# Vuong and Clarke tests

### Description

The Vuong and Clarke tests are likelihood-ratio-based tests that can be used for choosing between two non-nested models.

### Usage

1 | ```
VuongClarke(obj1, obj2, sig.lev = 0.05)
``` |

### Arguments

`obj1,obj2` |
Objects of the two fitted bivariate non-nested models. |

`sig.lev` |
Significance level used for testing. |

### Details

The Vuong (1989) and Clarke (2007) tests are likelihood-ratio-based tests for model selection that use the Kullback-Leibler information criterion. The implemented tests can be used for choosing between two bivariate models which are non-nested.

In the Vuong test, the null hypothesis is that the two models are equally close to the actual model, whereas
the alternative is that one model is closer. The test follows asymptotically a standard normal
distribution under the null. Assume that the critical region is *(-c,c)*, where *c* is typically set to 1.96. If the value
of the test is higher than *c* then we reject the null hypothesis
that the models are equivalent in favor of model `obj1`

. Viceversa if the value is smaller than *c*. If
the value falls in *[-c,c]* then we cannot discriminate between the two competing models given the data.

In the Clarke test, if the two models are statistically equivalent then the log-likelihood ratios of the
observations should be evenly distributed around zero
and around half of the ratios should be larger than zero. The test follows asymptotically a binomial distribution with
parameters *n* and 0.5. Critical values can be obtained as shown in Clarke (2007). Intuitively,
model `obj1`

is preferred over `obj2`

if the value of the test
is significantly larger than its expected value under the null hypothesis (*n/2*), and vice versa. If
the value is not significantly different from *n/2* then `obj1`

can be thought of as equivalent to `obj2`

.

### Value

It returns two decisions based on the tests and criteria discussed above.

### Author(s)

Maintainer: Giampiero Marra giampiero.marra@ucl.ac.uk

### References

Clarke K. (2007), A Simple Distribution-Free Test for Non-Nested Model Selection. *Political Analysis*, 15, 347-363.

Vuong Q.H. (1989), Likelihood Ratio Tests for Model Selection and Non-Nested Hypotheses. *Econometrica*, 57(2), 307-333.

### Examples

1 | ```
## see examples for SemiParBIVProbit
``` |