What are ARIMA Models

1. Formula
   Observation

   The slide shows the dependent variable as $Y_t$ in the AR(1) recursion and later in the general ARMA equation.

$Y_t$

Time-series observation at time $t$.

Observed series values used in autoregressive terms.

1. Formula
   Observation

   The slide shows $e_t$, $e_{t-1}$, $e_{t-2}$, and later $e_{t-q}$ in the ARMA equation.

$e_t$

Error term (innovation/shock) at time $t$.

Past errors used in moving-average terms.

1. Formula
   Observation

   The AR(1) recursion uses $\phi$ and $\phi^2$, and the general ARMA formula uses $\phi_1,\ldots,\phi_p$.

1. The video does not explicitly state stationarity restrictions on $\phi$.

2. The starred intercept $\omega^*$ is shown but not defined verbally in this clip.

$\phi$, $\phi_1$,$\ldots$,$\phi_p$

Autoregressive coefficients multiplying lagged observations.

Parameters of AR terms.

1. Formula
   Observation

   The ARMA slide displays $\theta_1 e_{t-1}+\cdots+\theta_q e_{t-q}$.

1. The video does not explicitly state invertibility restrictions on $\theta$.

2. No separate verbal definition of $\theta_i$ is given in this clip.

$\theta_1$,$\ldots$,$\theta_q$

Moving-average coefficients multiplying lagged errors.

Parameters of MA terms.

1. Audio
   Observation

   The speaker does not define $\omega$ or $\omega^*$ in words during this clip.

1. The exact relationship between $\omega$ and $\omega^*$ is not explained in the clip.

2. Whether $\omega^*$ equals $\omega/(1-\phi)$ is not stated here.

$\omega$,\ $\omega^*$

Constant/intercept term in the model equations.

Additive intercept in AR and ARMA representations.

1. Audio
   Observation

   The speaker says ARMA models combine AR and MA terms and typically have fewer terms.

1. The clip does not formally define $p$ and $q$ as integers beyond their appearance in the formula.

p,\ q

Order parameters: $p$ for autoregressive lags and $q$ for moving-average error lags.

Model orders in ARMA notation.

1. Formula
   Observation

   $Y_t = \omega + \phi_1 Y_{t-1} + \dots + \phi_p Y_{t-p} + \theta_1 e_{t-1} + \dots + \theta_q e_{t-q} + e_t$

$Y_t$

Time series variable at time t

Real numbers

1. Formula
   Observation

   $Y_t = \omega + \phi_1 Y_{t-1} + \dots + \phi_p Y_{t-p} + \theta_1 e_{t-1} + \dots + \theta_q e_{t-q} + e_t$

$\omega$

Constant term in the ARIMA model

Real numbers

1. Formula
   Observation

   $Y_t = \omega + \phi_1 Y_{t-1} + \dots + \phi_p Y_{t-p} + \theta_1 e_{t-1} + \dots + \theta_q e_{t-q} + e_t$

$\phi_i$

Coefficients of the autoregressive (AR) terms

Real numbers

1. Formula
   Observation

   $Y_t = \omega + \phi_1 Y_{t-1} + \dots + \phi_p Y_{t-p} + \theta_1 e_{t-1} + \dots + \theta_q e_{t-q} + e_t$

$\theta_j$

Coefficients of the moving average (MA) terms

Real numbers

1. Formula
   Observation

   $Y_t = \omega + \phi_1 Y_{t-1} + \dots + \phi_p Y_{t-p} + \theta_1 e_{t-1} + \dots + \theta_q e_{t-q} + e_t$

$e_t$

White noise error term at time t

Real numbers

1. Formula
   Observation

   $Y_t = \omega + \phi_1 Y_{t-1} + \dots + \phi_p Y_{t-p} + \theta_1 e_{t-1} + \dots + \theta_q e_{t-q} + e_t$

p

Order of the autoregressive (AR) part

Non-negative integers

1. Audio
   Observation

   The speaker says AR models forecast a series based solely on past values of the series, called lags.

2. Diagram
   Observation

   Slide text reads: 'AR – forecast a series based solely on the past values in the series – called lags.'

An autoregressive (AR) model forecasts a time series using only its own past observed values, referred to as lags.

1. Forecasting a time series

2. Uses past values of the same series

1. Audio
   Observation

   The speaker says MA models forecast a series based on past values of the errors in the series, called error lags.

2. Diagram
   Observation

   Slide text reads: 'MA – forecast a series based solely on the past errors in the series – called error lags.'

A moving-average (MA) model forecasts a time series using past error terms (shocks), called error lags.

1. Forecasting a time series

2. Uses past errors rather than past observations

1. Audio
   Observation

   The speaker says AR models are called long memory models because one can recursively look back over time and still see effects from lags.

2. Formula
   Observation

   The slide shows $Y_t = \omega + \phi Y_{t-1} + e_t$ and then $Y_t = \omega^* + \phi^2 Y_{t-2} + \phi e_{t-1} + e_t$.

1. The clip does not state the convergence condition needed for an infinite backward expansion.

AR models are described as long-memory models because the effect of earlier observations can be traced backward recursively through lagged dependence.

1. Applies to autoregressive time-series models

2. Assumes recursive substitution backward in time

1. Definition of AR models

1. Audio
   Observation

   The speaker says that instead of very large AR or MA models, one can combine them into ARMA models that typically have fewer terms but model just as well.

2. Formula
   Observation

   The slide title is 'Autoregressive Moving Average Models (ARMA)' and the text says 'Have both AR and MA terms together in the model.'

1. The claim that ARMA models 'typically have fewer terms but model just as well' is presented qualitatively, without proof in this clip.

An ARMA model combines autoregressive terms in past observations with moving-average terms in past errors within a single equation.

1. Time-series modeling

2. Includes both lagged observations and lagged errors

1. Definition of AR models
2. Definition of MA models

1. Diagram
   Observation

   The final slide lists 'Plotting patterns in correlation' and 'Automatic selection techniques' under 'Couple of different techniques to selecting models'.

2. Audio
   Observation

   The speaker begins to transition toward how to determine model order, but the sentence is cut off before explanation.

1. The clip ends before any method is explained in detail.

2. No specific correlation plots or selection criteria are shown yet.

The video introduces two ways to choose model order: plotting correlation patterns and using automatic selection techniques.

1. Used when deciding how many AR and MA terms to include

1. Definition of ARMA models

1. Formula
   Observation

   $Y_t = \omega + \phi_1 Y_{t-1} + \dots + \phi_p Y_{t-p} + \theta_1 e_{t-1} + \dots + \theta_q e_{t-q} + e_t$

The ARIMA model combines autoregressive (AR) and moving average (MA) terms to model time series data. The general form is given by the equation shown.

1. p and q are non-negative integers

2. $e_t$ is white noise

1. Audio
   Observation

   Three common techniques are called the MINIC, SCAN, and ESACF.

MINIC, SCAN, and ESACF are computational methods used to select appropriate AR and MA orders for an ARIMA model.

1. Used as starting points for model building

1. ARIMA Model Definition

1. Formula
   Observation

   Slide text states: 'Distribution depends only on difference in time, not location in time.'

2. Audio
   Observation

   Speaker explains that stationarity means the distribution of your data depends only on differences in time, not location in time.

In the context of time series, stationarity means that the statistical distribution of the data depends only on the difference in time between observations, rather than their absolute location in time.

1. Applies to time series data.

1. Formula
   Observation

   Slide text states: 'Essentially, you need to make your data stationary (typically done through differencing).'

2. Audio
   Observation

   Speaker says, 'Typically, we take differences in our data to make our data stationary.'

Differencing is a technique used to transform a non-stationary time series into a stationary one by computing the differences between consecutive observations.

1. Used when data exhibits non-stationarity.

1. Stationarity Definition

1. Audio
   Observation

   The speaker says that if time is not stopped, an AR model is an infinite moving average model.

2. Formula
   Observation

   The slide title changes to 'AR(1) = MA($\infty$)' and boxes the error-only part of the expanded expression.

1. The clip does not state the formal condition $|\phi|<1$ required for the infinite MA representation.

By recursively substituting lagged $Y$ terms backward, an AR(1) model can be rewritten as an infinite moving-average representation in past errors.

1. Start from an AR(1) equation of the form $Y_t = \omega + \phi Y_{t-1} + e_t$.

2. Continue the recursion far enough back in time.

3. The clip leaves unstated the usual convergence/stationarity condition needed for the infinite limit.

For the AR(1) model shown, repeated backward substitution yields an expression involving infinitely many lagged error terms if time is not stopped.

1. Audio
   Observation

   The speaker says AR and MA models are essentially opposites and that collecting short-memory MA effects makes the long-memory AR effect.

2. Diagram
   Observation

   The slide title reads 'Long Memory and Short Memory?' and the bottom text says 'Collection of a bunch of short memory effects!'

1. This is an intuitive characterization rather than a formal theorem statement in the clip.

The video presents AR and MA models as related opposites: many short-memory MA effects combine to produce the long-memory behavior associated with AR models.

1. Use the recursive expansion of an AR model into lagged error terms.

2. Interpret each lagged error contribution as a short-memory effect.

Qualitative claim about the relationship between AR and MA representations.

1. Audio
   Observation

   These techniques are just starting points though... They give you good models to start with that you can play around from there to potentially build better models.

Automatic selection techniques provide initial models that can be refined through trial and error.

1. Applicable to ARIMA model building

For all automatic selection techniques

1. Audio
   Observation

   The narrator explains looking back recursively until the first observation $Y_1$, then focusing on the right-hand side as a large moving-average term.

1. The algebraic derivation of $\omega^*$ from $\omega$ is skipped on screen.

2. The passage from finite $t$ to the infinite MA representation is asserted verbally rather than proved rigorously.

3. The clip does not mention the stationarity condition $|\phi|<1$.

1. Expression

   $$Y_t = \omega + \phi Y_{t-1} + e_t$$
   Explanation

   Begin with the AR(1) model shown on the slide.

   Justification

   Given model equation.

   Shown in the video
2. Expression

   $$Y_t = \omega^* + \phi^2 Y_{t-2} + \phi e_{t-1} + e_t$$
   Explanation

   Substitute the lagged dependence backward one more step so that $Y_t$ is expressed using $Y_{t-2}$ and earlier errors.

   Justification

   Recursive substitution of the AR relation into itself.

   Shown in the video
3. Expression

   $$\vdots$$
   Explanation

   Continue the same backward substitution repeatedly.

   Justification

   Iterating the recursion further back in time.

   Shown in the video
4. Expression

   $$Y_t = \omega^* + \phi^t Y_1 + \phi^{t-1} e_2 + \phi^{t-2} e_3 + \cdots + e_t$$
   Explanation

   After expanding all the way back to the first observation, the expression contains a term in $Y_1$ plus a weighted sum of past errors.

   Justification

   Finite backward expansion to the initial observation.

   Shown in the video
5. Expression

   $$\phi^{t-1} e_2 + \phi^{t-2} e_3 + \cdots + e_t$$
   Explanation

   The boxed portion isolates the accumulated error terms, which the speaker identifies as a large moving-average component.

   Justification

   Visual emphasis on the error-only part of the expanded equation.

   Shown in the video
6. Expression

   $$AR(1) = MA(\infty)$$
   Explanation

   If the recursion is not stopped at a finite initial observation, the error expansion continues indefinitely, giving an infinite moving-average representation.

   Justification

   Verbal conclusion by the speaker plus slide title change.

   Shown in the video

The AR(1) model can be rewritten as an infinite moving-average model in past errors, motivating the slide statement AR(1)=MA($\infty$).

1. Audio
   Observation

   The MINIC... builds many combinations of models across a grid search of AR and MA terms. It then finds the model with the lowest Bayesian information criterion.

1. Expression

   $$$$
   Explanation

   Construct a grid of possible AR and MA orders.

   Justification

   Grid search approach

   Shown in the video
2. Expression

   $$$$
   Explanation

   Fit ARIMA models for each combination in the grid.

   Justification

   Model fitting process

   Shown in the video
3. Expression

   $$$$
   Explanation

   Select the model with the lowest Bayesian Information Criterion (BIC).

   Justification

   Optimization criterion

   Shown in the video

MINIC selects the best ARIMA model based on BIC.

1. Audio
   Observation

   The SCAN... Look at the correlation matrix of your data. Then we're going to compare that with its lags. Then we're going to look at the eigenvalues from this correlation matrix... We want to find the combination of AR and MA terms that produce canonical correlations of approximately zero.

1. Expression

   $$$$
   Explanation

   Compute the correlation matrix of the time series data.

   Justification

   Correlation analysis

   Shown in the video
2. Expression

   $$$$
   Explanation

   Compare the correlation matrix with its lagged versions.

   Justification

   Lag comparison

   Shown in the video
3. Expression

   $$$$
   Explanation

   Extract eigenvalues (squared canonical correlations) from the comparison.

   Justification

   Eigenvalue decomposition

   Shown in the video
4. Expression

   $$$$
   Explanation

   Identify AR and MA orders where canonical correlations approach zero.

   Justification

   Zero-correlation criterion

   Shown in the video

SCAN identifies suitable ARIMA models by minimizing canonical correlations.

1. Audio
   Observation

   The ESACF... We know that the AR and MA terms are related. Essentially, we're filtering out AR terms until all we have left is the MA piece. If we filter and there's still AR correlations, then we haven't filtered enough ARs out.

1. Expression

   $$$$
   Explanation

   Filter out AR terms sequentially.

   Justification

   Sequential filtering

   Shown in the video
2. Expression

   $$$$
   Explanation

   Check remaining autocorrelations after each filter step.

   Justification

   Residual correlation check

   Shown in the video
3. Expression

   $$$$
   Explanation

   Continue filtering until only MA components remain.

   Justification

   Termination condition

   Shown in the video

ESACF determines ARIMA orders by isolating MA components through iterative AR filtering.

1. Formula
   Observation

   Slide displays ARIMA(1,1,1) with equations $Y_t - Y_{t-1} = W_t$ and $W_t = \omega + \phi_1 W_{t-1} + \theta_1 e_{t-1} + e_t$.

2. Audio
   Observation

   Speaker explains that for an ARIMA(1,1,1), we first take a difference ($d=1$) to go from Y to W, then have a lag of $W (p=1)$, and a lag of error ($q=1$).

1. Expression

   $$Y_t - Y_{t-1} = W_t$$
   Explanation

   Apply first-order differencing ($d=1$) to the original time series $Y_t$ to obtain the stationary series $W_t$.

   Justification

   Definition of differencing for stationarity.

   Shown in the video
2. Expression

   $$W_t = \omega + \phi_1 W_{t-1} + \theta_1 e_{t-1} + e_t$$
   Explanation

   Model the differenced series $W_t$ using an ARMA(1,1) process, incorporating one AR term ($p=1$) and one MA term ($q=1$).

   Justification

   Definition of ARMA(p,q) model applied to the differenced series.

   Shown in the video

An ARIMA(1,1,1) model consists of taking a first difference of the original series and then fitting an ARMA(1,1) model to the resulting differenced series.

1. Audio
   Observation

   Here's one at an AR3 MA0 model. Here is another at an AR2 MA2 model. And here is yet another at an AR0 MA3 model.

Identify valid ARIMA models using SCAN method.

1. Correlation matrix and its lags

Find AR and MA orders with near-zero canonical correlations.

1. Expression

   $$$$
   Explanation

   Examine correlation matrices for patterns.

   Justification

   Visual inspection

   Shown in the video
2. Expression

   $$$$
   Explanation

   Identify rectangles indicating low canonical correlations.

   Justification

   Pattern recognition

   Shown in the video

Valid models include AR(3,0), AR(2,2), and AR(0,3).

Models show minimal canonical correlations.

1. Formula
   Observation

   Slide explicitly shows the title 'ARIMA(1,1,1)' and the corresponding equations.

2. Audio
   Observation

   Speaker says, 'So if we had an ARIMA 1 1 1, we have the following...'

Construct the mathematical representation of an ARIMA(1,1,1) model.

1. $p=1$ (one AR term)

2. $d=1$ (one first difference)

3. $q=1$ (one MA term)

Write the equations defining the ARIMA(1,1,1) process.

1. Expression

   $$Y_t - Y_{t-1} = W_t$$
   Explanation

   Take the first difference of the original series $Y_t$.

   Justification

   $d=1$ indicates one order of differencing.

   Shown in the video
2. Expression

   $$W_t = \omega + \phi_1 W_{t-1} + \theta_1 e_{t-1} + e_t$$
   Explanation

   Write the ARMA(1,1) equation for the differenced series $W_t$.

   Justification

   $p=1$ and $q=1$ indicate one autoregressive term and one moving average term.

   Shown in the video

The ARIMA(1,1,1) model is defined by the system: $Y_t - Y_{t-1} = W_t$ and $W_t = \omega + \phi_1 W_{t-1} + \theta_1 e_{t-1} + e_t$.

Matches the visual equations and audio explanation provided in the video.

1. Diagram
   Observation

   A slide titled 'Remember AR and MA Models?' shows a time-series chart with y-axis 'Sales ($ Millions)' and x-axis dates from $1/7/2005$ to $5/19/2005$.

2. Animation
   Observation

   Around 21s and again around 28s, orange circles highlight points on the chart while the speaker reviews AR and MA definitions.

1. The chart illustrates lag/error concepts visually, but the exact numerical values of the highlighted points are not discussed verbally.

1. Line chart of sales over time

2. Orange highlight circles

3. Bullet text defining AR and MA

1. Static chart remains on screen while orange circles appear around selected points.

2. Slide focus shifts from AR bullet to MA bullet.

1. The axes remain 'Sales ($ Millions)' versus dates.

2. The textual definitions of AR and MA stay visible.

The visual supports the verbal distinction between using past observations (lags) and past errors (error lags) in forecasting.

1. Diagram
   Observation

   Slides progress from 'Relationship Between AR and MA' to 'Keep Doing Math...' to 'AR(1) = MA($\infty$)' to 'Long Memory and Short Memory?' while retaining the same recursive equations.

2. Animation
   Observation

   An orange curved arrow links the first AR equation to the substituted second equation; later an orange box highlights the error-sum portion.

1. The intermediate algebra producing $\omega^*$ is not shown step by step.

1. AR(1) equation

2. Substituted equation with $Y_{t-2}$

3. Ellipsis line

4. Expanded equation ending in $Y_1$ and error terms

5. Orange arrow and orange box

1. Equations accumulate downward as the recursion is extended.

2. Title changes from relationship framing to 'Keep Doing Math...', then to 'AR(1) = MA($\infty$)', then to 'Long Memory and Short Memory?'.

3. The error-only tail is boxed for emphasis.

1. The core model remains an AR(1) equation being rewritten.

2. The left-hand side stays $Y_t$ throughout.

The animation visually demonstrates that repeated substitution turns lagged observations into a growing weighted sum of past errors.

1. Diagram
   Observation

   Three grids labeled MINIC, SCAN, ESACF showing AR vs MA orders.

1. MINIC grid

2. SCAN grid

3. ESACF grid

1. Highlighting of specific cells in grids

1. Grid structure remains constant

Each grid represents a different method for evaluating ARIMA model orders.

1. Diagram
   Observation

   Three 5x5 grids labeled MINIC, SCAN, and ESACF are shown, with axes labeled AR (0-4) and MA (0-4). Different cells are shaded to indicate possible model orders.

1. MINIC grid

2. SCAN grid

3. ESACF grid

4. AR axis

5. MA axis

6. Shaded cells

1. Slide transitions from selection grids to the stationarity slide.

1. Axes remain AR (0-4) and MA (0-4) across all three grids.

The shaded regions in each grid represent the combinations of AR and MA orders suggested by the respective automatic selection technique. The speaker notes looking for upper right triangles and their points to identify models like ARMA(1,1), ARMA(2,2), etc.

1. Diagram
   Observation

   Slide displays 'ARIMA(p, d, q)' with arrows pointing to definitions for p, d, and q.

1. ARIMA(p, d, q) text

2. Arrows

3. Labels for p, d, q

1. Slide transitions from stationarity to ARIMA notation.

Visually decomposes the ARIMA acronym, linking each parameter (p, d, q) to its specific meaning in the model structure.

1. Diagram
   Observation

   Slide displays 'ARIMA(1,1,1)' with two equations below it, connected by arrows indicating the flow from $Y_t$ to $W_t$ and then the ARMA equation.

1. ARIMA(1,1,1) text

2. Differencing equation

3. ARMA equation

4. Connecting arrows

1. Slide transitions from general notation to specific example.

Illustrates the step-by-step construction of an ARIMA(1,1,1) model, showing how differencing creates a new series which is then modeled using ARMA terms.

1. Audio
   Observation

   The speaker explicitly contrasts AR models, which use past values called lags, with MA models, which use past errors called error lags.

2. Diagram
   Observation

   The slide separates the two definitions in parallel bullets.

One might think AR and MA models differ only in name, or that both use the same kind of past information.

AR models use past observed values of the series; MA models use past error terms (shocks) of the series.

1. Audio
   Observation

   The speaker says AR models are long-memory models and then explains that they can be viewed as collections of short-memory MA effects.

2. Diagram
   Observation

   The slide title 'Long Memory and Short Memory?' and the boxed error sum support this interpretation.

1. The clip gives intuition rather than a formal memory-decay analysis.

One might think 'long memory' means the model only depends on one distant lag.

Here long memory means the recursive AR structure lets many past shocks accumulate, producing an effectively infinite MA representation.

1. Audio
   Observation

   These techniques are just starting points though... They give you good models to start with that you can play around from there to potentially build better models.

Automatic selection techniques provide final optimal models.

They offer initial models that require further refinement.

1. Audio
   Observation

   The speaker says ARMA models combine AR and MA terms together.

2. Formula
   Observation

   The ARMA equation contains both $\phi_i Y_{t-i}$ and $\theta_j e_{t-j}$ terms.

The ARMA model includes autoregressive terms as one component of its general form.

1. Audio
   Observation

   The speaker says ARMA combines AR and MA models into one framework.

2. Formula
   Observation

   The ARMA equation includes lagged error terms with coefficients $\theta_1,\ldots,\theta_q$.

The ARMA model includes moving-average terms as the other component of its general form.

1. Formula
   Observation

   The slide title becomes 'AR(1) = MA($\infty$)' after the recursive expansion.

2. Audio
   Observation

   The speaker states that an AR model is an infinite moving average model if time is not stopped.

1. The formal condition for equivalence is omitted in the clip.

The long-memory AR representation is shown to be equivalent, under the video's recursive argument, to an infinite MA representation.

1. Diagram
   Observation

   The slide 'Deciding How Many p's and q's...' appears directly after the ARMA definition slide.

2. Audio
   Observation

   The speaker transitions from defining ARMA to asking how to determine the model order.

1. The actual selection procedures are not yet explained in this clip.

Once ARMA models are defined, the next issue is choosing the orders $p$ and $q$.

1. Audio
   Observation

   Deciding how many p's and q's... Couple of different techniques to selecting models...

Automatic selection techniques apply to determining parameters in ARIMA models.

1. Audio
   Observation

   Speaker connects making data stationary with taking differences.

2. Formula
   Observation

   Slide text links 'make your data stationary' with '(typically done through differencing)'.

Differencing is a method applied to achieve stationarity in time series data.

1. Formula
   Observation

   ARIMA(p,d,q) notation includes 'd' for first differences, and the ARIMA(1,1,1) example shows the differencing equation.

The derivation and understanding of the ARIMA model rely on the concept of differencing to handle non-stationary data.

1. Formula
   Observation

   The ARMA equation is displayed on screen.

2. Audio
   Observation

   The speaker defines ARMA as combining AR and MA terms.

1. Definition of ARMA models

1. Audio
   Observation

   The speaker calls AR models long memory models and explains recursive backward dependence.

2. Formula
   Observation

   The recursive expansion slides show $Y_t$ rewritten in terms of earlier $Y$ and $e$ values.

1. Long-memory interpretation of AR models
2. AR(1) can be represented as MA($\infty$)

1. Diagram
   Observation

   The slide separately defines AR and MA using lags versus error lags.

2. Audio
   Observation

   The speaker contrasts past values with past errors.

1. Definition of AR models
2. Definition of MA models

1. Diagram
   Observation

   The final slide lists two techniques for selecting models.

2. Audio
   Observation

   The speaker begins transitioning to how to determine model order before the clip ends.

1. Only the names of the techniques are present; no detailed method is shown.

1. Introduction to ARMA model selection techniques

1. Audio
   Observation

   How do I decide the number of AR and MA terms in an ARIMA model?

1. ARIMA Model Definition
2. Automatic Selection Techniques

1. Audio
   Observation

   Speaker defines stationarity.

2. Formula
   Observation

   Slide text defines stationarity.

1. Stationarity Definition

1. Audio
   Observation

   Speaker explains differencing to make data stationary.

2. Formula
   Observation

   Slide text mentions differencing for stationarity.

1. Differencing for Stationarity

1. Audio
   Observation

   Speaker defines p, d, and q.

2. Formula
   Observation

   Slide visually breaks down ARIMA(p,d,q).

1. p
2. d
3. q

1. Audio
   Observation

   Speaker walks through the ARIMA(1,1,1) example.

2. Formula
   Observation

   Slide shows the equations for ARIMA(1,1,1).

1. ARIMA(1,1,1) Example
2. Derivation of ARIMA(1,1,1) Model