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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

$y_t$ is, by definition, not integrated. 

None of the components of $y_t$ is correlated with any shocks that happened more than one time step ago. 

$\gamma_j$ can, in fact, be non-zero for $j > 1$.

The unconditional mean and variance from the present time period are different from two time periods ago. 

An MA($1$) process, $y_t$, is the weighted sum of the present realization of some white noise process plus its realization in the previous time step. Why is the autocovariance of the MA($1$), $\gamma_j$, equal to 0 for $j > 1$?

An MA(<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, <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>y</mi><mi>t</mi></msub></mrow><annotation encoding="application/x-tex">y_t</annotation></semantics></math>yt​, is the weighted sum of the present realization of some white noise process plus its realization in the previous time step. Why is the autocovariance of the MA(<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), <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>γ</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex">\gamma_j</annotation></semantics></math>γj​, equal to 0 for <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>j</mi><mo>&gt;</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">j &gt; 1</annotation></semantics></math>j&gt;1?

Suppose you've got a time series that shows an exponentially vanishing autocorrelation function (ACF) and partial autocorrelation function (PACF) significant up to lag 3. What type of linear process could it be following?

Suppose you’ve got a time series that shows an exponentially vanishing autocorrelation function (ACF) and partial autocorrelation function (PACF) significant up to lag 3. What type of linear process could it be following?

In the long run, the values of the series would converge around 3, the unconditional mean of the process.

In the long run, the values of the series would converge around $\frac{3}{1+1.2} = 1.36$, the unconditional mean of the process.

The values of the series would explode (in absolute value). They would not converge to any point over time.

Since the series is a random walk, the values would not converge to any point, wandering randomly. 

Suppose you've got an AR($1$) process with the following specification:

$y_t = 3 -1.2 y_{t-1} + \epsilon_t$

If you were to plot it on a chart, what would you expect to see? Assume that the variance of the random shock is 4.

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 the following specification:
<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>y</mi><mi>t</mi></msub><mo>=</mo><mn>3</mn><mo>−</mo><mn>1.2</mn><msub><mi>y</mi><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>+</mo><msub><mi>ϵ</mi><mi>t</mi></msub></mrow><annotation encoding="application/x-tex">y_t = 3 -1.2 y_{t-1} + \epsilon_t</annotation></semantics></math>yt​=3−1.2yt−1​+ϵt​
If you were to plot it on a chart, what would you expect to see? Assume that the variance of the random shock is 4.

**(True or False)** All AR($1$) can be represented as an MA($\infty$).

(True or False) All 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) can be represented as an MA(<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∞</mi></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>∞).

$(1 - 0.1L)y_{t-2} = (1 + 0.4)\epsilon_t$

What is the lag polynomial representation (LPR) of the following process?

$y_t = 0.1y_{t-2} + \epsilon_t + 0.4 \epsilon_{t-1}$

What is the lag polynomial representation (LPR) of the following process?
<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>y</mi><mi>t</mi></msub><mo>=</mo><mn>0.1</mn><msub><mi>y</mi><mrow><mi>t</mi><mo>−</mo><mn>2</mn></mrow></msub><mo>+</mo><msub><mi>ϵ</mi><mi>t</mi></msub><mo>+</mo><mn>0.4</mn><msub><mi>ϵ</mi><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">y_t = 0.1y_{t-2} + \epsilon_t + 0.4 \epsilon_{t-1}</annotation></semantics></math>yt​=0.1yt−2​+ϵt​+0.4ϵt−1​

$y_t$ is integrated of order 1; $z_t$ isn't. 

$z_t$ is integrated of order 1; $y_t$ isn't.

There are no differences in terms of their distribution, as both follow an ARIMA($1$,$1$,$1$) process. Their AR and MA parameters might not be the same.

There are no differences in terms of their distribution, as both follow an ARMA($1$,$1$) process. Their AR and MA parameters might not be the same.

Suppose $y_t$ is a time series that follows an ARIMA($1$,$1$,$1$) process. $z_t$, another time series, follows an ARMA($1$,$1$) process after taking first differences. Both have a random shock component in the form of white noise with $\sigma^2 = 1$. What are the differences between $y_t$ and $z_t$?

Suppose <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>y</mi><mi>t</mi></msub></mrow><annotation encoding="application/x-tex">y_t</annotation></semantics></math>yt​ is a time series that follows an ARIMA(<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,<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,<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. <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>z</mi><mi>t</mi></msub></mrow><annotation encoding="application/x-tex">z_t</annotation></semantics></math>zt​, another time series, follows an ARMA(<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,<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 after taking first differences. Both have a random shock component in the form of white noise with <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>σ</mi><mn>2</mn></msup><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\sigma^2 = 1</annotation></semantics></math>σ2=1. What are the differences between <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>y</mi><mi>t</mi></msub></mrow><annotation encoding="application/x-tex">y_t</annotation></semantics></math>yt​ and <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>z</mi><mi>t</mi></msub></mrow><annotation encoding="application/x-tex">z_t</annotation></semantics></math>zt​?

ARIMA models only apply to stationary data. Seasonal data isn't stationary. 

Seasonal data is, by definition, integrated, and ARIMA models cannot deal with integrated data.

Seasonal data does not follow a linear model by definition, whereas ARIMA models can only deal with linear relations. 

ARIMA processes only apply to filtered data; that is, after the trend and seasonal parts have been separated.

Why are standard ARIMA models unable to deal with seasonal data? 

Why are standard ARIMA models unable to deal with seasonal data?

**(True or False)** The seasonal period is the number of time steps that constitute a season of the data. 

(True or False) The seasonal period is the number of time steps that constitute a season of the data.

$p$ (lower case) is the order of the non-seasonal AR component.

$P$ (upper case) is the order of the seasonal AR component. 

$d$ (lower case) is the order of the seasonal MA component.

$D$ (upper case) is the order of the non-seasonal MA component.

**(Select all that apply.)** Which option is correct about the SARIMA($p$,$d$,$q$)($P$,$D$,$Q$)$s$ process?

(Select all that apply.) Which option is correct about the SARIMA(<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>p,<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math>d,<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi></mrow><annotation encoding="application/x-tex">q</annotation></semantics></math>q)(<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math>P,<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>D,<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Q</mi></mrow><annotation encoding="application/x-tex">Q</annotation></semantics></math>Q)<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>s</mi></mrow><annotation encoding="application/x-tex">s</annotation></semantics></math>s process?

What type of ARMA process does this code snippet produce? 

```python
from statsmodels.tsa.arima_process import ArmaProcess

ar_coeff = [1, -0.5, 0.9]   
ma_coeff = [1, 0.3]   
n = 200 
stdev = 4

arma = ArmaProcess(ar=ar_coeff, ma=ma_coeff)
series = arma.generate_sample(nsample=n, scale = stdev)
```


What type of ARMA process does this code snippet produce?
<pre><code class="language-python">from statsmodels.tsa.arima_process import ArmaProcess

ar_coeff = [1, -0.5, 0.9] 
ma_coeff = [1, 0.3] 
n = 200 
stdev = 4

arma = ArmaProcess(ar=ar_coeff, ma=ma_coeff)
series = arma.generate_sample(nsample=n, scale = stdev)
</code></pre>

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