Modeling a dynamic simultaneous macro-economic system for a developing economy (A case of Nigerian economy)

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1 AMERICAN JOURNAL OF SOCIAL AND MANAGEMENT SCIENCES ISSN Print: , ISSN Online: , doi: /ajsms , ScienceHuβ, Modeling a dynamic simultaneous macro-economic system for a developing economy (A case of Nigerian economy) 1 Godwin Emeka Nworuh (Prof.) and 2 Chinedu Chidinma Nwachukwu (Ph.D.) 1 Dean, School of Management Technology, Federal University of Technology, Owerri GSM 234- ( ) 2 Department of Project Management Technology, Federal University of Technology, Owerri. nedumchisma@yahoo.com, 234-( ) ABSTACT Understanding macro-economic variables is seen as an antidote to a successful statistical planning policy for a sustainable growth of an emerging economic system. Developing countries run the risk of depending wholly on some developed econometrics theories that has not only worked for developed countries but has stood the test of time without being properly informed or guided that a micro-economic variable that is developed for a particular economy may not function optimally in another economic environment. The environment of an economy, government policies, the level of capital input and output,the rate of interest and exchange rate, the consumption pattern etc influences a particular nation as distinct to another. This paper is aimed at developing an econometric model that will be applicable in a developing economy using Nigeria as a model. Keywords; Econometrics, modeling, simultaneous macroeconomic system, mathematical model, statistical theory, multiplier analysis, Dynamic. INTRODUCTION Modeling a simultaneous macro-economic system (model) of an economy is more of an art than science. This involves the understanding of an economy; the way it works, the way it is operated upon and the active key variables in the economy, the knowledge of the economic theory, the statistical theory and the econometric theory. The coordination of all these would produce an operational dynamic simultaneous macroeconomic system that can be used for policy and multiplier analysis. The Centre for Econometric and Allied Research (CEAR) has one too. The economic theory helps to reveal the standard properties of these macro-economic variables and their relationship/interdependent among themselves. It guides a model builder or an econometrician on the variables that will explain a particular endogenous variable in the economy being modeled. The understanding of the economy in the way it works, guides the econometrician to know the variables that are active in the economy. This is necessary since every economy is distinct on its operation, activities, emphasis and level. A variable that is active in this economy may not be active in another economy and the emphasis on one economy might be different from the emphasis on another one. The model builder should be conversant with the economy he is modeling. Statistical theory helps to evaluate the estimates of the estimated and simulated values of the model so that proper evaluation can be carried out. Lastly, the econometric theory guides the econometrician on the assumption of the model and the appropriate estimation methods to adopt, properties to check for and accurate measurements to be made in other to enforce desirable properties of macro-econometric model. There are four stages to follow so as to model an economy and they are: Stage A: Formulation of the maintained Hypothesis (specification of the model) Stage B: Testing of the maintained hypothesis (Estimation of the specified model) Stage C: Stage D: Evaluation of the estimated model Testing the forecasting ability of the model.

2 Stage A: SPECIFICATION OF THE MODEL The first step in any econometric modeling is to specify the model with which one will attempt the measurement of the variables. This stage involves the determination of the following in the model: i) The variables that should be included in the model ii) The expected size and signs of the structural parameter of the model; and C = f(din 1 ) (1) Where DIN = expected national disposable income i.e. C 1 = α 0 + α 1 DIN t + α 2 C t-1 + U t (2) For 0< α 2 < 1 iii) The expected mathematical form of the model (number of equations, linear or non-linear form of these equations) The Variables should be included in the model: Following the permanent income hypothesis, where present assumption is determined by expectations regarding the level of disposable income; this: The inclusion of lagged consumption term as viewed by Cramer (1971) could be interpreted as a stabilizing force imposing past standards on current consumption. The result specification of the private consumption is as follows: P com p t =α 0 + α 1 DIN t + α 2 P com p t-1 +α 1 GDR t + U t (3) The inclusion of Government direct Taxes is one of the independent (explanatory) variable in equation (2) produces more satisfactory parameter estimated of the structural coefficients. In addition, some studies already published any particular field about the economy being modeled provide additional knowledge about the factors determining the dependent variable. Again, it should be clear that the number of variables to be included in the model depends on the time period, the nature of the phenomenon being studied and the purpose of the research. Usually we introduce explicitly in the relationship only the most important (three, four or five) explanatory variables that have behaved well in terms of evaluation criteria. When a variable performs badly, it has to be replaced by an alternative variable from the list of explanatory variables of the dependent variable. It is also important to note that he influence of less important variables are taken care of by the random terms, U. Signs and Magnitude of Parameters: In most cases, from the economic theory, and other applied researches certain suggestions about the signs of the parameters and possible of their sizes are made. Then the model builder can base the specification of the variables on these properties. For example, let us consider the example discussed earlier on consumption. Consumption is splitted into two, private consumption and Government consumption, and the specification of private consumption is as follows: P com p t = α 0 + α 1 DIN t + α 2 P com p t-1 +α 3 GDR t + U t In this relationship, the coefficients α 1 and α 3 are the marginal propensity to consume and should be positive with values less than unity. Actually the real specification of the parameters should be 0 < α 1 < 1, 0 < α 2 < 1, and 0 < α 3 < 1 while the constant intercept (α 0 ) of the relationship is also expected to be positive. The meaning of this positive constant is that when income is zero, consumption will assume a positive value, people will spend past savings, will borrow or find other means for covering their needs. Again all lagged values lie between 0 and 1 i.e. 0 < α 2 <1. if this value is close to 1, the model will refuse to converge. More clarifications and explanations can be obtained from the structural macro-econometric model in the Appendix. These specified signs and sizes guide the model builder on the expected signs and sizes so that if the estimated model predicted wrongly, he will provide answers to them or reject them. He must find why such violations take place and proffer solutions to them so that the model could depicts reality (real life situations). Mathematical Form of the Model: In most cases macroeconomics does not explicitly state the mathematical form of the economic relationship of the variables. It is often helpful to plot the actual data on two-dimensionally diagrams, taking two variables at a time (the dependent and each one of the explanatory 132

3 variables in turn). Most often, the examination of such scatter diagrams throw some light on the form of the relationship and helps in deciding upon the choice of the mathematical form of the relationship connecting the economic variables. In view of this, the econometricians (the model builders) are to Gcom t = α 0 TCR 1 α1 PoP 1 α2 Gcom t-1 α3 experiment with various forms (linear and non-linear) and then choose from among the various results, the ones that are judged as most satisfactory on the basis of certain evaluation criteria. For example, Government consumption has two specifications: +α 3 U t OR Gcom t = α 0 + α 1 InTCR 1 + α 2 InPoP 1 α 3 InGcom t-1 + U t and Gcom t = α 0 + α 1 TCR 1 + α 2 PoP 1 α 3 Gcom t-1 + U t (5) After assessing these two equations, the non-linear specification is most preferred in terms of the criteria for assessing structural parameters. Sometimes, when the linear specifications are not doing well, a model builder also tries the non-linear specification. When the two are compared the one that best describes the relationship is selected to be included in the model. Again, it should be noted that there is no rule or theory that explicitly states whether a particular phenomenon should be studied with a single equation model or with a multi-equation model. It is the econometrician who must decide whether the phenomenon being studied can be adequately describe by a single equation or by a system of simultaneous equations. If an economic relationship is complex and we attempt to approximate it by a single equation model, we are almost certainly bound to obtain incorrect estimates of the parameters. Taking into account the complexity of an economy, like Nigeria, one should hardly expect to study this satisfactorily by using single-equation models, but only systems of simultaneous equations can best describe the relationship existing among the economic variables of the economy. It is interesting to remark that specification is the most important and most difficult stage of any macroeconometric studies. It is often the weakest point of most econometric applications. Some of the reasons for incorrect specification of macro-econometric models are easy to see in: 1) The specification, looseness of statements in economic theories; 2) The limitation on the knowledge of the factors which are operative in the economy being considered; 3) The formidable obstacles presented by data requirements in the estimation of large models. The most common errors of specification are the omission of some variables from the relationships, the omission of some equations and the mistaken mathematical form of the relationships. It should be noted that almost all the macro-econometric estimators (estimation methods) are sensitive to errors of specification. This means that the estimates of the coefficients obtained will be incorrect or unreliable if the model is not correctly specified. Stage B: Estimation of the Model: After the model has been specified (formulated) the econometrician must proceed with its estimation, in other words, he must obtain numerical estimates of the structural coefficients of the model. The estimation of the model is a purely technical stage which requires knowledge of the various macro-econometric estimates, the assumptions and the economic implications for the estimates of the structural parameters of the model. The stage of estimation includes the following steps. 1) Gathering of statistical observations (data) on the variables included in the model. 2) Estimation of the identification conditions of the relationship in which we are interested. 3) Examination of the aggregation problems involved in the regression relationship; 4) Examination of the degree of correlation between the explanatory variables, that is examination of the degree of multicollinearity; and 5) Choice of the appropriate econometric technique for the estimation of the relationship and critical examination of the assumptions of the chosen technique and 133

4 their economic implication for the estimates of the coefficients. Gathering data for the Estimation of the Model: The sources of data for modeling Nigerian economy are the publications of: 1. Federal Office of Statistics (FOS), Beam of Statistics Digest of Statistics, Annual Abstract of Statistics, Review of International Trade, Nigerian Trade Summary, Consumer Price Indices, Industrial Survey of Nigeria, and Economic Indicators. 2. Central Bank of Nigeria: Economic and Financial Review; Annual Reports and Statements of Accounts. 3. International Bodies; International Financial Statistics and Year Book of International Trade Statistics. 4. Others, First Bank Financial Review, Nigeria Capital market, etc. These data are in various forms and types. We shall discuss the following types of data used in econometric modeling. 5. Time Series: Time Series data give information about he numerical values of variables from period to period, such as weekly, monthly, quarterly or yearly. For example, the data on gross national income in the period, 1970 to 1999 forms a yearly time series data on the variable income, or gross domestic product (GDP) in the period 1960 to 1984 forms also a yearly time series data on the variable gross domestic product. 6. Cross-Section data: These data given information on the variables concerning individual agents (consumers or producers) at a given point in time. For example a cross-section sample of consumers is a sample of family budgets showing expenditures on various commodities by each family, as well as information on family income, family composition and other demographic, social or financial characteristics. Crosssection data may also refer to aggregate variables at different countries (or other regional entities) referring to the same time. Such data are usually called crosssection (or cross-country) samples and are used fro international comparative studies. Panel Data: These are surveys of a single (crosssection) sample in different periods of time. They record the behaviour of the same set of individual macroeconomic units over time. Example, standard of living index, industrial surveys and rural and urban consumption survey on a particular product. Data constructed by the econometrician: Dummy variables: Dummy variables are constructed to describe the development of variation under consideration. We assign to it arbitrary units in such a way as to approximate as best we can the variation in the factor which we want to express quantitatively. Dummy variables are constructed by econometricians, or model builders or researchers by assigning 0 or 1 values as the case may be to be mainly used as proxies for other variables which cannot be measured in any particular case for one reason or other. The Dummy variables are used in applied econometric research (studies) as: Proxies to quantify Factors: Dummy variables are commonly used as proxies for qualitative factors such as profession, religion, sex, region-rural and urban, etc. for example, we study the demand for a product with cross-section data, the factor Sex could be represented by a dummy variable, which might be assigned the value of one when the individual is a male and the value of zero when the consumer is a female. In this case, the coefficients of the dummy variable will be positive if female consumes less in real life applications. Again, suppose we have a sample of family budgets from all regions or State of a country; rural and urban and we want to estimate the demand of a particular product as a function of income. It is known that urban dwellers consumer more. Thus, region/state is an explanatory factor in this case. We may represent this factor by a dummy variable to which we may assign 1 for urban dwellers and 0 for rural dwellers. Dummy Variables As Proxies for Numerical Factors: Dummy variables may be used as proxies for quantitative factors when observations on these factors are not available or when it is convenient to do so. For example, suppose we want to measure the savings function or relationship from a crosssection sample of consumers. In most cases age is an important explanatory factor in consumption and savings pattern since people become more thrifty with age. We may approximate with a dummy variable as follows, we divide the consumers into 2 134

5 age groups, each group containing persons with more or less similar consumption and saving pattern. Group B: People who are above 35 years. assigns the value zero (0) if the person belong to the first group and the value of one (1) for the second group. On the assumption that people become more thrifty The savings relationship assumes the function as they grow old. The dummy variable for age S i = β 0 + β 1 X i + β 2 Z i + U t (10) Where X i = income and Z = Dummy variable, β 2 > 0 the value 0 for each of the civil war year and 1 for each of the other years. Use of Dummy Variables for Modelling a Shift of a Relationship Over Time: A shift of relationship implies that the constant intercept (β 0 ) changes in different periods, while the other coefficients remain constant. Such shifts may be taken into account by the introduction of dummy variable in the relationship. Suppose that we have data on government expenditure for the period During the period the civil ware broke out in Nigeria. The abnormal period was characterized by a shift in government expenditure. To capture this we might include a dummy variable say Z which would assume Where : Q 1 = 1 in every first quarter 0 for other quarters Q 2 = 1 in every second quarter (6) 0 for other quarters The Use of Dummy Variable to Capture Seasional Effects in Time Series: A frequently used application of dummy variables is the adjustment of time series for seasional variation. For example, in using quarterly series of retail, purchases, purchases in quarters during which the festive events occur will be higher than in the normal quarters. This seasional effect can be captured by including the three dummy variables Q 1, Q 2 and Q 3 as independent variables. Q 3 = 1 in every third quarter 0 for other quarters The fourth quarter is usually referred to as the base during which all other quarters are zero. This is to Y 1 = β 0 + β i X 1i + α 1 Q 1i + α 2 Q 2i + α 3 Q 3i + U (7) Incorporating a constant term, an independent variable in the context of these three dummy variables. The problem of Aggregation in Macroeconometric Models Aggregation problems arise as a result of the use of aggregative variables in macro-econometric models. Such aggregative variables may involve. (a) Aggregation over sub-set of variables. For example a variable such as Total Gross Domestic Product (TGPD) is the sum of GPD in agriculture, manufacturing, mining and quarrying, transport and in other sectors and services, telecommunication. Again Total avoid the dummy variable trap. The interpretation of a model such as Consumption which is the sum of Private Consumption, P com P and Government Consumption, G com, etc. (b) Aggregation over commodities. We may aggregate over the quantities of various commodities (using appropriate quantity index). For example, if we want to estimate the demand function for food with explanatory variables total income, the price of food and the price of other commodities, all variables will include a certain level of aggregation. (c) Aggregation over time periods. In many cases statistical sources published data which refer to a time period different (longer or shorter) than the unit time period required in econometric research for the functional 135

6 relationship among the econometric variables. For example, the production of most manufacturing commodities is completed is a period shorter than a year. If we use annual figures there may be some error in the coefficients of the production function. (d) Spatial aggregation. For example the population of a ward, (autonomous community), local Government, State, Zone, or Country or the world. The above sources of aggregation create various complications which may impart some aggregation bias in the estimates of the coefficients. It is the function, and to adjust the aggregation variables or the model accordingly whenever possible. Examination of the Degree of Correlation among the Explanatory Variables The multicollinearity: Most economic variables are correlated, in the sense that they tend to change simultaneously during the various phases of economic activity. Income, employment, consumption, investment, exports, imports, taxes, tends to grow in booms and decline economic variables due to growth and technological progress. If however, the degree of collinearity is high, the estimate of the parameters obtained might be misleading, because, in this condition it may not be computationally possible to separate the influence of each one explanatory variable. For example, prices and wages tend to increase together. If we include these two variables in the set of explanatory variables in a demand function, it is most possible that the estimated values of the coefficients will be inaccurate and will show a distorted influence of each individual explanatory variable on demand. In most cases, we use the correlation matrix to measure the extent of correlation of these explanatory variables and where this cannot be tolerated in the research, we increase the size of the sample, that is the number of time period or we use mixed estimation approach which will circumvent this problem. Multicollinearity is usually a less serious problem in simultaneous equation models than in simple equation models. Stage C Evaluation of Estimates: Once the model has been estimated, one should proceed with evaluation of the estimates of the structural parameters, that is with the determination of the plausibility and reliability of these results. The evaluation consists of deciding whether the estimates of the parameters are theoretically meaningful and statistically satisfactory. For these purpose we use various criteria which may be classified into three groups, the economic criteria which are determined by the statistical theory and lastly, econometric criteria, determined by econometric theory. Economic Criteria: These are determined by the principle of economic theory and refer to the sign and the sizes of the parameters of economic relationships, the signs and size of the estimated parameters have to be cross-checked whether they conform to the specification of the model carried on earlier in consonance with the economic theory. If the signs and sizes of these estimates do not conform to this, they should be rejected, unless there is good reason to believe that in the particular instance the principle of economic theory do not hold. In such cases the reason for accepting the estimates with wrong signs or magnitude must be stated clearly. However, in most cases the wrong sign or size of the parameters may be attributed to deficiencies of the empirical data used for the estimation of the model. In other words either the observations are not representative of the relationship or their number is inadequate. In general, if the economic criteria are not satisfied the estimates should be considered unsatisfactory. Statistical Criteria: These are determined by statistical theory and aim at the evaluation of the statistical reliability of the estimates of the parameters of the model. These criteria are the correlation coefficient, the standard error of the estimates, and students-values. Another important criterion is the square root of the correlation coefficient, R 2, the coefficient determination. It is a measure of the extent to which the explanatory variables are responsible for the changes in the dependent variable of the relationship. The standard error of the estimates is a measure of the dispersion of the estimates around the true parameter. The larger the standard error of a parameter, the less reliable it is and vice versa. The t- value measures the inclusion of the explanatory variables in the relationship. Other criteria are the F- ratio, Residual sum of squares, and standard error of regression (SER). These are used to assess the estimates of the structural parameters of the macroeconometric model which is in the attached. Note that 136

7 the statistical criteria are secondary to the economic criteria. Econometric Criteria: These are set by the econometrics and aim at the investigation of whether the assumptions of the econometric estimation method used are satisfied or not in any particular case. The criteria help to establish whether the estimates have the desirable properties of unbiasedness, consistency and efficiency. Again, one of the criteria is the assumption of non-autocorrelated random disturbances, we compute a statistic, known as the Durbin Watson d statistic, for a sample of 20 to 25, if the values lies around 2, we confirm that it does not have autocorrelation, see the estimated equations in the attached, It is clear that the econometrician must use all the above criteria, economic, statistical and econometric, before he can accept or reject the estimates of the structural parameters of the model. Again when the assumptions of an econometric model are not satisfied it is customary to respecify the model (i.e. introduce new variables or omit some others, transform the original variables, etc) so as to produce a new form with re-estimation of the new model and with re-application of all the tests. This process of respecification of the model and re-estimation will continue until the results pass all the economic, statistical and econometric tests. Stage D: Evaluation of the Forecasting Power of the Estimated Model: One way of establishing the forecasting power of the estimated model is to carry out an ex-ante forecast (to forecast for the period outside the estimation period) and compare it with the Y it, Y it+1.. Y it+s, i=1,2., n (8) ex-post forecasts (period within the estimation). Using tests of significance and simulation statistics, if the predicted estimates are significant and the simulation statistics are satisfactory, then the model can be operational and universally acceptable. Another way of establishing the stability of the estimates and the performance of the model outside the sample of data from which it has been estimated is to re-estimate the function with an expanded sample, that is a sample including additional observations. The original estimates will normally differ from the new estimates. The differences is tested for statistical significance or the simulation statistics of the two estimates are compared and if the simulation statistics of the re-estimated model is adequate, hence the model s forecasting ability is good and stable and can be used for policy and multiplier analysis. SIMULATION Definition: In our search for a whole system assessment of the estimates, after the estimated parameters of the equation have been obtained, a solution of the whole model is sought for. There are two types of simulation, these are the stochastic simulation, where artificial samples (Monte Carlo) are drawn, and deterministic simulation, where the actual real-life data and models are used. The deterministic simulation can be dynamic or static, both are considered here. Simulation is a mathematical solution of the system of simultaneous equations when the estimates of the structural parameters have been obtained. It involves search for a predicted Y-vector. which has n equations for same known values of the predetermined variables (h= d + g + q) Y i, t-1, Y j, i-2,.. Y j, 1-c for lagged endogenous variable J=1,2., d (9) X k, t+1, X j, i+2,.. X j, 1+g for exogenous variable k=1,2., g (10) X i, t-1, X j, i-2,.. X 1, 1-c for lagged exogenous variable s=1,2., q (11) All the estimated parameters; (β 1, β 2,.., β 1i, α 1, α 2,., α 1i ) are sued for these solution. values of the predetermined variables are substituted into the structural equations (structural model, see Appendix) and for the period t, we have that The Gauss Seidel Iterative Technique: This is one of the methods used in solving for these endogenous variables (simulation). By this method, the known G(Y 1i, b i ) = Z i (12) Where b i = β ( i.e. estimates of β) Z i = ø (Y it-λ, X sj, X s, i-λ, b s ) 137

8 b * s = α * (i.e. estimates of r) And all the other subscripts and variables are at previously defined in equation (2) The matrix form of equation (3) can be written as βy = Z (13) where b 11 b b 1n β = b 21 b b 2n Y 1 = y 1, y 2, y 3 b n1 b n2... b nn and Z 1 = Z 1, Z 2, Z n normalizing each of the n- vector operations by making Y 1 the subject of the i th equation where [b ij ] is the maximum element in the i th row of equation (14), we have Y = β 1 Y + β k Y + Z (14) b * n2... b 1n β 1 = b ; β R = 0 0 b * 2n b * 1n b * n2... b n,n b n,n-1 0 b ij = b ij / b ii ; I b ij I is the biggest in row i and Z * = Z/b 0 The restructuring of the matrix β into the lower (β * 1) and upper (β 1 ) triangular matrices is necessary because unlike the Jacobi, the Gauss Seidel algorithm uses the latest known value of Y i. Thus the first iteration in the algorithm can be stated as: Y (1) = β * 1 Y (1) +β * R Y (0) + Z * (15) Where Y (0) is the n-vector of initial values of the endogenous variable (Y). These are the current values if Y is a historical (static) solution is required. for a dynamic solution only one set of initial values is required. At the end of the k th iteration, we have Y (k) = β * 1 Y (k) +β * R Y (k-1) + Z * (16) OR Y (K) = (1-β * i ) -1 β * R Y (k-1) +(1-β * i ) -1 Z * (17) The last expression, it should be noted it is in the jacobi from; Y (K) = (β * i - β * R)Y (k-1) + Z * (18) The system converges (i.e. a solution is obtained) when Y (k) and Y (k-1) do not differ appreciably. The necessary and sufficient condition of convergence in equation (15) is that the characteristic roots of (1-β * i ) - 1 β * R be less than unity in absolute sense. This implies that each I0 j I in 138

9 I (1-β * i ) -1 β * R - 01 I = 0 be less than unity (19) In studying the properties of the estimators in relation to the models, some simulation statistics, which we discuss are used to assess the performance of the model ands also to compare the performance of the estimators. 8.0 SIMULATION STATISTICS These are the criteria used to assess the performance of the simulated values from the original II. data series based on the estimator or estimators. These include the following: I. Root-Mean Square Error (RMSE), which Is required to obtain a quantitative measure of how closely individual (endogenous) variables track their corresponding data series. If it defined as: RMSE = 1 (Y s i Y n i) (20) T i=1 T Where Y i s = simulated value of Y i Y i n = actual value of Y i, and T =is the number of periods in the simulation ii. The Mean Absolute Square Error which is as: T MASE = 1 (Y s i Y n i) (21) T i=1 iii. Theills Inequality coefficient is used in studying the accuracy of the forecast and is defined as U = 1 (Y s i Y n i) 2 T i=1 T (22) 1 (Y s i Y n i) (Y a i ) 2 T i=1 T i=1 Note that the numerator of equation (22) is the RMSE, but the scaling of the numerator is such that the value of U will always fall between 0 and 1. The smaller the value of u, the better is the forecast or the historical simulation. The Theills inequality coefficient in equation (22) is decomposed into three components, namely; bias, (U m ), variance (U s ) and covariance (U c ) proportions. Here bias proportion measures the extent to which the average values of the simulated and observed data (series) deviate from each other, hence a desirable value is U m = 0. The variance proportion measures the ability of an estimator to replicate the variability in the endogenous variables of interest. The most desirable value U s is zero and the covariance proportion U c measures what we might call unsystematic error, i.e. it represents the remaining error after U m and U s have been accounted for. The most desirable distribution of these components is as follows: U m = U S = 0 and U c = (23) Application The macro-econometric model developed for the economy of Nigeria was subjected to the simulation, and the simulation statistics obtained are as follows: in Table 4.1, which is ex-post forecasts and 4.2, which is based on ex-ante forecasts. In addition, figures illustrates the tracking ability of the model. From the results obtained, the Theills inequalities, bias proportion, variance proportions and covariance proportions are seen to be very good. This allows that the model replicates the real life situations and can be used for policy and multiplier analysis. REFERENCES Guyarati (1995) Basic Econometrics, Mc-Graw Hill Publication Nworuh G.E. (2002) Consistent Estimators of Non-Linear Simultaneous Equations: Econometric Model of the Nigerian Economy with Undersized Sample Journal of Finance and Investment Vol. 2, No 2, PP

10 Nworuh G.E, and Nwabueze J. C. (2005) Consistent Estimations of Non-Linear Simultaneous Equations: Econometric Model of the Nigerian Economy Journal of Nigeria Statistical Association (JNSA) Vol. 13; No 2. Nehlawi, J.E. (1977) Consistent Estimation of: Econometric Models with Undersized Samples: A study of the Trace (MK 111) Econometric model of the Canadian Economy Inter. Econ. Rev. Vol. 18p Olofin, S. et al (1985) An operational Econometric Model of the Nigerian Economy Journal of Empirical Economics Vol. 10, pp Pindyck, R.S. and Rubinfield (1981) Econometric Models and Economic Forecasts, Kogakusha, McGraw-Hill Please contact the authors for the details of the referred tables, graphs, etc from the body of the paper which should have formed the appendix. 140