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Time series are all around us, from stock prices and weather forecasts to economic trends and medical diagnoses. This course is designed to equip you to effectively model, interpret, and forecast time series.

In this course, you’ll learn time series analysis concepts, such as stochasticity, stationarity, and autocorrelation. You’ll analyze the time series by computing its various moments and by visualizing it using histogram and density plots. Next, you’ll decompose the time series into its trend, season, and cycle components. You’ll then learn about linear time series models, including autoregressive (AR) processes, moving average (MA) processes, ARMA, and ARIMA. You’ll fit these models on data and forecast the future and finish with evaluating the models using various goodness of fit criteria. 

By the end of this course, you’ll have a solid foundation in univariate time series analysis and the skills to explore, model, and forecast time series data using Python.

An Introduction to Time Series

$e_{T+h}$ is the estimated value of the random shock component, $\epsilon_{T+h}$, of an ARIMA model. 

$e_{T+h}$ is the difference between the actual value of the series, $y_{T+h}$, and its predicted value, $\hat y_{T+h}$.

$e_{T+h}$ is the output of the loss function that we use to get the best prediction for $y_{T+h}$.

$e_{T+h}$ is the difference between the predictions of two ARIMA models.

How would you define the forecast error of a model (also known as residual), $e_{T+h}$?

How would you define the forecast error of a model (also known as residual), <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>e</mi><mrow><mi>T</mi><mo>+</mo><mi>h</mi></mrow></msub></mrow><annotation encoding="application/x-tex">e_{T+h}</annotation></semantics></math>eT+h​?

**(True or False)** If a model is correctly specified, its residuals should be centered around 0.

(True or False) If a model is correctly specified, its residuals should be centered around 0.

The average of the squared residuals over the forecast horizon

The average of the square root of the residuals over the forecast horizon

The square root of the average of the squared residuals

The average of the square of the absolute value of the residuals

What does the mean squared error (MSE) loss function measure?

What does the mean squared error (MSE) loss function measure?

The unconditional expectation at step $T$, $E[y_{T+h}]$.

The conditional expectation at step $T$, $E_T[y_{T+h}]$.

The problem is intractable, so there is no optimal functional form for the forecast. 

What is the optimal forecast for $y_{T+h}$ under a quadratic loss?

What is the optimal forecast for <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>y</mi><mrow><mi>T</mi><mo>+</mo><mi>h</mi></mrow></msub></mrow><annotation encoding="application/x-tex">y_{T+h}</annotation></semantics></math>yT+h​ under a quadratic loss?

Assume that you’ve got an MA($3$) model. You are at time step $T$. For which forecast horizon, $h$, would you expect it to revert to its unconditional mean? 

Assume that you’ve got an MA(<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>3</mn></mrow><annotation encoding="application/x-tex">3</annotation></semantics></math>3) model. You are at time step <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>T. For which forecast horizon, <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math>h, would you expect it to revert to its unconditional mean?

Suppose you've got an AR($1$) process with $\phi = 0.5$ and $c = 1$. The last observation of the series is 1.5. What's the two-step-ahead conditional forecast?   

Suppose you’ve got an AR(<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math>1) process with <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϕ</mi><mo>=</mo><mn>0.5</mn></mrow><annotation encoding="application/x-tex">\phi = 0.5</annotation></semantics></math>ϕ=0.5 and <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>c</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">c = 1</annotation></semantics></math>c=1. The last observation of the series is 1.5. What’s the two-step-ahead conditional forecast?

$[\hat y_{T+h} - 1.96 \sigma, \hat y_{T+h} + 1.96 \sigma]$

where $\sigma$ is the standard deviation of the _random shock component_ of the model.

$[\hat y_{T+1} - 1.96, \hat y_{T+1} + 1.96]$

$[\hat y_{T+h} - 1.96 \sigma_e, \hat y_{T+h} + 1.96 \sigma_e]$

where $\sigma_e$ is the standard deviation of the _forecast error_.

$[\hat y_{T+h} - 0.025 \sigma, \hat y_{T+h} + 0.975 \sigma]$

where $\sigma$ is the standard deviation of the _random shock component_ of the model.

What is the 95% confidence interval of a one-step-ahead prediction $\hat y_{T+h}$? Assume the random shock component of the process is Gaussian white noise.

What is the 95% confidence interval of a one-step-ahead prediction <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mover accent="true"><mi>y</mi><mo>^</mo></mover><mrow><mi>T</mi><mo>+</mo><mi>h</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\hat y_{T+h}</annotation></semantics></math>y^​T+h​? Assume the random shock component of the process is Gaussian white noise.

```Python
res_frame = res.get_forecast().summary_frame()

lower_ci = res_frame['mean_ci_lower'].values
upper_ci = res_frame['mean_ci_upper'].values
```

```Python
res_frame = res.get_forecast(5).summary_frame()

lower_ci, upper_ci = res_frame.confidence_intervals()

```Python
res_frame = res.get_forecast(5).summary_frame()

lower_ci = res_frame['mean_ci_lower'].values
upper_ci = res_frame['mean_ci_upper'].values
```

```Python
res_frame = res.get_forecast(5).summary_frame()

lower_ci = -1.96 * res_frame['mean_se'].values + res_frame['mean'].values 
upper_ci = 1.96 * res_frame['mean_se'].values + res_frame['mean'].values
```

(**Select all that apply.**) You've got the following code to fit an ARIMA model on a series `y` using statsmodels:
 
```Python
import pandas as pd
from statsmodels.tsa.arima.model import ARIMA

y = pd.read_csv('my_series.csv')
mod = ARIMA(y.values, order=(1, 0, 1))
res = mod.fit()
```

How would you get the confidence intervals of a five-step-ahead forecast in two separate arrays?

(Select all that apply.) You’ve got the following code to fit an ARIMA model on a series <code>y</code> using statsmodels:
<pre><code class="language-Python">import pandas as pd
from statsmodels.tsa.arima.model import ARIMA

y = pd.read_csv('my_series.csv')
mod = ARIMA(y.values, order=(1, 0, 1))
res = mod.fit()
</code></pre>
How would you get the confidence intervals of a five-step-ahead forecast in two separate arrays?

Prediction with ARIMA models

Prediction with ARIMA models

Quiz yourself on prediction with ARIMA models.

Introduction to Time Series

The Basics of Time Series

Exploring Data

The Properties of Time Series

ARIMA Models

On Prediction

Choosing, Fitting, and Evaluating Models

Conclusion

Appendix

What have you learned?

Quiz