Instrumental Variables

A simple system.

y_{1}^{*} = X_{1} \beta_{1} + \alpha_{1} y_{2}^{*} + \epsilon_{1} y_{2}^{*} = X_{2} \beta_{2} + \alpha_{2} y_{1}^{*} + \epsilon_{2}

• X_{1} and X_{2} are sets of exogenous variables that satisfy rank and order conditions
• \epsilon_{1} and \epsilon_{2} are bivariate normal random errors with correlation \rho, and \beta and \alpha are parameters to be estimated.

Other Advances

Fractional integration methods. Switching Models Structural breaks: see Jong Hee Park’s work. Unit roots in the presence of breaks…

Equation Balance

How to Use It

A Summary Paper to Think About

Andy Phillips has a recent paper on inference in dynamic settings. It is in the box for day 5. The paper is:

How to avoid incorrect inferences (while gaining correct ones) in dynamic models

A Key Piece of Homework for the Weekend

A very short description

1. The Core Problem: Autocorrelated Errors

Consider the linear regression model: y_t = X_t' \beta + u_t, \quad t = 1, \dots, T

Classical OLS Assumptions Fail

• Under Gauss-Markov, \mathbb{E}[u u' | X] = \sigma^2 I_T.
• In time series, errors are often serially correlated: \mathbb{E}[u_t u_s | X] = \gamma_{|t-s|} \neq 0.

Consequences

• \hat{\beta}_{\text{OLS}} remains unbiased and consistent (assuming exogeneity \mathbb{E}[X_t u_t] = 0).
• The classical OLS covariance matrix formula is invalid: \text{Var}(\hat{\beta}_{\text{OLS}} | X) = (X'X)^{-1} \left( X' \Omega X \right) (X'X)^{-1} where \Omega = \mathbb{E}[u u' | X].
• Positive autocorrelation (\gamma_j > 0) causes classical OLS standard errors to be severely underestimated, leading to inflated t-statistics and over-rejection of H_0.

2. Taxonomy of Standard Error “Fixes”

3. The Newey-West (HAC) Estimator

To estimate the middle matrix \Phi = \frac{1}{T} X' \Omega X = \frac{1}{T} \sum_{t=1}^T \sum_{s=1}^T \mathbb{E}[X_t X_s' u_t u_s]:

1. Long-Run Variance Representation

\Phi = \Gamma_0 + \sum_{j=1}^{T-1} \left( \Gamma_j + \Gamma_j' \right), \quad \text{where } \Gamma_j = \frac{1}{T} \sum_{t=j+1}^{T} \mathbb{E}[u_t u_{t-j} X_t X_{t-j}']

2. The Bartlett Kernel (Newey & West, 1987)

Truncates lags at m and applies linearly decaying weights w(j, m): \hat{\Phi}_{\text{HAC}} = \hat{\Gamma}_0 + \sum_{j=1}^{m} \left( 1 - \frac{j}{m + 1} \right) \left( \hat{\Gamma}_j + \hat{\Gamma}_j' \right)

Key Theoretical Result: The Bartlett kernel guarantees that \hat{\Phi}_{\text{HAC}} is positive semi-definite (unlike simple unweighted truncated sample autocovariances).

4. Lag Selection & Alternative Kernels

Choosing the Truncation Bandwidth (m)

• Trade-off:
  • m too small \implies persistent autocorrelation omitted \implies downward bias in SE.
  • m too large \implies estimates higher lag covariances \implies increased estimator variance.
• Common Selection Rules:
  • Heuristic: m = \lfloor 4 (T/100)^{2/9} \rfloor or m = \lfloor T^{1/4} \rfloor
  • Data-driven Bandwidths: Andrews (1991) and Newey-West (1994) optimal plug-in selectors based on AR(1) approximations.

Other Non-Parametric Kernels

• Parzen Kernel: Cubic decay; smooth downweighting near boundaries.
• Quadratic Spectral (QS) Kernel: Asymptotically optimal under Mean Squared Error (MSE) criteria (Andrews, 1991).

5. Parametric Fixes: FGLS vs. HAC

Instead of adjusting standard errors post hoc, we can transform the model to eliminate correlation.

AR(1) Error Transformation (Cochrane-Orcutt / Prais-Winsten)

Assuming u_t = \rho u_{t-1} + e_t, where e_t \sim \text{i.i.d.}(0, \sigma_e^2): (y_t - \rho y_{t-1}) = (X_t - \rho X_{t-1})' \beta + e_t

Theoretical Comparison

• FGLS Advantage: Efficient if the AR(p) model is correctly specified.
• FGLS Fatal Flaw: Requires strict exogeneity (\mathbb{E}[u_t | X_1, \dots, X_T] = 0). If X_t contains lagged dependent variables (y_{t-1}), FGLS is inconsistent.
• HAC Advantage: Valid under predeterminedness (\mathbb{E}[u_t | X_t, X_{t-1}, \dots] = 0). Allows for dynamic regressors.

6. Empirical Implementation: OLS vs HAC

suppressPackageStartupMessages({
  library(sandwich)
  library(lmtest)
})

# 1. Simulate Time Series Data with AR(1) Errors (rho = 0.7)
set.seed(42)
T_obs <- 200
e <- rnorm(T_obs, mean = 0, sd = 1)
u <- numeric(T_obs)
for (t in 2:T_obs) {
  u[t] <- 0.7 * u[t - 1] + e[t]
}
x <- rnorm(T_obs, mean = 2, sd = 1)
y <- 1.5 + 2.0 * x + u

# 2. Fit OLS Model
model <- lm(y ~ x)

# 3. Naive OLS Standard Errors (Uncorrected)
cat("--- Standard OLS (Understates SE due to AR(1) errors) ---\n")

--- Standard OLS (Understates SE due to AR(1) errors) ---

print(coeftest(model))

t test of coefficients:

            Estimate Std. Error t value  Pr(>|t|)    
(Intercept)  1.43088    0.22317  6.4115 1.034e-09 ***
x            1.96745    0.10043 19.5895 < 2.2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

# 4. Newey-West HAC Standard Errors (Fixed Lag = 4)
cat("\n--- Newey-West HAC (Lag = 4) ---\n")

--- Newey-West HAC (Lag = 4) ---

print(coeftest(model, vcov = NeweyWest(model, lag = 4, prewhite = FALSE)))

t test of coefficients:

            Estimate Std. Error t value  Pr(>|t|)    
(Intercept) 1.430880   0.241621   5.922 1.385e-08 ***
x           1.967446   0.082786  23.765 < 2.2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

# 5. Newey-West Automatic Bandwidth (Andrews 1991)
cat("\n--- Newey-West HAC (Automatic Andrews Bandwidth) ---\n")

--- Newey-West HAC (Automatic Andrews Bandwidth) ---

print(coeftest(model, vcov = vcovHAC(model)))

t test of coefficients:

            Estimate Std. Error t value  Pr(>|t|)    
(Intercept) 1.430880   0.251564  5.6879 4.569e-08 ***
x           1.967446   0.067042 29.3464 < 2.2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

* ====================================================================
* Stata Code for Comparison (Run in Stata with simulated or real data)
* ====================================================================

* 1. Declare time series structure
* tsset time_var

* 2. Naive OLS
regress y x

* 3. Breusch-Godfrey Test for Serial Correlation
estat bgodfrey, lags(1 2 4)

* 4. Newey-West Standard Errors (Fixed Lag = 4)
newey y x, lag(4)

* 5. Automatic Bandwidth HAC (requires ivreg2 from SSC)
* ssc install ivreg2
ivreg2 y x, bw(auto) kernel(bartlett)

* 6. Quadratic Spectral Kernel (Andrews 1991 Optimal)
ivreg2 y x, bw(auto) kernel(qs)

7. Parametric Models & Diagnostics

suppressPackageStartupMessages({
  library(nlme)
  library(lmtest)
})

# 1. Breusch-Godfrey Test for Serial Correlation in Residuals
bg_test <- bgtest(model, order = 4)
cat("Breusch-Godfrey Test P-Value (Lag 4):", round(bg_test$p.value, 5), "\n\n")

Breusch-Godfrey Test P-Value (Lag 4): 0 

# 2. Feasible Generalized Least Squares (FGLS with AR(1) Errors)
fgls_model <- gls(y ~ x, correlation = corAR1(form = ~ 1))
cat("--- FGLS Estimation (Explicit AR(1) Error Model) ---\n")

--- FGLS Estimation (Explicit AR(1) Error Model) ---

summary(fgls_model)$tTable

               Value  Std.Error   t-value      p-value
(Intercept) 1.319721 0.25346574  5.206702 4.796002e-07
x           2.034128 0.05854733 34.743312 3.408735e-86

* ====================================================================
* Stata Parametric FGLS Implementation
* ====================================================================

* 1. Prais-Winsten AR(1) Regression (Preserves first observation)
prais y x, nolog

* 2. Cochrane-Orcutt AR(1) Regression (Drops first observation)
prais y x, corc nolog

* 3. Comparing Standard Errors Side-by-Side in Stata:
* quiet regress y x
* estimates store OLS
* quiet newey y x, lag(4)
* estimates store NW_4
* quiet prais y x, nolog
* estimates store FGLS
* estimates table OLS NW_4 FGLS, se p(%6.4f)

8. Applied Guidelines & Decision Tree

Takeaway: HAC standard errors correct inference (p-values, confidence intervals); they do not fix model misspecification or parameter inconsistency.

Applied Rules of Thumb

• Rule 1: Check Dynamic Specification First If lags of y_t or x_t belong in the true relationship, include them explicitly in X_t rather than sweeping dynamics into the error term u_t.
• Rule 2: Use HAC When T is Moderate to Large (T > 100) Newey-West / HAC standard errors are asymptotic. In small samples (T < 50), HAC standard errors can be severely biased downwards.
• Rule 3: Beware of FGLS with Lagged Dependent Variables Never use Cochrane-Orcutt or Prais-Winsten if X_t contains y_{t-1}. FGLS becomes inconsistent; use HAC or Dynamic OLS/GMM instead.