lmtest/0000755000175400001440000000000013462005102011746 5ustar zeileisuserslmtest/inst/0000755000175400001440000000000013462003317012731 5ustar zeileisuserslmtest/inst/CITATION0000644000175400001440000000126513047554551014104 0ustar zeileisuserscitHeader("To cite lmtest in publications use:") citEntry(entry="Article", title = "Diagnostic Checking in Regression Relationships", author = personList(as.person("Achim Zeileis"), as.person("Torsten Hothorn")), journal = "R News", year = "2002", volume = "2", number = "3", pages = "7--10", url = "https://CRAN.R-project.org/doc/Rnews/", textVersion = paste("Achim Zeileis, Torsten Hothorn (2002).", "Diagnostic Checking in Regression Relationships.", "R News 2(3), 7-10.", "URL https://CRAN.R-project.org/doc/Rnews/") ) lmtest/inst/doc/0000755000175400001440000000000013462003317013476 5ustar zeileisuserslmtest/inst/doc/lmtest-intro.Rnw0000644000175400001440000002767212623340541016646 0ustar zeileisusers\documentclass[a4paper]{article} \usepackage{a4,graphicx,color} \usepackage[authoryear,round,longnamesfirst]{natbib} \usepackage{hyperref} \definecolor{Red}{rgb}{0.7,0,0} \definecolor{Blue}{rgb}{0,0,0.8} \oddsidemargin 0.15 in \evensidemargin 0.35 in \marginparwidth 1 in \oddsidemargin 0.25 in \evensidemargin 0.25 in \marginparwidth 0.75 in \textwidth 5.875 in \begin{document} \SweaveOpts{engine=R,eps=FALSE} %\VignetteIndexEntry{Diagnostic Checking in Regression Relationships} %\VignetteDepends{lmtest, strucchange} %\VignetteKeywords{diagnostic checking, structural change, autocorrelation, heteroskedasticity} %\VignettePackage{lmtest} <>= library(lmtest) options(SweaveHooks=list(twofig=function() {par(mfrow=c(1,2))}, twofig2=function() {par(mfrow=c(2,1))}, onefig=function() {par(mfrow=c(1,1))})) @ \title{Diagnostic Checking in Regression Relationships} \author{\hfill Achim Zeileis$^\dag$ \hfill Torsten Hothorn$^\ddag$ \hfill \hfill \\ \dag {\it \small Institut f\"ur Statistik \& Wahrscheinlichkeitstheorie, Technische Universit\"at Wien, Austria}\\ \ddag {\it \small Institut f\"ur Medizininformatik, Biometrie und Epidemiologie, Universit\"at Erlangen-N\"urnberg, Germany} } \date{} \maketitle \section{Introduction} The classical linear regression model \begin{equation} \label{eq:model} y_i \quad = \quad x_i^\top \beta + u_i \qquad (i = 1, \dots, n) \end{equation} is still one of the most popular tools for data analysis despite (or due to) its simple structure. Although it is appropriate in many situations, there are many pitfalls that might affect the quality of conclusions drawn from fitted models or might even lead to uninterpretable results. Some of these pitfalls that are considered especially important in applied econometrics are heteroskedasticity or serial correlation of the error terms, structural changes in the regression coefficients, nonlinearities, functional misspecification or omitted variables. Therefore, a rich variety of diagnostic tests for these situations have been developed in the econometrics community, a collection of which has been implemented in the packages \texttt{lmtest} and \texttt{strucchange} covering the problems mentioned above. These diagnostic tests are not only useful in econometrics but also in many other fields where linear regression is used, which we will demonstrate with an application from biostatistics. As \cite{lmtest:Breiman:2001} argues it is important to assess the goodness-of-fit of data models, in particular not only using omnibus tests but tests designed for a certain direction of the alternative. These diagnostic checks do not have to be seen as pure significance procedures but also as an explorative tool to extract information about the structure of the data, especially in connection with residual plots or other diagnostic plots. As \cite{lmtest:Brown+Durbin+Evans:1975} argue for the recursive CUSUM test, these procedures can ``be regarded as yardsticks for the interpretation of data rather than leading to hard and fast decisions.'' Moreover, we will always be able to reject the null-hypothesis provided we have enough data at hand. The question is not whether the model is wrong (it always is!) but if the irregularities are serious. The package \texttt{strucchange} implements a variety of procedures related to structural change of the regression coefficients and was already introduced in \textsf{R} news by \cite{lmtest:Zeileis:2001} and described in more detail in \cite{lmtest:Zeileis+Leisch+Hornik:2002}. Therefore, we will focus on the package \texttt{lmtest} in the following. Most of the tests and the datasets contained in the package are taken from the book of \cite{lmtest:Kraemer+Sonnberger:1986}, which originally inspired us to write the package. Compared to the book, we implemented later versions of some tests and modern flexible interfaces for the procedures. Most of the tests are based on the OLS residuals of a linear model, which is specified by a formula argument. Instead of a formula a fitted model of class \verb/"lm"/ can also be supplied, which should work if the data are either contained in the object or still present in the workspace---however this is not encouraged. The full references for the tests can be found on the help pages of the respective function. We present applications of the tests contained in \texttt{lmtest} to two different data sets: the first is a macroeconomic time series from the U.S. analysed by \cite{lmtest:Stock+Watson:1996} and the second is data from a study on measurments of fetal mandible length discussed by \cite{lmtest:Royston+Altman:1994}. \section{U.S. macroeconomic data} \label{sec:macro} \cite{lmtest:Stock+Watson:1996} investigate the stability of 76 monthly macroeconomic time series from 1959 to 1993, of which we choose the department of commerce commodity price index time series \texttt{jocci} to illustrate the tests for heteroskedasticity and serial correlation. The data is treated with the same methodology as all other series considered by \cite{lmtest:Stock+Watson:1996}: they were transformed suitably (here by log first differences) and then an AR(6) model was fitted and analysed. The transformed series is denoted \texttt{dy} and is depicted together with a residual plot of the AR(6) model in Figure~\ref{macro-jocci}. \setkeys{Gin}{width=0.8\textwidth} \begin{figure}[htbp] \begin{center} <>= data(jocci) plot(jocci[,"dy"], ylab = "jocci (log first differences)") ar6.model <- dy ~ dy1 + dy2 + dy3 + dy4 + dy5 +dy6 jocci.fm <- lm(ar6.model, data = jocci) plot(time(jocci), residuals(jocci.fm), xlab = "Time", ylab = "AR(6) residuals") @ \caption{\label{macro-jocci} The jocci series and AR(6) residual plot} \end{center} \end{figure} Not surprisingly, an autoregressive model is necessary as the series itself contains serial correlation, which can be shown by the Durbin-Watson test <>= data(jocci) dwtest(dy ~ 1, data = jocci) @ or the Breusch-Godfrey test which also leads to a highly significant result. In the AR(6) model given by <>= ar6.model <- dy ~ dy1 + dy2 + dy3 + dy4 + dy5 +dy6 @ where the variables on the right hand side denote the lagged variables, there is no remaining serial correlation in the residuals: <>= bgtest(ar6.model, data = jocci) @ The Durbin-Watson test is biased in dynamic models and should therefore not be applied. The residual plot suggests that the variance of the error component increases over time, which is emphasized by all three tests for heteroskedasticity implemented in \texttt{lmtest}: the Breusch-Pagan test fits a linear regression model to the residuals and rejects if too much of the variance is explained by the auxiliary explanatory variables, which are here the squared lagged values: <>= var.model <- ~ I(dy1^2) + I(dy2^2) + I(dy3^2) + I(dy4^2) + I(dy5^2) + I(dy6^2) bptest(ar6.model, var.model, data = jocci) @ The Goldfeld-Quandt test \verb/gqtest()/ and the Harrison-McCabe test \verb/hmctest()/ also give highly significant $p$ values. Whereas the Breusch-Pagan test does not assume a particular timing of the change of variance, the Goldfeld-Quandt and the Harrison-McCabe tests suffer from the same problem as the Chow test for a change of the regression coefficients (available in package \texttt{strucchange}): the breakpoint has to be known in advance. By default it is taken to be after 50\% of the observations, which leads to a significant result for the present series. \section{The mandible data} \label{sec:mandible} \cite{lmtest:Royston+Altman:1994} discuss a linear regression model for data taken from a study of fetal mandible length by \cite{lmtest:Chitty+Campbell+Altman:1993}. The data comprises measurements of mandible \texttt{length} (in mm) and gestational \texttt{age} (in weeks) in 158 fetuses. The data (after log transformation) is depicted in Figure~\ref{mandible-data} together with the fitted values of a linear model \verb/length ~ age/ and a quadratic model \verb/length ~ age + I(age^2)/. %% just to remember: the Royston & Altman model is %% fm.ra <- lm(log(length) ~ I(1/age), data = mandible) \setkeys{Gin}{width=0.6\textwidth} \begin{figure}[htbp] \begin{center} <>= data(Mandible) mandible <- log(Mandible) attach(mandible) plot(mandible) fm <- lm(length ~ age) fm2 <- lm(length ~ age + I(age^2)) lines(age, fitted(fm), col = 2) lines(age, fitted(fm2), col = 4) @ \caption{\label{mandible-data} The mandible data} \end{center} \end{figure} Although by merely visually inspecting the raw data or the residual plots in Figure~\ref{mandible-res} a quadratic model seems to be more appropriate, we will first fit a linear model for illustrating some tests for nonlinearity and misspecified functional form. \setkeys{Gin}{width=0.8\textwidth} \begin{figure}[htbp] \begin{center} <>= plot(age, residuals(fm), ylab = "residuals (linear model)") plot(age, residuals(fm2), ylab = "residuals (quadratic model)") detach(mandible) @ \caption{\label{mandible-res} Residual plots for mandible models} \end{center} \end{figure} The suitable tests in \texttt{lmtest} are the Harvey-Collier test, which is essentially a $t$ test of the recursive residuals (standardized one step prediction errors), and the Rainbow test. Both try to detect nonlinearities when the data is ordered with respect to a specific variable. <>= data(Mandible) mandible <- log(Mandible) harvtest(length ~ age, order.by = ~ age, data = mandible) raintest(length ~ age, order.by = ~ age, data = mandible) @ Both lead to highly significant results, suggesting that the model is not linear in \texttt{age}. Another appropriate procedure is the RESET test, which tests whether some auxiliary variables improve the fit significantly. By default the second and third powers of the fitted values are chosen: <>= resettest(length ~ age, data = mandible) @ In our situation it would also be natural to consider powers of the regressor \texttt{age} as auxiliary variables <>= resettest(length ~ age, power = 2, type = "regressor", data = mandible) @ which also gives a highly significant $p$ value (higher powers do not have a significant influence). These results correspond to the better fit of the quadratic model which can both be seen in Figure~\ref{mandible-data} and \ref{mandible-res}. Although its residual plot does not look too suspicious several tests are able to reveal irregularities in this model as well. The Breusch-Pagan tests gives a $p$ value of \Sexpr{round(bptest(length ~ age + I(age^2), data = mandible)$p.value, digits = 3)} and the Rainbow test gives <>= raintest(length ~ age + I(age^2), order.by = ~ age, data = mandible) @ <>= if(require(strucchange)) { supF.pval <- round(sctest(length ~ age + I(age^2), data = mandible, to = 0.9, type = "supF")$p.value, digits = 3) } else { # warning("`strucchange' not available: p value set to NA") supF.pval <- NA } @ and finally an sup$F$ test from the \texttt{strucchange} package would also reject the null hypothesis of stability at 10\% level ($p = \Sexpr{supF.pval}$) in favour of a breakpoint after about 90\% of the observations. All three tests probably reflect that there is more variability in the edges (especially the right one) than in the middle which the model does not describe sufficiently. \section{Conclusions} \label{sec:conclusions} We illustrated the usefulness of a collection of diagnostic tests for various situations of deviations from the assumptions of the classical linear regression model. We chose two fairly simple data sets---an econometric and a biometric application---to demonstrate how the tests work, but they are also particularly helpful to detect irregularities in regressions with a larger number of regressors. \bibliography{lmtest} \bibliographystyle{abbrvnat} \end{document} lmtest/inst/doc/lmtest-intro.pdf0000644000175400001440000024724213462003320016637 0ustar zeileisusers%PDF-1.5 % 1 0 obj << /Type /ObjStm /Length 3297 /Filter /FlateDecode /N 63 /First 503 >> stream xZks۶~L /w:G؉"?N>0mWJI %qݑivgH&b6dI1Ôe:,dF*1X,,, % B*X`GP!?dA!3bRZQ^J&/%S$*&I͔(V!ޣ]`$ cfFajTABJQĔa6F;eFU˜*X$BE2Lt"q>h D#4er6F@`H j.Bh.rdJD( T"t  . 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data(Mandible) ## mandible <- log(Mandible) ## harvtest(length ~ age, order.by = ~ age, data = mandible) ## raintest(length ~ age, order.by = ~ age, data = mandible) ################################################### ### code chunk number 10: mandible-tests2 ################################################### resettest(length ~ age, data = mandible) ################################################### ### code chunk number 11: mandible-tests3 ################################################### resettest(length ~ age, power = 2, type = "regressor", data = mandible) ################################################### ### code chunk number 12: mandible-tests4 ################################################### raintest(length ~ age + I(age^2), order.by = ~ age, data = mandible) ################################################### ### code chunk number 13: mandible-supF ################################################### if(require(strucchange)) { supF.pval <- round(sctest(length ~ age + I(age^2), data = mandible, to = 0.9, type = "supF")$p.value, digits = 3) } else { # warning("`strucchange' not available: p value set to NA") supF.pval <- NA } lmtest/tests/0000755000175400001440000000000012220001273013104 5ustar zeileisuserslmtest/tests/Examples/0000755000175400001440000000000012220001273014662 5ustar zeileisuserslmtest/tests/Examples/lmtest-Ex.Rout.save0000644000175400001440000014740313461775572020421 0ustar zeileisusers R version 3.6.0 (2019-04-26) -- "Planting of a Tree" Copyright (C) 2019 The R Foundation for Statistical Computing Platform: x86_64-pc-linux-gnu (64-bit) R is free software and comes with ABSOLUTELY NO WARRANTY. You are welcome to redistribute it under certain conditions. Type 'license()' or 'licence()' for distribution details. Natural language support but running in an English locale R is a collaborative project with many contributors. Type 'contributors()' for more information and 'citation()' on how to cite R or R packages in publications. Type 'demo()' for some demos, 'help()' for on-line help, or 'help.start()' for an HTML browser interface to help. Type 'q()' to quit R. > pkgname <- "lmtest" > source(file.path(R.home("share"), "R", "examples-header.R")) > options(warn = 1) > library('lmtest') Loading required package: zoo Attaching package: ‘zoo’ The following objects are masked from ‘package:base’: as.Date, as.Date.numeric > > base::assign(".oldSearch", base::search(), pos = 'CheckExEnv') > base::assign(".old_wd", base::getwd(), pos = 'CheckExEnv') > cleanEx() > nameEx("ChickEgg") > ### * ChickEgg > > flush(stderr()); flush(stdout()) > > ### Name: ChickEgg > ### Title: Chickens, Eggs, and Causality > ### Aliases: ChickEgg > ### Keywords: datasets > > ### ** Examples > > ## Which came first: the chicken or the egg? > data(ChickEgg) > ## chickens granger-cause eggs? > grangertest(egg ~ chicken, order = 3, data = ChickEgg) Granger causality test Model 1: egg ~ Lags(egg, 1:3) + Lags(chicken, 1:3) Model 2: egg ~ Lags(egg, 1:3) Res.Df Df F Pr(>F) 1 44 2 47 -3 0.5916 0.6238 > ## eggs granger-cause chickens? > grangertest(chicken ~ egg, order = 3, data = ChickEgg) Granger causality test Model 1: chicken ~ Lags(chicken, 1:3) + Lags(egg, 1:3) Model 2: chicken ~ Lags(chicken, 1:3) Res.Df Df F Pr(>F) 1 44 2 47 -3 5.405 0.002966 ** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 > > ## To perform the same tests `by hand', you can use dynlm() and waldtest(): > if(require(dynlm)) { + ## chickens granger-cause eggs? + em <- dynlm(egg ~ L(egg, 1) + L(egg, 2) + L(egg, 3), data = ChickEgg) + em2 <- update(em, . ~ . + L(chicken, 1) + L(chicken, 2) + L(chicken, 3)) + waldtest(em, em2) + + ## eggs granger-cause chickens? + cm <- dynlm(chicken ~ L(chicken, 1) + L(chicken, 2) + L(chicken, 3), data = ChickEgg) + cm2 <- update(cm, . ~ . + L(egg, 1) + L(egg, 2) + L(egg, 3)) + waldtest(cm, cm2) + } Loading required package: dynlm Wald test Model 1: chicken ~ L(chicken, 1) + L(chicken, 2) + L(chicken, 3) Model 2: chicken ~ L(chicken, 1) + L(chicken, 2) + L(chicken, 3) + L(egg, 1) + L(egg, 2) + L(egg, 3) Res.Df Df F Pr(>F) 1 47 2 44 3 5.405 0.002966 ** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 > > > > cleanEx() detaching ‘package:dynlm’ > nameEx("Mandible") > ### * Mandible > > flush(stderr()); flush(stdout()) > > ### Name: Mandible > ### Title: Mandible Data > ### Aliases: Mandible > ### Keywords: datasets > > ### ** Examples > > data(Mandible) > lm(length ~ age, data=Mandible, subset=(age <= 28)) Call: lm(formula = length ~ age, data = Mandible, subset = (age <= 28)) Coefficients: (Intercept) age -11.953 1.773 > > > > > cleanEx() > nameEx("USDistLag") > ### * USDistLag > > flush(stderr()); flush(stdout()) > > ### Name: USDistLag > ### Title: US Macroeconomic Data > ### Aliases: USDistLag > ### Keywords: datasets > > ### ** Examples > > ## Willam H. Greene, Econometric Analysis, 2nd Ed. > ## Chapter 7 > ## load data set, p. 221, Table 7.7 > data(USDistLag) > > ## fit distributed lag model, p.221, Example 7.8 > usdl <- na.contiguous(cbind(USDistLag, lag(USDistLag, k = -1))) > colnames(usdl) <- c("con", "gnp", "con1", "gnp1") > > ## C(t) = b0 + b1*Y(t) + b2*C(t-1) + u > fm <- lm(con ~ gnp + con1, data = usdl) > summary(fm) Call: lm(formula = con ~ gnp + con1, data = usdl) Residuals: Min 1Q Median 3Q Max -11.6127 -3.8405 -0.8536 4.3857 12.6428 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -7.69575 11.44429 -0.672 0.510891 gnp 0.40015 0.06272 6.380 9.12e-06 *** con1 0.38073 0.09479 4.017 0.000996 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 6.837 on 16 degrees of freedom Multiple R-squared: 0.9976, Adjusted R-squared: 0.9973 F-statistic: 3302 on 2 and 16 DF, p-value: < 2.2e-16 > vcov(fm) (Intercept) gnp con1 (Intercept) 130.9718602 -0.438682677 0.549587054 gnp -0.4386827 0.003933498 -0.005896092 con1 0.5495871 -0.005896092 0.008984392 > > > > cleanEx() > nameEx("bgtest") > ### * bgtest > > flush(stderr()); flush(stdout()) > > ### Name: bgtest > ### Title: Breusch-Godfrey Test > ### Aliases: bgtest vcov.bgtest df.residual.bgtest > ### Keywords: htest > > ### ** Examples > > > ## Generate a stationary and an AR(1) series > x <- rep(c(1, -1), 50) > > y1 <- 1 + x + rnorm(100) > > ## Perform Breusch-Godfrey test for first-order serial correlation: > bgtest(y1 ~ x) Breusch-Godfrey test for serial correlation of order up to 1 data: y1 ~ x LM test = 0.0036887, df = 1, p-value = 0.9516 > ## or for fourth-order serial correlation > bgtest(y1 ~ x, order = 4) Breusch-Godfrey test for serial correlation of order up to 4 data: y1 ~ x LM test = 3.0822, df = 4, p-value = 0.5442 > ## Compare with Durbin-Watson test results: > dwtest(y1 ~ x) Durbin-Watson test data: y1 ~ x DW = 1.9762, p-value = 0.4924 alternative hypothesis: true autocorrelation is greater than 0 > > y2 <- filter(y1, 0.5, method = "recursive") > bgtest(y2 ~ x) Breusch-Godfrey test for serial correlation of order up to 1 data: y2 ~ x LM test = 19.907, df = 1, p-value = 8.128e-06 > bg4 <- bgtest(y2 ~ x, order = 4) > bg4 Breusch-Godfrey test for serial correlation of order up to 4 data: y2 ~ x LM test = 23.687, df = 4, p-value = 9.228e-05 > coeftest(bg4) z test of coefficients: Estimate Std. Error z value Pr(>|z|) (Intercept) 0.0049353 0.0905891 0.0545 0.9566 x -0.0057380 0.0904290 -0.0635 0.9494 lag(resid)_1 0.4619715 0.1027742 4.4950 6.957e-06 *** lag(resid)_2 -0.0401917 0.1136254 -0.3537 0.7235 lag(resid)_3 -0.1030000 0.1136071 -0.9066 0.3646 lag(resid)_4 -0.1184116 0.1049020 -1.1288 0.2590 --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 > > > > cleanEx() > nameEx("bondyield") > ### * bondyield > > flush(stderr()); flush(stdout()) > > ### Name: bondyield > ### Title: Bond Yield > ### Aliases: bondyield > ### Keywords: datasets > > ### ** Examples > > data(bondyield) > > ## page 134, fit Cook-Hendershott OLS model and Yawitz-Marshall OLS model > ## third and last line in Table 6.5 > > modelCH <- RAARUS ~ MOOD + EPI + EXP + RUS > lm(modelCH, data=bondyield) Call: lm(formula = modelCH, data = bondyield) Coefficients: (Intercept) MOOD EPI EXP RUS 0.525913 -0.001402 -0.005623 -0.157801 0.137761 > dwtest(modelCH, data=bondyield) Durbin-Watson test data: modelCH DW = 0.84484, p-value = 2.888e-08 alternative hypothesis: true autocorrelation is greater than 0 > ## wrong sign of RUS coefficient > > modelYM <- RAARUS ~ MOOD + Y + K > lm(modelYM, data=bondyield) Call: lm(formula = modelYM, data = bondyield) Coefficients: (Intercept) MOOD Y K -5.06477 -0.01313 0.04899 1.70792 > dwtest(modelYM, data=bondyield) Durbin-Watson test data: modelYM DW = 1.4387, p-value = 0.004938 alternative hypothesis: true autocorrelation is greater than 0 > ## coefficient of Y and K differ by factor 100 > > > ## page 135, fit test statistics in Table 6.6 b) > ################################################ > > ## Chow 1971(1) > if(require(strucchange, quietly = TRUE)) { + sctest(modelCH, point=c(1971,1), data=bondyield, type="Chow") } Chow test data: modelCH F = 13.775, p-value = 1.838e-08 > > ## Breusch-Pagan > bptest(modelCH, data=bondyield, studentize=FALSE) Breusch-Pagan test data: modelCH BP = 2.0667, df = 4, p-value = 0.7235 > bptest(modelCH, data=bondyield) studentized Breusch-Pagan test data: modelCH BP = 2.9784, df = 4, p-value = 0.5614 > > ## Fluctuation test > if(require(strucchange, quietly = TRUE)) { + sctest(modelCH, type="fluctuation", data=bondyield, rescale=FALSE)} RE test (recursive estimates test) data: modelCH RE = 17.758, p-value < 2.2e-16 > > ## RESET > reset(modelCH, data=bondyield) RESET test data: modelCH RESET = 5.349, df1 = 2, df2 = 53, p-value = 0.007655 > reset(modelCH, power=2, type="regressor", data=bondyield) RESET test data: modelCH RESET = 4.7726, df1 = 4, df2 = 51, p-value = 0.002393 > reset(modelCH, type="princomp", data=bondyield) RESET test data: modelCH RESET = 9.3371, df1 = 2, df2 = 53, p-value = 0.0003359 > > ## Harvey-Collier > harvtest(modelCH, order.by= ~ MOOD, data=bondyield) Harvey-Collier test data: modelCH HC = 2.3951, df = 54, p-value = 0.02012 > harvtest(modelCH, order.by= ~ EPI, data=bondyield) Harvey-Collier test data: modelCH HC = 0.17075, df = 54, p-value = 0.8651 > harvtest(modelCH, order.by= ~ EXP, data=bondyield) Harvey-Collier test data: modelCH HC = 0.87128, df = 54, p-value = 0.3875 > harvtest(modelCH, order.by= ~ RUS, data=bondyield) Harvey-Collier test data: modelCH HC = 1.5877, df = 54, p-value = 0.1182 > > ## Rainbow > raintest(modelCH, order.by = "mahalanobis", data=bondyield) Rainbow test data: modelCH Rain = 1.702, df1 = 30, df2 = 25, p-value = 0.08919 > > > ## page 136, fit test statistics in Table 6.6 d) > ################################################ > > ## Chow 1966(1) > if(require(strucchange, quietly = TRUE)) { + sctest(modelYM, point=c(1965,4), data=bondyield, type="Chow") } Chow test data: modelYM F = 2.8266, p-value = 0.03388 > > ## Fluctuation test > if(require(strucchange, quietly = TRUE)) { + sctest(modelYM, type="fluctuation", data=bondyield, rescale=FALSE) } RE test (recursive estimates test) data: modelYM RE = 19.733, p-value < 2.2e-16 > > ## RESET > reset(modelYM, data=bondyield) RESET test data: modelYM RESET = 2.1436, df1 = 2, df2 = 54, p-value = 0.1271 > reset(modelYM, power=2, type="regressor", data=bondyield) RESET test data: modelYM RESET = 4.1716, df1 = 3, df2 = 53, p-value = 0.01003 > reset(modelYM, type="princomp", data=bondyield) RESET test data: modelYM RESET = 1.9931, df1 = 2, df2 = 54, p-value = 0.1462 > > ## Harvey-Collier > harvtest(modelYM, order.by= ~ MOOD, data=bondyield) Harvey-Collier test data: modelYM HC = 0.24869, df = 55, p-value = 0.8045 > harvtest(modelYM, order.by= ~ Y, data=bondyield) Harvey-Collier test data: modelYM HC = 2.1227, df = 55, p-value = 0.0383 > harvtest(modelYM, order.by= ~ K, data=bondyield) Harvey-Collier test data: modelYM HC = 0.026056, df = 55, p-value = 0.9793 > > ## Rainbow > raintest(modelYM, order.by = "mahalanobis", data=bondyield) Rainbow test data: modelYM Rain = 1.2978, df1 = 30, df2 = 26, p-value = 0.2515 > > > > cleanEx() detaching ‘package:strucchange’, ‘package:sandwich’ > nameEx("bptest") > ### * bptest > > flush(stderr()); flush(stdout()) > > ### Name: bptest > ### Title: Breusch-Pagan Test > ### Aliases: bptest > ### Keywords: htest > > ### ** Examples > > ## 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 > bptest(y1 ~ x) studentized Breusch-Pagan test data: y1 ~ x BP = 11.099, df = 1, p-value = 0.0008635 > bptest(y2 ~ x) studentized Breusch-Pagan test data: y2 ~ x BP = 0.11809, df = 1, p-value = 0.7311 > > > > cleanEx() > nameEx("coeftest") > ### * coeftest > > flush(stderr()); flush(stdout()) > > ### Name: coeftest > ### Title: Inference for Estimated Coefficients > ### Aliases: coeftest coefci coeftest.default coeftest.survreg coeftest.glm > ### coeftest.mlm coeftest.breakpointsfull print.coeftest confint.coeftest > ### df.residual.coeftest coefci.default coefci.survreg coefci.glm > ### coefci.mlm > ### Keywords: htest > > ### ** Examples > > ## load data and fit model > data("Mandible", package = "lmtest") > fm <- lm(length ~ age, data = Mandible, subset=(age <= 28)) > > ## the following commands lead to the same tests: > summary(fm) Call: lm(formula = length ~ age, data = Mandible, subset = (age <= 28)) Residuals: Min 1Q Median 3Q Max -9.2013 -1.6592 -0.1217 1.3420 6.4351 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -11.9534 0.9762 -12.24 <2e-16 *** age 1.7727 0.0477 37.16 <2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 2.373 on 156 degrees of freedom Multiple R-squared: 0.8985, Adjusted R-squared: 0.8978 F-statistic: 1381 on 1 and 156 DF, p-value: < 2.2e-16 > (ct <- coeftest(fm)) t test of coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -11.953366 0.976227 -12.245 < 2.2e-16 *** age 1.772730 0.047704 37.161 < 2.2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 > > ## a z test (instead of a t test) can be performed by > coeftest(fm, df = Inf) z test of coefficients: Estimate Std. Error z value Pr(>|z|) (Intercept) -11.953366 0.976227 -12.245 < 2.2e-16 *** age 1.772730 0.047704 37.161 < 2.2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 > > ## corresponding confidence intervals > confint(ct) 2.5 % 97.5 % (Intercept) -13.88169 -10.02504 age 1.67850 1.86696 > coefci(fm) 2.5 % 97.5 % (Intercept) -13.88169 -10.02504 age 1.67850 1.86696 > ## which in this simple case is equivalent to > confint(fm) 2.5 % 97.5 % (Intercept) -13.88169 -10.02504 age 1.67850 1.86696 > > if(require("sandwich")) { + ## a different covariance matrix can be also used: + (ct <- coeftest(fm, df = Inf, vcov = vcovHC)) + + ## the corresponding confidence interval can be computed either as + confint(ct) + ## or based on the original model + coefci(fm, df = Inf, vcov = vcovHC) + + ## note that the degrees of freedom _actually used_ can be extracted + df.residual(ct) + ## which differ here from + df.residual(fm) + + ## vcov can also be supplied as a function with additional arguments + coeftest(fm, df = Inf, vcov = vcovHC, type = "HC0") + ## or as a matrix + coeftest(fm, df = Inf, vcov = vcovHC(fm, type = "HC0")) + } Loading required package: sandwich z test of coefficients: Estimate Std. Error z value Pr(>|z|) (Intercept) -11.953366 1.009817 -11.837 < 2.2e-16 *** age 1.772730 0.054343 32.621 < 2.2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 > > > > cleanEx() detaching ‘package:sandwich’ > nameEx("coxtest") > ### * coxtest > > flush(stderr()); flush(stdout()) > > ### Name: coxtest > ### Title: Cox Test for Comparing Non-Nested Models > ### Aliases: coxtest > ### Keywords: htest > > ### ** Examples > > ## Fit two competing, non-nested models for aggregate > ## consumption, as in Greene (1993), Examples 7.11 and 7.12 > > ## load data and compute lags > data(USDistLag) > usdl <- na.contiguous(cbind(USDistLag, lag(USDistLag, k = -1))) > colnames(usdl) <- c("con", "gnp", "con1", "gnp1") > > ## C(t) = a0 + a1*Y(t) + a2*C(t-1) + u > fm1 <- lm(con ~ gnp + con1, data = usdl) > > ## C(t) = b0 + b1*Y(t) + b2*Y(t-1) + v > fm2 <- lm(con ~ gnp + gnp1, data = usdl) > > ## Cox test in both directions: > coxtest(fm1, fm2) Cox test Model 1: con ~ gnp + con1 Model 2: con ~ gnp + gnp1 Estimate Std. Error z value Pr(>|z|) fitted(M1) ~ M2 2.8543 1.29978 2.1960 0.02809 * fitted(M2) ~ M1 -4.4003 0.78961 -5.5727 2.508e-08 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 > > ## ...and do the same for jtest() and encomptest(). > ## Notice that in this particular case they are coincident. > jtest(fm1, fm2) J test Model 1: con ~ gnp + con1 Model 2: con ~ gnp + gnp1 Estimate Std. Error t value Pr(>|t|) M1 + fitted(M2) -2.7041 0.76273 -3.5454 0.0029371 ** M2 + fitted(M1) 2.7436 0.52710 5.2051 0.0001067 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 > encomptest(fm1, fm2) Encompassing test Model 1: con ~ gnp + con1 Model 2: con ~ gnp + gnp1 Model E: con ~ gnp + con1 + gnp1 Res.Df Df F Pr(>F) M1 vs. ME 15 -1 12.569 0.0029371 ** M2 vs. ME 15 -1 27.093 0.0001067 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 > > > > cleanEx() > nameEx("currencysubstitution") > ### * currencysubstitution > > flush(stderr()); flush(stdout()) > > ### Name: currencysubstitution > ### Title: Currency Substitution > ### Aliases: currencysubstitution > ### Keywords: datasets > > ### ** Examples > > data(currencysubstitution) > > ## page 130, fit Miles OLS model and Bordo-Choudri OLS model > ## third and last line in Table 6.3 > > modelMiles <- logCUS ~ log((1+Iu)/(1+Ic)) > lm(modelMiles, data=currencysubstitution) Call: lm(formula = modelMiles, data = currencysubstitution) Coefficients: (Intercept) log((1 + Iu)/(1 + Ic)) 2.562 5.871 > dwtest(modelMiles, data=currencysubstitution) Durbin-Watson test data: modelMiles DW = 0.27024, p-value < 2.2e-16 alternative hypothesis: true autocorrelation is greater than 0 > > modelBordoChoudri <- logCUS ~ I(Iu-Ic) + Ic + logY > lm(modelBordoChoudri, data=currencysubstitution) Call: lm(formula = modelBordoChoudri, data = currencysubstitution) Coefficients: (Intercept) I(Iu - Ic) Ic logY -5.9782 -9.2158 -18.4563 0.8366 > dwtest(modelBordoChoudri, data=currencysubstitution) Durbin-Watson test data: modelBordoChoudri DW = 0.53514, p-value = 2.261e-13 alternative hypothesis: true autocorrelation is greater than 0 > > > ## page 131, fit test statistics in Table 6.4 b) > ################################################ > > if(require(strucchange, quietly = TRUE)) { + ## Fluctuation test + sctest(modelMiles, type="fluctuation", data=currencysubstitution, + rescale=FALSE) } RE test (recursive estimates test) data: modelMiles RE = 3.1591, p-value = 8.583e-09 > > ## RESET > reset(modelMiles, data=currencysubstitution) RESET test data: modelMiles RESET = 1.1308, df1 = 2, df2 = 57, p-value = 0.3299 > reset(modelMiles, power=2, type="regressor", data=currencysubstitution) RESET test data: modelMiles RESET = 2.1078, df1 = 1, df2 = 58, p-value = 0.1519 > reset(modelMiles, type="princomp", data=currencysubstitution) RESET test data: modelMiles RESET = 1.1308, df1 = 2, df2 = 57, p-value = 0.3299 > > ## Harvey-Collier > harvtest(modelMiles, order.by = ~log((1+Iu)/(1+Ic)), data=currencysubstitution) Harvey-Collier test data: modelMiles HC = 0.86911, df = 58, p-value = 0.3884 > > ## Rainbow > raintest(modelMiles, order.by = "mahalanobis", data=currencysubstitution) Rainbow test data: modelMiles Rain = 1.766, df1 = 31, df2 = 28, p-value = 0.06603 > > > ## page 132, fit test statistics in Table 6.4 d) > ################################################ > > if(require(strucchange, quietly = TRUE)) { + ## Chow 1970(2) + sctest(modelBordoChoudri, point=c(1970,2), data=currencysubstitution, + type="Chow") } Chow test data: modelBordoChoudri F = 14.572, p-value = 4.317e-08 > > ## Breusch-Pagan > bptest(modelBordoChoudri, data=currencysubstitution, studentize=FALSE) Breusch-Pagan test data: modelBordoChoudri BP = 14.514, df = 3, p-value = 0.002283 > bptest(modelBordoChoudri, data=currencysubstitution) studentized Breusch-Pagan test data: modelBordoChoudri BP = 10.736, df = 3, p-value = 0.01324 > > ## RESET > reset(modelBordoChoudri, data=currencysubstitution) RESET test data: modelBordoChoudri RESET = 0.69314, df1 = 2, df2 = 55, p-value = 0.5043 > reset(modelBordoChoudri, power=2, type="regressor", data=currencysubstitution) RESET test data: modelBordoChoudri RESET = 6.6775, df1 = 3, df2 = 54, p-value = 0.0006458 > reset(modelBordoChoudri, type="princomp", data=currencysubstitution) RESET test data: modelBordoChoudri RESET = 8.2945, df1 = 2, df2 = 55, p-value = 0.0007107 > > ## Harvey-Collier > harvtest(modelBordoChoudri, order.by = ~ I(Iu-Ic), data=currencysubstitution) Harvey-Collier test data: modelBordoChoudri HC = 0.75539, df = 56, p-value = 0.4532 > harvtest(modelBordoChoudri, order.by = ~ Ic, data=currencysubstitution) Harvey-Collier test data: modelBordoChoudri HC = 0.40259, df = 56, p-value = 0.6888 > harvtest(modelBordoChoudri, order.by = ~ logY, data=currencysubstitution) Harvey-Collier test data: modelBordoChoudri HC = 2.5016, df = 56, p-value = 0.01531 > > ## Rainbow > raintest(modelBordoChoudri, order.by = "mahalanobis", data=currencysubstitution) Rainbow test data: modelBordoChoudri Rain = 3.7711, df1 = 31, df2 = 26, p-value = 0.0004604 > > > > cleanEx() detaching ‘package:strucchange’, ‘package:sandwich’ > nameEx("dwtest") > ### * dwtest > > flush(stderr()); flush(stdout()) > > ### Name: dwtest > ### Title: Durbin-Watson Test > ### Aliases: dwtest > ### Keywords: htest > > ### ** Examples > > > ## 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 > dwtest(y1 ~ x) Durbin-Watson test data: y1 ~ x DW = 1.9762, p-value = 0.4924 alternative hypothesis: true autocorrelation is greater than 0 > > err2 <- filter(err1, 0.9, method="recursive") > y2 <- 1 + x + err2 > dwtest(y2 ~ x) Durbin-Watson test data: y2 ~ x DW = 0.45961, p-value = 7.862e-15 alternative hypothesis: true autocorrelation is greater than 0 > > > > > cleanEx() > nameEx("encomptest") > ### * encomptest > > flush(stderr()); flush(stdout()) > > ### Name: encomptest > ### Title: Encompassing Test for Comparing Non-Nested Models > ### Aliases: encomptest > ### Keywords: htest > > ### ** Examples > > ## Fit two competing, non-nested models for aggregate > ## consumption, as in Greene (1993), Examples 7.11 and 7.12 > > ## load data and compute lags > data(USDistLag) > usdl <- na.contiguous(cbind(USDistLag, lag(USDistLag, k = -1))) > colnames(usdl) <- c("con", "gnp", "con1", "gnp1") > > ## C(t) = a0 + a1*Y(t) + a2*C(t-1) + u > fm1 <- lm(con ~ gnp + con1, data = usdl) > > ## C(t) = b0 + b1*Y(t) + b2*Y(t-1) + v > fm2 <- lm(con ~ gnp + gnp1, data = usdl) > > ## Encompassing model > fm3 <- lm(con ~ gnp + con1 + gnp1, data = usdl) > > ## Cox test in both directions: > coxtest(fm1, fm2) Cox test Model 1: con ~ gnp + con1 Model 2: con ~ gnp + gnp1 Estimate Std. Error z value Pr(>|z|) fitted(M1) ~ M2 2.8543 1.29978 2.1960 0.02809 * fitted(M2) ~ M1 -4.4003 0.78961 -5.5727 2.508e-08 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 > > ## ...and do the same for jtest() and encomptest(). > ## Notice that in this particular case they are coincident. > jtest(fm1, fm2) J test Model 1: con ~ gnp + con1 Model 2: con ~ gnp + gnp1 Estimate Std. Error t value Pr(>|t|) M1 + fitted(M2) -2.7041 0.76273 -3.5454 0.0029371 ** M2 + fitted(M1) 2.7436 0.52710 5.2051 0.0001067 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 > encomptest(fm1, fm2) Encompassing test Model 1: con ~ gnp + con1 Model 2: con ~ gnp + gnp1 Model E: con ~ gnp + con1 + gnp1 Res.Df Df F Pr(>F) M1 vs. ME 15 -1 12.569 0.0029371 ** M2 vs. ME 15 -1 27.093 0.0001067 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 > > ## the encompassing test is essentially > waldtest(fm1, fm3, fm2) Wald test Model 1: con ~ gnp + con1 Model 2: con ~ gnp + con1 + gnp1 Model 3: con ~ gnp + gnp1 Res.Df Df F Pr(>F) 1 16 2 15 1 12.569 0.0029371 ** 3 16 -1 27.093 0.0001067 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 > > > > cleanEx() > nameEx("ftemp") > ### * ftemp > > flush(stderr()); flush(stdout()) > > ### Name: ftemp > ### Title: Femal Temperature Data > ### Aliases: ftemp > ### Keywords: datasets > > ### ** Examples > > data(ftemp) > plot(ftemp) > y <- window(ftemp, start = 8, end = 60) > if(require(strucchange)) { + bp <- breakpoints(y ~ 1) + plot(bp) + fm.seg <- lm(y ~ 0 + breakfactor(bp)) + plot(y) + lines(8:60, fitted(fm.seg), col = 4) + lines(confint(bp)) + } Loading required package: strucchange Loading required package: sandwich > > > > > cleanEx() detaching ‘package:strucchange’, ‘package:sandwich’ > nameEx("gqtest") > ### * gqtest > > flush(stderr()); flush(stdout()) > > ### Name: gqtest > ### Title: Goldfeld-Quandt Test > ### Aliases: gqtest > ### Keywords: htest > > ### ** Examples > > ## 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 > gqtest(y1 ~ x) Goldfeld-Quandt test data: y1 ~ x GQ = 5.4699, df1 = 48, df2 = 48, p-value = 1.404e-08 alternative hypothesis: variance increases from segment 1 to 2 > gqtest(y2 ~ x) Goldfeld-Quandt test data: y2 ~ x GQ = 1.2638, df1 = 48, df2 = 48, p-value = 0.2102 alternative hypothesis: variance increases from segment 1 to 2 > > > > cleanEx() > nameEx("grangertest") > ### * grangertest > > flush(stderr()); flush(stdout()) > > ### Name: grangertest > ### Title: Test for Granger Causality > ### Aliases: grangertest grangertest.default grangertest.formula > ### Keywords: htest > > ### ** Examples > > ## Which came first: the chicken or the egg? > data(ChickEgg) > grangertest(egg ~ chicken, order = 3, data = ChickEgg) Granger causality test Model 1: egg ~ Lags(egg, 1:3) + Lags(chicken, 1:3) Model 2: egg ~ Lags(egg, 1:3) Res.Df Df F Pr(>F) 1 44 2 47 -3 0.5916 0.6238 > grangertest(chicken ~ egg, order = 3, data = ChickEgg) Granger causality test Model 1: chicken ~ Lags(chicken, 1:3) + Lags(egg, 1:3) Model 2: chicken ~ Lags(chicken, 1:3) Res.Df Df F Pr(>F) 1 44 2 47 -3 5.405 0.002966 ** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 > > ## alternative ways of specifying the same test > grangertest(ChickEgg, order = 3) Granger causality test Model 1: egg ~ Lags(egg, 1:3) + Lags(chicken, 1:3) Model 2: egg ~ Lags(egg, 1:3) Res.Df Df F Pr(>F) 1 44 2 47 -3 0.5916 0.6238 > grangertest(ChickEgg[, 1], ChickEgg[, 2], order = 3) Granger causality test Model 1: ChickEgg[, 2] ~ Lags(ChickEgg[, 2], 1:3) + Lags(ChickEgg[, 1], 1:3) Model 2: ChickEgg[, 2] ~ Lags(ChickEgg[, 2], 1:3) Res.Df Df F Pr(>F) 1 44 2 47 -3 0.5916 0.6238 > > > > cleanEx() > nameEx("growthofmoney") > ### * growthofmoney > > flush(stderr()); flush(stdout()) > > ### Name: growthofmoney > ### Title: Growth of Money Supply > ### Aliases: growthofmoney > ### Keywords: datasets > > ### ** Examples > > data(growthofmoney) > > ## page 137, fit Hetzel OLS model > ## first/second line in Table 6.7 > > modelHetzel <- TG1.TG0 ~ AG0.TG0 > lm(modelHetzel, data=growthofmoney) Call: lm(formula = modelHetzel, data = growthofmoney) Coefficients: (Intercept) AG0.TG0 0.007322 0.324858 > dwtest(modelHetzel, data=growthofmoney) Durbin-Watson test data: modelHetzel DW = 2.9046, p-value = 0.9839 alternative hypothesis: true autocorrelation is greater than 0 > > > ## page 135, fit test statistics in Table 6.8 > ############################################# > > if(require(strucchange, quietly = TRUE)) { + ## Chow 1974(1) + sctest(modelHetzel, point=c(1973,4), data=growthofmoney, type="Chow") } Chow test data: modelHetzel F = 0.37876, p-value = 0.6911 > > ## RESET > reset(modelHetzel, data=growthofmoney) RESET test data: modelHetzel RESET = 7.9337, df1 = 2, df2 = 15, p-value = 0.004461 > reset(modelHetzel, power=2, type="regressor", data=growthofmoney) RESET test data: modelHetzel RESET = 1.5265, df1 = 1, df2 = 16, p-value = 0.2345 > reset(modelHetzel, type="princomp", data=growthofmoney) RESET test data: modelHetzel RESET = 7.9337, df1 = 2, df2 = 15, p-value = 0.004461 > > ## Harvey-Collier > harvtest(modelHetzel, order.by= ~ AG0.TG0, data=growthofmoney) Harvey-Collier test data: modelHetzel HC = 3.7768, df = 16, p-value = 0.001651 > > ## Rainbow > raintest(modelHetzel, order.by = "mahalanobis", data=growthofmoney) Rainbow test data: modelHetzel Rain = 7.1731, df1 = 10, df2 = 7, p-value = 0.007924 > > > ## Identification of outliers > ############################# > > ## Figure 6.1 > plot(modelHetzel, data=growthofmoney) > abline(v=0) > abline(h=0) > abline(coef(lm(modelHetzel, data=growthofmoney)), col=2) > > ## Table 6.7, last line > growthofmoney2 <- as.data.frame(growthofmoney[-c(5:6),]) > lm(modelHetzel, data=growthofmoney2) Call: lm(formula = modelHetzel, data = growthofmoney2) Coefficients: (Intercept) AG0.TG0 0.05673 0.17752 > dwtest(modelHetzel, data=growthofmoney2) Durbin-Watson test data: modelHetzel DW = 1.5019, p-value = 0.1351 alternative hypothesis: true autocorrelation is greater than 0 > > > > cleanEx() detaching ‘package:strucchange’, ‘package:sandwich’ > nameEx("harvtest") > ### * harvtest > > flush(stderr()); flush(stdout()) > > ### Name: harvtest > ### Title: Harvey-Collier Test > ### Aliases: harvtest > ### Keywords: htest > > ### ** Examples > > # 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 <- harvtest(y1 ~ x) > harv Harvey-Collier test data: y1 ~ x HC = 0.5401, df = 47, p-value = 0.5917 > ## calculate critical value vor 0.05 level > qt(0.95, harv$parameter) [1] 1.677927 > harvtest(y2 ~ x) Harvey-Collier test data: y2 ~ x HC = 7.9751, df = 47, p-value = 2.775e-10 > > > > cleanEx() > nameEx("hmctest") > ### * hmctest > > flush(stderr()); flush(stdout()) > > ### Name: hmctest > ### Title: Harrison-McCabe test > ### Aliases: hmctest > ### Keywords: htest > > ### ** Examples > > ## 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 > hmctest(y1 ~ x) Harrison-McCabe test data: y1 ~ x HMC = 0.15542, p-value < 2.2e-16 > hmctest(y2 ~ x) Harrison-McCabe test data: y2 ~ x HMC = 0.44287, p-value = 0.225 > > > > cleanEx() > nameEx("jocci") > ### * jocci > > flush(stderr()); flush(stdout()) > > ### Name: jocci > ### Title: U.S. Macroeconomic Time Series > ### Aliases: fyff gmdc ip jocci lhur pw561 > ### Keywords: datasets > > ### ** Examples > > data(jocci) > > dwtest(dy ~ 1, data = jocci) Durbin-Watson test data: dy ~ 1 DW = 1.0581, p-value < 2.2e-16 alternative hypothesis: true autocorrelation is greater than 0 > bgtest(dy ~ 1, data = jocci) Breusch-Godfrey test for serial correlation of order up to 1 data: dy ~ 1 LM test = 91.037, df = 1, p-value < 2.2e-16 > ar6.model <- dy ~ dy1 + dy2 + dy3 + dy4 + dy5 +dy6 > bgtest(ar6.model, data = jocci) Breusch-Godfrey test for serial correlation of order up to 1 data: ar6.model LM test = 0.19999, df = 1, p-value = 0.6547 > > var.model <- ~ I(dy1^2) + I(dy2^2) + I(dy3^2) + I(dy4^2) + I(dy5^2) + I(dy6^2) > bptest(ar6.model, var.model, data = jocci) studentized Breusch-Pagan test data: ar6.model BP = 22.377, df = 6, p-value = 0.001034 > > > > cleanEx() > nameEx("jtest") > ### * jtest > > flush(stderr()); flush(stdout()) > > ### Name: jtest > ### Title: J Test for Comparing Non-Nested Models > ### Aliases: jtest > ### Keywords: htest > > ### ** Examples > > ## Fit two competing, non-nested models for aggregate > ## consumption, as in Greene (1993), Examples 7.11 and 7.12 > > ## load data and compute lags > data(USDistLag) > usdl <- na.contiguous(cbind(USDistLag, lag(USDistLag, k = -1))) > colnames(usdl) <- c("con", "gnp", "con1", "gnp1") > > ## C(t) = a0 + a1*Y(t) + a2*C(t-1) + u > fm1 <- lm(con ~ gnp + con1, data = usdl) > > ## C(t) = b0 + b1*Y(t) + b2*Y(t-1) + v > fm2 <- lm(con ~ gnp + gnp1, data = usdl) > > ## Cox test in both directions: > coxtest(fm1, fm2) Cox test Model 1: con ~ gnp + con1 Model 2: con ~ gnp + gnp1 Estimate Std. Error z value Pr(>|z|) fitted(M1) ~ M2 2.8543 1.29978 2.1960 0.02809 * fitted(M2) ~ M1 -4.4003 0.78961 -5.5727 2.508e-08 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 > > ## ...and do the same for jtest() and encomptest(). > ## Notice that in this particular case they are coincident. > jtest(fm1, fm2) J test Model 1: con ~ gnp + con1 Model 2: con ~ gnp + gnp1 Estimate Std. Error t value Pr(>|t|) M1 + fitted(M2) -2.7041 0.76273 -3.5454 0.0029371 ** M2 + fitted(M1) 2.7436 0.52710 5.2051 0.0001067 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 > encomptest(fm1, fm2) Encompassing test Model 1: con ~ gnp + con1 Model 2: con ~ gnp + gnp1 Model E: con ~ gnp + con1 + gnp1 Res.Df Df F Pr(>F) M1 vs. ME 15 -1 12.569 0.0029371 ** M2 vs. ME 15 -1 27.093 0.0001067 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 > > > > cleanEx() > nameEx("lrtest") > ### * lrtest > > flush(stderr()); flush(stdout()) > > ### Name: lrtest > ### Title: Likelihood Ratio Test of Nested Models > ### Aliases: lrtest lrtest.formula lrtest.default > ### Keywords: htest > > ### ** Examples > > ## with data from Greene (1993): > ## load data and compute lags > data("USDistLag") > usdl <- na.contiguous(cbind(USDistLag, lag(USDistLag, k = -1))) > colnames(usdl) <- c("con", "gnp", "con1", "gnp1") > > fm1 <- lm(con ~ gnp + gnp1, data = usdl) > fm2 <- lm(con ~ gnp + con1 + gnp1, data = usdl) > > ## various equivalent specifications of the LR test > lrtest(fm2, fm1) Likelihood ratio test Model 1: con ~ gnp + con1 + gnp1 Model 2: con ~ gnp + gnp1 #Df LogLik Df Chisq Pr(>Chisq) 1 5 -56.069 2 4 -65.871 -1 19.605 9.524e-06 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 > lrtest(fm2, 2) Likelihood ratio test Model 1: con ~ gnp + con1 + gnp1 Model 2: con ~ gnp + gnp1 #Df LogLik Df Chisq Pr(>Chisq) 1 5 -56.069 2 4 -65.871 -1 19.605 9.524e-06 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 > lrtest(fm2, "con1") Likelihood ratio test Model 1: con ~ gnp + con1 + gnp1 Model 2: con ~ gnp + gnp1 #Df LogLik Df Chisq Pr(>Chisq) 1 5 -56.069 2 4 -65.871 -1 19.605 9.524e-06 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 > lrtest(fm2, . ~ . - con1) Likelihood ratio test Model 1: con ~ gnp + con1 + gnp1 Model 2: con ~ gnp + gnp1 #Df LogLik Df Chisq Pr(>Chisq) 1 5 -56.069 2 4 -65.871 -1 19.605 9.524e-06 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 > > > > cleanEx() > nameEx("moneydemand") > ### * moneydemand > > flush(stderr()); flush(stdout()) > > ### Name: moneydemand > ### Title: Money Demand > ### Aliases: moneydemand > ### Keywords: datasets > > ### ** Examples > > data(moneydemand) > moneydemand <- window(moneydemand, start=1880, end=1972) > > ## page 125, fit Allen OLS model (and Durbin-Watson test), > ## last line in Table 6.1 > > modelAllen <- logM ~ logYp + Rs + Rl + logSpp > lm(modelAllen, data = moneydemand) Call: lm(formula = modelAllen, data = moneydemand) Coefficients: (Intercept) logYp Rs Rl logSpp -15.48806 1.61700 -0.01885 -0.08354 0.08421 > dwtest(modelAllen, data = moneydemand) Durbin-Watson test data: modelAllen DW = 0.18142, p-value < 2.2e-16 alternative hypothesis: true autocorrelation is greater than 0 > > ## page 127, fit test statistics in Table 6.1 c) > ################################################ > > ## Breusch-Pagan > bptest(modelAllen, studentize = FALSE, data = moneydemand) Breusch-Pagan test data: modelAllen BP = 13.342, df = 4, p-value = 0.009719 > bptest(modelAllen, studentize = TRUE, data = moneydemand) studentized Breusch-Pagan test data: modelAllen BP = 17.069, df = 4, p-value = 0.001874 > > ## RESET > reset(modelAllen, data = moneydemand) RESET test data: modelAllen RESET = 92.749, df1 = 2, df2 = 86, p-value < 2.2e-16 > reset(modelAllen, power = 2, type = "regressor", data = moneydemand) RESET test data: modelAllen RESET = 68.197, df1 = 4, df2 = 84, p-value < 2.2e-16 > reset(modelAllen, type = "princomp", data = moneydemand) RESET test data: modelAllen RESET = 1.7024, df1 = 2, df2 = 86, p-value = 0.1883 > > ## Harvey-Collier tests (up to sign of the test statistic) > harvtest(modelAllen, order.by = ~logYp, data = moneydemand) Harvey-Collier test data: modelAllen HC = 5.5579, df = 87, p-value = 2.943e-07 > harvtest(modelAllen, order.by = ~Rs, data = moneydemand) Harvey-Collier test data: modelAllen HC = 1.4391, df = 87, p-value = 0.1537 > harvtest(modelAllen, order.by = ~Rl, data = moneydemand) Harvey-Collier test data: modelAllen HC = 0.62177, df = 87, p-value = 0.5357 > harvtest(modelAllen, order.by = ~logSpp, data = moneydemand) Harvey-Collier test data: modelAllen HC = 0.79431, df = 87, p-value = 0.4292 > > ## Rainbow test > raintest(modelAllen, order.by = "mahalanobis", data = moneydemand) Rainbow test data: modelAllen Rain = 1.1387, df1 = 47, df2 = 41, p-value = 0.3374 > > > if(require(strucchange, quietly = TRUE)) { + ## Chow (1913) + sctest(modelAllen, point=c(1913,1), data = moneydemand, type = "Chow") } Chow test data: modelAllen F = 58.314, p-value < 2.2e-16 > > if(require(strucchange, quietly = TRUE)) { + ## Fluctuation + sctest(modelAllen, type = "fluctuation", rescale = FALSE, data = moneydemand)} RE test (recursive estimates test) data: modelAllen RE = 9.5229, p-value < 2.2e-16 > > > > cleanEx() detaching ‘package:strucchange’, ‘package:sandwich’ > nameEx("petest") > ### * petest > > flush(stderr()); flush(stdout()) > > ### Name: petest > ### Title: PE Test for Linear vs. Log-Linear Specifications > ### Aliases: petest > ### Keywords: htest > > ### ** Examples > > if(require("AER")) { + ## Verbeek (2004), Section 3 + data("HousePrices", package = "AER") + + ### Verbeek (2004), Table 3.3 + hp_lin <- lm(price ~ . , data = HousePrices) + summary(hp_lin) + + ### Verbeek (2004), Table 3.2 + hp_log <- update(hp_lin, log(price) ~ . - lotsize + log(lotsize)) + summary(hp_log) + + ## PE test + petest(hp_lin, hp_log) + + + ## Greene (2003), Example 9.8 + data("USMacroG", package = "AER") + + ## Greene (2003), Table 9.2 + usm_lin <- lm(m1 ~ tbill + gdp, data = USMacroG) + usm_log <- lm(log(m1) ~ log(tbill) + log(gdp), data = USMacroG) + petest(usm_lin, usm_log) + ## matches results from Greene's errata + } Loading required package: AER Loading required package: car Loading required package: carData Loading required package: sandwich Loading required package: survival PE test Model 1: m1 ~ tbill + gdp Model 2: log(m1) ~ log(tbill) + log(gdp) Estimate Std. Error t value Pr(>|t|) M1 + log(fit(M1))-fit(M2) -209.35 26.7580 -7.8240 2.9e-13 *** M2 + fit(M1)-exp(fit(M2)) 0.00 0.0003 -0.1603 0.8728 --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 > > > > cleanEx() detaching ‘package:AER’, ‘package:survival’, ‘package:sandwich’, ‘package:car’, ‘package:carData’ > nameEx("raintest") > ### * raintest > > flush(stderr()); flush(stdout()) > > ### Name: raintest > ### Title: Rainbow Test > ### Aliases: raintest > ### Keywords: htest > > ### ** Examples > > x <- c(1:30) > y <- x^2 + rnorm(30,0,2) > rain <- raintest(y ~ x) > rain Rainbow test data: y ~ x Rain = 25.76, df1 = 15, df2 = 13, p-value = 3.169e-07 > ## critical value > qf(0.95, rain$parameter[1], rain$parameter[2]) [1] 2.53311 > > > > cleanEx() > nameEx("resettest") > ### * resettest > > flush(stderr()); flush(stdout()) > > ### Name: resettest > ### Title: RESET Test > ### Aliases: resettest reset > ### Keywords: htest > > ### ** Examples > > x <- c(1:30) > y1 <- 1 + x + x^2 + rnorm(30) > y2 <- 1 + x + rnorm(30) > resettest(y1 ~ x, power=2, type="regressor") RESET test data: y1 ~ x RESET = 153881, df1 = 1, df2 = 27, p-value < 2.2e-16 > resettest(y2 ~ x, power=2, type="regressor") RESET test data: y2 ~ x RESET = 0.0077266, df1 = 1, df2 = 27, p-value = 0.9306 > > > > cleanEx() > nameEx("unemployment") > ### * unemployment > > flush(stderr()); flush(stdout()) > > ### Name: unemployment > ### Title: Unemployment Data > ### Aliases: unemployment > ### Keywords: datasets > > ### ** Examples > > data(unemployment) > > ## data transformation > myunemployment <- window(unemployment, start=1895, end=1956) > time <- 6:67 > > ## page 144, fit Rea OLS model > ## last line in Table 6.12 > > modelRea <- UN ~ log(m/p) + log(G) + log(x) + time > lm(modelRea, data = myunemployment) Call: lm(formula = modelRea, data = myunemployment) Coefficients: (Intercept) log(m/p) log(G) log(x) time 86.9466 -13.8332 -5.7374 -9.5063 0.9158 > ## coefficients of logged variables differ by factor 100 > > ## page 143, fit test statistics in table 6.11 > ############################################## > > if(require(strucchange, quietly = TRUE)) { + ## Chow 1941 + sctest(modelRea, point=c(1940,1), data=myunemployment, type="Chow") } Chow test data: modelRea F = 2.7896, p-value = 0.02634 > > ## Breusch-Pagan > bptest(modelRea, data=myunemployment, studentize=FALSE) Breusch-Pagan test data: modelRea BP = 6.191, df = 4, p-value = 0.1853 > bptest(modelRea, data=myunemployment) studentized Breusch-Pagan test data: modelRea BP = 5.6325, df = 4, p-value = 0.2283 > > ## RESET (a)-(b) > reset(modelRea, data=myunemployment) RESET test data: modelRea RESET = 13.594, df1 = 2, df2 = 55, p-value = 1.595e-05 > reset(modelRea, power=2, type="regressor", data=myunemployment) RESET test data: modelRea RESET = 5.2705, df1 = 4, df2 = 53, p-value = 0.001195 > > ## Harvey-Collier > harvtest(modelRea, order.by = ~ log(m/p), data=myunemployment) Harvey-Collier test data: modelRea HC = 0.23146, df = 56, p-value = 0.8178 > harvtest(modelRea, order.by = ~ log(G), data=myunemployment) Harvey-Collier test data: modelRea HC = 1.8008, df = 56, p-value = 0.07711 > harvtest(modelRea, order.by = ~ log(x), data=myunemployment) Harvey-Collier test data: modelRea HC = 0.70681, df = 56, p-value = 0.4826 > harvtest(modelRea, data=myunemployment) Harvey-Collier test data: modelRea HC = 1.6957, df = 56, p-value = 0.0955 > > ## Rainbow > raintest(modelRea, order.by = "mahalanobis", data=myunemployment) Rainbow test data: modelRea Rain = 1.9521, df1 = 31, df2 = 26, p-value = 0.04274 > > > > cleanEx() detaching ‘package:strucchange’, ‘package:sandwich’ > nameEx("valueofstocks") > ### * valueofstocks > > flush(stderr()); flush(stdout()) > > ### Name: valueofstocks > ### Title: Value of Stocks > ### Aliases: valueofstocks > ### Keywords: datasets > > ### ** Examples > > data(valueofstocks) > lm(log(VST) ~., data=valueofstocks) Call: lm(formula = log(VST) ~ ., data = valueofstocks) Coefficients: (Intercept) MB RTPD RTPS XBC 4.724163 0.017505 -0.010756 -0.046852 0.001713 > > > > cleanEx() > nameEx("wages") > ### * wages > > flush(stderr()); flush(stdout()) > > ### Name: wages > ### Title: Wages > ### Aliases: wages > ### Keywords: datasets > > ### ** Examples > > data(wages) > > ## data transformation to include lagged series > mywages <- cbind(wages, lag(wages[,2], k = -1), lag(wages[,2], k = -2)) > colnames(mywages) <- c(colnames(wages), "CPI2", "CPI3") > mywages <- window(mywages, start=1962, end=1979) > > ## page 142, fit Nichols OLS model > ## equation (6.10) > > modelNichols <- w ~ CPI + CPI2 + CPI3 + u + mw > lm(modelNichols, data = mywages) Call: lm(formula = modelNichols, data = mywages) Coefficients: (Intercept) CPI CPI2 CPI3 u mw 4.27536 0.51882 0.12133 0.21404 -0.48786 0.03164 > > ## page 143, fit test statistics in table 6.11 > ############################################## > > if(require(strucchange, quietly = TRUE)) { + ## Chow 1972 + sctest(modelNichols, point=c(1971,1), data=mywages, type="Chow") } Chow test data: modelNichols F = 1.5372, p-value = 0.3074 > > ## Breusch-Pagan > bptest(modelNichols, data=mywages, studentize=FALSE) Breusch-Pagan test data: modelNichols BP = 3.6043, df = 5, p-value = 0.6077 > bptest(modelNichols, data=mywages) studentized Breusch-Pagan test data: modelNichols BP = 2.5505, df = 5, p-value = 0.7689 > > ## RESET (a)-(b) > reset(modelNichols, data=mywages) RESET test data: modelNichols RESET = 0.86419, df1 = 2, df2 = 10, p-value = 0.4506 > reset(modelNichols, power=2, type="regressor", data=mywages) RESET test data: modelNichols RESET = 8.3265, df1 = 5, df2 = 7, p-value = 0.00735 > > ## Harvey-Collier > harvtest(modelNichols, order.by = ~ CPI, data=mywages) Harvey-Collier test data: modelNichols HC = 2.0179, df = 11, p-value = 0.06866 > harvtest(modelNichols, order.by = ~ CPI2, data=mywages) Harvey-Collier test data: modelNichols HC = 4.1448, df = 11, p-value = 0.001631 > harvtest(modelNichols, order.by = ~ CPI3, data=mywages) Harvey-Collier test data: modelNichols HC = 2.2039, df = 11, p-value = 0.04975 > harvtest(modelNichols, order.by = ~ u, data=mywages) Harvey-Collier test data: modelNichols HC = 0.20839, df = 11, p-value = 0.8387 > > ## Rainbow > raintest(modelNichols, order.by = "mahalanobis", data=mywages) Rainbow test data: modelNichols Rain = 0.61074, df1 = 9, df2 = 3, p-value = 0.7512 > > > > cleanEx() detaching ‘package:strucchange’, ‘package:sandwich’ > nameEx("waldtest") > ### * waldtest > > flush(stderr()); flush(stdout()) > > ### Name: waldtest > ### Title: Wald Test of Nested Models > ### Aliases: waldtest waldtest.formula waldtest.default waldtest.lm > ### Keywords: htest > > ### ** Examples > > ## fit two competing, non-nested models and their encompassing > ## model for aggregate consumption, as in Greene (1993), > ## Examples 7.11 and 7.12 > > ## load data and compute lags > data(USDistLag) > usdl <- na.contiguous(cbind(USDistLag, lag(USDistLag, k = -1))) > colnames(usdl) <- c("con", "gnp", "con1", "gnp1") > > ## C(t) = a0 + a1*Y(t) + a2*C(t-1) + u > fm1 <- lm(con ~ gnp + con1, data = usdl) > > ## C(t) = b0 + b1*Y(t) + b2*Y(t-1) + v > fm2 <- lm(con ~ gnp + gnp1, data = usdl) > > ## Encompassing model > fm3 <- lm(con ~ gnp + con1 + gnp1, data = usdl) > > ## a simple ANOVA for fm3 vs. fm2 > waldtest(fm3, fm2) Wald test Model 1: con ~ gnp + con1 + gnp1 Model 2: con ~ gnp + gnp1 Res.Df Df F Pr(>F) 1 15 2 16 -1 27.093 0.0001067 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 > anova(fm3, fm2) Analysis of Variance Table Model 1: con ~ gnp + con1 + gnp1 Model 2: con ~ gnp + gnp1 Res.Df RSS Df Sum of Sq F Pr(>F) 1 15 406.9 2 16 1141.8 -1 -734.93 27.093 0.0001067 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 > ## as df = 1, the test is equivalent to the corresponding t test in > coeftest(fm3) t test of coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 6.334762 9.574496 0.6616 0.5182445 gnp 0.367170 0.048676 7.5432 1.763e-06 *** con1 1.044563 0.200682 5.2051 0.0001067 *** gnp1 -0.391718 0.110488 -3.5454 0.0029371 ** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 > > ## various equivalent specifications of the two models > waldtest(fm3, fm2) Wald test Model 1: con ~ gnp + con1 + gnp1 Model 2: con ~ gnp + gnp1 Res.Df Df F Pr(>F) 1 15 2 16 -1 27.093 0.0001067 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 > waldtest(fm3, 2) Wald test Model 1: con ~ gnp + con1 + gnp1 Model 2: con ~ gnp + gnp1 Res.Df Df F Pr(>F) 1 15 2 16 -1 27.093 0.0001067 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 > waldtest(fm3, "con1") Wald test Model 1: con ~ gnp + con1 + gnp1 Model 2: con ~ gnp + gnp1 Res.Df Df F Pr(>F) 1 15 2 16 -1 27.093 0.0001067 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 > waldtest(fm3, . ~ . - con1) Wald test Model 1: con ~ gnp + con1 + gnp1 Model 2: con ~ gnp + gnp1 Res.Df Df F Pr(>F) 1 15 2 16 -1 27.093 0.0001067 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 > > ## comparing more than one model > ## (equivalent to the encompassing test) > waldtest(fm1, fm3, fm2) Wald test Model 1: con ~ gnp + con1 Model 2: con ~ gnp + con1 + gnp1 Model 3: con ~ gnp + gnp1 Res.Df Df F Pr(>F) 1 16 2 15 1 12.569 0.0029371 ** 3 16 -1 27.093 0.0001067 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 > encomptest(fm1, fm2) Encompassing test Model 1: con ~ gnp + con1 Model 2: con ~ gnp + gnp1 Model E: con ~ gnp + con1 + gnp1 Res.Df Df F Pr(>F) M1 vs. ME 15 -1 12.569 0.0029371 ** M2 vs. ME 15 -1 27.093 0.0001067 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 > > ## using the asymptotic Chisq statistic > waldtest(fm3, fm2, test = "Chisq") Wald test Model 1: con ~ gnp + con1 + gnp1 Model 2: con ~ gnp + gnp1 Res.Df Df Chisq Pr(>Chisq) 1 15 2 16 -1 27.093 1.939e-07 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 > ## plugging in a HC estimator > if(require(sandwich)) waldtest(fm3, fm2, vcov = vcovHC) Loading required package: sandwich Wald test Model 1: con ~ gnp + con1 + gnp1 Model 2: con ~ gnp + gnp1 Res.Df Df F Pr(>F) 1 15 2 16 -1 9.7456 0.006998 ** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 > > > > ### *