Multivariate Time Series Analysis/VAR and SVAR Model Analysis.ipynb

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    "# Vector Autoregressive Models\n",
    "\n",
    "VAR Models are used to analyse the interdependencies among different variables of interest, i.e. how one endogenous variable responds to a shock in another. By analysis Autoregressive Models in a multivariate space, we can get a sense of how different variables interact with each other.\n",
    "\n",
    "**Main Goal of VAR Models:** Impulse Response Functions for analysing contemporaneous effects.\n",
    "\n",
    "-------------------------------\n",
    "Apply VAR (look at significance) $\\Rightarrow$ Impulse Response Function $\\Rightarrow$ Apply Restrictions $\\Rightarrow$ Fit SVAR $\\Rightarrow$ Impulse Response Function\n",
    "\n",
    "--------------------------------\n",
    "\n",
    "\n",
    "## 1. Reduced VAR Models\n",
    "\n",
    "The VAR model specifies the evolution of each variable as a linear function of the lagged values of all other variables.\n",
    "\n",
    "<p style=\"text-align:center\"> $y_t = A_1y_{t_1} + ... + A_p y_{t-p} + u_t$ </p>\n",
    "\n",
    "$A_i$ are $nxn$ coefficient matrices $\\forall i = 1, ..., p$\n",
    "\n",
    "$u_t$ is a vector white noise with a positive definite covariance matrix which assumes a lack of autocorelation but allows contemporaneous correlation between the components, i.e. $u_t$ has a non-diagonal covariance matrix.\n",
    "\n",
    "**VAR(1):** In two variables\n",
    "\n",
    "<p style = \"text-align:center\"> $\\begin{bmatrix} y_{1,t} \\\\ y_{2,t} \\end{bmatrix} = \\begin{bmatrix} c_1 \\\\ c_2 \\end{bmatrix} + \\begin{bmatrix} a_{1,1} & a_{1,2} \\\\ a_{2,1} & a_{2,2} \\end{bmatrix} \\begin{bmatrix} y_{1,t-1} \\\\ y_{2,t-1} \\end{bmatrix} + \\begin{bmatrix} e_{1,t} \\\\ e_{2,t}\\end{bmatrix}$ </p>\n",
    "\n",
    "<p style = \"text-align:center\"> $\\Leftrightarrow\n",
    "$ </p> \n",
    "\n",
    "<p style = \"text-align:center\"> $y_{1,t} = c_1 + a_{1,1}y_{1,t-1} + a_{1,2}y_{2, t-1} + e_{1,t}\n",
    "$ </p> \n",
    "\n",
    "<p style = \"text-align:center\"> $y_{2,t} = c_2 + a_{2,1}y_{1,t-1} + a_{2,2}y_{2, t-1} + e_{2,t}\n",
    "$ </p> \n",
    "\n",
    "### Writing VAR(P) as VAR(1) \n",
    "\n",
    "The transformation amounts to stacking the lags of the *VAR(p)* variable in the new *VAR(1)* dependent variable and appending identities to complete the number of equations.\n",
    "\n",
    "<p style=\"text-align:center\"> $y_t = c + A_1y_{t_1} + A_2 y_{t-2} + e_t$ </p>\n",
    "\n",
    "Can be recast as the VAR(1) model:\n",
    "\n",
    "<p style = \"text-align:center\"> $\\begin{bmatrix} y_{t} \\\\ y_{t-1} \\end{bmatrix} = \\begin{bmatrix} c \\\\ 0 \\end{bmatrix} + \\begin{bmatrix} A_{1} & A_{2} \\\\ I & 0 \\end{bmatrix} \\begin{bmatrix} y_{1,t-1} \\\\ y_{2,t-1} \\end{bmatrix} + \\begin{bmatrix} e_{t} \\\\ 0\\end{bmatrix}$ </p>\n",
    "\n",
    "### Absence of Contemporaneous Effects in a Reduced VAR Model\n",
    "\n",
    "**Features of Reduced VAR models:** Contemporaneous dependencies are not explicitly modeled. This allows us to estimate the model by OLS applied equation-to-equation.\n",
    "\n",
    "**Contemporaneous effects $\\nexists$:** are not modeled $\\Rightarrow$ such dependencies appear in the correlation structure of the error term, i.e. the covariance matrix of $u_t$\n",
    "\n",
    "<p style = \"text-align:center\"> $\\Sigma_{u_t} = E[u_t,u_t']$ </p>\n",
    "\n",
    "Given the assumptions of the error term $u_t$:\n",
    "\n",
    "$E(u_t) = 0$\n",
    "\n",
    "$E(u_t, u_t') = \\Sigma $ are the Contemporaneous effects\n",
    "\n",
    "$E(u_t, u_s') = 0$\n",
    "\n",
    "\n",
    "<p style = \"text-align:center\"> $var(u) = \\begin{bmatrix} var(u_1) & 0 & . & . & . &  0\n",
    "\\\\ 0 & var(u_2) & . & . & . &  0 \\\\ . & . & . & . & . & . \\\\ . & . & . & . & . & .  \\\\ 0 & 0 & . & . & . &  var(u_n)\\end{bmatrix}$</p>\n",
    "\n",
    "Simplified to: \n",
    "\n",
    "<p style = \"text-align:center\">  $\\Sigma_{u_t} = E[u_t,u_t'] = Var(u)$ </p>\n",
    "\n",
    "Since conteporaneous effects are not modelled in the reduced VAR form, they appear in the variance-covariance matrix structure of the residuals $\\Rightarrow$ the matrix is **Positive Semi-Definite**.\n",
    "\n",
    "**Contemporaneous Effects:** Variables can have effects on other variables (e.g. $y_{2t}$ can have an effect on $y_{1t}$). In the reduced form VAR, all right hand side variables are predetermined at time *t*, meaning there are no time *t* endogenous variables on the right side, no variable has a *direct* contemporaneous effect on other variables in the model.\n",
    "\n",
    "\n",
    "## 2. Structural VAR Models\n",
    "\n",
    "Assuming there are no contemporaneous effects would be misleading for the Impulse Response Function, i.e. changes in the processes followed by a shock hitting a particular variable.\n",
    "\n",
    "**SVAR models:** They explicitly model contemporaneous effects among variables:\n",
    "\n",
    "Multiply the reduced VAR by $B_0$:\n",
    "\n",
    "<p style = \"text-align:center\"> $B_0y_t = B_0A_1y_{t-1} + ... + B_0A_py_{y-p} + B_0e_t$ </p?\n",
    "\n",
    "Define:\n",
    "\n",
    "$B_i = B_0A_i$\n",
    "\n",
    "$\\epsilon_t = B_0e_t$\n",
    "\n",
    "$\\epsilon_t$ are the structural shocks that satisfy the same conditions as the above residuals\n",
    "\n",
    "**SVAR(p):**\n",
    "\n",
    "<p style = \"text-align:center\"> $B_0y_t = B_1y_{t-1} + ... + B_py_{y-p} + \\epsilon_t$ </p>\n",
    "\n",
    "**SVAR(1)** with two variables\n",
    "\n",
    "<p style = \"text-align:center\"> $\\begin{bmatrix} 1 & B_{0;1,2} \\\\ B_{0;2,1} & 1 \\end{bmatrix} \\begin{bmatrix} y_{1,t} \\\\ y_{2,t} \\end{bmatrix} = \\begin{bmatrix} c_{0;1} \\\\ c_{0;2} \\end{bmatrix} + \\begin{bmatrix} B_{1;1,1} & B_{1;1,2} \\\\ B_{1;2,1} & B_{1;2,2} \\end{bmatrix} \\begin{bmatrix} y_{1,t-1} \\\\ y_{2,t-1} \\end{bmatrix} + \\begin{bmatrix} \\epsilon_{1,t} \\\\ \\epsilon_{2,t}\\end{bmatrix}$ </p>\n",
    "\n",
    "Where:\n",
    "\n",
    "$\\Sigma = E(\\epsilon_t, \\epsilon_t') = \\begin{bmatrix} \\sigma^2_1 & 0 \\\\ 0 & \\sigma^2_2 \\end{bmatrix} $\n",
    "\n",
    "Therefore, the **structural form** can be obtained from the **reduced form** by multypling it with an appropriate parameter matrix $A$, which reflecte the **contemporaneous effects** among variables.\n",
    "\n",
    "**Features of the Structural Form:** Gives the underlying structural economic relationships\n",
    "\n",
    "1. **Error Terms are not Correlated:** The structural, economic shocks which drive the dynamics of the economic variables are assumed to be independent, which implies Zero correlation between the error terms and a desired property.\n",
    "\n",
    "2. **Variables can have a Contemporaneous Impact on other variables:** This is a desirable feature especially when using low frequency data.\n",
    "\n",
    "3. **Parameter Identification Problem:** Paraneters in $B$ cannot be estimated without additional restrictions.\n",
    "\n",
    "**Error terms in the Reduced Form VAR:** are composites of the structural shocks $e_t = B_0^-1\\epsilon_t$ therefore, the occurrence of one structural shock $\\epsilon_{i,t}$ can potentially lead to the occurrence of shocks in all error terms $e_{j,t}$ thus creating contemporaneous movement in all endogenous variables.\n",
    "\n",
    "This gives us the covariance matrix of the Reduced VAR:\n",
    "\n",
    "<p style=\"text-align:center\"> $\\Omega = E(e_t, e_t') = E(B_0^{-1}\\epsilon_t\\epsilon_t'(B_0^{-1})') = B_0^{-1}\\Sigma(B_0^-1)'$ </p>\n",
    "\n",
    "This can have non-zero off diagonal elements, thus allowing non-zero correlation between error terms.\n",
    "\n",
    "<p style = \"text-align:center\"> $var(u) = \\begin{bmatrix} \\sigma_1^2 & cov(u_1, u_2) & . & . & . &  cov(u_1, u_n)\n",
    "\\\\ cov(u_1, u_2) & \\sigma_2^2 & . & . & . &  cov(u_2, u_n) \\\\ . & . & . & . & . & . \\\\ . & . & . & . & . & .  \\\\ cov(u_1, u_n) & cov(u_2, u_n) & . & . & . &  \\sigma_n^2\\end{bmatrix}$ </p>"
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    "# VAR Implementation Example\n",
    "\n",
    "We build a three-component VAR model from the following components\n",
    "\n",
    "- Equity Return: This specifies the Microsoft price index from January 01, 2004 to March 03, 2014\n",
    "\n",
    "- Stock Index: specifies the S&P500 index from January 01, 2004 to March 03, 2014\n",
    "\n",
    "- US Treasury bond interest rates: from January 01, 2004 to March 03, 2014\n",
    "\n",
    "**Goal:** to make a forecast for the stock market index by using the additional variables and to identify impulse responses.\n",
    "\n",
    "We suppose there are hidden long-term relationships between a given stock, the stock markets as a whole and the bond market.\n",
    "\n",
    "## Procedure:\n",
    "\n",
    "### 1. Data Cleaning: \n",
    "\n",
    "- Transform Prices to Log-Returns (Take the first difference), i.e. Non-Stationary $\\Rightarrow$ Stationary\n",
    "\n",
    "- Check for NA Values and if present, omit them and merge the data\n",
    "\n",
    "### 2. Fit the VAR Model\n",
    "\n",
    "- Lower the frequency (if working with daily returns) to monthly or annualy\n",
    "\n",
    "- Run the VAR Model with k max lags\n",
    "\n",
    "- Evaluate the Model Output based on Stationarity and Parsimony\n",
    "\n",
    "    - Parsimony can be re-attained by re-estimating without non-significant variables or dropping a lag\n",
    "    \n",
    "### 3. Test for Autocorrelation and Contemporaneous Correlation in the Residuals\n",
    "\n",
    "- Obtain the residuals for each model\n",
    "\n",
    "- **A. Autocorrelations:** Plot the ACF/PACF for each residual \n",
    "\n",
    "- **B. Contemporaneous Correlations:** Plot the cross-correlograms to determine the amount of contemporaneous correlation\n",
    "\n",
    "### 4. Predictions with VAR\n",
    "\n",
    "- Specify how many n-period ahead and construct the forecast\n",
    "\n",
    "- Forecast Variance Decomposition, i.e. how much of the forecats error variance is due to shocks in each equation\n",
    "\n",
    "### 5. Impulse Response Function\n",
    "\n",
    "- Shows how a variable reacts (responds) to a shock (impulse) hitting any other variable in the system"
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     "text": [
      "Warning message:\n",
      "\"package 'vars' was built under R version 3.6.2\"Loading required package: MASS\n",
      "Loading required package: strucchange\n",
      "Warning message:\n",
      "\"package 'strucchange' was built under R version 3.6.2\"Loading required package: zoo\n",
      "Warning message:\n",
      "\"package 'zoo' was built under R version 3.6.2\"\n",
      "Attaching package: 'zoo'\n",
      "\n",
      "The following objects are masked from 'package:base':\n",
      "\n",
      "    as.Date, as.Date.numeric\n",
      "\n",
      "Loading required package: sandwich\n",
      "Warning message:\n",
      "\"package 'sandwich' was built under R version 3.6.2\"Loading required package: urca\n",
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      "\"package 'lmtest' was built under R version 3.6.2\"Warning message:\n",
      "\"package 'quantmod' was built under R version 3.6.2\"Loading required package: xts\n",
      "Warning message:\n",
      "\"package 'xts' was built under R version 3.6.2\"Registered S3 method overwritten by 'xts':\n",
      "  method     from\n",
      "  as.zoo.xts zoo \n",
      "Loading required package: TTR\n",
      "Warning message:\n",
      "\"package 'TTR' was built under R version 3.6.2\"Registered S3 method overwritten by 'quantmod':\n",
      "  method            from\n",
      "  as.zoo.data.frame zoo \n",
      "Version 0.4-0 included new data defaults. See ?getSymbols.\n",
      "'getSymbols' currently uses auto.assign=TRUE by default, but will\n",
      "use auto.assign=FALSE in 0.5-0. You will still be able to use\n",
      "'loadSymbols' to automatically load data. getOption(\"getSymbols.env\")\n",
      "and getOption(\"getSymbols.auto.assign\") will still be checked for\n",
      "alternate defaults.\n",
      "\n",
      "This message is shown once per session and may be disabled by setting \n",
      "options(\"getSymbols.warning4.0\"=FALSE). See ?getSymbols for details.\n",
      "\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "'MSFT'"
      ],
      "text/latex": [
       "'MSFT'"
      ],
      "text/markdown": [
       "'MSFT'"
      ],
      "text/plain": [
       "[1] \"MSFT\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "'SNP'"
      ],
      "text/latex": [
       "'SNP'"
      ],
      "text/markdown": [
       "'SNP'"
      ],
      "text/plain": [
       "[1] \"SNP\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "'DTB3'"
      ],
      "text/latex": [
       "'DTB3'"
      ],
      "text/markdown": [
       "'DTB3'"
      ],
      "text/plain": [
       "[1] \"DTB3\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "           MSFT.Open MSFT.High MSFT.Low MSFT.Close MSFT.Volume MSFT.Adjusted\n",
       "2004-01-02     27.58     27.77    27.33      27.45    44487700      17.81634\n",
       "2004-01-05     27.73     28.18    27.72      28.14    67333700      18.26419\n",
       "2004-01-06     28.19     28.28    28.07      28.24    46950800      18.32909\n",
       "2004-01-07     28.17     28.31    28.01      28.21    54298200      18.30962\n",
       "2004-01-08     28.39     28.48    28.00      28.16    58810800      18.27717\n",
       "2004-01-09     28.03     28.06    27.59      27.66    67079900      17.95264"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "library(\"vars\")\n",
    "library(\"quantmod\")\n",
    "\n",
    "# Get data for the Stock\n",
    "getSymbols(\"MSFT\", from =\"2004-01-02\", to = \"2014-03-31\")\n",
    "\n",
    "# Get data for the stock index\n",
    "getSymbols(\"SNP\", from = \"2004-01-02\", to = \"2014-03-31\")\n",
    "\n",
    "# Get data for the bondcmarket and subset it by the appropriate period\n",
    "getSymbols(\"DTB3\", src=\"FRED\")\n",
    "DTB3.sub <- DTB3['2004-01-02/2014-03-31']\n",
    "\n",
    "# We need to plot the data for the close-to-close daily return\n",
    "head(MSFT)"
   ]
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   "source": [
    "The components of the downloaded data can be reached by using the following:\n",
    "\n",
    "``` r\n",
    "Cl(MSFT) # Closing Price\n",
    "\n",
    "Op(MSFT) # Opening Price\n",
    "\n",
    "Hi(MSFT) # Daily Highest Price\n",
    "\n",
    "Lo(MSFT) # Daily Lowest Price\n",
    "\n",
    "ClCl(MSFT) # Close-to-Close daily Return\n",
    "\n",
    "Ad(MSFT) # Adjusted Close\n",
    "```\n",
    "\n",
    "### Adjusted Closing Price of MSFT Chart"
   ]
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P+un5GU9Fi7u4dLel+ESLBD0r9WPotI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\nrkJggg==",
      "text/plain": [
       "Plot with title \"[2004-01-02/2014-03-28]\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "chartSeries(Ad(MSFT))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As we can see the Adjusted close is non-stationary.\n",
    "\n",
    "## 1. Data Cleaning\n",
    "\n",
    "### Transform the Non-Stationary Series to Stationary Series\n",
    "\n",
    "We will work with log returns"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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pJuZpoESR54DUMSdZC4anwKIfEMJEF0nZB4LySRhSS7iIXE9Z5GqG5SgiR6IZE+\nrggh8V5IwkBSOyUPEkWQGNevs5CIRkMSKSS92RHm7m8D24NEIguJqWsWGhL3IIkcJH1xTV6/\ncJBEDyS1clUz5yGZuyvIzMhCUhf7LSTKQqIAEl0lJCpAIr0GRkASKSSRhaQupnVfqnDTJQJI\nYilIdBYk0QOJfEiUQhIeJKEh6SoYSHovIjxIwkESBpIoQaIRkISDJEJIwkLiDpIoQ6IMJNod\nJBFBEgEkcpBMjxLch0Q+JHVOxASFkEg/GEhDEuZrEANJUuDkIFEIiVdCMu6GIYkEks9FsmoB\nyfQ6w0hBErr/DkBiASThIJlNmQdJVEDSHw1AohQSeZDIQRIaEg1AIgNJDSGzDnYISV/35g6S\nyEASWUgiA4lzspDMyXoekrpxUxYaA4kCSPqSmIZkNnsaku70F4ekamALCrIkBiDxAiQyJc1R\n1WRIfBwkkYMkDCQi9XwZH5IIIJl23yekbgFTSJRCohSSNDMBEs9CojMgyWvFul+r/RLPQKJR\nkHjwyQhIdojpYOZlBhIvQzLT6oEk8pDEaEiiGhKZ0119DCLsKhJDkPRC7BOSCCDp5hiAROai\njjoQdpD07WODkIQHSW3YSG/JfUhc7VocJNURHCTyIPEsJN2fC5CMkAiSOmLph8SnQOIOkv7I\nQmIjINmS8jqeB8mrRQxJ11MvsoEkEkhmeaJzJBoJiV8lJG4giQSSCCGRgSR/K43kdaOJkISg\nHCTbJeXUBDc4JPc+SO7QSVuqh8R8SPaIJQuJeiHxMiQmXIMSd9XKQzLLSLqL67cWkrCQzDV1\n1dUtJBoBiSeQqAiJNCSRhUTqO4oIkt1HqUO/vUIiDUmYjU0tJNKQSH3Hr3dZZjfVABLZKwu6\nPj4kGg9J5CHRJEiUg2Q+TyAJfcol7ABVLfWrhKMg6SNvVVBC4n2QWA4SZSEZkkOQzK7FQHKN\nH0LiISQ6C9LdkR/vbvIDks8uAUl2wwAS6d9LK0EiB4lSSHKTxVNIpssxdVqWQqIUEgWQ9CUF\nykHyjkkGIdFZkKgMiVtIQZ8tQjJzUBcbPUhmt+FB4maIB4k5SE4RqbIaEg1DUjfAmQaKIFHu\n0C6GZJZzTkgP3e8G8MfsgOSzS0LiYyHptuVcHZ7YbRV5zar/jyCJWkh6/n5fdZD0FpSbAgEk\n8iBxXVMfkqciD4lVQOLxF7Kmi0WQBLnpppBEAIl7Y9q20JDsojNhfp2qu2M+gSRKkFxLqRtR\nLSSvgYQwNzdYSGbhPEjCQbIrW1fXfPdl95IGEilIYhKk9/z47vDuyN9nBiSfLQ7JbiS7G3bU\nktZA4laIMhRActvhPCSztZclhyCZw0FHxG7ghb5gZyBRBpKYHVKg3IckN08+JL3oRUhkIVEG\nkgggqWlrSHJ1cX/eASQeQ5LHHn2QeA6SiCFRPSTWANId/3D69yP/nBmQfLZNSKIfkrCQSJjp\nuB2ShaTunfAkmM2j11d1VcuQuBv9XEiqjq0tb5kAACAASURBVKMgqaawl9ybQFJXMNWFjSwk\nGg9JlZCQeC8k0RKS6yyTIN3z29O/7/h9ZkDy2UYgkfmK0eta3O0o7J7BrLkEkr0bgenLT2ri\nWUhmtam7UouQ9Fc/KSQRQ6KZIZlNQx4SNzMwExmEJGJIpHv7eEhqmsz18zwkCiC51eC2nGYN\nGUjeVTvzPXZrSJz7P8IByWdrgOQA2V5WgCT8rpWDxIqQdAn1FEXhQfIkeGuwBMn0sRIkkYUk\nhiDR+ZBEDpI9dPUhUQ6S2U45SJRAUtMxkGgsJGVJQxJeAzEfki5quoDFkINkVgizfUUW0PvU\nfUPibq/D7fIK01MjSLrN8pBs8xovASSykGRnMvMxvxenIbnidg1mIdlyASRzwWsQ0uAeyYNk\n7tWthcRJb+cbQOIxJNejzTzNOoshCQ+SXHJ3nOtBIvM1h4VE/ZC8dd0DiZvVqV0BUhaSIOFd\niU0hmbL6OqjuYTyFxO2h1wAkPRXTO5lwdfEh+QeGdklSSELfsdwEklUeQjKNMQiJLCTyIPGR\nkIS+cjoBkv1Bpo56j0+2nv5sbTdIIJmQ60nmXvccpOIh0sohvS3EhyTs8qaQyLZgCsmNYyEJ\nBemtB+mtnZLthR4kEUHyjv/0UNMlxGmqjo99ZafrQzKnegbSWwfpbQTprX7nQXorZ1YJyfRC\nNWl1tc9BMhOhbkYRJLl96FpKfvRWQ+IG0tsyJLPYRUh2mybevtUN263xtwkk27jdy7cVkNSa\nfmtX/1u3NXGQ3qpZvHUN5tZxqUO+5X78PnyMsXgDks9mg1Ty77bC3uYr6LqZPRKFh3ZlSHaP\nRO6LUbP38S8GFCEJc4nWm4HPpwoSGUjc3yOZyZqNq910WEjy3KFmj2RqriHZy+b6d7ttebuZ\nTiEleyR5U7he/q7GESSKIdlVl4XkfQHuvib1G4pbSGJoj8RMxb1brty3FLJ3RHskloE0fo+k\nrszdxlftbt1Vu9sLXrWbCIlsW5P7qA+S2VjmIJmV7ear+1kIyZPiIJGFXA2JnQtJ1ENidlmm\nQDLLrzqniCGRB4m5A/MBSIaQf3Jk17xwi+xYRJDMTMmemRlIZskcJGoI6bP8rugDv8sMSD5b\nHJI9tPMgmU7M8pC4vjnbhxTskYwJ/W2EgaTv0tT39WQhmS4gbD+2v5nLYkgWteLCA0imDnIG\nPIGkCZj68wSSq8A0SCwLiXmQbIPJvZa6pcdAkr/oyINzJPLm5CB5je6uuY6GxGJI4R7Jfflm\n2tRCciuCWSHDkMQ0SOu+s4EzB4mTv78o7pE0JNe1YkjmF1JZHyRBZt+RhcRsiRIk4d1YPAWS\nML8LMBqSrWkJkr5Jgwm3ydG9jSeQ1C3DGUhyoOBqamYPZbZRUyDZ5jSQzCI5SFSAxAJIrA4S\nbwzp8ChPmx5kf+bRAO/lZSAxD5Jbt6pN7CVVs0a4haSPq8xm3XbePkh6SszskfSIwvQzuy7s\nplyvZiIiB8ncQlQDSegFK+yRxkGyH4eQ/H2E+l40hKQqPQ6SXnYfkkVyFiQKINEkSNx8TZSD\nRDEkpiGJ8yHdyDu8Dw6SN8B7eSFIrpMMQyIDSYSQ7HjjINkvkQYgCdIjyXECSDQASR0cZSDJ\nQ6+JkFxXroQkNFv9Zz7ykEQGEhuEZHbvQ5CEhcRt1cuQRArJfm4gcXILkkBiJUh2whMgjczF\nIHnrVnVU8zsm5K+TDCTZF/sgmXWniHDzizM+JPPbSsOQyMDMQfL3jgVI6mC2DhLVQPIP7UJI\nLIJE8mkv3o04ChKzkIS9+1sd2klI6u4fta0xTTsEyflmpI7imDdL7kGiPCS3wCEkrhtQfzFo\nIQmPilpWBYldDSRRB8k2tv1VYg+SuVrrQyIFSWQh2V/hM78nSvqZa/bknBkTxBQyfZytHxxt\n+vkISEx9F2ruRFaQ7GoXPiSzEa+GpG726N7z9pBYAIm7uWtIfACSupFOb4GEg2RWVwzJfElg\nbt7KQtKPehiERAVIVJS0Y0h6fZiOaiBxPbT7Tz0SXq9H83QZMk8Y8CGxqZBUr1d3GkkUDpJ5\nwqjZy5UhUQUks4uthETCLHkGEtfNoJqHJ5CYgSRKkLpnP3Pz1U0Wkr31oRKSMa+fFEPmd5l9\nSFyYJ2WoI9QCJN0ARUhcjW0h2Z3eDiG5TuK25KR7vAdJN5IQ5qo2mS2xB8ncRilMx/YhUTUk\n7kFi3EJSPUk9xasAScSQ1CrT/TiCJBpAYjlIwkEy2xOuEUSQWA6Smh3TmwADSS++PhnJQFLr\nwt15LheEe5DUV2rmimAMyfu9THUUmELSHlUD8gSSufgTQSLSv0httx07hWR/66cGEhlI5m/X\nC92hHCRmIAm9DoS5DVPt5zxIFEKifkjdZWEPEiWQ9C7I7WnUXjAPSQxCohIkM99eSNxC4hYS\nU5dapkKyf/JjCJIYgMT0oR1zrVADSe33EkjqRnN7qd+HxCJIfOeQTPRmPQtJmB1zAkn+oRQF\niamuH0Pyv/EbhGQeda0hMQupW20+JNuhLSTztC6z3uV/rAQpOrRjDSGxfkimWQ0k5iDpw2Bz\nhiLPkSQknoMkIki8AEm+kb/JZyGxCJL9mwfq8qIsq86NPEjyBgq1C2NTIHlHKXuFZPfO1ZCY\nB4mrtpVnXEyNqSDpv+qj/i1C0ltb0o/gSCDpv34eQuLCrrgMJLM74goSr4JEGUhUDYnGQZJ/\ndcZBUjVLIMn1QH2QmIEkhJ5VCEmYJ48LDxKXp54aEh8BiTEDiSwk2wj2SED90aYeSLRTSDQI\niaeQ1B90VXAMJIogcQuJ692ThcQtJJ6FxBwkMQSJ+ZCED4kpSGwIEmUhsTwkMQISmwjJ3DzF\nzd+4FhqS3ysF2SfHVUDiDhKLIfEEEq+BxLOQuPnrZ0yv2wASXQckexBmIJE6VRHC3KZjIMnI\nZu+HRKIPkuorIoIkNCQKIHEFiQJIogISjYVE3iqfCklwC4k0JFYJids5kfljycK0uQdJBJC0\nG1aCxH1ILILERkNiASTVJagOEts5JNYLiQ9BUn/mVP3NFnW1QO7Xs5CYB0lORv3FOm6uXpjD\nCwtJza4PkrzY4UFiAST1pYa70ToDSRhIJg4Sr4bEAkiC9GFUERKLIZEPifVA0j2zDxLFkLhu\nqAIkvVRqPXK9/0wgMQeJca4PiqkISf+KhrgiSGQhyeMS/cfhLCT1tywNJOFDUrsTXoQkW1zN\nS/519AQS64Vk/rGzZETm/ohBSFzoj8ZDMtetzG9T6b0A56Yr10JiBhL3IYksJE7m2lsIiesm\n0M8ospA4N9ctNCTTaYU5rIwgUQiJ5yDxPCS5aplcOr1OuFBbXEWyBxK7AkjyUHsEJL9fO0i8\nHxLzIHXvSfa5KZAogOSOPhgLLjYw3fEVJNYHSVwSkvAhsQSSXUfMhyTUtzoKklqqiZC67Rov\nQiILSdRCYikkdQAbQGI7hKQOPITtgQ4SiyExA8lOz4ckFCRmILEMJOZDYhEkVgWJ69MpFkJi\nBUgigMRmhcTYWZCEB0n0QuLyV2bl0XgHSXdUNg0SGUiiBxLTnySQKIEkBiEx8lfo3iCRuQYs\nb9wqQWIJJGYO7SJIrBcSU78LxfKQeA8kFkPiQ5DUUVEIicZBotaQaAQk1guJzG0UahtjIdkL\nHfrLBOYg8QiSEDEkdR7rIIkIkjksyUMSPiQWQWL7hcT0xVOhdiwGEhsFSe6RuIPEAkjMQTLv\nmFmt7mPV2qpb90NitZBYBpJQ5+EOEgVX7SohcQdJFCExD5JqmQIkZiExD5K7ppODRAkko/IM\nSGbVVEJiGpK6JK6+MgghcXkX7HVAUn/lxEGiSZB4BpKwkESg4RxIvARJLoa8Rkg8C0nMD0l3\nuG66GpL+DsxCEnYX2A+JDCQRQWIlSEJD4gkkPZscJOZDEikk/WhJ9QcWhyExA4lZSHRVkFgf\nJCYhsQpIzKqphRTUQ19JrYXEEkjMQeIWkp4EMQ8S8yExC8lU4HxI+rr7ECRmIPEIkj37jCHp\ntaB2H7oVSO0LCpDIQGLjIAkHiWcgqdooSELNS84lA+kKDu1ECEltXkiXTyGxCBL1Q2I+JOE1\n3lKQeDtINBGS2/HoZmApJKEfA1SCxBwkwfQRlAeJDUOyE8pAYgaSXVlqNiVItn+Yc98QkiKk\nSl4bJN4HiUJIui/rEhYSG4DE+iFxC4mNgMRCSObDGJKpagSJhiCxPkjsLEjCh8TykFgOkhpa\ngKSametFyENiBpK+CBBAYh4kbg7tWALJWxfDkJgPiV0DJHU9bRQk70KCD8ms71GQ9BxjSMKU\ndyvP9oW2kHgCicZAYiMh8QwkebGY1UHi+qvuIUj2i78IEmO8DImFkNz6sxXXzV0FSTcEuTPi\n64LE6iHJNcooB8mWjRovgGRGcZBYCZJ9W4AkqiCxHCRqB4mPgcQykOyCakheyypI7CxIsrFi\nSARI/RkBKejdlZCYGdYPKW6yEZCizASJzQDJuzinbgwYgKSazd/oqNP7EiRWAUkMQ+K9kLxW\n9yCJKkjCQWJXDYl5kNgoSKwfEpsOyauK+p2jBJLohcQUJF4BifVB0l/nqtsVKyGx+SDpRrWQ\n1HdYZUhMQ+JzQWJMQeLXAokVIJGDFLTmIpBoBCS/FhEktywJJEbmkq8HiXmQWB8k3gSSbvQQ\nEq+CJCwklkBiZUjxtxekZt93juS3dwqJAdJ6IbHlIVEVJJoESRhIpsickGgyJDYNEgshkYXU\nrQ1+NZDshux8SEm3vyQkUYSk1ryFRAYSzQfJbzU5Xh0kuxNLILESJNYKEstCYhEkde6sILEA\nklkbTM1/55DYGiFx/boaUlCLzhXnF4bEpkGiAiQ+DpLIQvIbya4ADcmecgXtWwHJfKJ+eYIl\nkFTBq4TEPEjMQPJaly0EiQ9BUuunDxIbCUk0gMQMJBZDci3RApJrmoaQgsaOIZkX9ZCEKchj\nSMVsGJLZEU2BlE4mHNgAUjnyns0eSJSHxGJIoj0k1guJe5DMoqjbxlNIdBYkloXkr4ARkFz1\nTaUBKai6+VmCxC4Dic6E1E3Mq2gFJD4MiRwk8xWShaS1DENSeyS/FZtCYs0gsWpIwkLigJSD\nxDYGiVlIbAgSN5BYA0iiH5LeWdnlroVkvtrhLIQUNLZaAOF6dT0kfdfD0pB6sitIflOeDSmZ\nj/epa1ezqQog9dVaj9UGkuqL7snc3dGRvk1gI5B4DyQhUgx68g0hsQpIYniV7gBShsJ2IemJ\nDUHiOUgshMQuAInOgCTUN2UjIQn11WnYugVIttIeJOZBYjlIsiGuHlLYuj2QolwKkplYDSTB\n1CAfEquHxPKQ5NhW0zAkZiEJVVndZiMgueUX6rRPzroXkhk9C0l/WFwNESRmqszZlUPyGtc1\nYwxJZEYpZwBSMCf5Q7iD/SpIIoXEPEh+2emQmA+Je5CoLSReDSlcTllg1ZDUUl8vJHY+pOJ8\n4tmoH8Ktk3kgqY+Jq+u1XPdpC4mr27NHQ+IXhKQKRJCYgcR6IJmpWUi5DwHJ5QxI8RWfmSFx\nD9Lw5HshUbgNmASJ5SCpyxHjIdkG0DMNayx3jXw6JG9aISQ2DIkDUl3WD8msEe5KTYekP8xB\nMmrykJg6GO220ZWQmIFkOpOYC1Kw2PHIvZB4PyQxHyTOvB/XBilprVVDim8R8ieZQOJVkJgP\niddBYnlI8p+xkHRl7eznhsTmgxQWB6So9RaCVDGTIUj+JMj+xbISJFtjcido6lcr9L1AvZBY\nS0hsLKTgrb5v1oM0kLaQRFpFWRyQotbbCiRWA4mNgySvqJ8NKbmQYiDFHfAMSHwUpMx0zKwA\nyWXfkMqTzENiCpLsbXwAkkggsWuDVIzwSvmQSsX3Dqk/y0IS4yEVy50FyRRM90gshcQKkNQE\nl4bEAKnLfiH1JgOpImdDYvoP5xUhsW1C4lMgZSYNSGHOgtT/ftzYxUyDxFYFyVw+r4GULEUZ\nUjCqrlI8cjQAkACpHSRWA8ldaT4HEq+C5NUZkIpVadXn1w7pjEn1ZRFIVAFJvmRVkFgeEm8I\nibWANNx9z4XEAGkwgLQEJH4OpKQBRkMqTrolpN5L6W70Vn0ekMbl0pDYmiHZ+VUHkHoDSGVI\nbAokEUIiQCrXxp8WIFW001mZH5IYguSKJpCoFxIHJEAazLirdJMzPyQWQqLZIMVqAMlNC5AW\nypyQWAxJ2KHeD/OyBpK+4XWdkNiikHjmVWbagLRQtg+JJZ3vPEg8syCAVBFAGjnGxSCxEqQo\ngOSmdcWQls3skMyrFFJUbi2QcguyIKS+ANJ6c/WQCj0NkEbnuiHxCZDqUwsp6NEzQOLXCamq\nJoDUJGuF1D3augSJUhfJXC8DaYyK/KR7M+H0GZAWCiDlp7EXSMMBpCZZISSxRUjZcjUBpN4A\nkkwEicdDbS3YACQ2FlI8fUAqBZCaZD2QvBkMQRrdYwGpHEBqknVAimbgQ2KA5GYzfvLDAaQm\nGe1ok5DSEQDJBJDaZCFIbNOQhj4EpAMgjR1hKUhsfkhqbtmMa5bGkEqTAaQ9ZeLaPBtS9wOQ\n2geQLpQzIOmXDSCNTg6Snlu+uqOmDkgHQFookyDR/JAKawqQRgeQFslaIZWqO2rql4Z0TncD\npG0FkPIBpN4AUhxAygeQegNIccZAEv2QpjQuIBXHHezMd0d+vLvJDuAqgLRgBiAJ7xUgFWZT\nmsiskB6klcfcgHeAtHyaQGJXDamYOSG958d3h3dH/j4z4B2/dwUBaZlMgyTmg9SbbUGactZo\nMgTpjn84/fuRf84M+OSGAtJSAaRsVg/pnt8egn2PN+AT/wRIS6cdpElzj96vBlJuRquCpM+A\nOM8MuOcfvvDjHSAtmT1DYlcLSeYBkBYMIFVnM5A4/3g43NypAzxAWiZTIalfGgKkpiN4OQeS\nyo26Nj4PpLdIlBMK8yr3sfBeSUhvFST5wenfbqTTPyI3bs3co/dDK2jkChT5uUzM6MmcM1/u\nJwCkBhxjN8kA/RJ7pIWCPVJtVvE9koGkLtLdxlftvAGAtGwoRBCRSCExQKrNnF/IfpZfG33g\nd5kBR97dKKRNAdJCAaTarApS350Nd52mG/UNLSAtlYtCigNIdtyhvvzoLnGrYzg34OYoX6qd\nFSAtFECqzbog3cibvQ8Okjege/mo724ApIUCSLVZF6TaANJCAaTaAJINIKU5HxIvQxoLDJDs\nuK36PCAtlCaQSgGkqQGkrQWQagNINoCUho+FRPKcaBOQdACpdQApk0pIDJDaTKYugLS5jIHE\ns5Bqpz0cQDIBpM0FkJadTF0AaXMZDcle7gak+QJImwsgLTuZugDS5gJIy06mLoC0uQDSspOp\nCyBtLoC07GTqAkibCyAtO5m6ANLmMgYSAdJCAaTN5SxI3Y+GkIYCSKMDSEulF5LfYgmkYSiA\nNDWAtLlsCdK0AFLrAFImgLTsZOoCSJtLb2cHpOaTqQsgbS5jITFAWiCAtLkA0rKTqQsgbS6A\ntOxk6gJImwsgLTuZugDS5jIKUld4c5AaBZBsACkTQKoMINkAUiaAVBlAsgGkTHobBZBcAMkG\nkDIBpDUGkDYXQFpjAGlzqYfEAWmxANLmAkhrDCBtLoC0xgDS5gJIawwgbS6AtMYA0uYCSGsM\nIG0ugLTGANLmAkhrDCBtLoC0xgDS5jIGEgOkhQJImwsgrTGAtLkA0hoDSJtLdaMA0oIBpM0F\nkNYYQNpcAGmNAaTNBZDWGEDabwwkBkjzB5D2G0BaMIC03wh3TAdIcweQ9htAWjCAtN8A0oIB\npP2mg6RbEJDmDiDtN4C0YABpv/Eg2UFDo8xWmb0HkPYbwQBpsQDSfgNICwaQ9htAWjCAtN8A\n0oIBpP0GkBYMIO03EyAhUwNI+438FYp0EDJHAGm/AaQFA0j7DSAtGEDabwBpwQDSfgNIC+aa\nIC36V65XEEBaMIC03wDSggGk/QaQFgwg7TuAtFAAad8BpIUCSPsOIC0UQNp3AGmhANK+A0gV\nadHNAGnfAaSKANK4ABIgZbMMpLsjP97dBIM+8fQzQFplAKkii0B64F0e/UHvOE8/A6RVBpAq\nsgSk9/z47vDuyN97jo4aUvAZIK0ygFSRJSDd8Q+nfz/yz3bIJ/6gIQWfAdIqA0gVadEqQ5Du\n+e2hO5i7d5377qAhBZ8B0ioDSBVZApI2w125d/FA9WMDkK4xgFSRy0BKBgLSigNIFQEkZCiA\nVJErgPQWOS9RC4rL1GLladEq3E9gRQ049kA6Yo+0+mCPVJE590gGkroyd+tdtXOQgs8AaZUB\npIoscWj3WX5X9IHfZSAFnwHSKgNIFVkCUubOBgtpa3c2XGMAqSJLQDo8ymO8B8+Pe+F9Bkjr\nDCBVZBFIN/IO78CPfeF9BkjrDCBVZBFItQGkVQaQKgJIyFAAqSKAhAwFkCoCSMhQAKkigIQM\nBZAqAkhryXr7JyBVBJDWkvX2T0CqCCCtJZvpn5up6KIBpLVkM/1zMxVdNIC0lmymf26moosG\nkNaSzfTPzVR00QDSWrKZ/rmZii4aQFpLNtM/N1PRRQNIa8lm+udmKrpoAGkt2Uz/3ExFFw0g\nrSWb6Z+bqeiiAaS1ZDP9czMVXTSAtJZspn9upqKLBpDWks30z81UdNEA0lqymf65mYouGkBa\nSzbTPzdT0UUDSGvJZvrnZiq6aABpLdlM/9xMRRcNIK0lm+mfm6noogGktWQz/XMzFV00gLSW\nbKZ/bqaiiwaQ1pLN9M/NVHTRANJaspn+uZmKLhpAWks20z83U9FFA0hryWb652YqumgAaS3Z\nTP/cTEUXDSAhIwNIubToZoB0VQGkXAAJGRlAygWQkJEBpFwACRkZQMoFkJCRAaRcAAkZGUDK\nBZAQpEEACUEaBJAQpEEACUEaBJAQZCUBJARpEEBCkAYBJARpEEBCkAYBJARpEEBCkAYBJARp\nEEBCkAYBJARpEEBCkAYBJARpEEBCkAYBJARpEEBCkAYBJARpEEBCkAZZOSQE2Uha9fkQ0t2R\nH+9u8gM+PdqXSbEY0qW3MwhSl3kgPUijj9kBd/Ll8SZXDJCQjWYWSO/58d3h3ZG/zwx4x7+c\nDH3iXzLFAAnZamaBdMc/nP79yD9nBtyrOXbHlEkxQEK2mlkg3fPbQ7fvuS8OkJDSoYCEbDSz\nQOLc/5EbcMMfMkMBCdlqLgTpU3dUl4cUXFN8y9hb/I//1///ZSDdHu9zxZJgj4Qsl7P+VMdF\nIN0cH7LFAAm5YNYH6RgLiQY8POaLARJywawPkrocdxtftdMDbh8fbvPFAAm5YNYH6bP8gugD\nv8sN+MAfSsUACblg1gep786GW+sIdzYgq8r6IB0e5bVr74qCG/DFu1vWKwZIyKWzQkg38rbu\ng4PkBvi3nXvFAAm5dFYIqVUACVkugHTxnLUKkJUEkC4eQNpDrh3S3/ijff3If/AXbjTEH+7V\nz7+bMX86ncX99H380uRezw2Q9pBrh8SO/G/61V/4MVi4sZD+whWUv5ox9e/wRi9tvud/kT8B\naQ+5ekif+U/61U/2lVq4sZAe1Oh/Peox/8aPf+3e/S146eWnB/kDkPaQq4f0vd0PHfUewizc\nSEg/qh3S3/mDHvMneaT4I/8cvAxm/WP3A5D2kKuHdDrqUucqf+NyB/H3R/74d7lw3GDSrz7z\nozShd1yngse/e9N5VPuX06ca0r2E+Vd+H7xkP5ygPahZPjyaySNbDyD9wH+WP3+WoNTJTGci\nhvS5++AH+Xkn6d4WVPkbV6r+amV4P7yXf1ffKMvCf5fHeoC0hwDS6ZBOLUz340d9MvNjCunh\n+w6B/PfY8Tu9+v7BXef7if/VNksfpGNX7kd13eGvkiQO7fYQQDoR6E5VftT7mU7GD92eJoak\ndh5/0e/v5RnR9/zeTOWB24vbvZC4d439e7lHA6Q9BJBO+4WuOz/IPUrU+YNzJBa+17Ft4WbX\nC+l0jnX/17/6JQFpD1kjpL5HFnePPtFOhp6aXH3q8XjamXzP/RP/JpC8Q0b/6PHzsXtW7F/c\nAEDadETwY1qWf2Rx9yQ7bn62gvR3/pl95vZSHauDFLdFAkldqvuLu2r3F30g+MNPj/q7WUDa\nflYLqe8X+w7dTwOp/EvmIyF1e6NHdYpjzpHuPS5/y0G6D24nYrlzpM96Wj8FL/0iOEfaQVYL\nqe+RxafjugcN6VPPw4pHQuqufOtr4NFVu8fTfur7hxwkWfC0L7MXG35yty303tnw2E1bX7X7\nG67abT+rhdT7yGJ+d7CQPjWD9AO319LC75Hklz73OUi64NHeDPE3d9uCOcp7dF81uZc/qiNS\nqe4zvkfaflYLqfe5du/s23v+4Uuz35A9uvtV/3707mzorgz8nD1Hknc28J+9m4oe7ZezRsb3\n8pbv+KW8s0HtvfZ9Z8OV7Gm3CengQco9s+GCjyz+gf+vkeP8L/5D91Nc/pG5s/y/1+WK/hfq\npzhnGpeExPnH7rkNPQd4S2/oH34aLhNk53d/73W5omx9j6Ry0/Mn+5aG9Bf+/XAhL3v/faS9\nLleU1UIaemRx+NXRmh5Z/MPPo4r/vPPfkN3rckVZLaT+RxavGdLE7LXD7XW5oqwWUu8jix2d\nI+/uGdrDs7/32uH2ulxRVgup/84GC+mug3WjvqwFpDVmr8sVZbWQ+h5Z7EG6OcqhPV8kAdJl\nM+9yrabV1gup55HFHiQ59LHv7gZAumwAqTr7+H2kS2c1XaJxAKk6O4H01RN68tUbb8DLp3aA\n99mrF0QvXmcKPPvWjkjq55tkkj1ZTZdoHECqzj4gPaMuT92Ar+SAJ2/Cz751Q5MCX6sRX5GC\n9PqJKvC6av6r6RKNA0jV2QWk7+jJK/bqCX1nBryiF2+6ncuL8LMnp5dvntNXQYGX9OwNe/OC\nXskRn2hIL7pSJ24vqiqwmi7ROFcGic6Yxi4gfUXdkdk3Zq/C2HPVJB0K77NvJI439CQo8Ewa\ney0/O6Ei94n7MZSaLrGabjMigFSdaYDSRwAAIABJREFUXUB6Tt0R2Ct6Hg3vGHif6b1OVMCQ\nedb985V5+0QPfVJVAUBa39RHBJB0CruPNx0O77OnxL5+Io/pCgXYK/v2a31o9zWrCSCtb+oj\nAkg6BUgvu6M67zOi5/ICQljgqdxlfWdGNj9fdlcbnrysqwAgrW/qIwJIOnlIr588ZxGk7mLD\nC7uXUQW+pudv2KtnMaSv/Wt5QwGk9U19RABJJwvpzZNn0Wckz5Fem8vkugCTF7qfR5Bedod2\nJ3R1u6TeLkEVZZbKyEoAUnV2AelJDtKzp/FnkbdnBtQLevI1iyA9pe5U6o3/3VRPAGlKVtEi\nXQBJR12Ze+1ftXv99Nnr+LPnPiRbQOWVIdP+8jcgFad+Ts9tmGpIPQUu+sjitNg0SF/L74q+\nlRfaVL6lZ+ln6uVr+ZEr8ETue14ahcHl7zf0v6sqAEhTAkiDkOoeWZwWmwgpubPhtWXif3Y6\nO5L3MHwTFJB3L3z3tBsq24r00O4+u6/o/1ZVAJCmBJCGIFU+sjgpNhXS6Yymi7u48IJ0ws++\nti+9Am/UXXXPdYvaMyhVtq4rAdKUANIQpMpHFifFJkNSt2qrZSX5j4Pkfca+faZf+gVen1Q9\n/9a0qD0rUqMB0nwBpCFIlY8sTopNhtQqpuNQMqRuvGwAqTh1QOqHVPnI4qTYniBR/KpZp5wy\nIRH9nHFWY6YOSGdASt5GkGZ9ZDHVPaKX7DDBTv9NebQvxfNs9vjfivoUxxk57ryPLBZeO1/0\nf/PIYhoq21Pf9UEKULXa5lRsUrpk90gVm2XskaYEeyRAKoyXndOVQyo3+YYgUfAjl0s+sjgp\ndhakqpUOSLPPKg4gTYdU+cjipBgg1WQypLplOXdWcQBpOqTKRxYnxQCpJoDUOquFVPnI4nZ3\nNnit0R9Amn1WcbYFqbDEF4JU+cjiaCgg1c0BkFpnvZBGPLK41d+QBaTB0bcLaWZw64XUKjuD\ndEZ/AKQZA0iZ1si+ixtkaBrTIAVj7w3S5LqvHRIBUpiJkETuDdmCgKQzF6Sq6daW05MdVaMe\nSCIeZd+QsucggJSOvlVIZMtVVQKQ0gCSGf/SkOKHV9QvCkVT8qc+L6RSaUDqbZd9QhKXghTN\n1Yck2N4hmQHxKICUb8OkYBWkzKRF8G+wzgmQwinZaVAGUnaygHR2JkOiLUCq6dULQ4q6zrmQ\nXElAqs2lIVEdpGT4mZD8btIcEgGSKnI2pMw6894NQSJASsrOCkl3V7fOaTQkfxhlII06PIsm\nmxk3PpNrA8l2uxCSP3NAKuW6INkWXQ8k6iM2F6T8HCdBKkgpQSr36V5IlIwaQfKWm5LucHWQ\nxBYh+WOraQMSILVJW0ji4pDCFSHCn0tDCvvUOEjCLVS+KurHGiCJZNiWIPU9+9u91I8KmheS\nW/g1QIo6k/9zJZD8PhNDErYOZF+Gy+LGpbkhlZZ2EFI4Yh6SbBzhLYo3xWyWf/a3e/luZZDc\nupkTUlAf/2cZknBduFAfpjtEaU3PCSnsqWpc0QoS+ZD8X2FtCMl0ivVB6vsNWe9l3zNWl4ak\nVnkTSAKQWkAiO1/q/m0KyV8HQ5DEBSH1Pfvbe/mp56nfl4Ok+lcMicIxxkESun55SGIIEk2G\nREEZOy7Zf7OQXP1XC0ksCsmOsDSkvmd/ey8/8U8zQFItfjFIwofk96cQktexhP8zODAk12n3\nASleF+TNZz2QxGog9T0g0nt5zz98afCr5uMh+e0xLyTKQTK9b+OQhJgISY90EUgp6EFIwo3Q\n8wccLwsp9/CT0c/+JvWsbf+53d16Tp/XfBqmhlPmuc+n/0k+65rk/+TGEUx+ZuclmHuGOJH3\njGzRTUtCMvPoIMnJqvoRyeddq//VPLvp+T/187Dl9E/F5fxPo+qyIvcscmGXU8TPNlfLTrIM\nuXHJLosrT2ZaetmJgmd/k1oeYdrkVCehp2eGvfWmpacjbB28ZTPLTvI9mfZzyx3Unbr6d5C6\ntiW3zHIe2WeT2/bx1o0d5tdDP39ctq9sf1LP/xayIU/jC9knhGtXosz8LvTs7+Dlx+4BKD0H\neH17pGin0GqPJHr3SMS8zafbRI3dI7m9khpNuNGp/R6J9J7G7EFEcY9Edo+k6pHskeyeLdwj\nBcuT7JHcHphsA7bZIwU7muR15R7JHqZ2a2uTeySVm56/fXkxSHaVmSNk3QGykMxRhuxZChJR\nJSRKIdkO7kGypyQxJHKQBGsMyT96Ggfp9BFNhuQ2DKwGUnQMWg1Jf0oSkmmXlUHqe/Z3+sDv\nic/+pqjFLSQ6GxKdCUlMhiRSSKQhqVkPQQp3V+Mg0TVAErL9yX5qIJFYIaS+Z3+nD/yeAonU\nCvJYrAISjYNEESSRgUQWkshBMgPPhSSmQLJG7Auh10UESRhIoikkkUDSN3ALbyV5sxRbg9T3\n7G/v5ZHfHCY+RL8fkhgFyXaE5pBoBCQxDySz1zsLEhUgGR81kMzeVZcrQooPNEZAEruDVHln\nw12n6UZ9Q3sOJNXGzSCJEBIJux6nQ6LZILlmGAnJzYeZulAGki7VFpIIIHndnpLt4xhIFEMS\nfgudAckAWhpS77O/3cubo3zZ80VSE0j2ZQDJShGXgCQmQhK2s5qFmwZJ0XAXJ/OQaBIkUQeJ\nUkiiBpKGoyGJMyDp5eyDJC4Mqe/Z39HLx767G2ohUQkSpZBc51AD1UqXDU8RJLKQzMVgMh3W\nzcyDpHpXAInqIIkqSMJB0n2tBMmu+AQS6Z1NAImGIAk1JWH+8SC5DVECSeQgmUVNIIk2kITd\nc+s6T4REa4DUKotBEnWQxChIoh+Suk6u/QgfkjgLklelHkhywlWQTGm9QzSQSFj+ej7C3pyh\nFqMEicZBks2o1uTikMhCUi25U0h6RRcgmYMS2/GE0N++mQPgMyD5m6hKSCKAJM6BpItkIene\n5pZIzVZTmAZJOEhUDUkUIZGFJDKQzEZANzANQzIzVkvlQ6LGkOxGeT+QRAESnQWJipBoEBL1\nQRIOEtVA0juwWkhmZxtAEm0hiTwkGoKk2qUPEqWQhPn+wEHSa0XvZhwkIYSYBokKkEQWkrhi\nSFSEJMwRWg8kEUIifdOC6u6jIInzIYklIVE/JBFCEhEkcSYk1f1NFw4h0SUh0f4h2bMikYFk\njhZ6IYkhSKIMidpCEudDUp1MZCHRrJCkotGQKIJEQq/GAUjkIFENJKHahNTSApKDJHxIVIRE\nqhUDSLpnT4ckzoBEPiQxApI+xLCQ3HlKBSSqgyQcJPIgyZDaqNdCEhaSuCAkSiAJvbQpJAKk\nBBJ5kCgDiQJIao2ZlTcIiQwkKkASw5DENEi6ixpIlIGke3wISRQgUQtI4mxIQteuNSSykMRI\nSGqxASmARD4kMRWSKEAy4+u+Xg/JOLKQXM903YBSSCKEJKogiRSSPrGvhiRaQxJFSHZdCHWO\nGkISGUg0FpKqq65mBEm4xXaQaL+QyEISoyGZPjgCEllIogaSCCCZjnUeJFPtFJIYgJTZIwl1\ntqDskD9RDxIVIInRkEQESU9iEJIIIIkYEiWQhF5CUYBEPiQxCMkcJ1wLJEogCQuJAkjCQSJ9\n+GS6uIOke38JkhgNSYyEJEZCEhlIphXGQNIL4UHy62ma04ekdlAxJOFBEiEks/RkIFEEya2L\nAJIYBUlYSNQLiRwkXevrgCQ0JLWQQ5Dsmg8hmXXtQ7LH5cxekEggUQxJzAHJWdAIqBeSqkQG\nEukDWANJzAdJjIdkmrwaEvVBEuakJoUkspCoAIlykEy/2BUkuykRtv9XQzKHKT2QqB6SWacp\nJMpBsvu98yCROrAMIOntuwggkap8FpI+Q3e9JgNJBJB0GwaQqAkkUx+NOoVEFpLp4fostADJ\nrCrhVWAIEpnp2V7lQxIXgFT3yOK02JmQhO01ekObg2T6wBAky4I8SLaBmdqCkTmLtwiqIVEK\nyexuKAtJmE/7IFENJDM9ZpfXQhIxJAogUR0k22C1kKgtJG+b5yCRW4R6SG4HdQlIdY8sTouN\ngERlSORBEi0h+X3OQRJDkAwPFqwxc3HjfEgigUTeyAtAIlfgXEjxJqsPEpkFH4CkPh4LyazH\nS0Kq/MW+pNh5kExHV41nIHkrwvZKDxJ5UwghORYxJDEOkuXRD8lqaADJH3k0JHILMQMkc2hN\nISRTnwSStyGkEHkZkl1vCSSzVGZ6w5DEJSFVPrI4KdYakihAMt3WvC5BEhaS7cZlSKablyFR\nsMZqIPn2PUhq0cuQxGhIwjVAAEk0gkSDkEjXJwPJLkc1JLOyzoVE9p1plGUhVT6yOCnWFJIo\n75HGQDL9pATJrtASpHCb66C4ixNTIVEFJIogkW0j17PHQfKaMwvJtVMBkluSCJI6ZZwRkmm9\nKZDMCl4UUvUDIqNiy0GyfswkEkh2PRUgiSFIbm4TINkK+pD86VRAEh4kj4W/eF53KUISkyCZ\nXhxMNIZkoFhIYgiS7fQZSF4bA9Kh8pHFtr29JnAt6zq7WwORIte1AiwkbAv71PxO4c3Edkky\nIOwKsuvaW2Gmlv7cIkZ6vbpua2Ziu5HdUng92O+f9nOvI/qQbAP53dQbYhk5S2QXLzLnN6ud\nla28t/XwtzF+25mpmQVzHwtbeyvDtLOdD7n5ea3iNDoFbnqumb05e26df1Mj+1ZNbJ2PLM5B\nClD17pHsio/Xv+vjdlPnLqh6fdR1U7dWrDXLzdvukbdeyY3sNr/WkbfC3H4rWKXmY+bmyLwZ\nurXqxvJmaSfIXJ3tyvekuvkI4a62e5BM73QbAK9bMr+IbUALybaHvihh5h72U/mv/fbczNXj\nazcj5FUvcOmLsHrtf26l2UVwy+Jvj4RtFLcJUCvILZstYg9p7FZAraPN7JFWBsluUQ0ks8JJ\nf1M7CpLbWpYh2Y6VgaR7qjuB8yExsgt5SUhuPQxCMu08DpK/ppyOCkiqvPnOOILkZrgeSJWP\nLE6fXjwPJOYg2S9AfUjqCyjmrZ6FIJGDRFlIptPKf5nr6z4k4UYPIPln+SVIbgmJ7InDKEhe\nwctCoumQxGohVT6yOH168fKQ3C2dESSzzryurboPCyAF69tCsj01gmQ52R5ZhsTcubmDxDKQ\nLO8CJOunCIkFkJyntpBs3y1CYnapypDspqwEyX1eAYlZSN4MeyDRspAqH1mcFBsBicZAYllI\nbBCS6aIjIblpWEimX02DJDxIQkMyXds7MTkXkj/2eEgihcR8SG4nkIckAkhhw/ZB8nbJloxw\nC+G+066G5OhfGtICdzYMQWJmlTIPkmgNSTe97Ws5SGQguU44EZIoQ1JP9RgNiUWQmJn4ZEim\nHUZDoh5IFEDydhxFSObyQT8kUYLEVgKp8pHF/ssWkOztm1lI+lf8XCfNQjJr2XWXYIupILEz\nIJGBpHuYf69BHyTRDpKrHjNEQ0i68cjdRB1CIjcXvfTu6+kQkmmVXkjMjJJAMu2UQmK2JSdC\nIn3n37ohjXhk8bS/IWsb29wSOgiJxkNyN/GwXkhqBZqbTvXJhv6ljgwkv7q1kNTvWGQhUQaS\n+T2pcyCxaZCsCtu4ek7MNkIZku9lGJJpaOZBUlspykAy12r0QrNNQGqVAUiqu9j9gv1dZ7k+\n9YOjMpCYgcQsJFaCpMBlITF9B7OF5LmZAInMVYmRkNQADxJLIYkIkv3p7edUv8pB0nPzf2/J\n9VUS6ilfoyAxs7dykFgtJKb/EwEkxhwkpipn7jLXeAApD4l5kPT+Qf/erPpH/fangcQsJPtp\nERLzVjAxc4aVQBKTIDEDidVCojwkNi8k/YSuqZAogkRM3ZNeA0lvGwchqeegDUBSF3hZAknk\nITEiszV2kMzqlm2xN0hsHkim3XKQWAESq4VEtZDs7MZCYiEkZrtOColNhaR/PyUPiWUhUQKJ\nnQOJFSAxHxIrQBJVkPReX5FU9bsWSGwiJNYIEstCYoOQGBO1kFglJDYeEuuBJHogsTGQbJ9O\nILEsJL1oIyCxEBLrg6T2Wh4klofErgQSOxuSqIBktmEWEitCMuuXyHbXMiSWgcRqIbESJP2Y\nMBZAYgEkNYIHSW+gWQSJ9UCiCki6YhaSaANJrhS57tTaDCGxSkisHhLtGhIbgMT6ILEAEstD\nYn2Q1JpkASRWDYl5kHSnzEJSq1FBMh3QQGJ6g8zGQWIpJENGL5HZq6n3DrKGxAYhsRgSs5Co\nBhJrD4mFkFQpc8pWgMQA6UxITK8npseW3Vv1HQ2J+ZBYCIlSSKwSEgsgsSwks7fwIbEEEstC\nYpeFZBptEJKsYgCJDUAyE1WL50NigNQLiaWQmNXkILFhSKwWEuuHxHohMQdJ12o8JOYfdrEi\nJB3d1dXY+pAqB4mZvqkb0YfE7ClJBEmcDYk5SMxCogwkpnt3BhJrB4kiSCyCxAApgWT6SgKJ\n5SGxPkisDxJLIOlaR5BYDyRWhmQXsQzJtIxmYiExDYn1QjLLVw+JsQFIts8zc1KZg8RSSOxs\nSKwAiQCpISRWB4nNCUloSIwNQnJNYXo72WO6JpCEnXgekq5KPSQahMTMNZU8JNYPiQYgsRGQ\n6BoheW05BIm1gWR7qjAdLYLEdEkPEuuDxAJIVsd4SEEz5SGZ8ch06aaQWAJJmOqGkJgHialy\nlIPEKiBRCEnV2daNpZBsiQQSM5BYBElPpZDtQzKdaTwkVoDkyjhIpusNQGIeJOZBYpeCZOrG\nXOu4aU+FxAYgqQNg0QeJmDk68CHpz8qQWC8ks+XQdTZ1M62UQmIpJL2i1wKp7pHF+gknEyCx\ny0Bic0BiEyB5YC4JyXyShcTs9r0XklkXwlRwBCQWQ2JsIiQ1X7KQzEoi5uZ9AUh1jyx+1xqS\nfu0O/F2HdJCYt/KYgyRykNiskKgaEpsKydbNFEwh2X4cQ/LqkYXEAkisCpKt8nhIpuO7ytoV\nLxh5Vc1CMkOUmJGQ2KUgVf5iX9+jIRtB8oq7DaJ5zxpDsn3Qg2Tqqdb1ICQ2HpJJCZIFoTtP\nDyRbtAUkNgCJlSHZ3YjeLZqGYmMgmY3GMCS2WkiVjyz+1POw4oUgMQ8SCyCZMrNCYiVI/lJm\nIaUpQ/IbqA6SHTWBxHohmREGIJFbfrewlZDMvwkku1DjIDEfkm2xlUCqfGTxJ/7pTEiuyWog\n5dsBkGyJIiTmIEWLGEPyZ6zPavoheRUYB8lbrkpILIKky6wYUuVz7e75hy8Tf0M2Wg1nQXKj\nW0hsHkhsGiQ570mQ0gYib+p5SF57VUGyHW3XkJipUCmXhZR7ZkPVI4tP/5P3Wuj/1WtBb0VS\nnuJxoumIroyE1L0nfzyKpiPcdE7rWU3j9FOoaZzKE+npdtOSkPQ4EpKcnvm/G/9U326V6ffi\nrfDmJ4t3w4Uwy0DqZ/y/oKSu3iOe9f+nz4WbNkvaSfjL6b8XatlIj9ctommbrqPJYd50VJ3N\nMssO2JUXQZuGddXLLbx6dQ0j3qr1KXQ9THnhLZcIlsHWQ0MK1rVcfgnJltGQuvVm66Ta2Fs+\n28brfGTxx+65DT0HeGfskQbHiYbl9kjMbpWC4t7E7R6JBXskM6x6j8TMRrC4R2Iz75FcOfuh\n/97eCmAWMbdHYuHYbo9k9vDn75Gimgo3YlRzNmqPxJI9klfP1e+RVG56/mTfDiF5cy1BCmbX\nHFJpUWw5V8EeSOGhT9JM7SAx3T5XBcl8LVT5yOJQWStILNzsFseJhqnucHFIcY8ZASmta/Bh\nA0imBnlIQbXVyOEyD0GKagBItY8svkJIwVwtJD0kC8lO3MymHaRkZvNA8ob2QcrUqA+Sv1wF\nSEm/yEBiVZBylYtyyUcWH/nNYeKzv8P2WgckdllImbq6TIXktu6NIWXGSWs0DEmNUQ2JjYYU\njr00pMo7G+46TTfqG9qLQ1KvKRmYgeRnHKRwMoBUqKk3GTuFPkiZebuPWkLqySUfWXxzlC97\nvkiqhRS154Uh2e529ZCiWVCybPmaeiWuHNKIRxY/9t3dMBlS5TjxsLkgsbaQStkDpEyJK4bU\nKktACj6/CCQzZAlImeH597uAlH6sFgCQyu3VBlJacGOQCmkKyVZ2AFIyi9GQempa/gSQ0qwf\nUjKwMSQztX1Aqq1uRU3Ln0yH5EYGpOxHwx+UC06GZOqUh2SKppDyE18lpPKsAWlydgxpeBKA\n1D/rfCFAygWQ3MsRkPx3gNST6h7dD0kPA6Rie1UtMiBlhuffj4RUztohMUAabrChcSaMMhnS\nxEmvE9KIlNfLhInVBJDS9EFKGqeizLogjakBIE1NJSQGSD0NFucSkIa+T6yrwUogTQggjQsg\nuZeXglTOVEjuA0DqDyCNHOVKIU1fgDVCCrJqSH2PLO6ew1UqtltIZ+QykIZKjpp1PiuBxFYM\nqe+RxfJRxYVigFSauIM0KYDUm7VC6vvFvkP3k+eLzQ5pQvYNKVdu7CeTJwlIQ5D6Hll8Oq57\n0JCSYoBUDCC1n79LGVJ1Fn9k8YHfmcedJMV2B0lmH5DOmnU+gDQEqe+5dod38ecTnyI0pcEm\n5HxITQJI7efvkrjZAqTkbQSp9pHFySN7R5Qd8X/8eODs44Ip/Dxb5rz/RXHeVf8Llj6e+OL/\nFx6v3Pj/mn5BcX2ovK4L/68PUoBqxCZh33uk8ryrUrtHWjTYIwFSeSAgVQeQSpBqHlns/Uyf\nXgxI1xRAGoLU98hij0769OIdQCqWQeIAUgmSSd8jiz1ISbGzIM0UQJotgDQEqf/OBgup7Z0N\nM6WqMSFnSlYMieUH9GT5RxY7SNHQdUKqCiBNCSANQup7ZLEHKRwKSFcWQBqE1CqAtOesB1Ih\ngLR0AGlKVg9pTACpRQBpSgApF0BCRgaQcrlqSMiUAFIugISMDCDlAkjIyABSLoCErDKAdAAk\n5PwA0gGQkPMDSAdAQs4PIB0ACTk/gHQAJOT8bBtS3bO/ufmVWkBC5sqmIdU9+/sdICGzZ8uQ\nKp/93feMVUBC2mTLkCqf/f2p56nfgIS0yZYhVT77+xP/BEjIzNkypMpnf9/zD1/28avmyHqz\nW0gHD1Lu4SdTn/2N//F/7v+Zngkf/3/ZRxZ/7B6A0nOAhz0Scm6uYI+kctPzty8BCTk3W4Q0\n8tnfh9w7QEKaZsuQKp/9DUjIbnLJZ38feXfPUKuH6CPIBXPJZ3/fdbBu1Je1gIRsOpd89vfN\nUQ7t+SIJkJCN5OLP/n7su7sBkJCNBL+PhCANsnJIpfw/BLlIil2yVZ+fBVIx/+/tObng2Jut\nOMZW72bv2oC09llj7AZjA1K5bRYde7MVx9jq3exdG5DWPmuM3WBsQCq3zaJjb7biGFu9m71r\nA9LaZ42xG4wNSOW2WXTszVYcY6t3s3dtQFr7rDF2g7EBqdw2i4692YpjbPVu9q4NSGufNcZu\nMDYgldtm0bE3W3GMrd7N3rUBae2zxtgNxt4bJATZaQAJQRoEkBCkQQAJQRoEkBCkQQAJQRoE\nkBCkQQAJQRoEkBCkQQAJQRoEkBCkQQAJQRoEkBCkQQAJQRoEkBCkQQAJQRoEkBCkQQAJQRoE\nkBCkQQAJQRoEkBCkQQAJQRoEkBCkQQAJQRoEkBCkQQAJQRoEkBCkQQAJQRoEkBCkQQAJQRoE\nkBCkQQAJQRoEkBCkQQAJQRoEkBCkQQAJQRoEkBCkQQAJQRoEkBCkQQAJQRoEkBCkQc6CxPlQ\nibsjP97dqLIy3mefHs1nXjH/5SnvB+cwKWMqfvOF8y/v/M8GKs55uqzNMqbmXcIGzNf8NJxn\nR26aUTX3KnqIB2Rrnq6lhTMvpAfZoR5Pr94lnetOvj/eBMX8l6fcHC8EyavGUb701tFQxY2j\n46VrfogbMF9zuXJyI7fNmJp7FVUZrHmylpbOrJDe8+O7w7sjf98t8X342Tv+5abboHwJinkv\nu9zPs1kfU/G7roZ3Xu2rKn44fHAvL1TzLkED5mt+6H7y3MgXq7lX0cqaJ2tp8cwK6Y5/OP37\nkX/uGuFz+Nk9t5Pwinkv5c8LQfKqceQ34Qg1Fe92BfOs1BE1P8QNmK/5adU82O4YLkTTjKi5\nV1GVwZona2nxNID04bTZO96p97f3/OhWwz2/Peid0Sf+qTgJr5j38nC4te3UOiMqrkdIjtP6\nKi6nMOeZRmXN8w0Y1fzA70wPTBb7YjX3xqiquf58lqPpupwP6bM6I7iT7+Wh6ufgc7P8H77o\nFvRzwx+CYtzfFj3w2zkh1VVc5i7ZDPRWvFvRyaJeoObZBoxrfngX137GyyQj2lxVtLLmMula\nWjDnQ+L8ozyGkC8fbk77nsfgcwNJJmqaU+EPxf74+TThOSHVVfwgyyQs+ip+mG+HNKrm+QaM\nax6PNSuk6jY3Fa2s+SG/lhZMq3Mk3Tbvg6FBRzu14E28zbiVZxL5/ij33vOfIw1V/JRP98f4\ntKGv4gd5cjxLvUfVPN+ASc0PydsZIdXV3Kuol76aH7Jracm0gHT74fODbhtv6CGzYm7CS6s3\nx4eomPfysbvYOS+k+op/CbcAvRU/mDPiC9c824BpzdO3s0KqbXNd0cqaq3y54LFdA0gP9hui\nuG2OyYoJ19HDY1zMvfwi++KskEZU/CY8j+2ruPfBRWueb8C05mm154RU3eYP8bdZvTVXubng\n1YapkO5tW3zhj58+3GbbRl1huS1ciLl9fLiNi7mX9v6Axmt19oof5rruNbLmuQbM1dwbP13s\ni9Tcr2hlzQ+5d4tmKiTOzXGD/CffNp/lRvFDdxKoLvT76+iDvfLgFXMv54I0teJuAzlQ8UP5\nWv+iNc80YLbm3vjR0EvV3K9oZc3TtbR0pu+R7k7LpM7+3h/e5Q97gxsE7uTFBnvucOvaqucG\ngfabmPEV/3Kq+L2jUVHx+1nuVRlb80P4SanmXqm57mwYW/Pb2NFgzZO1tHimQnovt3Xd0tzp\n7d77tG0Oj/aa983RfoGg8sXbXrpi/stoWo0ytuL6Li63Zisq/jjLxe/RNY8+KdXcKxWNfKma\n+xWtrHm8lhbP5IsNHx75o9quf9DnAAAAj0lEQVQqnJby4f0Hd63Va5sbeaeuffnobTD8A4+o\nmPe97QwHvWMrfhhf8ZkO1UfXPKpKoeZeqWjkS9Xcr2hlzeO1tHjw+0gI0iCAhCANAkgI0iCA\nhCANAkgI0iCAhCANAkgI0iCAhCANAkgI0iCAhCANAkgI0iCAhCANAkgI0iCAhCANAkgI0iCA\nhCAN8v8BbYU+Dj0jAxMAAAAASUVORK5CYII=",
      "text/plain": [
       "Plot with title \"[2004-01-05/2014-03-28]\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Take the first difference of the Series\n",
    "\n",
    "MSFT.ret <- diff(log(MSFT))\n",
    "SNP.ret <- diff(log(SNP))\n",
    "\n",
    "# Check for NA values in the T-Bill Series\n",
    "\n",
    "#DTB3.sub\n",
    "\n",
    "chartSeries(MSFT.ret)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Omit the NA Values \n",
    "\n",
    "NA values are present in the T-Bill return series and merge all"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "               SNP.Open     SNP.High     SNP.Low    SNP.Close  SNP.Volume\n",
       "2004-01-05  0.008064578 -0.005770562 -0.04242747 -0.013314746 -0.02375409\n",
       "2004-01-06 -0.118159676 -0.085579151 -0.09182794 -0.119129388  0.41379155\n",
       "2004-01-07 -0.019395837 -0.050835815 -0.03658223 -0.025253296 -0.61272184\n",
       "2004-01-08 -0.006936467  0.019255349  0.03309995  0.030729030 -0.11941022\n",
       "2004-01-09  0.031966189  0.009601467  0.01270959 -0.001138347 -0.60947823\n",
       "2004-01-12 -0.089261129 -0.085799967 -0.09508060 -0.093034868  1.49271599\n",
       "           SNP.Adjusted     MSFT.Open    MSFT.High      MSFT.Low   MSFT.Close\n",
       "2004-01-05  -0.01331476  0.0054239873  0.014656205  0.0141691383  0.024825812\n",
       "2004-01-06  -0.11912920  0.0164524810  0.003542370  0.0125472521  0.003547396\n",
       "2004-01-07  -0.02525343 -0.0007097587  0.001060187 -0.0021398011 -0.001062923\n",
       "2004-01-08   0.03072915  0.0077793534  0.005987023 -0.0003570791 -0.001773958\n",
       "2004-01-09  -0.00113824 -0.0127615344 -0.014857048 -0.0147511219 -0.017915205\n",
       "2004-01-12  -0.09303494 -0.0129266071 -0.011830179 -0.0087368594 -0.003259101\n",
       "           MSFT.Volume MSFT.Adjusted DTB3\n",
       "2004-01-05  0.41444811   0.024825854 0.90\n",
       "2004-01-06 -0.36056061   0.003547158 0.90\n",
       "2004-01-07  0.14539083  -0.001062701 0.89\n",
       "2004-01-08  0.07983443  -0.001773974 0.86\n",
       "2004-01-09  0.13155893  -0.017915228 0.85\n",
       "2004-01-12 -0.18330087  -0.003259199 0.88"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "dataDaily <- na.omit(merge(SNP.ret, MSFT.ret, DTB3.sub), join = \"inner\")\n",
    "head(dataDaily)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 2. VAR Modelling\n",
    "\n",
    "### Lower Frequency Data\n",
    "\n",
    "A VAR model is usually done on lower frequency data, therefore, we will transform the daily data to monthly"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Warning message in to.period(x, \"months\", indexAt = indexAt, name = name, ...):\n",
      "\"missing values removed from data\"Warning message in to.period(x, \"months\", indexAt = indexAt, name = name, ...):\n",
      "\"missing values removed from data\"Warning message in to.period(x, \"months\", indexAt = indexAt, name = name, ...):\n",
      "\"missing values removed from data\""
     ]
    },
    {
     "data": {
      "text/plain": [
       "         SNP.ret.Close MSFT.ret.Close DTB3.sub.Close\n",
       "Jan 2004  -0.010037715   -0.009359320           0.90\n",
       "Feb 2004   0.003614909    0.001131473           0.94\n",
       "Mar 2004   0.029947885   -0.010772137           0.93\n",
       "Apr 2004   0.008738797   -0.013305690           0.96\n",
       "May 2004   0.023069712    0.001526097           1.06\n",
       "Jun 2004   0.018574697    0.002103015           1.31"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "SNP.M <- to.monthly(SNP.ret)$SNP.ret.Close\n",
    "MSFT.M <- to.monthly(MSFT.ret)$MSFT.ret.Close\n",
    "DTB3.M <- to.monthly(DTB3.sub)$DTB3.sub.Close\n",
    "\n",
    "dataMonthly <- merge(SNP.M, MSFT.M, DTB3.M)\n",
    "head(dataMonthly)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Fitting the Model\n",
    "\n",
    "A simple reduced VAR model may be fitted to the data. The parametereization shown allows a maximum of 4 lags in the equations and chooses the model with the lowest AIC."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "VAR Estimation Results:\n",
       "========================= \n",
       "Endogenous variables: SNP.ret.Close, MSFT.ret.Close, DTB3.sub.Close \n",
       "Deterministic variables: const \n",
       "Sample size: 120 \n",
       "Log Likelihood: 653.626 \n",
       "Roots of the characteristic polynomial:\n",
       "0.979 0.6956 0.6375 0.6375 0.5922 0.5922 0.2964 0.2918 0.2918\n",
       "Call:\n",
       "VAR(y = dataMonthly, lag.max = 4, ic = \"AIC\")\n",
       "\n",
       "\n",
       "Estimation results for equation SNP.ret.Close: \n",
       "============================================== \n",
       "SNP.ret.Close = SNP.ret.Close.l1 + MSFT.ret.Close.l1 + DTB3.sub.Close.l1 + SNP.ret.Close.l2 + MSFT.ret.Close.l2 + DTB3.sub.Close.l2 + SNP.ret.Close.l3 + MSFT.ret.Close.l3 + DTB3.sub.Close.l3 + const \n",
       "\n",
       "                   Estimate Std. Error t value Pr(>|t|)   \n",
       "SNP.ret.Close.l1   0.074634   0.098738   0.756  0.45134   \n",
       "MSFT.ret.Close.l1 -0.252464   0.113366  -2.227  0.02799 * \n",
       "DTB3.sub.Close.l1  0.015643   0.008528   1.834  0.06930 . \n",
       "SNP.ret.Close.l2  -0.111754   0.098346  -1.136  0.25829   \n",
       "MSFT.ret.Close.l2 -0.059053   0.116510  -0.507  0.61327   \n",
       "DTB3.sub.Close.l2 -0.034492   0.012730  -2.710  0.00782 **\n",
       "SNP.ret.Close.l3  -0.076812   0.097182  -0.790  0.43100   \n",
       "MSFT.ret.Close.l3 -0.006794   0.115425  -0.059  0.95317   \n",
       "DTB3.sub.Close.l3  0.019860   0.008555   2.321  0.02211 * \n",
       "const              0.002237   0.002475   0.904  0.36800   \n",
       "---\n",
       "Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1\n",
       "\n",
       "\n",
       "Residual standard error: 0.0202 on 110 degrees of freedom\n",
       "Multiple R-Squared: 0.1474,\tAdjusted R-squared: 0.07768 \n",
       "F-statistic: 2.114 on 9 and 110 DF,  p-value: 0.03412 \n",
       "\n",
       "\n",
       "Estimation results for equation MSFT.ret.Close: \n",
       "=============================================== \n",
       "MSFT.ret.Close = SNP.ret.Close.l1 + MSFT.ret.Close.l1 + DTB3.sub.Close.l1 + SNP.ret.Close.l2 + MSFT.ret.Close.l2 + DTB3.sub.Close.l2 + SNP.ret.Close.l3 + MSFT.ret.Close.l3 + DTB3.sub.Close.l3 + const \n",
       "\n",
       "                    Estimate Std. Error t value Pr(>|t|)  \n",
       "SNP.ret.Close.l1  -0.0564751  0.0894017  -0.632   0.5289  \n",
       "MSFT.ret.Close.l1  0.0342138  0.1026462   0.333   0.7395  \n",
       "DTB3.sub.Close.l1  0.0097897  0.0077213   1.268   0.2075  \n",
       "SNP.ret.Close.l2  -0.0205293  0.0890467  -0.231   0.8181  \n",
       "MSFT.ret.Close.l2 -0.0010123  0.1054932  -0.010   0.9924  \n",
       "DTB3.sub.Close.l2 -0.0242045  0.0115263  -2.100   0.0380 *\n",
       "SNP.ret.Close.l3   0.0867832  0.0879931   0.986   0.3262  \n",
       "MSFT.ret.Close.l3 -0.0110151  0.1045111  -0.105   0.9163  \n",
       "DTB3.sub.Close.l3  0.0134447  0.0077462   1.736   0.0854 .\n",
       "const             -0.0001593  0.0022406  -0.071   0.9435  \n",
       "---\n",
       "Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1\n",
       "\n",
       "\n",
       "Residual standard error: 0.01829 on 110 degrees of freedom\n",
       "Multiple R-Squared: 0.07048,\tAdjusted R-squared: -0.005567 \n",
       "F-statistic: 0.9268 on 9 and 110 DF,  p-value: 0.505 \n",
       "\n",
       "\n",
       "Estimation results for equation DTB3.sub.Close: \n",
       "=============================================== \n",
       "DTB3.sub.Close = SNP.ret.Close.l1 + MSFT.ret.Close.l1 + DTB3.sub.Close.l1 + SNP.ret.Close.l2 + MSFT.ret.Close.l2 + DTB3.sub.Close.l2 + SNP.ret.Close.l3 + MSFT.ret.Close.l3 + DTB3.sub.Close.l3 + const \n",
       "\n",
       "                  Estimate Std. Error t value Pr(>|t|)    \n",
       "SNP.ret.Close.l1  -1.13400    0.99198  -1.143 0.255450    \n",
       "MSFT.ret.Close.l1 -0.34895    1.13893  -0.306 0.759895    \n",
       "DTB3.sub.Close.l1  1.09668    0.08567  12.801  < 2e-16 ***\n",
       "SNP.ret.Close.l2   1.43575    0.98804   1.453 0.149033    \n",
       "MSFT.ret.Close.l2 -0.62803    1.17052  -0.537 0.592671    \n",
       "DTB3.sub.Close.l2  0.24737    0.12789   1.934 0.055660 .  \n",
       "SNP.ret.Close.l3  -3.76207    0.97635  -3.853 0.000196 ***\n",
       "MSFT.ret.Close.l3  1.24527    1.15963   1.074 0.285239    \n",
       "DTB3.sub.Close.l3 -0.35019    0.08595  -4.074 8.73e-05 ***\n",
       "const              0.02110    0.02486   0.849 0.397913    \n",
       "---\n",
       "Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1\n",
       "\n",
       "\n",
       "Residual standard error: 0.203 on 110 degrees of freedom\n",
       "Multiple R-Squared: 0.9886,\tAdjusted R-squared: 0.9877 \n",
       "F-statistic:  1062 on 9 and 110 DF,  p-value: < 2.2e-16 \n",
       "\n",
       "\n",
       "\n",
       "Covariance matrix of residuals:\n",
       "               SNP.ret.Close MSFT.ret.Close DTB3.sub.Close\n",
       "SNP.ret.Close      0.0004082      0.0001261      0.0002367\n",
       "MSFT.ret.Close     0.0001261      0.0003347     -0.0004633\n",
       "DTB3.sub.Close     0.0002367     -0.0004633      0.0412020\n",
       "\n",
       "Correlation matrix of residuals:\n",
       "               SNP.ret.Close MSFT.ret.Close DTB3.sub.Close\n",
       "SNP.ret.Close        1.00000         0.3411        0.05772\n",
       "MSFT.ret.Close       0.34107         1.0000       -0.12475\n",
       "DTB3.sub.Close       0.05772        -0.1248        1.00000\n",
       "\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "var1 <- VAR(dataMonthly, lag.max = 4, ic = \"AIC\")\n",
    "# Can also do the following for a more established model selection (it provides multiple ICs)\n",
    "#var1 <- VARselect(dataMonthly, lag.max=4)\n",
    "summary(var1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Get the VAR Estimation Results"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "VAR Estimation Results:\n",
       "======================= \n",
       "\n",
       "Estimated coefficients for equation SNP.ret.Close: \n",
       "================================================== \n",
       "Call:\n",
       "SNP.ret.Close = SNP.ret.Close.l1 + MSFT.ret.Close.l1 + DTB3.sub.Close.l1 + SNP.ret.Close.l2 + MSFT.ret.Close.l2 + DTB3.sub.Close.l2 + SNP.ret.Close.l3 + MSFT.ret.Close.l3 + DTB3.sub.Close.l3 + const \n",
       "\n",
       " SNP.ret.Close.l1 MSFT.ret.Close.l1 DTB3.sub.Close.l1  SNP.ret.Close.l2 \n",
       "      0.074633617      -0.252463853       0.015642893      -0.111753709 \n",
       "MSFT.ret.Close.l2 DTB3.sub.Close.l2  SNP.ret.Close.l3 MSFT.ret.Close.l3 \n",
       "     -0.059053153      -0.034492399      -0.076811541      -0.006794271 \n",
       "DTB3.sub.Close.l3             const \n",
       "      0.019859823       0.002236926 \n",
       "\n",
       "\n",
       "Estimated coefficients for equation MSFT.ret.Close: \n",
       "=================================================== \n",
       "Call:\n",
       "MSFT.ret.Close = SNP.ret.Close.l1 + MSFT.ret.Close.l1 + DTB3.sub.Close.l1 + SNP.ret.Close.l2 + MSFT.ret.Close.l2 + DTB3.sub.Close.l2 + SNP.ret.Close.l3 + MSFT.ret.Close.l3 + DTB3.sub.Close.l3 + const \n",
       "\n",
       " SNP.ret.Close.l1 MSFT.ret.Close.l1 DTB3.sub.Close.l1  SNP.ret.Close.l2 \n",
       "    -0.0564751434      0.0342138440      0.0097896785     -0.0205293345 \n",
       "MSFT.ret.Close.l2 DTB3.sub.Close.l2  SNP.ret.Close.l3 MSFT.ret.Close.l3 \n",
       "    -0.0010122809     -0.0242044652      0.0867831771     -0.0110151424 \n",
       "DTB3.sub.Close.l3             const \n",
       "     0.0134447199     -0.0001592819 \n",
       "\n",
       "\n",
       "Estimated coefficients for equation DTB3.sub.Close: \n",
       "=================================================== \n",
       "Call:\n",
       "DTB3.sub.Close = SNP.ret.Close.l1 + MSFT.ret.Close.l1 + DTB3.sub.Close.l1 + SNP.ret.Close.l2 + MSFT.ret.Close.l2 + DTB3.sub.Close.l2 + SNP.ret.Close.l3 + MSFT.ret.Close.l3 + DTB3.sub.Close.l3 + const \n",
       "\n",
       " SNP.ret.Close.l1 MSFT.ret.Close.l1 DTB3.sub.Close.l1  SNP.ret.Close.l2 \n",
       "      -1.13399603       -0.34894618        1.09668458        1.43575050 \n",
       "MSFT.ret.Close.l2 DTB3.sub.Close.l2  SNP.ret.Close.l3 MSFT.ret.Close.l3 \n",
       "      -0.62802988        0.24736759       -3.76206739        1.24526638 \n",
       "DTB3.sub.Close.l3             const \n",
       "      -0.35019382        0.02109879 \n",
       "\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "var1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Interpretation:\n",
    "\n",
    "**1. Stationarity:**\n",
    "\n",
    "All the roots of the characteristic polynomial are inside the unit circle therefore implying stationarity.\n",
    "\n",
    "**2. Parsimony:**\n",
    "\n",
    "Looking at the full output, there are many insignificant variables at different time horizons and in different equations. However, if we were to restrict these individually to zero and re-estimate the model it would mean we are imposing a restriction without any apparent economic reasoning behind it.\n",
    "\n",
    "- Solution: **Lag restriction Test** to remove either a variable from the model or reducing the lag structure of the whole model."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 3. Residual Test for Autocorrelation\n",
    "\n",
    "### Get the Plot of the VAR Model"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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lLeQVQVKZ65tUhrbwk2RG0KIq0DkVJ5Eak4hTfX4IsNiFQn9mGfxGT5ex2tRUqe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\nPiDYeInO7QAAAABJRU5ErkJggg==",
      "text/plain": [
       "Plot with title \"PACF Residuals\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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4rUjZtYuqpEymgZeSwkipRYkSuScxrdxqPKRNI76MVtODcccwuTl15TWfJEkkWd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\nSAAK/A+N2+LxBOFqnwAAAABJRU5ErkJggg==",
      "text/plain": [
       "Plot with title \"PACF Residuals\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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nbGINIMp0iTjYhUKimvkTzfBXndxkekemGxISNyIoWtTUA+WP7OyJpId/9FZWBU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\nAElFTkSuQmCC",
      "text/plain": [
       "Plot with title \"PACF Residuals\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot(var1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### View Cross-Correlograms\n",
    "\n",
    "Cross-correlograms are a measure of similarity of two series as a function of the displacement of one relative to the other. Also called the *sliding dot product*."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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DrFwmRN8KB0LwVEGhxjYbImeFC6lwoiOQakUHphHCndCyLFU3phHCndCyLFU3ph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\nAElFTkSuQmCC",
      "text/plain": [
       "Plot with title \"resid_2 & resid_3\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "residuals <- residuals(var1)\n",
    "resid_1 <- residuals[,1]\n",
    "resid_2 <- residuals[,2]\n",
    "resid_3 <- residuals[,3]\n",
    "\n",
    "# Plot the cross-correlograms\n",
    "par(mfcol=c(1,3))\n",
    "ccf(resid_1, resid_2, lag.max=4, type = \"correlation\", plot=TRUE)\n",
    "ccf(resid_1, resid_3, lag.max=4, type=\"correlation\", plot=TRUE)\n",
    "ccf(resid_2, resid_3, lag.max=4, type=\"correlation\", plot=TRUE)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "There are contemporaneous effects between the **first and second residuals** however they become non-significant at longer lags."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 4. Predictions with VAR Models\n",
    "\n",
    "Specify number of periods and desired confidence interval"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "$SNP.ret.Close\n",
       "              fcst       lower      upper         CI\n",
       " [1,] -0.003521937 -0.04312153 0.03607766 0.03959959\n",
       " [2,]  0.003016369 -0.03815964 0.04419238 0.04117601\n",
       " [3,]  0.003925111 -0.03835475 0.04620498 0.04227986\n",
       " [4,]  0.002631408 -0.03970186 0.04496468 0.04233327\n",
       " [5,]  0.001812762 -0.04072484 0.04435036 0.04253760\n",
       " [6,]  0.001725203 -0.04092518 0.04437559 0.04265039\n",
       " [7,]  0.002223603 -0.04044849 0.04489570 0.04267210\n",
       " [8,]  0.002472261 -0.04020299 0.04514751 0.04267525\n",
       " [9,]  0.002576089 -0.04011235 0.04526453 0.04268844\n",
       "[10,]  0.002557676 -0.04013795 0.04525330 0.04269563\n",
       "\n",
       "$MSFT.ret.Close\n",
       "               fcst       lower      upper         CI\n",
       " [1,]  0.0003098750 -0.03554527 0.03616502 0.03585514\n",
       " [2,] -0.0010486220 -0.03714845 0.03505120 0.03609982\n",
       " [3,] -0.0008919769 -0.03746788 0.03568392 0.03657590\n",
       " [4,] -0.0012164045 -0.03810162 0.03566881 0.03688521\n",
       " [5,] -0.0008566409 -0.03780414 0.03609086 0.03694750\n",
       " [6,] -0.0008123192 -0.03782626 0.03620162 0.03701394\n",
       " [7,] -0.0007339333 -0.03775190 0.03628403 0.03701796\n",
       " [8,] -0.0007753505 -0.03781561 0.03626491 0.03704026\n",
       " [9,] -0.0007680283 -0.03781481 0.03627875 0.03704678\n",
       "[10,] -0.0007518435 -0.03780732 0.03630363 0.03705547\n",
       "\n",
       "$DTB3.sub.Close\n",
       "            fcst      lower     upper        CI\n",
       " [1,] 0.09171963 -0.3061191 0.4895583 0.3978387\n",
       " [2,] 0.17378755 -0.4180679 0.7656430 0.5918555\n",
       " [3,] 0.25928097 -0.5642320 1.0827940 0.8235130\n",
       " [4,] 0.33080315 -0.6981900 1.3597963 1.0289931\n",
       " [5,] 0.37814641 -0.8300223 1.5863151 1.2081687\n",
       " [6,] 0.41374547 -0.9688467 1.7963377 1.3825922\n",
       " [7,] 0.44259649 -1.0945001 1.9796931 1.5370966\n",
       " [8,] 0.46924541 -1.2117662 2.1502570 1.6810116\n",
       " [9,] 0.49392453 -1.3181772 2.3060263 1.8121018\n",
       "[10,] 0.51596369 -1.4160326 2.4479599 1.9319963\n"
      ]
     },
     "execution_count": 76,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "var.pred <- predict(var1, n.ahead=10, ci=0.95)\n",
    "var.pred"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Forecast Variance Decomposition\n",
    "\n",
    "Which looks for how much of the forecast error variance is due to the shock in each equation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "Plot with title \"FEVD for DTB3.sub.Close\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "VAR_fevd <- fevd(var1, n.ahead=10)\n",
    "plot(VAR_fevd)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**a. SNP:** Explained almost completely by itself with some influences from the other variables after the first period.\n",
    "\n",
    "**b. MSFT:** Explained for the most part by itself and a part by the S&P 500 (increasing with time).\n",
    "\n",
    "**c. DTB3:** Explained almost completely by itself with some influences from the other variables (increasing with time)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 5. Impulse Response Function (Interpretation)\n",
    "\n",
    "Used to assess the dynamic path of each variable given a one SD shock by another variable.\n",
    "\n",
    "**Legend:**\n",
    "\n",
    "- Red Lines: 95% Confidence Interval\n",
    "\n",
    "- Black Lines: Shock (impulse)\n",
    "\n",
    "It is read as the movement of the black line across time, i.e.\n",
    "\n",
    "a. No Response = Stable line\n",
    "\n",
    "b. Increasing = Increasing line\n",
    "\n",
    "c. Decreasing = Decreasing line"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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MJ3ZcvFeMd2XGxdhnYqgk8m79iOi63L0E5F8MnkHdtxsXUZ2qkIPpm8Yzsuti5D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\nQEiAACEBAoQECBASIEBIgMB/+WHcEclHiq4AAAAASUVORK5CYII=",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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XHmdx/MvDvm1fqgYFBxDmhW5yxWbO4QZ43xg8Qu9Hw0s9asxfJ7cP6NCyi28cx/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\nAEECBAgSIECQAAGCBAgQJEDgP+lP0mTIUY46AAAAAElFTkSuQmCC",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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AkQICRAgJAAAUICBAgJECAkQICQAAFCAgQICRAgJECAkAABQgIECAkQICRAgJAA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\n7I4lm48AAAAASUVORK5CYII=",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "var.irf <- irf(var1, ci = 0.95)\n",
    "plot(var.irf)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "All the variables seem to respond in a delayed way by a shock in the US T-Bonds rates whereas there seem to be no major perturbations given by any other shocks in the other variables.\n",
    "\n",
    "Responses to their own shocks are normal therefore not factored in our analysis.\n",
    "\n",
    "Imposing restrictions:\n",
    "\n",
    "- S&P Index shocks do not have a contemporaneous effect on Microsoft\n",
    "\n",
    "- S&P Index shocks do not have a contemporaneous effect on interest rates\n",
    "\n",
    "- T-Bonds interest rates have minimal contemporaneous effects on Microsoft\n",
    "\n",
    "## Constructing the Structural VAR:\n",
    "\n",
    "The point of departure for a SVAR model is the already estimated reduced form of the VAR model (var1).\n",
    "\n",
    "We then amend it with the above **restrictions** following considerations on:\n",
    "\n",
    "- The **non-significance** of a majority of terms within the reduced VAR model\n",
    "\n",
    "- the **absence of contemporaneous effects** among some variables\n",
    "\n",
    "### Restrictions\n",
    "\n",
    "These restrictions, following our analysis of the impulse response function, enter into the SVAR model as $0$ in the $A$ matrix which is as follows:\n",
    "\n",
    "<p style = \"text-align:center\"> $\\begin{bmatrix} 1 & a_{12} & a_{13} \\\\ 0 & 1 & 0 \\\\ 0 & a_{32} & 1 \\end{bmatrix}$ </p>\n",
    "\n",
    "When setting up the $A$ matrix as a parameter for SVAR estimation, the positions of the to-be estimated parameters should take NA values:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Create the A-Matrix\n",
    "\n",
    "# Create an identity (3x3) matrix\n",
    "amat <- diag(3)\n",
    "\n",
    "# Populate the restriction values as NA values\n",
    "amat[2,1] <- NA\n",
    "amat[2,3] <- NA\n",
    "amat[3,1] <- NA"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Fit the SVAR \n",
    "\n",
    "Using the newly constructed A-Matrix obtained from the restrions of the impulse response function, and re-plot the irf."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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1ZzYl/13yfMu+GaRyRCeloSIc28UzBn3GbeRZq5izMPtHMPta/fjpCetXpIq/3M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\nICRAgJAAgf+OeQ5Wi8MQzwAAAABJRU5ErkJggg==",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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PckExJEfE8yIUHE9yQTEkR8TzIhQcT3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\nRK5CYII=",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "svar1 <- SVAR(var1, estmethod=\"direct\", Amat=amat)\n",
    "irf.svar <- irf(svar1)\n",
    "plot(irf.svar)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "After applying the restrictions we can cleary see that:\n",
    "\n",
    "- **Shocks in the T-Bond interest rates** have a clear influence on the dynamic path of both the **S&P Index** and **Microsoft Stock**."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Additional Elements from the VAR Model\n",
    "\n",
    "### a. Get the VAR Coefficients"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<dl>\n",
       "\t<dt>$SNP.ret.Close</dt>\n",
       "\t\t<dd><table>\n",
       "<thead><tr><th></th><th scope=col>Estimate</th><th scope=col>Std. Error</th><th scope=col>t value</th><th scope=col>Pr(&gt;|t|)</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><th scope=row>SNP.ret.Close.l1</th><td> 0.074633617</td><td>0.098738119 </td><td> 0.75587441 </td><td>0.451340139 </td></tr>\n",
       "\t<tr><th scope=row>MSFT.ret.Close.l1</th><td>-0.252463853</td><td>0.113365852 </td><td>-2.22698325 </td><td>0.027985083 </td></tr>\n",
       "\t<tr><th scope=row>DTB3.sub.Close.l1</th><td> 0.015642893</td><td>0.008527641 </td><td> 1.83437526 </td><td>0.069301064 </td></tr>\n",
       "\t<tr><th scope=row>SNP.ret.Close.l2</th><td>-0.111753709</td><td>0.098346061 </td><td>-1.13633132 </td><td>0.258287479 </td></tr>\n",
       "\t<tr><th scope=row>MSFT.ret.Close.l2</th><td>-0.059053153</td><td>0.116510204 </td><td>-0.50684962 </td><td>0.613274857 </td></tr>\n",
       "\t<tr><th scope=row>DTB3.sub.Close.l2</th><td>-0.034492399</td><td>0.012730067 </td><td>-2.70952221 </td><td>0.007817351 </td></tr>\n",
       "\t<tr><th scope=row>SNP.ret.Close.l3</th><td>-0.076811541</td><td>0.097182432 </td><td>-0.79038505 </td><td>0.431003608 </td></tr>\n",
       "\t<tr><th scope=row>MSFT.ret.Close.l3</th><td>-0.006794271</td><td>0.115425443 </td><td>-0.05886285 </td><td>0.953168150 </td></tr>\n",
       "\t<tr><th scope=row>DTB3.sub.Close.l3</th><td> 0.019859823</td><td>0.008555129 </td><td> 2.32139374 </td><td>0.022108015 </td></tr>\n",
       "\t<tr><th scope=row>const</th><td> 0.002236926</td><td>0.002474612 </td><td> 0.90395003 </td><td>0.367997322 </td></tr>\n",
       "</tbody>\n",
       "</table>\n",
       "</dd>\n",
       "\t<dt>$MSFT.ret.Close</dt>\n",
       "\t\t<dd><table>\n",
       "<thead><tr><th></th><th scope=col>Estimate</th><th scope=col>Std. Error</th><th scope=col>t value</th><th scope=col>Pr(&gt;|t|)</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><th scope=row>SNP.ret.Close.l1</th><td>-0.0564751434</td><td>0.089401653  </td><td>-0.631701337 </td><td>0.52889268   </td></tr>\n",
       "\t<tr><th scope=row>MSFT.ret.Close.l1</th><td> 0.0342138440</td><td>0.102646218  </td><td> 0.333318115 </td><td>0.73952866   </td></tr>\n",
       "\t<tr><th scope=row>DTB3.sub.Close.l1</th><td> 0.0097896785</td><td>0.007721285  </td><td> 1.267881981 </td><td>0.20751724   </td></tr>\n",
       "\t<tr><th scope=row>SNP.ret.Close.l2</th><td>-0.0205293345</td><td>0.089046667  </td><td>-0.230545795 </td><td>0.81809591   </td></tr>\n",
       "\t<tr><th scope=row>MSFT.ret.Close.l2</th><td>-0.0010122809</td><td>0.105493247  </td><td>-0.009595694 </td><td>0.99236124   </td></tr>\n",
       "\t<tr><th scope=row>DTB3.sub.Close.l2</th><td>-0.0242044652</td><td>0.011526339  </td><td>-2.099926540 </td><td>0.03802165   </td></tr>\n",
       "\t<tr><th scope=row>SNP.ret.Close.l3</th><td> 0.0867831771</td><td>0.087993068  </td><td> 0.986250159 </td><td>0.32617476   </td></tr>\n",
       "\t<tr><th scope=row>MSFT.ret.Close.l3</th><td>-0.0110151424</td><td>0.104511059  </td><td>-0.105396907 </td><td>0.91625282   </td></tr>\n",
       "\t<tr><th scope=row>DTB3.sub.Close.l3</th><td> 0.0134447199</td><td>0.007746174  </td><td> 1.735659359 </td><td>0.08542438   </td></tr>\n",
       "\t<tr><th scope=row>const</th><td>-0.0001592819</td><td>0.002240618  </td><td>-0.071088370 </td><td>0.94345651   </td></tr>\n",
       "</tbody>\n",
       "</table>\n",
       "</dd>\n",
       "\t<dt>$DTB3.sub.Close</dt>\n",
       "\t\t<dd><table>\n",
       "<thead><tr><th></th><th scope=col>Estimate</th><th scope=col>Std. Error</th><th scope=col>t value</th><th scope=col>Pr(&gt;|t|)</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><th scope=row>SNP.ret.Close.l1</th><td>-1.13399603 </td><td>0.99197596  </td><td>-1.1431689  </td><td>2.554504e-01</td></tr>\n",
       "\t<tr><th scope=row>MSFT.ret.Close.l1</th><td>-0.34894618 </td><td>1.13893399  </td><td>-0.3063796  </td><td>7.598948e-01</td></tr>\n",
       "\t<tr><th scope=row>DTB3.sub.Close.l1</th><td> 1.09668458 </td><td>0.08567324  </td><td>12.8007831  </td><td>1.591251e-23</td></tr>\n",
       "\t<tr><th scope=row>SNP.ret.Close.l2</th><td> 1.43575050 </td><td>0.98803713  </td><td> 1.4531341  </td><td>1.490334e-01</td></tr>\n",
       "\t<tr><th scope=row>MSFT.ret.Close.l2</th><td>-0.62802988 </td><td>1.17052383  </td><td>-0.5365375  </td><td>5.926706e-01</td></tr>\n",
       "\t<tr><th scope=row>DTB3.sub.Close.l2</th><td> 0.24736759 </td><td>0.12789306  </td><td> 1.9341752  </td><td>5.566049e-02</td></tr>\n",
       "\t<tr><th scope=row>SNP.ret.Close.l3</th><td>-3.76206739 </td><td>0.97634669  </td><td>-3.8532085  </td><td>1.964596e-04</td></tr>\n",
       "\t<tr><th scope=row>MSFT.ret.Close.l3</th><td> 1.24526638 </td><td>1.15962574  </td><td> 1.0738520  </td><td>2.852391e-01</td></tr>\n",
       "\t<tr><th scope=row>DTB3.sub.Close.l3</th><td>-0.35019382 </td><td>0.08594940  </td><td>-4.0744183  </td><td>8.734772e-05</td></tr>\n",
       "\t<tr><th scope=row>const</th><td> 0.02109879 </td><td>0.02486128  </td><td> 0.8486607  </td><td>3.979132e-01</td></tr>\n",
       "</tbody>\n",
       "</table>\n",
       "</dd>\n",
       "</dl>\n"
      ],
      "text/latex": [
       "\\begin{description}\n",
       "\\item[\\$SNP.ret.Close] \\begin{tabular}{r|llll}\n",
       "  & Estimate & Std. Error & t value & Pr(>\\textbar{}t\\textbar{})\\\\\n",
       "\\hline\n",
       "\tSNP.ret.Close.l1 &  0.074633617 & 0.098738119  &  0.75587441  & 0.451340139 \\\\\n",
       "\tMSFT.ret.Close.l1 & -0.252463853 & 0.113365852  & -2.22698325  & 0.027985083 \\\\\n",
       "\tDTB3.sub.Close.l1 &  0.015642893 & 0.008527641  &  1.83437526  & 0.069301064 \\\\\n",
       "\tSNP.ret.Close.l2 & -0.111753709 & 0.098346061  & -1.13633132  & 0.258287479 \\\\\n",
       "\tMSFT.ret.Close.l2 & -0.059053153 & 0.116510204  & -0.50684962  & 0.613274857 \\\\\n",
       "\tDTB3.sub.Close.l2 & -0.034492399 & 0.012730067  & -2.70952221  & 0.007817351 \\\\\n",
       "\tSNP.ret.Close.l3 & -0.076811541 & 0.097182432  & -0.79038505  & 0.431003608 \\\\\n",
       "\tMSFT.ret.Close.l3 & -0.006794271 & 0.115425443  & -0.05886285  & 0.953168150 \\\\\n",
       "\tDTB3.sub.Close.l3 &  0.019859823 & 0.008555129  &  2.32139374  & 0.022108015 \\\\\n",
       "\tconst &  0.002236926 & 0.002474612  &  0.90395003  & 0.367997322 \\\\\n",
       "\\end{tabular}\n",
       "\n",
       "\\item[\\$MSFT.ret.Close] \\begin{tabular}{r|llll}\n",
       "  & Estimate & Std. Error & t value & Pr(>\\textbar{}t\\textbar{})\\\\\n",
       "\\hline\n",
       "\tSNP.ret.Close.l1 & -0.0564751434 & 0.089401653   & -0.631701337  & 0.52889268   \\\\\n",
       "\tMSFT.ret.Close.l1 &  0.0342138440 & 0.102646218   &  0.333318115  & 0.73952866   \\\\\n",
       "\tDTB3.sub.Close.l1 &  0.0097896785 & 0.007721285   &  1.267881981  & 0.20751724   \\\\\n",
       "\tSNP.ret.Close.l2 & -0.0205293345 & 0.089046667   & -0.230545795  & 0.81809591   \\\\\n",
       "\tMSFT.ret.Close.l2 & -0.0010122809 & 0.105493247   & -0.009595694  & 0.99236124   \\\\\n",
       "\tDTB3.sub.Close.l2 & -0.0242044652 & 0.011526339   & -2.099926540  & 0.03802165   \\\\\n",
       "\tSNP.ret.Close.l3 &  0.0867831771 & 0.087993068   &  0.986250159  & 0.32617476   \\\\\n",
       "\tMSFT.ret.Close.l3 & -0.0110151424 & 0.104511059   & -0.105396907  & 0.91625282   \\\\\n",
       "\tDTB3.sub.Close.l3 &  0.0134447199 & 0.007746174   &  1.735659359  & 0.08542438   \\\\\n",
       "\tconst & -0.0001592819 & 0.002240618   & -0.071088370  & 0.94345651   \\\\\n",
       "\\end{tabular}\n",
       "\n",
       "\\item[\\$DTB3.sub.Close] \\begin{tabular}{r|llll}\n",
       "  & Estimate & Std. Error & t value & Pr(>\\textbar{}t\\textbar{})\\\\\n",
       "\\hline\n",
       "\tSNP.ret.Close.l1 & -1.13399603  & 0.99197596   & -1.1431689   & 2.554504e-01\\\\\n",
       "\tMSFT.ret.Close.l1 & -0.34894618  & 1.13893399   & -0.3063796   & 7.598948e-01\\\\\n",
       "\tDTB3.sub.Close.l1 &  1.09668458  & 0.08567324   & 12.8007831   & 1.591251e-23\\\\\n",
       "\tSNP.ret.Close.l2 &  1.43575050  & 0.98803713   &  1.4531341   & 1.490334e-01\\\\\n",
       "\tMSFT.ret.Close.l2 & -0.62802988  & 1.17052383   & -0.5365375   & 5.926706e-01\\\\\n",
       "\tDTB3.sub.Close.l2 &  0.24736759  & 0.12789306   &  1.9341752   & 5.566049e-02\\\\\n",
       "\tSNP.ret.Close.l3 & -3.76206739  & 0.97634669   & -3.8532085   & 1.964596e-04\\\\\n",
       "\tMSFT.ret.Close.l3 &  1.24526638  & 1.15962574   &  1.0738520   & 2.852391e-01\\\\\n",
       "\tDTB3.sub.Close.l3 & -0.35019382  & 0.08594940   & -4.0744183   & 8.734772e-05\\\\\n",
       "\tconst &  0.02109879  & 0.02486128   &  0.8486607   & 3.979132e-01\\\\\n",
       "\\end{tabular}\n",
       "\n",
       "\\end{description}\n"
      ],
      "text/markdown": [
       "$SNP.ret.Close\n",
       ":   \n",
       "| <!--/--> | Estimate | Std. Error | t value | Pr(>|t|) |\n",
       "|---|---|---|---|---|\n",
       "| SNP.ret.Close.l1 |  0.074633617 | 0.098738119  |  0.75587441  | 0.451340139  |\n",
       "| MSFT.ret.Close.l1 | -0.252463853 | 0.113365852  | -2.22698325  | 0.027985083  |\n",
       "| DTB3.sub.Close.l1 |  0.015642893 | 0.008527641  |  1.83437526  | 0.069301064  |\n",
       "| SNP.ret.Close.l2 | -0.111753709 | 0.098346061  | -1.13633132  | 0.258287479  |\n",
       "| MSFT.ret.Close.l2 | -0.059053153 | 0.116510204  | -0.50684962  | 0.613274857  |\n",
       "| DTB3.sub.Close.l2 | -0.034492399 | 0.012730067  | -2.70952221  | 0.007817351  |\n",
       "| SNP.ret.Close.l3 | -0.076811541 | 0.097182432  | -0.79038505  | 0.431003608  |\n",
       "| MSFT.ret.Close.l3 | -0.006794271 | 0.115425443  | -0.05886285  | 0.953168150  |\n",
       "| DTB3.sub.Close.l3 |  0.019859823 | 0.008555129  |  2.32139374  | 0.022108015  |\n",
       "| const |  0.002236926 | 0.002474612  |  0.90395003  | 0.367997322  |\n",
       "\n",
       "\n",
       "$MSFT.ret.Close\n",
       ":   \n",
       "| <!--/--> | Estimate | Std. Error | t value | Pr(>|t|) |\n",
       "|---|---|---|---|---|\n",
       "| SNP.ret.Close.l1 | -0.0564751434 | 0.089401653   | -0.631701337  | 0.52889268    |\n",
       "| MSFT.ret.Close.l1 |  0.0342138440 | 0.102646218   |  0.333318115  | 0.73952866    |\n",
       "| DTB3.sub.Close.l1 |  0.0097896785 | 0.007721285   |  1.267881981  | 0.20751724    |\n",
       "| SNP.ret.Close.l2 | -0.0205293345 | 0.089046667   | -0.230545795  | 0.81809591    |\n",
       "| MSFT.ret.Close.l2 | -0.0010122809 | 0.105493247   | -0.009595694  | 0.99236124    |\n",
       "| DTB3.sub.Close.l2 | -0.0242044652 | 0.011526339   | -2.099926540  | 0.03802165    |\n",
       "| SNP.ret.Close.l3 |  0.0867831771 | 0.087993068   |  0.986250159  | 0.32617476    |\n",
       "| MSFT.ret.Close.l3 | -0.0110151424 | 0.104511059   | -0.105396907  | 0.91625282    |\n",
       "| DTB3.sub.Close.l3 |  0.0134447199 | 0.007746174   |  1.735659359  | 0.08542438    |\n",
       "| const | -0.0001592819 | 0.002240618   | -0.071088370  | 0.94345651    |\n",
       "\n",
       "\n",
       "$DTB3.sub.Close\n",
       ":   \n",
       "| <!--/--> | Estimate | Std. Error | t value | Pr(>|t|) |\n",
       "|---|---|---|---|---|\n",
       "| SNP.ret.Close.l1 | -1.13399603  | 0.99197596   | -1.1431689   | 2.554504e-01 |\n",
       "| MSFT.ret.Close.l1 | -0.34894618  | 1.13893399   | -0.3063796   | 7.598948e-01 |\n",
       "| DTB3.sub.Close.l1 |  1.09668458  | 0.08567324   | 12.8007831   | 1.591251e-23 |\n",
       "| SNP.ret.Close.l2 |  1.43575050  | 0.98803713   |  1.4531341   | 1.490334e-01 |\n",
       "| MSFT.ret.Close.l2 | -0.62802988  | 1.17052383   | -0.5365375   | 5.926706e-01 |\n",
       "| DTB3.sub.Close.l2 |  0.24736759  | 0.12789306   |  1.9341752   | 5.566049e-02 |\n",
       "| SNP.ret.Close.l3 | -3.76206739  | 0.97634669   | -3.8532085   | 1.964596e-04 |\n",
       "| MSFT.ret.Close.l3 |  1.24526638  | 1.15962574   |  1.0738520   | 2.852391e-01 |\n",
       "| DTB3.sub.Close.l3 | -0.35019382  | 0.08594940   | -4.0744183   | 8.734772e-05 |\n",
       "| const |  0.02109879  | 0.02486128   |  0.8486607   | 3.979132e-01 |\n",
       "\n",
       "\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "$SNP.ret.Close\n",
       "                      Estimate  Std. Error     t value    Pr(>|t|)\n",
       "SNP.ret.Close.l1   0.074633617 0.098738119  0.75587441 0.451340139\n",
       "MSFT.ret.Close.l1 -0.252463853 0.113365852 -2.22698325 0.027985083\n",
       "DTB3.sub.Close.l1  0.015642893 0.008527641  1.83437526 0.069301064\n",
       "SNP.ret.Close.l2  -0.111753709 0.098346061 -1.13633132 0.258287479\n",
       "MSFT.ret.Close.l2 -0.059053153 0.116510204 -0.50684962 0.613274857\n",
       "DTB3.sub.Close.l2 -0.034492399 0.012730067 -2.70952221 0.007817351\n",
       "SNP.ret.Close.l3  -0.076811541 0.097182432 -0.79038505 0.431003608\n",
       "MSFT.ret.Close.l3 -0.006794271 0.115425443 -0.05886285 0.953168150\n",
       "DTB3.sub.Close.l3  0.019859823 0.008555129  2.32139374 0.022108015\n",
       "const              0.002236926 0.002474612  0.90395003 0.367997322\n",
       "\n",
       "$MSFT.ret.Close\n",
       "                       Estimate  Std. Error      t value   Pr(>|t|)\n",
       "SNP.ret.Close.l1  -0.0564751434 0.089401653 -0.631701337 0.52889268\n",
       "MSFT.ret.Close.l1  0.0342138440 0.102646218  0.333318115 0.73952866\n",
       "DTB3.sub.Close.l1  0.0097896785 0.007721285  1.267881981 0.20751724\n",
       "SNP.ret.Close.l2  -0.0205293345 0.089046667 -0.230545795 0.81809591\n",
       "MSFT.ret.Close.l2 -0.0010122809 0.105493247 -0.009595694 0.99236124\n",
       "DTB3.sub.Close.l2 -0.0242044652 0.011526339 -2.099926540 0.03802165\n",
       "SNP.ret.Close.l3   0.0867831771 0.087993068  0.986250159 0.32617476\n",
       "MSFT.ret.Close.l3 -0.0110151424 0.104511059 -0.105396907 0.91625282\n",
       "DTB3.sub.Close.l3  0.0134447199 0.007746174  1.735659359 0.08542438\n",
       "const             -0.0001592819 0.002240618 -0.071088370 0.94345651\n",
       "\n",
       "$DTB3.sub.Close\n",
       "                     Estimate Std. Error    t value     Pr(>|t|)\n",
       "SNP.ret.Close.l1  -1.13399603 0.99197596 -1.1431689 2.554504e-01\n",
       "MSFT.ret.Close.l1 -0.34894618 1.13893399 -0.3063796 7.598948e-01\n",
       "DTB3.sub.Close.l1  1.09668458 0.08567324 12.8007831 1.591251e-23\n",
       "SNP.ret.Close.l2   1.43575050 0.98803713  1.4531341 1.490334e-01\n",
       "MSFT.ret.Close.l2 -0.62802988 1.17052383 -0.5365375 5.926706e-01\n",
       "DTB3.sub.Close.l2  0.24736759 0.12789306  1.9341752 5.566049e-02\n",
       "SNP.ret.Close.l3  -3.76206739 0.97634669 -3.8532085 1.964596e-04\n",
       "MSFT.ret.Close.l3  1.24526638 1.15962574  1.0738520 2.852391e-01\n",
       "DTB3.sub.Close.l3 -0.35019382 0.08594940 -4.0744183 8.734772e-05\n",
       "const              0.02109879 0.02486128  0.8486607 3.979132e-01\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "coef(var1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### b. List of Fitted Values"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th scope=col>SNP.ret.Close</th><th scope=col>MSFT.ret.Close</th><th scope=col>DTB3.sub.Close</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><td> 0.007554302 </td><td>-0.0046100653</td><td>0.9587519    </td></tr>\n",
       "\t<tr><td> 0.004859807 </td><td>-0.0018847259</td><td>1.0070912    </td></tr>\n",
       "\t<tr><td> 0.003093812 </td><td> 0.0007861252</td><td>0.9635066    </td></tr>\n",
       "\t<tr><td> 0.002838880 </td><td>-0.0006318675</td><td>1.3447005    </td></tr>\n",
       "\t<tr><td>-0.002978245 </td><td>-0.0026881910</td><td>1.4597739    </td></tr>\n",
       "\t<tr><td> 0.002324267 </td><td>-0.0010164105</td><td>1.5659121    </td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{r|lll}\n",
       " SNP.ret.Close & MSFT.ret.Close & DTB3.sub.Close\\\\\n",
       "\\hline\n",
       "\t  0.007554302  & -0.0046100653 & 0.9587519    \\\\\n",
       "\t  0.004859807  & -0.0018847259 & 1.0070912    \\\\\n",
       "\t  0.003093812  &  0.0007861252 & 0.9635066    \\\\\n",
       "\t  0.002838880  & -0.0006318675 & 1.3447005    \\\\\n",
       "\t -0.002978245  & -0.0026881910 & 1.4597739    \\\\\n",
       "\t  0.002324267  & -0.0010164105 & 1.5659121    \\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "| SNP.ret.Close | MSFT.ret.Close | DTB3.sub.Close |\n",
       "|---|---|---|\n",
       "|  0.007554302  | -0.0046100653 | 0.9587519     |\n",
       "|  0.004859807  | -0.0018847259 | 1.0070912     |\n",
       "|  0.003093812  |  0.0007861252 | 0.9635066     |\n",
       "|  0.002838880  | -0.0006318675 | 1.3447005     |\n",
       "| -0.002978245  | -0.0026881910 | 1.4597739     |\n",
       "|  0.002324267  | -0.0010164105 | 1.5659121     |\n",
       "\n"
      ],
      "text/plain": [
       "  SNP.ret.Close MSFT.ret.Close DTB3.sub.Close\n",
       "1  0.007554302  -0.0046100653  0.9587519     \n",
       "2  0.004859807  -0.0018847259  1.0070912     \n",
       "3  0.003093812   0.0007861252  0.9635066     \n",
       "4  0.002838880  -0.0006318675  1.3447005     \n",
       "5 -0.002978245  -0.0026881910  1.4597739     \n",
       "6  0.002324267  -0.0010164105  1.5659121     "
      ]
     },
     "execution_count": 71,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "head(fitted(var1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### c. Coefficient matrices of the VMA Representation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>1</li>\n",
       "\t<li>0</li>\n",
       "\t<li>0</li>\n",
       "\t<li>0</li>\n",
       "\t<li>1</li>\n",
       "\t<li>0</li>\n",
       "\t<li>0</li>\n",
       "\t<li>0</li>\n",
       "\t<li>1</li>\n",
       "\t<li>0.0746336172703193</li>\n",
       "\t<li>-0.056475143426268</li>\n",
       "\t<li>-1.13399602901675</li>\n",
       "\t<li>-0.252463852956273</li>\n",
       "\t<li>0.0342138439767821</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 1\n",
       "\\item 0\n",
       "\\item 0\n",
       "\\item 0\n",
       "\\item 1\n",
       "\\item 0\n",
       "\\item 0\n",
       "\\item 0\n",
       "\\item 1\n",
       "\\item 0.0746336172703193\n",
       "\\item -0.056475143426268\n",
       "\\item -1.13399602901675\n",
       "\\item -0.252463852956273\n",
       "\\item 0.0342138439767821\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 1\n",
       "2. 0\n",
       "3. 0\n",
       "4. 0\n",
       "5. 1\n",
       "6. 0\n",
       "7. 0\n",
       "8. 0\n",
       "9. 1\n",
       "10. 0.0746336172703193\n",
       "11. -0.056475143426268\n",
       "12. -1.13399602901675\n",
       "13. -0.252463852956273\n",
       "14. 0.0342138439767821\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       " [1]  1.00000000  0.00000000  0.00000000  0.00000000  1.00000000  0.00000000\n",
       " [7]  0.00000000  0.00000000  1.00000000  0.07463362 -0.05647514 -1.13399603\n",
       "[13] -0.25246385  0.03421384"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "head(Phi(var1), 14)"
   ]
  }
 ],
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