ARDL Cointegration Test PART A COINTEGRATION ...

Prepared by Dr Kelly Wong Kai Seng and Associate Professor Dr. Law Siong Hook UNIVERSITI PUTRA MALAYSIA

ARDL Cointegration Test Pesaran et al. (2001) first conduct the bounds tests in the unrestricted model or namely an ARDL (p,p,p,p,p) model (see their paper, Equation 30), and secondly adopt the ARDL (p,q,r,s,v) approach to the estimation of the level relations. Reference Pesaran, M. H., Y. Shin, and R. Smith, 2001, Bounds testing approaches to the analysis of level relationships. Journal of Applied Econometrics, 16, pp. 289-326.

PART A COINTEGRATION TEST – ARDL BOUNDS TEST The VAR(p) model can be rewritten in vector ECM form as: 𝜌−1

∆𝑧𝑡 = 𝑎0 + 𝑎1 𝑡𝑟𝑒𝑛𝑑 + 𝛑𝑧𝑡−1 +

∀𝑖 ∆𝑧𝑡−𝑖 + 𝜀𝑡 𝑖=1

where ∆ = 1 – L is the difference operator, zt = f(yt, xt) ɛt=disturbance terms and assumed to be i.i.d~N (0, σ )1

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

𝜋𝑦𝑥 𝜋𝑥𝑥

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

1

𝜋𝑦𝑦 0

𝜋𝑦𝑥 𝜋𝑥𝑥

independent and identically distributed (i.i.d.)

1


Consequently, the conditional ECM can be written as following:

∆𝑦𝑡 = 𝑎0 + 𝑎1 𝑡𝑟𝑒𝑛𝑑 + 𝛑𝑦𝑦 𝑦𝑡−1 + 𝛑𝑦𝑥 .𝑥 𝑥𝑡−1 +

𝜌−1 𝑖=1 ∀𝑖 ∆𝑧𝑡−𝑖

+ 𝑤 ′ ∆𝑥𝑡 + 𝜀𝑡

(1)

If the 𝜋𝑦𝑦 ≠ 0 and 𝜋𝑦𝑥 .𝑥 = 0′, the yt is (trend) stationary, 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 and 𝜋𝑦𝑥 .𝑥 ≠ 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 y t 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 The basic steps in the ARDL Bound test methodology are: (i) (ii) (iii) (iv)

Identification of a tentative model; To estimate the Equation (1) by using Ordinary Least Square (OLS) technique; Diagnostic checking (if the model is found inadequate, we go back to step 1); Using Wald test (F-test) to test the null and alternative hypotheses are constructed as follows: H0 : 𝛑𝑦𝑦 = 𝛑𝑦𝑥 .𝑥 = 𝟎 (No long run levels relationship) H1 : 𝛑𝑦𝑦 ≠ 𝟎; 𝐚𝐧𝐝 𝛑𝑦𝑥 .𝑥 ≠ 𝟎 (Long run levels relationship exists)

(v)

To compare the computed F-statistic with the critical value.

2


If the Wald F-statistic fall

Conclusion

a. above the upper critical value

Cointegration

b. between the lower bound and upper bound critical value

Inconclusive

c. below the lower bound critical value

No Cointegration

Example 1:

Data file: Data FD Bound.xls (Annual Data from 1970 – 2004, 35 observations) Empirical Model: FD = f(FDI, RGDPC, K) Variables: Financial development (FD); foreign direct investment (FDI); Real GDP per capita (RGDPC) and capital (K).

The ARDL Bound cointegration test model: p

p

i 1

i 0

FDt  c  1 FDt 1   2 FDI t 1   3 RGDPCt 1   4 K t 1    1i FDt i    2i FDI t i p

p

i 0

i 0

   3i RGDPCt i    4i K t i   t where c FD FDI RGDPC K p

= constant = financial development (% of GDP) = foreign direct investment (% of GDP) = real GDP per capita (Malaysian ringgit, RM) = physical capital (% of GDP) = optimum lag length

Transfer the data from Excel to Eviews

3

(1)


Paste the Data on the Eview Open Eviews – File – New – Workfile

Fill out the start date and end date, then click OK

4


Select Quick – Empty Group

Paste your cursor here (Obs - First row)

5


Copy the data from Excel

6


Now, we are really to paste the data in Eviews – Paste (or Control V)

Cointegration Test – ARDL Bounds Test Step 1 and 2: Identification of a Tentative Model & Estimation of the Model in OLS First, we examine the Bounds test by selecting the higher lag length. In our example, the sample period is covering from 1970 – 2004 (35 observations). In order to avoid the over parameter problem, we start with the minimum lag order 1 and then increase to lag 2:

7


To estimate the ARDL bounds test equation, select ―Quick‖ – ―Estimate Equation‖

and insert the model specification

where d = change (First difference or ) -1 = lag one variable or t –1 c = constant term

8


The ARDL Bound cointegration test model: p

p

i 1

i 0

FDt  c  1 FDt 1   2 FDI t 1   3 RGDPCt 1   4 K t 1    1i FDt i    2i FDI t i p

p

i 0

i 0

   3i RGDPCt i    4i K t i   t

(2)

The minimum lag order (p) = 1. Therefore, the way we specify using Eviews: d(fd) c fd(-1) fdi(-1) rgdpc(-1) k(-1) d(fd(-1)) d(fdi) d(fdi(-1)) d(rgdpc) d(rgdpc(-1)) d(k) d(k(-1)) The estimated result: Dependent Variable: D(FD) Method: Least Squares Sample (adjusted): 1972 2004 Included observations: 33 after adjustments Variable

Coefficient

Std. Error

t-Statistic

Prob.

C FD(-1) FDI(-1) RGDPC(-1) K(-1) D(FD(-1)) D(FDI) D(FDI(-1)) D(RGDPC) D(RGDPC(-1)) D(K) D(K(-1))

-1.428741 -0.328412 0.075941 0.340279 0.083633 -0.249965 0.153498 -0.004147 -0.441956 -0.243550 0.085005 0.018578

1.607201 0.136995 0.067390 0.249277 0.110211 0.212433 0.094500 0.082880 0.296191 0.338869 0.106634 0.092836

-0.888962 -2.397249 1.126887 1.365063 0.758846 -1.176679 1.624317 -0.050041 -1.492130 -0.718715 0.797166 0.200112

0.3841 0.0259 0.2725 0.1867 0.4564 0.2525 0.1192 0.9606 0.1505 0.4802 0.4343 0.8433

R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood F-statistic Prob(F-statistic)

0.673030 0.501761 0.053367 0.059808 57.34146 3.929650 0.003439

Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion Hannan-Quinn criter. Durbin-Watson stat

9

0.035106 0.075605 -2.747967 -2.203783 -2.564866 2.116844


Step 3: Diagnostic Checks for the ARDL bounds model: I.

Perform diagnostic check for serial correlation using the Breusch-Godfrey LM test

Select ―View‖ – ―Residual Tests‖ – ―Serial Correlation LM Test‖:

Lag Specification: 2

Breusch-Godfrey Serial Correlation LM Test: F-statistic Obs*R-squared

0.900460 2.857102

Prob. F(2,14) Prob. Chi-Square(2)

0.4230 0.2397

The LM test indicates no serial correlation problem since the p-value is greater than 0.05.

10


The above empirical results (using p = 1) can be summarized as: Table 1: Optimal Lag-length Selection P

AIC

SBC

x SC (2)

x SC (4)

1

-2.7479

-2.2037

2.8571

?

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 (2) 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, 0.05 and 0.10 levels, respectively.

Step 4: Using the Robust Model to estimate the Cointegration Relationship: After specifying the optimum lag model, we proceed to the ARDL Cointegration Bounds test. The code used in Eviews for hypothesis testing – c(1) represents the first coefficient (constant) – c(2) represents the second coefficient FD(-1) – c(3) represents the third coefficient FDI(-1), etc. Sequencing is crucial here Dependent Variable: D(FD) Method: Least Squares Sample (adjusted): 1973 2004 Included observations: 32 after adjustments

C(1) C(2) C(3) C(4) C(5) C(6) C(7) C(8) c(9) C(10) C(11) C(12)

Variable

Coefficient

Std. Error

t-Statistic

Prob.

C FD(-1) FDI(-1) RGDPC(-1) K(-1) D(FD(-1)) D(FDI) D(FDI(-1)) D(RGDPC) D(RGDPC(-1)) D(K) D(K(-1))

-1.428741 -0.328412 0.075941 0.340279 0.083633 -0.249965 0.153498 -0.004147 -0.441956 -0.243550 0.085005 0.018578

1.607201 0.136995 0.067390 0.249277 0.110211 0.212433 0.094500 0.082880 0.296191 0.338869 0.106634 0.092836

-0.888962 -2.397249 1.126887 1.365063 0.758846 -1.176679 1.624317 -0.050041 -1.492130 -0.718715 0.797166 0.200112

0.3841 0.0259 0.2725 0.1867 0.4564 0.2525 0.1192 0.9606 0.1505 0.4802 0.4343 0.8433

11


According to Pesaran et al. (2001), if the coefficients among the lag 1 variables (level) are jointly fall above the upper bound critical value, this implies that there is a long-run cointegration relationship among the variables. In order to test this hypothesis, we need to restrict the coefficients of FD(-1) = FDI(-1) = RGDPC(-1) = K(-1) = 0

Select ―View‖ – ―Coefficient Tests‖ – ―Wald - Coefficient Restrictions‖.

Write the hypothesis testing restriction as C(2) = C(3) = C(4) = C(5) = 0.

12


The Empirical Result:

Wald Test: Equation: Untitled Test Statistic F-statistic Chi-square

Value

df

Probability

4.270800 17.08320

(4, 21) 4

0.0110 0.0019

Null Hypothesis: C(2)=C(3)=C(4)=C(5)=0 Null Hypothesis Summary: Normalized Restriction (= 0) C(2) C(3) C(4) C(5)

Value -0.328412 0.075941 0.340279 0.083633

Std. Err. 0.136995 0.067390 0.249277 0.110211

Compare the F-statistic with the Narayan (2005) Critical value (if the sample size is relative small, < 100 observations).

Restrictions are linear in coefficients.

Compare the F-statistic value with critical value provided by Pesaran et al. (2001). However, if the sample size is small (< 100 observations), then compare with the critical value provided by Narayan (2005) – see next page. Reference: Narayan, P. K. (2005) The saving and investment nexus in China: evidence from cointegration tests. Applied Economics, 37, 1979 – 1990.

13

Prepared by Dr Kelly Wong Kai Seng and Associate k = theProfessor dimensionDr. of Law Siong Hook UNIVERSITI xt = (FDIt, RGDPCtPUTRA , Kt) or 3MALAYSIA n = 35 (1970 – 2004)

The result can be summarized as: Table 2. F-statistics for testing the existence of long-run cointegration Model Model 1: FD = f (FDI, RGDPC, K)

F-statistic 4.2708*

Narayan (2005) Critical Value 1% 5% 10%

k = 3, n=35 Lower bound Upper bound 5.198 6.845 3.615 4.913 2.958 4.100

Notes: *, **, and *** denote significant at 10%, 5%, and 1% levels, respectively. Critical values are obtained from Narayan (2005) (Table Case III: Unrestricted intercept and no trend; pg. 1988).

In this example, The F-statistic > critical upper bound value at 10% significance level; there is a long-run cointegration relationship among financial development and it determinants, namely real GDP per capita, trade openness and FDI. 14


Sensitivity Analysis: Refer back to our LM test table on page 11: Table 1: Optimal Lag-length Selection P

AIC

SBC

x SC (2)

x SC (4)

1

-2.7479

-2.2037

2.8571

?

However, based on the above model (p = 1), if we perform the LM test with lag specification: 4, then there is a serial correlation problem.


2.056942 10.76260


0.1317 0.0294

P

AIC

SBC

x SC (2)

x SC (4)

1

-2.7479

-2.2037

2.8571

10.7626**

How to Rectify the Serial Correlation problem in this case? 1.

Increase the lag length to p = 2

d(fd) c fd(-1) fdi(-1) rgdpc(-1) k(-1) d(fd(-1)) d(fd(-2)) d(fdi) d(fdi(-1)) d(fdi(-2)) d(rgdpc) d(rgdpc(1)) d(rgdpc(-2)) d(k) d(k(-1)) d(k(-2))

15


Empirical result: Dependent Variable: D(FD) Method: Least Squares Sample (adjusted): 1973 2004 Included observations: 32 after adjustments Variable

Coefficient

Std. Error

t-Statistic

Prob.

C FD(-1) FDI(-1) RGDPC(-1) K(-1) D(FD(-1)) D(FD(-2)) D(FDI) D(FDI(-1)) D(FDI(-2)) D(RGDPC) D(RGDPC(-1)) D(RGDPC(-2)) D(K) D(K(-1)) D(K(-2))

-0.446954 -0.361417 0.089338 0.248175 0.010199 -0.251403 0.022809 0.096372 -0.088222 -0.213793 -0.665165 -0.280067 -0.068742 -0.006439 -0.034050 0.216657

1.415670 0.126350 0.052268 0.223279 0.095805 0.163554 0.167052 0.067955 0.073088 0.061415 0.223363 0.259844 0.258141 0.085835 0.087631 0.065100

-0.315719 -2.860434 1.709209 1.111501 0.106456 -1.537124 0.136540 1.418179 -1.207060 -3.481097 -2.977959 -1.077826 -0.266295 -0.075015 -0.388557 3.328069

0.7563 0.0113 0.1067 0.2828 0.9165 0.1438 0.8931 0.1753 0.2450 0.0031 0.0089 0.2971 0.7934 0.9411 0.7027 0.0043


0.875309 0.758411 0.036829 0.021702 71.33157 7.487813 0.000121


16

0.032207 0.074929 -3.458223 -2.725355 -3.215298 2.190115


Perform Serial Correlation tests (Lag Specification 2 and Lag Specification 4) again –


0.148720 0.665721


0.8632 0.7169


0.756296 6.442970


0.5730 0.1684

Table 1: Optimal Lag-length Selection x SC (2) SBC

x SC (4)

P

AIC

2

-3.4582

-2.7253

0.6657

6.4429

1

-2.7479

-2.2037

2.8571

10.7626**

The result showed that the auto-serial correlation was overcome after increased that lag order to 2. Hence, we can use this model to further test that cointegration relationship.

17


In order to test the cointegration hypothesis, we need to restrict again the coefficients of FD(-1) = FDI(-1) = RGDPC(-1) = K(-1) = 0

Select ―View‖ – ―Coefficient Tests‖ – ―Wald - Coefficient Restrictions‖.

Write the hypothesis testing restriction as C(2) = C(3) = C(4) = C(5) = 0.

18


The Empirical Result:

Wald Test: Equation: Untitled Test Statistic F-statistic Chi-square

Value

df

Probability

9.763378 39.05351

(4, 16) 4

0.0003 0.0000

Null Hypothesis: C(2)=C(3)=C(4)=C(5)=0 Null Hypothesis Summary: Normalized Restriction (= 0) C(2) C(3) C(4) C(5)

Value -0.361417 0.089338 0.248175 0.010199

Std. Err. 0.126350 0.052268 0.223279 0.095805

Compare the F-statistic with the Narayan (2005) Critical value (if the sample size is relative small, < 100 observations).


The result can be summarized as: Table 2. F-statistics for testing the existence of long-run cointegration Model Model 1: FD = f (FDI, RGDPC, K)

F-statistic 9.7633**

Narayan (2005) Critical Value 1% 5% 10%

k = 3, n=35 Lower bound Upper bound 5.198 6.845 3.615 4.913 2.958 4.100

Notes: *, **, and *** denote significant at 10%, 5%, and 1% levels, respectively. Critical values are obtained from Narayan (2005) (Table Case III: Unrestricted intercept and no trend; pg. 1988).

Hence, the lag 2 model shows that there is a long-run cointegration relationship among financial development and it determinants, namely FDI, real GDP per capita and physical capital. The Fstatistic is statistically significant at 5% level.

19


PART B ARDL Level Relation Based on the above example, where the model is: FD = f(FDI, RGDPC, K) The ARDL model can be written as follows: p

q

r

s

i 1

i 0

i 0

i 0

FDt  const   1 FDt i    2 FDI t i    3 RGDPCt i    4,t K t i   t

where FD const FDI RGDPC K p, q, r, s

t

= financial development (% of GDP) = constant = foreign direct investment (% of GDP) = real GDP per capita (RM) = physical capital (% of GDP) = optimum lag length = residual

Step 1: Identification of a Tentative Model The below criteria can be used to select the optimum lag of the above ARDL modeling: a) Akaike Information Criterion (AIC) b) Schwarz Bayesian Criterion (SBC) c) General to specific model Table below shows the ARDL models with different lag structure based on AIC and SBC selection criteria: Model 1 2 3 4 5 : :

ARDL (1,0,0,0) (1,1,0,0) (1,1,1,0) (1,1,1,1) (1,0,1,0) : (1,0,0,1)

Eviews Regression fd c fd(-1) fdi rgdpc k fd c fd(-1) fdi fdi(-1) rgdpc k fd c fd(-1) fdi fdi(-1) rgdpc rgdpc(-1) k fd c fd(-1) fdi fdi(-1) rgdpc rgdpc(-1) k k(-1) fd c fd(-1) fdi rgdpc rgdpc(-1) k ; fd c fd(-1) fdi rgdpc k k(-1)

The minimum AIC and SBC is Model with lag (1, 1, 1, 0)

20

AIC -2.448 -2.548 -2.945 -2.913 -2.999 : -2.698

SBC -2.224 -2.279 -2.630 -2.554 -2.727 : -2.429


Step 2: Autoregressive Distributed Lag Estimates 1. Based on the AIC and SBC, the selected lag length of (p, q, r, s) is (1, 0, 1, 0). The long-run OLS output is as follows:

Eviews output: Dependent Variable: FD Method: Least Squares Sample: 1972 2004 Included observations: 33

C(1) C(2) C(3) C(4) C(5) C(6)

Variable

Coefficient

Std. Error

t-Statistic

Prob.

C FD(-1) FDI RGDPC RGDPC(-1) K

-1.894067 0.668543 0.064902 -0.426630 0.819812 0.124131

0.987712 0.080099 0.051019 0.193312 0.167500 0.069896

-1.917631 8.346511 1.272110 -2.206952 4.894390 1.775955

0.0658 0.0000 0.2142 0.0360 0.0000 0.0870


0.979749 0.975999 0.049785 0.066920 55.48766 261.2521 0.000000


21

4.548579 0.321350 -2.999252 -2.727160 -2.907702 2.328328


Step 3: Diagnostic Tests 1. Serial Correlation The results of Breusch-Godfrey serial correlation LM test: Lag 2 Breusch-Godfrey Serial Correlation LM Test: F-statistic Obs*R-squared

1.073286 2.609423


0.3571 0.2713

Lag 4 Breusch-Godfrey Serial Correlation LM Test: F-statistic Obs*R-squared

1.772114 7.774380


0.1689 0.1002

The above LM test results indicated that the residuals are homoskedasticity (no serial correlation) since the p-values are greater than 5% significance level.

2. Stability Test - View – Stability Tests – Recursive Estimates

22


1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 78

80

82

84

86

88

90

CUSUM of Squares

92

94

96

98

00

02

04

5% Significance

Step 4: Compute Long-run Coefficients using the ARDL Approach 1. After obtaining the ARDL (1,0,1,0) model, the next step is to find the long run elasticities. i. Elasticity of FDI According to Pesaran et al. (2001), the long run elasticities can be obtained as follow:

23

Prepared by Dr Kelly Wong Kai Seng and Associate Professor Dr. Law Siong Hook UNIVERSITI PUTRA MALAYSIA q

Elasticity FDI 

 i 0

FDI

p

1    FD i 1

=

Sum of the independent coefficien t(s) FDI 1 - sum of the dependent coefficien t(s)

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

24


Insert (c(3))/(1-c(2))=0 in the Wald Test empty box

Eviews Output: Wald Test: Equation: Untitled Test Statistic t-statistic F-statistic Chi-square

Value

df

Probability

1.337456 1.788789 1.788789

27 (1, 27) 1

0.1922 0.1922 0.1811

Value

Std. Err.

0.195807

0.146403

Null Hypothesis: (C(3))/(1-C(2))=0 Null Hypothesis Summary: Normalized Restriction (= 0) C(3) / (1 - C(2))

Delta method computed using analytic derivatives.

The elasticity is 0.1958 and the standard error is 0.1464 with t-stat 1.337. The p-value of F-statistic is served as whether the long-run elasticity of FDI is significant. In this case, the FDI is not significant to determine that long-run financial development.

25


ii. Elasticity of RGDPC r

Elasticity RGDPC 

 i 1

RGDPC

p

1    FD i 1

Wald Test: Equation: Untitled Test Statistic t-statistic F-statistic Chi-square

Value

df

Probability

5.731905 32.85473 32.85473

27 (1, 27) 1

0.0000 0.0000 0.0000

Value

Std. Err.

1.186228

0.206952

Null Hypothesis: (C(4)+C(5))/(1-C(2))=0 Null Hypothesis Summary: Normalized Restriction (= 0) (C(4) + C(5)) / (1 - C(2))


iii. Elasticity of K s

Elasticity K 

 i 0 p

K

1    FD i 1

26



Value

df

Probability

2.393230 5.727551 5.727551

27 (1, 27) 1

0.0239 0.0239 0.0167

Value

Std. Err.

0.374503

0.156484

Null Hypothesis: (C(6)) / (1 - C(2))=0 Null Hypothesis Summary: Normalized Restriction (= 0) C(6) / (1 - C(2))


iv. Long-run coefficient of Constant Term

27



Value

df

Probability

-3.010062 9.060471 9.060471

27 (1, 27) 1

0.0056 0.0056 0.0026

Value

Std. Err.

-5.714376

1.898425

Null Hypothesis: (C(1)) / (1 - C(2))=0 Null Hypothesis Summary: Normalized Restriction (= 0) C(1) / (1 - C(2))


The coefficients of all variables using the ARDL approach are: FDI = 0.195807 RGDPC = 1.186228*** K = 0.374503** Constant = -5.714376*** Therefore, the long-run relation model can be written as follows: FDt = - 5.714376+ 0.195807 FDIt + 1.186228 RGDPCt + 0.389052 Kt

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Summary ARDL (1,0,1,0) Model: 𝐹𝐷𝑡 = 𝑐 + 𝛼1 𝐹𝐷𝑡−1 + 𝛽1 𝐹𝐷𝐼𝑡 + 𝛽2 𝑅𝐺𝐷𝑃𝐶𝑡 + 𝛽3 𝑅𝐺𝐷𝑃𝐶𝑡−1 + 𝛽4 𝐾𝑡 + 𝜀𝑡 Based on the above ARDL model, we can estimate the long run elasticity as following: where 𝜑𝐹𝐷𝐼 =

𝛽1 1 − 𝛼1

𝜑𝑅𝐺𝐷𝑃𝐶 = 𝜑𝐾 =

𝛽2 + 𝛽3 1 − 𝛼1

𝛽4 1 − 𝛼1

𝜑𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡 =

𝑐 1 − 𝛼1

Step 5: Error Correction Representation for the Selected ARDL Model

After obtaining the long-run relation, the next step is to estimate the short-run Error-correction Model (ECM). a.

Compute the value of Error-correction Term (ECT), which represents the residuals from long-run cointegration model.

Recap the ARDL (1,1,1,0) Model: 𝐹𝐷𝑡 = 𝑐 + 𝛼1 𝐹𝐷𝑡−1 + 𝛽1 𝐹𝐷𝐼𝑡 + 𝛽2 𝑅𝐺𝐷𝑃𝐶𝑡 + 𝛽3 𝑅𝐺𝐷𝑃𝐶𝑡−1 + 𝛽4 𝐾𝑡 + 𝜀𝑡

The short run dynamic model can be transformed using the above ARDL model:

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where 𝐹𝐷𝑡 = ∆𝐹𝐷𝑡 + 𝐹𝐷𝑡−1

;

𝐹𝐷𝑡−𝑖 = 𝐹𝐷𝑡−1 −

𝑝−1 𝑖=0 ∆𝐹𝐷𝑡−𝑖 𝑞−1

𝑅𝐺𝐷𝑃𝐶𝑡 = ∆𝑅𝐺𝐷𝑃𝐶𝑡 + 𝑅𝐺𝐷𝑃𝐶𝑡−1 ; 𝑅𝐺𝐷𝑃𝐶𝑡−𝑗 = 𝑅𝐺𝐷𝑃𝐶𝑡−1 −

∆𝑅𝐺𝐷𝑃𝐶𝑡−𝑗 𝑗 =0

𝐹𝐷𝐼𝑡 = ∆𝐹𝐷𝐼𝑡 + 𝐹𝐷𝐼𝑡−1 ; 𝐹𝐷𝐼𝑡−𝑗 = 𝐹𝐷𝐼𝑡−1 − 𝐾𝑡 = ∆𝐾𝑡 + 𝐾𝑡−1 ;

𝑞 −1 𝑗 =0 ∆𝐹𝐷𝐼𝑡−𝑗

𝑞 −1 𝑗 =0 ∆𝐾𝑡−𝑗

𝐾𝑡−𝑗 = 𝐾𝑡−1 −

𝐹𝐷𝑡 = 𝑐 + 𝛼1 𝐹𝐷𝑡−1 + 𝛽1 𝐹𝐷𝐼𝑡 + 𝛽2 𝑅𝐺𝐷𝑃𝐶𝑡 + 𝛽3 𝑅𝐺𝐷𝑃𝐶𝑡−1 + 𝛽4 𝐾𝑡 + 𝜀𝑡

Transform: ∆𝐹𝐷𝑡 + 𝐹𝐷𝑡−1 = 𝑐 + 𝛼1 𝐹𝐷𝑡−1 + 𝛽1 (∆𝐹𝐷𝐼𝑡 + 𝐹𝐷𝐼𝑡−1 ) + 𝛽2 (∆𝑅𝐺𝐷𝑃𝐶𝑡 + 𝑅𝐺𝐷𝑃𝐶𝑡−1 ) + 𝛽3 𝑅𝐺𝐷𝑃𝐶𝑡−1 + 𝛽4 (∆𝐾𝑡 + 𝐾𝑡−1 ) + 𝜀𝑡 ∆𝐹𝐷𝑡 = 𝑐 − 𝐹𝐷𝑡−1 + 𝛼1 𝐹𝐷𝑡−1 + 𝛽1 ∆𝐹𝐷𝐼𝑡 + 𝛽1 𝐹𝐷𝐼𝑡−1 + 𝛽2 ∆𝑅𝐺𝐷𝑃𝐶𝑡 + 𝛽2 𝑅𝐺𝐷𝑃𝐶𝑡−1 + 𝛽3 𝑅𝐺𝐷𝑃𝐶𝑡−1 + 𝛽4 ∆𝐾𝑡 + 𝛽4 𝐾𝑡−1 + 𝜀𝑡 ∆𝐹𝐷𝑡 = 𝑐 − (1 − 𝛼1 )𝐹𝐷𝑡−1 + 𝛽1 ∆𝐹𝐷𝐼𝑡 + 𝛽1 𝐹𝐷𝐼𝑡−1 + 𝛽2 ∆𝑅𝐺𝐷𝑃𝐶𝑡 + 𝛽2 𝑅𝐺𝐷𝑃𝐶𝑡−1 + 𝛽3 𝑅𝐺𝐷𝑃𝐶𝑡−1 + 𝛽4 ∆𝐾𝑡 + 𝛽4 𝐾𝑡−1 + 𝜀𝑡 ∆𝐹𝐷𝑡 = 𝑐 − (1 − 𝛼1 )𝐹𝐷𝑡−1 + 𝛽1 𝐹𝐷𝐼𝑡−1 + 𝛽2 + 𝛽3 𝑅𝐺𝐷𝑃𝐶𝑡−1 + 𝛽4 𝐾𝑡−1 + 𝛽1 ∆𝐹𝐷𝐼𝑡 + 𝛽2 ∆𝑅𝐺𝐷𝑃𝐶𝑡 + 𝛽4 ∆𝐾𝑡 + 𝜀𝑡

Speed of adjustment

∆𝐹𝐷𝑡 = 𝑐 − 1 − 𝛼1

ECT

𝛽2 + 𝛽3 𝛽4 𝑅𝐺𝐷𝑃𝐶𝑡−1 − 𝐾 + 1 1 − 𝛼1 1 − 𝛼1 𝑡−1 𝛽1 ∆𝐹𝐷𝐼𝑡 + 𝛽2 ∆𝑅𝐺𝐷𝑃𝐶𝑡 + 𝛽4 ∆𝐾𝑡 +𝜀𝑡 𝛽

𝐹𝐷𝑡−1 − 1−𝛼1 𝐹𝐷𝐼𝑡−1 −

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where 𝐸𝐶𝑇𝑡−1 = 𝐹𝐷𝑡−1 −

𝛽1 𝛽2 + 𝛽3 𝛽4 𝐹𝐷𝐼𝑡−1 − 𝑅𝐺𝐷𝑃𝐶𝑡−1 − 𝐾 1 − 𝛼1 1 − 𝛼1 1 − 𝛼1 𝑡−1

Hence, the short-run dynamic model can be rewritten on the following ECT model:

∆𝐹𝐷𝑡 = 𝑐 − 1 − 𝛼1 𝐸𝐶𝑇𝑡−1 + 𝛽1 ∆𝐹𝐷𝐼𝑡 + 𝛽2 ∆𝑅𝐺𝐷𝑃𝐶𝑡 + 𝛽4 ∆𝐾𝑡 +𝜀𝑡 Furthermore, we using Wald Test (F-test) to compute the short-run coefficient from the previous ARDL (1,0,1,0) model; Dependent Variable: FD Method: Least Squares Sample: 1972 2004 Included observations: 33 Variable

Coefficient

Std. Error

t-Statistic

Prob.

C FD(-1) FDI RGDPC RGDPC(-1) K

-1.894067 0.668543 0.064902 -0.426630 0.819812 0.124131

0.987712 0.080099 0.051019 0.193312 0.167500 0.069896

-1.917631 8.346511 1.272110 -2.206952 4.894390 1.775955

0.0658 0.0000 0.2142 0.0360 0.0000 0.0870


0.979749 0.975999 0.049785 0.066920 55.48766 261.2521 0.000000


Based on the previous ARDL (1,0,1,0) model, we can compute that short-run coefficient by using Wald-Test.

4.548579 0.321350 -2.999252 -2.727160 -2.907702 2.328328

Go to View – Coefficient Dignostics – Wald Test. Now, you need to key in that formula as shown in the short-run dynamic model. For example: the coefficient of ECT is –(1-c(2))

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Eviews Output: Wald Test: Equation: Untitled Test Statistic t-statistic F-statistic Chi-square

Value

df

Probability

- 4.138109 17.12394 17.12394

27 (1, 27) 1

0.0003 0.0003 0.0000

Value

Std. Err.

-0.331457

0.080099

Null Hypothesis: 1-C(2)=0 Null Hypothesis Summary: Normalized Restriction (= 0) 1 - C(2)

As we know that speed of adjustment is negative (1-α1), hence using Wald test to compute this value and obtain the standard error and tstatistic.


c ECTt-1 d(FDI) d(RGDPC) d(K)

Short-Run Dynamic ECT Model Elasticities Std. Error t-Statistic -1.8941* 0.9877 -1.9176 -0.3314*** 0.0801 -4.1381

Prob. 0.0658 0.003

After that, we continue to compute the rest of coefficient, which based on that Short run ECT model, and fill in to the Table as showed on the above. At the end, you will get the result as following:

c ECTt-1 d(FDI) d(RGDPC) d(K)

Short-Run Dynamic ECT Model Elasticities Std. Error t-Statistic -1.8941* 0.9877 -1.9176 -0.3314*** 0.0801 -4.1381 0.0649 0.0510 1.2721 -0.4266** 0.1933 -2.2069 0.1241* 0.0698 1.7759

Prob. 0.0658 0.003 0.2142 0.0360 0.0870

The ECT can be obtained as follows: The long-run Equation is FDt = - 5.714376 + 0.195807 FDIt + 1.186228 RGDPCt + 0.389052 Kt Hence, the ECT equation is: ECTt = FDt + 5.714376 - 0.195807 FDIt - 1.186228 RGDPCt - 0.389052 Kt 32