Hypothesis_Testing.ipynb

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  "metadata": {
    "colab": {
      "name": "Hypothesis Testing",
      "provenance": [],
      "include_colab_link": true
    },
    "kernelspec": {
      "name": "python3",
      "display_name": "Python 3"
    }
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  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "view-in-github",
        "colab_type": "text"
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      "source": [
        "<a href=\"https://colab.research.google.com/github/TheHouseOfVermeulens/Financial-Engineering/blob/master/Hypothesis_Testing.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
      ]
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    {
      "cell_type": "code",
      "metadata": {
        "id": "sS6_UkRFUSuC",
        "colab_type": "code",
        "colab": {}
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      "source": [
        "# Useful Libraries\n",
        "import pandas as pd\n",
        "import numpy as np\n",
        "import matplotlib.pyplot as plt\n",
        "from scipy.stats import t\n",
        "import scipy.stats"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "4lWzNMBtU-AO",
        "colab_type": "text"
      },
      "source": [
        "a. One tail test.\n",
        "\n",
        "Using the techniques laid out in lecture, verify if we can state that the returns of TSLA are greater than 0.\n",
        "\n",
        "    Start by stating the null and alternative hypothesis\n",
        "        Are we dealing with a one or two tailed test? Why?\n",
        "    Calculate the mean differences, and the Z-test using the formula provided in class.\n",
        "        Recall: This is a one parameter test, use the appropriate Z-test\n",
        "    Use the stat library to calculate the associated p value with your t statistic.\n",
        "        Compare your found p-value to the set α\n",
        "\n",
        "        value, and conclude.\n",
        "\n",
        "Useful Formulas:\n",
        "\n",
        "Test statistic=X¯∗μ−θ0s∗X¯=X¯μ−0sn−−⎷"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "5wN7lddzU4u9",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        "prices1 = get_pricing('TSLA', start_date = '2015-01-01', end_date = '2016-01-01', fields = 'price')\n",
        "returns_sample_tsla = prices1.pct_change()[1:]\n",
        "\n",
        "print ('Tesla return sample mean', returns_sample_tsla.mean())\n",
        "print ('Tesla return sample standard deviation', returns_sample_tsla.std())\n",
        "print ('Tesla return sample size', len(returns_sample_tsla))"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "P-_cygzPVOHD",
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      "source": [
        "\n",
        "\n",
        "```\n",
        "Tesla return sample mean 0.0006611637458\n",
        "Tesla return sample standard deviation 0.024566905352\n",
        "Tesla return sample size 251\n",
        "```\n",
        "\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "cKcyDSaQVTm6",
        "colab_type": "text"
      },
      "source": [
        "\n",
        "One tail hypotheses.\n",
        "\n",
        "    Null hypothesis: No difference in average returns in both population, the mean is less than 0\n",
        "    Althernative hypothesis: There is a difference in average returns of both populations, the mean is not less than 0\n",
        "\n"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "lMr12RCOVKAP",
        "colab_type": "code",
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      },
      "source": [
        "# Testing\n",
        "\n",
        "## Z- Statistic: \n",
        "test_stat = (returns_sample_tsla.mean() - 0) / \\\n",
        "    ( returns_sample_tsla.std() / np.sqrt( len(returns_sample_tsla) ) )\n",
        "print ('t-statistic is:', test_stat)\n",
        "\n",
        "## Finding the p-value for one tail test\n",
        "p_val = (1 - t.cdf(test_stat, len(returns_sample_tsla) - 1))\n",
        "print ('p-value is: ', p_val)"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "GO05Crg2Vfcg",
        "colab_type": "text"
      },
      "source": [
        "\n",
        "\n",
        "```\n",
        "t-statistic is: 0.426378639576\n",
        "p-value is:  0.335099432105\n",
        "```\n",
        "\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "a1uv-9ZUVj-c",
        "colab_type": "text"
      },
      "source": [
        "Comparing p-value with different α\n",
        "values\n",
        "\n",
        "With α=0.01\n",
        ", our p-value is greater than our α value, we thus fail to reject the null hypothesis.\n",
        "With α=0.5, our p-value is greater than our α value, we thus fail to reject the null hypothesis.\n",
        "With α=0.1, our p-value is greater than our α value, we thus fail to reject the null hypothesis."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "iSj2SaRmVUnG",
        "colab_type": "code",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 483
        },
        "outputId": "72d4451f-633f-4ecf-9696-06c820d6893f"
      },
      "source": [
        "## Graph for visualization.\n",
        "\n",
        "x = np.linspace(-3, 3, 100)\n",
        "norm_pdf = lambda x: (1/np.sqrt(2 * np.pi)) * np.exp(-x * x / 2)\n",
        "y = norm_pdf(x)\n",
        "\n",
        "fig, ax = plt.subplots(1, 1, sharex=True)\n",
        "ax.plot(x, y)\n",
        "\n",
        "ax.fill_between(x, 0, y, where = x > 1.282, label = 'Confidence region alpha = 0.1')\n",
        "ax.fill_between(x, 0, y, where = x > 1.645, label = 'Confidence region alpha = 0.05', color = 'green')\n",
        "ax.fill_between(x, 0, y, where = x > 2.326, label = 'Confidence region alpha = 0.01', color = 'red')\n",
        "plt.axvline(p_val, linestyle = 'dashed', label = 'Location of P Value')\n",
        "\n",
        "plt.title('Graph of rejection region and P-value')\n",
        "plt.xlabel('x')\n",
        "plt.ylabel('p(x)')\n",
        "plt.legend();"
      ],
      "execution_count": 4,
      "outputs": [
        {
          "output_type": "error",
          "ename": "NameError",
          "evalue": "ignored",
          "traceback": [
            "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
            "\u001b[0;31mNameError\u001b[0m                                 Traceback (most recent call last)",
            "\u001b[0;32m<ipython-input-4-ba708a711a7a>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[1;32m     10\u001b[0m \u001b[0max\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mfill_between\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;36m0\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0my\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mwhere\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mx\u001b[0m \u001b[0;34m>\u001b[0m \u001b[0;36m1.645\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mlabel\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m'Confidence region alpha = 0.05'\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mcolor\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m'green'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     11\u001b[0m \u001b[0max\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mfill_between\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;36m0\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0my\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mwhere\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mx\u001b[0m \u001b[0;34m>\u001b[0m \u001b[0;36m2.326\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mlabel\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m'Confidence region alpha = 0.01'\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mcolor\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m'red'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 12\u001b[0;31m \u001b[0mplt\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0maxvline\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mp_val\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mlinestyle\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m'dashed'\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mlabel\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m'Location of P Value'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     13\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     14\u001b[0m \u001b[0mplt\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mtitle\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'Graph of rejection region and P-value'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
            "\u001b[0;31mNameError\u001b[0m: name 'p_val' is not defined"
          ]
        },
        {
          "output_type": "display_data",
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Y47+AiMwAZgDExcV5mt0Y44C2EWFs/P3FXvmsR5at5G9rUrzyWc2R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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "BF8XWay3Vv-i",
        "colab_type": "text"
      },
      "source": [
        "In the graph above, we can clearly see the rejection region for all three values of α\n",
        "\n",
        "are bellow the found p-value.\n",
        "\n",
        "##b. Two tailed test.\n",
        "\n",
        "Using the techniques laid out in lecture, verify if we can state that the returns of TSLA are equal to 0.\n",
        "Two tail hypotheses.\n",
        "\n",
        "    Null hypothesis: No difference in average returns in both population, the mean is 0\n",
        "    Althernative hypothesis: There is a difference in average returns of both populations, the mean is not 0\n",
        "\n"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "DcJezbv3Vq09",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        "## Finding the p-value for two tailed test. \n",
        "p_val = 2*(1 - t.cdf(test_stat, len(returns_sample_tsla) - 1))\n",
        "print ('p-value is: ', p_val)"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "SOj6--bDWAGj",
        "colab_type": "text"
      },
      "source": [
        "\n",
        "\n",
        "```\n",
        "p-value is:  0.670198864209\n",
        "```\n",
        "\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "26OniSjHWExT",
        "colab_type": "text"
      },
      "source": [
        "With α=0.01, our p-value is greater than our α value, we thus fail to reject the null hypothesis.\n",
        "With α=0.5, our p-value is greater than our α value, we thus fail to reject the null hypothesis.\n",
        "With α=0.1, our p-value is greater than our α value, we thus fail to reject the null hypothesis."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "K1QjouEHV78Z",
        "colab_type": "code",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 483
        },
        "outputId": "d1b609c2-1ef5-4ee7-94ff-289f2465a9ca"
      },
      "source": [
        "x = np.linspace(-3, 3, 100)\n",
        "norm_pdf = lambda x: (1/np.sqrt(2 * np.pi)) * np.exp(-x * x / 2)\n",
        "y = norm_pdf(x)\n",
        "\n",
        "fig, ax = plt.subplots(1, 1, sharex=True)\n",
        "ax.plot(x, y)\n",
        "\n",
        "ax.fill_between(x, 0, y, where = x > 1.645, label = 'Confidence region at alpha = 0.1')\n",
        "ax.fill_between(x, 0, y, where = x < -1.645)\n",
        "\n",
        "ax.fill_between(x, 0, y, where = x > 1.96, label = 'Confidence region at alpha = 0.05', color = 'green')\n",
        "ax.fill_between(x, 0, y, where = x < -1.96, color = 'green')\n",
        "\n",
        "ax.fill_between(x, 0, y, where = x > 2.576, label = 'Confidence region at alpha = 0.05', color='red')\n",
        "ax.fill_between(x, 0, y, where = x < -2.576, color = 'red')\n",
        "plt.axvline(p_val, linestyle = 'dashed', label = 'Location of P Value')\n",
        "\n",
        "plt.title('Graph of rejection region and P-value')\n",
        "plt.xlabel('x')\n",
        "plt.ylabel('p(x)')\n",
        "plt.legend();"
      ],
      "execution_count": 6,
      "outputs": [
        {
          "output_type": "error",
          "ename": "NameError",
          "evalue": "ignored",
          "traceback": [
            "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
            "\u001b[0;31mNameError\u001b[0m                                 Traceback (most recent call last)",
            "\u001b[0;32m<ipython-input-6-b1e37ff28d78>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[1;32m     14\u001b[0m \u001b[0max\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mfill_between\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;36m0\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0my\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mwhere\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mx\u001b[0m \u001b[0;34m>\u001b[0m \u001b[0;36m2.576\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mlabel\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m'Confidence region at alpha = 0.05'\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mcolor\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;34m'red'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     15\u001b[0m \u001b[0max\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mfill_between\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;36m0\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0my\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mwhere\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mx\u001b[0m \u001b[0;34m<\u001b[0m \u001b[0;34m-\u001b[0m\u001b[0;36m2.576\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mcolor\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m'red'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 16\u001b[0;31m \u001b[0mplt\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0maxvline\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mp_val\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mlinestyle\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m'dashed'\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mlabel\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m'Location of P Value'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     17\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     18\u001b[0m \u001b[0mplt\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mtitle\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'Graph of rejection region and P-value'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
            "\u001b[0;31mNameError\u001b[0m: name 'p_val' is not defined"
          ]
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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sTcnhG+ePYuRAu6DFdK15541k1KCePPFhOi0e26t3ihV9mPnVB3vpEenmvq/Y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\nAElFTkSuQmCC\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "9BOFF_meWFt_",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        ""
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "axjglYp9WVkT",
        "colab_type": "text"
      },
      "source": [
        "##a. Critical Values.\n",
        "\n",
        "Find the critical values associated with α=1%,5%,10%\n",
        "and graph the rejection regions on a plot for a two tailed test."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "KpZs8O5_WYIH",
        "colab_type": "code",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 69
        },
        "outputId": "22f4dae7-6b9b-4fe5-f2ef-d71c3d715e0e"
      },
      "source": [
        "# For alpha = 10%\n",
        "alpha = 0.1\n",
        "f = 1 - (alpha/2)\n",
        "print ('alpha = 10%: f = ', f)\n",
        "\n",
        "# For alpha = 5%\n",
        "alpha = 0.05\n",
        "f = 1 - (alpha/2)\n",
        "print ('alpha = 5%: f = ', f)\n",
        "\n",
        "# For alpha = 1%\n",
        "alpha = 0.01\n",
        "f = 1 - (alpha/2)\n",
        "print ('alpha = 1%: f = ', f)"
      ],
      "execution_count": 8,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "alpha = 10%: f =  0.95\n",
            "alpha = 5%: f =  0.975\n",
            "alpha = 1%: f =  0.995\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "WW-BiULvWmM_",
        "colab_type": "text"
      },
      "source": [
        "Using the z-table above, we find that for"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "cJwXOYOhWbv_",
        "colab_type": "code",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 295
        },
        "outputId": "a564ecd9-1380-4a51-9c6e-c930d3220526"
      },
      "source": [
        "# Plot a standard normal distribution and mark the critical regions with shading\n",
        "x = np.linspace(-3, 3, 100)\n",
        "norm_pdf = lambda x: (1/np.sqrt(2 * np.pi)) * np.exp(-x * x / 2)\n",
        "y = norm_pdf(x)\n",
        "\n",
        "fig, ax = plt.subplots(1, 1, sharex=True)\n",
        "ax.plot(x, y)\n",
        "\n",
        "# Value for alpha = 1%\n",
        "ax.fill_between(x, 0, y, where =  x > 1.645, label = 'alpha = 10%')\n",
        "ax.fill_between(x, 0, y, where = x < -1.645)\n",
        "\n",
        "# Value for alpha = 5%\n",
        "ax.fill_between(x, 0, y, where = x > 1.96, color = 'red', label = 'alpha = 5%')\n",
        "ax.fill_between(x, 0, y, where = x < -1.96, color = 'red')\n",
        "\n",
        "#Value for alpha = 10%\n",
        "ax.fill_between(x, 0, y, where = x > 2.575, facecolor='green', label = 'alpha = 1%')\n",
        "ax.fill_between(x, 0, y, where = x < -2.575, facecolor='green')\n",
        "\n",
        "plt.title('Rejection regions for a two-tailed hypothesis test at 90%, 95%, 99% confidence')\n",
        "plt.xlabel('x')\n",
        "plt.ylabel('p(x)')\n",
        "plt.legend();"
      ],
      "execution_count": 9,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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Q9957j44dO3L33Xeza9cuAJ5++mnuvPNOXn31VYYMGcKzzz7L0KFDS2PRmFJg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\nXLzNzMwy5r+7MIoVdz7aWwAAAABJRU5ErkJggg==\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "icpBS4-UWnOb",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        ""
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "YUAJvx9tWu19",
        "colab_type": "text"
      },
      "source": [
        "##b. Mean T-Test\n",
        "\n",
        "Run a T-test on the SPY returns, to determine if the mean returns is 0.01.\n",
        "\n",
        "Find the two critical values for a 90% two tailed z -test. \n",
        "\n",
        "Use the formula above to run a t-test on the sample data.\n",
        "Conclude about the test results."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "JgtHEs7MWwfy",
        "colab_type": "code",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 34
        },
        "outputId": "103cc4c2-148d-48c2-cda7-15963d16c92c"
      },
      "source": [
        "# Calculating Critical Values probability\n",
        "\n",
        "alpha = 0.1\n",
        "v = 1 - (alpha/2)\n",
        "print (v)"
      ],
      "execution_count": 10,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "0.95\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "arKrli1aW-Hq",
        "colab_type": "text"
      },
      "source": [
        "From the previous question, we find a critical value of 1.645."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "M4nUJEuHW7Uz",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        "data = get_pricing('SPY', start_date = '2016-01-01', end_date = '2017-01-01', fields = 'price')\n",
        "returns_sample = data.pct_change()[1:]\n",
        "\n",
        "# Running the T-test.\n",
        "n = len(returns_sample)\n",
        "test_statistic = ((returns_sample.mean() - 0) /\n",
        "                (returns_sample.std()/np.sqrt(n)))\n",
        "print ('t test statistic: ', test_statistic)"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "TO7MEPARXIAn",
        "colab_type": "text"
      },
      "source": [
        "\n",
        "\n",
        "```\n",
        "t test statistic:  1.05168962394\n",
        "```\n",
        "\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "YhGXIrvRXN8I",
        "colab_type": "text"
      },
      "source": [
        "We find that −1.645<1.05<1.645. We thus conclude that we fail to reject our H0"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "h0mfS13_XUmG",
        "colab_type": "text"
      },
      "source": [
        "##Mean p-value test\n",
        "\n",
        "Given the returns data above, use the p-value to determine the results of the previous hypothesis test."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "QQVqhIiuW-6h",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        "# Running p-value test. \n",
        "\n",
        "alpha = 0.1\n",
        "p_val = 2 * (1 - t.cdf(test_statistic, n - 1))\n",
        "print ('p-value is: ', p_val)\n",
        "\n",
        "if p_val > alpha: \n",
        "    print ('p-value is greater than our significant level, we thus fail to reject the null hypothesis.')\n",
        "else: \n",
        "    print ('p-value is less than or equal to our significal level, we thus reject the null hypothesis.')\n"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "vAvNUHMgXi_u",
        "colab_type": "text"
      },
      "source": [
        "\n",
        "\n",
        "```\n",
        "p-value is:  0.293957454918\n",
        "p-value is greater than our significant level, we thus fail to reject the null hypothesis.```\n",
        "\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "ow5fjnnhXn2N",
        "colab_type": "text"
      },
      "source": [
        "As we can see above, our p-value is greater than our significant level, α=0.1, we thus fail to reject the null hypothesis."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "oodIytwGXPSu",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        ""
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "i1gR9ZsTXt21",
        "colab_type": "text"
      },
      "source": [
        "#Multiple Variables Tests.\n",
        "##a. Hypothesis testing on Means.\n",
        "\n",
        "State the hypothesis tests for comparing two means\n",
        "Find the test statistic along with the degrees of freedom for the following two assets. Assume variance is different (We assume XLF to be a safer buy than GS.\n",
        "Use the t-table to conclude about your hypothesis test. Pick α=10%\n",
        "\n",
        "Useful Formulas:\n",
        "\n",
        "t=X¯1−X¯2(s2pn1+s2pn2)1/2\n",
        "t=X¯1−X¯2(s21n1+s22n2)1/2\n",
        "df=(s21n1+s22n2)2(s21/n1)2/(n1−1)+(s22/n2)2/(n2−1)\n",
        "\n",
        "note: one formula for t involves equal variance, the other does not. Use the right one given the information above\n",
        "\n",
        "Answer:\n",
        "Hypothesis Tests when comparing two means:\n",
        "1.H0:μ1−μ2=θ0, HA:μ1−μ2≠θ0\n",
        "2.H0:μ1−μ2≤θ0, HA:μ1−μ2>θ0\n",
        "3.H0:μ1−μ2≥θ0, HA:μ1−μ2<θ0"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "8EJsruqRXo1U",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        "# Data Collection\n",
        "alpha = 0.1\n",
        "symbol_list = ['XLF', 'MCD']\n",
        "start = '2015-01-01'\n",
        "end = '2016-01-01'\n",
        "pricing_sample = get_pricing(symbol_list, start_date = start, end_date = end, fields='price')\n",
        "pricing_sample.columns = map(lambda x: x.symbol, pricing_sample.columns)\n",
        "returns_sample = pricing_sample.pct_change()[1:]\n",
        "\n",
        "\n",
        "# Sample mean values\n",
        "mu_xlf, mu_gs = returns_sample.mean()\n",
        "s_xlf, s_gs = returns_sample.std()\n",
        "n_xlf = len(returns_sample['XLF'])\n",
        "n_gs = len(returns_sample['MCD'])\n",
        "\n",
        "test_statistic = ((mu_xlf - mu_gs) - 0)/((s_xlf**2/n_xlf) + (s_gs**2/n_gs))**0.5\n",
        "df = ((s_xlf**2/n_xlf) + (s_gs**2/n_gs))**2/ \\\n",
        "     (((s_xlf**2 / n_xlf)**2 /(n_xlf-1))+((s_gs**2 / n_gs)**2/(n_gs-1)))\n",
        "\n",
        "print ('t test statistic: ', test_statistic)\n",
        "print ('Degrees of freedom (modified): ', df)\n",
        "print ('p-value: ', 2 * (1 - t.cdf(test_statistic, df)))"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "wT0Hx9k-YGD3",
        "colab_type": "text"
      },
      "source": [
        "\n",
        "\n",
        "```\n",
        "t test statistic:  -1.12060529448\n",
        "Degrees of freedom (modified):  496.271031856\n",
        "p-value:  1.73700210188```\n",
        "\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "17TYA1CYYKPk",
        "colab_type": "text"
      },
      "source": [
        "With a confidence level of 90%, our test statistic belongs to the range -1.645, 1.645. Since our test statistic is above these values we accept the null hypothesis and determine that the difference between XLF and MCD returns is significantly different from 0."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "YipOPi1EYOQ5",
        "colab_type": "text"
      },
      "source": [
        "\n",
        "##b. Hypothesis Testing on Variances.\n",
        "\n",
        "State the hypothesis tests for comparing two means.\n",
        "Calculate the returns and compare their variances.\n",
        "Calculate the F-test using the variances\n",
        "Check that both values have the same degrees of freedom.\n",
        "\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "Nk-tPWweYXlC",
        "colab_type": "text"
      },
      "source": [
        "Answer:\n",
        "\n",
        "1.H0:σ21=σ22, HA:σ21≠σ22\n",
        "2.H0:σ21≤σ22, HA:σ21>σ22\n",
        "3.H0:σ21≥σ22, HA:σ21<σ22"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "MVaZEL5eX8nb",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        "\n",
        "\n",
        "# Data\n",
        "symbol_list = ['XLF', 'MCD']\n",
        "start = \"2015-01-01\"\n",
        "end = \"2016-01-01\"\n",
        "pricing_sample = get_pricing(symbol_list, start_date = start, end_date = end, fields = 'price')\n",
        "pricing_sample.columns = map(lambda x: x.symbol, pricing_sample.columns)\n",
        "returns_sample = pricing_sample.pct_change()[1:]\n",
        "\n",
        "# Take returns from above, MCD and XLF, and compare their variances\n",
        "xlf_std_dev, mcd_std_dev = returns_sample.std()\n",
        "print ('XLF standard deviation is: ', xlf_std_dev)\n",
        "print ('MCD standard deviation is: ', mcd_std_dev)\n",
        "\n",
        "# Calculate F-test with MCD.std > XLF.std\n",
        "test_statistic = (mcd_std_dev / xlf_std_dev)**2\n",
        "print (\"F Test statistic: \", test_statistic)\n",
        "\n",
        "#degree of freedom \n",
        "df1 = len(returns_sample['XLF']) - 1\n",
        "df2 = len(returns_sample['MCD']) - 1\n",
        "print (df1)\n",
        "print (df2)\n",
        "\n",
        "# Calculate critical values. \n",
        "from scipy.stats import f\n",
        "upper_crit_value = f.ppf(0.975, df1, df2)\n",
        "lower_crit_value = f.ppf(0.025, df1, df2)\n",
        "print ('Upper critical value at a = 0.05 with df1 = {0} and df2 = {1}: '.format(df1, df2), upper_crit_value)\n",
        "print ('Lower critical value at a = 0.05 with df1 = {0} and df2 = {1}: '.format(df1, df2), lower_crit_value)\n",
        "\n"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "cRssgwi7YsSU",
        "colab_type": "text"
      },
      "source": [
        "\n",
        "\n",
        "```\n",
        "XLF standard deviation is:  0.0110764148949\n",
        "MCD standard deviation is:  0.0120820314786\n",
        "F Test statistic:  1.18982063302\n",
        "250\n",
        "250\n",
        "Upper critical value at a = 0.05 with df1 = 250 and df2 = 250:  1.28208064948\n",
        "Lower critical value at a = 0.05 with df1 = 250 and df2 = 250:  0.779982133263```\n",
        "\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "E4YUcAz9Yx0j",
        "colab_type": "text"
      },
      "source": [
        "We can see that our F-test statistic is bellow the Upper Critical Value, thus we accept the null hypothesis and reject the alternative and conclude that the variances of XLF and MCD indeed do not differ."
      ]
    }
  ]
}