Welcome to the World of Time Series!
Hello future FRMs! Welcome to one of the most important chapters in the Quantitative Analysis section. Today, we are diving into Stationary Time Series. If you’ve ever looked at a chart of stock prices or interest rates and wondered, "Can I predict what happens next?", you are already thinking about time series.
Don't worry if this seems a bit "math-heavy" at first. Think of a time series simply as a sequence of data points collected over time (like daily stock returns). Our goal is to find patterns in the past to understand the future. Let’s get started!
1. What is a Time Series?
A time series is a set of observations measured at successive points in time, usually at equal intervals (daily, monthly, quarterly). In the FRM exam, we care about the properties of these sequences so we can build reliable models.
2. The Concept of Stationarity
Imagine a lake. If the water is calm and the waves stay within a certain height, it's predictable. If a giant storm comes and the water level starts rising indefinitely, it's hard to predict. Stationarity is like that calm lake.
For a time series to be covariance stationary (the standard we use), it must satisfy three specific "rules":
- Constant Mean: The average value \( E(Y_t) \) does not change over time.
- Constant Variance: The "spread" or volatility \( Var(Y_t) \) stays the same.
- Constant Covariance: The relationship between two observations (e.g., today and yesterday) depends only on the distance (lag) between them, not the specific point in time.
Why is this important? If a series isn't stationary (e.g., it has a trend), our standard statistical tests might give us "garbage" results. We need stationarity to ensure the past is actually a reliable guide to the future.
Quick Tip: If you see a chart where the data is "drifting" upward or the swings are getting wider and wider, it is NOT stationary.
Key Takeaway: Covariance stationarity means the mean, variance, and autocovariance are stable over time.
3. White Noise: The Purest Randomness
White Noise is the simplest form of a stationary process. Imagine the static on an old TV—it's completely random. In a white noise process:
- The mean is zero.
- The variance is constant.
- There is zero correlation between different observations.
In our models, we call the "random error" term \( \epsilon_t \) white noise. It represents the "shocks" to the system that we can't predict.
4. Autoregressive (AR) Models
The term "Auto" means "Self." So, an Autoregressive model is one where the variable is regressed on its own past values.
The AR(1) Model
The most common version is the AR(1) model, where today's value depends on yesterday's value:
\( Y_t = \phi_0 + \phi_1 Y_{t-1} + \epsilon_t \)
Where:
- \( Y_t \) is the value today.
- \( \phi_0 \) is a constant (intercept).
- \( \phi_1 \) is the coefficient (how much yesterday matters).
- \( \epsilon_t \) is the random shock (white noise).
The Stationarity Condition for AR(1)
For an AR(1) model to be stationary, the coefficient \( |\phi_1| \) must be less than 1.
- If \( \phi_1 = 1 \), we have a "Random Walk" (not stationary).
- If \( \phi_1 > 1 \), the series explodes to infinity (not stationary).
Mean Reversion
Stationary AR(1) models are mean-reverting. If a shock pushes the price up, it will eventually pull back toward its long-run average (the "mean level").
The Mean Level formula: \( \mu = \frac{\phi_0}{1 - \phi_1} \)
Key Takeaway: AR models use past values. For AR(1) to be stationary, the "slope" coefficient must be between -1 and 1.
5. Moving Average (MA) Models
Don't let the name confuse you! This isn't the "moving average" you see in technical analysis charts. In time series, a Moving Average (MA) model means today's value depends on past random shocks (errors).
The MA(1) Model
\( Y_t = \mu + \epsilon_t + \theta_1 \epsilon_{t-1} \)
Where:
- \( \mu \) is the mean of the series.
- \( \epsilon_t \) is today's shock.
- \( \theta_1 \) is the weight given to yesterday's shock.
Did you know? MA models are always stationary, regardless of the value of \( \theta_1 \). This is because they are just a weighted sum of stationary white noise terms.
Analogy: Think of an AR model as "Memory of the Price" and an MA model as "Memory of the Shock." If a surprise news event happened yesterday, an MA model calculates how much that specific surprise still affects the price today.
6. ARMA Models
Sometimes, past values AND past shocks both matter. We combine them into an ARMA(p, q) model.
- p = number of AR (past value) lags.
- q = number of MA (past error) lags.
Example: An ARMA(1,1) would look like:
\( Y_t = \phi_0 + \phi_1 Y_{t-1} + \epsilon_t + \theta_1 \epsilon_{t-1} \)
The Principle of Parsimony: In the FRM world, "less is more." We want the simplest model that explains the data well. Adding too many lags makes the model overfit (it looks great on old data but fails on new data).
7. Model Selection: AIC and BIC
How do we decide if an ARMA(1,1) is better than an ARMA(2,1)? We use information criteria. These are like "scores" for models.
- AIC (Akaike Information Criterion)
- BIC (Bayesian Information Criterion)
The Goal: Choose the model with the LOWEST AIC or BIC value.
Important Difference: BIC is "stricter" than AIC. It applies a larger penalty for adding extra parameters. If you are worried about overcomplicating your model, BIC is your best friend.
Memory Aid: BIC = Bigger penalty for complexity.
8. Common Pitfalls to Avoid
1. Confusing AR and MA: Remember, AR uses \( Y_{t-1} \) (the actual data), while MA uses \( \epsilon_{t-1} \) (the error).
2. Stationarity check: Always check if \( |\phi| < 1 \) for AR models. If \( \phi = 1 \), it’s a Unit Root, and your statistics are invalid!
3. Lag Selection: Don't just pick the model with the highest R-squared. Use AIC or BIC because they account for the "cost" of adding more variables.
Quick Summary Review
Stationarity: Constant mean, variance, and autocovariance.
AR(1): \( Y_t \) depends on \( Y_{t-1} \). Stationary if \( |\phi_1| < 1 \).
MA(1): \( Y_t \) depends on \( \epsilon_{t-1} \). Always stationary.
Selection: Use the lowest AIC or BIC.
Parsimony: Keep models simple to avoid overfitting.
You've made it through! Time series can be tricky, but by focusing on these core properties and the logic behind the models, you'll be well-prepared for the exam. Keep practicing the formulas and you'll master this!