![](\"media/cover.png\")
"
]
},
{
"cell_type": "markdown",
"id": "b1359e9e",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"## Overview\n",
"\n",
"- In the previous chapters, we covered stationarity, smoothing, trend, seasonality, and autocorrelation, and built two different models: \n",
"\n",
"> **MA models**: The current value of the series depends linearly on the series' mean and a set of prior (observed) white noise error terms.\n",
"\n",
"> **AR models**: The current value of the series depends linearly on its own previous values and on a stochastic term (an imperfectly predictable term).\n",
"\n",
"- In this chapter we will review these concepts and combine the AR and MA models into three more complicated ones."
]
},
{
"cell_type": "markdown",
"id": "29644aac",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"In particular, we will cover:\n",
"\n",
"1. Autoregressive Moving Average (ARMA) models.\n",
"2. Autoregressive Integrated Moving Average (ARIMA) models.\n",
"3. SARIMA models (ARIMA model for data with seasonality).\n",
"4. Selecting the best model."
]
},
{
"cell_type": "code",
"execution_count": 1,
"id": "c765c6ff",
"metadata": {
"slideshow": {
"slide_type": "skip"
},
"tags": [
"hide-input"
]
},
"outputs": [],
"source": [
"import sys\n",
"\n",
"# Install dependencies if the notebook is running in Colab\n",
"if 'google.colab' in sys.modules:\n",
" !pip install -U -qq tsa-course pmdarima numpy==1.26"
]
},
{
"cell_type": "code",
"execution_count": 2,
"id": "876fd91c",
"metadata": {
"slideshow": {
"slide_type": "skip"
},
"tags": [
"hide-input"
]
},
"outputs": [],
"source": [
"# Imports\n",
"import warnings\n",
"warnings.filterwarnings(\"ignore\")\n",
"from statsmodels.tools.sm_exceptions import ConvergenceWarning, ValueWarning\n",
"warnings.simplefilter('ignore', ConvergenceWarning)\n",
"warnings.simplefilter('ignore', ValueWarning)\n",
"warnings.simplefilter('ignore', UserWarning)\n",
"import time\n",
"from itertools import product\n",
"import matplotlib.pyplot as plt\n",
"import numpy as np\n",
"import pandas as pd\n",
"import statsmodels as ss\n",
"import seaborn as sns\n",
"from tqdm.notebook import tqdm\n",
"from scipy import stats\n",
"from statsmodels.tsa.arima.model import ARIMA\n",
"from statsmodels.graphics.tsaplots import month_plot, plot_acf, plot_pacf\n",
"from statsmodels.graphics.gofplots import qqplot\n",
"from statsmodels.tsa.stattools import adfuller, acf, pacf\n",
"from statsmodels.tsa.seasonal import STL, seasonal_decompose\n",
"from statsmodels.tsa.statespace.sarimax import SARIMAX\n",
"from statsmodels.tools.eval_measures import mse\n",
"from statsmodels.tsa.statespace.tools import diff\n",
"import pmdarima as pm\n",
"from tsa_course.lecture1 import fft_analysis\n",
"np.random.seed(0) "
]
},
{
"cell_type": "markdown",
"id": "891afa5a",
"metadata": {
"slideshow": {
"slide_type": "-"
}
},
"source": [
"---"
]
},
{
"cell_type": "markdown",
"id": "161403b8",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"## ARMA\n",
"\n",
"The ARMA model (also known as the *Box-Jenkins* approach) combines two models:\n",
"\n",
"- An autoregressive (AR) model of order $p$.\n",
"- A moving average (MA) model of order $q$."
]
},
{
"cell_type": "markdown",
"id": "f1bd5a33",
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"source": [
"- When we have autocorrelation between outcomes and their ancestors, there will be a pattern in the time series. \n",
"- This relationship can be modeled using an ARMA model. \n",
"- It allows us to predict the future with a confidence level proportional to the strength of the relationship and the proximity to known values (prediction weakens the further out we go)."
]
},
{
"cell_type": "markdown",
"id": "e2f1bf76",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"```{note}\n",
"- ARMA models assume the time series is stationary.\n",
"- A good rule of thumb is to have at least 100 observations when fitting an ARMA model.\n",
"```"
]
},
{
"cell_type": "markdown",
"id": "dd8fdaab",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"### Example data\n",
"\n",
"- In the following, we'll look at the monthly average temperatures between 1907-1972."
]
},
{
"cell_type": "code",
"execution_count": 3,
"id": "fb49cacb",
"metadata": {},
"outputs": [],
"source": [
"# load data and convert to datetime\n",
"monthly_temp = pd.read_csv('https://zenodo.org/records/10951538/files/arima_temp.csv?download=1', \n",
" skipfooter=2, \n",
" header=0, \n",
" index_col=0, \n",
" names=['month', 'temp'],\n",
" engine='python')\n",
"monthly_temp.index = pd.to_datetime(monthly_temp.index)"
]
},
{
"cell_type": "markdown",
"id": "6a82c7bb",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"- This is how the data looks like."
]
},
{
"cell_type": "code",
"execution_count": 4,
"id": "3c4bb8d6",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"| \n", " | temp | \n", "
|---|---|
| month | \n", "\n", " |
| 1907-01-01 | \n", "33.3 | \n", "
| 1907-02-01 | \n", "46.0 | \n", "
| 1907-03-01 | \n", "43.0 | \n", "
| 1907-04-01 | \n", "55.0 | \n", "
| 1907-05-01 | \n", "51.8 | \n", "
| \n", " | temp | \n", "
|---|---|
| count | \n", "792.000000 | \n", "
| mean | \n", "53.553662 | \n", "
| std | \n", "15.815452 | \n", "
| min | \n", "11.200000 | \n", "
| 25% | \n", "39.675000 | \n", "
| 50% | \n", "52.150000 | \n", "
| 75% | \n", "67.200000 | \n", "
| max | \n", "82.400000 | \n", "
| \n", " | temp | \n", "Seasonally_Differenced | \n", "
|---|---|---|
| month | \n", "\n", " | \n", " |
| 1908-01-01 | \n", "35.6 | \n", "2.3 | \n", "
| 1908-02-01 | \n", "35.2 | \n", "-10.8 | \n", "
| 1908-03-01 | \n", "48.1 | \n", "5.1 | \n", "
| 1908-04-01 | \n", "50.0 | \n", "-5.0 | \n", "
| 1908-05-01 | \n", "46.8 | \n", "-5.0 | \n", "
| ... | \n", "... | \n", "... | \n", "
| 1972-08-01 | \n", "75.6 | \n", "-4.9 | \n", "
| 1972-09-01 | \n", "64.1 | \n", "-1.7 | \n", "
| 1972-10-01 | \n", "51.7 | \n", "0.6 | \n", "
| 1972-11-01 | \n", "40.3 | \n", "-1.5 | \n", "
| 1972-12-01 | \n", "30.3 | \n", "-0.3 | \n", "
780 rows × 2 columns
\n", "| Dep. Variable: | temp | No. Observations: | 750 | \n", "
|---|---|---|---|
| Model: | SARIMAX(2, 1, 2)x(0, 1, [1], 12) | Log Likelihood | -2026.066 | \n", "
| Date: | Thu, 03 Apr 2025 | AIC | 4066.132 | \n", "
| Time: | 16:20:27 | BIC | 4098.350 | \n", "
| Sample: | 01-01-1907 | HQIC | 4078.557 | \n", "
| - 06-01-1969 | \n", " | ||
| Covariance Type: | opg | \n", " |
| coef | std err | z | P>|z| | [0.025 | 0.975] | \n", "|
|---|---|---|---|---|---|---|
| intercept | -6.589e-05 | 0.000 | -0.370 | 0.712 | -0.000 | 0.000 | \n", "
| ar.L1 | -0.7653 | 0.169 | -4.525 | 0.000 | -1.097 | -0.434 | \n", "
| ar.L2 | 0.1790 | 0.042 | 4.273 | 0.000 | 0.097 | 0.261 | \n", "
| ma.L1 | -0.0549 | 0.180 | -0.305 | 0.760 | -0.408 | 0.298 | \n", "
| ma.L2 | -0.9427 | 0.172 | -5.492 | 0.000 | -1.279 | -0.606 | \n", "
| ma.S.L12 | -0.9813 | 0.039 | -25.144 | 0.000 | -1.058 | -0.905 | \n", "
| sigma2 | 13.3636 | 0.935 | 14.293 | 0.000 | 11.531 | 15.196 | \n", "
| Ljung-Box (L1) (Q): | 0.03 | Jarque-Bera (JB): | 201.91 | \n", "
|---|---|---|---|
| Prob(Q): | 0.86 | Prob(JB): | 0.00 | \n", "
| Heteroskedasticity (H): | 0.77 | Skew: | -0.55 | \n", "
| Prob(H) (two-sided): | 0.04 | Kurtosis: | 5.32 | \n", "
| \n", " | d0_D0_s12 | \n", "d0_D1_s12 | \n", "d0_D2_s12 | \n", "d1_D0_s12 | \n", "d2_D0_s12 | \n", "d1_D1_s12 | \n", "d2_D1_s12 | \n", "d1_D2_s12 | \n", "d2_D2_s12 | \n", "
|---|---|---|---|---|---|---|---|---|---|
| month | \n", "\n", " | \n", " | \n", " | \n", " | \n", " | \n", " | \n", " | \n", " | \n", " |
| 1907-01-01 | \n", "33.3 | \n", "NaN | \n", "NaN | \n", "NaN | \n", "NaN | \n", "NaN | \n", "NaN | \n", "NaN | \n", "NaN | \n", "
| 1907-02-01 | \n", "46.0 | \n", "NaN | \n", "NaN | \n", "12.7 | \n", "NaN | \n", "NaN | \n", "NaN | \n", "NaN | \n", "NaN | \n", "
| 1907-03-01 | \n", "43.0 | \n", "NaN | \n", "NaN | \n", "-3.0 | \n", "-15.7 | \n", "NaN | \n", "NaN | \n", "NaN | \n", "NaN | \n", "
| 1907-04-01 | \n", "55.0 | \n", "NaN | \n", "NaN | \n", "12.0 | \n", "15.0 | \n", "NaN | \n", "NaN | \n", "NaN | \n", "NaN | \n", "
| 1907-05-01 | \n", "51.8 | \n", "NaN | \n", "NaN | \n", "-3.2 | \n", "-15.2 | \n", "NaN | \n", "NaN | \n", "NaN | \n", "NaN | \n", "
| ... | \n", "... | \n", "... | \n", "... | \n", "... | \n", "... | \n", "... | \n", "... | \n", "... | \n", "... | \n", "
| 1972-08-01 | \n", "75.6 | \n", "-4.9 | \n", "-4.9 | \n", "-3.4 | \n", "-10.3 | \n", "-2.3 | \n", "1.7 | \n", "1.1 | \n", "8.4 | \n", "
| 1972-09-01 | \n", "64.1 | \n", "-1.7 | \n", "-2.5 | \n", "-11.5 | \n", "-8.1 | \n", "3.2 | \n", "5.5 | \n", "2.4 | \n", "1.3 | \n", "
| 1972-10-01 | \n", "51.7 | \n", "0.6 | \n", "2.7 | \n", "-12.4 | \n", "-0.9 | \n", "2.3 | \n", "-0.9 | \n", "5.2 | \n", "2.8 | \n", "
| 1972-11-01 | \n", "40.3 | \n", "-1.5 | \n", "3.4 | \n", "-11.4 | \n", "1.0 | \n", "-2.1 | \n", "-4.4 | \n", "0.7 | \n", "-4.5 | \n", "
| 1972-12-01 | \n", "30.3 | \n", "-0.3 | \n", "3.2 | \n", "-10.0 | \n", "1.4 | \n", "1.2 | \n", "3.3 | \n", "-0.2 | \n", "-0.9 | \n", "
792 rows × 9 columns
\n", "| \n", " | Test Statistic | \n", "p-value | \n", "#Lags Used | \n", "No. of Obs. Used | \n", "Critical Value (1%) | \n", "Critical Value (5%) | \n", "Critical Value (10%) | \n", "
|---|---|---|---|---|---|---|---|
| d0_D0_s12 | \n", "-6.481466 | \n", "1.291867e-08 | \n", "21 | \n", "770 | \n", "-3.438871 | \n", "-2.865301 | \n", "-2.568773 | \n", "
| d0_D1_s12 | \n", "-12.658082 | \n", "1.323220e-23 | \n", "12 | \n", "767 | \n", "-3.438905 | \n", "-2.865316 | \n", "-2.568781 | \n", "
| d0_D2_s12 | \n", "-10.416254 | \n", "1.751310e-18 | \n", "14 | \n", "753 | \n", "-3.439064 | \n", "-2.865386 | \n", "-2.568818 | \n", "
| d1_D0_s12 | \n", "-12.302613 | \n", "7.391771e-23 | \n", "21 | \n", "769 | \n", "-3.438882 | \n", "-2.865306 | \n", "-2.568775 | \n", "
| d2_D0_s12 | \n", "-15.935084 | \n", "7.651998e-29 | \n", "17 | \n", "772 | \n", "-3.438849 | \n", "-2.865291 | \n", "-2.568767 | \n", "
| d1_D1_s12 | \n", "-11.846173 | \n", "7.390517e-22 | \n", "20 | \n", "758 | \n", "-3.439006 | \n", "-2.865361 | \n", "-2.568804 | \n", "
| d2_D1_s12 | \n", "-18.352698 | \n", "2.234953e-30 | \n", "21 | \n", "756 | \n", "-3.439029 | \n", "-2.865371 | \n", "-2.568810 | \n", "
| d1_D2_s12 | \n", "-12.221559 | \n", "1.104309e-22 | \n", "20 | \n", "746 | \n", "-3.439146 | \n", "-2.865422 | \n", "-2.568837 | \n", "
| d2_D2_s12 | \n", "-15.080476 | \n", "8.468854e-28 | \n", "20 | \n", "745 | \n", "-3.439158 | \n", "-2.865427 | \n", "-2.568840 | \n", "
| \n", " | order | \n", "seasonal_order | \n", "MAPE | \n", "MSE | \n", "AIC | \n", "BIC | \n", "
|---|---|---|---|---|---|---|
| 140 | \n", "(3, 1, 2) | \n", "(1, 1, 1, 12) | \n", "0.069031 | \n", "16.776397 | \n", "3860.352438 | \n", "3896.749644 | \n", "
| 45 | \n", "(1, 1, 2) | \n", "(1, 1, 1, 12) | \n", "0.069051 | \n", "16.774613 | \n", "3855.644633 | \n", "3882.942537 | \n", "
| 129 | \n", "(3, 1, 1) | \n", "(1, 1, 1, 12) | \n", "0.069079 | \n", "16.741025 | \n", "3856.094704 | \n", "3887.942260 | \n", "
| 135 | \n", "(3, 1, 2) | \n", "(0, 1, 1, 12) | \n", "0.069124 | \n", "16.792337 | \n", "3859.551130 | \n", "3891.398685 | \n", "
| 81 | \n", "(2, 1, 1) | \n", "(1, 1, 1, 12) | \n", "0.069124 | \n", "16.802775 | \n", "3857.201338 | \n", "3884.499242 | \n", "
| 39 | \n", "(1, 1, 2) | \n", "(0, 1, 1, 12) | \n", "0.069130 | \n", "16.781163 | \n", "3854.503774 | \n", "3877.252028 | \n", "
| 123 | \n", "(3, 1, 1) | \n", "(0, 1, 1, 12) | \n", "0.069167 | \n", "16.746320 | \n", "3854.845307 | \n", "3882.143212 | \n", "
| 93 | \n", "(2, 1, 2) | \n", "(1, 1, 1, 12) | \n", "0.069172 | \n", "16.805176 | \n", "3860.077582 | \n", "3891.925137 | \n", "
| 79 | \n", "(2, 1, 1) | \n", "(1, 0, 1, 12) | \n", "0.069201 | \n", "17.185380 | \n", "3965.090220 | \n", "3992.490254 | \n", "
| 75 | \n", "(2, 1, 1) | \n", "(0, 1, 1, 12) | \n", "0.069203 | \n", "16.811203 | \n", "3856.176397 | \n", "3878.924651 | \n", "
| \n", " | order | \n", "seasonal_order | \n", "MAPE | \n", "MSE | \n", "AIC | \n", "BIC | \n", "
|---|---|---|---|---|---|---|
| 129 | \n", "(3, 1, 1) | \n", "(1, 1, 1, 12) | \n", "0.069079 | \n", "16.741025 | \n", "3856.094704 | \n", "3887.942260 | \n", "
| 123 | \n", "(3, 1, 1) | \n", "(0, 1, 1, 12) | \n", "0.069167 | \n", "16.746320 | \n", "3854.845307 | \n", "3882.143212 | \n", "
| 45 | \n", "(1, 1, 2) | \n", "(1, 1, 1, 12) | \n", "0.069051 | \n", "16.774613 | \n", "3855.644633 | \n", "3882.942537 | \n", "
| 140 | \n", "(3, 1, 2) | \n", "(1, 1, 1, 12) | \n", "0.069031 | \n", "16.776397 | \n", "3860.352438 | \n", "3896.749644 | \n", "
| 39 | \n", "(1, 1, 2) | \n", "(0, 1, 1, 12) | \n", "0.069130 | \n", "16.781163 | \n", "3854.503774 | \n", "3877.252028 | \n", "
| 135 | \n", "(3, 1, 2) | \n", "(0, 1, 1, 12) | \n", "0.069124 | \n", "16.792337 | \n", "3859.551130 | \n", "3891.398685 | \n", "
| 81 | \n", "(2, 1, 1) | \n", "(1, 1, 1, 12) | \n", "0.069124 | \n", "16.802775 | \n", "3857.201338 | \n", "3884.499242 | \n", "
| 93 | \n", "(2, 1, 2) | \n", "(1, 1, 1, 12) | \n", "0.069172 | \n", "16.805176 | \n", "3860.077582 | \n", "3891.925137 | \n", "
| 75 | \n", "(2, 1, 1) | \n", "(0, 1, 1, 12) | \n", "0.069203 | \n", "16.811203 | \n", "3856.176397 | \n", "3878.924651 | \n", "
| 87 | \n", "(2, 1, 2) | \n", "(0, 1, 1, 12) | \n", "0.069289 | \n", "16.825118 | \n", "3859.565045 | \n", "3886.862949 | \n", "
| \n", " | order | \n", "seasonal_order | \n", "MAPE | \n", "MSE | \n", "AIC | \n", "BIC | \n", "
|---|---|---|---|---|---|---|
| 3 | \n", "(1, 0, 1) | \n", "(0, 1, 1, 12) | \n", "0.080283 | \n", "61.491561 | \n", "3846.181890 | \n", "3864.386212 | \n", "
| 9 | \n", "(1, 0, 1) | \n", "(1, 1, 1, 12) | \n", "0.080153 | \n", "61.471329 | \n", "3846.879649 | \n", "3869.635051 | \n", "
| 15 | \n", "(1, 0, 2) | \n", "(0, 1, 1, 12) | \n", "0.080286 | \n", "61.482998 | \n", "3847.914835 | \n", "3870.670236 | \n", "
| 51 | \n", "(2, 0, 1) | \n", "(0, 1, 1, 12) | \n", "0.080283 | \n", "61.485310 | \n", "3847.985625 | \n", "3870.741026 | \n", "
| 99 | \n", "(3, 0, 1) | \n", "(0, 1, 1, 12) | \n", "0.080295 | \n", "61.449767 | \n", "3848.292676 | \n", "3875.599158 | \n", "
| 21 | \n", "(1, 0, 2) | \n", "(1, 1, 1, 12) | \n", "0.080146 | \n", "61.461761 | \n", "3848.568559 | \n", "3875.875041 | \n", "
| 57 | \n", "(2, 0, 1) | \n", "(1, 1, 1, 12) | \n", "0.080145 | \n", "61.464316 | \n", "3848.650344 | \n", "3875.956826 | \n", "
| 105 | \n", "(3, 0, 1) | \n", "(1, 1, 1, 12) | \n", "0.080096 | \n", "61.430629 | \n", "3849.069448 | \n", "3880.927010 | \n", "
| 117 | \n", "(3, 0, 2) | \n", "(1, 1, 1, 12) | \n", "0.080078 | \n", "61.396978 | \n", "3849.127920 | \n", "3885.536562 | \n", "
| 63 | \n", "(2, 0, 2) | \n", "(0, 1, 1, 12) | \n", "0.080282 | \n", "61.469826 | \n", "3849.361800 | \n", "3876.668282 | \n", "
| \n", " | order | \n", "seasonal_order | \n", "MAPE | \n", "MSE | \n", "AIC | \n", "BIC | \n", "
|---|---|---|---|---|---|---|
| 3 | \n", "(1, 0, 1) | \n", "(0, 1, 1, 12) | \n", "0.080283 | \n", "61.491561 | \n", "3846.181890 | \n", "3864.386212 | \n", "
| 9 | \n", "(1, 0, 1) | \n", "(1, 1, 1, 12) | \n", "0.080153 | \n", "61.471329 | \n", "3846.879649 | \n", "3869.635051 | \n", "
| 15 | \n", "(1, 0, 2) | \n", "(0, 1, 1, 12) | \n", "0.080286 | \n", "61.482998 | \n", "3847.914835 | \n", "3870.670236 | \n", "
| 51 | \n", "(2, 0, 1) | \n", "(0, 1, 1, 12) | \n", "0.080283 | \n", "61.485310 | \n", "3847.985625 | \n", "3870.741026 | \n", "
| 27 | \n", "(1, 1, 1) | \n", "(0, 1, 1, 12) | \n", "0.069317 | \n", "16.836785 | \n", "3855.816702 | \n", "3874.015305 | \n", "
| 99 | \n", "(3, 0, 1) | \n", "(0, 1, 1, 12) | \n", "0.080295 | \n", "61.449767 | \n", "3848.292676 | \n", "3875.599158 | \n", "
| 21 | \n", "(1, 0, 2) | \n", "(1, 1, 1, 12) | \n", "0.080146 | \n", "61.461761 | \n", "3848.568559 | \n", "3875.875041 | \n", "
| 57 | \n", "(2, 0, 1) | \n", "(1, 1, 1, 12) | \n", "0.080145 | \n", "61.464316 | \n", "3848.650344 | \n", "3875.956826 | \n", "
| 63 | \n", "(2, 0, 2) | \n", "(0, 1, 1, 12) | \n", "0.080282 | \n", "61.469826 | \n", "3849.361800 | \n", "3876.668282 | \n", "
| 39 | \n", "(1, 1, 2) | \n", "(0, 1, 1, 12) | \n", "0.069130 | \n", "16.781163 | \n", "3854.503774 | \n", "3877.252028 | \n", "
| \n", " | order | \n", "seasonal_order | \n", "MAPE | \n", "MSE | \n", "AIC | \n", "BIC | \n", "
|---|---|---|---|---|---|---|
| 140 | \n", "(3, 1, 2) | \n", "(1, 1, 1, 12) | \n", "0.069031 | \n", "16.776397 | \n", "3860.352438 | \n", "3896.749644 | \n", "
| 129 | \n", "(3, 1, 1) | \n", "(1, 1, 1, 12) | \n", "0.069079 | \n", "16.741025 | \n", "3856.094704 | \n", "3887.942260 | \n", "
| 3 | \n", "(1, 0, 1) | \n", "(0, 1, 1, 12) | \n", "0.080283 | \n", "61.491561 | \n", "3846.181890 | \n", "3864.386212 | \n", "
| 3 | \n", "(1, 0, 1) | \n", "(0, 1, 1, 12) | \n", "0.080283 | \n", "61.491561 | \n", "3846.181890 | \n", "3864.386212 | \n", "
| \n", " | co2 | \n", "
|---|---|
| 1965-01-01 | \n", "319.32 | \n", "
| 1965-02-01 | \n", "320.36 | \n", "
| 1965-03-01 | \n", "320.82 | \n", "
| 1965-04-01 | \n", "322.06 | \n", "
| 1965-05-01 | \n", "322.17 | \n", "
| ... | \n", "... | \n", "
| 1980-08-01 | \n", "337.19 | \n", "
| 1980-09-01 | \n", "335.49 | \n", "
| 1980-10-01 | \n", "336.63 | \n", "
| 1980-11-01 | \n", "337.74 | \n", "
| 1980-12-01 | \n", "338.36 | \n", "
192 rows × 1 columns
\n", "