14.1 Nested Model Tests

Nested models are those where the restricted model is a special case of the unrestricted model. In other words, the restricted model can be derived from the unrestricted model by imposing constraints on certain parameters, typically setting them equal to zero. This structure allows us to formally test whether the additional variables in the unrestricted model significantly improve the model’s explanatory power. The following tests help compare these models:

Wald Test: Assesses the significance of individual coefficients or groups of coefficients.
Likelihood Ratio Test: Compares the goodness-of-fit between restricted and unrestricted models.
F-Test: Evaluates the joint significance of multiple coefficients.
Chow Test: Evaluates whether the coefficients of a regression model are the same across different groups or time periods.

Consider the following models:

\[ \begin{aligned} \textbf{Unrestricted Model:} \quad & y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \beta_3 x_3 + \epsilon \\ \textbf{Restricted Model:} \quad & y = \beta_0 + \beta_1 x_1 + \epsilon \end{aligned} \]

The unrestricted model includes all potential explanatory variables: \(x_1\), \(x_2\), and \(x_3\).
The restricted model is nested within the unrestricted model, containing a subset of variables (in this case, excluding \(x_2\) and \(x_3\)).

Our goal is to test the null hypothesis that the restrictions are valid:

\[ H_0: \beta_2 = \beta_3 = 0 \quad \text{(restrictions are valid)} \]

against the alternative hypothesis:

\[ H_1: \text{At least one of } \beta_2, \beta_3 \neq 0 \quad \text{(restrictions are invalid)} \]

To conduct this test, we use the following methods:

14.1.1 Wald Test

The Wald Test assesses whether certain linear restrictions on the parameters of the model are valid. It is commonly used when testing the joint significance of multiple coefficients.

Consider a set of linear restrictions expressed as:

\[ H_0: R\boldsymbol{\beta} = r \]

where:

\(R\) is a \(q \times k\) restriction matrix,
\(\boldsymbol{\beta}\) is the \(k \times 1\) vector of parameters,
\(r\) is a \(q \times 1\) vector representing the hypothesized values (often zeros),
\(q\) is the number of restrictions being tested.

For example, if we want to test \(H_0: \beta_2 = \beta_3 = 0\), then:

\[ R = \begin{bmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}, \quad r = \begin{bmatrix} 0 \\ 0 \end{bmatrix} \]

The Wald statistic is calculated as:

\[ W = (R\hat{\boldsymbol{\beta}} - r)' \left[ R \, \hat{\text{Var}}(\hat{\boldsymbol{\beta}}) \, R' \right]^{-1} (R\hat{\boldsymbol{\beta}} - r) \]

Where:

\(\hat{\boldsymbol{\beta}}\) is the vector of estimated coefficients from the unrestricted model,
\(\hat{\text{Var}}(\hat{\boldsymbol{\beta}})\) is the estimated covariance matrix of \(\hat{\boldsymbol{\beta}}\).

Distribution and Decision Rule

Under \(H_0\), the Wald statistic follows a \(\chi^2\) distribution with \(q\) degrees of freedom:

\[ W \sim \chi^2_q \]

Decision Rule:
- Reject \(H_0\) if \(W > \chi^2_{q,\alpha}\), where \(\alpha\) is the significance level.
- A large Wald statistic indicates that the restrictions are invalid.

14.1.2 Likelihood Ratio Test

The Likelihood Ratio Test compares the goodness-of-fit between the restricted and unrestricted models. It evaluates whether the additional parameters in the unrestricted model significantly improve the likelihood of observing the data.

Same as before:

\[ H_0: \beta_2 = \beta_3 = 0 \quad \text{vs.} \quad H_1: \text{At least one of } \beta_2, \beta_3 \neq 0 \]

The LR statistic is calculated as:

\[ LR = -2 \left( \ln L_R - \ln L_U \right) \]

Where:

\(L_R\) is the maximized likelihood of the restricted model,
\(L_U\) is the maximized likelihood of the unrestricted model.

Distribution and Decision Rule

Under \(H_0\), the LR statistic follows a \(\chi^2\) distribution with \(q\) degrees of freedom:

\[ LR \sim \chi^2_q \]

Decision Rule:
- Reject \(H_0\) if \(LR > \chi^2_{q,\alpha}\).
- A large LR statistic suggests that the unrestricted model provides a significantly better fit.

Connection to OLS

In the case of linear regression with normally distributed errors, the LR statistic can be expressed in terms of the sum of squared residuals (SSR):

\[ LR = n \ln \left( \frac{SSR_R}{SSR_U} \right) \]

where \(n\) is the sample size.

14.1.3 F-Test (for Linear Regression)

The F-Test is commonly used in linear regression to evaluate the joint significance of multiple coefficients. It compares the fit of the restricted and unrestricted models using their sum of squared residuals.

Again:

\[ H_0: \beta_2 = \beta_3 = 0 \quad \text{vs.} \quad H_1: \text{At least one of } \beta_2, \beta_3 \neq 0 \]

The F-statistic is calculated as:

\[ F = \frac{(SSR_R - SSR_U) / q}{SSR_U / (n - k)} \]

Where:

\(SSR_R\) = Sum of Squared Residuals from the restricted model,
\(SSR_U\) = Sum of Squared Residuals from the unrestricted model,
\(q\) = Number of restrictions (here, 2),
\(n\) = Sample size,
\(k\) = Number of parameters in the unrestricted model.

Distribution and Decision Rule

Under \(H_0\), the F-statistic follows an \(F\)-distribution with \((q, n - k)\) degrees of freedom:

\[ F \sim F_{q, n - k} \]

Decision Rule:
- Reject \(H_0\) if \(F > F_{q, n - k, \alpha}\).
- A large F-statistic indicates that the restricted model fits significantly worse, suggesting the excluded variables are important.

14.1.4 Chow Test

The Chow Test evaluates whether the coefficients of a regression model are the same across different groups or time periods. It is often used to detect structural breaks in the data.

Key Question:
Should we run two different regressions for two groups, or can we pool the data and use a single regression?

Chow Test Procedure

Estimate the regression model for the pooled data (all observations).
Estimate the model separately for Group 1 and Group 2.
Compare the sum of squared residuals (SSR) from these models.

The test statistic follows an F-distribution:

\[ F = \frac{(SSR_P - (SSR_1 + SSR_2)) / k}{(SSR_1 + SSR_2) / (n_1 + n_2 - 2k)} \]

Where:

\(SSR_P\) = Sum of Squared Residuals for the pooled model
\(SSR_1\), \(SSR_2\) = SSRs for Group 1 and Group 2 models
\(k\) = Number of parameters
\(n_1\), \(n_2\) = Number of observations in each group

Interpretation:

A significant F-statistic suggests structural differences between groups, implying separate regressions are more appropriate.
A non-significant F-statistic indicates no structural break, supporting the use of a pooled model.

# Load necessary libraries
library(car)        # For Wald Test
library(lmtest)     # For Likelihood Ratio Test
library(strucchange)  # For Chow Test

# Simulated dataset
set.seed(123)
n <- 100
x1 <- rnorm(n)
x2 <- rnorm(n)
x3 <- rnorm(n)
epsilon <- rnorm(n)
y <- 2 + 1.5 * x1 + 0.5 * x2 - 0.7 * x3 + epsilon

# Creating a group variable (simulating a structural break)
group <- rep(c(0, 1), each = n / 2)  # Group 0 and Group 1

# ----------------------------------------------------------------------
# Wald Test
# ----------------------------------------------------------------------
unrestricted_model <- lm(y ~ x1 + x2 + x3)    # Unrestricted model
restricted_model <- lm(y ~ x1)                # Restricted model

wald_test <- linearHypothesis(unrestricted_model, c("x2 = 0", "x3 = 0"))
print(wald_test)
#> Linear hypothesis test
#> 
#> Hypothesis:
#> x2 = 0
#> x3 = 0
#> 
#> Model 1: restricted model
#> Model 2: y ~ x1 + x2 + x3
#> 
#>   Res.Df    RSS Df Sum of Sq      F    Pr(>F)    
#> 1     98 182.26                                  
#> 2     96 106.14  2    76.117 34.421 5.368e-12 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

# ----------------------------------------------------------------------
# Likelihood Ratio Test
# ----------------------------------------------------------------------
lr_test <- lrtest(unrestricted_model, restricted_model)
print(lr_test)
#> Likelihood ratio test
#> 
#> Model 1: y ~ x1 + x2 + x3
#> Model 2: y ~ x1
#>   #Df  LogLik Df  Chisq Pr(>Chisq)    
#> 1   5 -144.88                         
#> 2   3 -171.91 -2 54.064  1.821e-12 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

# ----------------------------------------------------------------------
# F-Test (for Linear Regression)
# ----------------------------------------------------------------------
SSR_U <- sum(residuals(unrestricted_model)^2)  # SSR for unrestricted model
SSR_R <- sum(residuals(restricted_model)^2)    # SSR for restricted model
q <- 2                                        # Number of restrictions
n <- length(y)                                # Sample size
k <- length(coef(unrestricted_model))         # Number of parameters in unrestricted model

# F-statistic formula
F_statistic <- ((SSR_R - SSR_U) / q) / (SSR_U / (n - k))
p_value_F <- pf(F_statistic, df1 = q, df2 = n - k, lower.tail = FALSE)

cat("F-statistic:", F_statistic, "\n")
#> F-statistic: 34.42083
cat("P-value:", p_value_F, "\n")
#> P-value: 5.367912e-12

# ----------------------------------------------------------------------
# Chow Test (Proper Use of the Group Variable)
# ----------------------------------------------------------------------
# Pooled model (all data)
pooled_model <- lm(y ~ x1 + x2 + x3)

# Separate models for Group 0 and Group 1
model_group0 <- lm(y[group == 0] ~ x1[group == 0] + x2[group == 0] + x3[group == 0])
model_group1 <- lm(y[group == 1] ~ x1[group == 1] + x2[group == 1] + x3[group == 1])

# Calculating SSRs
SSR_pooled <- sum(residuals(pooled_model)^2)
SSR_group0 <- sum(residuals(model_group0)^2)
SSR_group1 <- sum(residuals(model_group1)^2)

# Chow Test formula
k_chow <- length(coef(pooled_model))  # Number of parameters (including intercept)
n0 <- sum(group == 0)                 # Sample size for Group 0
n1 <- sum(group == 1)                 # Sample size for Group 1

F_chow <- ((SSR_pooled - (SSR_group0 + SSR_group1)) / k_chow) /
          ((SSR_group0 + SSR_group1) / (n0 + n1 - 2 * k_chow))

# Corresponding p-value
p_value_chow <-
    pf(
        F_chow,
        df1 = k_chow,
        df2 = (n0 + n1 - 2 * k_chow),
        lower.tail = FALSE
    )

cat("Chow Test F-statistic:", F_chow, "\n")
#> Chow Test F-statistic: 0.3551197
cat("P-value:", p_value_chow, "\n")
#> P-value: 0.8398657

Interpretation of the Results

Wald Test
- Null Hypothesis (\(H_0\)): \(\beta_2 = \beta_3 = 0\) (the coefficients for \(x_2\) and \(x_3\) are zero).
- Decision Rule:
  - Reject \(H_0\) if p-value < 0.05: \(x_2\) and \(x_3\) are jointly significant.
  - Fail to reject \(H_0\) if p-value ≥ 0.05: \(x_2\) and \(x_3\) do not significantly contribute to the model.
Likelihood Ratio Test (LR Test)
- Null Hypothesis (\(H_0\)): The restricted model fits the data as well as the unrestricted model.
- Decision Rule:
  - Reject \(H_0\) if p-value < 0.05: The unrestricted model fits better, indicating \(x_2\) and \(x_3\) improve the model.
  - Fail to reject \(H_0\) if p-value ≥ 0.05: Adding \(x_2\) and \(x_3\) doesn’t improve the model significantly.
F-Test (for Linear Regression)
- Null Hypothesis (\(H_0\)): \(\beta_2 = \beta_3 = 0\) (similar to the Wald Test).
- Decision Rule:
  - Reject \(H_0\) if p-value < 0.05: The excluded variables are significant.
  - Fail to reject \(H_0\) if p-value ≥ 0.05: The excluded variables are not significant.
Chow Test (Using the group Variable)
- Null Hypothesis (\(H_0\)): No structural break exists; the regression coefficients are the same across Group 0 and Group 1.
- Decision Rule:
  - Reject \(H_0\) if p-value < 0.05: A structural break exists—model coefficients differ significantly between the groups.
  - Fail to reject \(H_0\) if p-value ≥ 0.05: No significant structural break detected; the model coefficients are stable across both groups.