Cross-section data has no yesterday. Time series does — and that memory is the whole game. This lab builds the Box–Jenkins family one letter at a time: AR (memory of past values), MA (memory of past shocks), ARMA (both), and the I that makes a drifting series sit still long enough to be modelled. You'll tune coefficients with sliders, fire shocks and watch them echo, read ACF/PACF plots like a diagnostician, and forecast at the end.
Every model here is a recipe that turns an invisible stream of random shocks εt (mean 0, constant variance, no autocorrelation — "white noise") into the series you actually observe. The letters only differ in how long the recipe remembers.
Today is a weighted echo of its own past levels. One shock keeps ringing forever, decaying like φh.
Today is a weighted sum of the last few shocks. Memory is finite: after q periods the shock is gone, exactly.
A short-lived shock kick (MA) on top of a long decaying tail (AR). Most real series need a bit of each.
The series wanders with no fixed mean. Difference it d times (Δyt = yt − yt−1) and model the result.
These models assume the mean, variance and autocorrelations don't depend on when you look — otherwise there is no stable thing to estimate, and the usual t-stats and R² mislead you (the classic spurious-regression trap). For an AR(1), stationarity means |φ| < 1: shocks die out. At φ = 1 you have a unit root — a random walk that never returns to a mean, whose variance grows with t. The fix is the I: difference it, then model the differences. Over-differencing has its own tell, and you'll meet it in Level 5.
"Cuts off" means: inside the ±1.96/√n band from that lag onwards — statistically indistinguishable from zero. Mixed ARMA can't be read off the plots; that's what AIC/BIC comparisons are for.
Tell apart no-memory noise, a random walk, and a near-unit-root AR — by eye, then by ACF.
Same shocks, one slider. Watch persistence, half-life and variance move together — and the series stop reverting as φ → 1.
The clean AR-vs-MA distinction: infinite decaying memory versus memory that ends at lag q.
Three mystery processes. Read the cut-offs, name the model. This is the skill that pays.
Difference a random walk into stationarity — then difference once too often and meet ρ₁ ≈ −0.5.
Two sliders, one hidden truth. Match the target's correlogram and see why mixed models are hard to pin down.
Compute 1- and 3-step forecasts by hand, then watch them decay to μ as the interval widens to ±1.96σy.
One mystery series, five calls: d, then p, then q, name the model, and forecast one step ahead.
All three panels have roughly the same scale of random variation. Only one is white noise: zero mean, constant variance, and zero autocorrelation at every lag. Which one?
One AR(1): yt = φ·yt−1 + εt. The shock sequence
εt is frozen — every change you see comes from φ alone.
Your task: tune φ so that a shock's effect has a half-life of
3 periods, then lock it in.
Both models below are driven by the same single unit shock at t = 5 and nothing else. Pick a model, fire the shock, and watch how long it survives. This is an impulse response function — the cleanest way to feel the AR/MA difference.
Q1 · In the MA(1), how long does a shock keep moving the series after it lands?
Three mystery processes, one at a time. Read the two correlograms and name the model. Rule of thumb: PACF cuts off at p for an AR, ACF cuts off at q for an MA.
Dashed lines are the ±1.96/√n significance band (n = 250). Stems inside the band ≈ zero.
Here is a series of 160 observations. Before any ARMA model is allowed near it, one question has to be answered.
Q1 · Is this series stationary?
A hidden yt = φ·yt−1 + εt + θ·εt−1 generated the target below. Your series uses the same shocks, so if you find the right φ and θ the two paths will lie on top of each other. Fit on the ACF — that's what the data actually identifies.
A fitted AR(1) in deviation form: (yt − μ) = φ(yt−1 − μ) + εt. The optimal forecast sets all future shocks to their expectation, zero. Work out the next step and the three-step-ahead point forecast.
One mystery series. Five calls, in the order a real analyst makes them: how much differencing, then p, then q, then the ARIMA label, then one step ahead.
Final score: 0 / 0