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Structural & Reduced Form Models

Structural Model

A structural model is a complete set of equations that explains how both endogenous and exogenous variables interact structurally in a system. The parameters associated with the model is known as structural parameters. The structural parameters are represented by Latin letters, i.e., α, β, γ, ...

Consider the following SEM:

$$

${\mathrm{Y}}_{\mathrm{1 }}= {\mathrm{\alpha }}_{\mathrm{1 }}{\mathrm{X}}_{\mathrm{1 }}+ {\mathrm{\alpha }}_{\mathrm{2 }}{\mathrm{Y}}_{\mathrm{2 }}+ {\mathrm{ϵ}}_1$

$$\begin{gathered} \text{ \\ Y2 = β1 Y1 + β2 X2 + ϵ2} \end{gathered}$$

$$\begin{gathered} \text{ \\ <br>} \end{gathered}$$

$$

Where:

α1, α2, β1 and β2 are the structural parameters.

e.g. Keynesian income determination model;

$$

Ct = β0 + β1 Yt + ut

Yt = Ct + It

Transformed the observed variables to the left-hand side as:

$$

Ct - β0 - β1 Yt = ut

Yt - Ct - It

The table of structural parameters can be obtained as: consider the above SEM model.

Reduced Form Models

An equation in which the endogenous variables are expressed as functions of the endogenous variables is known as a reduced form model of a structural model. The reduced form parameters are represented by πij and OLS method is used to estimate these parameters.

 Let us we taken the following structural model

Y1 = α0 + α1 X1 + α2 Y2 + u1

Y2 = β0 + β1 Y1 + u2

In this model (Y1,Y2 ) are endogenous variables and (X1) is exogenous variable.

Now to transform the endogenous variables is a function of exogenous variable. Substitute Y2 (from equation 2) into equation 1.

                                                    Y1 = α0 + α1 X1 + α2 Y2 + u1

                                        Y1 = α0 + α1 X1 + α2 (β0 + β1 Y1 + u2 ) + u1

                                      Y1 = α0 + α1 X1 + (α2 β0 +α2  β1 Y1 +α2  u2 ) + u1

                                      Y1- α2  β1 Y1  = α0 +α2 β0  + α1 X1  + u1 + (α2  u2 )

                                    (1- α2  β1 )Y1  = α0 +α2 β0  + α1 X1 + (α2  u2  + u1 )

$$

Substitute Y1 (from eq 1) in equation (2) as:

Y2 = β0 + β1 Y1 + u2

Y2 = β0 + β1 (α0 + α1 X1 + α2 Y2 + u1) + u2

Y2 = β0 +  α0 β1 + α1 β1 X1 + α2 β1  Y2 + β1 u1 + u2

Y2 - α2 β1  Y2 = β0 +  α0 β1 + α1 β1 X1 +  + β1 u1 + u2

(1- α2 β1) Y2 = β0 +  α0 β1 + α1 β1 X1 +  + (β1 u1 + u2)

From equation – A and equation – B, it is clear endogenous variables as a function of exogenous variable that’s translated in to a single equation model. Now the OLS method to apply and estimate the parameters of A and B.

Checking

Now apply OLS method on model (A) to find the estimates π10 of  and π11 as:

Practice Question – 1:

Convert the following structural model into reduced form model.

Qd = α1 P + α2 I + u1 

Qs = β1 P + u2

Solution: Consider at equilibrium

$$

Qd = Qs

In the model, Q and P are endogenous variables and Income “I” is exogenous variable. The two equilibrium values, for P and Q determine at the same time.

α1 P + α2 I + u1 = β1 P + u2

α1 P -  β1 P = - α2 I - u1 + u2

(α1 -  β1) P = - α2 I  + u2 - u1

The reduced form of the above structural model:

P = π11 I + ν1 → 2

The OLS method can be applied to estimate the parameter of equation – 2.

Substitute P (from equ. 1) in supply equation (or in demand equation)

Qs = β1 P + u2

Qs = β1 P + u2

Practice Question – 2:

Find the reduced form of the following structural model.

Qd = α0 + α1 P + α2 Y + u1 →Demand Equ.

Qs = β0 + β1 P + β2 R + u2 →Supply Equ.

Qd=Qs→at equilibrium

Solution: The Qd=Qs = Q. The endogenous variables are Qd, Qs, and P.

P is endogenous variable because in some cases demand influences the price.

Qd = Qs

α0 + α1 P + α2 Y + u1 =  β0 + β1 P + β2 R + u2

 α1 P - β1 P  =  β0 - α0  + β2 R  - α2 Y+ u2 -  u1

(α1  - β1)P  =  β0 - α0  + β2 R  - α2 Y+ u2 -  u1

P = π10 + π11 Y + π12 R + v1

Now substitute the value of P in supply equation.

Q = β0 + β1 P + β2 R + u2

Q = π20 + π21 Y + π22 R + v2

Now we can use OLS method to find the estimates of the above reduced form model parameters.

Practice Question – 3:

Identify the exogenous and endogenous variables in the following model and also find the reduced form of the model.

$$

Ct = a0 + a1 Yt + u1

$$\begin{gathered} \text{ \\ <br>} \end{gathered}$$

$$\begin{gathered} \text{ \\ It = b0 + b1 Yt + b2 Yt-1 + u2} \end{gathered}$$

$$\begin{gathered} \text{ \\ <br>} \end{gathered}$$

$$\begin{gathered} \text{ \\ Yt = Ct + It + Gt \\ } \end{gathered}$$

$$\begin{gathered} \text{ \\ <br>} \end{gathered}$$

$$\begin{gathered} \text{ \\ <b><span style="background: white; color: \#385623; font-family: "Times New Roman",serif; font-size: 12pt; line-height: 107\%; mso-ansi-language: EN-US; mso-bidi-language: AR-SA; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-US; mso-fareast-theme-font: minor-fareast; mso-font-kerning: 0pt; mso-ligatures: none; mso-themecolor: accent6; mso-themeshade: 128;">Solution:</span></b><span style="background: white; color: \#242729; font-family: "Times New Roman",serif; font-size: 12pt; line-height: 107\%; mso-ansi-language: EN-US; mso-bidi-language: AR-SA; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-US; mso-fareast-theme-font: minor-fareast; mso-font-kerning: 0pt; mso-ligatures: none;">The endogenous variables are </span>} \end{gathered}$$Ct, It  and  Yt and the exogenous variables are Yt-1, Gt

$$

$$

Ct = a0 + a1 Yt + u1 →1 

$$\begin{gathered} \text{ \\ <br>} \end{gathered}$$

$$\begin{gathered} \text{ \\ It = b0 + b1 Yt + b2 Yt-1 + u2 →2} \end{gathered}$$

$$\begin{gathered} \text{ \\ <br>} \end{gathered}$$

$$\begin{gathered} \text{ \\ Yt = Ct + It + Gt →3 \\ } \end{gathered}$$

$Substitute the value of  Ct(from eq. 1) and It  (from eq.2) in equation (3)                                                                Yt = Ct + It + Gt                                      Yt = ($a0 + a1 Yt + u1) + (b0 + b1 Yt + b2 Yt-1 + u2)+ Gt

$$\begin{gathered} \text{ \\ <msub style="text-align: center;"><mi> Y</mi><mi>t - </mi></msub>} \end{gathered}$$a1 Yt - b1 Yt$$\begin{gathered} \text{ \\ <msub style="text-align: center;"><mi> </mi></msub><mo style="text-align: center;">= </mo>} \end{gathered}$$a0 + b0 +  b2 Yt-1 + Gt + u1 + u2 

                               (1$$\begin{gathered} \text{ \\ <msub style="text-align: center;"><mi> - </mi></msub>} \end{gathered}$$a1 - b1 )Yt$$\begin{gathered} \text{ \\ <msub style="text-align: center;"><mi> </mi></msub><mo style="text-align: center;">= </mo>} \end{gathered}$$a0 + b0 +  b2 Yt-1 + Gt + u1 + u2 

Now to obtain the reduced form of equation – 1. Substitute the value of Yt  in equation – 1.

                                                    Ct = a0 + a1 Yt + u1 →1 

Now to obtain the reduced form of equation – 1. Substitute the value of Yt  in equation – 2.

The OLS method can be applied to estimate the parameters of the above reduced form equations.