Panel Data Analysis Codes in R.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# <font color=DeepSkyBlue >**Panel Data of Individual Wages** </font>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### ***by Xiaoxuan Ma***"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Agenda\n",
    "* [<font color=DeepSkyBlue>**Overview**</font>](#1)\n",
    "* [<font color=DeepSkyBlue>**Descriptive facts**</font>](#2)\n",
    "* [<font color=DeepSkyBlue>**Distribution of log wage**</font>](#3)\n",
    "* [<font color=DeepSkyBlue>**Examining and cleaning outlier**</font>](#4)\n",
    "* [<font color=DeepSkyBlue>**Panel Data Models**</font>](#5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Overview<a name=\"1\"></a>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Panel data characteristics"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "1.Panel data provide information on individual behavior, both across individuals and over time – they have both **cross-sectional** and **time-series** dimensions."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "2.Panel data include **N individuals** observed at **T regular time periods**."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "3.Panel data can be balanced when all individuals are observed in all time periods."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Data for this project"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "#install.packages(\"plm\")\n",
    "library(plm)\n",
    "library(dplyr)\n",
    "library(moments)\n",
    "\n",
    "data('Wages', package = 'plm')\n",
    "pd <- pdata.frame(Wages, index=595) #convert to a pdata.frame.\n",
    "attach(pd)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th></th><th scope=col>exp</th><th scope=col>wks</th><th scope=col>bluecol</th><th scope=col>ind</th><th scope=col>south</th><th scope=col>smsa</th><th scope=col>married</th><th scope=col>sex</th><th scope=col>union</th><th scope=col>ed</th><th scope=col>black</th><th scope=col>lwage</th><th scope=col>id</th><th scope=col>time</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><th scope=row>1-1</th><td> 3     </td><td>32     </td><td>no     </td><td>0      </td><td>yes    </td><td>no     </td><td>yes    </td><td>male   </td><td>no     </td><td> 9     </td><td>no     </td><td>5.56068</td><td>1      </td><td>1      </td></tr>\n",
       "\t<tr><th scope=row>1-2</th><td> 4     </td><td>43     </td><td>no     </td><td>0      </td><td>yes    </td><td>no     </td><td>yes    </td><td>male   </td><td>no     </td><td> 9     </td><td>no     </td><td>5.72031</td><td>1      </td><td>2      </td></tr>\n",
       "\t<tr><th scope=row>1-3</th><td> 5     </td><td>40     </td><td>no     </td><td>0      </td><td>yes    </td><td>no     </td><td>yes    </td><td>male   </td><td>no     </td><td> 9     </td><td>no     </td><td>5.99645</td><td>1      </td><td>3      </td></tr>\n",
       "\t<tr><th scope=row>1-4</th><td> 6     </td><td>39     </td><td>no     </td><td>0      </td><td>yes    </td><td>no     </td><td>yes    </td><td>male   </td><td>no     </td><td> 9     </td><td>no     </td><td>5.99645</td><td>1      </td><td>4      </td></tr>\n",
       "\t<tr><th scope=row>1-5</th><td> 7     </td><td>42     </td><td>no     </td><td>1      </td><td>yes    </td><td>no     </td><td>yes    </td><td>male   </td><td>no     </td><td> 9     </td><td>no     </td><td>6.06146</td><td>1      </td><td>5      </td></tr>\n",
       "\t<tr><th scope=row>1-6</th><td> 8     </td><td>35     </td><td>no     </td><td>1      </td><td>yes    </td><td>no     </td><td>yes    </td><td>male   </td><td>no     </td><td> 9     </td><td>no     </td><td>6.17379</td><td>1      </td><td>6      </td></tr>\n",
       "\t<tr><th scope=row>1-7</th><td> 9     </td><td>32     </td><td>no     </td><td>1      </td><td>yes    </td><td>no     </td><td>yes    </td><td>male   </td><td>no     </td><td> 9     </td><td>no     </td><td>6.24417</td><td>1      </td><td>7      </td></tr>\n",
       "\t<tr><th scope=row>2-1</th><td>30     </td><td>34     </td><td>yes    </td><td>0      </td><td>no     </td><td>no     </td><td>yes    </td><td>male   </td><td>no     </td><td>11     </td><td>no     </td><td>6.16331</td><td>2      </td><td>1      </td></tr>\n",
       "\t<tr><th scope=row>2-2</th><td>31     </td><td>27     </td><td>yes    </td><td>0      </td><td>no     </td><td>no     </td><td>yes    </td><td>male   </td><td>no     </td><td>11     </td><td>no     </td><td>6.21461</td><td>2      </td><td>2      </td></tr>\n",
       "\t<tr><th scope=row>2-3</th><td>32     </td><td>33     </td><td>yes    </td><td>1      </td><td>no     </td><td>no     </td><td>yes    </td><td>male   </td><td>yes    </td><td>11     </td><td>no     </td><td>6.26340</td><td>2      </td><td>3      </td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{r|llllllllllllll}\n",
       "  & exp & wks & bluecol & ind & south & smsa & married & sex & union & ed & black & lwage & id & time\\\\\n",
       "\\hline\n",
       "\t1-1 &  3      & 32      & no      & 0       & yes     & no      & yes     & male    & no      &  9      & no      & 5.56068 & 1       & 1      \\\\\n",
       "\t1-2 &  4      & 43      & no      & 0       & yes     & no      & yes     & male    & no      &  9      & no      & 5.72031 & 1       & 2      \\\\\n",
       "\t1-3 &  5      & 40      & no      & 0       & yes     & no      & yes     & male    & no      &  9      & no      & 5.99645 & 1       & 3      \\\\\n",
       "\t1-4 &  6      & 39      & no      & 0       & yes     & no      & yes     & male    & no      &  9      & no      & 5.99645 & 1       & 4      \\\\\n",
       "\t1-5 &  7      & 42      & no      & 1       & yes     & no      & yes     & male    & no      &  9      & no      & 6.06146 & 1       & 5      \\\\\n",
       "\t1-6 &  8      & 35      & no      & 1       & yes     & no      & yes     & male    & no      &  9      & no      & 6.17379 & 1       & 6      \\\\\n",
       "\t1-7 &  9      & 32      & no      & 1       & yes     & no      & yes     & male    & no      &  9      & no      & 6.24417 & 1       & 7      \\\\\n",
       "\t2-1 & 30      & 34      & yes     & 0       & no      & no      & yes     & male    & no      & 11      & no      & 6.16331 & 2       & 1      \\\\\n",
       "\t2-2 & 31      & 27      & yes     & 0       & no      & no      & yes     & male    & no      & 11      & no      & 6.21461 & 2       & 2      \\\\\n",
       "\t2-3 & 32      & 33      & yes     & 1       & no      & no      & yes     & male    & yes     & 11      & no      & 6.26340 & 2       & 3      \\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "| <!--/--> | exp | wks | bluecol | ind | south | smsa | married | sex | union | ed | black | lwage | id | time | \n",
       "|---|---|---|---|---|---|---|---|---|---|\n",
       "| 1-1 |  3      | 32      | no      | 0       | yes     | no      | yes     | male    | no      |  9      | no      | 5.56068 | 1       | 1       | \n",
       "| 1-2 |  4      | 43      | no      | 0       | yes     | no      | yes     | male    | no      |  9      | no      | 5.72031 | 1       | 2       | \n",
       "| 1-3 |  5      | 40      | no      | 0       | yes     | no      | yes     | male    | no      |  9      | no      | 5.99645 | 1       | 3       | \n",
       "| 1-4 |  6      | 39      | no      | 0       | yes     | no      | yes     | male    | no      |  9      | no      | 5.99645 | 1       | 4       | \n",
       "| 1-5 |  7      | 42      | no      | 1       | yes     | no      | yes     | male    | no      |  9      | no      | 6.06146 | 1       | 5       | \n",
       "| 1-6 |  8      | 35      | no      | 1       | yes     | no      | yes     | male    | no      |  9      | no      | 6.17379 | 1       | 6       | \n",
       "| 1-7 |  9      | 32      | no      | 1       | yes     | no      | yes     | male    | no      |  9      | no      | 6.24417 | 1       | 7       | \n",
       "| 2-1 | 30      | 34      | yes     | 0       | no      | no      | yes     | male    | no      | 11      | no      | 6.16331 | 2       | 1       | \n",
       "| 2-2 | 31      | 27      | yes     | 0       | no      | no      | yes     | male    | no      | 11      | no      | 6.21461 | 2       | 2       | \n",
       "| 2-3 | 32      | 33      | yes     | 1       | no      | no      | yes     | male    | yes     | 11      | no      | 6.26340 | 2       | 3       | \n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "    exp wks bluecol ind south smsa married sex  union ed black lwage   id time\n",
       "1-1  3  32  no      0   yes   no   yes     male no     9 no    5.56068 1  1   \n",
       "1-2  4  43  no      0   yes   no   yes     male no     9 no    5.72031 1  2   \n",
       "1-3  5  40  no      0   yes   no   yes     male no     9 no    5.99645 1  3   \n",
       "1-4  6  39  no      0   yes   no   yes     male no     9 no    5.99645 1  4   \n",
       "1-5  7  42  no      1   yes   no   yes     male no     9 no    6.06146 1  5   \n",
       "1-6  8  35  no      1   yes   no   yes     male no     9 no    6.17379 1  6   \n",
       "1-7  9  32  no      1   yes   no   yes     male no     9 no    6.24417 1  7   \n",
       "2-1 30  34  yes     0   no    no   yes     male no    11 no    6.16331 2  1   \n",
       "2-2 31  27  yes     0   no    no   yes     male no    11 no    6.21461 2  2   \n",
       "2-3 32  33  yes     1   no    no   yes     male yes   11 no    6.26340 2  3   "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "head(pd, 10)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A panel of 595 individuals from 1976 to 1982, taken from the Panel Study of Income Dynamics (PSID).\n",
    "\n",
    "The data are organized as a stacked time series/balanced panel.\n",
    "\n",
    "total number of observations : 4165\n",
    "\n",
    "observation : individuals\n",
    "\n",
    "country : United States"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Checking if there is any nan in the dataframe."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1] \"number of missing values:  0\"\n"
     ]
    }
   ],
   "source": [
    "print(paste(\"number of missing values: \",sum(is.na(pd))))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Topics overview"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "I use Jupyter Notebook to show the codes and results, and use PowerPoint to summarize and present the results."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Due to the limited time provided, I will focus on the follow questions:  \n",
    "1) Highlight three descriptive facts from the data with supporting analysis and graphs.   \n",
    "2) Pick a continuous variable of interest, what is the distribution of this variable?   \n",
    "3) Continuing from above, how would you examine and clean outlier?  \n",
    "5) Build a model from data, share with us the insights and results.  \n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Also, I'll briefly talk about my thoughts about some other questions."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Descriptive facts from the data<a name=\"2\"></a>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 1. Means and Variations of variables "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    " **There are four continuous variables:**   \n",
    " years of full-time work experience,   \n",
    " weeks worked,  \n",
    " years of education,  \n",
    " logarithm of wage.\n",
    "   \n",
    " Let's use basic statistics to summary them.   "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Means**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th scope=col>mean(exp)</th><th scope=col>mean(wks)</th><th scope=col>mean(ed)</th><th scope=col>mean(lwage)</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><td>19.85</td><td>46.81</td><td>12.85</td><td>6.68 </td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{r|llll}\n",
       " mean(exp) & mean(wks) & mean(ed) & mean(lwage)\\\\\n",
       "\\hline\n",
       "\t 19.85 & 46.81 & 12.85 & 6.68 \\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "mean(exp) | mean(wks) | mean(ed) | mean(lwage) | \n",
       "|---|\n",
       "| 19.85 | 46.81 | 12.85 | 6.68  | \n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "  mean(exp) mean(wks) mean(ed) mean(lwage)\n",
       "1 19.85     46.81     12.85    6.68       "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pd_mean <- pd %>% summarise(mean(exp), mean(wks), mean(ed), mean(lwage))\n",
    "pd_mean <- round(pd_mean,2)\n",
    "pd_mean"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Overall variation: variation over time and individuals.**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th scope=col>sd(exp)</th><th scope=col>sd(wks)</th><th scope=col>sd(ed)</th><th scope=col>sd(lwage)</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><td>10.97</td><td>5.13 </td><td>2.79 </td><td>0.46 </td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{r|llll}\n",
       " sd(exp) & sd(wks) & sd(ed) & sd(lwage)\\\\\n",
       "\\hline\n",
       "\t 10.97 & 5.13  & 2.79  & 0.46 \\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "sd(exp) | sd(wks) | sd(ed) | sd(lwage) | \n",
       "|---|\n",
       "| 10.97 | 5.13  | 2.79  | 0.46  | \n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "  sd(exp) sd(wks) sd(ed) sd(lwage)\n",
       "1 10.97   5.13    2.79   0.46     "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pd_sd_overall <- pd %>% summarise(sd(exp), sd(wks), sd(ed), sd(lwage))\n",
    "pd_sd_overall <- round(pd_sd_overall,2)\n",
    "pd_sd_overall"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Between variation: variation between individuals.**\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th scope=col>exp</th><th scope=col>wks</th><th scope=col>ed</th><th scope=col>lwage</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><td>10.79</td><td>3.28 </td><td>2.79 </td><td>0.39 </td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{r|llll}\n",
       " exp & wks & ed & lwage\\\\\n",
       "\\hline\n",
       "\t 10.79 & 3.28  & 2.79  & 0.39 \\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "exp | wks | ed | lwage | \n",
       "|---|\n",
       "| 10.79 | 3.28  | 2.79  | 0.39  | \n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "  exp   wks  ed   lwage\n",
       "1 10.79 3.28 2.79 0.39 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pd_sd_between <- pd %>% group_by(id) %>% \n",
    "                 summarise_at(vars(exp, wks, ed, lwage),mean) %>% \n",
    "                 summarise_at(vars( exp, wks, ed, lwage),sd)\n",
    "pd_sd_between <- round(pd_sd_between,2)\n",
    "pd_sd_between"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Within variation: variation within individuals over time.**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th scope=col>sd(exp)</th><th scope=col>sd(wks)</th><th scope=col>sd(ed)</th><th scope=col>sd(lwage)</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><td>1.98</td><td>3.94</td><td>0   </td><td>0.24</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{llll}\n",
       " sd(exp) & sd(wks) & sd(ed) & sd(lwage)\\\\\n",
       "\\hline\n",
       "\t 1.98 & 3.94 & 0    & 0.24\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "sd(exp) | sd(wks) | sd(ed) | sd(lwage) | \n",
       "|---|\n",
       "| 1.98 | 3.94 | 0    | 0.24 | \n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "     sd(exp) sd(wks) sd(ed) sd(lwage)\n",
       "[1,] 1.98    3.94    0      0.24     "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pd_sd_within <- (pd_sd_overall**2 - pd_sd_between**2)**0.5\n",
    "pd_sd_within <- round(pd_sd_within,2)\n",
    "pd_sd_within"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 2. Average log wages of males and females over time."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th scope=col>time</th><th scope=col>lwage</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><td>1       </td><td>6.423298</td></tr>\n",
       "\t<tr><td>2       </td><td>6.515570</td></tr>\n",
       "\t<tr><td>3       </td><td>6.655162</td></tr>\n",
       "\t<tr><td>4       </td><td>6.752830</td></tr>\n",
       "\t<tr><td>5       </td><td>6.842012</td></tr>\n",
       "\t<tr><td>6       </td><td>6.915456</td></tr>\n",
       "\t<tr><td>7       </td><td>7.004088</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{r|ll}\n",
       " time & lwage\\\\\n",
       "\\hline\n",
       "\t 1        & 6.423298\\\\\n",
       "\t 2        & 6.515570\\\\\n",
       "\t 3        & 6.655162\\\\\n",
       "\t 4        & 6.752830\\\\\n",
       "\t 5        & 6.842012\\\\\n",
       "\t 6        & 6.915456\\\\\n",
       "\t 7        & 7.004088\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "time | lwage | \n",
       "|---|---|---|---|---|---|---|\n",
       "| 1        | 6.423298 | \n",
       "| 2        | 6.515570 | \n",
       "| 3        | 6.655162 | \n",
       "| 4        | 6.752830 | \n",
       "| 5        | 6.842012 | \n",
       "| 6        | 6.915456 | \n",
       "| 7        | 7.004088 | \n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "  time lwage   \n",
       "1 1    6.423298\n",
       "2 2    6.515570\n",
       "3 3    6.655162\n",
       "4 4    6.752830\n",
       "5 5    6.842012\n",
       "6 6    6.915456\n",
       "7 7    7.004088"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "male_wage_mean <- pd %>% group_by(time) %>% filter(sex == 'male') %>% summarise_at(vars(lwage),mean) \n",
    "male_wage_mean"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th scope=col>time</th><th scope=col>lwage</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><td>1       </td><td>5.995927</td></tr>\n",
       "\t<tr><td>2       </td><td>6.068366</td></tr>\n",
       "\t<tr><td>3       </td><td>6.136134</td></tr>\n",
       "\t<tr><td>4       </td><td>6.248850</td></tr>\n",
       "\t<tr><td>5       </td><td>6.348618</td></tr>\n",
       "\t<tr><td>6       </td><td>6.458892</td></tr>\n",
       "\t<tr><td>7       </td><td>6.530366</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{r|ll}\n",
       " time & lwage\\\\\n",
       "\\hline\n",
       "\t 1        & 5.995927\\\\\n",
       "\t 2        & 6.068366\\\\\n",
       "\t 3        & 6.136134\\\\\n",
       "\t 4        & 6.248850\\\\\n",
       "\t 5        & 6.348618\\\\\n",
       "\t 6        & 6.458892\\\\\n",
       "\t 7        & 6.530366\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "time | lwage | \n",
       "|---|---|---|---|---|---|---|\n",
       "| 1        | 5.995927 | \n",
       "| 2        | 6.068366 | \n",
       "| 3        | 6.136134 | \n",
       "| 4        | 6.248850 | \n",
       "| 5        | 6.348618 | \n",
       "| 6        | 6.458892 | \n",
       "| 7        | 6.530366 | \n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "  time lwage   \n",
       "1 1    5.995927\n",
       "2 2    6.068366\n",
       "3 3    6.136134\n",
       "4 4    6.248850\n",
       "5 5    6.348618\n",
       "6 6    6.458892\n",
       "7 7    6.530366"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "female_wage_mean <- pd %>% group_by(time) %>% filter(sex == 'female') %>% summarise_at(vars(lwage),mean) \n",
    "female_wage_mean"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAtAAAALQCAMAAACOibeuAAAAOVBMVEUAAAAAAP9NTU1oaGh8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\nQtAQhaAhCkFDFIKGKAQNUQgaohA0RPkf6TiUGV65bp4AAAAASUVORK5CYII=",
      "text/plain": [
       "Plot with title \"Mean Log Wage: Male vs Female\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "options(repr.plot.width=6, repr.plot.height=6)\n",
    "plot(male_wage_mean$lwage,\n",
    "     type = \"o\",col = 'blue',lwd=2, \n",
    "     xlab = \"time\", ylab = \"log wage\",ylim=c(5.9,7.1),\n",
    "     main = 'Mean Log Wage: Male vs Female')\n",
    "lines(female_wage_mean, type = \"o\", col = 'red',lwd=2)\n",
    "grid(nx=NA,ny=NULL,lty=1,lwd=1,col=\" gray\")\n",
    "legend(\"bottomright\",pch=c(15,15),legend=c('Male','Female'),col=c('blue','red'))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 3. Samples' distribution characteristics"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, I use categorical variables to discover samples' distribution characteristics. Taking blue collar, sex, black at time 7 as an example."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "t7 <- pd[which(time == 7),]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "      exp             wks        bluecol        ind        south      smsa    \n",
       " Min.   : 7.00   Min.   : 5.00   no :290   Min.   :0.000   no :421   no :213  \n",
       " 1st Qu.:13.00   1st Qu.:46.00   yes:305   1st Qu.:0.000   yes:174   yes:382  \n",
       " Median :21.00   Median :48.00             Median :0.000                      \n",
       " Mean   :22.85   Mean   :46.45             Mean   :0.405                      \n",
       " 3rd Qu.:32.00   3rd Qu.:49.00             3rd Qu.:1.000                      \n",
       " Max.   :51.00   Max.   :52.00             Max.   :1.000                      \n",
       "                                                                              \n",
       " married       sex      union           ed        black         lwage      \n",
       " no :116   male  :528   no :377   Min.   : 4.00   no :552   Min.   :5.677  \n",
       " yes:479   female: 67   yes:218   1st Qu.:12.00   yes: 43   1st Qu.:6.685  \n",
       "                                  Median :12.00             Median :6.985  \n",
       "                                  Mean   :12.85             Mean   :6.951  \n",
       "                                  3rd Qu.:16.00             3rd Qu.:7.208  \n",
       "                                  Max.   :17.00             Max.   :8.537  \n",
       "                                                                           \n",
       "       id      time   \n",
       " 1      :  1   1:  0  \n",
       " 2      :  1   2:  0  \n",
       " 3      :  1   3:  0  \n",
       " 4      :  1   4:  0  \n",
       " 5      :  1   5:  0  \n",
       " 6      :  1   6:  0  \n",
       " (Other):589   7:595  "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "summary(t7)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "  male female \n",
       "   276     29 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "t7_blue <- t7[which(t7$bluecol == 'yes'),]\n",
    "table(t7_blue$sex)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       " no yes \n",
       "261  15 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "t7_blue_male <- t7_blue[which(t7_blue$sex == 'male'),]\n",
    "table(t7_blue_male$black)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       " no yes \n",
       " 16  13 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "t7_blue_female <- t7_blue[which(t7_blue$sex == 'female'),]\n",
    "table(t7_blue_female$black)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "  male female \n",
       "   252     38 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "t7_nonblue <- t7[which(t7$bluecol == 'no'),]\n",
    "table(t7_nonblue$sex)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       " no yes \n",
       "239  13 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "t7_nonblue_male <- t7_nonblue[which(t7_nonblue$sex == 'male'),]\n",
    "table(t7_nonblue_male$black)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       " no yes \n",
       " 36   2 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "t7_nonblue_female <- t7_nonblue[which(t7_nonblue$sex == 'female'),]\n",
    "table(t7_nonblue_female$black)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Distribution of log wage<a name=\"3\"></a>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Taking the log wage at time 7 as an example, let's try to find it's distribution."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [],
   "source": [
    "lwage7 <- t7$lwage"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAtAAAALQCAMAAACOibeuAAAAPFBMVEUAAAAA/wBNTU1oaGh8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\nGrDifyIim7ZUPKYqAAAAAElFTkSuQmCC",
      "text/plain": [
       "Plot with title \"Distribution of Log Wage of Time 7\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "options(repr.plot.width=6, repr.plot.height=6)\n",
    "par(yaxs='i',las=1)\n",
    "hist(lwage7,\n",
    "     freq=FALSE,\n",
    "     col='grey',border='black',\n",
    "     xlab='log wage of time 7',\n",
    "     main='Distribution of Log Wage of Time 7')\n",
    "box(bty='l')\n",
    "lines(density(lwage7,na.rm=T),col='red',lwd=2)\n",
    "abline(v=mean(lwage7), col='green' ,lwd=2)\n",
    "abline(v=median(lwage7), col='yellow' ,lwd=2)\n",
    "grid(nx=NA,ny=NULL,lty=1,lwd=1,col=\" gray\")\n",
    "legend(\"topright\",pch=c(15,15,15),legend=c('Density','Mean','Median'),col=c('red','green','yellow'))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "-0.114000506077867"
      ],
      "text/latex": [
       "-0.114000506077867"
      ],
      "text/markdown": [
       "-0.114000506077867"
      ],
      "text/plain": [
       "[1] -0.1140005"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "skewness(lwage7)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "3.39365097004502"
      ],
      "text/latex": [
       "3.39365097004502"
      ],
      "text/markdown": [
       "3.39365097004502"
      ],
      "text/plain": [
       "[1] 3.393651"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "kurtosis(lwage7)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAlgAAAJYCAMAAACJuGjuAAAAMFBMVEUAAABNTU1oaGh8fHyM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\nOL/eQZJKpgAAAABJRU5ErkJggg==",
      "text/plain": [
       "Plot with title \"Normal Q-Q Plot for Log Wage of Time 7\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "options(repr.plot.width=5, repr.plot.height=5)\n",
    "qqnorm(lwage7, main = 'Normal Q-Q Plot for Log Wage of Time 7')\n",
    "qqline(lwage7)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "\tShapiro-Wilk normality test\n",
       "\n",
       "data:  lwage7\n",
       "W = 0.99239, p-value = 0.003932\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "nortest <- shapiro.test(lwage7)\n",
    "nortest"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Examining and cleaning outlier<a name=\"4\"></a>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAeAAAALQCAMAAAB/mFOXAAAAM1BMVEUAAABNTU1oaGh8fHyM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\nxUGwOAgWB8HiIFgcBIuDYHH+B2zNfFrsG/xnAAAAAElFTkSuQmCC",
      "text/plain": [
       "Plot with title \"Boxplot of Log Wage of Time 7\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "options(repr.plot.width=4, repr.plot.height=6)\n",
    "boxplot(lwage7,col=\"grey\",border=\"black\",\n",
    "        ylab='log wage of time 7',\n",
    "        main='Boxplot of Log Wage of Time 7')\n",
    "grid(nx=NA,ny=NULL,lty=1,lwd=1,col=\" gray\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1] \"number of outliers:  15\"\n"
     ]
    }
   ],
   "source": [
    "outliers <- boxplot.stats(lwage7)$out\n",
    "print(paste(\"number of outliers: \",length(outliers)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1] \"number of samples after cleaning outliers:  580\"\n"
     ]
    }
   ],
   "source": [
    "m_lwage7 <- lwage7[! lwage7 %in% outliers]\n",
    "print(paste(\"number of samples after cleaning outliers: \",length(m_lwage7)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Panel Data Models<a name=\"5\"></a>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The type of categorical variables is factor."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [],
   "source": [
    "mpd <- pd\n",
    "mpd$sex <- factor(mpd$sex,labels=c(0,1))\n",
    "mpd$married <- factor(mpd$married,labels=c(0,1))\n",
    "mpd$black <- factor(mpd$black,labels=c(0,1))\n",
    "detach(pd)\n",
    "attach(mpd)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Pooling Model\n",
       "\n",
       "Call:\n",
       "plm(formula = lwage ~ exp + wks + ed + sex + married + black, \n",
       "    data = mpd, model = \"pooling\")\n",
       "\n",
       "Balanced Panel: n=595, T=7, N=4165\n",
       "\n",
       "Residuals :\n",
       "      Min.    1st Qu.     Median    3rd Qu.       Max. \n",
       "-2.2034348 -0.2376862  0.0033105  0.2524530  1.9345178 \n",
       "\n",
       "Coefficients :\n",
       "               Estimate  Std. Error  t-value  Pr(>|t|)    \n",
       "(Intercept)  5.30448293  0.06570553  80.7312 < 2.2e-16 ***\n",
       "exp          0.01182958  0.00054972  21.5191 < 2.2e-16 ***\n",
       "wks          0.00405787  0.00113382   3.5790 0.0003489 ***\n",
       "ed           0.07374622  0.00214177  34.4323 < 2.2e-16 ***\n",
       "sex1        -0.34976966  0.02662526 -13.1368 < 2.2e-16 ***\n",
       "married1     0.06150482  0.02186638   2.8128 0.0049349 ** \n",
       "black1      -0.15159601  0.02316837  -6.5432 6.751e-11 ***\n",
       "---\n",
       "Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1\n",
       "\n",
       "Total Sum of Squares:    886.9\n",
       "Residual Sum of Squares: 580.16\n",
       "R-Squared:      0.34586\n",
       "Adj. R-Squared: 0.34492\n",
       "F-statistic: 366.411 on 6 and 4158 DF, p-value: < 2.22e-16"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pooling <- plm(lwage ~ exp + wks + ed + sex + married + black, data=mpd, model= \"pooling\")\n",
    "summary(pooling)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Oneway (individual) effect Between Model\n",
       "\n",
       "Call:\n",
       "plm(formula = lwage ~ exp + wks + ed + sex + married + black, \n",
       "    data = mpd, model = \"between\")\n",
       "\n",
       "Balanced Panel: n=595, T=7, N=4165\n",
       "Observations used in estimation: 595\n",
       "\n",
       "Residuals :\n",
       "     Min.   1st Qu.    Median   3rd Qu.      Max. \n",
       "-0.949431 -0.194872  0.031572  0.218650  0.818082 \n",
       "\n",
       "Coefficients :\n",
       "              Estimate Std. Error t-value  Pr(>|t|)    \n",
       "(Intercept)  5.1686396  0.1981128 26.0894 < 2.2e-16 ***\n",
       "exp          0.0085564  0.0011917  7.1797 2.119e-12 ***\n",
       "wks          0.0078176  0.0037787  2.0688  0.038997 *  \n",
       "ed           0.0712038  0.0045413 15.6793 < 2.2e-16 ***\n",
       "sex1        -0.3015372  0.0603729 -4.9946 7.784e-07 ***\n",
       "married1     0.1242968  0.0525362  2.3659  0.018308 *  \n",
       "black1      -0.1389058  0.0491550 -2.8259  0.004875 ** \n",
       "---\n",
       "Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1\n",
       "\n",
       "Total Sum of Squares:    92.322\n",
       "Residual Sum of Squares: 52.576\n",
       "R-Squared:      0.43051\n",
       "Adj. R-Squared: 0.4247\n",
       "F-statistic: 74.0853 on 6 and 588 DF, p-value: < 2.22e-16"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "between <- plm(lwage ~ exp + wks + ed + sex + married + black, data=mpd, model= \"between\")\n",
    "summary(between)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Oneway (individual) effect Random Effect Model \n",
       "   (Swamy-Arora's transformation)\n",
       "\n",
       "Call:\n",
       "plm(formula = lwage ~ exp + wks + ed + sex + married + black, \n",
       "    data = mpd, model = \"random\")\n",
       "\n",
       "Balanced Panel: n=595, T=7, N=4165\n",
       "\n",
       "Effects:\n",
       "                  var std.dev share\n",
       "idiosyncratic 0.02356 0.15349 0.215\n",
       "individual    0.08605 0.29334 0.785\n",
       "theta:  0.806  \n",
       "\n",
       "Residuals :\n",
       "      Min.    1st Qu.     Median    3rd Qu.       Max. \n",
       "-2.0459196 -0.1149687  0.0021092  0.1184320  2.0643465 \n",
       "\n",
       "Coefficients :\n",
       "               Estimate  Std. Error t-value  Pr(>|t|)    \n",
       "(Intercept)  4.29832874  0.09299567 46.2207 < 2.2e-16 ***\n",
       "exp          0.05235392  0.00107787 48.5716 < 2.2e-16 ***\n",
       "wks          0.00151232  0.00076914  1.9662 0.0493366 *  \n",
       "ed           0.10749796  0.00579063 18.5641 < 2.2e-16 ***\n",
       "sex1        -0.32979262  0.05512963 -5.9821 2.388e-09 ***\n",
       "married1    -0.07301560  0.02313775 -3.1557 0.0016126 ** \n",
       "black1      -0.22763295  0.06302307 -3.6119 0.0003076 ***\n",
       "---\n",
       "Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1\n",
       "\n",
       "Total Sum of Squares:    264.98\n",
       "Residual Sum of Squares: 163.2\n",
       "R-Squared:      0.38408\n",
       "Adj. R-Squared: 0.38319\n",
       "F-statistic: 432.144 on 6 and 4158 DF, p-value: < 2.22e-16"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "random <- plm(lwage ~ exp + wks + ed + sex + married + black, data=mpd, model= \"random\")\n",
    "summary(random)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "\tHausman Test\n",
       "\n",
       "data:  lwage ~ exp + wks + ed + sex + married + black\n",
       "chisq = 8945.9, df = 6, p-value < 2.2e-16\n",
       "alternative hypothesis: one model is inconsistent\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "phtest(random, between)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "***Thank you for reading!***"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "R",
   "language": "R",
   "name": "ir"
  },
  "language_info": {
   "codemirror_mode": "r",
   "file_extension": ".r",
   "mimetype": "text/x-r-source",
   "name": "R",
   "pygments_lexer": "r",
   "version": "3.4.3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}