econometrics with r

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Econometrics using R

Rajat Tayal

Rmetrics Panel Data Workshop

February 16-February 18, 2011

Indira Gandhi Institute of Development Research, Mumbai

February 17, 2012

Rajat Tayal Econometrics using R
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Outline of the presentation

Linear regression

Simple linear regressionMultiple linear regression

Regression diagnostics

Leverage and standardized residualsDeletion diagnosticsThe function influence.measures()Testing for heteroskedasticityTesting for functional form

Testing for autocorrelationRobust standard errors and tests

Linear regression with panel data

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Part I

Linear regression

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Introduction

The linear regression model, typically estimated by ordinary leastsquares (OLS), is the workhorse of applied econometrics. Themodel is

yi = xTi + i, i = 1, 2, . . . , n. (1)

y = X+ (2)For cross-sections:

E(|X) = 0 (3)

Var(|X) =

2

I (4)

For time series:E(j|xi) = 0, i j. (5)

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Introduction

We estimate the OLS by:

= (XTX)1XTy (6)

The corresponding fitted values are: y = X, the residuals are = y y and the residual sum of squares is T.

In R, models are typically fitted by calling a model-fitting function,in this case lm(), with a formula object describing the model and adata.frame object containing the variables used in the formula.

fm < lm(formula, data, . . .)

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The first example

Data from Stock & Watson (2007) on subscriptions to economicsjournals at US libraries for the year 2000:

> install.packages("AER", dependencies=TRUE)

> library(AER)

> data("Journals") ; names("Journals")

> journals journals$citeprice summary(journals)

subs price citeprice

Min. : 2.0 Min. : 20.0 Min. : 0.005223

1st Qu.: 52.0 1st Qu.: 134.5 1st Qu.: 0.464495

Median : 122.5 Median : 282.0 Median : 1.320513

Mean : 196.9 Mean : 417.7 Mean : 2.548455

3rd Qu.: 268.2 3rd Qu.: 540.8 3rd Qu.: 3.440171

Max. :1098.0 Max. :2120.0 Max. :24.459459

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The first example

In view of the wide range of the variables, combined with aconsiderable amount of skewness, it is useful to takelogarithms.

The goal is to estimate the effect of the price per citation onthe number of library subscriptions.

To explore this issue quantitatively, we will fit a linearregression model,

log(subs)i = 1 + 2log(citeprice)i + i (7)

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The first example





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The first example





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The first example

Here, the formula of interest is log(subs) log(citeprice). Thiscan be used both for plotting and for model fitting:

> plot(log(subs) ~ log(citeprice), data = journals)> jour_lm abline(jour_lm)

abline() extracts the coefficients of the fitted model and adds thecorresponding regression line to the plot.

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The first example

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The first example

The function lm() returns a fitted-model object, here stored asjour lm.It is an object of class lm.

> class(jour_lm)

[1] "lm"

> names(jour_lm)

[1] "coefficients" "residuals" "effects" "rank" "fitted

[7] "qr" "df.residual" "xlevels" "call" "terms" "model"

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The first example

> summary(jour_lm)Call:

lm(formula = log(subs) ~ log(citeprice), data = journals)

Residuals:

Min 1Q Median 3Q Max

-2.72478 -0.53609 0.03721 0.46619 1.84808

Coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept) 4.76621 0.05591 85.25


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Generic functions for fitted (linear) model objects

Function Function Descriptionprint() simple printed displaysummary() standard regression outputcoef() (or coefficients()) extracting the regression coefficientsresiduals() (or resid()) extracting residuals

fitted() (or fitted.values()) extracting fitted valuesanova() comparison of nested modelspredict() predictions for new dataplot() diagnostic plotsconfint() confidence intervals for the regression

coefficients

deviance() residual sum of squaresvcov() (estimated) variance-covariance matrixlogLik() log-likelihood (assuming normally distributed

errors)AIC() information criteria including AIC, BIC/SBC

(assuming normally distributed errors)Rajat Tayal Econometrics using R
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The first example

It is instructive to take a brief look at what the summary()method returns for a fitted lm object:

> jour_slm class(jour_slm)

[1] "summary.lm"> names(jour_slm)

[1] "call" "terms" "residuals" "coefficients" "aliased" "sigma"

[7] "df" "r.squared" "adj.r.squared" "fstatistic" "cov.unscale

> jour_slm$coefficients

Estimate Std. Error t value Pr(>|t|)(Intercept) 4.7662121 0.05590908 85.24934 2.953913e-146

log(citeprice) -0.5330535 0.03561320 -14.96786 2.563943e-33

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Analysis of variance

> anova(jour_lm)

Analysis of Variance Table

Response: log(subs)

Df Sum Sq Mean Sq F value Pr(>F)

log(citeprice) 1 125.93 125.934 224.04 < 2.2e-16 ***

Residuals 178 100.06 0.562

---

Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1 1

The ANOVA table breaks the sum of squares about the mean (forthe dependent variable, here log(subs)) into two parts: a part thatis accounted for by a linear function of log(citeprice) and a partattributed to residual variation.

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Point and Interval estimates

To extract the estimated regression coefficients , the functioncoef() can be used:

> coef(jour_lm)

(Intercept) log(citeprice)4.7662121 -0.5330535

> confint(jour_lm, level = 0.95)

2.5 % 97.5 %

(Intercept) 4.6558822 4.8765420

log(citeprice) -0.6033319 -0.4627751

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Prediction

Two types of predictions:1 the prediction of points on the regression line and2 the prediction of a new data value.

The standard errors of predictions for new data take intoaccount both the uncertainty in the regression line and thevariation of the individual points about the line.

Thus, the prediction interval for prediction of new data islarger than that for prediction of points on the line. Thefunction predict() provides both types of standard errors.

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Prediction

> predict(jour_lm, newdata = data.frame(citeprice = 2.11),

interval = "confidence")

fit lwr upr

4.368188 4.247485 4.48889

> predict(jour_lm, newdata = data.frame(citeprice = 2.11),interval = "prediction")

fit lwr upr

4.368188 2.883746 5.852629

The point estimates are identical (fit) but the intervals differ.The prediction intervals can also be used for computing andvisualizing confidence bands.

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Prediction

The prediction intervals can also be used for computing andvisualizing confidence bands.

> lciteprice jour_pred plot(log(subs) ~ log(citeprice), data = journals)

> lines(jour_pred[, 1] ~ lciteprice, col = 1)

> lines(jour_pred[, 2] ~ lciteprice, col = 1, lty = 2)

> lines(jour_pred[, 3] ~ lciteprice, col = 1, lty = 2)

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Prediction

Figure: Scatterplot with prediction intervals for the journals data



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Plotting lm objects

The plot() method for class lm() provides six types of diagnosticplots, four of which are shown by default.We set the graphical parameter mfrow to c(2, 2) using the par()

function, creating a 2 * 2 matrix of plotting areas to see all fourplots simultaneously:

> par(mfrow = c(2, 2))

> plot(jour_lm)

> par(mfrow = c(1, 1))

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Plotting lm objects

Figure: Diagnostic plots for the journals data

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Testing a linear hypothesis

The standard regression output as provided by summary() onlyindicates individual significance of each regressor and jointsignificance of all regressors in the form of t and F statistics,respectively. Often it is necessary to test more general hypotheses.This is possible using the function linear.hypothesis() from the carpackage.Suppose we want to test the hypothesis that the elasticity of thenumber of library subscriptions with respect to the price percitation equals 0.5.

H0 : 2 = 0.5 (8)

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The standard regression output as provided by summary() onlyindicates individual significance of each regressor and jointsignificance of all regressors in the form of t and F statistics,respectively. Often it is necessary to test more general hypotheses.This is possible using the function linear.hypothesis() from the carpackage.Suppose we want to test the hypothesis that the elasticity of thenumber of library subscriptions with respect to the price percitation equals 0.5.

H0 : 2 = 0.5 (8)

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> linear.hypothesis(jour_lm, "log(citeprice) = -0.5")

Linear hypothesis test

Hypothesis:

log(citeprice) = - 0.5

Model 1: restricted model

Model 2: log(subs) ~ log(citeprice)

Res.Df RSS Df Sum of Sq F Pr(>F)

1 179 100.54

2 178 100.06 1 0.48421 0.8614 0.3546

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Part II

Regression diagnostics

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Review

> data("Journals")

> journals journals$citeprice journals$age jour_lm


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Testing for heteroskedasticity

For cross-section regressions, the assumption Var(i|xi) = 2

is typically in doubt. A popular test for checking thisassumption is the Breusch-Pagan test (Breusch and Pagan1979).

For our model fitted to the journals data, stored in jour lm,

the diagnostic plots in suggest that the variance decreaseswith the fitted values or, equivalently, it increases with theprice per citation.

Hence, the regressor log(citeprice) used in the main modelshould also be employed for the auxiliary regression.

Under H0, the test statistic of the Breusch-Pagan testapproximately follows a 2q distribution, where q is thenumber of regressors in the auxiliary regression (excluding theconstant term).


T i f h k d i i
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T i f h k d i i
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T ti f h t k d ti it
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The function bptest() implements all these flavors of theBreusch-Pagan test. By default, it computes the studentizedstatistic for the auxiliary regression utilizing the original

regressors X.> bptest(jour_lm)

studentized Breusch-Pagan test

data: jour_lm

BP = 9.803, df = 1, p-value = 0.001742


T ti f h t k d ti it
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Alternatively, the White test picks up the heteroskedasticity.It uses the original regressors as well as their squares andinteractions in the auxiliary regression, which can be passed as

a second formula to bptest().> bptest(jour_lm, ~ log(citeprice) + I(log(citeprice)^2),

+ data = journals)

studentized Breusch-Pagan test

data: jour_lm

BP = 10.912, df = 2, p-value = 0.004271


T ti th f ti l f
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Testing the functional form

The assumption E(|X) = 0 is crucial for consistency of theleast-squares estimator.

A typical source for violation of this assumption is a

misspecification of the functional form; e.g., by omittingrelevant variables.

One strategy for testing the functional form is to constructauxiliary variables and assess their significance using a simple

F test. This is what Ramseys RESET does.


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The function resettest() defaults to using second and thirdpowers of the fitted values as auxiliary variables.

> resettest(jour_lm)RESET test

data: jour_lm

RESET = 1.4409, df1 = 2, df2 = 176, p-value = 0.2395


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The rainbow test (Utts 1982) takes a different approach totesting the functional form. It fits a model to a subsample(typically the middle 50%) and compares it with the modelfitted to the full sample using an F test.

> raintest(jour_lm, order.by = ~ age, data = journals)Rainbow test

data: jour_lm

Rain = 1.774, df1 = 90, df2 = 88, p-value = 0.003741

This appears to be the case, signaling that the relationshipbetween the number of subscriptions and the price percitation also depends on the age of the journal.


Testing for autocorrelation
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Let us reconsider the first model for the US consumptionfunction.

> library(dynlm)

> data("USMacroG")

> consump1 dwtest(consump1)

Durbin-Watson test

data: consump1

DW = 0.0866, p-value < 2.2e-16

alternative hypothesis: true autocorrelation is greater

than 0


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Further tests for autocorrelation are the Box-Pierce test andthe Ljung-Box test, both being implemented in the functionBox.test() in base R.

> Box.test(residuals(consump1), type = "Ljung-Box")

Box-Ljung test

data: residuals(consump1)

X-squared = 176.0698, df = 1, p-value < 2.2e-16


Multiple linear regression
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Labour economics example: Estimation of wage equation

> data("CPS1988")

> summary(CPS1988)

wage education experience ethnicity

Min. : 50.05 Min. : 0.00 Min. :-4.0 cauc:25923

1st Qu.: 308.64 1st Qu.:12.00 1st Qu.: 8.0 afam: 2232

Median : 522.32 Median :12.00 Median :16.0

Mean : 603.73 Mean :13.07 Mean :18.2

3rd Qu.: 783.48 3rd Qu.:15.00 3rd Qu.:27.0

Max. :18777.20 Max. :18.00 Max. :63.0

The model of interest is

log(wage) = 1 + 2exp+ 3exp2 + 4education + 5ethnicity+

(9)


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> cps_lm summary(cps_lm)

Call:

lm(formula = log(wage) ~ experience + I(experience^2) + education +

ethnicity, data = CPS1988)

Residuals:

Min 1Q Median 3Q Max

-2.9428 -0.3162 0.0580 0.3756 4.3830Coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept) 4.321e+00 1.917e-02 225.38


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Comparison of models

With more than a single explanatory variable, it is interesting totest for the relevance of subsets of regressors.

> cps_noeth anova(cps_noeth, cps_lm)

Analysis of Variance Table

Model 1: log(wage) ~ experience + I(experience^2) + education

Model 2: log(wage) ~ experience + I(experience^2) + education + ethnicity

Res.Df RSS Df Sum of Sq F Pr(>F)

1 28151 9719.6

2 28150 9598.6 1 121.02 354.91 < 2.2e-16 ***

---

Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1 1

This reveals that the effect of ethnicity is significant at any reasonable level.


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> cps_noeth waldtest(cps_lm, . ~ . - ethnicity) from the

package lmtest

Wald test

Model 1: log(wage) ~ experience + I(experience^2) +

education + ethnicity

Model 2: log(wage) ~ experience + I(experience^2) + education

Res.Df Df F Pr(>F)

1 28150

2 28151 -1 354.91 < 2.2e-16 ***

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Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1 1

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Part III

Linear regression with panel data


Introduction


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Introduction

There has been considerable interest in panel dataeconometrics over the last two decades.

The package plm (Croissant and Millo 2008) contains therelevant fitting functions and methods for specifications in R.

Two types of panel data models:1 Statis linear models2 Dynamic linear models


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Grunfeld data comprising 20 annual observations on the threevariables real gross investment (invest), real value of the firm(value), and real value of the capital stock (capital) for 11 largeUS firms for the years 1935-1954.

> data("Grunfeld", package = "AER")

> library("plm")

> gr pgr gr_pool


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One-way panel regression


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y p g

A two-way model could have been estimated upon setting

effect = twoways.

If fixed effects need to be inspected, a fixef() method and anassociated summary() method are available.

To check whether the fixed effects are really needed, we

compare the fixed effects and the pooled OLS fits by means ofpFtest().

> gr_fe pFtest(gr_fe, gr_pool)

F test for individual effects

data: invest ~ value + capital

F = 56.8247, df1 = 2, df2 = 55, p-value = 4.148e-14

alternative hypothesis: significant effects


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y p g

It is also possible to fit a random-effects using the same fittingfunction upon setting model = random and selecting amethod for estimating the variance components.

Four methods are available: Swamy-Arora, Amemiya,Wallace-Hussain, and Nerlove.

> gr_re summary(gr_re)


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Oneway (individual) effect Random Effect Model

(Wallace-Hussains transformation)

Call:plm(formula = invest ~ value + capital, data = pgr, model = "random",

random.method = "walhus")

Balanced Panel: n=3, T=20, N=60

Effects:

var std.dev share

idiosyncratic 4389.31 66.25 0.352

individual 8079.74 89.89 0.648theta: 0.8374

Residuals :

Min. 1st Qu. Median 3rd Qu. Max.

-187.00 -32.90 6.96 31.40 210.00

Coefficients :

Estimate Std. Error t-value Pr(>|t|)(Intercept) -109.976572 61.701384 -1.7824 0.08001 .

value 0.104280 0.014996 6.9539 3.797e-09 ***

capital 0.344784 0.024520 14.0613 < 2.2e-16 ***

Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1 1

TSS: 1988300 RSS: 257520

R-Squared : 0.87048 Adj. R-Squared : 0.82696

F-statistic: 191.545 on 2 and 57 DF, p-value: < 2.22e-16Rajat Tayal Econometrics using R

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A comparison of the regression coefficients shows that fixed-and randomeffects methods yield rather similar results forthese data.

To check whether the random effects are really needed, aLagrange multiplier test is available in plmtest(), defaulting to

the test proposed by Honda (1985).> plmtest(gr_pool)

Lagrange Multiplier Test - (Honda)


normal = 15.4704, p-value < 2.2e-16

alternative hypothesis: significant effects


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Random-effects methods are more efficient than the

fixed-effects estimator under more restrictive assumptions,namely exogeneity of the individual effects. It is thereforeimportant to test for endogeneity, and the standard approachemploys a Hausman test. The relevant function phtest()requires two panel regression objects, in our case yielding

> phtest(gr_re, gr_fe)

Hausman Test


chisq = 0.0404, df = 2, p-value = 0.98

alternative hypothesis: one model is inconsistent

In line with the rather similar estimates presented above,endogeneity does not appear to be a problem here.

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Part IV

Useful resources


Econometrics Task View in R
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Linear regression models: stats, lmtest, car, sandwich, AER.

Non linear regression models: stats.

Quantile regression: quantreg.

Panel data: plm

Linear structural equation model: sem.


Books on Econometrics in R
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Kleiber C. & A. Zeileis (2008), Applied Econometrics withR, Springer.

Vinod H.R. (2008), Hands-on Econometrics using R, WorldScientific Publishing.

Hatekar N.R. (2010), Principles of Econometrics: Anintroduction using R, Sage Publications.

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Thank You

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econometrics with r

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