Heteroskedasticity-Based Identification: Theory and Methods

Introduction

This document summarizes the heteroskedasticity-based identification strategy for models with endogenous regressors, as developed by Lewbel (2012), Klein and Vella (2010), Rigobon (2003) and Prono (2014), and proposes an application to a time-series context. The method provides a way to construct valid instruments from the model’s data when traditional external instruments are unavailable and heteroskedasticity is present.

Lewbel (2012)

Structural Forms and Reduced-Form Residuals

Let (Y_1, Y_2) be an endogenous vector, X a vector of exogenous variables (including a constant).

The models we consider are:

Triangular: Y_1 = X^\top\beta_1 + \gamma_1 Y_2 + \varepsilon_1, Y_2 = X^\top\beta_2 + \varepsilon_2

Simultaneous: Y_1 = X^\top\beta_1 + \gamma_1 Y_2 + \varepsilon_1, Y_2 = X^\top\beta_2 + \gamma_2 Y_1 + \varepsilon_2

with \gamma_1\gamma_2 \neq 1.

Projecting each Y_j on X yields reduced-form residuals W_j := Y_j - X^\top(\mathbb{E}[XX^\top])^{-1}\mathbb{E}[XY_j]. A short calculation gives

W_1 = \frac{\varepsilon_1 + \gamma_1\varepsilon_2}{1-\gamma_1\gamma_2}, \qquad W_2 = \frac{\varepsilon_2 + \gamma_2\varepsilon_1}{1-\gamma_1\gamma_2},

with the triangular case obtained by setting \gamma_2 = 0.

Core Assumptions for Lewbel-Style Identification

The method relies on the following key assumptions:

(LW1) Strict exogeneity. \mathbb{E}[\varepsilon_j \mid X] = 0 for j = 1,2.

(LW2) Covariance restriction. \text{Cov}(Z, \varepsilon_1\varepsilon_2) = 0. This holds automatically if the errors have a common factor structure \varepsilon_j := a_j u + \eta_j where the common factor u is mean independent of Z, i.e., \mathbb{E}[u \mid Z] = 0.

(LW3) Instrument relevance via heteroskedasticity.

Triangular: \text{Cov}(Z, \varepsilon_2^2) \neq 0.
Simultaneous: the r \times 2 matrix \Phi_W := [\text{Cov}(Z, W_1^2) \; \text{Cov}(Z, W_2^2)] has rank 2.

(LW4) Normalization (simultaneous case). The parameter space for (\gamma_1, \gamma_2) precludes the observationally equivalent pair (1/\gamma_2, 1/\gamma_1).

Triangular System: Closed-Form Identification and 2SLS

The coefficient \gamma_1 has closed-form:

\gamma_1 = \frac{\text{Cov}(Z, W_1W_2)}{\text{Cov}(Z, W_2^2)}. \tag{1}

Remark [Why (1) identifies \gamma_1]. Under (LW2) the numerator simplifies to \text{Cov}(Z, \varepsilon_1\varepsilon_2) + \gamma_1\text{Cov}(Z, \varepsilon_2^2) = \gamma_1\text{Cov}(Z, \varepsilon_2^2). Because the denominator is non-zero by (LW3), \gamma_1 is point identified.

Feasible Two-Step 2SLS

For the triangular system:

Generate residuals. Regress Y_2 on X via OLS and store the residuals \hat{\varepsilon}_2 := Y_2 - X^\top\hat{\beta}_2^{\text{OLS}}.
Construct the heteroskedasticity-based instrument. The generated instrument is IV := (Z - \bar{Z})\hat{\varepsilon}_2, where \bar{Z} is the sample mean of Z.
First stage. Regress the endogenous variable Y_2 on the exogenous variables and the generated instrument, [X, IV], to obtain fitted values \hat{Y}_2.
Second stage. Regress Y_1 on [X, \hat{Y}_2] to estimate (\beta_1, \gamma_1).

# Generate example data
set.seed(123)
params <- list(
  beta1_0 = 0.5, beta1_1 = 1.5, gamma1 = -0.8,
  beta2_0 = 1.0, beta2_1 = -1.0,
  alpha1 = -0.5, alpha2 = 1.0, delta_het = 1.2
)
data <- generate_lewbel_data(n_obs = 1000, params = params)

# Manual 2SLS implementation
# Step 1: Generate residuals from first stage
first_stage <- lm(Y2 ~ Xk, data = data)
e2_hat <- residuals(first_stage)

# Step 2: Construct Lewbel instrument
z_demeaned <- data$Z - mean(data$Z)
lewbel_iv <- z_demeaned * e2_hat

# Step 3: Run 2SLS using ivreg
suppressPackageStartupMessages(library(ivreg))
tsls_model <- ivreg(Y1 ~ Xk + Y2 | Xk + lewbel_iv, data = data)

# Display results
cat("True gamma1:", params$gamma1, "\n")
#> True gamma1: -0.8
cat("Estimated gamma1:", round(coef(tsls_model)["Y2"], 3), "\n")
#> Estimated gamma1: -0.709
cat("Standard error:", round(sqrt(diag(vcov(tsls_model)))["Y2"], 3), "\n")
#> Standard error: 0.1

GMM Moment Conditions

The identification strategy can be expressed more generally using GMM. The core idea is to form moment conditions that are zero at the true parameter values \theta.

Triangular and Simultaneous Systems

For the triangular system, the parameter vector is \theta := (\beta_1^\top, \gamma_1, \beta_2^\top)^\top. The structural errors are defined as \varepsilon_1(\theta) := Y_1 - X^\top\beta_1 - Y_2\gamma_1 and \varepsilon_2(\theta) := Y_2 - X^\top\beta_2.

For the simultaneous system, the parameter vector is \theta := (\beta_1^\top, \gamma_1, \beta_2^\top, \gamma_2)^\top. The structural errors are \varepsilon_1(\theta) := Y_1 - X^\top\beta_1 - Y_2\gamma_1 and \varepsilon_2(\theta) := Y_2 - X^\top\beta_2 - Y_1\gamma_2.

The vector of moments is:

Q(\theta) := \begin{pmatrix} X \cdot \varepsilon_1(\theta) \\ X \cdot \varepsilon_2(\theta) \\ (Z - \bar{Z}) \cdot \varepsilon_1(\theta) \cdot \varepsilon_2(\theta) \end{pmatrix}. \tag{2}

Note that for the triangular case, \gamma_2 = 0.

The instruments are constructed directly from the exogenous data:

For the two mean equations, the instruments are the exogenous variables X.
For the covariance restriction, the instrument is the de-meaned heteroskedasticity driver Z - \bar{Z}.

Since the errors are functions of data and parameters, the entire moment vector is observable for any candidate \theta.

GMM Estimation

Estimation proceeds by finding the parameter vector \hat{\theta} that minimizes the sample analog of the moment conditions, \bar{Q}(\theta)^\top W\bar{Q}(\theta), where W is a weighting matrix.

Note that in the case of a simultaneous system, Assumption (LW3) requires Z to contain at least two elements. Sometimes we can use a constant as an instrument (and include it in Z - \bar{Z}).

Note. Let Assumptions (LW1), (LW2), (LW3), and (LW4) hold in the simultaneous equation model, replacing \text{Cov}(Z, \varepsilon_1\varepsilon_2) in Assumption (LW2) with \mathbb{E}[(Z - \bar{Z})\varepsilon_1\varepsilon_2] and replacing \text{Cov}(Z, W_j^2) with \mathbb{E}[(Z - \bar{Z})W_j^2] in Assumption (LW3), for j = 1,2. Then, all the parameters are identified.

This note can be used when \mathbb{E}[\varepsilon_1\varepsilon_2] = 0.

Set Identification Under a Relaxed Covariance Restriction

When assumption (LW2) is weakened to allow for some correlation, |\text{Corr}(Z, \varepsilon_1\varepsilon_2)| \leq \tau|\text{Corr}(Z, \varepsilon_2^2)|, \tau \in [0,1), the parameter \gamma_1 in the triangular model is set-identified rather than point-identified.

Theorem 1 [Bounds for \gamma_1 with \tau > 0]. \gamma_1 is contained in the closed interval whose endpoints are the (real) roots of the quadratic equation in \gamma_1:

\frac{\text{Cov}(Z,W_1W_2)^2}{\text{Cov}(Z,W_2^2)^2} - \frac{\text{Var}(W_1W_2)}{\text{Var}(W_2^2)}\tau^2 + 2\left(\frac{\text{Cov}(W_1W_2,W_2^2)}{\text{Var}(W_2^2)}\tau^2 - \frac{\text{Cov}(Z,W_1W_2)}{\text{Cov}(Z,W_2^2)}\right)\gamma_1 + (1-\tau^2)\gamma_1^2 = 0.

The interval collapses to the point estimate from (1) when \tau = 0 and widens as \tau \to 1.

Identification of other parameters.

The remaining parameters are identified conditional on a value of \gamma_1 from its identified set.

The parameter \beta_2 is always point-identified by OLS: \hat{\beta}_2 := (\mathbb{E}[X^\top X])^{-1}\mathbb{E}[X^\top Y_2].
For each value \gamma_{1,k} in the identified interval for \gamma_1, there is a corresponding identified value for \beta_1, given by \beta_{1,k} := (\mathbb{E}[X^\top X])^{-1}\mathbb{E}[X^\top(Y_1 - Y_2\gamma_{1,k})].

The result is an identified set of parameter pairs (\beta_1, \gamma_1) corresponding to the interval for \gamma_1.

This set-identification cannot be used in the simultaneous system.

Klein and Vella (2010)

Semiparametric Control Function Approach

An alternative to the IV/GMM framework for the triangular system is the semiparametric control function approach of Klein and Vella (2010). This method addresses endogeneity by directly augmenting the structural equation with a non-constant control term.

The Model and General Control Function

The starting point is a general statistical decomposition of the first structural error via a linear projection onto the second error, conditional on the exogenous variables:

\varepsilon_1 = A(X)\varepsilon_2 + \eta_1, \qquad \text{where} \quad \mathbb{E}[\eta_1 | X, \varepsilon_2] = 0.

This decomposition holds by defining A(X) as the coefficient from a conditional linear projection, given by:

A(X) := \frac{\text{Cov}(\varepsilon_1, \varepsilon_2 \mid X)}{\text{Var}(\varepsilon_2 \mid X)}.

Substituting this into the primary structural equation yields the augmented regression model:

Y_1 = X^\top\beta_1 + \gamma_1 Y_2 + A(X)\varepsilon_2 + \eta_1. \tag{4}

Identification hinges on giving the generally intractable function A(X) an estimable structure.

Assumptions for Identification

The key assumptions which provide this structure are:

(KV1) Exogeneity. \mathbb{E}[\varepsilon_j | X] = 0 for j = 1,2.

(KV2) Constant Conditional Correlation. The correlation between the structural errors is constant conditional on X: \text{Corr}(\varepsilon_1, \varepsilon_2 | X) = \rho_0.

(KV3) Heteroskedasticity and Variation. The conditional variances, denoted \text{Var}(\varepsilon_1|X) = S_1^2(X) and \text{Var}(\varepsilon_2|X) = S_2^2(X), are non-constant functions of X. The ratio of the conditional standard deviations, S_1(X)/S_2(X), is not constant (it has different values for different X).

Identification via Control Function

The Klein and Vella (2010) assumptions transform the inestimable general form of A(X) into a specific, identifiable structure. The derivation proceeds as follows:

A(X) = \frac{\text{Cov}(\varepsilon_1, \varepsilon_2 \mid X)}{\text{Var}(\varepsilon_2 \mid X)} = \frac{\text{Corr}(\varepsilon_1, \varepsilon_2 \mid X) S_1(X) S_2(X)}{S_2^2(X)} = \frac{\rho_0 S_1(X) S_2(X)}{S_2^2(X)} = \rho_0 \frac{S_1(X)}{S_2(X)}

While the original form is inestimable due to the unknown conditional covariance function, this final form depends only on the parameter \rho_0 and the conditional standard deviation functions, which can be estimated nonparametrically. Assumption (KV3) is crucial as it ensures this ratio is not constant. If the ratio were a constant, say c, the model would collapse into a simple linear form Y_1 = X^\top\tilde{\beta}_1 + Y_2\tilde{\gamma}_1 + \eta_1, where an estimator could only identify the two composite parameters \tilde{\beta}_1 and \tilde{\gamma}_1, leaving the three structural parameters (\beta_1, \gamma_1, \rho_0) unidentified from the two equations:

\hat{\tilde{\beta}}_1 = \beta_1 - \rho_0 c \beta_2 and \hat{\tilde{\gamma}}_1 = \gamma_1 + \rho_0 c.

Estimation Strategy

The estimable regression model is:

Y_1 = X^\top\beta_1 + \gamma_1 Y_2 + \rho_0 \frac{S_1(X)}{S_2(X)}\varepsilon_2 + \eta_1.

The parameters (\beta_1, \gamma_1, \rho_0) and the unknown variance functions are estimated jointly via a multi-step Semiparametric Least Squares (SLS) procedure.

First-step residual. Estimate the second structural equation Y_2 = X^\top\beta_2 + \varepsilon_2 by OLS to obtain consistent residuals \hat{\varepsilon}_2.
Second-error variance. Estimate the conditional variance function S_2^2(X) = \text{Var}(\varepsilon_2|X) by nonparametrically regressing the squared residuals \hat{\varepsilon}_2^2 on X, for instance using kernel regression. This yields the estimator \hat{S}_2(X).
Main equation SLS. The primary parameters (\beta_1, \gamma_1, \rho_0) are estimated by minimizing the sum of squared residuals of the main equation. This is a nested procedure where for each candidate parameter vector (\beta_{1,k}, \gamma_{1,k}, \rho_{0,k}):
1. The structural error \hat{\varepsilon}_{1,k} = Y_1 - X^\top\beta_{1,k} - Y_2\gamma_{1,k} is computed.
2. The variance function \hat{S}_{1,k}(X) is estimated by nonparametrically regressing \hat{\varepsilon}_{1,k}^2 on X.
3. The objective function \sum_{i=1}^N \left( Y_{1i} - X_i^\top\beta_{1,k} - Y_{2i}\gamma_{1,k} - \rho_{0,k}\frac{\hat{S}_{1,k}(X_i)}{\hat{S}_2(X_i)}\hat{\varepsilon}_{2i} \right)^2 is evaluated.
The final parameter estimates are those which minimize this objective function.

Fully Parametric Estimation

The most direct simplification replaces the unknown variance functions S_j(X) with specific parametric forms, converting the semiparametric problem into a standard parametric one. A common specification that ensures positivity is the exponential model:

S_j^2(X) = \text{Var}(\varepsilon_j|X) = \exp(X^\top\delta_j), \quad \text{for } j=1,2.

The full parameter vector for the triangular system is now finite-dimensional: \theta := (\beta_1^\top, \gamma_1, \beta_2^\top, \rho_0, \delta_1^\top, \delta_2^\top)^\top. Estimation can proceed via Maximum Likelihood or GMM.

Maximum Likelihood Estimation (MLE). This approach requires a full distributional assumption for the structural errors. Assuming (\varepsilon_1, \varepsilon_2) are conditionally bivariate normal given X, the distribution is:

\begin{pmatrix} \varepsilon_1 \\ \varepsilon_2 \end{pmatrix} \mid X \sim \mathcal{N} \left( \begin{pmatrix} 0 \\ 0 \end{pmatrix}, \Sigma(X; \theta) \right),

where the conditional covariance matrix \Sigma(X; \theta) is a function of the parameters:

\Sigma(X; \theta) = \begin{pmatrix} \exp(X^\top\delta_1) & \rho_0 \exp(\frac{1}{2}X^\top(\delta_1+\delta_2)) \\ \rho_0 \exp(\frac{1}{2}X^\top(\delta_1+\delta_2)) & \exp(X^\top\delta_2) \end{pmatrix}.

The structural errors are functions of the mean parameters: \varepsilon_1(\theta) = Y_1 - X^\top\beta_1 - Y_2\gamma_1 and \varepsilon_2(\theta) = Y_2 - X^\top\beta_2. The MLE estimator \hat{\theta}_{\text{MLE}} is the value of \theta that maximizes the log-likelihood function, \mathcal{L}(\theta) = \sum_{i=1}^N \log f(\varepsilon_{1i}(\theta), \varepsilon_{2i}(\theta) \mid X_i; \theta), where f(\cdot) is the bivariate normal PDF.

Generalized Method of Moments (GMM). GMM provides a more robust alternative that does not require a full distributional assumption. A set of moment conditions sufficient to identify \theta is given by \mathbb{E}[Q(\theta)]=0, where:

Q(\theta) := \begin{pmatrix} X \cdot \varepsilon_1(\theta) \\ X \cdot \varepsilon_2(\theta) \\ X \cdot (\varepsilon_1(\theta)^2 - \exp(X^\top\delta_1)) \\ X \cdot (\varepsilon_2(\theta)^2 - \exp(X^\top\delta_2)) \\ X \cdot (\varepsilon_1(\theta)\varepsilon_2(\theta) - \rho_0 \exp(\frac{1}{2}X^\top(\delta_1+\delta_2))) \end{pmatrix}.

The first two sets of moments identify the mean parameters (\beta_1, \gamma_1, \beta_2). The next two sets use the assumed variance structure to identify the variance parameters (\delta_1, \delta_2), and the final set identifies the correlation parameter \rho_0. The GMM estimator \hat{\theta}_{\text{GMM}} minimizes the standard quadratic form \bar{Q}(\theta)^\top W \bar{Q}(\theta) for a given weighting matrix W.

Time-Series Examples

The core idea can be adapted to other contexts by properly defining the heteroskedasticity-generating variable Z.

Prono (2014)

Model

In a time-series setting, the structural model is a triangular system:

\begin{align} Y_{1t} &= X_t^\top\beta_1 + \gamma_1 Y_{2t} + \varepsilon_{1t} \\ Y_{2t} &= X_t^\top\beta_2 + \varepsilon_{2t} \end{align}

The key insight is that the error variance may be time-varying and predictable. Prono (2014) assumes \varepsilon_{2t} follows a GARCH process, where its conditional variance is a function of past errors and variances:

\text{Var}(\varepsilon_{2t} \mid \mathcal{F}_{t-1}) := \sigma_{2t}^2 = \omega + \alpha\varepsilon_{2,t-1}^2 + \beta\sigma_{2,t-1}^2.

Procedure

The fitted conditional variance from the GARCH model serves as the heteroskedasticity driver.

Estimate the second equation by OLS to get residuals \hat{\varepsilon}_{2t} := Y_{2t} - X_t^\top\hat{\beta}_2.
Fit a GARCH(1,1) model to the residuals \hat{\varepsilon}_{2t} to obtain the series of fitted conditional variances, \hat{\sigma}_{2t}^2.
This fitted variance is the heteroskedasticity driver: set Z_t := \hat{\sigma}_{2t}^2.
Construct the generated instrument: IV_t := (Z_t - \bar{Z})\hat{\varepsilon}_{2t}.
Proceed with 2SLS as in the Feasible Two-Step 2SLS section, using [X_t, IV_t] as instruments for Y_{2t} in the first structural equation. Use HAC-robust standard errors.

# Example with Prono method (requires tsgarch package)
if (requireNamespace("tsgarch", quietly = TRUE)) {
  # Generate data with GARCH errors
  config <- create_prono_config(n = 500)
  data <- generate_prono_data(
    n = config$n,
    beta1 = config$beta1,
    beta2 = config$beta2,
    gamma1 = config$gamma1,
    garch_params = config$garch_params
  )

  # Run Prono simulation
  result <- run_single_prono_simulation(config, return_details = TRUE)

  cat("True gamma1:", config$gamma1, "\n")
  cat("Estimated gamma1:", round(result$gamma1_iv, 3), "\n")
}

Rigobon (2003)

Model Framework

The core idea of Rigobon (2003) is that identification can be achieved if the error variances differ across observable, discrete regimes, s \in \{1, \ldots, S\} (e.g., pre- and post-policy change, or high- vs. low-volatility periods). The structural parameters of the model are assumed to be constant across these regimes. The structural equations are:

Y_1 = X^\top\beta_1 + \gamma_1 Y_2 + \varepsilon_1 and Y_2 = X^\top\beta_2 + \gamma_2 Y_1 + \varepsilon_2

The key assumption is that the variance-covariance matrix of the structural errors, \Sigma_\varepsilon, is different in at least two regimes. Let \Sigma_{\varepsilon, s} denote this matrix in regime s.

\Sigma_{\varepsilon, s} := \begin{pmatrix} \sigma_{1,s}^2 & \sigma_{12,s} \\ \sigma_{12,s} & \sigma_{2,s}^2 \end{pmatrix}, \qquad \text{with } \Sigma_{\varepsilon, s} \neq \Sigma_{\varepsilon, s'} \text{ for some } s \neq s'.

This regime-based heteroskedasticity can be exploited using either the Lewbel or Klein-Vella identification strategy, depending on further assumptions about the error structure.

Lewbel-Style IV Procedure

Assumptions

This approach applies to the simultaneous system. It relies on the core Lewbel assumptions, specialized to the regime context. The key identifying restrictions are:

Covariance Restriction: The structural error covariance is constant across regimes: \sigma_{12,s} = \sigma_{12} for all s.
Instrument Relevance: At least one structural error variance is not constant across regimes: \sigma_{j,s}^2 \neq \sigma_{j,s'}^2 for some j, s, s'.

Let D_s be a dummy variable for regime s. The instrument vector Z is formed by the set of centered dummies, Z := [D_1 - p_1, \ldots, D_{S-1}-p_{S-1}]^\top, where p_s is the population proportion in regime s. Under these conditions, the Lewbel assumptions (LW2) and (LW3) hold. Specifically, \text{Cov}(Z, \varepsilon_1\varepsilon_2) = 0 because both Z and \varepsilon_1\varepsilon_2 are functions of the regime, but \varepsilon_1\varepsilon_2 has a constant mean across regimes while Z does not.

Procedure

The regime indicators are used to generate the instrument. For the triangular case (\gamma_2 = 0):

Generate residuals. Regress Y_2 on X by OLS to get residuals \hat{\varepsilon}_2.
Construct instruments. The heteroskedasticity drivers are the centered dummy variables, Z_s := D_s - \bar{D}_s. The generated instruments are IV_s := Z_s\hat{\varepsilon}_2.
Proceed with 2SLS, using [X, IV_1, \ldots, IV_{S-1}] as instruments for Y_2.

# Generate data with regime-based heteroskedasticity
params <- list(
  beta1_0 = 0.5, beta1_1 = 1.5, gamma1 = -0.8,
  beta2_0 = 1.0, beta2_1 = -1.0,
  alpha1 = -0.5, alpha2 = 1.0,
  regime_probs = c(0.4, 0.6),
  sigma2_regimes = c(1.0, 2.5) # Variance is 2.5x higher in regime 2
)
data <- generate_rigobon_data(n_obs = 1000, params = params)

# Run Rigobon estimation
result <- run_rigobon_estimation(data)

cat("True gamma1:", params$gamma1, "\n")
#> True gamma1: -0.8
cat("Estimated gamma1:", round(result$tsls$estimates["gamma1"], 3), "\n")
#> Estimated gamma1: -0.832

Klein & Vella-Style Control Function Procedure

Assumptions

This approach applies to the triangular system (\gamma_2=0). It relies on the core K&V assumptions, specialized to the regime context.

Constant Conditional Correlation: The structural error correlation is constant across regimes: \text{Corr}(\varepsilon_1, \varepsilon_2 \mid s) = \rho_0 for all s.
Heteroskedasticity and Variation: The structural error variances are not constant across regimes. Crucially for identification, the ratio of the standard deviations, \sigma_{1,s}/\sigma_{2,s}, must be different across at least two regimes.

Procedure

The regime indicators are used to construct a piecewise control function. The simplest case is where the conditional variances are constant within each regime, i.e., \text{Var}(\varepsilon_j \mid s) = \sigma_{js}^2.

Estimate second-equation residuals. Regress Y_2 on X to get residuals \hat{\varepsilon}_2.
Estimate regime-specific variances. For each regime s, compute the sample variance of the residuals, \hat{\sigma}_{2s}^2 = \frac{1}{N_s}\sum_{i \in s} \hat{\varepsilon}_{2i}^2. For any candidate (\beta_{1k}, \gamma_{1k}), compute \hat{\varepsilon}_{1k} and its regime-specific variances \hat{\sigma}_{1ks}^2.
Estimate via nonlinear least squares (NLS). The main equation is specified with an interacted control function: Y_1 = X^\top\beta_1 + \gamma_1 Y_2 + \rho_0 \sum_{s=1}^S D_s \left( \frac{\sigma_{1s}}{\sigma_{2s}} \hat{\varepsilon}_2 \right) + \eta_1 The parameters (\beta_1, \gamma_1, \rho_0) and the variance ratios (\sigma_{1s}/\sigma_{2s}) for s=1,\ldots,S are estimated jointly via NLS.

Comparison of Approaches

The Lewbel and K&V approaches use the same observable phenomenon—a shift in the error variance structure across regimes—but rely on different, non-nested assumptions to achieve identification.

Core Assumption: Lewbel requires the covariance of the structural errors to be constant across regimes. K&V requires the correlation to be constant. If both variances change across regimes, these two assumptions are mutually exclusive.
Applicability: The Lewbel approach is more general as it applies to simultaneous systems. The K&V control function approach, as formulated, is restricted to triangular systems.
Estimation: Lewbel’s method leads to a standard linear IV (2SLS) procedure. K&V’s method requires a more complex nonlinear least squares estimation to handle the variance ratios and correlation parameter.

The choice between the two depends on which assumption about the stability of the error structure—constant covariance or constant correlation—is more plausible in a given economic application.

Time-Series Variant with Log-Linear Conditional Variances

Model

The structural model is a simultaneous system:

\begin{align} Y_{1t} &= X_t^\top\beta_1 + \gamma_1 Y_{2t} + \varepsilon_{1t} \\ Y_{2t} &= X_t^\top\beta_2 + \gamma_2 Y_{1t} + \varepsilon_{2t} \end{align}

The key assumption is that the conditional variances are an explicit log-linear function of the exogenous variables X_t:

\log\sigma_{jt}^2 := X_t^\top\delta_j, \quad \text{where} \quad \varepsilon_{jt} \mid \mathcal{F}_{t-1} \sim (0, \sigma_{jt}^2) \quad \text{for } j=1,2.

and \mathcal{F}_{t-1} denotes time (t-1)-information. Since (\delta_1, \delta_2) are assumed non-zero, X_t is correlated with \varepsilon_{jt}^2, satisfying the instrument relevance condition (LW3). The heteroskedasticity driver is defined as the centered exogenous variables, Z_t := X_t - \mathbb{E}[X_t].

Moment vector

The full set of parameters is \theta := (\beta_1^\top, \beta_2^\top, \gamma_1, \gamma_2, \delta_1^\top, \delta_2^\top)^\top. The moment conditions implied by the model are \mathbb{E}[Q_t(\theta)] = 0, where:

Q_t(\theta) := \begin{pmatrix} X_t \cdot \varepsilon_{1t}(\theta) \\ X_t \cdot \varepsilon_{2t}(\theta) \\ Z_t \cdot \varepsilon_{1t}(\theta)\varepsilon_{2t}(\theta) \\ Z_t \cdot (\varepsilon_{1t}(\theta)^2 - e^{X_t^\top\delta_1}) \\ Z_t \cdot (\varepsilon_{2t}(\theta)^2 - e^{X_t^\top\delta_2}) \end{pmatrix}.

Instruments as Functions of Observables

The instruments are derived from the exogenous variables X_t. The terms inside the expectation are functions of these instruments and the structural errors (e.g., \varepsilon_{1t}(\theta) := Y_{1t} - X_t^\top\beta_1 - Y_{2t}\gamma_1), which are themselves functions of the observable data and the parameter vector \theta.

For the first two mean equations, the instruments are X_t.
For the error product and variance specification moments, the instrument is Z_t := X_t - \mathbb{E}[X_t], estimated in-sample as X_t - \bar{X}.

Therefore, the sample average of Q_t(\theta) is a computable criterion function for any candidate parameter values. As in the Simultaneous system section, Z_t must have at least 2 entries (including perhaps a constant, if Corollary 1 applies).

Other Notes

References

Klein, R., Vella, F., 2010. Estimating a class of triangular simultaneous equations models without exclusion restrictions. Journal of Econometrics 154, 154–164. https://doi.org/10.1016/j.jeconom.2009.05.005

Lewbel, A., 2012. Using heteroscedasticity to identify and estimate mismeasured and endogenous regressor models. Journal of Business & Economic Statistics 30, 67–80. https://doi.org/10.1080/07350015.2012.643126

Prono, T., 2014. The role of conditional heteroskedasticity in identifying and estimating linear triangular systems, with applications to asset pricing models that include a mismeasured factor. Journal of Applied Econometrics 29, 800–824. https://doi.org/https://doi.org/10.1002/jae.2340

Rigobon, R., 2003. Identification through heteroskedasticity. The Review of Economics and Statistics 85, 777–792. https://doi.org/10.1162/003465303772815727

2025-06-25

Introduction

Lewbel (2012)

Structural Forms and Reduced-Form Residuals

Core Assumptions for Lewbel-Style Identification

Triangular System: Closed-Form Identification and 2SLS

Feasible Two-Step 2SLS

GMM Moment Conditions

Triangular and Simultaneous Systems

GMM Estimation

Set Identification Under a Relaxed Covariance Restriction

Klein and Vella (2010)

Semiparametric Control Function Approach

The Model and General Control Function

Assumptions for Identification

Identification via Control Function

Estimation Strategy

Fully Parametric Estimation

Time-Series Examples

Prono (2014)

Rigobon (2003)

Lewbel-Style IV Procedure

Klein & Vella-Style Control Function Procedure

Comparison of Approaches

Time-Series Variant with Log-Linear Conditional Variances

Moment vector

Other Notes

See Also

References