tptest: Universal Turning Point and Inflection Point Tests

Overview

The tptest package implements tests for U-shaped and inverse U-shaped relationships in regression analysis. It provides a comprehensive framework for detecting turning points and inflection points in time series and panel data.

Key Features

• Sasabuchi (1980) / Lind-Mehlum (2010) Test: Rigorous test for U-shape or inverse U-shape
• Multiple Functional Forms: Quadratic, cubic, inverse, and log-quadratic specifications
• Delta-Method Standard Errors: Proper inference for turning point estimates
• Fieller Confidence Intervals: Robust intervals when denominator is uncertain
• Simonsohn (2018) Two-Lines Test: Alternative U-shape validation
• Parametric Bootstrap: Bootstrap confidence intervals

Installation

# Install from CRAN (when available)
install.packages("tptest")

# Install development version from GitHub
devtools::install_github("muhammedalkhalaf/tptest")

Usage

library(tptest)

# Simulate data with U-shaped relationship
set.seed(42)
n <- 200
x <- runif(n, 1, 10)
y <- 50 - 8*x + 0.5*x^2 + rnorm(n, sd = 5)
dat <- data.frame(y = y, x = x, x_sq = x^2)

# Fit quadratic model
fit <- lm(y ~ x + x_sq, data = dat)

# Test for U-shape
result <- tptest(fit, vars = c("x", "x_sq"), data = dat)
print(result)

Output

==========================================
  Turning Point Test (Lind & Mehlum 2010)
==========================================

Model form: Quadratic: y = b1*x + b2*x^2
Data interval: [1.023, 9.987]

Detected shape: U shape
Turning point (x*): 8.234
  Delta-method SE:  0.5123
  95% CI:         [7.230, 9.238]

------------------------------------------
Sasabuchi (1980) Test
------------------------------------------
                Lower bound    Upper bound
Interval            1.0230         9.9870
Slope              -6.2456         1.7532
t-value           -12.4532         3.4521
P>|t|               0.0000         0.0003

Overall test: t = 3.4521, p = 0.000312 ***
-> Strong evidence of U shape (p < 0.01)
------------------------------------------
*** p<0.01, ** p<0.05, * p<0.10

Environmental Kuznets Curve Example

# Load example data
data(ekc)

# Fit model
fit <- lm(emissions ~ gdp + gdp_sq, data = ekc)

# Test for inverse U-shape
result <- tptest(fit, vars = c("gdp", "gdp_sq"), 
                 fieller = TRUE, data = ekc)
summary(result)
plot(result)

References

• Lind, J. T., & Mehlum, H. (2010). With or without U? The appropriate test for a U-shaped relationship. Oxford Bulletin of Economics and Statistics, 72(1), 109-118. https://doi.org/10.1111/j.1468-0084.2009.00569.x

• Sasabuchi, S. (1980). A test of a multivariate normal mean with composite hypotheses determined by linear inequalities. Biometrika, 67(2), 429-439.

• Fieller, E. C. (1954). Some problems in interval estimation. Journal of the Royal Statistical Society: Series B, 16(2), 175-185. https://doi.org/10.1111/j.2517-6161.1954.tb00159.x

• Simonsohn, U. (2018). Two lines: A valid alternative to the invalid testing of U-shaped relationships with quadratic regressions. Advances in Methods and Practices in Psychological Science, 1(4), 538-555.

License

GPL-3

Author