paper/paper.md

In FedericoGarza/lineartestr: Linear Specification Testing

title: 'lineartestr: An R package for testing the linear specification of a model' tags: - R - linear-specification - linear-regression - dominguez-lobato-test - wild-bootstrap - reset-test authors: - name: Federico Garza Ramírez orcid: 0000-0001-7015-8186 affiliation: "1" # (Multiple affiliations must be quoted) #- name: Author Without ORCID # affiliation: 2 affiliations: - name: Instituto Tecnológico Autónomo de México index: 1 #- name: Institution 2 # index: 2 date: 15 Jun 2020 bibliography: paper.bib

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Summary

Many econometric models use the underlying assumption that the relationship between endogenous and exogenous variables is linear. This is the case of models such as difference-in-differences, fixed effects, regression discontinuity design among others that are particularly useful inferring causal relationships [@angrist_mostly_2008]. As a part of the hypothesis imposed to the models, the linear relationship between variables must be contrasted in order to guarantee the validity of the research results. To this end we have developed lineartestr for R [@r] with a novel and robust approach.

Statement of need

The package lineartestr implements a novel approach developed by Domínguez and Lobato [@lobato] that tests the linear specification of a model trough a bootstrap test with the main function dominguez_lobato_test(). This approach generalizes well known specification tests such as Ramsey's RESET (also implemented with the function reset_test()) and as the authors conclude this new test is more robust than others. This is the main difference from other R packages that implement linearity tests such as lmtest [@lmtest] and car [@car]. Therefore this new package provides to researchers with a new robust tool to their research arsenal.

Domínguez-Lobato test

The Domínguez-Lobato approach tests the linear specification of a model. It means that tests the null hypothesis,

$$ H_0: y = \beta^T x + u. $$

Where $y$ is the target variable, $x$ are control variables and $u$ is the model error. In their work, Domínguez and Lobato proposed two statistics to test this hypothesis [@lobato]. Consider a collection ${(y_i, x_i)}$ of size n from $(y, x)$. First, the Cramer von Mises (CvM) statistic given by

$$ C_n = \frac{1}{n^2} \sum_{l=1}^n \left[ \sum_{i=1}^n \hat{u}_i I(\hat{y}_i \leq \hat{y}_l)\right]. $$

Where $\hat{y}_j$ are the fitted values of the endogenous variable and $\hat{u}_i = y_i - \hat{y}_i$. Second, the Kolmogorov Smirnov (KS) statistic given by

$$ K_n = \max_l \left| \sum_{i=1}^n \hat{u}_i I(\hat{y}_i \leq \hat{y}_l) \right|. $$

With this statistics the authors propose a wild bootstrap test described by the following steps:

1. Fit the linear model $y_i = \alpha^T x_i + u_i$ and collect the fitted values $\hat{y}_i$.
2. With the actual residuals $\hat{u}_i = y_i - \hat{y}_i$ calculate $C_n$ or $K_n$.
3. Generate a collection ${V^b_i}$ of size $n$ of bounded random variables independent and identically distributed with mean zero and unit variance. With this observations construct a new endogenous variable,

$$ y^b_i = \hat{y}_i + \hat{u}_i*V^b_i. $$

1. Adjust a new model $y^b_i = \beta^{bT} x_i + u^b_i$. With $\hat{u}^b_i = y^b_i - \hat{y}^b_i$ calculate $C^b_n$ or $K^b_n$ as corresponds.

2. Generate a collection ${C^b_n}$ or ${K^b_n}$ of size $B$ repeating 2. Each collection ${V^b_i}$ is independent of each other.

3. Calculate the $(1-\alpha)$-quantile from ${C^b_n}$ or ${K^b_n}$: $C_{[1-\alpha]}$ or $K_{[1-\alpha]}$. Finally reject the null hypothesis at the $\alpha$ nominal level when $C_n > C_{[1-\alpha]}$ and when $K_n > K_{[1-\alpha]}$, respectively.

The algorithm works because the statistic $C_n$ and $C^b_n$ share the same asymptotic distribution under the null hypothesis (for almost all samples) [@lobato].

lineartestr package

The lineartestr package includes the function dominguez_lobato_test which performs the algorithm developed by Domínguez and Lobato [@lobato]. This function receives the four main parameters of the algorithm:

1. model: a fitted linear model. The package can handle any fitted linear model compatible with the update() function such as stats::lm() [@r] or lfe::felm() [@lfe]. Also the function can receive ARMA models (which are linear models) fitted from forecast::Arima() [@forecast].

2. distribution: a function name from which the collection ${V^b_i}$ will be calculated. By default, the package uses a standard normal random variable ('rnorm' that calls stats::rnorm [@r]) but other random variables with mean zero and unit variance can be used. In particular, the package includes three special functions that satisfies this requirements: 'rmammen_cont', 'rmammen_point' [@mammen] and 'rrademacher' [@rrademacher].

3. statistic: 'cvm_value' to use the CvM statistic or 'kmv_value' to use the KS statistic.

4. times: number of bootstrap repetitions ($B$).

Parallel processing

The wild bootstrap approach of this test can be time consuming. However, lineartestr can process this repetitions trough parallel processing using the parallel package [@r]. This can be done as simply as setting the n_cores parameter of the dominguez_lobato_test function with the number of desired workers to carry out this tasks.

Ramsey's RESET test

The Ramsey's RESET (Regression Equation Specification Error Test) test [@ramsey] is a widely known tool that can be used too to test the specification of a linear model. This method works as follows,

1. Fit the linear model $y_i = \alpha^T x_i + u_i$ and collect the fitted values $\hat{y}_i$.

2. Choose $k\geq3$ and estimate the following,

$$ y_i = \alpha^T x_i + \gamma_1 \hat{y}i^2 + ... + \gamma{k-1} \hat{y}_i^k + u_i. $$

1. Use a Wald test [@wald] to contrast the following null hypothesis. If this null hypothesis is rejected, then the hypothesis of the model's linearity is also rejected,

$$ H_0: \gamma_1 = \gamma_2 =...= \gamma_{k-1} = 0. $$

For completeness the RESET test is also implemented trough the reset_test function which uses the custom wald_test function to carry out the null hypothesis test. Like the dominguez_lobato_test function, it receives a fitted linear model.

Plot functions

The package lineartestr also contains special functions to plot each of the tests (plot_dl_test for the dominguez_lobato_test). This plots can be useful to get a visual description of the statistic's distribution, the statistic's value and the test's critical values.

Acknowledgements

We thank Daniela Baltazar Cerros and Kin Gutiérrez Olivares for their support and feedback.

References

FedericoGarza/lineartestr documentation built on Jan. 31, 2021, 11:11 a.m.