Explaining the ARIMA model

1. Formula
   Observation

   The white-noise equation is written as $y_t = C + ε_t$, and $y_t$ is labeled “obs value.”

$y_t$

Observed value at time t.

Time-series observation index t.

1. Formula
   Observation

   The constant term in the white-noise equation is written as C and labeled “constant.”

2. Audio
   Observation

   The speaker says this constant is fit to the data and, in the white noise process, ends up being the mean value.

C

Constant fitted to the data; in the white-noise process it corresponds to the mean value.

1. Formula
   Observation

   The error term in the white-noise equation is written as $ε_t$ and labeled “error.”

2. Formula
   Observation

   An arrow points from $ε_t$ to the expression $ε_t = f$(mean, sd).

$ε_t$

Error term at time t, describing variation around the constant.

1. Formula
   Observation

   The board shows $ε_t = f$(mean, sd).

2. Audio
   Observation

   The speaker says the error at time t is simply a function of the mean and the standard deviation.

f(mean, sd)

Function indicating that the error term depends on the mean and standard deviation.

1. Formula
   Observation

   The speaker adds + $β y_{t-1}$ to the white-noise equation.

2. Audio
   Observation

   The speaker calls β a coefficient added to incorporate autoregression.

β

Coefficient multiplying the one-time-step lagged observation in the AR(1) modification.

1. Formula
   Observation

   The added autoregressive term is written as $β y_{t-1}$.

2. Audio
   Observation

   The speaker describes $y_{t-1}$ as the value or observation at t minus one, one time step back.

$y_{t-1}$

Lagged observed value at time t-1, used as a predictor of $y_t$.

1. Formula
   Observation

   The first lag coefficient is later annotated with subscript 1, giving $β_1 y_{t-1}$.

2. Audio
   Observation

   The speaker says there are two coefficients and they are not identical coefficients.

$β_1$

Separate coefficient for the lag of one in the AR(2) model.

1. Formula
   Observation

   The second lag term is written as $β_2 y_{t-2}$.

2. Audio
   Observation

   The speaker says there is a separate coefficient fitted for the lag of two.

$β_2$

Separate coefficient for the lag of two in the AR(2) model.

1. Formula
   Observation

   The second autoregressive term is written as $β_2 y_{t-2}$.

2. Audio
   Observation

   The speaker describes adding a coefficient times y at t minus two.

$y_{t-2}$

Lagged observed value at time t-2, used as an additional predictor in the AR(2) model.

1. Formula
   Observation

   The board writes “AR1 order 1” beneath “Autoregressive model.”

2. Audio
   Observation

   The speaker says the model is sometimes referred to as an AR1 model or an order of one.

AR1

Autoregressive model of order 1, using one lag.

1. Audio
   Observation

   The speaker says turning the model into an AR2 means an autoregressive model with an order of two, two time lags.

2. Formula
   Observation

   The equation is extended by a second lagged term $β_2 y_{t-2}$.

AR2

Autoregressive model of order 2, using two lags.

1. Formula
   Observation

   The moving-average modification is written as + $θ ε_{t-1}$.

2. Audio
   Observation

   The speaker says another coefficient is added times the error at t minus one.

θ

Coefficient multiplying the previous error term in the moving-average model.

1. Formula
   Observation

   The board title reads “White noise,” with the equation $y_t = C + ε_t$.

2. Formula
   Observation

   Labels identify $y_t$ as “obs value,” C as “constant,” and $ε_t$ as “error.”

3. Formula
   Observation

   An arrow connects $ε_t$ to $ε_t = f$(mean, sd).

4. Audio
   Observation

   The speaker explains that the observed value equals a constant plus an error, and that the constant is fit to the data and becomes the mean value in the white noise process.

The white noise model is presented as the simplest baseline time-series model: an observation at time t equals a fitted constant plus an error term. The constant represents the mean level of the series, while the error captures variation around that constant. The error is described as depending on the mean and standard deviation.

1. Used as the starting point before adding autoregressive or moving-average structure.

2. C is the fitted constant or mean value.

3. $ε_t$ is the error term at time t.

1. Audio
   Observation

   The speaker says they will make one simple modification to turn the white-noise equation into something that incorporates autoregression.

2. Formula
   Observation

   The equation is modified by adding + $β y_{t-1}$.

3. Audio
   Observation

   The speaker says the value at the previous time step, $y_{t-1}$, is one of the predictors of the value of y at time t.

4. Formula
   Observation

   The board writes “Autoregressive model.”

An autoregressive model is formed by adding a lagged observed value to the white-noise equation. In the first-order case, the previous observation $y_{t-1}$ is multiplied by a coefficient β and added as a predictor of the current observation $y_t$.

1. Starts from the white-noise model.

2. Adds one lagged observation $y_{t-1}$.

3. β is the coefficient for the one-step lag.

1. White noise model

1. Formula
   Observation

   The board writes “AR1 order 1.”

2. Audio
   Observation

   The speaker says AR1 refers to an order of one.

3. Audio
   Observation

   The speaker explains that the one refers to the number of lags incorporated into the model, here t minus one.

The order of an autoregressive model is the number of lagged observations included in the model. An AR1 model includes one lag, $y_{t-1}$; an AR2 model includes two lags, $y_{t-1}$ and $y_{t-2}$.

1. The order counts lags of the observed variable y.

2. AR1 uses one lag.

3. AR2 uses two lags.

1. Autoregressive model from white noise

1. Audio
   Observation

   The speaker says that to turn the model into an AR2, or an autoregressive model with an order of two, they add a coefficient times y at t minus two.

2. Formula
   Observation

   The equation is extended to include + $β_2 y_{t-2}$.

3. Audio
   Observation

   The speaker notes that there are now two coefficients and they are not identical coefficients.

4. Formula
   Observation

   The first coefficient is annotated as $β_1$ and the second as $β_2$.

5. Audio
   Observation

   The speaker says the model is no longer a white noise process but an autoregressive model.

The AR2 model extends the AR1 model by adding a second lagged observation $y_{t-2}$. Each lag receives its own coefficient: $β_1$ for $y_{t-1}$ and $β_2$ for $y_{t-2}$. Adding more orders adds more lag terms.

1. Requires two lagged observations.

2. $β_1$ and $β_2$ are separate coefficients.

3. The model is no longer treated as a white noise process once autoregressive lags are added.

1. Autoregressive model from white noise
2. Order of an autoregressive model

1. Audio
   Observation

   The speaker says they will turn a white noise model into a moving average model.

2. Formula
   Observation

   A new equation begins as $y_t = C + ε_t$.

3. Audio
   Observation

   The speaker says they will add the influence of the error from previous time steps.

4. Formula
   Observation

   The equation is extended to $y_t = C + ε_t + θ ε_{t-1}$.

5. Formula
   Observation

   The board writes “Moving Average Model.”

A moving average model starts again from the white-noise core equation and adds the influence of previous error terms. In the first-order case shown, the previous error $ε_{t-1}$ is multiplied by a coefficient θ and added to the current equation.

1. Starts from the white-noise model.

2. Adds a lagged error term rather than a lagged observation.

3. θ is the coefficient for the previous error $ε_{t-1}$.

1. White noise model

1. Audio
   Observation

   The speaker explains adding error information from previous time steps to create an MA-2 model.

2. Formula
   Observation

   The equation $y_t = c + \varepsilon_t + \Theta_1 \varepsilon_{t-1} + \Theta_2 \varepsilon_{t-2}$ is written on the board.

A Moving Average (MA) model predicts the current value based on a constant, the current error, and past errors. The order of the model corresponds to the number of past error terms included.

1. Requires defining the order of the model.

2. Uses separate coefficients for each lagged error term.

1. Audio
   Observation

   The speaker states that autoregressive and moving average are not mutually exclusive and can be combined into a single model.

2. Formula
   Observation

   The equation $y_t = c + \varepsilon_t + \Theta_1 y_{t-1} + \beta_1 y_{t-1}$ is written under the ARMA heading.

1. The written formula uses $y_{t-1}$ for both the moving average and autoregressive components, which contradicts the standard definition where the MA component uses lagged errors $\varepsilon_{t-1}$.

An ARMA model combines both autoregressive and moving average components into a single equation to fit time series data that exhibits both types of correlation structures.

1. Combines elements of AR and MA models.

1. Moving Average Model

1. Audio
   Observation

   The speaker introduces the ARIMA model, explaining that the integrated component allows fitting a trend by modifying the left-hand side of the equation.

2. Formula
   Observation

   The equation $y_t - y_{t-1} = c + \varepsilon_t + \Theta_1 y_{t-1} + \beta_1 y_{t-1}$ is written under the ARIMA heading.

1. The written formula again uses $y_{t-1}$ for the moving average component instead of $\varepsilon_{t-1}$.

An ARIMA model extends the ARMA model by including an integrated component, which involves taking differences of the data to remove trends before applying the autoregressive and moving average components.

1. Used when the time series data contains a trend.

2. Modifies the left-hand side of the equation by taking differences.

1. Autoregressive Moving Average Model

1. Formula
   Observation

   Whiteboard shows the ARIMA equation: $y_t - y_{t-1} = c + \varepsilon_t + \Theta_1 y_{t-1} + \Phi_1 y_{t-1}$

2. Audio
   Observation

   Speaker explains that the 'I' stands for integrated component and involves differencing.

1. The handwritten equation on the board contains a likely error, using $y_{t-1}$ instead of $\varepsilon_{t-1}$ for the moving average term.

The ARIMA (AutoRegressive Integrated Moving Average) model is an extension of the ARMA model that includes differencing to account for trends in time series data. The 'I' represents the integrated component, which changes the values being fit by taking differences of the series.

1. Used for non-stationary time series data with trends.

1. ARMA Model Definition

1. Formula
   Observation

   Whiteboard shows the ARMA equation: $y_t = c + \varepsilon_t + \Theta_1 \varepsilon_{t-1} + \Phi_1 y_{t-1} + \Phi_2 y_{t-2}$

The ARMA (AutoRegressive Moving Average) model combines autoregressive and moving average components to fit time series data.

1. Used for stationary time series data.

1. Audio
   Observation

   Speaker states that the order for the 'I' or differencing can be greater than one, meaning differences go further back in time on the left-hand side of the equation.

Differencing is applied to remove trends from time series data. The order of differencing, denoted by d, indicates how many times the difference operation is applied. A higher order means taking differences of previous differences.

1. d can be greater than 1.

1. ARIMA Model Definition

1. Formula
   Observation

   The board displays $ε_t = f$(mean, sd).

2. Audio
   Observation

   The speaker says the error at time t is simply a function of the mean and the standard deviation.

In the white-noise model shown, the error term at time t is described as a function of the mean and standard deviation.

1. The model being discussed is the white-noise model $y_t = C + ε_t$.

For the error term $ε_t$ in the displayed white-noise model.

1. Audio
   Observation

   The speaker says the value at the previous time step, $y_{t-1}$, is one of the predictors of the value of y at time t.

2. Formula
   Observation

   The equation contains the added term $β y_{t-1}$.

In the AR1 model, the previous observation $y_{t-1}$ is used as a predictor of the current observation $y_t$.

1. The model has been modified from white noise by adding $β y_{t-1}$.

For the one-lag autoregressive model shown.

1. Audio
   Observation

   The speaker says the one refers to the number of lags incorporated into the model.

2. Formula
   Observation

   The board writes “AR1 order 1.”

The order number in AR1 or AR2 refers to the number of time lags incorporated into the autoregressive model.

1. The model is an autoregressive model built from lagged y values.

For the autoregressive orders discussed in the clip.

1. Audio
   Observation

   The speaker says, “And now, this is no longer a white noise process, but an autoregressive model.”

2. Formula
   Observation

   The equation has been extended to include $β_1 y_{t-1}$ and $β_2 y_{t-2}$.

Once lagged observations are added to the white-noise equation, the resulting process is described as an autoregressive model rather than a white noise process.

1. The equation includes lagged y terms such as $y_{t-1}$ and $y_{t-2}$.

For the modified model shown after adding autoregressive terms.

1. Audio
   Observation

   The speaker says they will add the influence of the error from previous time steps to create a moving average model.

2. Formula
   Observation

   The equation is written as $y_t = C + ε_t + θ ε_{t-1}$.

A moving average model is created by adding the influence of error terms from previous time steps to the white-noise core equation.

1. The starting equation is $y_t = C + ε_t$.

2. A lagged error term $ε_{t-1}$ is added with coefficient θ.

For the first-order moving-average example shown.

1. Audio
   Observation

   The speaker says they will take the white-noise equation and make one simple modification to incorporate autoregression.

2. Formula
   Observation

   The board adds + $β y_{t-1}$ to $y_t = C + ε_t$.

3. Formula
   Observation

   The board labels the result “Autoregressive model.”

1. Expression

   $$y_t = C + \varepsilon_t$$
   Explanation

   Begin with the white-noise model shown on the board.

   Justification

   This is the starting equation displayed under the title “White noise.”

   Shown in the video
2. Expression

   $$y_t = C + \varepsilon_t + \beta y_{t-1}$$
   Explanation

   Add a coefficient β multiplied by the previous observation $y_{t-1}$.

   Justification

   The speaker states that this simple modification incorporates autoregression.

   Shown in the video
3. Expression

   $$\text{Autoregressive model}$$
   Explanation

   Name the modified equation an autoregressive model.

   Justification

   The speaker says the modification creates an autoregressive model where $y_{t-1}$ predicts $y_t$.

   Shown in the video

The white-noise model becomes an AR1 autoregressive model by adding the lagged observation term $β y_{t-1}$.

1. Audio
   Observation

   The speaker says that to turn the model into an AR2, they add a coefficient times y at t minus two.

2. Formula
   Observation

   The equation is extended with + $β_2 y_{t-2}$.

3. Formula
   Observation

   The first coefficient is annotated as $β_1$ and the second as $β_2$.

4. Audio
   Observation

   The speaker says the two coefficients are not identical.

1. Expression

   $$y_t = C + \varepsilon_t + \beta y_{t-1}$$
   Explanation

   Start from the AR1 equation already written on the board.

   Justification

   This is the model after the first autoregressive modification.

   Shown in the video
2. Expression

   $$y_t = C + \varepsilon_t + \beta y_{t-1} + \beta_2 y_{t-2}$$
   Explanation

   Add a second lagged observation term involving $y_{t-2}$.

   Justification

   The speaker says an AR2 model has two time lags.

   Shown in the video
3. Expression

   $$y_t = C + \varepsilon_t + \beta_1 y_{t-1} + \beta_2 y_{t-2}$$
   Explanation

   Distinguish the two lag coefficients as $β_1$ and $β_2$.

   Justification

   The speaker explicitly says the coefficients are not identical and each lag gets a separate coefficient.

   Shown in the video

The AR2 model contains two lagged observations, $y_{t-1}$ and $y_{t-2}$, with separate coefficients $β_1$ and $β_2$.

1. Audio
   Observation

   The speaker says they will turn a white noise model into a moving average model.

2. Formula
   Observation

   A new equation is written beginning with $y_t = C + ε_t$.

3. Formula
   Observation

   The term + $θ ε_{t-1}$ is added.

4. Formula
   Observation

   The board labels the result “Moving Average Model.”

1. Expression

   $$y_t = C + \varepsilon_t$$
   Explanation

   Restart from the core white-noise equation.

   Justification

   The speaker says they will start with the core equation that came through white noise.

   Shown in the video
2. Expression

   $$y_t = C + \varepsilon_t + \theta \varepsilon_{t-1}$$
   Explanation

   Add a coefficient θ multiplied by the previous error $ε_{t-1}$.

   Justification

   The speaker says the influence of the error from previous time steps is added to create a moving average model.

   Shown in the video
3. Expression

   $$\text{Moving Average Model}$$
   Explanation

   Label the resulting equation as a moving average model.

   Justification

   The board writes “Moving Average Model” beneath the equation.

   Shown in the video

The moving average model shown is obtained by adding a lagged error term $θ ε_{t-1}$ to the white-noise equation.

1. Audio
   Observation

   The speaker explains combining the elements of the separate equations into the same equation.

2. Formula
   Observation

   The speaker writes the ARMA equation by adding terms from the AR and MA equations.

1. The derivation step where the speaker writes $\Theta_1 y_{t-1}$ instead of $\Theta_1 \varepsilon_{t-1}$ is mathematically inconsistent with the definition of an MA component.

1. Expression

   $$y_t = c + \varepsilon_t + \Theta_1 \varepsilon_{t-1} + \Theta_2 \varepsilon_{t-2}$$
   Explanation

   Start with the general form of a Moving Average model.

   Justification

   Definition of MA model.

   Shown in the video
2. Expression

   $$y_t = c + \varepsilon_t + \beta_1 y_{t-1} + \beta_2 y_{t-2}$$
   Explanation

   Start with the general form of an Autoregressive model.

   Justification

   Definition of AR model.

   Shown in the video
3. Expression

   $$y_t = c + \varepsilon_t + \Theta_1 y_{t-1} + \beta_1 y_{t-1}$$
   Explanation

   Combine the constant, current error, and lagged terms into a single equation to form the ARMA model.

   Justification

   The speaker states that AR and MA components can be combined into a single model.

   Shown in the video

The ARMA model is formed by combining the autoregressive and moving average components into a single equation.

1. Audio
   Observation

   The speaker explains that the integrated component modifies the left-hand side of the equation to flatten out the data.

2. Formula
   Observation

   The speaker writes $y_t - y_{t-1}$ on the left side of the ARIMA equation.

1. Expression

   $$y_t = c + \varepsilon_t + \Theta_1 y_{t-1} + \beta_1 y_{t-1}$$
   Explanation

   Start with the ARMA model equation.

   Justification

   Definition of ARMA model.

   Shown in the video
2. Expression

   $$y_t - y_{t-1} = c + \varepsilon_t + \Theta_1 y_{t-1} + \beta_1 y_{t-1}$$
   Explanation

   Modify the left-hand side by taking the first difference of the time series to account for a trend.

   Justification

   The speaker explains that the integrated component allows fitting a trend by taking differences.

   Shown in the video

The ARIMA model is formed by taking the difference of the time series on the left-hand side of the ARMA equation to handle trends.

1. Diagram
   Observation

   The board shows the heading “White noise,” the equation $y_t = C + ε_t$, labels “obs value,” “constant,” and “error,” and an arrow from $ε_t$ to $ε_t = f$(mean, sd).

1. Heading “White noise”

2. Equation $y_t = C + ε_t$

3. Label “obs value” under $y_t$

4. Label “constant” under C

5. Label “error” under $ε_t$

6. Arrow from $ε_t$ to $ε_t = f$(mean, sd)

1. No equation terms change during this interval.

2. The speaker gestures while explaining the existing labels.

1. The white-noise equation remains $y_t = C + ε_t$.

2. The error-dependence expression remains $ε_t = f$(mean, sd).

The visual layout establishes the baseline model before any autoregressive or moving-average modifications are made.

1. Animation
   Observation

   The presenter writes + $β y_{t-1}$ at the end of the white-noise equation.

2. Animation
   Observation

   The presenter writes “Autoregressive model” below the modified equation.

1. Original equation $y_t = C + ε_t$

2. Added term + $β y_{t-1}$

3. Handwritten label “Autoregressive model”

1. A new lagged observation term is appended to the equation.

2. A new model name is written below the equation.

1. The original terms C and $ε_t$ remain in place.

2. The error-dependence expression $ε_t = f$(mean, sd) remains visible above.

The animation shows the conceptual transition from a white-noise model to a first-order autoregressive model by adding one lagged y term.

1. Animation
   Observation

   The presenter writes “AR1” and then “order 1” beneath “Autoregressive model.”

1. Text “AR1”

2. Text “order 1”

3. Term $β y_{t-1}$ in the equation

1. The model is given the shorthand name AR1.

2. The order label “order 1” is added.

1. The equation still contains only one lagged observation term, $y_{t-1}$.

2. The coefficient is still written as β at this stage.

The written label connects the single lag $y_{t-1}$ to the terminology “order 1.”

1. Animation
   Observation

   The presenter adds + $β_2 y_{t-2}$ to the equation.

2. Animation
   Observation

   The presenter annotates the first coefficient as $β_1$ and the second as $β_2$.

3. Animation
   Observation

   The presenter draws a large X over the “White noise” heading and circles “Autoregressive model.”

1. Term $β y_{t-1}$

2. New term $β_2 y_{t-2}$

3. Subscripts 1 and 2 on β

4. Crossed-out “White noise” heading

5. Circled “Autoregressive model” label

1. A second lagged observation term is added.

2. The two β coefficients are distinguished by subscripts.

3. The original “White noise” heading is crossed out.

4. The “Autoregressive model” label is circled.

1. The base structure $y_t = C + ε_t$ remains part of the equation.

2. The error-dependence expression remains visible.

The visual edits emphasize that adding multiple lagged observations changes the model classification from white noise to autoregressive.

1. Animation
   Observation

   The presenter writes a new lower equation $y_t = C + ε_t$.

2. Animation
   Observation

   The presenter appends + $θ ε_{t-1}$.

3. Animation
   Observation

   The presenter writes “Moving Average Model” below the new equation.

1. New equation $y_t = C + ε_t$

2. Added term + $θ ε_{t-1}$

3. Label “Moving Average Model”

1. A separate lower equation is started from the white-noise core.

2. A lagged error term is appended.

3. The new model is labeled as a moving average model.

1. The upper autoregressive equation remains visible.

2. The constant C and current error $ε_t$ remain in the new equation.

The visual contrast shows that the moving-average modification uses a previous error $ε_{t-1}$, whereas the autoregressive modification used previous observations $y_{t-1}$ and $y_{t-2}$.

1. Animation
   Observation

   The speaker writes mathematical equations on a digital whiteboard throughout the video.

1. Whiteboard

2. Digital pen

3. Mathematical equations

1. Equations for MA, ARMA, and ARIMA models are sequentially written on the board.

1. The speaker remains visible in the top right corner.

The visual progression of writing equations illustrates the hierarchical development of time series models from simple moving averages to complex integrated models.

1. Animation
   Observation

   The speaker writes 'order' below '$d=1$' on the whiteboard.

1. Whiteboard

2. Marker

1. The word 'order' is added to the board.

1. The rest of the equations remain unchanged.

This visual action emphasizes that 'd' represents the order of differencing in the ARIMA model.

1. Audio
   Observation

   The speaker says, “Oh, except that now I've got two coefficients. They are not identical coefficients.”

2. Formula
   Observation

   The board distinguishes $β_1$ and $β_2$ by adding subscripts.

When extending from AR1 to AR2, one might assume the same coefficient β multiplies both $y_{t-1}$ and $y_{t-2}$.

The video explicitly corrects this by stating that the two coefficients are not identical and by writing separate coefficients $β_1$ and $β_2$ for the two lags.

1. Audio
   Observation

   The speaker says the model is no longer a white noise process but an autoregressive model.

2. Animation
   Observation

   The presenter crosses out the “White noise” heading and circles “Autoregressive model.”

After adding lagged observations to the white-noise equation, the process might still be called a white noise process.

The video marks the transition visually and verbally: once autoregressive lag terms are present, the model is described as an autoregressive model rather than white noise.

1. Formula
   Observation

   The speaker writes $\Theta_1 y_{t-1}$ in the ARMA equation.

2. Audio
   Observation

   The speaker refers to it as a 'moving average process'.

In the written ARMA and ARIMA equations, the moving average component is incorrectly represented using lagged observed values ($y_{t-1}$) instead of lagged error terms ($\varepsilon_{t-1}$).

A true moving average component should use past error terms, so the term should be $\Theta_1 \varepsilon_{t-1}$, not $\Theta_1 y_{t-1}$.

1. Formula
   Observation

   The handwritten ARIMA equation on the board is $y_t - y_{t-1} = c + \varepsilon_t + \Theta_1 y_{t-1} + \Phi_1 y_{t-1}$.

The moving average term in the handwritten ARIMA equation incorrectly uses $y_{t-1}$ instead of the lagged error term $\varepsilon_{t-1}$.

In a standard ARIMA(p,d,q) model, the moving average component should involve past error terms ($\varepsilon$), not past observed values (y). The correct form for the right side should be $c + \varepsilon_t + \Theta_1 \varepsilon_{t-1} + \Phi_1 (y_{t-1} - y_{t-2})$.

1. Audio
   Observation

   The speaker says they will take the white-noise equation and modify it to incorporate autoregression.

2. Formula
   Observation

   The AR1 equation is formed by appending $β y_{t-1}$ to $y_t = C + ε_t$.

The autoregressive model is derived directly from the white-noise model by adding a lagged observation term.

1. Audio
   Observation

   The speaker says that to turn the model into an AR2, they add a coefficient times y at t minus two.

2. Formula
   Observation

   The AR2 equation contains the AR1 lag term plus an additional $β_2 y_{t-2}$ term.

The AR2 model generalizes the AR1 model by adding a second lagged observation with its own coefficient.

1. Audio
   Observation

   The speaker explains that the order refers to the number of lags incorporated into the model.

2. Formula
   Observation

   The board labels AR1 as order 1 and the later equation includes two lags for AR2.

The definition of autoregressive order applies to both AR1 and AR2 by counting how many lagged y terms are included.

1. Audio
   Observation

   The speaker says they will start with the core equation that came through white noise and add previous error influence to create a moving average model.

2. Formula
   Observation

   The moving-average equation begins as $y_t = C + ε_t$ and then adds $θ ε_{t-1}$.

The moving-average model shown is derived from the same white-noise core equation by adding a lagged error term.

1. Formula
   Observation

   The autoregressive equations add lagged observations such as $y_{t-1}$ and $y_{t-2}$.

2. Formula
   Observation

   The moving-average equation adds a lagged error $ε_{t-1}$.

3. Audio
   Observation

   The speaker contrasts adding autoregression with adding the influence of error from previous time steps.

Autoregressive models use lagged observed values as predictors, while the moving-average model shown uses lagged error terms.

1. Audio
   Observation

   The speaker states that autoregressive and moving average are not mutually exclusive and can be combined.

The ARMA model contains the moving average component as part of its structure.

1. Audio
   Observation

   The speaker introduces the ARIMA model as a step up in complexity from the ARMA model.

The ARIMA model generalizes the ARMA model by adding an integrated component to handle trends.

1. Audio
   Observation

   Speaker contrasts the integrated component with adding more terms to the right, implying ARIMA builds upon ARMA.

The ARIMA model generalizes the ARMA model by adding an integrated (differencing) component to handle non-stationary data with trends.

1. Audio
   Observation

   Speaker explains that the 'I' or integrated component involves differencing.

Differencing is the specific mathematical operation applied within the integrated component of the ARIMA model.

1. Formula
   Observation

   The board shows $y_t = C + ε_t$ under the heading “White noise.”

2. Audio
   Observation

   The speaker explains observed value, constant, and error.

1. White noise model

1. Formula
   Observation

   The equation is modified by adding $β y_{t-1}$.

2. Audio
   Observation

   The speaker says this modification incorporates autoregression.

1. White noise model
2. Autoregressive model from white noise
3. Deriving the AR1 model from white noise

1. Formula
   Observation

   The board writes “AR1 order 1.”

2. Audio
   Observation

   The speaker says the one refers to the number of lags incorporated into the model.

1. Order of an autoregressive model

1. Audio
   Observation

   The speaker says the two coefficients are not identical.

2. Formula
   Observation

   The board labels the coefficients $β_1$ and $β_2$.

1. AR2 model
2. Assuming all autoregressive lag coefficients are identical

1. Formula
   Observation

   The autoregressive model uses lagged y terms, while the moving-average model uses $θ ε_{t-1}$.

2. Audio
   Observation

   The speaker introduces the moving average model by adding the influence of error from previous time steps.

1. Autoregressive model from white noise
2. Moving average model
3. s0-rel-ar-contrasts-ma

1. Audio
   Observation

   The speaker says the model is no longer a white noise process but an autoregressive model.

2. Animation
   Observation

   The “White noise” heading is crossed out and “Autoregressive model” is circled.

1. AR2 model
2. Treating an autoregressive model as still white noise
3. Adding autoregressive lags changes the process type

1. Audio
   Observation

   The speaker explains how to build an MA-2 model by adding error information from previous time steps.

1. Moving Average Model

1. Audio
   Observation

   The speaker explains combining AR and MA elements into a single equation.

1. Autoregressive Moving Average Model

1. Audio
   Observation

   The speaker explains that the integrated component modifies the left-hand side to fit a trend.

1. Autoregressive Integrated Moving Average Model

1. Formula
   Observation

   ARIMA equation shown on the board.

1. ARIMA Model Definition

1. Audio
   Observation

   Speaker discusses the order of differencing.

1. Differencing Order