Elliott-Muller Test for Time-Varying Coefficients

Computes the \(\hat{qLL}\) statistic of Elliott and Muller (2006) for testing the null hypothesis that regression coefficients are constant over time against the alternative of general time variation.

Usage

qll_hat(y, X, Z = NULL, L = 0L)

Arguments

y: Numeric vector of length T; the dependent variable.
X: A T x k matrix of regressors linked to potentially time-varying coefficients.
Z: A T x d matrix of regressors with constant coefficients. Use NULL (default) if all coefficients may vary.
L: Integer; lag truncation for the Newey-West estimator of the variance. Default 0 (no correction).

Value

A list with class "qll_hat" containing:

statistic: The \(\hat{qLL}\) test statistic.
k: Number of potentially time-varying coefficients.
n: Number of observations.

Details

The test is based on optimal invariant statistics for the null of constant coefficients against local alternatives. The \(\hat{qLL}\) statistic has non-standard critical values that depend on k; see Table 1 in Elliott and Muller (2006).

Selected critical values (5\

k = 1: -5.91
k = 2: -10.64
k = 3: -15.78
k = 4: -20.62
k = 5: -25.87

Reject the null when \(\hat{qLL}\) is below the critical value.

References

Elliott, G. and Muller, U.K. (2006). Efficient Tests for General Persistent Time Variation in Regression Coefficients. Review of Economic Studies, 73(4), 907-940.

Examples

set.seed(42)
n <- 200
x <- matrix(rnorm(n * 2), n, 2)
y <- x %*% c(0.5, -0.3) + rnorm(n)
qll_hat(y, x)
#> 
#> ╭────────────────────────────────────────────────────╮
#> │ Elliott-Muller Test for Time-Varying Coefficients  │
#> │             (Elliott and Muller, 2006)             │
#> ├────────────────────────────────────────────────────┤
#> │ H0: Constant coefficients (beta(t) = beta)         │
#> │ H1: Time-varying coefficients                      │
#> ├┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┤
#> │ Test Results:                                      │
#> │  qLL-hat statistic: -13.8782                       │
#> │  5% critical value: -10.64                         │
#> │  Decision (5%): Rejected ***                       │
#> ├┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┄┤
#> │ Details:                                           │
#> │  Observations (T): 200                             │
#> │  Time-varying coefficients (k): 2                  │
#> │ Note: Non-standard distribution.                   │
#> │ Reject when qLL-hat < critical value.              │
#> ╰────────────────────────────────────────────────────╯
#>