CHAPTER ONE 1.1 INTRODUCTION
A casual look at the published empirical work in business and econometric will reveal that many economic relationships are of single –equation type. In such models, one variable (the dependent variable Y) is expressed as a linear function of one or more other variables (the explanatory variables, the X’s). An implicit assumption is that the cause and effect relationship, if any , between Y and X’s is unidirectional. The explanatory variables are the cause and the dependent variable is the effect .
However, there are situations where there is a two- way or simultaneous relationships between Y and some of the X’s which makes the distinction between the dependent and the explanatory variables of dubious value. It is better to lump together a set of variables that can be determined simultaneously by the remaining set of variables- precisely what is done in simultaneous equation models. In such models, there is more than one equation - one for each of the mutually or jointly dependent or endogenous variables. And unlike the single equation models, in the simultaneous equation models, one may estimate the parameters of a single equation without taking into account information provided by the other equation in the system. In a simultaneous equation system, variables that appear only on the right – hand side of the equation are called exogenous or predetermined variables. They are truly independent or non-stochastic because they remain fixed. Variables that appear on the right-hand side and also used to express the equations are referred to as endogenous variables. Unlike exogenous variables, endogenous variables change value as the simultaneous system of the equation grinds out equilibrium solution. They are endogenous variables because their valuables are determined within the system of the equations.
As a consequence of the endogenous variables appearing as explanatory variables, such an endogenous explanatory variables becomes stochastic and is usually correlated with the disturbance term of the equation in which it appears as an explanatory variable. In this situation, the classical Ordinary Least Squares (OLS) method may not be applied because the estimators thus obtained are not consistent and that is, they do not converge to their true population values no matter how large the sample size. This is simultaneity bias.
In this research, our interest is to consider a simultaneous system of equation that could represent a truly economic situation. We therefore consider the following two-equation having the structural form: y1t = β12y2t + Г 11x1t + Г 12x2t + u1t . . . . 1.1 y2t = β21y1t + Г 21x1t + Г 23x3t + u2t . . . . 1.2
Where,
y1t & y2t are the endogenous variables x1t , x2t and x3t are the exogenous variable u1t & u2t are the disturbance terms β12, β21, Г 11, Г 12, Г 21, Г 23 are the structural parameters
1.2 ASSUMPTIONS ABOUT ERROR TERMS
1 u is random
2 u has zero mean i.e. E (ui) = 0
3 u has constant variance, E(ui2) = δu
4 u is serially independent, E(uiuj) = 0 for i ≠ j
5 u is normally distributed, u ͠ N(0,δ2) 6 u is independent of the exogenous variables of the model, E(uiuj) = 0
1.3 CHOICE OF ECONOMETRIC TECHNIQUES
The problem of choice of technique arises from the fact that any relationship of economic theory is almost certain to belong to a system of simultaneous techniques. The choice of estimation techniques depends on many factors such as: 1. The purpose for which we embark on the estimation of the model. 2. The identification condition of the equations of our model. 3. The presence of (other) endogenous variables among the set of explanatory variables in any particular equation. 4. The available information concerning the other equation of the system. 5. The importance, which the researcher attributes to the various statistical properties of the parameter estimates. 6. The availability of the data. 7. The computational complexity of the technique.
The choice of estimation method depends also on whether we are interested in the values of the structural parameters or of the reduced form coefficients. However, for the purpose of this research, we are interested in both the structural parameters and its reduced form.

1.4 SCOPE AND OBJECTIVES OF THE STUDY In simultaneous equation, it is important to study the small sample properties of various estimators and therefore we have to work with finite sample. It is rather almost impossible to obtain real world samples in which the exogenous variables are held constant. Hence, if we are to judge the small sample performance of alternative estimators of structural parameters, we must abstract from all influences not directly related to such estimators. Obviously, it is inappropriate to use real world data. We therefore use artificial models and generate through them, artificial data. Basic theory behind random number generation with computers offer a simple example of Monte Carlo simulation to understand the properties of different statistics computed from sample data. In other words, we will test – drive estimators, figuring out how different recipes perform under different circumstances. The procedure is not too complex: in each case, we will set up an artificial environment, in which the values of important parameters and the nature of the chance process are specified, then the computer will display the results of the experiments. This study, is designed to: 1. Determine the performance of the parameter estimates using only the upper triangular matrices. 2. Compare OLS estimator to other simultaneous estimating techniques of small samples. To know whether OLS is sufficiently inferior to simultaneous equations techniques to warrant its complete exclusion from serious consideration as an estimation procedure for structural parameters. 3. Performance of simultaneous estimation methods of small sample properties.

1.5 DEFINITION OF BASIC TERMS 1. Exogenous variables: Non-stochastic variables whose values are determined outside the model. 2. Endogenous variables: Variables that are jointly dependent whole values are determined by simultaneous interaction of the relations in the model. 3. Exactly Identified: If an equation has a unique statistical form, it is exactly identified. A system is exactly identified if all its equations are identified. 4. Reduced form equation: An equation from a simultaneous equation model, which expresses the value of the dependent variable {Y} as a function of independent {X} variables alone. 5. Simultaneous equation model: A model with more than one equation in which dependent variable {Y’s} appear as requestors {in the right hand side of the equation}. 6. Simultaneous Bias: This occurs when an estimates sampling destitution is not centered on the time parameter value because it was obtained by estimating a single equation which is actually part of a simultaneous equation model {SEM}. The bias is due to correlation between endogenous right hand side variables and the error term. 7. Structural Equation: One of the equations in a simultaneous equation models. The typical structural equation includes dependent endogenous variables on both sides of the equation. 8. Structural model: It is a complete system of equation, which describes the structure of the relationships of the economic variables. 9. Structural parameters: It expresses the direct effect of each explanatory variable on the dependent variables.
1.6 LIMITATION OF THE STUDY
This study suffered from the effect of poor computing environment. Monte Carlo approach being computationally demanding requires uninhibited access to computing facilities (hardware and software) which were not readily available to us.
This study was done with the aid of my supervisor’s computing system , another major set back is power failure.
1.7 ORGANIZATION OF THE WORK Monte Carlo methods comprises that branch of experimental mathematics, which is concerned with utilization of random normal deviates. The random deviates are generated to have zero mean and unit standard deviation. They are usually assumed to be independently normally distributed.
The random number and deviates thus generated are subsequently transformed to have prescribed characteristics, which are of interest to the investigator

CHAPTER TWO REVIEW OF LITERATURE
2.1 MONTE CARLO STUDIES OF SYSTEM ESTIMATORS The behavior of the system estimators is now available using Monte Carlo studies. The most important of these are Summers [1965], Cragg [1967], and Mosback and World [1970]. The concentration of nearly all the Monte Carlo studies has been on small sample systems. The main findings of the studies is that in the absence of departures from “textbook” assumptions, the finite – sample behavior of the system estimators broadly in line with that indicated by the analytical asymptotic results. The Monte Carlo method is the nearest thing to a controlled laboratory type experiment in econometrics. It may be briefly described as follows: the experimenter artificially sets up a system and generates fixed values from uniform [0, 1] distribution for all exogenous variables which are assumed for the parameters. Values are then generated for the random disturbances for some specified sample size. Values are also calculated for the endogenous variables at each point using the values generated for the random disturbances. Following this, pretending as if the parameters are unknown and using the values of the endogenous and predetermined variables at each point, one or more estimating techniques are applied to obtain an associated estimate of parameters. This process of generating values for the endogenous variables undertaking parameter estimation is repeated or replicated several times. The set of estimates obtained for each parameter by a given estimator can then be used to infer properties for the given sample size. Nearly all the properties of estimators studied have been asymptotic properties. In practice, one has to work with finite samples and it is therefore important to study the sample properties of various estimates. This is often impossible by analytical methods and so numerous Monte Carlo studies have been made in an attempt to shed light on these questions. The essence of a Monte Carlo study is that various sets of parameter values are specified for postulated distributions underlying a mode; repeated numerical drawings from the resultant distributions generate a large number of “samples” and the sampling distributions of the estimates are studied in relation to the true parameter values and to theoretical expectations about asymptotic distributions. The results are of course conditional in the numerical values used to generate the samples, but a range of such studies can build up variable information
2.2 HISTORY OF MONTE CARLO METHOD Monte Carlo simulation is a general term that has many meanings. The word “simulation” signifies that we build an artificial model of a real system to study and understand the system. The “Monte Carlo” part of the name alludes to the randomness inherit in the analysis. It is a method of analysis based on artificially recreating a chance process [usually with a computer] running it many times and directly observing the results. The name “Monte Carlo” was coined by [Physicist Nicholas] Metropolis [inspired by[Stamlaw] Ulam’s interest in Poker], during the Manhattan project of World War II, because of the similarity of Statistical Simulation to games of chance and because of the capital of Monaco was a center for garnishing and similar pursuits. Monte Carlo is now used routinely in many diverse fields. Monte Carlo methods constitute a fascinating, exacting and often indispensable craft with a range of applications that is already very wide yet far from fully exposed. The Monte Carlo method provides heuristic solutions to a variety of mathematical problems by performing statistical sampling experiments on a computer. The method applies to problem with no probabilistic content as well as to those with inherit probabilistic structure. Among all numerical methods that rely on a N – point evaluation in M – dimensional space to produce an approximate solution, the Monte Carlo method has absolute error of estimate that decrease as N superscript ½ whereas, in the absence of exploitable special structure all others have errors that decrease as N superscript – 1/M at best. The method is called after the city in the Monaco principality, because of the roulette, a simple random number generator. The name and the systematic development of the Monte Carlo method dates from about 1944. In 1899, Lord Rayleigh showed that a one – dimensional random walk without absorbing barriers could provide an approximate solution to a parabolic differential equation. In 1931, Kolmogorov showed the relationship between Markov stochastic processes and certain integro – differential equations. In early part of the twentieth century, British Statistical Schools indulged in a fair amount of unsophisticated Monte Carlo work. The real use of Monte Carlo methods as a researcher tool stems from work on the atomic bomb during the second World War. This work involves a direct simulation of the probabilistic problems concerned with random Newton diffusion in fissile material, but, even at an early stage of these investigations, von Neumann and Ulam refined this particular “Russian roulette” and “Splitting” methods (Curtiss 1951). However, the systematic development of these ideas had to wait the work of Harris and Herman Kalm in 1948. The possibility of applying Monte Carlo method to deterministic problems was noticed by Fermi, von Neumann and Ulma and popularized by them in the immediate post –wars years.
About 1948, Fermi, Metropolis and Ulma obtained Monte Carlo estimates for the Eigen values of Schrodinger equation. In about 1970, the newly developing theory of computational complexity began to provide a more precise and persuasive rationale for employing the Monte Carlo method. The theory identified a class of problems for which the time to evaluate the exact solution to a problem within the class grow, at least, exponentially with M. The question to be resolved was whether or not the Monte Carlo method could estimate the solution to a problem in this intractable class 0 within a specified statistical accuracy in time bounded above by a polynomial in M. Numerous examples now support this contention. Karp (1985) shows this property for estimating reliability in a planar multiterminal network with randomly failing edges. Dryer (1989) establishes it for estimating the volume of a convex body in M-dimensional space. Brooder (1986) and Jerrum and Sinclair (1988) establish the property for estimating the permanent of a matrix, or equivalently, the number of perfect matching in a bipartite graph.

2.3 IDENTIFICATION CONDITION OF THE MODEL
Liu (1960) argues that most econometric models are incorrectly specified with very few explanatory variables appearing in each equation. The resulting over identification is fictitious, because there are too many variables missing from each equation. In the actual economic world, the relationships include many under identified and hence, it is pointless to attempt the estimation of individual coefficients, because they would not be reliable. Under these circumstances, the only meaningful measurement would be the estimation of the reduced-form models with a large number of predetermined variables.
The proponents of the recursive systems express the opposite view. For example, Wold and Jureen suggest that economic theory usually defines the most important explanatory variables of the relationship. There may be many other minor determinants of a dependent variable, but they may be well excluded from the function, simply because each out of them has a negligible effect. Their influence as a whole may be well reflected by the random variable. Under these circumstances, economic relationships are over-identified, but recursive models may be adequately present them. In this case, each question may be estimated by OLS without any of its defects, if one proceeds methodically with the estimation, that is, one starts with the equation, which includes only exogenous variables and then substitutes gradually in the next equations.

2.4 IMPLICATIONS OF THE IDENTIFICATION STATE OF A MODEL
Identification is closely related to the estimation of the model. If an equation (or a model) is under identified, it is impossible to estimate all its parameters with any econometric technique. If an equation is identified, its coefficients can in general be statistically estimated. In particular: (a) If the equation is exactly identified, the appropriate method to be used for estimation is the method of two stage least square (2SLS) (b) If the equation is over-identified, Indirect Least Squares cannot be applied because it will not yield unique estimates of the structural parameters
There are various other methods, which can be used in this case, for example, Limited Information Maximum Likelihood(LIML)
2.4.1 RULES FOR IDENTIFICATION Identification may be established either by the examination of the specification of the structural model or by the examination of the reduced form of the model. 1. The order condition for identification
This condition is based on a counting rule of the variables included and excluded from the particular equation. It is a necessary but not sufficient condition for identification of an equation. It may be stated in two different, but equivalent ways as follows:
Definition 1
In a model of M simultaneous equations, in order for an equation to be identified, it must exclude at least M-1 variables (endogenous as well as predetermined) appearing in the model. If it excluded exactly M-1 variables, the equation is just or exactly identified. If it excludes more than m-1 variables, it is over identified.
Definition 2 In a model of M simultaneous equations, in order for an excluded from the definition must not be less than the number of endogenous variables included in that question less 1 i.e. K-k > m-1 m - Number of endogenous variables k - Predetermined variables in a particular equation.
K - Total number of variables in the model [endogenous and exogenous]

2. Rank condition for identification
The order condition stated above is a necessary but not sufficient condition for identification i.e. even if it is satisfied; it may happen that an equation is not identified. Therefore, we need both a necessary and sufficient condition for identification. This is provided by the rank condition of identification, which may be stated as:
“In a model containing M equations in M endogenous variables, an equation is identified if and only if at least one non-zero determinant of order [M-1][M-1] can be constructed from the coefficients of the variables [both endogenous and predetermined] excluded from that particular equation, but included in the other equations of the model”
To apply the rank condition, one may proceed as follows: 1. Write down the system in a tabular form 2. Strike out the coefficients of the row in which the equation under consideration 3. Also, strike out the columns corresponding to those coefficient in 2 which are non – zero 4. The entries left in the table will then give only the coefficients of the variables included in the system, but not in the equation under consideration. 5. From these entries, form all possible matrices, like A, of order M-1 and obtain the corresponding determinants.
If at least one non – vanishing or non – zero determinants can be found, the equation is just or over identified. The rank of the matrix, say A, in this case is exactly equal to M-1. If all the possible [M-1][M-1] determinants are zero, the rank of the matrix A is less than M-1 and the equation under investigation is not identified. The following general principles of identifiability of a structural equation in a system of M simultaneous equations 1. If K-k > m-1 and the rank of the matrix A is M-1, the equation is over identified. 2. IF K-k = m-1 and the rank of the matrix A is M-1, the equation is exactly identified. 3. If K-k > m-1 and the rank of the matrix A is les than M-1, the equation is under identified. 4. If K-k < m-1, the structural equation is under identified. The rank of the matrix A is bound to be less than M-1
Notice an interesting feature of the reduced – form equations. Since only the predetermined variables and stochastic disturbances appear on the right – sides of these equations and since the predetermined variables are assumed to be uncorrelated with the disturbance terms, the Ordinary Least Squares method can be applied to estimate the coefficient of the reduced – form equations [II’s]. From the estimated reduced – form coefficients, one may estimate the structural coefficients [the β’s]. This procedure is known as Indirect Least Square [ILS] and the estimated structural coefficients are called ILS estimates.
2.5 METHODS OF ESTIMATING SIMULTANEOUS MODELS Economists have a genuine interest in structural parameters since many of them correspond to fundamental parameters in economic theory. To some extent, this also true of reduced – form parameters for some of these are important impact multipliers. However, the major use of reduced – form system is the production of forecasts of endogenous variables conditional on specified endogenous variables. Such forecasts depend upon all the reduced – form coefficients and their accuracy reflects the performance of the whole model. Summers has given this aspect major prominence in his study. Given any set of estimates of the structural parameters B and Г , the corresponding reduced – form coefficients can be derived from ∏ = -B-1Г. In addition, the reduced – form coefficients may be estimated by the direct application of ordinary least squares {OLS} to each reduced – form equation separately. An interesting effect of variations in the B and Σ matrices on the estimates of the structural coefficients may be noted. The Σ matrix is the variance – covariance matrix of the structural disturbances. In the derivation of simultaneous equation estimators, no restrictions were placed on the off – diagonal elements of Σ. in other words, the disturbances in various equations may be contemporaneously correlated. Cragg (1966) has compared the effects of a structure 1 in which Σ has off – diagonal elements and a structure 2 in which the off – diagonal elements of Σ are very small. His main conclusion is, “the experiment using structure did not produce the same ranking of the estimators. Differences in the rankings of the full model methods from the consistent k – class estimators were negligible. The inferiority of OLS was not as pronounced.” This is of course a very plausible result. Quandt also found that as the sparseness {number of zero coefficients} of the B matrix increased the mean bias of k – class estimators {including OLS and 2SLS}decreased. However, research by Nagar and others have provided some evidence that 2SLS may have advantage over LIML in small samples.
2.6 LIMITED INFORMATION METHODS OR SINGLE– EQUATION METHODS
The limited Information Maximum Likelihood approach is to maximize the likelihood function for the (m-1) endogenous variables. This approach was developed by Anderson and Rubin. The application of the method requires one to know in addition to the specification of the single equation being estimated, merely the predetermined variables appearing in the other equations of the model as in 2SLS. The mathematical development of the LIML is complicated and lengthy. As Klein (1974) puts it, “Single equation methods in the context of a simultaneous system, may be less sensitive to specification error in the sense that those parts of the system that are correctly specified may not affected appreciably by errors in specification in another part”.
Sawa (1969) and Richardson (1968) derived the exact distribution of the OLS and 2SLS estimators in an equation with two endogenous variables. Mariano and McDonald (1979) consider the 2SLS estimator, (and LIML) in the just identified case, while Holly and Phillips (1979) used an asymptotic expansion to approximate the distribution of 2SLS estimator. Anderson and Sawa (1973) derived an alternative form of the exact distributions of OLS and 2SLS and present approximations as well. Anderson (1977) examines the expansions further and Anderson and Sawa (1979) present tables of the exact distribution and examine the accuracy of the approximation. Additional work on approximation of instrumental variables estimation is done by Mikhali (1971). Mariano and Sawa (1972) obtained the exact distribution of LIML estimator for the two endogenous variables case while Anderson (1974) provides an approximation. Anderson and Sawa (1979) compare the2 SLS and LIML estimators and note substantial differences in small sample properties. Carter (1976), Phillips (1980) and Mariano (1977) consider the distribution of instrumental variables estimation.
2.7 RECURSIVE MODELS AND ORDINARY LEAST SQUARES Because of the interdependence between the stochastic disturbance terms and the explanatory variables(s), the OLS method is inappropriate for the estimation of an equation in a system of simultaneous equations. if applied erroneously, then, the estimators are not only biased (in small samples), but also inconsistent; that is, the bias does not disappear no matter how large the sample size. There is however, one situation where OLS can be applied approximately even in the context of simultaneous equations. This is the case of the recursive, triangular or causal models. To see the nature of these models, consider the following two – equations system given in * and ** where, as usual, the Y’s and the X’s are, respectively, the endogenous and exogenous variables. The disturbances are such that cov (U 1t,U2t)= 0. That is, the same period disturbances in different equations are uncorrelated (technically, this is the assumption of zero contemporaneous correlation). This equation satisfies the critical assumption of the classical assumption of the classical OLS, namely, uncorrelatedness between the explanatory variables and the stochastic disturbances. Hence, OLS can be applied straightforwardly to these equations. Thus, in the recursive system of OLS can be applied to each equation separately.

2.8 REDUCED – FORM METHOD OR INDIRECT LEAST SQUARES The reduced – form method is a single equation method in that it is applied to one equation of a system at a time. It is inappropriate when the equations of the structural system contain both predetermined and endogenous variables among the set of explanatory variables, provided the equations of the system are exactly identified. The reduced – form method, which is also known as Indirect Least Squares [ILS] method may be outlined as follows:
Step 1: Obtain the reduced – form of the structural model rewriting the equations in such a way that the endogenous variables are expressed as a function of the predetermined variables and the stochastic disturbances. Consider the equation 1.1 and 1.2, the model may be written in matrix in matrix form as Byt + Гxt = Ut t = 1, 2 , . . n …………………2.1 Where, B is G x G matrix of coefficients of endogenous variables. Г is a G x K matrix of coefficient of predetermined variables. and yt xt and ut are column vectors of G, K and G element respectively. Yt = y1t Xt = X1t Ut = u1t y2t X2t u2t yGt Xkt uGt If we assumed that B matrix is non-singular, the reduced –form of the model may be written as: yt = ∏xt + Vt …………………………. 2.2 where ∏ is a G X K matrix of a reduced-form coefficients
Vt is a column vector of G reduced-form coefficients disturbances
∏ = -B-1Г
Vt = B-1Ut
Step 2: If the other usual assumptions about the disturbances term of the reduced – form equations are satisfied, we apply OLS to each equations of the reduced – from system and we obtain estimates of the reduced – form coefficients.
Step 3: Use the estimates f the reduced form coefficients in solving the system of coefficients’ relationships for the structural parameters. These estimates will be unique, if the structural model is exactly identified.

2.8.1 ASSUMPTIONS OF THE REDUCED – FORM METHOD
1. The structural equations must be exactly identified
2. The random variable of the reduced – form equations must satisfy the six stochastic assumptions of the OLS. The random variable v of the reduced – form equation, which is a linear combination of the random variables of the structural model [u’s] and the structural parameters, must have the following properties: * It is random * It has zero mean i.e. E[vi] = 0 * It has constant variance, E[vi 2] = δv * It is serially independent, E[vivj] = 0 for i ≠ j * It is normally distributed, v ͠ N(0,δ2v)
It is independent of the exogenous variables of the model, The exogenous variables of the model must not be perfectly collinear. the macro – variables must be properly aggregated
2.9 INSTRUMENTAL VARAIBLES Instrumental variables estimation is a technique with considerable intuitive appeal due to its simplicity; on the other hand, its importance in empirical applications is rather exaggerated. An instrument is a variable that is independent of (or minimally uncorrlelated with) the disturbances of the equation and is correlated (presumably highly) with the explanatory variables appearing therein. All instrumental variables estimators, no matter the choice of instruments, are unbiased and consistent. If a choice is to be made, the criterion must be based on efficiency considerations. It is intuitively clear that instruments replace the explanatory variables of the model in forming the moment matrix to be inverted in the process of estimation. Hence, it would be desirable to have instruments that are highly correlated with the variables they replace.
2.10 TWO STAGE LEAST SQUARES [2SLS] This method was developed by Theil and independently by Basmann. It is a single – equation method, being applied to one of the system at a time. It has provided satisfactory results for the estimates of the structural parameters and has been accepted as the most important of the single – equation techniques for the estimation of over identified models. Two stages least squares, like other simultaneous equation techniques, aims at the elimination as far as possible of the simultaneous equation bias. The existence of endogenous variables in the set of explanatory variables of the function is the source of the bias. This method boils down the application of OLS in two stages: in the first stage, we apply least squares to the reduced – form equations in order to obtain an estimate of the exact and the random components of the endogenous variables. In the second stage, we replace the endogenous variables appearing in the right – hand side of the equation with their estimated value and we apply OLS to the transformed original equation to obtain estimates of the structural parameters.
2.10.1 ASSUMPRIONS OF 2SLS 1. The disturbance term u of the original structural equations must satisfy the usual stochastic assumptions of zero mean, constant variance and zero covariance. 2. The error term of the reduced – form equations v1 must satisfy the usual stochastic assumptions. 3. The explanatory variables not perfectly multicollinear and all macro – variables are properly aggregated. 4. It is assumed that the specification of the model is correct so far as the endogenous variables are concerned. 5. It is assumed that the sample is large enough, and in particular that the number of observations is greater than the number of predetermined variables in the structural system.

CHAPTER THREE
3.0 METHODOLOGY This chapter focuses on the methodology employed for the research work. It begins by discussing the Monte Carlo Approach and design of our Monte Carlo experiment. It covers a general framework of the study by describing the assumed model, giving the statement of the values of the parameters, procedures for generating the variables and the disturbance terms as well as the estimation procedures. It ends with a statement of the criteria for evaluating the performance of the estimators. 3.1 THE MONTE-CARLO APPROACH (MCA) The MCA involves an attempt to study the small sample properties of estimators. It is applied to solve problems in both pure and sciences and econometrics. The MCA has been applied not only to the choice of alternative estimators but also in determining the impact of sample size, serial correlation, multicollinearity and other factors in the different possible estimators in a given study. This approach creates a “laboratory environment” where controlled experiments on estimators are performed. To date, the sample properties of the various techniques have been studied from simulated data in Monte Carlo Studies and not with direct application of the techniques to actual observations. This approach is due to the fact that actual observations on economic variables are usually infected by multicollinearity, autocorrelation, error of measurements and some other economic “diseases” and in some cases simultaneously. Studies on small sample properties of estimators are usually based on the assumption of the simultaneous occurrence of all the problems. By the Monte Carlo approach, the econometrician can generate data sets and stochastic terms, which are free of all but one problems listed above and therefore generate data resembling those obtained from controlled experiments. In a Monte Carlo experiment, the experimental artificially sets up a system and specifies values for all the explanatory variables and for the parameters, values are then generated for the random disturbances for some specified sample sizes. Using these values, estimates are then computed for the endogenous variables of each sample point. Next, pretending as if the parameters are unknown, using only the values of the endogenous and explanatory variables at each sample point, several estimators are separately applied to obtain an associated estimate of parameters. The process of generating values for the disturbances, calculating values for the endogenous and undertaking parameter estimation is then replicated a large number of times. This, however, builds up the empirical distribution of the estimators, so that we can clearly determine how close to or far it may be from parameter it is estimating. A typical Monte- Carlo econometric experiment takes the following Y = F(X,β) +U Where U ͠ N(0,δ2) and also satisfies other classical assumption of squares estimation. Numerical values are assigned to all the parameters embodied vector β. The variances δ2 is also assigned a numerical value, and or the assumed δ2 normal deviates are selected and used in generating disturbance term. A random sample of size T is X selected and the numerical vector F(X,β) are computed. The vector Y is then obtained by computing F(X,β) + U. The regression of Y on X is performed to produce estimates β of β is repeated many times or replicated using the same size making possible for the sampling distribution of β to be constructed. The empirical distribution so obtained is then utilized in evaluating precision of β and in making other comparisons especially of the relative performance of different estimators of β. The procedure described above is then repeated for different samples sizes and replicated to investigate asymptotic effects and the stability of the results.
3.2 IDENTIFIABILITY OF THE EQUATION Recall the equation 1.1 and 1.2 Y1t = β12y2t + ϒ11x1t + ϒ12x2t + u1t . . . . 3.1 Y2t = β21y1t + ϒ21x1t + ϒ23x3t + u2t . . . . 3.2
Using ORDER consideration, K-k ≥ m-1 and rewriting the above equation, we have -Y1t+ β12y2t + ϒ11x1t + ϒ12x2t + u1t = 0 . . . . 3.3 - Y1t + β21y1t + ϒ21x1t + ϒ23x3t + u2t =0 . . . . 3.4

Equation Number of Predetermined No of Endogenous Identified variable excluded K-k variable included less one (m-1)
1 3-2 = 1 2-1 = 1 Exactly
2 3-2 = 1 2-1 = 1 Exactly

Likewise considering the RANK condition
Equation Y1 Y2 X1 X2 X3
1 1 β12 Г11 Г12 0
2 β21 1 Г21 0 Г31

Consider the first equation which exclude X3 for the equation to be identified, we must obtain at least one non-zero determinant of order m-1 from the coefficients of the variables excluded from this equation, but included in other equations. To obtain the determinant we first obtain the relevant matrix of coefficient of variables.

4.3 REDUCED FORM Y1t = β12y2t + Г11x1t + Г12x2t + u1t Y2t = β21y1t + Г21x1t + Г23x3t + u2t

Hence in Matrix form: X1t u1t 1 β12 Y1t = Г11 Г12 0 x2t u2t Β21 1 Y2t Г21 0 Г23 x3t Therefore the reduced form can be given as βyt = Гxt + Ut yt = β-1Гxt + β-1ut yt = ∏x + vt where :
∏ = β-1Гt and V = β-1u

1 β12 -1 X Г11 Г12 0 1 β21 Г21 0 Г23

-1 1 β12 Г11 Г 12 0
1- β12 β21 β21 1 Г21 0 Г23

β-1Гx = -1 1 β12 Г11 Г12 0 x1t 1- β12 β21 β21 1 Г21 0 Г23 x2t X3t = - 1 Г 11 x1t + β12 Г21 x1t + Г12 x2t + β12 Г23 X3t 1- β12 β21 β21 Г11 x1t + Г21 x1t + β21 Г12 x2t + Г23 X3t

V = β-1u = -1 1 β12 u1t 1 - β12 β21 β21 1 u2t

= -1 U1t + β12u2t 1 - β12 β21 β21u1t + u2t

yt = β-1Гx + β-1u = -1 Г11 x1t + β12 Г21 x1t + Г12 x2t + β12 Г23 X3t + -1 U1t + β12u2t 1- β12 β21 β211 Г11 x1t + Г21 x1t + β21 Г12 x2t + Г 23 X3t 1- β12 β21 β21 u1t + u2t

Y1t = -1 Г11 x1t + β12 Г21 x1t + Г12 x2t + β12 Г23 X3t + U1t + β12u2t 1 - β12 β21 β211Г11 x1t + Г21 x1t + β21 Г12 x2t + Г23 X3t + β21 u1t + u2t Y2t

Y1t = -1 Г 11 x1t + β12 Г21 x1t + Г12 x2t + β12 Г23 X3t + U1t + β12u2t 1 - β12 β21

Y2t = -1 β21 Г11 x1t + Г21 x1t + β21 Г12 x2t + Г23 X3t + β21 u1t + u2t 1- β12 β21

3.4 GENERATION OF MONTE CARLO DATA
The main task is the generation of stochastic dependent (endogenous) variables Yit ( i= 1,2. t =1,2, , , n) which are subsequently used in estimating the parameters of the model.
To achieve this, the following have to be assumed; 1. Values of the predetermined variables X1t, X2t,X3t t=( 1, 2 - - - - n) 2. Values of the parameter β12, β21, Г11, Г12, Г21, Г23 3. Values of the element Ω
The most complex step in generating stochastic dependent variables is the simulation of the error Uit (i= 1,2 t=1,2, , n)
To set up Monte-Carlo experiment, we proceed as follows: 1. The sample size M is specified as M= 15, 20 2. Numerical values are assigned arbitrarily to each of the structural parameters as follows : β12 = 1.8 β21 = 1.5 Г11, = 1.5 Г12,= 1.0 Г21, = 0.5 Г23= 2.0 3. The covariance matrix of the disturbance is specified arbitrarily as follows: Ω = δ11 δ 12 = 7 3 δ21 δ22 3 5 where δ12 = δ21 4. Values of the predetermined variables X1t, X2t, X3t are generated from a pool of uniformly (0,1) distributed random numbers (Kimenta 2006) using Microsoft Excel Package such that the correlation coefficients ᵨ(X1, X2), ᵨ(X1, X3) and ᵨ(X2, X3),are of the following magnitudes: a. Highly positively correlated r>+0.05 b. Highly negatively correlated r<-0.05 c. Feebly negatively and positively correlated -0.05<r<+0.05
Consequently, there are three sets of X’s in each category of multicollinearity group. We perform the correlation matrices to ascertain the usefulness of data set. 3.5 GENERATING THE RANDOM DISTURBANCE TERM (U) A two stage process was used to generate the values of the random disturbances. The first stage involves drawing independent series of normal deviates of the required length. In the second stage, the series are then transformed into series of the random disturbances in such a way as to guarantee that they the covariance which are of the same s those set out in the model design.
There are two random disturbances (one for each equation) in our model; hence we generate two independent series denoted as ξ1t , ξ2t . Thereafter, each of the series ξ1t and ξ2t is standardized to have zero mean and unit variance. To compute random disturbance terms that behave as described above, we used the method presented in Nagar (1969) for transforming M independent series of normal deviates of length N into M series of random variables with zero means and a specified covariance matrix.
At the third stage, the error terms are then transformed to be assumed distributed as N (O, Σ) where E[εt] = 0 and cov [εtεt’] = δtt. Since Σ is a positive definite matrix, we decompose it by a non- singular triangular matrix P such that Σ = PP’ U1t = D* D* β12 U2t D* β21 D*
Where D* = [ 1- β12 β21]-1
These steps are discussed below: First Stage: Select εit (i= 1,2 t= 1,2, , , N)
Second stage: Standardized ε using εit such that εt* ͠ N(μt, δ2t).
Third Stage: Selected only pairs of it which are not correlated These pairs are then transformed and are assumed to be normally distributed as
N (0,ø) where ø = ø X In since ø is a non –singular, it can be shown that ø = ø X I can be decomposed by a non –singular matrix P and P’ such that
Let ξ1t = (ξ1t, ξ2t) then by construction:
E (ξt) =0, Cov (ξ1t, ξ2t) = δn I
Since Σ by definition is a positive definite matrix, there exists a non singular matrix P such that
Σ = PP’
Consider now the random vectors:
Ut = P ξt
By construction, the vectors
{ Ut : t =1,2, , , N} have the properties
E(Ut) =0
Σ = Cov (Ut, Ut’) = Ω In . . . . . . . . . . . . . . . . . . where is the Kronecter product or direct product. In this fashion, the desired error terms having the prescribed variance-covariance matrix is obtained. According to this approach, specific values have to be assigned to the structural parameters: δ1= β12 δ2 = β12 ϒ11 ϒ21 say δ0 = δ01 δ02 putting β0*= 1 - β012 - β021 1 C*0 = ϒ011 0 0 ϒ021
The reduced form of the system can be written as:
(Y1t, Y1t,) = (X1t, X2t) C*0 B1*-1 + (U1t, U2t) B0*-1 …. …………………i since all quantities on the right hand of the last equation are known, vectors of the predetermined variables { [Y1t, Y2t]: t = 1, 2, . . ., N} can be generated. At the end of the third step, the data set obtained { [Y1t, Y2t]: t = 1, 2, . . ., N} has been generated by a model described in equation 1.1 and 1.2 with fixed parameter vectors δ0 and error terms that are “intertemporarily independent” and distributed as N (0,Σ), the matrix Σ being known.
The next step at this stage is to estimate the known parameters as if they were not known using different estimation methods and comparing the performances of the estimators.
Consequently, if N samples of { [ξt, ξ2t]: t = 1, 2, . . ., N} are selected and are used to generate N random vectors (U1,U2) each of size N, we have M samples of the form { [U1t, U2t]: t = 1, 2, . . ., N}.
Also, selecting a random sample of the size N on the predetermined variables yields { [Xit, X2t]: t = 1, 2, . . ., N}. These in conjunction with the fixed parameter vector δ0 will determine M samples of the endogenous variables i.e. { [Y1t, Y2t]: t = 1, 2, . . ., N} using the relation of i. Both the parameter vector δ0 as well a the set of predetermined variables { [Xit, X2t]: t = 1, 2, . . ., N} is common to all M samples.
Processing the N data sets yield N estimates of the parameter vector δ, since each estimate corresponds to a random vector { [U1t, U2t]: t = 1, 2, . . ., N} this estimate is treated as an “observation” from a population whose distribution is the distribution of the given estimator for samples of size T.
It is rather difficult if not impossible to generate { [ξt, ξ2t]: t = 1, 2, . . ., N} such that Cov(ξt, ξ2t) = 0. Consequently, several studies such as Wegner (1958) and Nagar (1960) remained silent on this condition. Therefore, the first step here is to compute the correlation coefficient between the pairs of normal deviates generated.
From equation (1.1) and Σ being a positive definite matrix, we can decompose it by a non- singular upper triangle matrix such that Ω = pp’
Let P = P11 P12 = S11 S12 P21 P22 S21 S22
Then,
P11 P12 P11 P21 = δ11 δ12 P21 P22 P21 P22 δ21 δ22
Hence, P211 + P212 = δ11 i P11 P21 + P12 P22 = δ12 ii P21 P11 + P22 P12 = δ21 iii P221 + P222 = δ22 iv
Σ = S11 0 S11 S12 = 7 3 S12 S22 0 S22 3 5
S211 = 7
S11 S12 = 3
S11S12= 3
S11S12= 3
S12 2 + S222 = 5
Solving the equation above:
S11= √7 = 2.645751311
S12= 3/2.645751311 = 1.133893419
S222 = 5 – S122
S222 = √ 5-1.1338934192
S222 = √5-1.285714286
S22= √3.714285714 S22=1.927248074
The upper triangular matrix values are thus:
P = 2.645751311 1.133893419 0 1.927248074 The above results are based on using the upper triangular matrix P. Later in the study the lower triangular matrix p* = S11 0 S12 S22
Will be used and the results of the approach will be compared
3.6 GENERATION OF THE ENDOGENOUS VARIABLE
With the numerical values already assigned to the structural parameters, we have values required in equation for the generation of the endogenous variables. Considering the upper triangular matrix U1t, u2t define as

U1t = 2.645751311 1.133893419 ξ1t U2t 0 1.927248074 ξ2t

U1t = 2.645751311 ξ1t + 1.133893419 ξ2t
U2t = 1.927248074 ξ2t The above equation produces the disturbance terms U1t, U2t which are supposed to have zero mean and the variance- covariance matrix specified for. We then generate the two endogenous variable (y1t, y2t). The y’s are obtained using the X’s , U’s and the earlier assigned parameters of β and ϒ. Thus, with the numerical values already assigned to the structural parameters as in equation * and **, we have all values required in the equation for the generation of the endogenous values. This step (which involves generation of two endogenous variables is sub divided into two parts to enhance ease computation of y’s Recall : yt = ∏x + vt = β-1Гxt + β-1ut Component A This first part involves the computation of the reduced form parameters ∏ = β-1Г x1t ∏xt = -1 1 β12 -ϒ11 -ϒ12 0 x2t 1- β12 β21 β21 1 -ϒ21 0 -ϒ23 x3t So, = -1 -ϒ11x1t - β12ϒ21x1t - ϒ12x2t - β12ϒ23x3t 1- β12 β21 -β21ϒ11x1t -ϒ21x1t - β21ϒ12x2t -ϒ23x3t

= -1 1 1.8 -1.5 -1.0 0 X1t 1.7 1.5 1 -0.5 0 -2.0 X2t X3t ∏x1t = 1.411764706X1t + 0.588235294X2t + 2.1176470589X3t
∏x2t = 1.617647059X1t + 0.882352941X2t + 1.176470588X3t Component B The second part involves the computation of Vt = β-1Ut β-1Ut = -1 1 1.5 U1t 1.7 1.8 1 U2t So , V1t = -0.588235294U1t - 0.882352941U2t V2t = -1.058823529U1t - 0.588235294U2t These two components were then combined to obtain y’s. Thus, Y1t =1.411764706X1t + 0.588235294X2t + 2.1176470589X3t -0.588235294U1t - 0.882352941U2t ----- + Y2t = 1.617647059X1t + 0.882352941X2t + 1.176470588X3t -1.058823529U1t - 0.588235294U2t---------- ++ The value earlier generated for X1t, X2t, X3t, U1t, U2t are then substitute into equations +, ++ to obtain numerical values for y1t, and y2t

3.7 ESTIMATION OF THE STRUCTURAL PARAMETERS At this junction, we are in possession of numerical for the exogenous and endogenous variables as well as the disturbance terms. The structural parameters are then estimated using the three estimation method {OLS, 2SLS and LIML} that were earlier discussed. The Statistical Application Software Version 9.2 (SAS 9.2) and Econometric Time series Software version were used to estimate the parameters see appendix 1
3.8 CRITERIA FOR EVALUATING THE PERFORMANCE OF THE ESTIMATORS We define the following criteria for comparative evaluation of the performance of our estimators 1. Average or Mean values of parameter estimates
Let YI be the estimates of the parameter obtained in the ith replication. We compute Y = 1 R R Σ Yi where R = Number of Replications i =1 2. Bias of Estimates
Bias (ϴi) = (ϴ - ϴ0) where ϴ0 is the true parameter value CHAPTER FOUR RESULT OF THE MONTE CARLO EXPERIMENT 5.0 INTRODUCTION
The results of our study on assessing the performance of three estimators are presented hereunder considering the two levels of triangular matrices, upper and lower . We provide results of the estimators using: 1. Average or mean values of parameter estimates. 2. Absolute bias of parameter estimates
The following estimation techniques were employed in estimating the parameters of our model. 1. Ordinary Least Squares (OLS) 2. Two Stage Least Squares Estimation (2SLS) 3. Limited Information Maximum Likelihood (LIML)
In theory and as confirmed by Johnston 2006, when an equation is just identified, estimates of parameters obtained by 2SLS and LIML are identical. The results obtained in this study show that 2SLS and LIML estimators have yielded virtually identical results while OLS have yielded result that are clearly different from those of the two estimators. Consequently, three estimates obtained from estimators shall be compared.
Meanwhile, since 2SLS and LIML have same results, it shall be denoted as 2SLIML.

4.1 PERFOMANCE EVALUATION OF ESTIMATORS USING AVERAGE OF PARAMETER ESTIMATES
Tables 4.1.1 and 4.1.2 represent the averages of estimates at two sample sizes n and number of replication r= 50. These average values are presented on parameter bias under both levels of triangular matrices for the two equations considered. The assumed true parameter values as assigned in chapter three are shown in parenthesis below the parameters. These average values are presented for each of the three estimators.
For instance, table 4.1.1 highlights average of parameter estimates at sample size N=15 over 50 replication using Ordinary Least Square Method (OLS), While table 4.1.2 represent average of parameter estimates at sample size N =20 over 50 replication using Ordinary Least Square Method (OLS). In table 4.1.1, the first entry 0.9474 represents the average of 15 estimates of parameter β12 under P1 and 1.0763 under P2. Each of the 15 estimates came out of OLS estimation of β12 of sample size 15. The same process is repeated for all parameters over three estimators across P1 and P2 and for the two sample sizes.
Best estimators are those whose average estimators are closest to the true parameter value. For example in table 4.1.1 when n=15, best estimates are obtained from 2SLIML for β12 whose actual value is 1.8 for P1 and P2 triangular matrices followed by OLS except for the parameter ϒ23 where 2SLIML performed best in P2. Also from table 4.1.2, when n = 20, best estimates are obtained from 2SLMIL for β12 whose actual value is 1.8 for P1 and P2 triangular matrices followed by OLS except for parameter ϒ12 where OLS performed best in P1 and 2SLIML in P2.
TABLE 4.1.1 PERFORMANCE EVALUATION OF ESTIMATORS USING AVERAGE OF PARAMETER ESTIMATES, N=15

| EQUATION 1 | EQUATION 2 | ESTIMATORS | | β12=1.8 | ϒ11=1.5 | ϒ12=1.0 | β21=1.5 | ϒ21=0.5 | ϒ23=2.0 | OLS | P1 | 0.9474 | -0.7495 | 0.3020 | 1.0497 | -0.3297 | 0.9684 | | P2 | 1.0763 | -0.3931 | 0.4418 | 0.9237 | -0.6589 | 0.7668 | 2SLS | P1 | 1.2177 | -0.5134 | 0.7836 | 1.0694 | -0.2469 | 1.0333 | | P2 | 1.3318 | -0.0536 | 0.8386 | 1.0966 | -0.5118 | 1.4230 | LIML | P1 | 1.2177 | -0.5134 | 0.7836 | 1.0694 | -0.2469 | 1.0333 | | P1 | 1.3318 | -0.0536 | 0.8386 | 1.0966 | -0.5118 | 1.4230 |

TABLE 4.1.2 PERFORMANCE EVALUATION OF ESTIMATORS USING AVERAGE OF PARAMETER ESTIMATES, N=20 | EQUATION 1 | EQUATION 2 | ESTIMATORS | | β12=1.8 | ϒ11=1.5 | ϒ12=1.0 | β21=1.5 | ϒ21=0.5 | ϒ23=2.0 | OLS | P1 | 0.9480 | -0.3000 | -0.1100 | -1.0498 | -0.3463 | 0.9814 | | P2 | 1.0695 | -0.0063 | 0.0228 | 0.9233 | -0.5581 | 0.6921 | 2SLS | P1 | 0.9677 | 0.0733 | -0.3987 | 1.1976 | -0.1292 | 1.3051 | | P2 | 0.9677 | 0.0733 | -0.3987 | 1.1976 | -0.1292 | 1.3051 | LIML | P1 | 0.9677 | 0.0733 | -0.3987 | 1.1976 | -0.1292 | 1.3051 | | P1 | 1.2800 | 0.3640 | 0.1532 | 0.9541 | -0.4251 | 0.7053 |

4.2 PERFORMANCE EVALUATION OF ESTIMATORS WHEN SAMPLE SIZE INCRESES AND NUMBER OF REPLICATION KEPT CONSTANT USING AVERAGE OF PARAMETER ESTIMATE FOR P1 AND P2
We deduce from the table below that 2SLIML still performed as the best estimator as sample size increases. We deduce that 2SLIML estimators are superior to other estimators in estimating all the parameters at P1 and P2. This superiority is displayed in some cases by OLS (Y21 at P1). 2SLIML is noticeably overwhelming superior in estimating the parameters of the endogenous variables.
TABLE 4.2.1 SUMMARY OF RESULTS IN TABLE 4.1.1 AND 4.1.2 | n =15 | n =20 | | P1 | P2 | P1 | P2 | β12 | 2SLIML | 2SLIML | 2SLIML | 2SLIML | ϒ11 | 2SLIML | 2SLIML | 2SLIML | 2SLIML | ϒ12 | 2SLIML | 2SLIML | | 2SLIML | Β21 | 2SLIML | 2SLIML | 2SLIML | 2SLIML | ϒ21 | 2SLIML | 2SLIML | 2SLIML | 2SLIML |
4.3 PERFORMANCE EVALUATION OF ESTIMATION USING ABSOLUTE BIAS OF PARAMETER ESTIMATES.
Tables 4.3.1 and 4.3.2 present the absolute bias of estimates of the two samples sizes, replicated 50 times. These absolute biases are presented on parameter basis. For instance, table 4.2.1 highlights absolute biases of parameter estimates at sample size n = 15. In this table, the first entry 0.8526 represents the average of 15 OLS biases of parameter B21 estimates. The same process is repeated for the six parameters over the four estimators. Best estimators are those with the least absolute biases From tables 4.2.1 to 4.2.2, we can deduce the following: * For parameter β12, 2SLIML generated the least Bias when sample size N = 15 and ILS when sample size N = 20 over both P1 and P2 * For parameter y11, when n = 15, OLS has the least absolute bias over P1 and 2SLIML over P2 while as n= 20 * For parameter y21, OLS generated the least absolute bias when n=15 for P1 and P * For parameter β21,2SLIML generated the least bias when sample size n = 15 and when n = 20 over P1 * For parameters y21, and y23, for both sample sizes, n = 15 and 20 and over both P1 and P2
The summary of the result in tables 4.1, 4.2 and 4.3

TABLE 4.3.1 PERFORMANCE EVALUATION OF ESTIMATORS USING ABSOLUTE BIAS OF PARAMETER ESTIMATES n =15 | EQUATION 1 | EQUATION 2 | ESTIMATORS | | β12=1.8 | ϒ11=1.5 | ϒ12=1.0 | β21=1.5 | ϒ21=0.5 | ϒ23=2.0 | OLS | P1 | 0.8526 | 2.2495 | 0.1980 | 0.4503 | 1.3298 | 1.0316 | | P2 | 0.7237 | 1.8931 | 0.0582 | 0.5763 | 1.6589 | 1.2332 | 2SLS | P1 | 0.5823 | 2.0134 | 0.2836 | 0.4306 | 1.2469 | 0.9667 | | P2 | 0.4682 | 1.5536 | 0.3387 | 0.4034 | 1.5118 | 0.5770 | LIML | P1 | 0.5823 | 2.0134 | 0.2836 | 0.4306 | 1.2469 | 0.9667 | | P1 | 0.4682 | 1.5536 | 0.3387 | 0.4034 | 1.5118 | 0.5770 |

TABLE 4.3.2 PERFORMANCE EVALUATION OF ESTIMATORS USING ABSOLUTE BIAS OF PARAMETER ESTIMATES, n = 20 | EQUATION 1 | EQUATION 2 | ESTIMATORS | | β12=1.8 | ϒ11=1.5 | ϒ12=1.0 | β21=1.5 | ϒ21=0.5 | ϒ23=2.0 | OLS | P1 | 0.8520 | 1.8000 | 0.6100 | 0.4502 | 1.3463 | 1.019 | | P2 | 0.7350 | 1.5063 | 0.4772 | 0.5767 | 1.5581 | 1.3080 | 2SLS | P1 | 0.8322 | 1.4267 | 0.8987 | 0.3024 | 1.1292 | 0.1949 | | P2 | 0.5200 | 1.1360 | 0.3468 | 0.5459 | 1.4251 | 1.2947 | LIML | P1 | 0.8322 | 1.4267 | 0.8987 | 0.3024 | 1.1292 | 0.1949 | | P1 | 0.5200 | 1.1360 | 0.3468 | 0.5459 | 1.4251 | 1.2947 |

T SUMMARY OF RESULTS IN TABLE 4.3.1 and 4.3.2 | n =15 | n =20 | | P1 | P2 | P1 | P2 | β12 =1.8 | 2SLIML | 2SLIML | 2SLIML | 2SLIML | ϒ11= 1.5 | OLS | OLS | OLS | 2SLIML | ϒ12= 1.0 | OLS | OLS | 2SLIML | 2SLIML | β21=1.5 | 2SLIML | 2SLIML | 2SLIML | OLS | ϒ21 | 2SLIML | 2SLIML | 2SLIML | 2SLIML | ϒ23 | 2SLIML | 2SLIML | 2SLIML | 2SLIML | Hence, from the summary of the results in the table above, we can infer that from the model given, 2SLS and LIML which is being referred to 2SLIML performed best in 18 cases out of 24 case considered, while OLS performed best in 6 cases .

| EQUATION 1 | EQUATION 2 | ESTIMATORS | | β12=1.8 | ϒ11=1.5 | ϒ12=1.0 | β21=1.5 | ϒ21=0.5 | ϒ23=2.0 | OLS | P1 | 0.8532 | 2.3127 | 0.5615 | 0.4506 | 1.3849 | 1.1104 | | P2 | 0.7245 | 1.9928 | 0.5417 | 0.5765 | 1.7016 | 1.2919 | 2SLS | P1 | 1.0954 | 3.3352 | 2.2959 | 0.8026 | 1.7689 | 2.5207 | | P2 | 1.0716 | 2.9822 | 2.0547 | 0.8553 | 2.3413 | 1.9322 | LIML | P1 | 1.0954 | 3.3352 | 2.2959 | 0.8026 | 1.7689 | 2.5207 | | P1 | 1.0716 | 2.9822 | 2.0547 | 0.8553 | 2.3413 | 1.9322 | PERFORMANCE EVALUATION OF ESTIMATORS USING ROOT MEAN SQUARE ERROR n=15

PERFORMANCE EVALUATION OF ESTIMATORS USING ROOT MEAN SQUARE ERROR, n =20 | EQUATION 1 | EQUATION 2 | ESTIMATORS | | β12=1.8 | ϒ11=1.5 | ϒ12=1.0 | β21=1.5 | ϒ21=0.5 | ϒ23=2.0 | OLS | P1 | 0.8527 | 1.8412 | 0.6989 | 0.4504 | 1.3840 | 1.0708 | | P2 | 0.7366 | 1.5524 | 0.5612 | 0.5769 | 1.6025 | 1.3422 | 2SLS | P1 | 1.3985 | 2.8000 | 2.5138 | 0.6422 | 1.6150 | 1.4870 | | P2 | 1.1697 | 2.8629 | 3.0434 | 0.7129 | 1.6967 | 1.7694 | LIML | P1 | 1.3985 | 2.8000 | 2.5138 | 0.6422 | 1.6150 | 1.4870 | | P1 | 1.1697 | 2.8629 | 3.0434 | 0.7129 | 1.6967 | 1.7694 |

PERFROMANCE EVALUATION OF ESTIMATORS USING ROOT MEAN SQUARE ERROR OF PARAMETER ESTRIMATES Tables 4.3.1 and 4.3.2 present the Root Mean Square Error of parameter estimates at various sample sizes of N and number of replication kept fixed. These values for the Root Mean Square Error are obtained from the Mean Square error (MSE). For instance, in table 4.3.1, the first entry .0853 represents the square root of Mean Square Error of sample size N = 15 over p1 The smaller the Root Mean Square Error, the better the performance of the estimator.From tables 4.3.1 to 4.3.2, we can deduce the following: * For parameter y11, ILS has the smallest Root Mean Square Error over p1 and OLS over P2 when sample size N = 15 and 20 respectively. * For parameter y21, b21, b12, y32, when N = 15, OLS has the smallest Root mean Square Error over P1 and P2, when sample size N = 15 and 20 respectively.
The summary of the results in tables 4.3.1 and 4.3.2 are presented in table 4.3a
From table 4.1.1, we can deduce that as the sample size increases, N = 20, the values of the parameter increases across upper triangular matrix while it was not so across the lower triangular matrix. For the upper and lower triangular matrices, some of the parameters performed better as the sample size increases CHAPTER FIVE SUMMARY OF FINDINGS, CONCLUSIONS AND RECOMMENDATIONS 6.0 INTRODUCTION
Two major problems a researcher always encounters are the problems of estimation of parameters and inference making. Given the nature of economic phenomena, it is almost certain that any equation will belong to a wider system of simultaneous equations.
There are questions of great practical importance to which in the general case, no analytical answer can be given. Nonetheless, some tentative results can be obtained in terms of Monte Carlo methods. Perhaps, the simplest way of explaining these methods is by analogy with numerical values, which has been done in this research work.
An entirely similar situation prevails with respect to the situation problem we are considering here in order to form a reasonable impression of the small sample distribution of a given estimator, we may observe a number of estimates from various samples of given size. By observing the behavior of the resulting estimates, we may infer some of the properties of the small distribution.
In this study, the comparative performance of four estimation methods under two levels of triangular matrices of a two – equation simultaneous model, both exactly identifies are assessed.
For each level of triangular matrices, we set the sample sizes N = 15 and 20. Our pairs of normal deviates based on the sample size N were generated, each replicated 50 times. These were also standardized and appropriately transformed to conform to the variance – covariance matrix E assumed in the model under consideration. Numerical values were assigned to each of the structural parameters of the model and numerical values were generated for both endogenous and exogenous variables of the model.
The result presented below is valid under the condition of just identified model.
FINDINGS AND CONCLUSIONS
Three performance evaluation criteria namely, Average of parameter estimates in comparing with the true values, absolute Bias of estimates and Root Mean Square error of estimates, were applied to the estimates generated by the three estimation techniques were considered, by identified by levels of triangular matrices, sample size and number of replication.
By our results, 2SLS and LIML estimators have yielded virtually identical parameter estimates. This is expected since the two equations in the model are just identified.
The following are the major findings of this study made on the use of indices Average of parameter estimates, Absolute Bias of estimates and Root Mean Square Error of estimates
On comparing the Average of the parameter estimates, it was observed that 2SLIML are the best estimators then followed by OLS.
In addition, as sample sizes increases to 20, 2SLIML still performed as the best estimators followed by OLS, across the triangular matrices, the result remain the same.
In the use of Absolute Bias of parameter estimates 2SLIML performed very well when compared to OLS, also there is no remarkable asymptotic behaviour in the performances of the estimators. The performance of OLS confirmed its properties of inconsistency as an estimator of simultaneous equations model. That is, with the increase in the sample sizes, the bias will remain.
The followings are the main findings based on the Root Mean Square Error of parameter estimates.
OLS performed best and it is clearly shown that 2SLS and LIML did not perform at all.
In conclusion, our study has shown that Monte Carlo studies give us these reliable results based on the performance of the model in question.
2SLS and LIML overwhelmingly performed best in the average of the parameter estimates followed by OLS, this means that OLS is not sufficiently inferior to simultaneous equation techniques. The choice of OLS may be sensible even for every simple models conforming to the assumptions under which the simultaneous equations estimators were derived and the experiments conducted.
There is no asymptotic effect on the performance of the estimators, this is attributable to the fact that increasing the sample size of a data would not yield any remarkable asymptotic effect. Finally, since a Monte Carlo Simulation techniques was used the scope of generalization is not as wide as in studies based on then analytical approach AREA OF FURTHER RESEARCH The researcher may further his research by studying the effect of exclusion of structural equations with exactly identified equation.

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