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PALESTINE MONETARY AUTHORITY
PMA WORKING PAPER
WP/11/03
THE AUTOREGRESSIVE DISTRIBUTED LAG APPROACH TO CO-INTEGRATION TESTING: APPLICATION TO OPT INFLATION
Saed Khalil and Michel Dombrecht
® Palestine Monetary Authority WP/11/03
ii
PMA Working Paper
Research and Monetary Policy Department
The Autoregressive Distributed Lag Approach to Co-integration Testing: Application to OPT Inflation
Prepared by Saed Khalil* and Michel Dombrecht†
November 2011
Abstract This paper aims at estimating and forecasting the inflation rate in the OPT using the ARDL approcach to co-integration testing as an alternative to the Johansen’s co-integration testing used in the inflation report. Findings show that the results of the ARDL and the FMOLS approaches are very close. And when comparing the ARDL with the Johansen test for co-integration, the results are close and not contradicoty. Hence, ARDL and the FMOLS approches along with the Johansen testing for co-integration are suggested to be used in the inflation report for Palestine.
© November, 2011 All Rights Reserved. Suggested Citation: Palestine Monetary Authority (PMA), 2011. The Autoregressive Distributed Lag Approach to Co-integration Testing: Application to OPT Inflation. Ramallah – Palestine All Correspondence should be directed to: Palestine Monetary Authority (PMA) P. O. Box 452, Ramallah, Palestine. Tel.: 02-2409920 Fax: 02-2409922 E-mail: [email protected] www.pma.ps
JEL Classification Numbers: C18, C50, E31 Keywords: Econometric methodology, Macroeconometrics, Inflation, . Authors’ E-mail addresses: [email protected]; [email protected]
* Palestine Monetary Authority. † University of Antwerp and Hogeschool Universiteit Brussels, Belgium.
iii
Contents
I. INTRODUCTION .................................................................................................................................................................. 1
II. METHODOLOGY ................................................................................................................................................................. 1
II.1 TESTING PROCEDURE ............................................................................................................................................................ 2
II.2 MODELING PROCEDURE ....................................................................................................................................................... 3
III. APPLICATION TO OPT INFLATION ........................................................................................................................ 4
III.1 CO-INTEGRATION.................................................................................................................................................................. 4
III.2 ARDL ..................................................................................................................................................................................... 6
III.3 FMOLS .................................................................................................................................................................................. 9
IV. CONCLUSION ................................................................................................................................................................... 10
REFERENCES ........................................................................................................................................................................... 10
1
I. Introduction
This technical paper aims at estimating and forecasting inflation in the Occupied
Palestinian Territory (OPT) using the Autoregressive Distributed Lag (ARDL) approach to
co-integration testing. In the inflation report produced by the Palestine Monetary Authority
(PMA) Johansen’s con-integration testing is used to find the long-run relationships between
inflation rate and its determinants and the Vector Error Correction Model (VECM) is used to
estimate the inflation rate in OPT for 2011 on a quarterly basis.
In this paper we use an alternative approach to Johansen’s co-integration to estimate
inflation rate in OPT. We will build an ARDL model to estimate inflation rate in OPT and
compare the obtained results with those in the inflation report.
In addition to estimating inflation rate for OPT we provide a detailed procedure for
testing an ARDL model, which will be useful for future uses at PMA.
This paper found that the results of the ARDL and the FMOLS approaches are very
close. And when comparing the ARDL approach with the Johansen test of co-integration
results are close and not contradictory.
The rest of the paper is organized as follows. Section II provides the methodology of
estimating the OPT CPI in 2011. Section III presents the results of the estimation based on
three alternative methodologies, Johansen’s co-integration testing, ARDL approach to co-
integration, and the fully modified ordinary least squares (FMOLS) approach. Section IV
concludes.
II. Methodology
The most popular single equation testing for co-integration between a set of I(1)
variables relied on the Engle-Granger (1987) and Phillips-Ouliaris (1990) residual based tests.
Also Hansen’s instability test (1995), Park’s added variables test (1992) and the stochastic
common trends approach of Stock and Watson (1988) are well known. System co-integration
testing is mostly based on Johansen’s (1991, 1995) system based reduced rank approach.
Saed Khalil and Michel Dombrecht
2
Recently, also the so called Autoregressive Distributed Lag (ARDL) test is found in
applied empirical papers. This test is based on Pesaran, Shin (1999) and Pesaran, Shin, Smith
(2001). This technique is reported to offer several advantages. The test is based on a single
ARDL equation, rather than on a VAR as in Johansen, thus reducing the number of
parameters to be estimated. Also unlike the Johansen approach the restrictions on the
number of lags can be applied to each variable separately. The ARDL approach also does not
require pre-testing for the order of integration (0 or 1) of the variables used in the model.
II.1 Testing Procedure
Pesaran, Shin and Smith (PSS 2001) developed a new approach to co-integration
testing which is applicable irrespective of whether the regressor variables are I(0), I(1) or
mutually co-integrated.
The starting point of their test is a data generating process represented by a general
VAR of order p which is rewritten in vector ECM form involving a vector z of variables. They
focus on the conditional modeling of the dependent scalar variable y. To that end, the vector z
is partitioned into the scalar y and vector x of dependent variables. Under the assumption
that there is no feedback from y to x, the model can be written as the following conditional
ECM model for ∆y:
∑−
=−−− +∆+∆+++=∆
1
1
''1.1
'p
ittititxyxtyyt uxzxywcy ωψππ (1)
Where,
w is a set of deterministic variables like the constant term, trend, seasonal dummies,
etc..
c is a vector of coefficients of deterministic variables
ut is the residual term.
To test the absence of a level relationship between y and x, the approach uses a Wald
or F-statistic to test for the joint hypothesis that all coefficients of all (lagged) levels in the
ECM equation are zero. PSS distinguishes five cases according to how the deterministic
ARDL Approach to Co‐integration Testing: Application to OPT Inflation
3
components are specified: no intercepts, no trends; restricted intercepts, no trends;
unrestricted intercepts, no trends; unrestricted intercepts, restricted trends, unrestricted
intercepts, unrestricted trends.
The resulting conditional ECM’s may be interpreted as autoregressive distributed
models of orders (p, p,… p), i.e. ARDL (p, p,…p) models. PSS publishes tabulated asymptotic
critical value bounds for the F-statistic for all 5 conditional ECM models. If the computed F-
statistic from exclusion of levels in the conditional EMS’s fall outside the critical value
bounds, the test allows a conclusive inference without needing to know the integration/co-
integration status of the underlying regressors. But if the F-statistic falls inside the bounds,
inference is inconclusive and knowledge of the order of integration of the underlying
variables is required before conclusive inferences can be made. If the computed F-statistic lies
below the 0.05 lower bound, the hypothesis that there is no level relationship is not rejected
at the 5 percent level. If the statistic falls within the 0.05 bounds, the test is inconclusive and
when the F-statistic lies above the 0.05 upper bound, the hypothesis of no level relationship
is conclusively rejected.
In addition to the F-test, PSS also tabulates asymptotic critical value bounds of the t-
statistic for testing the significance of the coefficient on the lagged dependent variable in the
conditional ECM. Concerning the use of the F and t-statistics, PSS suggests the following
procedure: test H0 using the bounds procedure based on the Wald or F-statistic. If H0 is not
rejected, proceed no further. If H0 is rejected test the coefficient of the lagged dependent
variables using the bounds procedure based on the t-statistic. A large value of t confirms the
existence of a level relationship between y and x;
II.2 Modeling Procedure
Testing for the existence of a level relationship as in the previous section requires that
the coefficients of the lagged changes remain unrestricted. But for the subsequent estimation
of the ECM model, a more parsimonious approach is recommended, such as the ARDL
approach to the estimation of the level relations discussed in Pesaran and Shin (1999). In
practical terms an ARDL (p, p,… p) model is selected from a broader search analysis testing
the lag orders using information criteria such as AIC or SBC. In this respect, it is interesting
that PSS note that the ARDL estimation procedure is directly comparable with the semi-
Saed Khalil and Michel Dombrecht
4
parametric Fully Modified OLS approach (FMOLS) of Phillips and Hansen (1990). From the
parsimonious ARDL specification the specification of the estimated levels relationship is then
derived, as well as the associated ECM model.
III. Application to OPT Inflation
III.1 Co-integration
In the inflation report we built a Vector Error Correction Model (VECM) which
encompasses the long run equilibrium relationship between CPI in OPT and the cost of
imports indicator (CIM, see Michel and Khalil 2011). But simultaneously it also incorporates
the short term dynamics that drive the CPI back to its long run equilibrium path after shocks
have temporarily driven CPI apart from its long run equilibrium. That a long-run co-
integrated relationship between CPI and the cost of imports exists is confirmed by the formal
Johansen co-integration test reported in table 1.
Table 1 Co-integration tests between CPI and CIM in PT (Variables expressed in logarithms)
Unrestricted Co-integration Rank Test (Trace) Sample: 1997Q4 – 2010Q3
Lags interval (in first difference): 1 to 2
Hypothesized no. of Co-integration relation(s)
Eigenvalue Trace Statistic 0.05 Critical Value Prob.**
None* 0.3415 29.8201 20.2618 0.0018
At most 1 0.1442 8.0958 9.1645 0.0795 Unrestricted Co-integration Rank Test (Max. Eigenvalue) Sample: 1997Q4 – 2010Q3
Lags interval (in first difference): 1 to 2
Hypothesized no. of Co-integration relation(s)
Eigenvalue Max-Eigen. Statistic
0.05 Critical Value Prob.**
None* 0.3415 21.7244 15.8921 0.0054 At most 1 0.1442 8.0958 9.1645 0.0795
* denotes rejection of the hypothesis at the 0.05 level. ** MacKinnon-Haug-Michelis (1999) p-values.
Besides the long run relationship between CPI and CIM, the estimated VECM also
contains the world food and beverage price index (WFOBEV) and delayed reactions of CPI to
shocks in the cost of imports (the long run effect of a change in the cost of imports is not
attained in one single quarter, but may take more time to be realized).
ARDL Approach to Co‐integration Testing: Application to OPT Inflation
5
CPI in OPT, as shown in equation 2, depends upon cost of imports, and World food and
beverages price index, but the latter is considered to be an exogenous variable in the VECM.
To estimate equation 2 VECM is used and the results are in table 2.
LCPI = f(LCIM, LWFOBEV) (2)
Where,
LCPI is the log of the consumer price index (CPI),
LCIM is the log of cost of imports (CIM), and
LWFOBEV is the log of World food and beverage price index (WFOBEV).
Table 2 VECM of CPI in PT3
Co-integrating equation: CointEq1
LCPI-1 1.0000
LCIM-1 -0.9438
[-6.4481]
C -0.2584
Error Correction: D(LCPI_PT) D(LCIM_TIM)
CointEq1 -0.1462 -0.0021
[-3.0008] [-0.0257]
∆LCPI-1 0.1671 -0.0850
[ 1.0205] [-0.3098]
∆LCPI-2 -0.4270 -0.4052
[-2.7200] [-1.5417]
∆LCPI-3 0.1841 0.45123
[ 1.2441] [ 1.8209]
∆LCIM-1 0.1615 0.1370
[ 1.5517] [ 0.7858]
∆LCIM-2 0.0317 0.1341
[ 0.2986] [ 0.7547]
∆LCIM-3 -0.1211 -0.1824
[-1.1341] [-1.0197]
C -0.1685 -0.0083
[-3.1513] [-0.0924]
LWFOBEV 0.0386 0.0032
[ 3.3232] [ 0.1628]
3 t-statistic between [ ].
Saed Khalil and Michel Dombrecht
6
Results show that there is a long run relationship between CPI in OPT and the cost of
imports in OPT. The long run coefficient in this relationship is significant and close to unity.
The results illustrate the crucial dependence of CPI in OPT on the exchange rates. Apart from
the cost of imports, the CPI is also affected by the movements of the world market prices of
food and beverages. This influence is relatively small, but can exert a pronounced influence on
CPI in times when these world prices are very volatile, as has been the case since 2007.
In the short term, inflation in OPT absorbs the shocks in import costs and world
prices mentioned above. Imported inflation shocks take time before the CPI level has fully
reacted and this drives inflation in the short run. The coefficient on the error correcting term
is negative, statistically significant and smaller than one and these conditions are an
additional confirmation of the existence of a long run equilibrium condition between CPI and
imported cost shocks.
III.2 ARDL
As an alternative we re-estimated this basic quarterly inflation model using the ARDL
technique along the lines suggested by PSS (2001).
The analysis starts from the assumption that the CPI in OPT can be modeled by a log-
linear VAR (p) model, augmented with deterministics such as a constant, seasonal dummies
and a time trend.
Let z = (LCPI, LCIM, and LWFOBEV)’. Then using proofs in PSS (2001) the following
conditional ECM was estimated:
∑−
=−
−−−
+∆+∆+∆+
+++=∆1
121
'
131211'
p
itttiti
tttt
uLWFOBEVLCIMz
LWFOBEVLCIMLCPIwcLCPI
ωωψ
πππ (3)
To determine the appropriate length of p, we estimated this conditional ECM by OLS
for p=1, 2, 3, 4, 5 over the period 1997Q1 to 2010Q4. We found the constant, time trend and the
first seasonal dummy to be insignificant for all lags and therefore decided to estimate the
ECM without these three deterministic variables. Table 3 reports the Akaike (AIC) and
Schwarz Bayesian (SBC) Information Criteria for all lags up to four.
ARDL Approach to Co‐integration Testing: Application to OPT Inflation
7
Table 3 Information Criteria for selecting the lag order of the CPI equation
P AIC SBC
1 -6.42 -6.17
2 -6.55 -6.18
3 -6.53 -6.04
4 -6.52 -5.92
5 -6.43 -5.71
These are the statistics calculated in E-views which deviate from the formulas in PSS,
although both are essentially based on the Log Likelihood parameter. The optimal lags are
those that maximize the absolute values of the AIC and SBC as presented in table 3. Based on
these results, both the AIC and SBC would suggest p = 2. We then calculated the F-statistics
to test the null that there are no level effects in the conditional ECM. Also the t-statistics on
the lagged dependent variable are reported in table 4.
Table 4 F- and t-statistics for testing the existence of a levels CPI equation
P F-statistic t-statistic
1 14.89 -2.26
2 7.26 -1.99
3 8.58 -2.31
4 8.04 -3.08
5 5.05 -2.34
These values have to be compared to the asymptotic critical value bounds as reported
in PSS (2001) for k = 2. The 95 percent critical bounds for the F-statistic for k = 2 in the case of
no intercept and no trend are (2.72, 3.83) and for the t-statistic (-1.95, -3.02).
The F-statistic rejects conclusively the null hypothesis that there exists no level CPI
equation for all lags reported in table 4. On the other hand the results of the application of
the bounds t-test are less clear cut. Only p = 4 rejects the null, while for all other lags, the test
is inconclusive. We therefore interpret the results in support of the existence of a level CPI
equilibrium relationship, which confirms the findings when applying the Johansen approach.
The search for parsimonious model can be undertaken along the lines of PSS, which
involves estimating all possible combinations for p in the conditional ECM. This would
Saed Khalil and Michel Dombrecht
8
require the estimation of (5)3 equations. A selection among those would be based on the AIC.
Our exercise would point to an underlying ARDL (1, 2, 2) model as follows:
∆LCPI = - 0.0099 SD2 - 0.0077 SD3 - 0.0549 LCPI(-1) + 0.0366 LCIM(-1)
+ 0.0202 LWFOBEV-1 + 0.3822 ∆LCIM + 0.0392 ∆LWFOBEV
+ 0.1728 ∆LCIM-1 + 0.0412 ∆LWFOBEV-1 (4)
Where,
SD2 and SD3 are seasonal dummies.
The error correction term v̂ can be written as follows:
LWFOBEVLCIMLCPIv1
2
1
2ˆππ
ππ
−−= (5)
Where, π1 = 0.0549, π2 = 0.0366, and π3 = 0.0202. The long-run relationship between LCPI and
LCIM and LWFOBEV can be written as follows:
LCPI = 0.67 LCIM + 0.37 LWFOBEV (6)
As can be seen this result implies that the sum of both cost-push factors (cost of
imports and world food and beverages index) is very close to one which conforms with
theoretical expectation. Compared to the co-integration vector that was obtained in the
VECM estimated in the 2011 inflation report, we now have a more prominent effect of the
world food and beverages price index at the expense of the more broader cost of imports
which reflects mainly CPI developments in the countries from which OPT mainly imports.
The conditional ECM is reported in table 5.
Table 5 Equilibrium correction of the ARDL (1, 2, 2) CPI equation
Regressor Coefficient Standard error p-value
1ˆ−v -0.0549 0.0090 0.0000
SD2 -0.0100 0.0029 0.0014
SD3 -0.0077 0.0028 0.0088
∆LCIM 0.3822 0.0714 0.0000
ARDL Approach to Co‐integration Testing: Application to OPT Inflation
9
∆LWFOBEV 0.0392 0.0188 0.0426
∆LCIM(-1) 0.1728 0.0737 0.0234
∆LWFOBEV(-1) 0.0412 0.0202 0.0469
2R = 0.5566 AIC = -6.6616 SBC = -6.4038
III.3 FMOLS
As an alternative, we estimated the levels CPI equation with the FMOLS (equation
7)4. The results as, shown in the equation are very close to the results obtained using ARDL.
LCPI = 0.75 LCIM + 0.24 LWFOBEV (7)
[12.934] [4.231]
The error correction term FMOLSv̂ can be written as follows:
LWFOBEVLCIMLCPIvFMOLS 24.075.0ˆ −−= (8)
As shown in table 6 using this co-integration vector in a parsimonious ECM yields
results that are again close to the final ECM of the ARDL (1, 2, 2) model.
Table 6 Equilibrium correction of the FMOLS CPI equation
Regressor Coefficient Standard error p-value
constant 0.0092 0.0019 0.0000
1ˆ −FMOLSv -0.0561 0.0265 0.0396
SD2 -0.0101 0.0031 0.0022
SD3 -0.0077 0.0030 0.0121
∆LCIM 0.3859 0.0730 0.0000
∆LWFOBEV 0.0363 0.0206 0.0838
∆LCIM(-1) 0.1696 0.0760 0.0307
∆LWFOBEV(-1) 0.0464 0.0211 0.0331 2R = 0.5357 AIC = -6.6000 SBC = -6.3054
4 T-statistics are between [ ]
Saed Khalil and Michel Dombrecht
10
IV. Conclusion
The ARDL estimation procedure is based on two basic steps. First, the existence of a
level relationship is tested using an unrestricted ARDL specification. Secondly, when the
existence of such a level relationship cannot be rejected, a more parsimonious ARDL lag
model is selected using information criteria to determine the optimal lag orders of the
independent variables in the ARDL equation. The second step involves the estimation of the
conditional ECM model from which the co-integration vector and the short term dynamics
can be obtained.
We have applied this procedure to the inflation model presented in the PMA’s 2011
inflation report. The results obtained in that report using the Johansen maximum likelihood co-
integration test and the corresponding vector error correction estimation are broadly
confirmed by the estimation of an ARDL (1, 2, 2) model.
Furthermore the results from the ARDL approach are very much in line with those
obtained using Phillips and Hansen’s Fully Modified OLS.
REFERENCES
Engle, F. and C. Granger, 1987, “Cointegration and Error Correction Representation: Estimation and Testing”, Econometrica, Vol. 55.
Hansen B., 1995, “Rethinking the univariate approach to unit root testing: using covariates to increase power”, Econometric Theory Vol. 11.
Johansen, S., 1991, “Estimation and Hypothesis Testing of Cointegrating Vectors in Gaussian Vector Autoregressive Models”, Econometrica, Vol. 59.
–––––, 1995, Likelihood-based Inference in Cointegrated Vector Autoregressive Models Oxford: Oxford University Press.
MacKinnon, J., A. Haug, and L. Michelis, "Numerical Distribution Functions of Likelihood Ratio Tests for Cointegration", Journal of Applied Econometrics, Vol. 14.
Michel, D. And S. Khalil, 2011, “Effective Exchange Rates For Palestine”, PMA.
Park, J., 1992, “Canonical Cointegrating Regressions”, Econometrica, Vol. 60.
ARDL Approach to Co‐integration Testing: Application to OPT Inflation
11
Pesaran, H. and Y. Shin, 1999, “An Autoregressive Distributed Lag Modelling Approach to Cointegration Analysis”, In S. Strom (eds.) Econometrics and Economic Theory in the 20th Century: The Ragnar Frisch Centennial Symposium Cambridge University Press.
Pesaran, H., Y. Shin, and R. Smith, 2001,“Bounds Testing Approaches to the Analysis of Level Relationships”, Journal of Applied Econometrics, special issue in honour of J. Sargan on the theme “Studies in Empirical Macroeconometrics”, (eds.) D. Hendry and M. Pesaran, Vol.16.
Phillips, P. and B. Hansen, 1990, “Statistical Inference in Instrumental Variables Regression with I(1) Processes”, Review of Economic Studies, Vol. 57.
Phillips, P. and S. Ouliaris, 1990, “Asymptotic properties of residual based tests for cointegration”, Econometrica, Vol. 58.
PMA, 2011 Inflation Report for Palestine.
Stock, J. and M. Watson, 1988, “Testing for common trends”, Journal of the American Statistical Association, Vol. 83.