Rational Exp Reference Manual

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Package ‘RationalExp’
November 4, 2018
Title Rationalizing Rational Expectations ? Tests and Deviations
Version 0.9.8.9000
Description This package implements a test of the rational expectations hypothesis from D’Haultfoeuille, Gaillac, and Maurel (2018, DGM hereafter) based on the marginal distributions of realizations and subjective beliefs. This test (function test below) can be used in cases where realizations and subjective beliefs are observed in two different datasets that cannot be matched, or when they are observed in the same dataset. The package also computes the estimator of the minimal deviations from rational expectations than can be rationalized by the data (function estimDev below).
Depends R (>= 3.0.0)
License GPL-3
Encoding UTF-8
LazyData true
Suggests knitr,
rmarkdown
VignetteBuilder knitr
RoxygenNote 6.1.0
Imports MASS,
sfsmisc,
snowfall,
stats,
matlab

R topics documented:
boot_stat . .
c_cube . . .
c_fun . . .
estimDev .
inverse . . .
S1 . . . . .
test . . . . .
test_base . .
T_stat . . .
which.min2

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2

boot_stat

boot_stat

Compute the bootstrap test statistic for parallel implementation

Description
This is an internal function to separately compute the bootsrap test statsitic.
Usage
boot_stat(u, Y_tilde, X, D, epsilon, N3, p, prec, N, sample_mat,
generalized, weights, y_grid, phi_n, M_bar, DX)

Arguments
u

bootstrap index;

Y_tilde

the vector stacking the realisations y then the anticipated values psi of respective
sizes n_y and n_p.

X

the matrix of covariates. Set to a vector of 1 by default (in which case the test
without covariates is performed).

D

the vector stacking the dummies for the dataset of realisation : n_y ones then
n_p zeros

epsilon

the parameter epsilonon in Section 3 of DGM. Default value is 0.05.

N3

equals to N if covariates, to 1 other wise.

p

the parameter p in Section 3 of DGM. Default is 0.05.

prec

the number of points to be tested. Default is 30.

N

the total numeber of obs

sample_mat

matrix of bootrap indexes

generalized

"Add" if additive shocks for the generalized test

weights

survey weights

y_grid

the grid points. Default is quantile(Y_tilde,seq(0,1,length.out=30)).

phi_n

the GMS function in DGM

M_bar

the quantilty bar m in section 2 of DGM

DX

the total number of covariates

Details
By default, the test is implemented without covariates. To perform the test with covariates, one
has to indicate in X a non-constant vector or matrix. Also, one can perform the « generalized »
tests allowing for aggregate shocks by using the dummy variable generalized. Survey weights can
be added. The user can modify the number of cores used by R to reduce the computational time.
Tuning parameters used in the test can also be modified.

c_cube

3

c_cube

Instrumental functions computations

Description
This function defines, for each specified value of r_n the set of indicator funtions h(X_i) which are
the key elements for the RE test with co covariates
Usage
c_cube(X_adj, N, DX, r_n)

Arguments
X_adj
N
DX
r_n

the standardised version of the covariates X
the size of X
the number of covariates
the parameter indexing the number of instrumental function, which is chosen
according the the rule used in AS y default.

Value
a list containing, in order:
-X_adj
-r_n
-g_col
-Q_AR
-G_X

the standardised version of the covariates X
the parameter indexing the number of instrumental function, which is chosen
according the the rule used in AS y default.
a vector containing part of the weights
a matrix with the weights that enter the statistic T
a binary matrix indexing the observations X that fall into the hypercubes indexed
by h.

c_fun

Compute the difference between mean of subvectors of two vectors

Description
Compute the difference between mean of subvectors of two vectors
Usage
c_fun(i, i_t, y, z)

Arguments
i
i_t
y
z

starting index
final index
first vector of elements
second vector of elements

4

inverse

Value
a real, the difference between means of subvectors of two vectors

estimDev

Estimation of the minimal deviations from rational expectations with
unconstrained information set g*

Description
This function estimates of the minimal deviations from rational expectations with unconstrained
information set. Both vectors should have the same length. If not, one can randomly select a subset
of the longer vector with length equal to that of the shorter one. The function returns a function via
the approxfun of the package stats. This function can then be evaluated directly on a desired grid.
Usage
estimDev(psi, y)

Arguments
psi

vector of subjective expectations

y

vector of realisations of an individual outcome.

inverse

Inverse the function f

Description
This function implements the numerical inverse of the function f.
Usage
inverse(f, lower = -3, upper = 3)

Arguments
f

the function to be inverted

lower

a lower bound for the inverse

upper

an lower bound for the inverse

S1

5

S1

Core part of the Statistic T

Description
This function implements the core part of the Cramer-von-Mises test statistic T, denoted by S in
AS.
Usage
S1(m_bar, sigma_bar, M1, N_k, p)

Arguments
m_bar

the sample vector of moments for a specified vector $(h_a,r,y)$

sigma_bar

the sample covariance matrix of m_bar

M1

number of inequality moments

N_k

index of the $ h_a,r$ function considered

p

parameter p in the statistic

Value
a real number with the statistic evaluated

test

Implementation of the RE test with possible survey weights (direct and
with parallel computing)

Description
This function performs the test of rational expectations described in Section 3 of D’Haultfoeuille
et al. (2018). By default, the test is implemented without covariates. To perform the test with
covariates, one has to indicate in X a non-constant vector or matrix. Also, one can perform the «
generalized » tests allowing for aggregate shocks by using the dummy variable generalized. Survey
weights can be added. The user can modify the number of cores used by R to reduce the computational time. Tuning parameters used in the test can also be modified.
Usage
test(Y_tilde, D, X = matrix(1, length(Y_tilde), 1),
weights = rep(1/length(Y_tilde), length(Y_tilde)),
generalized = "No", nbCores = 1, tuningParam = NULL)

6

test

Arguments
Y_tilde

the vector stacking the realisations y then the anticipated values psi of respective
sizes n_y and n_p.

D

the vector stacking the dummies for the dataset of realisation : n_y ones then
n_p zeros

X

the matrix of covariates. Set to a vector of 1 by default (in which case the test
without covariates is performed).

weights

the vector of survey weights. Uniform by default.

generalized

whether a generalized test should be performed or not: "Add" for additive
shocks (default), "Mult" for multiplicative shocks. Set by default to "No" (no
generalized test).

nbCores

the number of cores used by the program. To reduce the computational time,
this function can use several cores, in which case the library snowfall should be
loaded first. By default nbCores is set to 1.

tuningParam

a dictionnary (see the example below for modification of the default parameters)
containing:
- the parameter p in Section 3 of DGM. Default is0.05.
- epsilon the parameter epsilonon in Section 3 of DGM. Default value is 0.05
and p is set to 0 if a generalized test is performed.
- B the number of bootstrap samples. Default value is 500.
- grid_y: the number of points to be tested.
Default is quantile(Y_tilde,seq(0,1,length.out=30)).
- c: the parameter c inSection 3 of DGM. Default is 0.3.
- kappa : the parameter kappapa in Section 3 of DGM. Default is 0.001.
Default values are associated with the test without covariates.

Value
a list containing, in order:
- N, the number of observations
- cv01, the 1% critical value
- cv05, the 5% critical value
- cv10, the 10% critical value
- T_n, the Test ststistic
- B, the number of bootstrap samples
- p_value, the p-value
- T_reps, the vector of bootstraped test statitics.
References
D’Haultfoeuille X, Gaillac C, Maurel A (2018). “Rationalizing Rational Expectations? Tests and
Deviations.” CREST Working paper
Andrews D, Shi X (2017). “Inference Based on Many Conditional Moment Inequalities.” Journal
of Econometrics, 196(2), 275–287.
Andrews DW, Kim W, Shi X (2017). “Commands for testing conditional moment inequalities and
equalities.” The Stata journal, 17(1).

test_base

7

Examples

## The RE test without covariates
n_p=1200
n_y=n_p
N <- n_y + n_p
rho <-0.29
sig=0.1
u=1
b=0.10
a=2
psi <-rnorm(n_p,0,u)
pp_y <- runif(n_y,0,1)
zeta <- rnorm(n_y,a,sig)
zeta1 <- rnorm(n_y,-a,sig)
pp1_y <- 1*(pp_y 1-b)
pp3_y <- 1*(pp_y <=(1-b) & pp_y >=b)
psi_y <-rnorm(n_y,0,u)
y = rho*psi_y+ pp1_y*zeta + pp2_y*zeta1
D <- rbind(matrix(1,n_y,1),matrix(0,n_p,1))
Y_tilde <- rbind(matrix(y,n_y,1),matrix(psi,n_p,1))
res <- test(Y_tilde ,D)

test_base

The test statistic for the RE test with survey weights

Description
This is an internal function used in the function test to compute the test statistic with survey weights.
Usage
test_base(Y_tilde, X, D, data_test, epsilon, B, N3, c, kappa, p, N,
weights)

Arguments
Y_tilde

the vector stacking the realisations y then the anticipated values psi of respective
sizes n_y and n_p.

X

the matrix of covariates. Set to a vector of 1 by default (in which case the test
without covariates is performed).

D

the vector stacking the dummies for the dataset of realisation : n_y ones then
n_p zeros

data_test

the matrix of sample moments

epsilon

the parameter epsilonon inSection 3

8

T_stat
B

the number of bootstrap samples

N3

a parameter equal to 1 if no covariates, to N otherwise

c

the parameter c in Section 3

kappa

the parameter kappapa in Section 3

p

the parameter p in Section 3. Equals 0.0 if generalized RE test.

N

total number of observations

weights

the vector of survey weights. Uniform by default.

Details
By default, the test is implemented without covariates. To perform the test with covariates, one
has to indicate in X a non-constant vector or matrix. Also, one can perform the « generalized »
tests allowing for aggregate shocks by using the dummy variable generalized. Survey weights can
be added. The user can modify the number of cores used by R to reduce the computational time.
Tuning parameters used in the test can also be modified.
Value
a list containing, in order:
- T_n : the test statistic
- phi_n: the vector of coresponding GMS functions
- M_bar : the matrix of M_bar in Section 3
References
D’Haultfoeuille X, Gaillac C, Maurel A (2018). “Rationalizing Rational Expectations? Tests and
Deviations.” CREST Working paper
Andrews D, Shi X (2017). “Inference Based on Many Conditional Moment Inequalities.” Journal
of Econometrics, 196(2), 275–287.
Andrews DW, Kim W, Shi X (2017). “Commands for testing conditional moment inequalities and
equalities.” The Stata journal, 17(1).

T_stat

Computation of the test statistic

Description
This function implements the Computation of the test statistic T given in section 3. "Statistical
tests" of "Rationalizing Rational Expectations? Tests and Deviations".
Usage
T_stat(m_bar, Sigma_bar, prob_weight, N_g, N_k, p)

which.min2

9

Arguments
m_bar

the moments m_bar for the different instrumental functions h considered

Sigma_bar

the matrix of all the variances of the moments m_bar for the different instrumental functions h considered

prob_weight

vector of weigths for the test statistic

N_g

number of instrumental functions h considered

N_k

number of moments

p

the parameter p in the Statistic.

Value
a real T which is the test statistic

which.min2

Find the min of a list starting from the end

Description
Find the min of a list starting from the end
Usage
which.min2(x, last.index = FALSE, ...)

Arguments
x

list of elements

last.index

starting from the last index (=TRUE). Default is false

...

hypotetical additional elements



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