# Returns to Scale (RTS) Test

The non-parametric Returns to Scale test (Simar and Wilson, 2002) is based on the bootstrap radial DEA model. It tests the hypothesis that the technology exhibits constant returns to scale (CRS) against the alternative that it is variable returns to scale (VRS):

\begin{aligned} H_0 & \text{: Technology is CRS} \\ H_1 & \text{: Tecnhology is VRS} \end{aligned}

The test can be performed with the deartstest function, indicating the number of bootstrap replications in the nreps parameter. A random number generator can be specified in the rng parameter for reproducibility.

using DataEnvelopmentAnalysis
using StableRNGs

X = [2, 4, 3, 5, 6]
Y = [1, 2, 3, 4, 5]

ioboot = deartstest(X, Y, orient = :Input, nreps = 200, rng = StableRNG(1234567))
DEA Returns to Scale (RTS) Test
--------------------------------

H0: Globally CRS
H1: VRS

Bootstrap replications: 200
Bandwidth = 0.30976

Scale efficiency: 0.8029
Critical value (α = 0.05): 0.6746
p-value: 0.335

Number of bootstrap replications

The example above uses 200 bootstrap replications for illustrative purposes. In practice, at least 1000 replications are recommended.

We reject the null hypothesis if the estimated scale efficiency is less than the critical value. Alternatively, we can guide our decision using the calculated p-value. In the example above, we do not reject the null hypothesis of constant returns to scale.

### deaboot Function Documentation

DataEnvelopmentAnalysis.deartstestFunction
deartstest(X, Y)

Compute the DEA Returns to Scale (RTS) test using the bootstrap radial model for inputs X and outputs Y.

Optional Arguments

• nreps=200: number of bootstrap replications.
• rng=default_rng(): random number generator.
• orient=:Input: chooses the radially oriented input mode. For the radially oriented output model choose :Output.
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