2.1 Classification of multiple-equation models

A multiple-equation model is a system of equations consisting of many (at least two) equations, which describe a given economic system or its part called a subsystem. The model contains a G of endogenous variables: Y₁, …, Y_g, …, Y_G with statistical observations y_1t, …, y_gt, …, y_Gt. The endogenous variable is characterized by the fact that in one of the model’s equations it acts as a dependable variable; however, it can also act as an explanatory variable. This type of a model also contains exogenous variables X₁, …, X_j, …, X_k with observations x_t1, …, x_tj, …, x_tk. Exogenous variables in the equations act solely as explanatory variables. Endogenous variables without anytime-delays will be called the model’s total interdependent variables. An alternative group of predetermined variables Z₁, …, Z_j, …, Z_K (with observations z_t1, …, z_tj, …, z_tK) is formed by exogenous variables and delayed endogenous variables, which in the model’s equations appear as explanatory variables.

A system of G equations in a multiple-equation model in a structural form¹ can be written as follows:

(2.1)

In the above G equations, some random components as well as structural parameters associated with the total interdependent variables have appeared. Additionally, the parameters α_gj (g = 1, …, G; j = 0, 1, …, K), associated with the predetermined variables, have appeared. In practice, it is natural for only some of the total interdependent variables and the predetermined variables acting as explanatory variables to occur in individual equations. This means that a significant part of parameters β_gg and α_gj (g, g′ = 1, …, G; j = 0, 1, …, K) takes zero values. What is more, the parameters indicate the explanatory variable of the gth equation.

A multiple-equation model can also be written in as a matrix

(2.2)

where

Matrix B contains the model’s structural parameters along its total interdependent variables. Matrix A contains the model’s structural parameters occurring along the predetermined variables. Vector Y contains the model’s total interdependent variables, vector Z contains the model’s predetermined variables, and vector η holds the random components of the equations of the structural-form’s model.

If we consider the mechanism of interrelations between the total interdependent variables, a multiple-equation model can belong to one of three classes. The manner of those interrelations between the total interdependent variables allows distinction of the following:

• simple models,
• recursive models,
• systems of interdependent equations.

In simple models, there are no direct relations between the total interdependent variables. This means that none of the total interdependent variables act as explanatory variables in any of the equations. The following system of equations can be an example of a simple model:

(2.3)

In the model 2.3, total interdependent variables (y_1t, y_2t, y_3t) are not related to each other. None of them acts as an explanatory variable. Only the time-delayed endogenous variables y_2t−1 and y_3t−1, which belong in the group of predetermined variables, act as explanatory variables.

A recursive model is characterized by a chain (recursive) nature of relations between the total interdependent variables. This chain character of those relations signifies their one-directionality, with a possibility of indicating the chain’s beginning and its end. The following can be an example of such a chain denoting the model’s recursivity:

The beginning of the chain is formed by the variable y_1t, while the variable y_4t is its end. A recursive model of the relations between total interdependent variables, which are presented in the above diagram, can have the following form:

(2.4)

A system of interdependent equations is characterized by mutual multilateral relations between the total interdependent variables. There may be two kinds of such relations: direct feedback or indirect feedback, also called a closed cycle of relations between the total interdependent variables. Feedback is based on a simultaneous self-impact of such a pair of variables. For example, the variables y_gt and y_g′t (for g, g′ = 1, …, G; g ≠ g′) are linked by feedback when

For example, indirect feedback (a closed cycle of relations) of the total interdependent variables occurs in the following situation:

Both kinds of relations can occur in a model simultaneously. This often occurs in large multiple-equation models. However, appearance of one of the above indicated mechanisms is enough for the model to form a system of interdependent equations. The following can be an example of such a model with direct feedback:

(2.5)

Model 2.5 contains feedback between the variables y_2t and y_3t. The variable y_3t influences the variable y_2t, while acting as an explanatory variable in the second equation. What is more, the variable y_2t influences the variable y_3t, acting as an explanatory variable in the third equation. As such, the requirement of a direct feedback is met.

In the model 2.5, the first equation draws attention. In that equation’s set of explanatory variables, only predetermined variables occur. Thereby, the equation has the nature of a simple model. Such an equation in a system of interdependent equations, in which only predetermined variables are explanatory variables, is called a detached equation.

Let us consider the following model:

(2.6)

In the above model, the total interdependent variables form a closed cycle of relations, which has the form

Thereby, model 2.6 is a system of interdependent equations and can be classified as one of the most complicated econometric models.

2.2 A reduced form of the model

Having a model in a structural form, written in a matrix form as Equation 2.2, its left-side multiplication by a matrix B⁻¹ can be performed. As a result we get the following:

The ratio of matrices B⁻¹B = I, where I is the unit matrix of a G degree. By moving the expression B⁻¹ AZ to the right side, we obtain

By substituting C = −B⁻¹ A and ε = B⁻¹η we arrive at a reduced form of the model

(2.7)

where²

Matrix C contains the structural parameters of the equations of the reduced-form’s model, while ε is the vector of the random components of the reduced-form’s equations.

Reduced-form’s equations (in the number of G) can be written as follows:

(2.8)

Based on the above, it can be inferred that each of the reduced-form’s equations contains an identical set of explanatory variables. At the same time, all predetermined variables of the entire multiple-equation model make up a set of explanatory variables of each reduced-form equation. For instance, a system of reduced-form’s equations for the model 2.6 form will be written as follows³:

(2.9)

2.3 Identification of the model

Application of a multiple-equation model requires determining whether it has a correct form, as far as the relationships between its reduced- and structural-forms are concerned. Let us consider the equation

(2.10)

which combines the structural-form with the reduced one. Left-side multiplication of both sides of Equation 2.10 by matrix B results in an identification equation:

(2.11)

The model is identifiable when, based on the components of the matrix C, it is possible to solve the system of linear equations, with regard to the components of matrices B and A. This means that it is necessary to solve the G(K + 1) system of linear equations.⁴

While attempting to solve the G(K + 1) system of linear equations, three options can be encountered:

1. There is only one solution of the system of equations. We, then, can speak of an explicit solution. In such cases, the multiple-equation model is identifiable explicitly;
2. There are many solutions of the G(K + 1) system of equations. The system’s solution is ambiguous. This means that the model is identifiable ambiguously. Such model has a correct form. It is also called an overidentified model.
3. There is no solution of the system. In this case, the multiple-equation model is unidentifiable, which signifies its faulty construction. Such a model needs to be reconstructed (respecified) in such a way that it can be identifiable at least ambiguously.

In the empirical model’s identification test, two identifiability conditions, which arise from the need to impose the so-called zero-limits onto some of the structural parameters, must be met. This means that some of each equation’s structural parameters must take zero-values. Thus, practically speaking, some of the total interdependent variables and some of the predetermined ones should not occur in the set of explanatory variables of specific equations of that form. The identification test is done for each equation separately. The necessary condition for identifiability of the gth equation (g = 1, …, G) is for the number of the entire model’s variables, which are not present in that equation (L_g), to be at least equal to the number G − 1, that is:

(2.12)

The second condition, which is a necessary and a sufficient requirement, is for the matrix sequence W_g (g = 1, …, G) to be equal to G − 1, that is⁵:

(2.13)

If the condition 2.13 is met, then the gth equation is identifiable explicitly, when L_g = G − 1. However, if the condition 2.13 is met, the gth equation is identifiable ambiguously (overidentified), when L_g > G − 1.

The gth equation is not identifiable if L_g < G − 1, or . This means that the entire model is not identifiable and requires reconstruction. When all model equations are identifiable, it is identifiable explicitly, provided that each of its equations is identifiable explicitly. A multiple-equation model is identifiable ambiguously, if all of its equations are identifiable, or at least one of them is overidentified. Let us consider the model 2.6

First, all variables of the model, that is, y_1t, y_2t, y_3t, x_t0, x_t1, x_t2, x_t3, t, y_3t−1, should be specified. It can be seen that in the first equation the variables y_2t, x_t2, x_t3, and t are absent. This means that L₁ = 4 is the number of variables (L₁), which are not present in the first equation. As such, the necessary condition is met, since L₁ = 4 > G − 1 = 2. Similarly, in the second equation, the variables y, y_3t, x_t1, x_t3, or y_3t−1, are absent, which means that L₂ = 4. It means that the second equation can be identifiable. In the third equation, the variables y_1t, x_t1, x_t2, t, and y_3t−1 are absent, as a result of which L₃ = 5. As such, the second and the third equations can be identifiable.

It is therefore necessary to construct matrices W₁, W₂, and W₃, which will contain the coefficients of the variables that are absent in a given equation. Matrix W₁ will thus contain the structural parameters of the variables y_2t, x_t2, x_t3, and t from the second and the third equation, that is

(2.14)

As long as the parameters α₂₂, α₂₄, α₃₂, α₃₃, and β₃₂ are different from zero, the sequence of that matrix r(W₁) = 2 = G − 1. Thus, the condition 2.13 is met, as a result of which the first equation is identifiable. Since the inequality L₁ = 4 > G − 1 = 2 occurs, the first equation is identifiable ambiguously.

Analogically, matrices W₂ and W₃ are as follows:

and

It can be demonstrated that r(W₂) = r(W₃) = 2 = G − 1. What is more, L₂ = 4 > G − 1 and L₃ = 5 > G − 1. Equations: the second and the third are thus identifiable ambiguously. Model 2.6 is therefore identifiable ambiguously (overidentified), which signifies its correct construction and enables further work on it in the subsequent stages.

2.4 Estimation of the parameters of a multiple-equation econometric model

The methods for estimation of the parameters of a multiple-equation model, essentially, are divided into two groups. The first group entails assessment methods for each equation separately, same as in a single-equation model. The second group entails estimation methods for the parameters of all equations simultaneously, called the total estimation methods.

In econometric literature, the prevailing viewpoint is that the parameters of equations in simple and recursive models can be assessed using the ordinary least squares method (OLS). This means that during estimation, each equation in those models can be treated as a single-equation model.

The OLS method does not provide consistent estimators of the parameters of the structural-form’s equations of the systems of interdependent equations. This inconsistency results from the correlation of the total interdependent variables, which in the equations are explanatory, with the parallel random components. In this case, the condition of the OLS applicability, written as Equation 1.18, is not met. When this occurs, it is necessary to seek other methods of structural parameters’ estimation in the models with interdependent equations. At the same time, it should be remembered that transition to the estimation stage is only possible when the model is identifiable explicitly or ambiguously.

It is noticeable that the reduced-form of the system of interdependent equations has characteristics of a simple model. Therefore, if there are no particularly unfavorable conditions, the parameters of the equations of the reduced-form’s model can be assessed using the OLS method, for each equation separately.

Let us consider a case, when the system of interdependent equations is identifiable explicitly. This means that the identification equation 2.11 BC = −A has an explicit solution. Let us consider the model

(2.15)

The above model is a system of interdependent equations, in which each equation is explicitly identifiable. A reduced form of that model has the following form:

(2.16)

An identification equation for the above structural- and reduced-forms of the model has the following form:

(2.17)

Let us suppose that using the OLS method and applying all available statistical data, the parameters of each equation of the reduced-form’s model were assessed, the following empirical equations were obtained:

(2.18)

In Equation 2.18, some residuals are present and they are respectively marked by the symbols e_1t and e_2t. The empirical identification equation can be written as

(2.19)

In the matrices of the system 2.19, the symbols b₁₂ and b₂₁ represent estimations of the parameters β₁₂ and β₂₁, while a₁₀, a₁₁, a₂₀, a₂₂ represent estimations of the parameters α₁₀, α₁₁, α₂₀, α₂₂ in the system of structural-form 2.15.

A system of six linear equations with six unknowns results from the matrix 2.19:

(2.20)

The solution of the system of Equation 2.20 reveals estimations of the structural parameters in the system of Equation 2.15. They reach the following numerical values:

The above estimations of the parameters of the structural-form’s equations were assessed using the indirect least squares method (ILS). As a result, empirical equations of the model 2.15 can be written as follows:

(2.21)

Bases on the above, the ILS method is easy in application. It takes place in two stages: in the first, the OLS method is used to estimate the parameters of equations of the reduced-form’s model, while in the second, the system of linear equations, obtained from the matrix identification equation, is solved. This drawback of this method is the lack of a matrix of variances and covariances of the structural parameters’ estimations for the empirical structural-form’s equations. This prevents assignation of the average parameter assessment errors for the equations of such a model. As a result, there is no opportunity to test the significance of the explanatory variables in each of the structural-form’s empirical equations.

If the system of interdependent equations is identifiable ambiguously, then the ILS method cannot be applied. In that case, the most commonly used estimation procedure is the double least squares method (2LS). It involves a twofold application of the least squares method. In the first step, the OLS method is used to estimate the parameters of the equations in the reduced-form’s model. Based on the empirical reduced-form equations, the theoretical values of the total interdependent variables, which at the same time are devoid of their random part, are assigned. Next, in the structural-form’s equations, the total interdependent explanatory variables are substituted with their theoretical values obtained from the reduced-form’s empirical equations. The parameters of such modified structural-form’s equations can be assessed using the ordinary least squares method.

As an example, we are going to follow the 2OLS method applied to the system of interdependent equation 2.5:

A system of the reduced-form’s equations of this model is as follows:

After assessing the parameters of the above equations using the OLS method, we will get the following reduced-form’s empirical equations:

In the above system of empirical equations of the reduced-form’s model 2.3, the symbols , , represent the theoretical values of each equation’s total interdependent variables, which resulted from calculations after application of the OLS method.

Now, the second step of this estimation procedure can be performed. Where the actual amounts of the total interdependent variables in structural-form’s equations act as the explanatory variables, they are replaced with their theoretical amounts obtained from the reduced-form’s empirical equations. As such, let us consider a new system of structural-form’s equations:

(2.22)

Parameters of each equation in the system of Equation 2.22 can be assessed using the OLS method. Attention draws the first equation, whose set of explanatory variables has no total interdependent variables; meaning that the parameters of the detached equation in the system of interdependent equations can be directly estimated using the ordinary least square method.

In the second equation, the explanatory variable y_3t has been substituted with a variable in the form of theoretical values calculated from the reduced-form . At the same time, in the second equation, there is no correlation of the total interdependent explanatory variable that is nonrandom, with the random component. Thus, estimation of the parameters in the second equation, which is modified by the OLS method, is permitted. A similar change has occurred in the third equation, in which y_2t was substituted by , thus enabling estimation of the equation’s parameters using the OLS method.

2.5 Forecasts estimation based on multiple-equation models

A model composed of many equations, for which fundamental assumptions of econometric prediction theory are met, can become a predictor. A forecast estimation procedure based on a multiple-equation model, ultimately, can be conducted very similarly to the one used during a construction of the forecasts that are based on a single-equation model. Predictive techniques differ for each class of multiple-equation models. However, similarities in the way the forecasts are achieved can be found, regardless of the multiple-equation model’s class.

Regardless of the multiple-equation model’s class, the key issue in econometric prediction is to determine the values of exogenous variables for each forecasted period. This requirement does not concern the time-delayed endogenous variables. The values of the endogenous variables, which are delayed by 1 period, are known for the first forecasted period T = t₀₀ + 1 as the y_en (g = 1, G) amounts for n = t₀. In the subsequent forecasted periods T = t₀ + 2, T = t₀ + 3, T = t₀ + τ the values are obtained by a sequential reference from the already estimated forecasts of endogenous variables. This procedure is called a sequential prediction.

Another common feature of the predictions that are based on various classes of multiple-equation models is the possibility of forecast estimation that is based on each of the equations separately. The procedure, thus, can be reduced to the same one, which takes place during a simultaneous prediction based on the G single-equation models.

Positive results of a simple model’s verification allow extrapolation outside the statistical sample, on the status quo principle. Then, a vector forecasts for the vector of the total interdependent variables is reached:

(2.23)

where the vector’s components are made up by the forecasts of each of the forecasted variables.

A forecast in the form of the above vector is obtained by substituting the values of explanatory variables for the forecasted period T in the empirical multiple-equation model. Thus, vector Y_Tp emerges by combining the G forecasts, which arose independently based on each equation separately. Forecasting based on each equation is identical to that in a single-equation model. A prediction from a simple model is thus a G-fold prediction based on a single-equation model.

In a recursive model, each of the equations can be considered separately, identically as in a single-equation model. It is necessary to forecast each equation in the correct order. Such procedure is called a chain prediction. In a recursive model, each endogenous variable is numbered accordingly to its causal order. In such ordering, the variable y_it depends on the predetermined variables and solely on those total interdependent variables y_lt, for which indicators 1 and i fulfill inequality l < i. Chain prediction, in this case, involves construction of forecasts for each individual component of the forecasted vector Y_Tp in a recursive manner, according to the order of the variables, which is reflected by the model. A specific feature of a chain forecast is that: if the model indicates that the forecasted variable Y_iT depends on any other simultaneous variable Y_1T, Y_1T, …, Y_lT, while l < i, then during the forecasting , the prognoses referring to the total interdependent variables occurring earlier in the chain are used.

If the forecasted period does not occur directly after the period t₀ and is in the h > 1 time-unit distance from it, then the chain prediction involves an h-fold repetition of the above procedure. As such, we get the h number of forecasted vectors for the subsequent periods. The sequence of those vectors marks the expected paths of individual forecasted variables. Thus, it provides information on the expected manner of achieving the amounts forecasted for the last period. Therefore, it can be stated that when h > 1, the chain prediction, in relation to each forecasted variable, generates a sequence of forecasts for the subsequent periods, which signifies a sequential prediction. By combining a chain prediction with a sequential prediction, we get multiple forecast vectors, which can be written in a form of a suitable forecast matrix.

During a chain prediction, it is worth to determine the matrix of the correlation coefficients of the random components from each equation of the model:

Matrix ρ of a G×G size contains the elements, which are the coefficients of a linear correlation between the random components of the g-th and g′-th equations, while g, g′ = 1,…,G and g ≠ g′. In practice, based on the residuals of the model’s equations, the coefficients of the correlation are estimated and their assessments are obtained.

Very small correlation coefficients – as far as the module is concerned – suggest that individual equations are independent of each other. When are close to +1, it can be inferred that the residuals of the gth and g′th equations, simultaneously, took the values of the same sign. However, when the is close to −1, it can be assumed that the signs of the examined residuals of the equations numbered g and g′ generally were different.

Let us consider the prediction technique from the following recursive model 2.4:

A predictor according the OLS method for the model 2.4 will be written as follows:

(2.24)

Constructing forecasts for forecasted variables should begin with the so-called initial equation, which is the first equation in the model 2.4. Each equation, in which only the predetermined variables⁶ are the explanatory variables, is an initial equation of the recursive model. Estimation of the forecast will require determining the value of the exogenous variable x_T1 in the forecasted period T. The time variable will reach a T value in the forecasted period. In contrast, the time-delayed variable y_3T−1 will be known from the observation set for the T = n + 1 as y_3n, or when T > n + 1, we will use the forecast estimated earlier on. When estimating forecasts , we will apply a procedure analogical to that in the first equation, with one difference – it will be necessary to use the predetermined forecasts .

Continuing the chain proceeding, we eventually arrive at the fourth equation, after obtaining a prediction from the third equation, where the forecast was estimated. The forecast is going to be obtained using the forecasts and estimated earlier on. As a result, we have a vector forecast of type 2.23 in the following form:

(2.25)

where each of the components of the vector Y_Tp were calculated in the same way as the forecast from a single-equation model. Each time, using a formula 2. 25, we also calculate average prediction errors

(2.26)

Prediction from a system of interdependent equations can be done in two ways. In the first method, equations of the structural-form’s model are used, while in the second, inference into the future is based on equations of the reduced-form. These methods do not replace each other, and their applicability depends on the type of questions, which are posed and need to be answered by performing such inference into the future.⁷

Structural equations can be used when existence of causal interrelations in the stochastic total interdependent variables is omitted in considerations and when the aim is to estimate the effect of one-side interdependence of those variables. In such cases, the procedure is close to that which is applied in case of simple equations. At the same time, the values of those endogenous variables, which in equations act as explanatory variables, are assessed for the forecasted period T, using the same methods as for exogenous variables.

A prediction based on structural-form’s equations, respecting only one side of the multiple-sided interrelations between the total interdependent variables, has the nature of inference into the future only for very short periods. Only during a very short time-period, it is allowed to abstract from the other aspects of interdependence between the total interdependent variables. In longer periods, interdependencies between endogenous variables play an important role and their omission can distort the sense as well as the results of a predictive testing.

With this in view, the second way of inference into the future – based on equations of a reduced-form’s model – has greater practical importance. In this method, forecasting can be regarded as a conditional mathematical expectation, where in the condition some predetermined variables occur. Forecasting is based on each of the reduced-form’s equations separately. The procedure is the same as in the case of a simple model, because the reduced-form has characteristics of a simple model.

If parameters of the reduced-form’s equations were estimated directly, then the variances and covariances of structural parameters’ estimations for each equation of that form are known. It is easy to determine the prediction variances for each equation; however, it is more difficult when the reduced-form was determined from an empirical structural-form. It is worth noting that the reduced-form’s equations, each of which contains all the predetermined variables, usually are characterized by a presence of statistically insignificant explanatory variables. Therefore, it is worthwhile to determine the average prediction errors for the forecasts from the systems of interdependent equations that were obtained from the reduced-form’s equations, from the matrix of variances and covariance’s of structural parameters’ estimations that are obtained from the structural-form’s equations.

Prediction based on equations of the reduced-form of the model, in certain sense, has optimal properties, provided that an appropriate method was used to estimate the parameters. Prediction based on equations of the reduced-form is optimal, in such sense that it provides smaller average prediction errors than other methods using the same information resources.⁸

Let us consider a prediction from the system of interdependent equations, based on the model 2.5:

A prediction from the first equation of the above system for the T = n + 1 period can be performed independently of other equations, since it is a detached equation. The forecasts and ought to be estimated on the basis of a predictor from the reduced-form

where the symbols ĉ_gj (g = 2, 3; j = 0, 1, …, 5) represent the estimations of the parameters of the second and third equation from the reduced-form that were obtained using the OLS method. The following will be the predictor for the first interdependent variable:

in which the symbols a₁₀, a₁₁, a₁₄, a₁₅ represent the structural parameters’ estimations of the equation that were obtained using the OLS method.

It can be noticed that in the subsequent forecasted periods (T = n + 2, n + 3, …, n + τ) it becomes necessary to apply sequential forecasting. The delayed variable , which appears in each equation of the considered predictor, forces the forecast estimation of the third total interdependent variable to be done in the first instance. This will allow using the forecast as each equation’s explanatory variable – in each equation of the predictor Y_Tp – in subsequent periods.

Forecasts from a system of interdependent equations can also be partially estimated from the reduced-form’s equations as well as from the structural-form’s equations. Let us consider the following system of equations:

(2.27)

A closed cycle of relations between the total interdependent variables can be noticed:

(2.28)

which signifies a system of interdependent equations. Forecasting of the above model can be done using a mixed technique: partially from the reduced-form and partially from the structural one, applying a technique of chain prediction, which is specific for a recursive model. Applying in the following structural-form’s predictor:

(2.29)

in the forecasting is not possible immediately. Lack of the initial equation forming the “loop” 2.29 is an obstacle resultant from the closed cycle of relations. The “loop” can be eliminated using a reduced-form’s equation for forecast estimation , that is

(2.30)

Knowledge of the forecast allows application of a chain prediction technique to the subsequent equations of the structural-form’s predictor. We can thus estimate the forecast from the following equation:

Having the forecasts and allows estimation of the forecast on the basis of the following equation:

Having the forecast , it is possible to estimate the forecast on the basis of the following equation:

The technique of predicting subsequent forecasted T periods should take into account the necessity for sequential proceedings resultant from the occurrence of the delayed endogenous variables and . Ultimately, a prediction from a system of interdependent equations can connect a prediction from reduced-form’s equations with a sequential and chain prediction.