 0, 0, 0, 0)

ARDL Model – Equation (1):

it

r

i

tit

q

i

tit

p

i

t UNRGDPCRPCCconstRPCC0

,3

0

,2

1

,1

ttit

v

i

tit

u

i

tit

s

i

t DUMLLTAXWAGE

,7

0

,6

0

,5

0

,4

where

RPCC = real Consumption Per capita

Const = constant

RGDPC = real GDP Per capita

UN = unemployment

WAGE = Wages

TAX = Individual Tax

LL = Liquid Liability

DUM = 1 for crisis and 0 for otherwise; the crisis is refer to the Malaysian oil crisis at 1973,

1974, 1980 and 1981; commodities crisis at 1985 to 1986; and 1997/98 for Asian financial

Crisis.

p, q, r, s, u, v = optimum lag length uses in model

t = residual

Such based on the AIC or SBC criteria, the selected lag length for this model (p, q, r, s, u, v) is (1, 1,

0, 0, 0, 0). This can use Microfit or Rats programming code to obtain the optimum lag base on

such listed criteria.


2. After selected lag length, using Eviews to estimate the Long-run OLS. The Eviews output

is showed as following:

Dependent Variable: RPCC


Date: 06/28/09 Time: 21:23


Included observations: 34 after adjustments

Variable Coefficient Std. Error t-Statistic Prob.

C 1.380580 1.142938 1.207922 0.2384

RPCC(-1) 0.688508 0.175616 3.920539 0.0006

RGDPC 0.981537 0.179936 5.454926 0.0000

RGDPC(-1) -0.866497 0.237380 -3.650252 0.0012

UN 0.010227 0.036818 0.277762 0.7835

WAGE -0.019567 0.047700 -0.410202 0.6852

TAX 0.071253 0.047672 1.494647 0.1475

LL -0.095664 0.036619 -2.612439 0.0150

DUM 0.015761 0.016593 0.949891 0.3513

R-squared 0.992724 Mean dependent var 7.949910







3. After the ARDL model, we have using the ―Wald Test‖ to compute the long run

elasticities and it standard error.

According to Pesaran et al. (2001) the Long run elasticities should compute as follow:

p

i

t

q

i

t

RGDPCforesElasticiti

,1

,2

1

__

= Sum of the independent coefficients (RGDPC)

1 – Sum of the dependent coefficients

The coefficient for the

variables in the Eviews

output is shows as:

c(1)

c(2)

c(3)

c(4)

c(5)

c(6)

c(7)

c(8)

c(9)


4. Go to ―View‖ – ―Coefficient Test‖ – ―Wald Test‖

Eviews Output:

Wald Test:

Equation: Untitled

Test Statistic Value df Probability

F-statistic 0.507750 (1, 25) 0.4827

Chi-square 0.507750 1 0.4761

Null Hypothesis Summary:

Normalized Restriction (= 0) Value Std. Err.

(C(3) + C(4)) / (1 - C(2)) 0.369318 0.518293

Delta method computed using analytic derivatives.

For others variables elasticities are showed as: constant (4.4322), RGDPC (0.36932), UN (0.032831),

WAGE (-0.062815), TAX (0.22875), LL (-0.30712), DUM (0.050599).

Type in the code:

(c(3)+c(4))/(1-c(2))=0

In the Wald Test windows

The value of elasticities is

shows as 0.369318. The

standard error is

0.518293. However, the t-

statistic need compute by

the user, where t-stat =

coefficient / std. Err.

The probability 0.4827

also represent as p-value

for the computed

elasticities.


5. After computed the all long-run elasticities, we need proceed into short-run Error

correction Model.

a. Firstly, compute the values of Error Correction Term (ECT). Based on the

knowledge the ECT is represent as a long-run steady point for the model or more

statistically the ECT is a residual from long-run cointegration model. Long-run

Cointegration Model (Equation 2):

tp

i

t

s

i

t

tp

i

t

r

i

t

tp

i

t

q

i

t

p

i

t

t WAGEUNRGDPCconst

RPCC

1

,1

0

,4

1

,1

0

,3

1

,1

0

,2

,1 1111

tp

i

t

tp

i

t

v

i

t

tp

i

t

u

i

t

ECTDUMLLTAX

1

,1

7

1

,1

0

,6

1

,1

0

,5

111

(2)

After some mathematical adjustment, the Error correction term equation is shows as:

tp

i

t

s

i

t

tp

i

t

r

i

t

tp

i

t

q

i

t

t WAGEUNRGDPCRPCCECT

1

,1

0

,4

1

,1

0

,3

1

,1

0

,2

111

tp

i

t

v

i

t

tp

i

t

u

i

t

LLTAX

1

,1

0

,6

1

,1

0

,5

11

(3)

Therefore, based on the previous example model and using the calculated elasticities, the Long-run

Cointegrated Model is shows as following:

RPCC = 4.4322 + 0.36932 RGDPC + 0.032831 UN – 0.062815 WAGE + 0.22875 TAX – 0.30712 LL +

0.050599 DUM

Hence, the ECT equation shows as:

ECT = RPCC– 0.36932*RGDPC – 0.032831*UN + 0.062815*WAGE – 0.22875*TAX + 0.30712*LL

So, generate this ECT equation in the Eviews before the Short-run dynamic model.


b. Type in the ECT equation on the upper blank box of Eviews and then “Enter”.

6. After generated the ECT series, now we have go to “Quick” – “Estimate Equation” – choose the

method “TSLS”.

Type in this equation on the top blank

box:

ECT = RPCC – 0.36932*RGDPC –

0.032831*UN + 0.062815*WAGE –

0.22875*TAX + 0.30712*LL

Furthermore, press the “Enter” and the

ECT will shows in the workfile windows.


7. Now the Eviews shows you two boxes, one is for Equation specification and other one is for

instrument list.

a. The Equation Specification is refer to the Short-run dynamic model, which is:

it

q

i

tit

p

i

ttt RGDPCRPCCECTconstRPCC1

0

,2

1

1

,11

it

u

i

tit

s

i

tit

r

i

t TAXWAGEUN1

0

,5

1

0

,4

1

0

,3

ttit

v

i

t DUMLL

,7

1

0

,6

For our Example:

The Equation Specification code is follows:

D(RPCC) C ECT(-1) D(RGDPC) D(UN) D(WAGE)

D(TAX) D(LL) DUM

b. The instrument list refers to the endogenous for ECT models. For our Example, the Eviews code to

represent the exogenous variables for our ECT model is:

C RPCC(-1) RGDPC RGDPC(-1) UN WAGE TAX

LL

Type in this equation on the

Equation Specification box:

D(RPCC) C ECT(-1) D(RGDPC)

D(UN) D(WAGE) D(TAX) D(LL)

DUM

Furthermore, type in the

instrument list:

C RPCC(-1) RGDPC RGDPC(-1) UN

WAGE TAX LL


After that click “Options” and tick that “Heteroskedasticity consistent coefficient covariance” and

“Newey-West” and then click ok.

8. After that you should get the Short-run dynamic results as follows:


Method: Two-Stage Least Squares

Date: 06/29/09 Time: 10:19


Included observations: 34 after adjustments

Newey-West HAC Standard Errors & Covariance (lag truncation=3)

Instrument list: C RPCC(-1) RGDPC RGDPC(-1) UN WAGE TAX LL

Variable Coefficient Std. Error t-Statistic Prob.

C 1.380587 1.160771 1.189371 0.2450

ECT(-1) -0.311494 0.259603 -1.199883 0.2410

D(RGDPC) 0.981566 0.410089 2.393546 0.0242

D(UN) 0.010237 0.146174 0.070031 0.9447

D(WAGE) -0.019561 0.058823 -0.332532 0.7422

D(TAX) 0.071255 0.046388 1.536072 0.1366

D(LL) -0.095667 0.049627 -1.927704 0.0649

DUM 0.015762 0.018952 0.831679 0.4132

R-squared 0.768700 Mean dependent var 0.031574


S.E. of regression 0.029595 Sum squared resid 0.022773


Prob(F-statistic) 0.000001 Second-Stage SSR 0.022774

Click on “Options” and tick

the box of

“Heteroskedasticity

consistent coefficient

covariance” and “Newey-

West”.

And then click “ok”.


As compared to the Microfit Output:

Error Correction Representation for the Selected ARDL Model

ARDL(1,1,0,0,0,0) selected based on Schwarz Bayesian Criterion

*******************************************************************************

Dependent variable is dRPCC

34 observations used for estimation from 1971 to 2004

*******************************************************************************

Regressor Coefficient Standard Error T-Ratio[Prob]

dRGDPC .98154 .17994 5.4549[.000]

dUN .010227 .036818 .27776[.783]

dWAGE -.019567 .047700 -.41020[.685]

dTAX .071253 .047672 1.4946[.147]

dLL -.095664 .036619 -2.6124[.015]

dINPT 1.3806 1.1429 1.2079[.238]

dDUM .015761 .016593 .94989[.351]

ecm(-1) -.31149 .17562 -1.7737[.088]

*******************************************************************************

List of additional temporary variables created:

dRPCC = RPCC-RPCC(-1)

dRGDPC = RGDPC-RGDPC(-1)

dUN = UN-UN(-1)

dWAGE = WAGE-WAGE(-1)

dTAX = TAX-TAX(-1)

dLL = LL-LL(-1)

dINPT = INPT-INPT(-1)

dDUM = DUM-DUM(-1)

ecm = RPCC -.36932*RGDPC -.032831*UN + .062815*WAGE -.22875*TAX + .30

712*LL -4.4322*INPT -.050599*DUM

*******************************************************************************

R-Squared .76870 R-Bar-Squared .69468

S.E. of Regression .030182 F-stat. F( 7, 26) 11.8689[.000]

Mean of Dependent Variable .031574 S.D. of Dependent Variable .054622

Residual Sum of Squares .022774 Equation Log-likelihood 76.0008

Akaike Info. Criterion 67.0008 Schwarz Bayesian Criterion 60.1322

DW-statistic 1.7895

*******************************************************************************

R-Squared and R-Bar-Squared measures refer to the dependent variable

dRPCC and in cases where the error correction model is highly

restricted, these measures could become negative.

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