The Indirect Least Squares Method (Ils)

1. Core Concept Overview & Intuition

1.1 Introduction

When a structural model exhibits double causation — that is, when an endogenous variable appears on the right-hand side of an equation while also being determined elsewhere in the system — ordinary least squares (OLS) estimators become both inconsistent and biased. The source of the failure is the correlation between the stochastic disturbance term and the endogenous regressor(s). Theil and Basmann independently developed procedures that deliver consistent estimates in this setting.

The estimation methods available for simultaneous equation models fall into two broad groups:

1. Single-equation methods, also called limited information methods, which estimate one structural equation at a time using only the information needed to establish the identifiability of that equation. The most popular members are ordinary least squares (OLS), indirect least squares (ILS), the instrumental variable method (IV), and two-stage least squares (2SLS).
2. Systems methods, also called full information methods, which estimate all equations jointly, exploiting parameter restrictions across all equations together with the variances and covariances of the residuals. These include three-stage least squares (3SLS), full-information least generalized variance (FI/LGV), and full-information maximum likelihood (FI/ML).

The indirect least squares (ILS) method, sometimes called the reduced form method, is a single-equation procedure reserved for exactly identified equations. Its central intuition is a change of vantage point: instead of regressing an endogenous variable on other endogenous variables, ILS transforms the structural model so that every endogenous variable is regressed on the predetermined variables of the system alone. Because predetermined variables are uncorrelated with the stochastic terms, the double-causation problem is eliminated. OLS is then applied to these transformed (reduced form) equations, and the resulting coefficient estimates are mapped back, through a system of parameter equations, onto the structural parameters.

1.2 Learning Outcomes

By the end of this unit the learner will be able to:

1. Discuss the indirect least squares estimation method.
2. Discuss the theoretical properties of estimators obtained using the indirect least squares estimation method.

1.3 Identification and Choice of Estimation Method

Identification status dictates which estimation technique is admissible. The correspondence is summarized below.

The logic is arithmetic: an under-identified equation yields fewer estimating equations than unknowns, so no unique solution exists. An exactly identified equation yields exactly as many estimating equations as structural unknowns, so a unique solution exists and ILS delivers it. An over-identified equation yields more estimating equations than unknowns, so ILS would produce conflicting estimates of the same parameter, and a method that imposes the restrictions systematically (2SLS) is required.

1.4 Equation Bias

The motivating problem is equation bias, also described as simultaneous equation bias. In a structural equation such as a consumption function, the regressor $Y_t$ is itself endogenous, being jointly determined with $C_t$ through the identity $Y_t = C_t + I_t$. Consequently $\operatorname{Cov}(Y_t, u_t) \neq 0$, violating the OLS assumption of zero covariance between regressor and disturbance. OLS applied directly to the structural equation therefore produces biased and inconsistent estimates. ILS removes this bias at its root by replacing the endogenous regressor with a combination of predetermined variables that are uncorrelated with the disturbance.

2. Mathematical Formalization & Core Definitions

2.1 Notation and Symbols

• $C_t$: consumption at period $t$ (endogenous).
• $Y_t$: income at period $t$ (endogenous).
• $I_t$: investment at period $t$ (predetermined/exogenous).
• $u_t$: structural stochastic disturbance term.
• $\beta_1, \beta_2$: structural parameters (intercept and slope).
• $\pi_{ij}$: reduced form parameters, indexed so that the first subscript denotes the equation and the second the regressor.
• $w_{1t}, w_{2t}$: reduced form disturbance terms.
• $G$: number of endogenous variables in the system.
• $A_Q$: matrix of variables excluded from a given equation (used in the rank condition).

2.2 Assumptions of the ILS Method

> Assumption 1 (Exact identification). The structural equation must be exactly identified. If the structural equation is under-identified, the structural parameters cannot be estimated, since the number of estimating equations would be less than the number of unknowns to be estimated.

> Assumption 2 (OLS conditions on the reduced form). The stochastic terms in the reduced form equations must satisfy the usual assumptions of ordinary least squares, because ILS applies OLS to the reduced form equations. These are:
> 1. Randomness: $u_i$ is random and may take positive or negative values.
> 2. Zero mean: $E(u_i) = 0$.
> 3. Homoscedasticity: the variance of $u_i$ is constant, $E(u_i^2) = \sigma^2$.
> 4. Normality: $u_i \sim N(0, \sigma^2)$.
> 5. Zero covariance with the regressor: $\operatorname{Cov}(X_i, u_i) = 0$, since $X_i$ is a fixed constant.
> 6. Non-autocorrelation: $E(u_i u_j) = 0$ for $i \neq j$.

> Assumption 3 (No strong collinearity). There should not be strong collinearity among the exogenous variables.

2.3 The Process of ILS

> Step 1. Construct the reduced form model from the structural model. In the reduced form the explanatory variables are predetermined alone and are therefore uncorrelated with the error term, so double causation is avoided.

> Step 2. Write the reduced form model in two forms: the initial form, whose parameters are a mixture of structural parameters, and a second form whose parameters are denoted by $\pi$'s.

> Step 3. Form the system of parameter equations linking the two sets of parameters.

> Step 4. Apply OLS to the reduced model in its second form to obtain estimates of the $\pi$'s.

> Step 5. Substitute the estimates of the $\pi$'s into the system of parameter equations and solve for the structural parameters. The values obtained are unique if the structural model is exactly identified.

2.4 The Rank Condition

The rank condition is necessary and sufficient for identification. For an equation to be identified, one must be able to form at least one non-zero determinant of order $G-1$ from the matrix of variables excluded from that equation. In a two-equation system, $G - 1 = 2 - 1 = 1$, so a single non-zero scalar (a $1 \times 1$ determinant) suffices.