7 Reagan: linear regression with AR(1) disturbances
Ninety-six monthly observations on presidential job approval ratings for Ronald Reagan are modeled via linear regression, with a correction for first-order serial correlation among the disturbances.5 Note the marginal model for the first observation, and the conditioning on the lagged observation for months 2 through 96. A uniform prior over the stationary (-1,1) interval is employed for the residual AR(1) parameter.
\[ \begin{aligned}[t] y_i &= \mu_i + \epsilon_i + \theta \epsilon_{i - 1} ,\\ \mu_i &= \alpha + x_i' \beta , \\ \epsilon_i &\sim \mathsf{Normal}(0, \sigma^2) , \end{aligned} \] for \(i \in 1, \dots, N\). Weakly informative priors for each parameter are used, \[ \begin{aligned}[t] \alpha &\sim \mathsf{Normal}(0, 10), \\ \beta_k &\sim \mathsf{Normal}(0, 2.5), & k \in 1, \dots, K, \\ \sigma &\sim \mathsf{HalfCauchy}(0, 5), \\ \theta &= 2 \theta^{*} - 1 , \\ \theta^{*} &\sim \mathsf{Beta}(1, 1) . \end{aligned} \]
data("ReaganApproval", package = "bayesjackman")
ReaganApproval
#> # A tibble: 96 x 3
#> app infl unemp
#> <dbl> <dbl> <dbl>
#> 1 51. 11.8 7.50
#> 2 55. 11.4 7.40
#> 3 60. 10.6 7.40
#> 4 67. 10.1 7.20
#> 5 68. 9.79 7.50
#> 6 59. 9.70 7.50
#> # ... with 90 more rowsreagan_data <- within(list(), {
y <- ReaganApproval$app
N <- length(y)
X <- model.matrix(~ 0 + infl + unemp, data = ReaganApproval) %>% scale()
K <- ncol(X)
alpha_loc <- 0
alpha_scale <- 10
beta_loc <- rep(0, K)
beta_scale <- rep(2.5 * sd(y), K)
sigma_scale <- 5 * sd(y)
theta_a <- 1
theta_b <- 1
})mod_regar1 <- stan_model("stan/regar1.stan")
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#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/src/stan/services/optimize/bfgs.hpp:11:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/src/stan/optimization/bfgs.hpp:9:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/src/stan/optimization/lbfgs_update.hpp:6:
#> In file included from /Users/jrnold/Library/R/3.4/library/BH/include/boost/circular_buffer.hpp:54:
#> In file included from /Users/jrnold/Library/R/3.4/library/BH/include/boost/circular_buffer/details.hpp:20:
#> In file included from /Users/jrnold/Library/R/3.4/library/BH/include/boost/move/move.hpp:30:
#> In file included from /Users/jrnold/Library/R/3.4/library/BH/include/boost/move/iterator.hpp:27:
#> /Users/jrnold/Library/R/3.4/library/BH/include/boost/move/detail/iterator_traits.hpp:29:1: warning: inline namespaces are a C++11 feature [-Wc++11-inline-namespace]
#> BOOST_MOVE_STD_NS_BEG
#> ^
#> /Users/jrnold/Library/R/3.4/library/BH/include/boost/move/detail/std_ns_begin.hpp:18:34: note: expanded from macro 'BOOST_MOVE_STD_NS_BEG'
#> #define BOOST_MOVE_STD_NS_BEG _LIBCPP_BEGIN_NAMESPACE_STD
#> ^
#> /Applications/Xcode.app/Contents/Developer/Toolchains/XcodeDefault.xctoolchain/usr/include/c++/v1/__config:439:52: note: expanded from macro '_LIBCPP_BEGIN_NAMESPACE_STD'
#> #define _LIBCPP_BEGIN_NAMESPACE_STD namespace std {inline namespace _LIBCPP_NAMESPACE {
#> ^
#> In file included from filed1046087152.cpp:8:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/src/stan/model/model_header.hpp:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math.hpp:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math/rev/mat.hpp:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math/rev/core.hpp:44:
#> /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math/rev/core/set_zero_all_adjoints.hpp:14:17: warning: unused function 'set_zero_all_adjoints' [-Wunused-function]
#> static void set_zero_all_adjoints() {
#> ^
#> In file included from filed1046087152.cpp:8:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/src/stan/model/model_header.hpp:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math.hpp:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math/rev/mat.hpp:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math/rev/core.hpp:45:
#> /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math/rev/core/set_zero_all_adjoints_nested.hpp:17:17: warning: 'static' function 'set_zero_all_adjoints_nested' declared in header file should be declared 'static inline' [-Wunneeded-internal-declaration]
#> static void set_zero_all_adjoints_nested() {
#> ^
#> In file included from filed1046087152.cpp:8:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/src/stan/model/model_header.hpp:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math.hpp:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math/rev/mat.hpp:12:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math/prim/mat.hpp:58:
#> /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math/prim/mat/fun/autocorrelation.hpp:17:14: warning: function 'fft_next_good_size' is not needed and will not be emitted [-Wunneeded-internal-declaration]
#> size_t fft_next_good_size(size_t N) {
#> ^
#> 19 warnings generated.
data {
// number of observations
// need at least two to estimates
int N;
// response
vector[N] y;
// regression design matrix
int K;
matrix[N, K] X;
// priors
real alpha_loc;
real alpha_scale;
vector[K] beta_loc;
vector[K] beta_scale;
real sigma_scale;
real theta_a;
real theta_b;
}
parameters {
// regression coefficients
real alpha;
vector[K] beta;
// error scale
real sigma;
// lag coefficients
real theta_raw;
}
transformed parameters {
// observation means
vector[N] mu;
// error terms
vector[N] epsilon;
// lag coefficient;
real theta;
// convert range of theta from (0, 1) to (-1, 1)
theta = (2. * theta_raw - 1.);
// regression
mu = alpha + X * beta;
// construct errors
epsilon[1] = y[1] - mu[1];
for (i in 2:N) {
epsilon[i] = y[i] - mu[i] - theta * epsilon[i - 1];
}
}
model {
alpha ~ cauchy(alpha_loc, alpha_scale);
beta ~ cauchy(beta_loc, beta_scale);
theta_raw ~ beta(theta_a, theta_b);
sigma ~ cauchy(0, sigma_scale);
for (i in 2:N) {
y[i] ~ normal(mu[i] + theta * epsilon[i - 1], sigma);
}
}
summary(reagan_fit, par = c("alpha", "beta", "theta", "sigma"))$summary
#> mean se_mean sd 2.5% 25% 50% 75% 97.5% n_eff
#> alpha 53.195 0.013180 0.8336 51.576 52.62510 53.199 53.758 54.784 4000
#> beta[1] 0.613 0.014395 0.9104 -1.162 0.00317 0.612 1.240 2.396 4000
#> beta[2] -3.690 0.013022 0.8236 -5.328 -4.23172 -3.695 -3.130 -2.069 4000
#> theta 0.675 0.000881 0.0557 0.555 0.64104 0.678 0.713 0.775 4000
#> sigma 4.906 0.006004 0.3797 4.230 4.63457 4.879 5.153 5.700 4000
#> Rhat
#> alpha 0.999
#> beta[1] 1.000
#> beta[2] 1.001
#> theta 1.000
#> sigma 1.0007.1 Cochrane-Orcutt/Prais-Winsten
An AR(1) error model can also be estimated Prais-Winsten estimation: \[ \begin{aligned}[t] y_1 &\sim \mathsf{Normal}\left(\alpha + x_1' \beta, \frac{\sigma ^ 2}{1 - \theta ^ 2} \right), \\ y_i &\sim \mathsf{Normal}\left(\theta y_{i - 1} + \alpha (1 - \theta) + \beta (X_i - \theta X_{i - 1}), \sigma ^ 2 \right) & i = 2, \dots, N \end{aligned} \]
mod_pw <- stan_model("stan/pw.stan")
#> In file included from filed1062414e11.cpp:8:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/src/stan/model/model_header.hpp:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math.hpp:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math/rev/mat.hpp:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math/rev/core.hpp:12:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math/rev/core/gevv_vvv_vari.hpp:5:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math/rev/core/var.hpp:7:
#> In file included from /Users/jrnold/Library/R/3.4/library/BH/include/boost/math/tools/config.hpp:13:
#> In file included from /Users/jrnold/Library/R/3.4/library/BH/include/boost/config.hpp:39:
#> /Users/jrnold/Library/R/3.4/library/BH/include/boost/config/compiler/clang.hpp:200:11: warning: 'BOOST_NO_CXX11_RVALUE_REFERENCES' macro redefined [-Wmacro-redefined]
#> # define BOOST_NO_CXX11_RVALUE_REFERENCES
#> ^
#> <command line>:6:9: note: previous definition is here
#> #define BOOST_NO_CXX11_RVALUE_REFERENCES 1
#> ^
#> In file included from filed1062414e11.cpp:8:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/src/stan/model/model_header.hpp:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math.hpp:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math/rev/mat.hpp:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math/rev/core.hpp:14:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math/rev/core/matrix_vari.hpp:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math/rev/mat/fun/Eigen_NumTraits.hpp:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/RcppEigen/include/Eigen/Dense:1:
#> In file included from /Users/jrnold/Library/R/3.4/library/RcppEigen/include/Eigen/Core:531:
#> /Users/jrnold/Library/R/3.4/library/RcppEigen/include/Eigen/src/Core/util/ReenableStupidWarnings.h:10:30: warning: pragma diagnostic pop could not pop, no matching push [-Wunknown-pragmas]
#> #pragma clang diagnostic pop
#> ^
#> In file included from filed1062414e11.cpp:8:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/src/stan/model/model_header.hpp:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math.hpp:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math/rev/mat.hpp:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math/rev/core.hpp:14:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math/rev/core/matrix_vari.hpp:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math/rev/mat/fun/Eigen_NumTraits.hpp:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/RcppEigen/include/Eigen/Dense:2:
#> In file included from /Users/jrnold/Library/R/3.4/library/RcppEigen/include/Eigen/LU:47:
#> /Users/jrnold/Library/R/3.4/library/RcppEigen/include/Eigen/src/Core/util/ReenableStupidWarnings.h:10:30: warning: pragma diagnostic pop could not pop, no matching push [-Wunknown-pragmas]
#> #pragma clang diagnostic pop
#> ^
#> In file included from filed1062414e11.cpp:8:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/src/stan/model/model_header.hpp:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math.hpp:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math/rev/mat.hpp:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math/rev/core.hpp:14:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math/rev/core/matrix_vari.hpp:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math/rev/mat/fun/Eigen_NumTraits.hpp:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/RcppEigen/include/Eigen/Dense:3:
#> In file included from /Users/jrnold/Library/R/3.4/library/RcppEigen/include/Eigen/Cholesky:12:
#> In file included from /Users/jrnold/Library/R/3.4/library/RcppEigen/include/Eigen/Jacobi:29:
#> /Users/jrnold/Library/R/3.4/library/RcppEigen/include/Eigen/src/Core/util/ReenableStupidWarnings.h:10:30: warning: pragma diagnostic pop could not pop, no matching push [-Wunknown-pragmas]
#> #pragma clang diagnostic pop
#> ^
#> In file included from filed1062414e11.cpp:8:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/src/stan/model/model_header.hpp:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math.hpp:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math/rev/mat.hpp:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math/rev/core.hpp:14:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math/rev/core/matrix_vari.hpp:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math/rev/mat/fun/Eigen_NumTraits.hpp:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/RcppEigen/include/Eigen/Dense:3:
#> In file included from /Users/jrnold/Library/R/3.4/library/RcppEigen/include/Eigen/Cholesky:43:
#> /Users/jrnold/Library/R/3.4/library/RcppEigen/include/Eigen/src/Core/util/ReenableStupidWarnings.h:10:30: warning: pragma diagnostic pop could not pop, no matching push [-Wunknown-pragmas]
#> #pragma clang diagnostic pop
#> ^
#> In file included from filed1062414e11.cpp:8:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/src/stan/model/model_header.hpp:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math.hpp:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math/rev/mat.hpp:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math/rev/core.hpp:14:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math/rev/core/matrix_vari.hpp:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math/rev/mat/fun/Eigen_NumTraits.hpp:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/RcppEigen/include/Eigen/Dense:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/RcppEigen/include/Eigen/QR:17:
#> In file included from /Users/jrnold/Library/R/3.4/library/RcppEigen/include/Eigen/Householder:27:
#> /Users/jrnold/Library/R/3.4/library/RcppEigen/include/Eigen/src/Core/util/ReenableStupidWarnings.h:10:30: warning: pragma diagnostic pop could not pop, no matching push [-Wunknown-pragmas]
#> #pragma clang diagnostic pop
#> ^
#> In file included from filed1062414e11.cpp:8:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/src/stan/model/model_header.hpp:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math.hpp:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math/rev/mat.hpp:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math/rev/core.hpp:14:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math/rev/core/matrix_vari.hpp:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math/rev/mat/fun/Eigen_NumTraits.hpp:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/RcppEigen/include/Eigen/Dense:5:
#> In file included from /Users/jrnold/Library/R/3.4/library/RcppEigen/include/Eigen/SVD:48:
#> /Users/jrnold/Library/R/3.4/library/RcppEigen/include/Eigen/src/Core/util/ReenableStupidWarnings.h:10:30: warning: pragma diagnostic pop could not pop, no matching push [-Wunknown-pragmas]
#> #pragma clang diagnostic pop
#> ^
#> In file included from filed1062414e11.cpp:8:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/src/stan/model/model_header.hpp:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math.hpp:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math/rev/mat.hpp:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math/rev/core.hpp:14:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math/rev/core/matrix_vari.hpp:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math/rev/mat/fun/Eigen_NumTraits.hpp:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/RcppEigen/include/Eigen/Dense:6:
#> In file included from /Users/jrnold/Library/R/3.4/library/RcppEigen/include/Eigen/Geometry:58:
#> /Users/jrnold/Library/R/3.4/library/RcppEigen/include/Eigen/src/Core/util/ReenableStupidWarnings.h:10:30: warning: pragma diagnostic pop could not pop, no matching push [-Wunknown-pragmas]
#> #pragma clang diagnostic pop
#> ^
#> In file included from filed1062414e11.cpp:8:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/src/stan/model/model_header.hpp:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math.hpp:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math/rev/mat.hpp:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math/rev/core.hpp:14:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math/rev/core/matrix_vari.hpp:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math/rev/mat/fun/Eigen_NumTraits.hpp:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/RcppEigen/include/Eigen/Dense:7:
#> In file included from /Users/jrnold/Library/R/3.4/library/RcppEigen/include/Eigen/Eigenvalues:58:
#> /Users/jrnold/Library/R/3.4/library/RcppEigen/include/Eigen/src/Core/util/ReenableStupidWarnings.h:10:30: warning: pragma diagnostic pop could not pop, no matching push [-Wunknown-pragmas]
#> #pragma clang diagnostic pop
#> ^
#> In file included from filed1062414e11.cpp:8:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/src/stan/model/model_header.hpp:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math.hpp:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math/rev/mat.hpp:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math/rev/core.hpp:36:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math/rev/core/operator_unary_plus.hpp:7:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math/prim/scal/fun/constants.hpp:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/BH/include/boost/math/constants/constants.hpp:13:
#> In file included from /Users/jrnold/Library/R/3.4/library/BH/include/boost/math/tools/convert_from_string.hpp:15:
#> In file included from /Users/jrnold/Library/R/3.4/library/BH/include/boost/lexical_cast.hpp:32:
#> In file included from /Users/jrnold/Library/R/3.4/library/BH/include/boost/lexical_cast/try_lexical_convert.hpp:42:
#> In file included from /Users/jrnold/Library/R/3.4/library/BH/include/boost/lexical_cast/detail/converter_lexical.hpp:52:
#> In file included from /Users/jrnold/Library/R/3.4/library/BH/include/boost/container/container_fwd.hpp:61:
#> /Users/jrnold/Library/R/3.4/library/BH/include/boost/container/detail/std_fwd.hpp:27:1: warning: inline namespaces are a C++11 feature [-Wc++11-inline-namespace]
#> BOOST_MOVE_STD_NS_BEG
#> ^
#> /Users/jrnold/Library/R/3.4/library/BH/include/boost/move/detail/std_ns_begin.hpp:18:34: note: expanded from macro 'BOOST_MOVE_STD_NS_BEG'
#> #define BOOST_MOVE_STD_NS_BEG _LIBCPP_BEGIN_NAMESPACE_STD
#> ^
#> /Applications/Xcode.app/Contents/Developer/Toolchains/XcodeDefault.xctoolchain/usr/include/c++/v1/__config:439:52: note: expanded from macro '_LIBCPP_BEGIN_NAMESPACE_STD'
#> #define _LIBCPP_BEGIN_NAMESPACE_STD namespace std {inline namespace _LIBCPP_NAMESPACE {
#> ^
#> In file included from filed1062414e11.cpp:8:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/src/stan/model/model_header.hpp:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math.hpp:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math/rev/mat.hpp:12:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math/prim/mat.hpp:83:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math/prim/mat/fun/csr_extract_u.hpp:6:
#> In file included from /Users/jrnold/Library/R/3.4/library/RcppEigen/include/Eigen/Sparse:26:
#> In file included from /Users/jrnold/Library/R/3.4/library/RcppEigen/include/Eigen/SparseCore:66:
#> /Users/jrnold/Library/R/3.4/library/RcppEigen/include/Eigen/src/Core/util/ReenableStupidWarnings.h:10:30: warning: pragma diagnostic pop could not pop, no matching push [-Wunknown-pragmas]
#> #pragma clang diagnostic pop
#> ^
#> In file included from filed1062414e11.cpp:8:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/src/stan/model/model_header.hpp:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math.hpp:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math/rev/mat.hpp:12:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math/prim/mat.hpp:83:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math/prim/mat/fun/csr_extract_u.hpp:6:
#> In file included from /Users/jrnold/Library/R/3.4/library/RcppEigen/include/Eigen/Sparse:27:
#> In file included from /Users/jrnold/Library/R/3.4/library/RcppEigen/include/Eigen/OrderingMethods:71:
#> /Users/jrnold/Library/R/3.4/library/RcppEigen/include/Eigen/src/Core/util/ReenableStupidWarnings.h:10:30: warning: pragma diagnostic pop could not pop, no matching push [-Wunknown-pragmas]
#> #pragma clang diagnostic pop
#> ^
#> In file included from filed1062414e11.cpp:8:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/src/stan/model/model_header.hpp:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math.hpp:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math/rev/mat.hpp:12:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math/prim/mat.hpp:83:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math/prim/mat/fun/csr_extract_u.hpp:6:
#> In file included from /Users/jrnold/Library/R/3.4/library/RcppEigen/include/Eigen/Sparse:29:
#> In file included from /Users/jrnold/Library/R/3.4/library/RcppEigen/include/Eigen/SparseCholesky:43:
#> /Users/jrnold/Library/R/3.4/library/RcppEigen/include/Eigen/src/Core/util/ReenableStupidWarnings.h:10:30: warning: pragma diagnostic pop could not pop, no matching push [-Wunknown-pragmas]
#> #pragma clang diagnostic pop
#> ^
#> In file included from filed1062414e11.cpp:8:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/src/stan/model/model_header.hpp:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math.hpp:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math/rev/mat.hpp:12:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math/prim/mat.hpp:83:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math/prim/mat/fun/csr_extract_u.hpp:6:
#> In file included from /Users/jrnold/Library/R/3.4/library/RcppEigen/include/Eigen/Sparse:32:
#> In file included from /Users/jrnold/Library/R/3.4/library/RcppEigen/include/Eigen/SparseQR:35:
#> /Users/jrnold/Library/R/3.4/library/RcppEigen/include/Eigen/src/Core/util/ReenableStupidWarnings.h:10:30: warning: pragma diagnostic pop could not pop, no matching push [-Wunknown-pragmas]
#> #pragma clang diagnostic pop
#> ^
#> In file included from filed1062414e11.cpp:8:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/src/stan/model/model_header.hpp:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math.hpp:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math/rev/mat.hpp:12:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math/prim/mat.hpp:83:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math/prim/mat/fun/csr_extract_u.hpp:6:
#> In file included from /Users/jrnold/Library/R/3.4/library/RcppEigen/include/Eigen/Sparse:33:
#> In file included from /Users/jrnold/Library/R/3.4/library/RcppEigen/include/Eigen/IterativeLinearSolvers:46:
#> /Users/jrnold/Library/R/3.4/library/RcppEigen/include/Eigen/src/Core/util/ReenableStupidWarnings.h:10:30: warning: pragma diagnostic pop could not pop, no matching push [-Wunknown-pragmas]
#> #pragma clang diagnostic pop
#> ^
#> In file included from filed1062414e11.cpp:638:
#> In file included from /Users/jrnold/Library/R/3.4/library/rstan/include/rstan/rstaninc.hpp:3:
#> In file included from /Users/jrnold/Library/R/3.4/library/rstan/include/rstan/stan_fit.hpp:36:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/src/stan/services/optimize/bfgs.hpp:11:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/src/stan/optimization/bfgs.hpp:9:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/src/stan/optimization/lbfgs_update.hpp:6:
#> In file included from /Users/jrnold/Library/R/3.4/library/BH/include/boost/circular_buffer.hpp:54:
#> In file included from /Users/jrnold/Library/R/3.4/library/BH/include/boost/circular_buffer/details.hpp:20:
#> In file included from /Users/jrnold/Library/R/3.4/library/BH/include/boost/move/move.hpp:30:
#> In file included from /Users/jrnold/Library/R/3.4/library/BH/include/boost/move/iterator.hpp:27:
#> /Users/jrnold/Library/R/3.4/library/BH/include/boost/move/detail/iterator_traits.hpp:29:1: warning: inline namespaces are a C++11 feature [-Wc++11-inline-namespace]
#> BOOST_MOVE_STD_NS_BEG
#> ^
#> /Users/jrnold/Library/R/3.4/library/BH/include/boost/move/detail/std_ns_begin.hpp:18:34: note: expanded from macro 'BOOST_MOVE_STD_NS_BEG'
#> #define BOOST_MOVE_STD_NS_BEG _LIBCPP_BEGIN_NAMESPACE_STD
#> ^
#> /Applications/Xcode.app/Contents/Developer/Toolchains/XcodeDefault.xctoolchain/usr/include/c++/v1/__config:439:52: note: expanded from macro '_LIBCPP_BEGIN_NAMESPACE_STD'
#> #define _LIBCPP_BEGIN_NAMESPACE_STD namespace std {inline namespace _LIBCPP_NAMESPACE {
#> ^
#> In file included from filed1062414e11.cpp:8:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/src/stan/model/model_header.hpp:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math.hpp:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math/rev/mat.hpp:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math/rev/core.hpp:44:
#> /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math/rev/core/set_zero_all_adjoints.hpp:14:17: warning: unused function 'set_zero_all_adjoints' [-Wunused-function]
#> static void set_zero_all_adjoints() {
#> ^
#> In file included from filed1062414e11.cpp:8:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/src/stan/model/model_header.hpp:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math.hpp:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math/rev/mat.hpp:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math/rev/core.hpp:45:
#> /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math/rev/core/set_zero_all_adjoints_nested.hpp:17:17: warning: 'static' function 'set_zero_all_adjoints_nested' declared in header file should be declared 'static inline' [-Wunneeded-internal-declaration]
#> static void set_zero_all_adjoints_nested() {
#> ^
#> In file included from filed1062414e11.cpp:8:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/src/stan/model/model_header.hpp:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math.hpp:4:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math/rev/mat.hpp:12:
#> In file included from /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math/prim/mat.hpp:58:
#> /Users/jrnold/Library/R/3.4/library/StanHeaders/include/stan/math/prim/mat/fun/autocorrelation.hpp:17:14: warning: function 'fft_next_good_size' is not needed and will not be emitted [-Wunneeded-internal-declaration]
#> size_t fft_next_good_size(size_t N) {
#> ^
#> 19 warnings generated.
data {
// number of observations
// need at least two to estimates
int N;
// response
vector[N] y;
// regression design matrix
int K;
matrix[N, K] X;
// priors
real alpha_loc;
real alpha_scale;
vector[K] beta_loc;
vector[K] beta_scale;
real sigma_scale;
real theta_a;
real theta_b;
}
parameters {
// regression coefficients
real alpha;
vector[K] beta;
// error scale
real sigma;
// lag coefficients
real theta_raw;
}
transformed parameters {
// observation means
vector[N] mu;
// lag coefficient;
real theta;
// convert range of theta from (0, 1) to (-1, 1)
theta = (2. * theta_raw - 1.);
// regression
mu[1] = alpha + dot_product(beta, X[1, ]);
mu[2:N] = alpha * (1 - theta) + (X[2:N, ] - theta * X[1:(N - 1), ]) * beta;
}
model {
alpha ~ cauchy(alpha_loc, alpha_scale);
beta ~ cauchy(beta_loc, beta_scale);
theta_raw ~ beta(theta_a, theta_b);
sigma ~ cauchy(0, sigma_scale);
y[1] ~ normal(mu[1], sigma / sqrt(1 + theta ^ 2));
y[2:N] ~ normal(mu[2:N], sigma);
}
summary(reagan_fit2, par = c("alpha", "beta", "theta", "sigma"))$summary
#> mean se_mean sd 2.5% 25% 50% 75% 97.5% n_eff
#> alpha 48.366 0.16737 6.476 36.705 43.689 48.130 52.6559 61.579 1497
#> beta[1] 0.561 0.01305 0.682 -0.806 0.139 0.566 1.0225 1.848 2733
#> beta[2] -3.473 0.01755 0.780 -5.133 -3.970 -3.433 -2.9177 -2.066 1978
#> theta -0.119 0.00407 0.152 -0.450 -0.218 -0.103 -0.0102 0.136 1394
#> sigma 6.938 0.01048 0.514 6.016 6.582 6.915 7.2597 8.043 2402
#> Rhat
#> alpha 1
#> beta[1] 1
#> beta[2] 1
#> theta 1
#> sigma 1