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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 Sets: Exact sets for the turning point, including the two-ray case when the denominator coefficient is not significant
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 and Mehlum 2010)
==========================================

Model form: Quadratic: y = b1*x + b2*x^2 
Data interval: [1.002, 9.9]
Distribution: t(197) 

Fitted shape on the interval: U shape 
Turning point (x*): 8.0909
  Delta-method SE:  0.336815
  95% CI:         [7.42667, 8.75513]

------------------------------------------
Sasabuchi (1980) Test
------------------------------------------
                Lower bound    Upper bound
Interval             1.0022         9.9000
Slope               -6.8191         1.7403
t-value            -12.9977         3.3549
P (one-sided)        0.0000         0.0005

Tested alternative: U shape 
Overall test: t = 3.3549, p = 0.000476 ***
-> Strong evidence of U shape (p < 0.01) 
------------------------------------------
*** p<0.01, ** p<0.05, * p<0.10

The interval is taken from the estimation sample of the model unless min and max are supplied. The t distribution with the residual degrees of freedom is used for lm-type models; the normal distribution is used for glm objects and when coefficients are passed through coefs.

Functional forms

`form`	Model	Turning point	Notes
`quadratic`	`y = b1x + b2x^2`	`-b1/(2*b2)`	`vars = c("x", "x_sq")`
`inverse`	`y = b1*x + b2/x`	`sqrt(b2/b1)`	`vars = c("x", "x_inv")`; U shape when `b1 > 0`, inverse U when `b1 < 0`
`logquadratic`	`y = b1ln(x) + b2ln(x)^2`	`exp(-b1/(2*b2))`	`vars` names the `ln(x)` and `ln(x)^2` regressors; bounds are on the `ln(x)` scale unless `bounds_scale = "levels"`; the turning point and its intervals are reported in levels of `x`
`cubic`	`y = b1x + b2x^2 + b3*x^3`	roots of the slope	the slope is not monotone across the inflection point, so the two-endpoint test is applied on each monotone sub-interval (a package extension, not part of Lind and Mehlum 2010)

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. and 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

These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.
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