Design of Cointegrating Vectors and its Empirical Implications

Design of Cointegrating Vectors and its Empirical Implications *

**Olowofeso O. E. and ***Fagbohun, A. B.

**Industrial Mathematics and Computer Science DepartmentFederal University of Technology,

P. M. B. 704, Akure, Nigeria.Email: olowofeso@hotmail.com

***Mathematics Department Obafemi Awolowo University

Ile Ife, Nigeria.

ABSTRACT

This work examines the design of cointegrating vectors by evaluating the performance of standard tests for unit roots/ stationary. It shows the behaviour of the estimated coefficients of the models with little variation in the number of replications. The maximum likelihood estimates were obtained by using Johansen’s algorithm. The system design indicates from the Monte Carlo experiments that for the cointegrating system to be efficient and consistent, the method must incorporate all prior knowledge about the presence of unit roots. This eliminates the median bias, the nonsymmentry; part of the nuisance parameter dependencies and increase efficiency.

Empirically, the vector error correction estimates shows that the higher the lag interval the better the error correction model and the cointegrating equation provided the sample size of the series is not too short.

Keywords: Vector autoregressive, Monte-Carlo, Johansen’s algorithm, OLS and ML estimates.

1. INTRODUCTION2.

The last century has witnessed tremendous explosion of research on unit roots and cointegration and related topics. However, so many problems remain to be resolved. It was also noticed that most real

1

* This research was completed while the first author was a Visiting Lecturer at the University of Nairobi, Kenya in 2000-2001. The financial assistant of ANSTI UNESCO is gratefully acknowledged.

economic variables have changed radically in mean and variance so that their first two moments are far from constant. The consequences for the statistical properties of estimators and tests are profound as evidenced by the substantial literature on spurious regressions. To overcome such problems, some researchers have suggested differencing the data to remove random walk and trend-like components although others have seriously argued that this loses valuable ‘long-run’ information in the data. The concepts of cointegrated series is one means of effecting a resolution of such debates (Engle and Granger (1987)). The work of Olowofeso and Fagbohun (2001) suggests that simply using first differences of the data in the regression generally will not suffice to uncover the true relations. Shibata (1976), Morimune and Mantani (1995) also argued that among the order selection procedures for autoregressive processes, Akaike Information Criteria (AIC) is found inconsistent, but BIC is found consistent for the univariate stationary process. Lutkepohl (1985) studied the small sample properties of some statistical criteria for the stationary vector autoregressive (VAR) processes where the average frequency of choosing a correct lag order was calculated across one thousand processes. Morimune et al (1995) also estimates the rank of cointegration when the order of vector autoregressive is unknown. Toda (1993) studied the finite sample properties of the Likelihood ratio (LR) tests for cointegration when the lag order of the VAR process with a constant term is known. In this work of designing of cointegrating vectors, the research methodology is in phases as explained in section 2. The system design for the numerical and analytical analysis are presented in section 3. The simulated and empirical results are summarised in section 4 while section 5 concludes.

2. METHODOLOGICAL FRAMEWORK

The research methodology of this work is:

Phase I of the Experiment

A time series analysis package (TSAP) was developed for this study. All programmes are written in Microsoft Visual Basic 6.0 Enterprise and some routines are partially adopted from White Kenneth’s SHAZAM computer program.

All the simulations are performed on a Pentium Processor Intel MMX with 32 MB RAM with Virtual memory of 32- bit – PC.

Phase II of the Experiment

Monte Carlo design was used here. This method uses random numbers to obtain solution which is very close to the optimal but not necessarily the exact solution. This technique yields solution that converges to optimal or correct solution as the number of simulated trials leads to infinity.The first procedure here was to define the problem and structure in a quantitative format. This design helps us to evaluate the performance of standard tests for unit roots/ stationary and examining the behaviour of the estimated coefficients of the models with a little variation in the numbers of replications.

The variables yt and xt are generated by a data-generating process [DGP] that was used by Banerjee et al (1986), Engle and Granger (1987), Hansen and Phillips (1990), and Gonzalo (1994), among others. For the sake of brevity, the first presentation of this work focuses on the bivariate systems Xt = (yt , x1t). However, the experiments can be extended for n 2 and this will be examined in subsequent phases of this research work.

The DGP for the bivariate case is defined as:

yt - x1t = ut

b1yt - b2x1t = t ,ut = ut – 1 + wt

t = t – 1 + t

t = t + t – 1 ,

and

wt = iid N 0 1 t 0 2

For other researchers and reader(s) to understand the results of this work, we must note that DGP can be written with moving average (MA) errors (model used in Gonzalo (1994)). The approach of simulation in Haug (1996) and Gonzalo (1998) are carefully followed. For this experiment, all samples are of size 100 and 5,000 replications are used for each test statistic.

The following values for the parameters are considered for the DGP formulated.

b1 = ( 0, 1) ,b2 = -1 , = (0.5, 0.75, 1.00), = (-0.7, 0, 0.7), = (-0.6, 0, 0.6),

= (0.25, 1, 4).

To choose the lag length n in the Monte Carlo study, we applied a data-dependent lag selection criterion; Akaike Information Criterion (AIC). This is a commonly used criterion in empirical research in connection with the Augmented Dickey Fuller (ADF) test. A rule of thumb, used by some researchers is n = int [4 (T/100)1/4 ] which is equal to 4 when T = 100. In particular, the variances of the error terms are restricted to be one in this work.

Phase III of the Experiment

The Johansen’s algorithms were used here and we also verify that it indeed calculates the maximum like likelihood estimates. The algorithm is presented in the chart below.

The five sets of assumptions used for this algorithm for cointegration equation and VAR specification are:

i. no deterministic trend in the VAR and no intercept or trend in the cointegrating equation.

ii. no deterministic trend in the VAR and an intercept but no trend in the cointegrating equation.

iii linear trend in the VAR and intercept but no trend in the cointegrating equation.

iv. linear trend in the VAR and intercept and trend in the cointegrating equation.

v. quadratic trend in the VAR and intercept and trend in the cointegrating equation.

3. SYSTEM DESIGN

The Time Series Analysis Package (TSAP) was developed. The package can sufficiently be used for unit roots and cointegration tests. It has the ability to estimate the coefficients and the standard errors of time series model (both the single-equation based regression involving the variables that are potentially cointegrated and the systems based test) of any specified lag variables used in any time series analysis.

The package is implemented by a powerful, flexible, efficient easy to learn and use, programming language, called the Visual Basic version 6.0 Enterprise Edition.

,

Calculate the Auxiliary Regressions

Calculate the Canonical Correlations

Calculate the Maximum Likelihood Estimates of

the Parameters

Estimate a (p-1)th order VAR for

yt

Estimate second battery of

regressions, by regressing the scalar

yi, t-1 on a constant

yt-1, yt-2, …,yt-p+1 for

all i = 1, 2, …, n.

Calculate the sample variance-

covariance matrices of the OLS residuals and

Find eigenvalues of the matrix

with eigenvalues ordered 1 , 2, … , n

Johansen’s Algorithm

Hardware Requirements of the system include a minimum of a micro computer with one 3.5 floppy disk drive, 133 MHz clock speed, 32MB RAM, 1.2GB hard disk with four free expansion slots, A VGA monitor and A Light Amplification Simulation from Emission of Radiation (LAZER) printer. The Software Requirements include: A Microsoft Window Operating System, any Visual Basic Programming Language (With the exception of the DOS version), Microsoft Excel, Microsoft Word.

The procedures for the installation are

presented below:

i. Insert the installation diskette on “A” drive;

ii. At the desktop, enter through the “start menu”, to “programs”, and click on “window explorer”;

iii. Select “A” drive and directory box;

iv. Click on INSTALL on the list of files that are explored – the files that are needed for proper running of this package will be transferred from the installation diskette into the appropriate directories.

v. This package can now be loaded by:a. Click the Start Menub. Go through Programc. Look for TSAP and click on it.

As regards the menu system the system has a user interface that consists of the login system, welcome screen, the main menu and a number of submenus through which the user interact with the system.

Access is gained into the system by following this procedure described above. A Login menu is displayed on the screen immediately after loading the package. The system allows the user three trials in the login procedures, after which it terminates the access process if the user fails to supply correct name and correct password. When the login is correct, and the number of trials is less that or equal to three, the welcoming window appears on the screen. The welcoming window contains the name of the package and version. There is provision made in the system to disallowing the welcoming screen coming up, that is, if the user does not want the welcoming screen to appear at another period of loading this package, he can uncheck the box that is checked in the form view. Clicking on Ok on the welcoming screen activates the main menu of this package.

4. SIMULATED AND EMPIRICAL RESULTS

The results of the simulated experiments

and empirical analyses are presented in this section. The first kind of data generating process (DGP) focuses on the bivariate system as discussed in section 2.

Table 1:Characteristics of the empirical distribution of estimators of the cointegrating vectora,b

yt - xt = zt, zt = zt-1 + ezt , ezt 0 1

DGP iid N , a1yt – a2xt = wt, wt = wt-1 + ewt , ext 0 2

Parameters = 1, a1 = 0, a2 = -1, = 0.8, = -0.5, T = 100

t OLS NLS (0) NLS (4) MLECM (0) MLECM (4)

-10

0.50.75

0.250.512

-0.2221-0.1120-0.0651-0.0252

0.0788-0.0014-0.0400-0.0891

Bias in mean0.09720.0054-0.0379-0.0067

0.01030.00510.00260.0015

0.11840.05910.02970.0149

-10

0.50.75

0.250.512

-0.1785-0.0883-0.0442-0.0221

0.04220.0029-0.0150-0.0241

Bias in median0.04930.0041-0.0132-0.0265

0.00450.00220.00130.0004

0.00990.00510.00250.0012

-10

0.50.75

0.250.512

0.39280.19700.09830.0492

0.39300.18870.10860.1163

IQR(50)

0.40780.18620.10650.1184

0.37500.18750.09370.0469

0.40960.20480.10240.0512

-10

0.50.75

0.250.512

0.37850.18940.09470.0474

0.46840.21320.12560.6258

SD0.52190.23340.14160.9837

0.48880.24460.12230.0611

1.66610.83350.41660.2083

-10

0.50.75

0.250.512

14.326.643.473.7

15.228.851.653.2

Prob.(-< .05)12.227.847.951.2

15.029.654.778.5

15.226.049.174.8

a 400 replications were used to calculate the Monte Carlo numerical summaries. b1 is defined by E(yt xt, zt-1) = 1xt + ( - 1)zt-1.

Table 2: Characteristics of the empirical distribution of estimators of the cointegrating vector.a, b

yt - xt = zt, zt = zt-1 + ezt , ezt 0 1 DGP iid N , a1yt – a2xt = wt, wt = wt-1 + ewt , ewt 0 2

Parameters = 1, a1 = 0, a2 = -1, = 0.8, = -0.5, T = 300

t OLS NLS (0) NLS (4) MLECM(0) MLECM (4)

-10

0.50.75

0.250.512

-0.1133-0.0566-0.0283-0.0142

0.0187-0.0052-0.0171-0.0284

Bias in mean0.0196-0.0046-0.0177-0.0257

0.00180.00090.00050.0002

0.00130.00060.00030.0002

-10

0.50.75

0.250.512

-0.0817-0.0408-0.0204-0.0102

0.00860.0033-0.0078-0.0108

Bias in median0.0048-0.0030-0.0071-0.0094

0.00250.00120.00060.0003

-0.0012-0.0006-0.0003-0.0001

-10

0.50.75

0.250.512

0.15780.07890.03960.0197

0.14390.06830.04380.0424

IQR(50)

0.14170.06940.04530.0453

0.13780.06890.03440.0172

0.14060.07030.03520.0176

-10

0.50.75

0.250.512

0.15420.07720.03870.0194

0.13920.06600.04520.0821

SD0.14660.06940.04880.2030

0.13790.06900.03460.0173

0.14890.07440.03720.0186

-10

0.50.75

0.250.512

27.753.478.095.7

40.265.581.880.2

Prob. (- < 0.05)37.865.282.077.6

37.464.288.998.2

37.264.888.397.8

a 500 replications were used to calculate the Monte Carlo numerical summaries.b1 is defined by E(yt xt, zt-1) = 1xt + ( - 1)zt-1.

Table 3 Characteristics of the empirical distribution of estimators of the cointegrating vector.a, b

yt - xt = zt, zt = zt-1 + ezt , ezt 0 1 DGP iid N , a1yt – a2xt = wt, wt = wt-1 + ewt , ewt 0 2

Parameters = 1, a1 = 0, a2 = -1, = 0.8, = -0.5, T = 100

t OLS NLS (0) NLS (4) MLECM (0) MLECM (4)

-0.7-0.4

00.40.7

0.250.5124

-1.0117-0.5422-0.2359-0.0940-0.0383

0.7443-0.0023-0.0534-0.1205-0.2559

Bias in mean-0.22420.4562-0.0421-0.0472-0.0948

0.91870.06300.04350.01020.0034

1.70600.17590.09730.0039-0.0246

-0.7-0.4

00.40.7

0.250.5124

-1.0379-0.5146-0.1982-0.0751-0.0295

0.04130.0036-0.0296-0.0493-0.0500

Bias in median0.03830.0078-0.0258-0.0531-0.0305

-0.0183-0.00020.00220.00110.0005

-0.0309-0.00140.00290.00140.0009

-0.7-0.4

00.40.7

0.250.5124

0.65160.49370.25940.11380.0543

0.81420.37350.20810.21680.2000

IQR(50)

0.88480.38270.20560.22430.1811

0.72810.37120.18790.09380.0469

0.82690.41390.20650.10260.0514

-0.7-0.4

00.40.7

0.250.5124

0.42480.34800.20480.10170.0493

10.85941.60070.20080.79152.6322

SD17.62146.44670.23950.86816.9943

39.79961.25400.37320.13010.0615

35.67121.97050.97390.29780.6236

-0.7-0.4

00.40.7

0.250.5124

0.24.413.537.167.2

7.812.827.232.431.2

Prob. (- < 0.05)6.414.226.229.034.0

8.415.229.654.678.7

8.014.826.249.876.2

a 500 replications were used to calculate the Monte Carlo numerical summaries.b1 is defined by E(yt xt, zt-1) = 1xt + f (zt-1).

Table 4:Characteristics of the empirical distribution of estimators of the cointegrating vector.a, b

yt - xt = zt, zt = zt-1 + ezt , ezt 0 1

DGP iid N , a1yt – a2xt = wt, wt = wt-1 + ewt , ewt 0 2

Parameters = 1, a1 = 1, a2 = -1, = 0.8, = -0.5, T = 300

t OLS NLS (0) NLS (4) MLECM (0)

MLECM (4)

-0.7-0.4

00.40.7

0.250.5124

-0.7011-0.3210-0.1249-0.0476-0.0192

0.1149-0.0014-0.0305-0.0474-0.3787

Bias in mean0.17370.0003-0.0306-0.0830-0.1089

0.29600.00890.00160.0001-0.0001

-0.17400.01380.00350.00100.0003

-0.7-0.4

00.40.7

0.250.5124

-0.6315-0.2519-0.0942-0.0338-0.0133

0.0173-0.0063-0.0151-0.0213-0.0200

Bias in median0.0097-0.0059-0.0141-0.0188-0.0144

-0.0050-0.0025-0.0012-0.0006-0.0003

-0.0028-0.0012-0.0006-0.0003-0.0001

-0.7-0.4

00.40.7

0.250.5124

0.59980.33360.13930.05500.0239

0.29680.13550.08540.08160.0922

IQR(50)

0.28830.13770.08840.09040.0900

0.27480.13740.06880.03440.0172

0.28110.14030.07020.03510.0176

-0.7-0.4

00.40.7

0.250.5124

0.37820.23890.11410.04970.0222

0.81970.13210.08030.10498.7760

SD1.93260.13810.08550.04221.9424

5.90840.16220.07240.03520.0174

5.00640.17620.07790.03780.0187

-0.7-0.4

00.40.7

0.250.5124

0.85.628.664.491.8

20.437.860.461.056.8

Prob. (- < 0.05)20.238.060.860.254.4

19.237.064.288.898.2

19.637.465.487.097.8

a 500 replications were used to calculate the Monte Carlo numerical summaries.b1 is defined by E(yt xt, zt-1) = 1xt + f (zt-1).

From the simulated results of tables 1-4, we pick as the best estimator the one that has the smaller bias in median and smaller inter-quartile range (IQR). The bias and dispersion of all the empirical distributions, except that of non linear least squares (NLS) decrease as

the ratio of signal-noise increases and as the difference between the short-run and long run multiplier (1 ) decreases. In the case of NLS, its performance become worse when 1 increases.

Johansen cointegration test (summary of all the 5 sets of assumptions)Series: FGESC FGEES FGEALags interval: 1 to 1

Table 5Data Trend: None None Linear Linear Quadratic

Rank or No Intercept Intercept Intercept Intercept Intercept

No. of Ces No Trend No Trend No Trend Trend Trend

Log Likelihood by Model and Rank

0 -687.7468 -687.7468 -685.8481 -685.8481 -678.5142

1 -663.4499 -657.7337 -656.9505 -655.9358 -651.7629

2 -654.6010 -647.5576 -647.1575 -645.9488 -642.5753

3 -650.2578 -642.5055 -642.5055 -640.5190 -640.5190

Akaike Information Criteria by Model and Rank

0 45.65280 45.65280 45.58623 45.58623 45.11705

1 44.00118 43.60244 43.59381 43.54334 43.28362

2 43.49385 43.02150 43.01656 42.97640 42.75120

3 43.32028 42.82011 42.82011 42.74703 42.74703

Schwarz Criteria by Model and Rank

0 45.79678 45.79678 45.77820 45.77820 45.35702

1 44.24115 43.85841 43.88177 43.84730 43.61958

2 43.82981 43.38945 43.40051 43.39235 43.18315

3 43.75223 43.30005 43.30005 43.27497 43.27497

L.R. Test:Rank = 3 Rank = 3 Rank = 3 Rank = 2 Rank = 3

Source: Data Analysis

Table 6 :Vector Autoregression Estimates For The Three Selected Variables

FGESC FGEES FGEAFGESC(-1) -0.137358 -4.209682 -1.046668 (0.20423) (4.20684) (0.35917) (-0.67255) (-1.00067) (-2.91412)FGESC (-2) -0.009983 -7.149684 -0.428498 (0.22442) (4.62270) (0.39468) (-0.04448) (-1.54665) (-1.08570)FGEES (-1) 0.008429 0.160163 -0.032560 (0.00991) (0.20410) (0.01743) (0.85070) (0.78472) (-1.86847)FGEES (-2) 0.043687 -0.407862 0.039801 (0.01031) (0.21234) (0.01813) (4.23792) (-1.92081) (2.19545)FGEA(-1) 0.329803 9.222666 1.135662 (0.14007) (2.88510) (0.24632) (2.35462) (3.19665) (4.61045)FGEA(-2) 0.400594 2.443357 1.109553 (0.21334) (4.39442) (0.37519) (1.87772) (0.55601) (2.95734)C 378.7756 2170.860 483.7263 (106.853) (2200.98) (187.914) (3.54482) (0.98632) (2.57418)R-squared 0.990256 0.891455 0.987497Adj. R-squared 0.987333 0.858891 0.983746Sum sq. resids 3324418. 1.41E+09 10281568S.E. equation 407.7019 8397.884 716.9926Log likelihood -196.5443 -278.2247 -211.7866Akaike AIC 12.23949 18.28988 13.36854Schwarz SC 12.57544 18.62584 13.70450Mean dependent 2726.659 13777.46 3987.844S.D. dependent 3622.489 22355.93 5623.889Determinant Residual Covariance 1.88E+18Log Likelihood -642.5055Akaike Information Criteria 42.59788Schwarz Criteria 42.93384

Standard errors and t-statistics in parenthesesSource: Data Analysis

Table 7: Vector Error Correction Estimates at Lag Interval (1 1) Of the Selected Variables.

Cointegrating Eq: CointEq1 CointEq2FGESC(-1) 1.000000 0.000000FGEES(-1) 0.000000 1.000000FGEA(-1) -0.797497 -2.509799 (0.02509) (0.34267) (-31.7825) (-7.32417)C 325.9439 -3660.762Error Correction: D(FGESC) D(FGEES) D(FGEA)CointEq1 -0.963788 -9.569912 -1.771807 (0.18032) (3.54455) (0.31426) (-5.34493) (-2.69989) (-5.63800)CointEq2 0.050265 -1.265750 0.010234 (0.01280) (0.25167) (0.02231) (3.92610) (-5.02945) (0.45865)D(FGESC (-1)) -0.114019 5.940789 0.628899 (0.21872) (4.29941) (0.38119) (-0.52130) (1.38177) (1.64984)D(FGEES(-1)) -0.039272 0.450909 -0.046937 (0.01031) (0.20266) (0.01797) (-3.80914) (2.22491) (-2.61223)D(FGEA(-1)) -0.353040 -1.979747 -1.186406 (0.21919) (4.30873) (0.38201) (-1.61063) (-0.45947) (-3.10566)C 785.9331 -230.5723 1245.753 (111.123) (2184.38) (193.668) (7.07261) (-0.10556) (6.43241)R-squared 0.849227 0.649072 0.733895Adj. R-squared 0.813328 0.565518 0.670537Sum sq. resids 3756566. 1.45E+09 11410259S.E. equation 422.9469 8313.963 737.1199Log likelihood -198.1942 -278.6122 -213.1928Akaike AIC 12.28762 18.24451 13.39863Schwarz SC 12.57559 18.53248 13.68659Mean dependent 460.8704 1960.267 738.0111S.D. dependent 978.9189 12613.11 1284.206Determinant Residual Covariance 2.66E+18Log Likelihood -647.1575Akaike Information Criteria 43.01656Schwarz Criteria 43.40051

Standard errors and t-statistics in parenthesesSource: Data Analysis

Table 8: Vector Error Correction Estimates At Lag Interval (1 2) Of The Selected Variables.

Cointegrating Eq: CointEq1 CointEq2FGESC (-1) 1.000000 0.000000FGEES (-1) 0.000000 1.000000FGEA(-1) -0.999507 -3.772111

(0.13173) (1.07212) (-7.58760) (-3.51835)

@TREND(70) 186.7213 -52.17491C -1875.873 1260.307Error Correction: D(FGESC ) D(FGEES) D(FGEA)CointEq1 -0.397865 -0.088340 -0.760994

(0.20255) (4.33390) (0.23294)(-1.96424) (-0.02038) (-3.26688)

CointEq2 0.040810 -1.217993 -0.070083(0.02985) (0.63869) (0.03433)(1.36715) (-1.90702) (-2.04151)

D(FGESC (-1)) –0.606012 -2.664288 0.167156(0.23443) (5.01584) (0.26960)(-2.58509) (-0.53117) (0.62002)

D(FGESC (-2)) -0.320436 -6.071229 0.212222 (0.22941) (4.90860) (0.26383) (-1.39676) (-1.23686) (0.80439)

D(FGEES(-1)) -0.034798 0.494827 0.025244(0.02113) (0.45216) (0.02430)(-1.64667) (1.09437) (1.03872)

D(FGEES(-2)) 0.020948 0.220544 0.074152(0.02034) (0.43518) (0.02339)(1.02995) (0.50679) (3.17024)

D(FGEA(-1)) 0.318009 6.969873 -0.766973(0.30476) (6.52080) (0.35049)(1.04346) (1.06887) (-2.18831)

D(FGEA(-2)) 0.468498 4.437119 -0.254418(0.25920) (5.54595) (0.29809)(1.80747) (0.80007) (-0.85350)

C -442.5706 3737.783 -1315.256(404.864) (8662.57) (465.603)(-1.09314) (0.43149) (-2.82484)

@TREND(70) 56.99238 -401.9276 151.0872(41.7666) (893.649) (48.0326)(1.36454) (-0.44976) (3.14551)

R-squared 0.860063 0.616777 0.892043Adj. R-squared 0.781349 0.401214 0.831318Sum sq. resids 3459439. 1.58E+09 4575309.

S.E. equation 464.9892 9949.034 534.7493Log likelihood -190.2731 -269.9167 -193.9075Akaike AIC 12.56775 18.69418 12.84732Schwarz SC 13.05163 19.17807 13.33120Mean dependent 477.4885 2032.788 764.6423S.D. dependent 994.4138 12857.16 1302.013Determinant Residual Covariance 8.38E+17Log Likelihood -608.1839Akaike Information Criteria 42.19283Schwarz Criteria 42.77349

Standard errors and t-statistics in parentheses. Source: Data Analysis

Table 9:Vector Error Correction Estimates At Lag Interval (1 3) Of The Selected Variables.

Cointegrating Eq: CointEq1 CointEq2FGESC (-1) 1.000000 0.000000FGEES (-1) 0.000000 1.000000FGEA(-1) -0.894953 -6.984749 (0.10700) (0.97489) (-8.36380) (-7.16468)@TREND(70) 146.9608 1586.073C -1661.250 -13664.93Error Correction: D(FGESC) D(FGEES) D(FGEA)CointEq1 -1.932359 -2.640775 -1.192127 (0.41960) (11.4099) (0.69989) (-4.60522) (-0.23144) (-1.70331)CointEq2 0.168622 -1.878591 -0.034180 (0.03684) (1.00185) (0.06145) (4.57676) (-1.87513) (-0.55620)D(FGESC (-1)) 0.533706 -0.351253 0.532773 (0.36315) (9.87481) (0.60572) (1.46967) (-0.03557) (0.87957)D(FGESC (-2)) 0.115500 -1.403971 0.534267 (0.24252) (6.59468) (0.40452) (0.47625) (-0.21289) (1.32075)D(FGESC (-3)) -0.127955 -9.583555 -0.380365 (0.17623) (4.79202) (0.29394) (-0.72608) (-1.99990) (-1.29401)D(FGEES (-1)) -0.164908 0.699472 -0.029423 (0.03286) (0.89362) (0.05481) (-5.01803) (0.78274) (-0.53676)D(FGEES(-2)) -0.090518 0.825181 0.053958

(0.02873) (0.78122) (0.04792)

(-3.15071) (1.05628) (1.12601)D(FGEES(-3)) -0.086408 -0.091195 -0.053583 (0.02206) (0.59986) (0.03680) (-3.91694) (-0.15203) (-1.45624)D(FGEA(-1)) 0.080830 -0.291110 -0.727864 (0.37142) (10.0998) (0.61952) (0.21762) (-0.02882) (-1.17488)D(FGEA(-2)) 0.301274 -9.681406 -0.746726 (0.34524) (9.38782) (0.57585) (0.87266) (-1.03127) (-1.29674)D(FGEA(-3)) 0.497700 -11.53421 -0.307602 (0.24646) (6.70177) (0.41109) (2.01941) (-1.72107) (-0.74827)C -250.8046 -24848.77 -2277.284 (450.327) (12245.4) (751.135) (-0.55694) (-2.02923) (-3.03179)@TREND(70) 41.52616 2515.663 240.5402 (44.9884) (1223.34) (75.0396) (0.92304) (2.05639) (3.20551)R-squared 0.953623 0.796660 0.924449Adj. R-squared 0.907247 0.593321 0.848899Sum sq. resids 1135426. 8.40E+08 3158928.S.E. equation 307.6018 8364.400 513.0731Log likelihood -169.5189 -252.0923 -182.3092Akaike AIC 11.76364 18.36951 12.78687Schwarz SC 12.39746 19.00333 13.42068Mean dependent 496.6520 2109.436 794.2280S.D. dependent 1010.008 13116.22 1319.912Determinant Residual Covariance 1.20E+17Log Likelihood -560.4730Akaike Information Criteria 40.52421Schwarz Criteria 41.25553

Standard errors and t-statistics in parenthesesSource: Data Analysis

Table 10: Vector Error Correction Estimates At Lag Interval (1 4) Cointegrating Eq: CointEq1 CointEq2FGESC (-1) 1.000000 0.000000FGEES (-1) 0.000000 1.000000FGEA(-1) -1.106985 -11.36352 (0.09689) (1.47330) (-11.4251) (-7.71297)@TREND(70) 274.5688 4049.619C -3042.312 -38693.74Error Correction: D(FGESC) D(FGEES) D(FGEA)CointEq1 -1.742827 45.53271 -1.722945 (0.55039) (17.4571) (1.61271) (-3.16653) (2.60826) (-1.06835)CointEq2 0.072158 -4.986282 0.011112 (0.04189) (1.32863) (0.12274) (1.72258) (-3.75295) (0.09053)D(FGESC (-1)) 0.595113 -43.70619 0.763390 (0.43271) (13.7244) (1.26788) (1.37533) (-3.18455) (0.60210)D(FGESC(-2)) 0.037707 -38.57335 0.688556 (0.31900) (10.1179) (0.93470) (0.11820) (-3.81239) (0.73666)D(FGESC (-3)) 0.035886 -32.90572 -0.277617 (0.20202) (6.40763) (0.59195) (0.17763) (-5.13539) (-0.46899)D(FGESC (-4)) -0.406798 -24.81860 -0.471885 (0.15614) (4.95234) (0.45750) (-2.60537) (-5.01149) (-1.03144)D(FGEES (-1)) -0.086141 3.260112 -0.086125 (0.03751) (1.18986) (0.10992) (-2.29624) (2.73992) (-0.78352)D(FGEES (-2)) -0.037233 2.882681 -0.022074 (0.03620) (1.14825) (0.10608) (-1.02848) (2.51050) (-0.20810)D(FGEES (-3)) -0.023019 1.984223 -0.067596 (0.02593) (0.82245) (0.07598) (-0.88771) (2.41257) (-0.88967)D(FGEES (-4)) 0.015556 0.477347 -0.081057 (0.01830) (0.58042) (0.05362) (0.85009) (0.82242) (-1.51170)D(FGEA(-1)) -0.652251 10.84158 -1.281669 (0.26501) (8.40536) (0.77650) (-2.46127) (1.28984) (-1.65058)D(FGEA(-2)) 0.107985 13.43842 -0.502876

(0.22365) (7.09360) (0.65532) (0.48284) (1.89444) (-0.76738)D(FGEA(-3)) -0.097587 -4.957313 -0.779524 (0.19296) (6.12014) (0.56539) (-0.50574) (-0.81000) (-1.37875)D(FGEA(-4)) -0.710011 -0.927996 -0.296992 (0.16230) (5.14792) (0.47557) (-4.37457) (-0.18027) (-0.62449)C -1671.069 -46274.23 -4287.120 (402.089) (12753.3) (1178.17) (-4.15597) (-3.62840) (-3.63880)@TREND(70) 173.6969 4461.245 419.1470 (38.1291) (1209.37) (111.723) (4.55550) (3.68891) (3.75167)R-squared 0.990489 0.943290 0.951727Adj. R-squared 0.972656 0.836958 0.861216Sum sq. resids 232538.7 2.34E+08 1996486.S.E. equation 170.4915 5407.598 499.5606Log likelihood -144.1995 -227.1645 -170.0006Akaike AIC 10.51209 17.42584 12.66218Schwarz SC 11.29746 18.21121 13.44755Mean dependent 504.1083 2188.304 821.7167S.D. dependent 1031.027 13392.26 1340.970Determinant Residual Covariance 4.39E+15Log Likelihood -498.3891Akaike Information Criteria 37.51879Schwarz Criteria 38.40233Standard errors and t-statistics in parentheses. Source: Data Analysis

Empirically, the three sets of data collected from Central Bank of Nigeria Statistical bulletin (1999 edition) on Federal Government Expenditure on Social and community services (FGESC), Expenditure on Economic services (FGEES) and Expenditure on Administration (FGEA) from 1971 to1999 shows that the use of Augmented Dickey-Fuller test and Phillips-Perron procedure; at 5% significant level, the null hypothesis of a unit root cannot be rejected. The Expenditure on Social and community services for full sample as well as no inflation on expenditure targeting samples display unit root behaviour

according to the unit root tests (For summarise this paper, all the results are not presented here but they are available on request). The Johansen cointegration test that summarizes all the five set s of assumption is presented on table 5. It was discovered that the Akaike information criteria (AIC) and Schwarz criteria (SC) produce interesting results. We noticed from the test that as the number of cointegration increases the log likelihood decreases in all the five sets of assumptions. The apriori expectation concerning the characteristics of SC and AIC was achieved. The likelihood ratio test has rank 3 in all except linear deterministic trend

with intercept and trend in the model. Other interesting results obtained from the lag interval of (1, 1) to lag interval (1, 4) for vector error correction estimates of the selected variables are as shown in tables 7-10, but one good inference that can be drawn from these comparative analyses is that the higher the lag interval, the better the error correction model and the cointegrating equation provided the sample size of the series is not too short.

5. CONCLUSION

From the numerical, analytical and empirical analyses considered in this work, we observed from the Monte Carlo experiment that for the system to be efficient and consistent, the method must incorporate all prior knowledge about the presence of unit roots; this eliminates the median bias, the nonsymmentry; part of the nuisance parameter dependencies and increase efficiency; it must also be flexible enough to capture the dynamic of the system and finally, it must be full system estimation; this eliminates the simultaneous equation bias (violation of assumption six of ordinary least squares (OLS)) and increase efficiency.

The methods compared for the characterisation of the empirical distribution of estimates of the cointegrating vectors shows that only maximum likelihood (ML) in an error correction model (MLECM) satisfies the requirements. The OLS and NLS are not as consistent as the ML in an ECM. Phillips (1991) and Gonzalo (1994) shows that this approach ensures that

coefficients estimates are symmentrically distributed and median unbiased, and that the hypotheses testing may be conducted using standard asymptotic chi-squared tests. None of the other two methods analysed offer these properties (see tables 1 to 4).

The three estimates from tables 1 to 4 are compared in terms of the mean, median, standard deviation (SD) and interquartile range (IQR) of the empirical distributions. Based on the median and IQR, we conclude that all the estimators, except OLS which improves a normalisation before the estimation, are ratios of random variables; therefore their finite sample moments may not exists, thus making the comparison in terms of the second moments very unfair. Hence, the best estimator is the one that has smaller bias in median and smaller IQR.

Empirically, whilst the results on expenditures are revealing, it is worth sounding a note of caution. We should note that the secondary data used for this work are yearly expenditures for the country. Monthly data would have displayed more better results but this was not possible to collect at the time of this work. Complete picture would have been obtained by using vector multivariate design which provides statistically more complete and robust results, and analysis which makes it possible to distinguish the different types of common periodicities and trend in the data. The analysis of a large set of variables and secondly, the appropriate solution to be offered when there are not enough degree of freedom to choose the right number of lags will be a very good exercise for future research.

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