Regression Tests

Description

A collection and description of functions to test linear regression models, including tests for higher serial correlations, for heteroskedasticity, for autocorrelations of disturbances, for linearity, and functional relations.

The methods are:

"bg" Breusch--Godfrey test for higher order serial correlation,
"bp" Breusch--Pagan test for heteroskedasticity,
"dw" Durbin--Watson test for autocorrelation of disturbances,
"gq" Goldfeld--Quandt test for heteroskedasticity,
"harv" Harvey--Collier test for linearity,
"hmc" Harrison--McCabe test for heteroskedasticity,
"rain" Rainbow test for linearity, and
"reset" Ramsey's RESET test for functional relation.

There is nothing new, it's just a wrapper to the underlying test functions from R's contributed package lmtest. The functions are available as "Builtin" functions. Nevertheless, the user can still install and use the original functions from R's lmtest package.

Usage

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lmTest(formula, method = c("bg", "bp", "dw", "gq", "harv", "hmc", 
    "rain", "reset"), data = list(), ...)
    
bgTest(formula, order = 1, type = c("Chisq", "F"), data = list())
bpTest(formula, varformula = NULL, studentize = TRUE, data = list())
dwTest(formula, alternative = c("greater", "two.sided", "less"),
    iterations = 15, exact = NULL, tol = 1e-10, data = list())
gqTest(formula, point=0.5, order.by = NULL, data = list())
harvTest(formula, order.by = NULL, data = list())
hmcTest(formula, point = 0.5, order.by = NULL, simulate.p = TRUE, 
    nsim = 1000, plot = FALSE, data = list()) 
rainTest(formula, fraction = 0.5, order.by = NULL, center = NULL, 
    data = list())
resetTest(formula, power = 2:3, type = c("fitted", "regressor", "princomp"), 
    data = list())

Arguments

alternative

[dwTest] -
a character string specifying the alternative hypothesis, either "greater", "two.sided", or "less".

center

[rainTest] -
a numeric value. If center is smaller than 1 it is interpreted as percentages of data, i.e. the subset is chosen that n*fraction observations are around observation number n*center. If center is greater than 1 it is interpreted to be the index of the center of the subset. By default center is 0.5. If the Mahalanobis distance is chosen center is taken to be the mean regressor, but can be specified to be a k-dimensional vector if k is the number of regressors and should be in the range of the respective regressors.

data

an optional data frame containing the variables in the model. By default the variables are taken from the environment which lmTest and the other tests are called from.

exact

[dwTest] -
a logical flag. If set to FALSE a normal approximation will be used to compute the p value, if TRUE the "pan" algorithm is used. The default is to use "pan" if the sample size is < 100.

formula

a symbolic description for the linear model to be tested.

fraction

[rainTest] -
a numeric value, by default 0.5. The percentage of observations in the subset is determined by fraction*n if n is the number of observations in the model.

iterations

[dwTest] -
an integer specifying the number of iterations when calculating the p-value with the "pan" algorithm. By default 15.

method

the test method which should be applied.

nsim

[hmcTest] -
an integer value. Determins how many runs are used to simulate the p value, by default 1000.

order

[bgTest] -
an integer. The maximal order of serial correlation to be tested. By default 1.

order.by

[gqTest][harvTest] -
a formula. A formula with a single explanatory variable like ~ x. Then the observations in the model are ordered by the size of x. If set to NULL, the default, the observations are assumed to be ordered (e.g. a time series).
[rainTest] -
either a formula or a string. A formula with a single explanatory variable like ~ x. The observations in the model are ordered by the size of x. If set to NULL, the default, the observations are assumed to be ordered (e.g. a time series). If set to "mahalanobis" then the observations are ordered by their Mahalanobis distance of the data.

plot

[hmcTest] -
a logical flag. If TRUE the test statistic for all possible breakpoints is plotted, the default is FALSE.

point

[gqTest][hmcTest] -
a numeric value. If point is smaller than 1 it is interpreted as percentages of data, i.e. n*point is taken to be the (potential) breakpoint in the variances, if n is the number of observations in the model. If point is greater than 1 it is interpreted to be the index of the breakpoint. By default 0.5.

power

[resetTest] -
integers, by default 2:3. A vector of positive integers indicating the powers of the variables that should be included. By default it is tested for a quadratic or cubic influence of the fitted response.

simulate.p

[hmcTest] -
a logical. If TRUE, the default, a p-value will be assessed by simulation, otherwise the p-value is NA.

studentize

[bpTest] -
a logical value. If set to TRUE Koenker's studentized version of the test statistic will be used. By default set to TRUE.

tol

[dwTest] -
the tolerance value. Eigenvalues computed have to be greater than tol=1e-10 to be treated as non-zero.

type

[bgTest] -
the type of test statistic to be returned. Either "Chisq" for the Chi-squared test statistic or "F" for the F test statistic.
[resetTest] -
a string indicating whether powers of the "fitted" response, the "regressor" variables (factors are left out) or the first principal component, "princomp", of the regressor matrix should be included in the extended model.

varformula

[bpTest] -
a formula describing only the potential explanatory variables for the variance, no dependent variable needed. By default the same explanatory variables are taken as in the main regression model.

...

[regTest] -
additional arguments passed to the underlying lm test. Some of the tests can specify additional optional arguments like for alternative hypothesis, the type of test statistic to be returned, or others. All the optional arguments have default settings.

Details

bg – Breusch Godfrey Test:

Under H_0 the test statistic is asymptotically Chi-squared with degrees of freedom as given in parameter. If type is set to "F" the function returns the exact F statistic which, under H_0, follows an F distribution with degrees of freedom as given in parameter. The starting values for the lagged residuals in the supplementary regression are chosen to be 0.
[lmtest:bgtest]

bp – Breusch Pagan Test:

The Breusch–Pagan test fits a linear regression model to the residuals of a linear regression model (by default the same explanatory variables are taken as in the main regression model) and rejects if too much of the variance is explained by the additional explanatory variables. Under H_0 the test statistic of the Breusch-Pagan test follows a chi-squared distribution with parameter (the number of regressors without the constant in the model) degrees of freedom.
[lmtest:bptest]

dw – Durbin Watson Test:

The Durbin–Watson test has the null hypothesis that the autocorrelation of the disturbances is 0; it can be tested against the alternative that it is greater than, not equal to, or less than 0 respectively. This can be specified by the alternative argument. The null distribution of the Durbin-Watson test statistic is a linear combination of chi-squared distributions. The p value is computed using a Fortran version of the Applied Statistics Algorithm AS 153 by Farebrother (1980, 1984). This algorithm is called "pan" or "gradsol". For large sample sizes the algorithm might fail to compute the p value; in that case a warning is printed and an approximate p value will be given; this p value is computed using a normal approximation with mean and variance of the Durbin-Watson test statistic.
[lmtest:dwtest]

gq – Goldfeld Quandt Test:

The Goldfeld–Quandt test compares the variances of two submodels divided by a specified breakpoint and rejects if the variances differ. Under H_0 the test statistic of the Goldfeld-Quandt test follows an F distribution with the degrees of freedom as given in parameter.
[lmtest:gqtest]

harv - Harvey Collier Test:

The Harvey-Collier test performs a t-test (with parameter degrees of freedom) on the recursive residuals. If the true relationship is not linear but convex or concave the mean of the recursive residuals should differ from 0 significantly.
[lmtest:harvtest]

hmc – Harrison McCabe Test:

The Harrison–McCabe test statistic is the fraction of the residual sum of squares that relates to the fraction of the data before the breakpoint. Under H_0 the test statistic should be close to the size of this fraction, e.g. in the default case close to 0.5. The null hypothesis is reject if the statistic is too small.
[lmtest:hmctest]

rain – Rainbow Test:

The basic idea of the Rainbow test is that even if the true relationship is non-linear, a good linear fit can be achieved on a subsample in the "middle" of the data. The null hypothesis is rejected whenever the overall fit is significantly inferious to the fit of the subsample. The test statistic under H_0 follows an F distribution with parameter degrees of freedom.
[lmtest:raintest]

reset – Ramsey's RESET Test

RESET test is popular means of diagnostic for correctness of functional form. The basic assumption is that under the alternative, the model can be written by the regression y=X * beta + Z * gamma. Z is generated by taking powers either of the fitted response, the regressor variables or the first principal component of X. A standard F-Test is then applied to determin whether these additional variables have significant influence. The test statistic under H_0 follows an F distribution with parameter degrees of freedom.
[lmtest:reset]

Value

A list with class "htest" containing the following components:

statistic

the value of the test statistic.

parameter

the lag order.

p.value

the p-value of the test.

method

a character string indicating what type of test was performed.

data.name

a character string giving the name of the data.

alternative

a character string describing the alternative hypothesis.

Note

The underlying lmtest package comes wit a lot of helpful examples. We highly recommend to install the lmtest package and to study the examples given therein.

Author(s)

Achim Zeileis and Torsten Hothorn for the lmtest package,
Diethelm Wuertz for the Rmetrics R-port.

References

Breusch, T.S. (1979); Testing for Autocorrelation in Dynamic Linear Models, Australian Economic Papers 17, 334–355.

Breusch T.S. and Pagan A.R. (1979); A Simple Test for Heteroscedasticity and Random Coefficient Variation, Econometrica 47, 1287–1294

Durbin J. and Watson G.S. (1950); Testing for Serial Correlation in Least Squares Regression I, Biometrika 37, 409–428.

Durbin J. and Watson G.S. (1951); Testing for Serial Correlation in Least Squares Regression II, Biometrika 38, 159–178.

Durbin J. and Watson G.S. (1971); Testing for Serial Correlation in Least Squares Regression III, Biometrika 58, 1–19.

Farebrother R.W. (1980); Pan's Procedure for the Tail Probabilities of the Durbin-Watson Statistic, Applied Statistics 29, 224–227.

Farebrother R.W. (1984); The Distribution of a Linear Combination of chi^2 Random Variables, Applied Statistics 33, 366–369.

Godfrey, L.G. (1978); Testing Against General Autoregressive and Moving Average Error Models when the Regressors Include Lagged Dependent Variables, Econometrica 46, 1293–1302.

Goldfeld S.M. and Quandt R.E. (1965); Some Tests for Homoskedasticity Journal of the American Statistical Association 60, 539–547.

Harrison M.J. and McCabe B.P.M. (1979); A Test for Heteroscedasticity based on Ordinary Least Squares Residuals Journal of the American Statistical Association 74, 494–499.

Harvey A. and Collier P. (1977); Testing for Functional Misspecification in Regression Analysis, Journal of Econometrics 6, 103–119.

Johnston, J. (1984); Econometric Methods, Third Edition, McGraw Hill Inc.

Kraemer W. and Sonnberger H. (1986); The Linear Regression Model under Test, Heidelberg: Physica.

Racine J. and Hyndman R. (2002); Using R To Teach Econometrics, Journal of Applied Econometrics 17, 175–189.

Ramsey J.B. (1969); Tests for Specification Error in Classical Linear Least Squares Regression Analysis, Journal of the Royal Statistical Society, Series B 31, 350–371.

Utts J.M. (1982); The Rainbow Test for Lack of Fit in Regression, Communications in Statistics - Theory and Methods 11, 1801–1815.

Examples

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## bg | dw -
   # Generate a Stationary and an AR(1) Series:
   x = rep(c(1, -1), 50)
   y1 = 1 + x + rnorm(100)
   # Perform Breusch-Godfrey Test for 1st order serial correlation:
   lmTest(y1 ~ x, "bg")
   # ... or for fourth order serial correlation:
   lmTest(y1 ~ x, "bg", order = 4)    
   # Compare with Durbin-Watson Test Results:
   lmTest(y1 ~ x, "dw")
   y2 = filter(y1, 0.5, method = "recursive")
   lmTest(y2 ~ x, "bg") 
   
## bp -
   # Generate a Regressor:
   x = rep(c(-1, 1), 50)
   # Generate heteroskedastic and homoskedastic Disturbances
   err1 = rnorm(100, sd = rep(c(1, 2), 50))
   err2 = rnorm(100)
   # Generate a Linear Relationship:
   y1 = 1 + x + err1
   y2 = 1 + x + err2
   # Perform Breusch-Pagan Test
   bp = lmTest(y1 ~ x, "bp")
   bp
   # Calculate Critical Value for 0.05 Level
   qchisq(0.95, bp$parameter)
   lmTest(y2 ~ x, "bp")
   
## dw -
   # Generate two AR(1) Error Terms 
   # with parameter rho = 0 (white noise) 
   # and rho = 0.9 respectively
   err1 = rnorm(100)
   # Generate Regressor and Dependent Variable
   x = rep(c(-1,1), 50)
   y1 = 1 + x + err1
   # Perform Durbin-Watson Test:
   lmTest(y1 ~ x, "dw")
   err2 = filter(err1, 0.9, method = "recursive")
   y2 = 1 + x + err2
   lmTest(y2 ~ x, "dw")
   
## gq -
   # Generate a Regressor:
   x = rep(c(-1, 1), 50)
   # Generate Heteroskedastic and Homoskedastic Disturbances:
   err1 = c(rnorm(50, sd = 1), rnorm(50, sd = 2))
   err2 = rnorm(100)
   # Generate a Linear Relationship:
   y1 = 1 + x + err1
   y2 = 1 + x + err2
   # Perform Goldfeld-Quandt Test:
   lmTest(y1 ~ x, "gq")
   lmTest(y2 ~ x, "gq")
   
## harv -
   # Generate a Regressor and Dependent Variable:
   x = 1:50
   y1 = 1 + x + rnorm(50)
   y2 = y1 + 0.3*x^2
   # Perform Harvey-Collier Test:
   harv = lmTest(y1 ~ x, "harv")
   harv
   # Calculate Critical Value vor 0.05 level:
   qt(0.95, harv$parameter)
   lmTest(y2 ~ x, "harv")
   
## hmc -
   # Generate a Regressor:
   x = rep(c(-1, 1), 50)
   # Generate Heteroskedastic and Homoskedastic Disturbances:
   err1 = c(rnorm(50, sd = 1), rnorm(50, sd = 2))
   err2 = rnorm(100)
   # Generate a Linear Relationship:
   y1 = 1 + x + err1
   y2 = 1 + x + err2
   # Perform Harrison-McCabe Test:
   lmTest(y1 ~ x, "hmc")
   lmTest(y2 ~ x, "hmc")
   
## rain -
   # Generate Series:
   x = c(1:30)
   y = x^2 + rnorm(30, 0, 2)
   # Perform rainbow Test
   rain = lmTest(y ~ x, "rain")
   rain
   # Compute Critical Value:
   qf(0.95, rain$parameter[1], rain$parameter[2]) 
   
## reset -
   # Generate Series:
   x = c(1:30)
   y1 = 1 + x + x^2 + rnorm(30)
   y2 = 1 + x + rnorm(30)
   # Perform RESET Test:
   lmTest(y1 ~ x , "reset", power = 2, type = "regressor")
   lmTest(y2 ~ x , "reset", power = 2, type = "regressor")