Package 'tptest'

Environmental Kuznets Curve Data

Description

Simulated data representing the Environmental Kuznets Curve (EKC) hypothesis, which posits an inverse U-shaped relationship between environmental degradation and economic development.

Usage

ekc

Format

A data frame with 500 observations and 6 variables:

country

Country identifier (1-50)

year

Year (2000-2009)

gdp

GDP per capita (thousands of dollars)

gdp_sq

Squared GDP per capita

emissions

CO2 emissions (tons per capita)

emissions_log

Log of CO2 emissions

Details

This is simulated panel data designed to demonstrate the tptest package. The true data generating process follows an inverse U-shape with:

• Turning point at GDP ~ 25,000 USD per capita

• Country-specific fixed effects

• Year-specific trends

Value

A data frame containing Environmental Kuznets Curve data with columns for income and emissions.

Source

Simulated data for demonstration purposes.

References

Grossman, G. M., & Krueger, A. B. (1995). Economic growth and the environment. Quarterly Journal of Economics, 110(2), 353-377.

Examples

data(ekc)
head(ekc)

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

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

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Fieller Confidence Interval for Ratio of Coefficients

Description

Computes the Fieller (1954) confidence interval for the turning point, which is more appropriate when the denominator coefficient has high uncertainty.

Usage

fieller_ci(b1, b2, s11, s12, s22, level = 0.95, form = "quadratic")

Arguments

b1	First coefficient
b2	Second coefficient
s11	Variance of b1
s12	Covariance of b1 and b2
s22	Variance of b2
level	Confidence level
form	Functional form ("quadratic" or "inverse")

Value

A list with elements lo, hi, and type.

References

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

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Universal Turning Point and Inflection Point Test

Description

Tests for U-shaped or inverse U-shaped relationships using the Sasabuchi (1980) test as extended by Lind and Mehlum (2010). Supports quadratic, cubic, log-quadratic, and inverse functional forms.

Usage

tptest(
  model = NULL,
  vars,
  coefs = NULL,
  vcov_mat = NULL,
  min = NULL,
  max = NULL,
  form = c("auto", "quadratic", "cubic", "inverse", "logquadratic"),
  level = 0.95,
  delta = TRUE,
  fieller = FALSE,
  twolines = FALSE,
  bootstrap = FALSE,
  breps = 1000,
  data = NULL,
  depvar = NULL
)

Arguments

model	A fitted model object (e.g., from lm, glm). Alternatively, coefficients can be provided directly via coefs.
vars	Character vector of length 2 or 3 specifying the variable names: c("x", "x_sq") for quadratic, c("x", "x_sq", "x_cu") for cubic.
coefs	Named numeric vector of coefficients. If provided, model is not required. Must include names matching vars.
vcov_mat	Variance-covariance matrix for the coefficients. Required when coefs is provided.
min	Lower bound of the data interval. If NULL, extracted from the model data.
max	Upper bound of the data interval. If NULL, extracted from the model data.
form	Functional form: "auto" (default), "quadratic", "cubic", "inverse", or "logquadratic".
level	Confidence level for intervals (default 0.95).
delta	Logical; compute delta-method SE and CI (default TRUE).
fieller	Logical; compute Fieller confidence interval (default FALSE).
twolines	Logical; perform Simonsohn (2018) two-lines test (default FALSE).
bootstrap	Logical; compute parametric bootstrap CI (default FALSE).
breps	Number of bootstrap replications (default 1000).
data	Optional data frame for two-lines test.
depvar	Name of dependent variable for two-lines test.

Details

The function implements several approaches for testing non-monotonic relationships:

Sasabuchi (1980) / Lind-Mehlum (2010) Test: Tests whether the relationship is U-shaped (or inverse U-shaped) by examining slopes at the interval boundaries. The null hypothesis is monotonicity or opposite U-shape.

Functional Forms:

• Quadratic: $y = \beta_1 x + \beta_2 x^2$; turning point at $x^* = -\beta_1 / (2\beta_2)$

• Cubic: $y = \beta_1 x + \beta_2 x^2 + \beta_3 x^3$; up to two turning points

• Inverse: $y = \beta_1 x + \beta_2 / x$; turning point at $x^* = \sqrt{\beta_2 / \beta_1}$

• Log-quadratic: $\ln(y) = \beta_1 \ln(x) + \beta_2 [\ln(x)]^2$

Value

An object of class "tptest" containing:

tp

Turning point estimate

tp_se

Delta-method standard error

tp_ci

Confidence interval for turning point

shape

Detected shape ("U shape" or "Inverse U shape")

model_form

Functional form used

sasabuchi

List with Sasabuchi test results

fieller

Fieller interval (if requested)

twolines

Two-lines test results (if requested)

bootstrap

Bootstrap results (if requested)

coefficients

Named vector of relevant coefficients

vcov

Variance-covariance matrix

bounds

Data interval bounds

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. doi: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. doi: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.

Examples

# 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)


# With Fieller interval and two-lines test
result2 <- tptest(fit, vars = c("x", "x_sq"),
                  fieller = TRUE, twolines = TRUE,
                  data = dat, depvar = "y")
summary(result2)

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Methods for tptest Objects

Description

Print, summary, plot, and accessor methods for tptest objects.

Usage

## S3 method for class 'tptest'
print(x, ...)

## S3 method for class 'tptest'
summary(object, ...)

## S3 method for class 'tptest'
plot(
  x,
  main = NULL,
  xlab = "x",
  ylab = "Marginal Effect",
  col.line = "steelblue",
  col.tp = "red",
  col.ci = grDevices::rgb(0.2, 0.4, 0.8, 0.2),
  lwd = 2,
  n = 200,
  ...
)

## S3 method for class 'tptest'
coef(object, ...)

## S3 method for class 'tptest'
confint(object, parm = NULL, level = NULL, ...)

Arguments

x	A tptest object
...	Additional arguments (currently ignored)
object	A tptest object
main	Plot title
xlab	X-axis label
ylab	Y-axis label
col.line	Color for the fitted curve
col.tp	Color for the turning point marker
col.ci	Color for confidence band
lwd	Line width
n	Number of points for plotting curve
parm	Parameter specification (currently ignored)
level	Confidence level (default uses level from tptest object)

Value

For print and summary: the input object, returned invisibly. For plot: no return value, called for side effects (generates plots).

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Simonsohn (2018) Two-Lines Test

Description

Performs the two-lines test proposed by Simonsohn (2018) as an alternative to quadratic regression for testing U-shaped relationships.

Usage

twolines_test(data, x_var, y_var, split_point, form = "quadratic")

Arguments

data	Data frame containing the variables
x_var	Name of the x variable (character)
y_var	Name of the y variable (character)
split_point	Point at which to split the data (usually the turning point)
form	Functional form (for log-quadratic, split is in log-space)

Value

A list with test results including slopes, t-values, p-values, and whether the test confirms the U-shape.

References

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.

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