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difference(Close)computes the first difference of theClosevariable.
Stationarity & Differencing
What Does “Stationary” Mean?
Have you ever heard the word “stationary” before?
- In everyday language, stationary means not moving — fixed in place.
- For a time series, what would that mean?
Think about it
If a time series is “not moving”, what statistical properties would you expect it to have?
Are These Series Stationary?
Which of the following six series are stationary (i.e., which look more stable)?
Look at these two series:
The second series was produced directly from the first. Can you figure out how?
Differencing
The daily return is simply today’s price minus yesterday’s price — the change between consecutive observations. This operation is called differencing, and it is how we stabilize the mean of a non-stationary series.
Random walks & stationarity
- Google’s stock price follows a random walk.
- By definition, random walks are non-stationary, but their first differences are stationary.
Second Differences
Sometimes the first-differenced series is still non-stationary. We can difference again:
\begin{align*} y''_t &= y'_t - y'_{t-1} \\ &= (y_t-y_{t-1}) - (y_{t-1} - y_{t-2}) \\ &= y_t - 2y_{t-1} + y_{t-2} \end{align*}
- Second differences have T - 2 observations.
- They represent the change in the changes — acceleration rather than velocity.
- Economic and business series almost never require more than two differences.
Needing three or more differences usually signals something else is wrong — an outlier, a structural break, or a transformation that should have been applied first.
Seasonal Differencing
If the series has seasonality, we take a seasonal difference — the change relative to the same period in the previous cycle:
y'_t = y_t - y_{t-m}
where m is the seasonal period (m = 12 for monthly data, m = 4 for quarterly, …).
- Seasonal differences represent season-over-season change1 at each point.
- Also called lag-m differences.
- After seasonal differencing, any remaining non-seasonal trend can be removed with a first difference.
Backshift Notation
The backshift operator B provides a compact way to write differencing operations — and, as we will see next class, to write out full model equations cleanly.
It is defined simply as:
By_t = y_{t-1} ; B(By_t) = B^2 y_t = y_{t-2}
That is, applying B to a series shifts it back one period.
First Differences
Recall the first difference:
y'_t = y_t - y_{t-1}
We can rewrite y_{t-1} = By_t, so:
y'_t = y_t - By_t = (1 - B)y_t
Second Differences
Recall:
y''_t = y_t - 2y_{t-1} + y_{t-2}
Using the backshift operator turns to:
y''_t = y_t - 2By_t + B^2 y_t = (1 - 2B + B^2)y_t
which gives a perfect square trinomial2
y''_t = (1-B)^2 y_t
General d-level Differences
The pattern generalizes naturally:
(1 - B)^d y_t
where d is the number of times we difference the series.
Seasonal Differences
A seasonal difference shifts back m periods:
y_t - y_{t-m} = y_t - B^m y_t = (1 - B^m)y_t
And when both are needed — as with mexretail — the two operators simply multiply:
(1-B)(1-B^m)y_t
which is exactly the expression we expanded by hand in the previous section.
| Operation | Backshift form |
|---|---|
| First difference | (1 - B)\,y_t |
| Second difference | (1 - B)^2\,y_t |
| d-th difference | (1 - B)^d\,y_t |
| Seasonal difference | (1 - B^m)\,y_t |
| Both together | (1 - B)^d(1 - B^m)\,y_t |
Order of differencing — revisited
Using backshift notation, it becomes trivial to see that the order doesn’t matter. Both orderings produce the same expression, since multiplication is commutative:
(1-B)(1-B^m)y_t = (1-B^m)(1-B)y_t
No algebra needed.
ACF Shapes for Different Patterns
ACF and PACF Together
gg_tsdisplay() with plot_type = "partial" shows the time series, ACF, and PACF together — this is the standard diagnostic display for the rest of the module.
Autoregressive Models — AR(p)
An autoregressive model of order p forecasts y_t as a weighted sum of its own past values:
y_t = c + \phi_1 y_{t-1} + \phi_2 y_{t-2} + \cdots + \phi_p y_{t-p} + \varepsilon_t
- Structurally identical to multiple linear regression — except the predictors are lagged values of the series itself.
- p is the number of past values used.
- \varepsilon_t is white noise.
Signature in ACF/PACF
- PACF cuts off sharply after lag p — this is how you read the order.
- ACF decays exponentially or sinusoidal.
Moving Average Models — MA(q)
A moving average model of order q forecasts y_t as a weighted sum of past forecast errors:
y_t = c + \varepsilon_t + \theta_1 \varepsilon_{t-1} + \theta_2 \varepsilon_{t-2} + \cdots + \theta_q \varepsilon_{t-q}
- The predictors are past errors, not past values of the series.
- q is the number of past errors used.
- \varepsilon_t is white noise.
MA models ≠ moving average smoothing
Do not confuse MA models with the moving average smoothing used in decomposition. Smoothing estimates the trend-cycle from past observed values. MA models use past errors to describe the correlation structure of the series.
Signature in ACF/PACF
- ACF cuts off sharply after lag q — the mirror image of the AR signature.
- PACF decays exponentially or sinusoidal.
You cannot identify a model from the time plot alone
Now that you have seen AR and MA models: their time plots can look very similar to each other. The ACF and PACF are what distinguish them. This is fundamentally different from ETS, where the form of trend and seasonality in the time plot directly guides model selection — one of the key differences we will revisit when comparing these two families.
Footnotes
most commonly YoY (year-over-year), but remember there can be daily, weekly, or more seasonal patterns too.
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