{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "(Bayesian Vector Autoregressive Models)=\n", "# Bayesian Vector Autoregressive Models\n", "\n", ":::{post} November, 2022\n", ":tags: time series, vector autoregressive model, hierarchical model\n", ":category: intermediate\n", ":author: Nathaniel Forde\n", ":::" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [ { "name": "stderr", "output_type": "stream", "text": [ "/Users/nathanielforde/mambaforge/envs/myjlabenv/lib/python3.11/site-packages/pymc/sampling/jax.py:39: UserWarning: This module is experimental.\n", " warnings.warn(\"This module is experimental.\")\n" ] } ], "source": [ "import os\n", "\n", "import arviz as az\n", "import matplotlib.pyplot as plt\n", "import numpy as np\n", "import pandas as pd\n", "import pymc as pm\n", "import statsmodels.api as sm\n", "\n", "from pymc.sampling_jax import sample_blackjax_nuts" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "RANDOM_SEED = 8927\n", "rng = np.random.default_rng(RANDOM_SEED)\n", "az.style.use(\"arviz-darkgrid\")\n", "%config InlineBackend.figure_format = 'retina'" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## V(ector)A(uto)R(egression) Models \n", "\n", "In this notebook we will outline an application of the Bayesian Vector Autoregressive Modelling. We will draw on the work in the PYMC Labs [blogpost](https://www.pymc-labs.io/blog-posts/bayesian-vector-autoregression/) (see {cite:t}`vieira2022BVAR`). This will be a three part series. In the first we want to show how to fit Bayesian VAR models in PYMC. In the second we will show how to extract extra insight from the fitted model with Impulse Response analysis and make forecasts from the fitted VAR model. In the third and final post we will show in some more detail the benefits of using hierarchical priors with Bayesian VAR models. Specifically, we'll outline how and why there are actually a range of carefully formulated industry standard priors which work with Bayesian VAR modelling. \n", "\n", "In this post we will (i) demonstrate the basic pattern on a simple VAR model on fake data and show how the model recovers the true data generating parameters and (ii) we will show an example applied to macro-economic data and compare the results to those achieved on the same data with statsmodels MLE fits and (iii) show an example of estimating a hierarchical bayesian VAR model over a number of countries. \n", "\n", "## Autoregressive Models in General\n", "\n", "The idea of a simple autoregressive model is to capture the manner in which past observations of the timeseries are predictive of the current observation. So in traditional fashion, if we model this as a linear phenomena we get simple autoregressive models where the current value is predicted by a weighted linear combination of the past values and an error term. \n", "\n", "$$ y_t = \\alpha + \\beta_{y0} \\cdot y_{t-1} + \\beta_{y1} \\cdot y_{t-2} ... + \\epsilon $$\n", "\n", "for however many lags are deemed appropriate to the predict the current observation. \n", "\n", "A VAR model is kind of generalisation of this framework in that it retains the linear combination approach but allows us to model multiple timeseries at once. So concretely this mean that $\\mathbf{y}_{t}$ as a vector where:\n", "\n", "$$ \\mathbf{y}_{T} = \\nu + A_{1}\\mathbf{y}_{T-1} + A_{2}\\mathbf{y}_{T-2} ... A_{p}\\mathbf{y}_{T-p} + \\mathbf{e}_{t} $$\n", "\n", "where the As are coefficient matrices to be combined with the past values of each individual timeseries. For example consider an economic example where we aim to model the relationship and mutual influence of each variable on themselves and one another.\n", "\n", "$$ \\begin{bmatrix} gdp \\\\ inv \\\\ con \\end{bmatrix}_{T} = \\nu + A_{1}\\begin{bmatrix} gdp \\\\ inv \\\\ con \\end{bmatrix}_{T-1} + \n", " A_{2}\\begin{bmatrix} gdp \\\\ inv \\\\ con \\end{bmatrix}_{T-2} ... A_{p}\\begin{bmatrix} gdp \\\\ inv \\\\ con \\end{bmatrix}_{T-p} + \\mathbf{e}_{t} $$\n", "\n", "This structure is compact representation using matrix notation. The thing we are trying to estimate when we fit a VAR model is the A matrices that determine the nature of the linear combination that best fits our timeseries data. Such timeseries models can have an auto-regressive or a moving average representation, and the details matter for some of the implication of a VAR model fit. \n", "\n", "We'll see in the next notebook of the series how the moving-average representation of a VAR lends itself to the interpretation of the covariance structure in our model as representing a kind of impulse-response relationship between the component timeseries. \n", "\n", "### A Concrete Specification with Two lagged Terms\n", "\n", "The matrix notation is convenient to suggest the broad patterns of the model, but it is useful to see the algebra is a simple case. Consider the case of Ireland's GDP and consumption described as: \n", "\n", "$$ gdp_{t} = \\beta_{gdp1} \\cdot gdp_{t-1} + \\beta_{gdp2} \\cdot gdp_{t-2} + \\beta_{cons1} \\cdot cons_{t-1} + \\beta_{cons2} \\cdot cons_{t-2} + \\epsilon_{gdp}$$\n", "$$ cons_{t} = \\beta_{cons1} \\cdot cons_{t-1} + \\beta_{cons2} \\cdot cons_{t-2} + \\beta_{gdp1} \\cdot gdp_{t-1} + \\beta_{gdp2} \\cdot gdp_{t-2} + \\epsilon_{cons}$$\n", "\n", "In this way we can see that if we can estimate the $\\beta$ terms we have an estimate for the bi-directional effects of each variable on the other. This is a useful feature of the modelling. In what follows i should stress that i'm not an economist and I'm aiming to show only the functionality of these models not give you a decisive opinion about the economic relationships determining Irish GDP figures. \n", "\n", "### Creating some Fake Data" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [], "source": [ "def simulate_var(\n", " intercepts, coefs_yy, coefs_xy, coefs_xx, coefs_yx, noises=(1, 1), *, warmup=100, steps=200\n", "):\n", " draws_y = np.zeros(warmup + steps)\n", " draws_x = np.zeros(warmup + steps)\n", " draws_y[:2] = intercepts[0]\n", " draws_x[:2] = intercepts[1]\n", " for step in range(2, warmup + steps):\n", " draws_y[step] = (\n", " intercepts[0]\n", " + coefs_yy[0] * draws_y[step - 1]\n", " + coefs_yy[1] * draws_y[step - 2]\n", " + coefs_xy[0] * draws_x[step - 1]\n", " + coefs_xy[1] * draws_x[step - 2]\n", " + rng.normal(0, noises[0])\n", " )\n", " draws_x[step] = (\n", " intercepts[1]\n", " + coefs_xx[0] * draws_x[step - 1]\n", " + coefs_xx[1] * draws_x[step - 2]\n", " + coefs_yx[0] * draws_y[step - 1]\n", " + coefs_yx[1] * draws_y[step - 2]\n", " + rng.normal(0, noises[1])\n", " )\n", " return draws_y[warmup:], draws_x[warmup:]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "First we generate some fake data with known parameters." ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "data": { "text/html": [ "
\n", " | x | \n", "y | \n", "
---|---|---|
0 | \n", "34.606613 | \n", "30.117581 | \n", "
1 | \n", "34.773803 | \n", "23.996700 | \n", "
2 | \n", "35.455237 | \n", "29.738941 | \n", "
3 | \n", "33.886706 | \n", "27.193417 | \n", "
4 | \n", "31.837465 | \n", "26.704728 | \n", "
\n", " | mean | \n", "sd | \n", "hdi_3% | \n", "hdi_97% | \n", "mcse_mean | \n", "mcse_sd | \n", "ess_bulk | \n", "ess_tail | \n", "r_hat | \n", "
---|---|---|---|---|---|---|---|---|---|
alpha[x] | \n", "8.607 | \n", "1.765 | \n", "5.380 | \n", "12.076 | \n", "0.029 | \n", "0.020 | \n", "3823.0 | \n", "4602.0 | \n", "1.0 | \n", "
alpha[y] | \n", "17.094 | \n", "1.778 | \n", "13.750 | \n", "20.431 | \n", "0.027 | \n", "0.019 | \n", "4188.0 | \n", "5182.0 | \n", "1.0 | \n", "
lag_coefs[x, 1, x] | \n", "1.333 | \n", "0.062 | \n", "1.218 | \n", "1.450 | \n", "0.001 | \n", "0.001 | \n", "5564.0 | \n", "4850.0 | \n", "1.0 | \n", "
lag_coefs[x, 1, y] | \n", "-0.120 | \n", "0.069 | \n", "-0.247 | \n", "0.011 | \n", "0.001 | \n", "0.001 | \n", "3739.0 | \n", "4503.0 | \n", "1.0 | \n", "
lag_coefs[x, 2, x] | \n", "-0.711 | \n", "0.097 | \n", "-0.890 | \n", "-0.527 | \n", "0.002 | \n", "0.001 | \n", "3629.0 | \n", "4312.0 | \n", "1.0 | \n", "
lag_coefs[x, 2, y] | \n", "0.267 | \n", "0.073 | \n", "0.126 | \n", "0.403 | \n", "0.001 | \n", "0.001 | \n", "3408.0 | \n", "4318.0 | \n", "1.0 | \n", "
lag_coefs[y, 1, x] | \n", "0.838 | \n", "0.061 | \n", "0.718 | \n", "0.948 | \n", "0.001 | \n", "0.001 | \n", "5203.0 | \n", "5345.0 | \n", "1.0 | \n", "
lag_coefs[y, 1, y] | \n", "-0.800 | \n", "0.069 | \n", "-0.932 | \n", "-0.673 | \n", "0.001 | \n", "0.001 | \n", "3749.0 | \n", "5131.0 | \n", "1.0 | \n", "
lag_coefs[y, 2, x] | \n", "0.094 | \n", "0.097 | \n", "-0.087 | \n", "0.277 | \n", "0.002 | \n", "0.001 | \n", "3573.0 | \n", "4564.0 | \n", "1.0 | \n", "
lag_coefs[y, 2, y] | \n", "-0.004 | \n", "0.074 | \n", "-0.145 | \n", "0.133 | \n", "0.001 | \n", "0.001 | \n", "3448.0 | \n", "4484.0 | \n", "1.0 | \n", "
noise_chol_corr[0, 0] | \n", "1.000 | \n", "0.000 | \n", "1.000 | \n", "1.000 | \n", "0.000 | \n", "0.000 | \n", "8000.0 | \n", "8000.0 | \n", "NaN | \n", "
noise_chol_corr[0, 1] | \n", "0.021 | \n", "0.072 | \n", "-0.118 | \n", "0.152 | \n", "0.001 | \n", "0.001 | \n", "6826.0 | \n", "5061.0 | \n", "1.0 | \n", "
noise_chol_corr[1, 0] | \n", "0.021 | \n", "0.072 | \n", "-0.118 | \n", "0.152 | \n", "0.001 | \n", "0.001 | \n", "6826.0 | \n", "5061.0 | \n", "1.0 | \n", "
noise_chol_corr[1, 1] | \n", "1.000 | \n", "0.000 | \n", "1.000 | \n", "1.000 | \n", "0.000 | \n", "0.000 | \n", "7813.0 | \n", "7824.0 | \n", "1.0 | \n", "
\n", " | country | \n", "iso2c | \n", "iso3c | \n", "year | \n", "GDP | \n", "CONS | \n", "GFCF | \n", "dl_gdp | \n", "dl_cons | \n", "dl_gfcf | \n", "more_than_10 | \n", "time | \n", "
---|---|---|---|---|---|---|---|---|---|---|---|---|
1 | \n", "Australia | \n", "AU | \n", "AUS | \n", "1971 | \n", "4.647670e+11 | \n", "3.113170e+11 | \n", "7.985100e+10 | \n", "0.039217 | \n", "0.040606 | \n", "0.031705 | \n", "True | \n", "1 | \n", "
2 | \n", "Australia | \n", "AU | \n", "AUS | \n", "1972 | \n", "4.829350e+11 | \n", "3.229650e+11 | \n", "8.209200e+10 | \n", "0.038346 | \n", "0.036732 | \n", "0.027678 | \n", "True | \n", "2 | \n", "
3 | \n", "Australia | \n", "AU | \n", "AUS | \n", "1973 | \n", "4.955840e+11 | \n", "3.371070e+11 | \n", "8.460300e+10 | \n", "0.025855 | \n", "0.042856 | \n", "0.030129 | \n", "True | \n", "3 | \n", "
4 | \n", "Australia | \n", "AU | \n", "AUS | \n", "1974 | \n", "5.159300e+11 | \n", "3.556010e+11 | \n", "8.821400e+10 | \n", "0.040234 | \n", "0.053409 | \n", "0.041796 | \n", "True | \n", "4 | \n", "
5 | \n", "Australia | \n", "AU | \n", "AUS | \n", "1975 | \n", "5.228210e+11 | \n", "3.759000e+11 | \n", "8.255900e+10 | \n", "0.013268 | \n", "0.055514 | \n", "-0.066252 | \n", "True | \n", "5 | \n", "
... | \n", "... | \n", "... | \n", "... | \n", "... | \n", "... | \n", "... | \n", "... | \n", "... | \n", "... | \n", "... | \n", "... | \n", "... | \n", "
366 | \n", "United States | \n", "US | \n", "USA | \n", "2016 | \n", "1.850960e+13 | \n", "1.522497e+13 | \n", "3.802207e+12 | \n", "0.016537 | \n", "0.023425 | \n", "0.021058 | \n", "True | \n", "44 | \n", "
367 | \n", "United States | \n", "US | \n", "USA | \n", "2017 | \n", "1.892712e+13 | \n", "1.553075e+13 | \n", "3.947418e+12 | \n", "0.022306 | \n", "0.019885 | \n", "0.037480 | \n", "True | \n", "45 | \n", "
368 | \n", "United States | \n", "US | \n", "USA | \n", "2018 | \n", "1.947957e+13 | \n", "1.593427e+13 | \n", "4.119951e+12 | \n", "0.028771 | \n", "0.025650 | \n", "0.042780 | \n", "True | \n", "46 | \n", "
369 | \n", "United States | \n", "US | \n", "USA | \n", "2019 | \n", "1.992544e+13 | \n", "1.627888e+13 | \n", "4.248643e+12 | \n", "0.022631 | \n", "0.021396 | \n", "0.030758 | \n", "True | \n", "47 | \n", "
370 | \n", "United States | \n", "US | \n", "USA | \n", "2020 | \n", "1.924706e+13 | \n", "1.582501e+13 | \n", "4.182801e+12 | \n", "-0.034639 | \n", "-0.028277 | \n", "-0.015619 | \n", "True | \n", "48 | \n", "
370 rows × 12 columns
\n", "\n", " | country | \n", "iso2c | \n", "iso3c | \n", "year | \n", "GDP | \n", "CONS | \n", "GFCF | \n", "dl_gdp | \n", "dl_cons | \n", "dl_gfcf | \n", "more_than_10 | \n", "time | \n", "
---|---|---|---|---|---|---|---|---|---|---|---|---|
0 | \n", "Ireland | \n", "IE | \n", "IRL | \n", "1971 | \n", "3.314234e+10 | \n", "2.897699e+10 | \n", "8.317518e+09 | \n", "0.034110 | \n", "0.043898 | \n", "0.085452 | \n", "True | \n", "1 | \n", "
1 | \n", "Ireland | \n", "IE | \n", "IRL | \n", "1972 | \n", "3.529322e+10 | \n", "3.063538e+10 | \n", "8.967782e+09 | \n", "0.062879 | \n", "0.055654 | \n", "0.075274 | \n", "True | \n", "2 | \n", "
2 | \n", "Ireland | \n", "IE | \n", "IRL | \n", "1973 | \n", "3.695956e+10 | \n", "3.280221e+10 | \n", "1.041728e+10 | \n", "0.046134 | \n", "0.068340 | \n", "0.149828 | \n", "True | \n", "3 | \n", "
3 | \n", "Ireland | \n", "IE | \n", "IRL | \n", "1974 | \n", "3.853412e+10 | \n", "3.381524e+10 | \n", "9.207243e+09 | \n", "0.041720 | \n", "0.030416 | \n", "-0.123476 | \n", "True | \n", "4 | \n", "
4 | \n", "Ireland | \n", "IE | \n", "IRL | \n", "1975 | \n", "4.071386e+10 | \n", "3.477232e+10 | \n", "8.874887e+09 | \n", "0.055024 | \n", "0.027910 | \n", "-0.036765 | \n", "True | \n", "5 | \n", "
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<xarray.Dataset>\n", "Dimensions: (chain: 1, draw: 500, time: 49, equations: 2)\n", "Coordinates:\n", " * chain (chain) int64 0\n", " * draw (draw) int64 0 1 2 3 4 5 6 7 ... 492 493 494 495 496 497 498 499\n", " * time (time) int64 2 3 4 5 6 7 8 9 10 11 ... 42 43 44 45 46 47 48 49 50\n", " * equations (equations) <U7 'dl_gdp' 'dl_cons'\n", "Data variables:\n", " obs (chain, draw, time, equations) float64 0.4478 -0.1462 ... 0.00633\n", "Attributes:\n", " created_at: 2023-02-21T19:20:42.614050\n", " arviz_version: 0.14.0\n", " inference_library: pymc\n", " inference_library_version: 5.0.1
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<xarray.Dataset>\n", "Dimensions: (time: 49, equations: 2)\n", "Coordinates:\n", " * time (time) int64 2 3 4 5 6 7 8 9 10 11 ... 42 43 44 45 46 47 48 49 50\n", " * equations (equations) <U7 'dl_gdp' 'dl_cons'\n", "Data variables:\n", " data_obs (time, equations) float64 0.04613 0.06834 ... 0.1264 0.05461\n", "Attributes:\n", " created_at: 2023-02-21T19:20:42.614628\n", " arviz_version: 0.14.0\n", " inference_library: pymc\n", " inference_library_version: 5.0.1
\n", " | dl_gdp | \n", "dl_cons | \n", "
---|---|---|
const | \n", "0.034145 | \n", "0.006996 | \n", "
L1.dl_gdp | \n", "0.324904 | \n", "0.330003 | \n", "
L1.dl_cons | \n", "0.076629 | \n", "0.305824 | \n", "
L2.dl_gdp | \n", "0.137721 | \n", "-0.053677 | \n", "
L2.dl_cons | \n", "-0.278745 | \n", "0.033728 | \n", "
\n", " | dl_gdp | \n", "dl_cons | \n", "
---|---|---|
dl_gdp | \n", "1.000000 | \n", "0.435807 | \n", "
dl_cons | \n", "0.435807 | \n", "1.000000 | \n", "
\n", " | mean | \n", "sd | \n", "hdi_3% | \n", "hdi_97% | \n", "mcse_mean | \n", "mcse_sd | \n", "ess_bulk | \n", "ess_tail | \n", "r_hat | \n", "
---|---|---|---|---|---|---|---|---|---|
alpha[dl_gdp] | \n", "0.033 | \n", "0.011 | \n", "0.012 | \n", "0.053 | \n", "0.000 | \n", "0.000 | \n", "6683.0 | \n", "5919.0 | \n", "1.0 | \n", "
alpha[dl_cons] | \n", "0.007 | \n", "0.007 | \n", "-0.007 | \n", "0.020 | \n", "0.000 | \n", "0.000 | \n", "7651.0 | \n", "5999.0 | \n", "1.0 | \n", "
lag_coefs[dl_gdp, 1, dl_gdp] | \n", "0.321 | \n", "0.170 | \n", "0.008 | \n", "0.642 | \n", "0.002 | \n", "0.002 | \n", "6984.0 | \n", "6198.0 | \n", "1.0 | \n", "
lag_coefs[dl_gdp, 1, dl_cons] | \n", "0.071 | \n", "0.273 | \n", "-0.447 | \n", "0.582 | \n", "0.003 | \n", "0.003 | \n", "7376.0 | \n", "5466.0 | \n", "1.0 | \n", "
lag_coefs[dl_gdp, 2, dl_gdp] | \n", "0.133 | \n", "0.190 | \n", "-0.228 | \n", "0.488 | \n", "0.002 | \n", "0.002 | \n", "7471.0 | \n", "6128.0 | \n", "1.0 | \n", "
lag_coefs[dl_gdp, 2, dl_cons] | \n", "-0.235 | \n", "0.269 | \n", "-0.748 | \n", "0.259 | \n", "0.003 | \n", "0.002 | \n", "8085.0 | \n", "5963.0 | \n", "1.0 | \n", "
lag_coefs[dl_cons, 1, dl_gdp] | \n", "0.331 | \n", "0.106 | \n", "0.133 | \n", "0.528 | \n", "0.001 | \n", "0.001 | \n", "7670.0 | \n", "6360.0 | \n", "1.0 | \n", "
lag_coefs[dl_cons, 1, dl_cons] | \n", "0.302 | \n", "0.170 | \n", "-0.012 | \n", "0.616 | \n", "0.002 | \n", "0.001 | \n", "7963.0 | \n", "6150.0 | \n", "1.0 | \n", "
lag_coefs[dl_cons, 2, dl_gdp] | \n", "-0.054 | \n", "0.118 | \n", "-0.279 | \n", "0.163 | \n", "0.001 | \n", "0.001 | \n", "8427.0 | \n", "6296.0 | \n", "1.0 | \n", "
lag_coefs[dl_cons, 2, dl_cons] | \n", "0.048 | \n", "0.170 | \n", "-0.259 | \n", "0.378 | \n", "0.002 | \n", "0.002 | \n", "8669.0 | \n", "6264.0 | \n", "1.0 | \n", "
noise_chol_corr[0, 0] | \n", "1.000 | \n", "0.000 | \n", "1.000 | \n", "1.000 | \n", "0.000 | \n", "0.000 | \n", "8000.0 | \n", "8000.0 | \n", "NaN | \n", "
noise_chol_corr[0, 1] | \n", "0.416 | \n", "0.123 | \n", "0.180 | \n", "0.633 | \n", "0.001 | \n", "0.001 | \n", "9155.0 | \n", "6052.0 | \n", "1.0 | \n", "
noise_chol_corr[1, 0] | \n", "0.416 | \n", "0.123 | \n", "0.180 | \n", "0.633 | \n", "0.001 | \n", "0.001 | \n", "9155.0 | \n", "6052.0 | \n", "1.0 | \n", "
noise_chol_corr[1, 1] | \n", "1.000 | \n", "0.000 | \n", "1.000 | \n", "1.000 | \n", "0.000 | \n", "0.000 | \n", "7871.0 | \n", "8000.0 | \n", "1.0 | \n", "
<xarray.Dataset>\n", "Dimensions: (chain: 4, draw: 2000, equations: 3,\n", " lags: 2, cross_vars: 3,\n", " omega_global_dim_0: 6,\n", " noise_chol_Australia_dim_0: 6,\n", " noise_chol_Canada_dim_0: 6,\n", " noise_chol_Chile_dim_0: 6,\n", " ...\n", " betaX_United States_dim_1: 3,\n", " noise_chol_United States_corr_dim_0: 3,\n", " noise_chol_United States_corr_dim_1: 3,\n", " noise_chol_United States_stds_dim_0: 3,\n", " omega_United States_dim_0: 3,\n", " omega_United States_dim_1: 3)\n", "Coordinates: (12/73)\n", " * chain (chain) int64 0 1 2 3\n", " * draw (draw) int64 0 1 2 ... 1997 1998 1999\n", " * equations (equations) <U7 'dl_gdp' ... 'dl_gfcf'\n", " * lags (lags) int64 1 2\n", " * cross_vars (cross_vars) <U7 'dl_gdp' ... 'dl_g...\n", " * omega_global_dim_0 (omega_global_dim_0) int64 0 1 2 3 4 5\n", " ... ...\n", " * betaX_United States_dim_1 (betaX_United States_dim_1) int64 0...\n", " * noise_chol_United States_corr_dim_0 (noise_chol_United States_corr_dim_0) int64 ...\n", " * noise_chol_United States_corr_dim_1 (noise_chol_United States_corr_dim_1) int64 ...\n", " * noise_chol_United States_stds_dim_0 (noise_chol_United States_stds_dim_0) int64 ...\n", " * omega_United States_dim_0 (omega_United States_dim_0) int64 0...\n", " * omega_United States_dim_1 (omega_United States_dim_1) int64 0...\n", "Data variables: (12/80)\n", " alpha_hat_location (chain, draw) float64 0.01867 ... 0...\n", " beta_hat_location (chain, draw) float64 0.03707 ... 0...\n", " lag_coefs_Australia (chain, draw, equations, lags, cross_vars) float64 ...\n", " alpha_Australia (chain, draw, equations) float64 0....\n", " lag_coefs_Canada (chain, draw, equations, lags, cross_vars) float64 ...\n", " alpha_Canada (chain, draw, equations) float64 0....\n", " ... ...\n", " noise_chol_United Kingdom_stds (chain, draw, noise_chol_United Kingdom_stds_dim_0) float64 ...\n", " omega_United Kingdom (chain, draw, omega_United Kingdom_dim_0, omega_United Kingdom_dim_1) float64 ...\n", " betaX_United States (chain, draw, betaX_United States_dim_0, betaX_United States_dim_1) float64 ...\n", " noise_chol_United States_corr (chain, draw, noise_chol_United States_corr_dim_0, noise_chol_United States_corr_dim_1) float64 ...\n", " noise_chol_United States_stds (chain, draw, noise_chol_United States_stds_dim_0) float64 ...\n", " omega_United States (chain, draw, omega_United States_dim_0, omega_United States_dim_1) float64 ...\n", "Attributes:\n", " created_at: 2023-02-21T19:24:14.259388\n", " arviz_version: 0.14.0
<xarray.Dataset>\n", "Dimensions: (chain: 4, draw: 2000, obs_Australia_dim_2: 49,\n", " obs_Australia_dim_3: 3, obs_Canada_dim_2: 22,\n", " obs_Canada_dim_3: 3, obs_Chile_dim_2: 49,\n", " obs_Chile_dim_3: 3, obs_Ireland_dim_2: 49,\n", " obs_Ireland_dim_3: 3, obs_New Zealand_dim_2: 41,\n", " obs_New Zealand_dim_3: 3,\n", " obs_South Africa_dim_2: 49,\n", " obs_South Africa_dim_3: 3,\n", " obs_United Kingdom_dim_2: 49,\n", " obs_United Kingdom_dim_3: 3,\n", " obs_United States_dim_2: 46,\n", " obs_United States_dim_3: 3)\n", "Coordinates: (12/18)\n", " * chain (chain) int64 0 1 2 3\n", " * draw (draw) int64 0 1 2 3 4 ... 1996 1997 1998 1999\n", " * obs_Australia_dim_2 (obs_Australia_dim_2) int64 0 1 2 3 ... 46 47 48\n", " * obs_Australia_dim_3 (obs_Australia_dim_3) int64 0 1 2\n", " * obs_Canada_dim_2 (obs_Canada_dim_2) int64 0 1 2 3 4 ... 18 19 20 21\n", " * obs_Canada_dim_3 (obs_Canada_dim_3) int64 0 1 2\n", " ... ...\n", " * obs_South Africa_dim_2 (obs_South Africa_dim_2) int64 0 1 2 ... 46 47 48\n", " * obs_South Africa_dim_3 (obs_South Africa_dim_3) int64 0 1 2\n", " * obs_United Kingdom_dim_2 (obs_United Kingdom_dim_2) int64 0 1 2 ... 47 48\n", " * obs_United Kingdom_dim_3 (obs_United Kingdom_dim_3) int64 0 1 2\n", " * obs_United States_dim_2 (obs_United States_dim_2) int64 0 1 2 ... 43 44 45\n", " * obs_United States_dim_3 (obs_United States_dim_3) int64 0 1 2\n", "Data variables:\n", " obs_Australia (chain, draw, obs_Australia_dim_2, obs_Australia_dim_3) float64 ...\n", " obs_Canada (chain, draw, obs_Canada_dim_2, obs_Canada_dim_3) float64 ...\n", " obs_Chile (chain, draw, obs_Chile_dim_2, obs_Chile_dim_3) float64 ...\n", " obs_Ireland (chain, draw, obs_Ireland_dim_2, obs_Ireland_dim_3) float64 ...\n", " obs_New Zealand (chain, draw, obs_New Zealand_dim_2, obs_New Zealand_dim_3) float64 ...\n", " obs_South Africa (chain, draw, obs_South Africa_dim_2, obs_South Africa_dim_3) float64 ...\n", " obs_United Kingdom (chain, draw, obs_United Kingdom_dim_2, obs_United Kingdom_dim_3) float64 ...\n", " obs_United States (chain, draw, obs_United States_dim_2, obs_United States_dim_3) float64 ...\n", "Attributes:\n", " created_at: 2023-02-21T19:31:34.868405\n", " arviz_version: 0.14.0\n", " inference_library: pymc\n", " inference_library_version: 5.0.1
<xarray.Dataset>\n", "Dimensions: (chain: 4, draw: 2000)\n", "Coordinates:\n", " * chain (chain) int64 0 1 2 3\n", " * draw (draw) int64 0 1 2 3 4 5 ... 1994 1995 1996 1997 1998 1999\n", "Data variables:\n", " lp (chain, draw) float64 -2.64e+03 -2.628e+03 ... -2.604e+03\n", " diverging (chain, draw) bool False False False ... False False False\n", " energy (chain, draw) float64 -2.532e+03 -2.521e+03 ... -2.494e+03\n", " tree_depth (chain, draw) int64 6 6 6 6 6 6 7 7 7 ... 6 6 6 5 6 6 6 6 6\n", " n_steps (chain, draw) int64 63 63 63 63 63 63 ... 31 63 63 63 63 63\n", " acceptance_rate (chain, draw) float64 0.8533 0.9783 ... 0.9693 0.9407\n", "Attributes:\n", " created_at: 2023-02-21T19:24:14.274886\n", " arviz_version: 0.14.0
<xarray.Dataset>\n", "Dimensions: (chain: 1, draw: 500,\n", " omega_global_dim_0: 6,\n", " betaX_Canada_dim_0: 22,\n", " betaX_Canada_dim_1: 3,\n", " noise_chol_South Africa_dim_0: 6,\n", " omega_New Zealand_dim_0: 3,\n", " ...\n", " noise_chol_New Zealand_corr_dim_0: 3,\n", " noise_chol_New Zealand_corr_dim_1: 3,\n", " noise_chol_United States_corr_dim_0: 3,\n", " noise_chol_United States_corr_dim_1: 3,\n", " noise_chol_Chile_dim_0: 6,\n", " noise_chol_Ireland_stds_dim_0: 3)\n", "Coordinates: (12/73)\n", " * chain (chain) int64 0\n", " * draw (draw) int64 0 1 2 3 ... 497 498 499\n", " * omega_global_dim_0 (omega_global_dim_0) int64 0 1 2 3 4 5\n", " * betaX_Canada_dim_0 (betaX_Canada_dim_0) int64 0 1 ... 21\n", " * betaX_Canada_dim_1 (betaX_Canada_dim_1) int64 0 1 2\n", " * noise_chol_South Africa_dim_0 (noise_chol_South Africa_dim_0) int64 ...\n", " ... ...\n", " * noise_chol_New Zealand_corr_dim_0 (noise_chol_New Zealand_corr_dim_0) int64 ...\n", " * noise_chol_New Zealand_corr_dim_1 (noise_chol_New Zealand_corr_dim_1) int64 ...\n", " * noise_chol_United States_corr_dim_0 (noise_chol_United States_corr_dim_0) int64 ...\n", " * noise_chol_United States_corr_dim_1 (noise_chol_United States_corr_dim_1) int64 ...\n", " * noise_chol_Chile_dim_0 (noise_chol_Chile_dim_0) int64 0 ... 5\n", " * noise_chol_Ireland_stds_dim_0 (noise_chol_Ireland_stds_dim_0) int64 ...\n", "Data variables: (12/80)\n", " omega_global (chain, draw, omega_global_dim_0) float64 ...\n", " betaX_Canada (chain, draw, betaX_Canada_dim_0, betaX_Canada_dim_1) float64 ...\n", " noise_chol_South Africa (chain, draw, noise_chol_South Africa_dim_0) float64 ...\n", " omega_New Zealand (chain, draw, omega_New Zealand_dim_0, omega_New Zealand_dim_1) float64 ...\n", " lag_coefs_United States (chain, draw, equations, lags, cross_vars) float64 ...\n", " betaX_United States (chain, draw, betaX_United States_dim_0, betaX_United States_dim_1) float64 ...\n", " ... ...\n", " noise_chol_Chile (chain, draw, noise_chol_Chile_dim_0) float64 ...\n", " z_scale_alpha_United States (chain, draw) float64 0.2131 ... 0....\n", " noise_chol_Ireland_stds (chain, draw, noise_chol_Ireland_stds_dim_0) float64 ...\n", " beta_hat_location (chain, draw) float64 -0.1648 ... 0...\n", " rho (chain, draw) float64 0.4541 ... 0....\n", " z_scale_alpha_Canada (chain, draw) float64 0.06378 ... 0...\n", "Attributes:\n", " created_at: 2023-02-21T19:22:35.383920\n", " arviz_version: 0.14.0\n", " inference_library: pymc\n", " inference_library_version: 5.0.1
<xarray.Dataset>\n", "Dimensions: (chain: 1, draw: 500, obs_Ireland_dim_0: 49,\n", " obs_Ireland_dim_1: 3,\n", " obs_South Africa_dim_0: 49,\n", " obs_South Africa_dim_1: 3,\n", " obs_United States_dim_0: 46,\n", " obs_United States_dim_1: 3, obs_Chile_dim_0: 49,\n", " obs_Chile_dim_1: 3, obs_Canada_dim_0: 22,\n", " obs_Canada_dim_1: 3, obs_New Zealand_dim_0: 41,\n", " obs_New Zealand_dim_1: 3,\n", " obs_United Kingdom_dim_0: 49,\n", " obs_United Kingdom_dim_1: 3,\n", " obs_Australia_dim_0: 49, obs_Australia_dim_1: 3)\n", "Coordinates: (12/18)\n", " * chain (chain) int64 0\n", " * draw (draw) int64 0 1 2 3 4 5 ... 495 496 497 498 499\n", " * obs_Ireland_dim_0 (obs_Ireland_dim_0) int64 0 1 2 3 ... 45 46 47 48\n", " * obs_Ireland_dim_1 (obs_Ireland_dim_1) int64 0 1 2\n", " * obs_South Africa_dim_0 (obs_South Africa_dim_0) int64 0 1 2 ... 46 47 48\n", " * obs_South Africa_dim_1 (obs_South Africa_dim_1) int64 0 1 2\n", " ... ...\n", " * obs_New Zealand_dim_0 (obs_New Zealand_dim_0) int64 0 1 2 3 ... 38 39 40\n", " * obs_New Zealand_dim_1 (obs_New Zealand_dim_1) int64 0 1 2\n", " * obs_United Kingdom_dim_0 (obs_United Kingdom_dim_0) int64 0 1 2 ... 47 48\n", " * obs_United Kingdom_dim_1 (obs_United Kingdom_dim_1) int64 0 1 2\n", " * obs_Australia_dim_0 (obs_Australia_dim_0) int64 0 1 2 3 ... 46 47 48\n", " * obs_Australia_dim_1 (obs_Australia_dim_1) int64 0 1 2\n", "Data variables:\n", " obs_Ireland (chain, draw, obs_Ireland_dim_0, obs_Ireland_dim_1) float64 ...\n", " obs_South Africa (chain, draw, obs_South Africa_dim_0, obs_South Africa_dim_1) float64 ...\n", " obs_United States (chain, draw, obs_United States_dim_0, obs_United States_dim_1) float64 ...\n", " obs_Chile (chain, draw, obs_Chile_dim_0, obs_Chile_dim_1) float64 ...\n", " obs_Canada (chain, draw, obs_Canada_dim_0, obs_Canada_dim_1) float64 ...\n", " obs_New Zealand (chain, draw, obs_New Zealand_dim_0, obs_New Zealand_dim_1) float64 ...\n", " obs_United Kingdom (chain, draw, obs_United Kingdom_dim_0, obs_United Kingdom_dim_1) float64 ...\n", " obs_Australia (chain, draw, obs_Australia_dim_0, obs_Australia_dim_1) float64 ...\n", "Attributes:\n", " created_at: 2023-02-21T19:22:35.399482\n", " arviz_version: 0.14.0\n", " inference_library: pymc\n", " inference_library_version: 5.0.1
<xarray.Dataset>\n", "Dimensions: (obs_Australia_dim_0: 49, obs_Australia_dim_1: 3,\n", " obs_Canada_dim_0: 22, obs_Canada_dim_1: 3,\n", " obs_Chile_dim_0: 49, obs_Chile_dim_1: 3,\n", " obs_Ireland_dim_0: 49, obs_Ireland_dim_1: 3,\n", " obs_New Zealand_dim_0: 41,\n", " obs_New Zealand_dim_1: 3,\n", " obs_South Africa_dim_0: 49,\n", " obs_South Africa_dim_1: 3,\n", " obs_United Kingdom_dim_0: 49,\n", " obs_United Kingdom_dim_1: 3,\n", " obs_United States_dim_0: 46,\n", " obs_United States_dim_1: 3)\n", "Coordinates: (12/16)\n", " * obs_Australia_dim_0 (obs_Australia_dim_0) int64 0 1 2 3 ... 46 47 48\n", " * obs_Australia_dim_1 (obs_Australia_dim_1) int64 0 1 2\n", " * obs_Canada_dim_0 (obs_Canada_dim_0) int64 0 1 2 3 4 ... 18 19 20 21\n", " * obs_Canada_dim_1 (obs_Canada_dim_1) int64 0 1 2\n", " * obs_Chile_dim_0 (obs_Chile_dim_0) int64 0 1 2 3 4 ... 45 46 47 48\n", " * obs_Chile_dim_1 (obs_Chile_dim_1) int64 0 1 2\n", " ... ...\n", " * obs_South Africa_dim_0 (obs_South Africa_dim_0) int64 0 1 2 ... 46 47 48\n", " * obs_South Africa_dim_1 (obs_South Africa_dim_1) int64 0 1 2\n", " * obs_United Kingdom_dim_0 (obs_United Kingdom_dim_0) int64 0 1 2 ... 47 48\n", " * obs_United Kingdom_dim_1 (obs_United Kingdom_dim_1) int64 0 1 2\n", " * obs_United States_dim_0 (obs_United States_dim_0) int64 0 1 2 ... 43 44 45\n", " * obs_United States_dim_1 (obs_United States_dim_1) int64 0 1 2\n", "Data variables:\n", " obs_Australia (obs_Australia_dim_0, obs_Australia_dim_1) float64 ...\n", " obs_Canada (obs_Canada_dim_0, obs_Canada_dim_1) float64 0....\n", " obs_Chile (obs_Chile_dim_0, obs_Chile_dim_1) float64 -0.0...\n", " obs_Ireland (obs_Ireland_dim_0, obs_Ireland_dim_1) float64 ...\n", " obs_New Zealand (obs_New Zealand_dim_0, obs_New Zealand_dim_1) float64 ...\n", " obs_South Africa (obs_South Africa_dim_0, obs_South Africa_dim_1) float64 ...\n", " obs_United Kingdom (obs_United Kingdom_dim_0, obs_United Kingdom_dim_1) float64 ...\n", " obs_United States (obs_United States_dim_0, obs_United States_dim_1) float64 ...\n", "Attributes:\n", " created_at: 2023-02-21T19:22:35.401625\n", " arviz_version: 0.14.0\n", " inference_library: pymc\n", " inference_library_version: 5.0.1
\n", " | mean | \n", "sd | \n", "hdi_3% | \n", "hdi_97% | \n", "mcse_mean | \n", "mcse_sd | \n", "ess_bulk | \n", "ess_tail | \n", "r_hat | \n", "
---|---|---|---|---|---|---|---|---|---|
rho | \n", "0.974 | \n", "0.006 | \n", "0.962 | \n", "0.984 | \n", "0.000 | \n", "0.000 | \n", "749.0 | \n", "1354.0 | \n", "1.00 | \n", "
alpha_hat_location | \n", "0.022 | \n", "0.002 | \n", "0.018 | \n", "0.026 | \n", "0.000 | \n", "0.000 | \n", "1642.0 | \n", "3651.0 | \n", "1.00 | \n", "
alpha_hat_scale | \n", "0.047 | \n", "0.012 | \n", "0.027 | \n", "0.069 | \n", "0.000 | \n", "0.000 | \n", "2933.0 | \n", "1097.0 | \n", "1.00 | \n", "
beta_hat_location | \n", "0.029 | \n", "0.009 | \n", "0.014 | \n", "0.046 | \n", "0.000 | \n", "0.000 | \n", "855.0 | \n", "2218.0 | \n", "1.00 | \n", "
beta_hat_scale | \n", "0.186 | \n", "0.064 | \n", "0.077 | \n", "0.305 | \n", "0.003 | \n", "0.002 | \n", "312.0 | \n", "647.0 | \n", "1.01 | \n", "
z_scale_alpha_Ireland | \n", "0.264 | \n", "0.147 | \n", "0.070 | \n", "0.508 | \n", "0.002 | \n", "0.002 | \n", "2074.0 | \n", "854.0 | \n", "1.00 | \n", "
z_scale_alpha_United States | \n", "0.132 | \n", "0.066 | \n", "0.045 | \n", "0.245 | \n", "0.001 | \n", "0.001 | \n", "8457.0 | \n", "5680.0 | \n", "1.00 | \n", "
z_scale_beta_Ireland | \n", "0.508 | \n", "0.326 | \n", "0.096 | \n", "1.067 | \n", "0.014 | \n", "0.010 | \n", "592.0 | \n", "1294.0 | \n", "1.00 | \n", "
z_scale_beta_United States | \n", "0.213 | \n", "0.125 | \n", "0.050 | \n", "0.445 | \n", "0.005 | \n", "0.003 | \n", "646.0 | \n", "1423.0 | \n", "1.00 | \n", "
alpha_Ireland[dl_gdp] | \n", "0.035 | \n", "0.008 | \n", "0.020 | \n", "0.048 | \n", "0.000 | \n", "0.000 | \n", "3446.0 | \n", "4001.0 | \n", "1.00 | \n", "
alpha_Ireland[dl_cons] | \n", "0.014 | \n", "0.006 | \n", "0.003 | \n", "0.024 | \n", "0.000 | \n", "0.000 | \n", "1597.0 | \n", "4464.0 | \n", "1.00 | \n", "
alpha_Ireland[dl_gfcf] | \n", "0.024 | \n", "0.011 | \n", "0.002 | \n", "0.046 | \n", "0.000 | \n", "0.000 | \n", "6768.0 | \n", "5217.0 | \n", "1.00 | \n", "
alpha_United States[dl_gdp] | \n", "0.021 | \n", "0.003 | \n", "0.015 | \n", "0.026 | \n", "0.000 | \n", "0.000 | \n", "2322.0 | \n", "3773.0 | \n", "1.00 | \n", "
alpha_United States[dl_cons] | \n", "0.020 | \n", "0.003 | \n", "0.015 | \n", "0.025 | \n", "0.000 | \n", "0.000 | \n", "2164.0 | \n", "3321.0 | \n", "1.00 | \n", "
alpha_United States[dl_gfcf] | \n", "0.024 | \n", "0.005 | \n", "0.015 | \n", "0.034 | \n", "0.000 | \n", "0.000 | \n", "4455.0 | \n", "5176.0 | \n", "1.00 | \n", "
omega_global_corr[0, 0] | \n", "1.000 | \n", "0.000 | \n", "1.000 | \n", "1.000 | \n", "0.000 | \n", "0.000 | \n", "8000.0 | \n", "8000.0 | \n", "NaN | \n", "
omega_global_corr[0, 1] | \n", "0.980 | \n", "0.019 | \n", "0.944 | \n", "1.000 | \n", "0.000 | \n", "0.000 | \n", "2393.0 | \n", "2273.0 | \n", "1.00 | \n", "
omega_global_corr[0, 2] | \n", "0.916 | \n", "0.033 | \n", "0.854 | \n", "0.977 | \n", "0.001 | \n", "0.001 | \n", "915.0 | \n", "763.0 | \n", "1.00 | \n", "
omega_global_corr[1, 0] | \n", "0.980 | \n", "0.019 | \n", "0.944 | \n", "1.000 | \n", "0.000 | \n", "0.000 | \n", "2393.0 | \n", "2273.0 | \n", "1.00 | \n", "
omega_global_corr[1, 1] | \n", "1.000 | \n", "0.000 | \n", "1.000 | \n", "1.000 | \n", "0.000 | \n", "0.000 | \n", "7937.0 | \n", "7898.0 | \n", "1.00 | \n", "
omega_global_corr[1, 2] | \n", "0.879 | \n", "0.045 | \n", "0.794 | \n", "0.960 | \n", "0.001 | \n", "0.001 | \n", "1345.0 | \n", "3018.0 | \n", "1.00 | \n", "
omega_global_corr[2, 0] | \n", "0.916 | \n", "0.033 | \n", "0.854 | \n", "0.977 | \n", "0.001 | \n", "0.001 | \n", "915.0 | \n", "763.0 | \n", "1.00 | \n", "
omega_global_corr[2, 1] | \n", "0.879 | \n", "0.045 | \n", "0.794 | \n", "0.960 | \n", "0.001 | \n", "0.001 | \n", "1345.0 | \n", "3018.0 | \n", "1.00 | \n", "
omega_global_corr[2, 2] | \n", "1.000 | \n", "0.000 | \n", "1.000 | \n", "1.000 | \n", "0.000 | \n", "0.000 | \n", "8042.0 | \n", "7890.0 | \n", "1.00 | \n", "
lag_coefs_Ireland[dl_gdp, 1, dl_gdp] | \n", "0.012 | \n", "0.080 | \n", "-0.151 | \n", "0.160 | \n", "0.001 | \n", "0.001 | \n", "5559.0 | \n", "3646.0 | \n", "1.00 | \n", "
lag_coefs_Ireland[dl_gdp, 1, dl_cons] | \n", "0.007 | \n", "0.086 | \n", "-0.161 | \n", "0.174 | \n", "0.001 | \n", "0.002 | \n", "4661.0 | \n", "3294.0 | \n", "1.00 | \n", "
lag_coefs_Ireland[dl_gdp, 1, dl_gfcf] | \n", "0.057 | \n", "0.037 | \n", "-0.009 | \n", "0.128 | \n", "0.001 | \n", "0.000 | \n", "4699.0 | \n", "4249.0 | \n", "1.00 | \n", "
lag_coefs_Ireland[dl_gdp, 2, dl_gdp] | \n", "-0.005 | \n", "0.087 | \n", "-0.176 | \n", "0.152 | \n", "0.002 | \n", "0.002 | \n", "3099.0 | \n", "2447.0 | \n", "1.00 | \n", "
lag_coefs_Ireland[dl_gdp, 2, dl_cons] | \n", "0.003 | \n", "0.088 | \n", "-0.171 | \n", "0.171 | \n", "0.001 | \n", "0.001 | \n", "4239.0 | \n", "2997.0 | \n", "1.00 | \n", "
lag_coefs_Ireland[dl_gdp, 2, dl_gfcf] | \n", "0.104 | \n", "0.041 | \n", "0.027 | \n", "0.180 | \n", "0.001 | \n", "0.001 | \n", "1785.0 | \n", "4172.0 | \n", "1.00 | \n", "
lag_coefs_Ireland[dl_cons, 1, dl_gdp] | \n", "0.132 | \n", "0.081 | \n", "-0.000 | \n", "0.287 | \n", "0.002 | \n", "0.002 | \n", "1109.0 | \n", "1022.0 | \n", "1.00 | \n", "
lag_coefs_Ireland[dl_cons, 1, dl_cons] | \n", "0.158 | \n", "0.110 | \n", "-0.004 | \n", "0.376 | \n", "0.003 | \n", "0.002 | \n", "1227.0 | \n", "2390.0 | \n", "1.00 | \n", "
lag_coefs_Ireland[dl_cons, 1, dl_gfcf] | \n", "-0.031 | \n", "0.028 | \n", "-0.082 | \n", "0.023 | \n", "0.001 | \n", "0.000 | \n", "1756.0 | \n", "4031.0 | \n", "1.00 | \n", "
lag_coefs_Ireland[dl_cons, 2, dl_gdp] | \n", "0.018 | \n", "0.069 | \n", "-0.120 | \n", "0.147 | \n", "0.001 | \n", "0.001 | \n", "5040.0 | \n", "3039.0 | \n", "1.00 | \n", "
lag_coefs_Ireland[dl_cons, 2, dl_cons] | \n", "0.048 | \n", "0.073 | \n", "-0.095 | \n", "0.189 | \n", "0.001 | \n", "0.001 | \n", "5765.0 | \n", "3448.0 | \n", "1.00 | \n", "
lag_coefs_Ireland[dl_cons, 2, dl_gfcf] | \n", "0.071 | \n", "0.025 | \n", "0.021 | \n", "0.116 | \n", "0.000 | \n", "0.000 | \n", "5169.0 | \n", "5742.0 | \n", "1.00 | \n", "
lag_coefs_Ireland[dl_gfcf, 1, dl_gdp] | \n", "0.091 | \n", "0.118 | \n", "-0.094 | \n", "0.332 | \n", "0.003 | \n", "0.002 | \n", "2232.0 | \n", "1385.0 | \n", "1.00 | \n", "
lag_coefs_Ireland[dl_gfcf, 1, dl_cons] | \n", "0.058 | \n", "0.099 | \n", "-0.136 | \n", "0.254 | \n", "0.002 | \n", "0.002 | \n", "5377.0 | \n", "2440.0 | \n", "1.00 | \n", "
lag_coefs_Ireland[dl_gfcf, 1, dl_gfcf] | \n", "0.064 | \n", "0.076 | \n", "-0.076 | \n", "0.216 | \n", "0.001 | \n", "0.001 | \n", "4693.0 | \n", "3703.0 | \n", "1.00 | \n", "
lag_coefs_Ireland[dl_gfcf, 2, dl_gdp] | \n", "0.050 | \n", "0.092 | \n", "-0.121 | \n", "0.232 | \n", "0.001 | \n", "0.001 | \n", "6475.0 | \n", "3196.0 | \n", "1.00 | \n", "
lag_coefs_Ireland[dl_gfcf, 2, dl_cons] | \n", "0.024 | \n", "0.096 | \n", "-0.154 | \n", "0.221 | \n", "0.001 | \n", "0.002 | \n", "5052.0 | \n", "3012.0 | \n", "1.00 | \n", "
lag_coefs_Ireland[dl_gfcf, 2, dl_gfcf] | \n", "-0.048 | \n", "0.093 | \n", "-0.239 | \n", "0.102 | \n", "0.002 | \n", "0.002 | \n", "1904.0 | \n", "2237.0 | \n", "1.00 | \n", "
lag_coefs_United States[dl_gdp, 1, dl_gdp] | \n", "0.033 | \n", "0.044 | \n", "-0.043 | \n", "0.122 | \n", "0.001 | \n", "0.001 | \n", "2364.0 | \n", "1396.0 | \n", "1.00 | \n", "
lag_coefs_United States[dl_gdp, 1, dl_cons] | \n", "0.031 | \n", "0.042 | \n", "-0.044 | \n", "0.119 | \n", "0.001 | \n", "0.001 | \n", "3946.0 | \n", "2353.0 | \n", "1.00 | \n", "
lag_coefs_United States[dl_gdp, 1, dl_gfcf] | \n", "-0.008 | \n", "0.031 | \n", "-0.072 | \n", "0.043 | \n", "0.001 | \n", "0.001 | \n", "1330.0 | \n", "2254.0 | \n", "1.00 | \n", "
lag_coefs_United States[dl_gdp, 2, dl_gdp] | \n", "0.036 | \n", "0.042 | \n", "-0.037 | \n", "0.121 | \n", "0.001 | \n", "0.001 | \n", "3903.0 | \n", "2216.0 | \n", "1.00 | \n", "
lag_coefs_United States[dl_gdp, 2, dl_cons] | \n", "0.048 | \n", "0.047 | \n", "-0.033 | \n", "0.141 | \n", "0.001 | \n", "0.001 | \n", "1335.0 | \n", "1481.0 | \n", "1.00 | \n", "
lag_coefs_United States[dl_gdp, 2, dl_gfcf] | \n", "0.027 | \n", "0.028 | \n", "-0.027 | \n", "0.076 | \n", "0.000 | \n", "0.000 | \n", "3760.0 | \n", "1877.0 | \n", "1.00 | \n", "
lag_coefs_United States[dl_cons, 1, dl_gdp] | \n", "0.024 | \n", "0.041 | \n", "-0.060 | \n", "0.102 | \n", "0.001 | \n", "0.001 | \n", "5468.0 | \n", "2430.0 | \n", "1.00 | \n", "
lag_coefs_United States[dl_cons, 1, dl_cons] | \n", "0.044 | \n", "0.047 | \n", "-0.031 | \n", "0.141 | \n", "0.001 | \n", "0.001 | \n", "1768.0 | \n", "1738.0 | \n", "1.00 | \n", "
lag_coefs_United States[dl_cons, 1, dl_gfcf] | \n", "0.007 | \n", "0.031 | \n", "-0.053 | \n", "0.060 | \n", "0.001 | \n", "0.001 | \n", "1909.0 | \n", "2312.0 | \n", "1.00 | \n", "
lag_coefs_United States[dl_cons, 2, dl_gdp] | \n", "0.039 | \n", "0.044 | \n", "-0.033 | \n", "0.131 | \n", "0.001 | \n", "0.001 | \n", "2644.0 | \n", "1843.0 | \n", "1.00 | \n", "
lag_coefs_United States[dl_cons, 2, dl_cons] | \n", "0.036 | \n", "0.042 | \n", "-0.040 | \n", "0.116 | \n", "0.001 | \n", "0.001 | \n", "4171.0 | \n", "1978.0 | \n", "1.00 | \n", "
lag_coefs_United States[dl_cons, 2, dl_gfcf] | \n", "0.031 | \n", "0.027 | \n", "-0.018 | \n", "0.084 | \n", "0.000 | \n", "0.000 | \n", "4192.0 | \n", "2149.0 | \n", "1.00 | \n", "
lag_coefs_United States[dl_gfcf, 1, dl_gdp] | \n", "0.039 | \n", "0.045 | \n", "-0.045 | \n", "0.129 | \n", "0.001 | \n", "0.001 | \n", "3809.0 | \n", "1798.0 | \n", "1.00 | \n", "
lag_coefs_United States[dl_gfcf, 1, dl_cons] | \n", "0.034 | \n", "0.045 | \n", "-0.046 | \n", "0.125 | \n", "0.001 | \n", "0.001 | \n", "4896.0 | \n", "2476.0 | \n", "1.00 | \n", "
lag_coefs_United States[dl_gfcf, 1, dl_gfcf] | \n", "0.075 | \n", "0.057 | \n", "-0.005 | \n", "0.189 | \n", "0.002 | \n", "0.002 | \n", "746.0 | \n", "1317.0 | \n", "1.00 | \n", "
lag_coefs_United States[dl_gfcf, 2, dl_gdp] | \n", "0.016 | \n", "0.047 | \n", "-0.078 | \n", "0.102 | \n", "0.001 | \n", "0.001 | \n", "4171.0 | \n", "1975.0 | \n", "1.00 | \n", "
lag_coefs_United States[dl_gfcf, 2, dl_cons] | \n", "0.018 | \n", "0.047 | \n", "-0.078 | \n", "0.104 | \n", "0.001 | \n", "0.001 | \n", "4508.0 | \n", "1892.0 | \n", "1.00 | \n", "
lag_coefs_United States[dl_gfcf, 2, dl_gfcf] | \n", "0.015 | \n", "0.043 | \n", "-0.072 | \n", "0.091 | \n", "0.001 | \n", "0.001 | \n", "2937.0 | \n", "2034.0 | \n", "1.00 | \n", "
\n", " | GDP | \n", "CONS | \n", "GFCF | \n", "
---|---|---|---|
GDP | \n", "1.000 | \n", "0.980 | \n", "0.916 | \n", "
CONS | \n", "0.980 | \n", "1.000 | \n", "0.879 | \n", "
GFCF | \n", "0.916 | \n", "0.879 | \n", "1.000 | \n", "