Welcome to Time-Series Analysis!
Hello there! Welcome to one of the most practical chapters in the CFA Level II Quantitative Methods curriculum. If you’ve ever looked at a stock chart or a graph of GDP over time, you’ve looked at a time series. In this chapter, we move beyond simple regression to understand how data points relate to their own past values.
Don't worry if this seems a bit "math-heavy" at first. We are going to break these complex models down into simple stories. By the end of this, you’ll be able to spot trends, identify when a model is "broken," and understand how analysts predict future values. Let's dive in!
1. Linear vs. Log-Linear Trend Models
The simplest way to model time series is to see if the data grows consistently over time. We have two main "flavors" here:
Linear Trend Models
In a Linear Trend Model, we assume the variable changes by a constant amount each period.
Formula: \(y_t = b_0 + b_1(t) + \epsilon_t\)
Example: A small coffee shop that gains exactly 10 new customers every month. Whether they have 100 or 1,000 customers, they always add 10 more next month.
Log-Linear Trend Models
In a Log-Linear Trend Model, we assume the variable changes by a constant percentage each period. This is much more common in finance (like compound interest).
Formula: \(ln(y_t) = b_0 + b_1(t) + \epsilon_t\)
Example: A tech startup growing at 10% every month. As the company gets bigger, the 10% growth represents more and more actual people.
How to Choose?
Look at your data! If the data plots as a straight line, use Linear. If the data looks like it’s curving upward (exponential growth), use Log-Linear. If you see "Heteroskedasticity" (the dots getting further from the line as time goes on) in a linear model, a log-linear model often fixes it.
Quick Tip: If the growth is in percentages, "Log" is your best friend!
Key Takeaway: Use Linear for constant amount changes; use Log-Linear for constant percentage changes.
2. Autoregressive (AR) Models
Sometimes, the best way to predict tomorrow is to look at today. An Autoregressive (AR) Model uses past values of a variable to predict its future values.
A first-order AR model, known as AR(1), looks like this:
\(x_t = b_0 + b_1 x_{t-1} + \epsilon_t\)
This says: "The value today (\(x_t\)) depends on a constant (\(b_0\)), plus a fraction of what happened yesterday (\(b_1 x_{t-1}\)), plus some random noise (\(\epsilon_t\))."
The Golden Rule: Covariance Stationarity
For an AR model to be valid, the time series must be Covariance Stationary. Think of this as the "stability" of the data. To be stationary, three things must stay constant over time:
1. Expected Value (Mean): The data centers around a specific level.
2. Variance: The "swing" or volatility of the data stays the same.
3. Covariance: The relationship between different periods is stable.
Did you know? If a series isn't stationary, our statistical tests (like t-tests) become unreliable. It’s like trying to measure a toddler who won’t stop jumping!
Key Takeaway: AR models use past values to predict future ones, but they only work if the data is "stationary" (stable over time).
3. Detecting Problems in AR Models
In simple regression, we used the Durbin-Watson test to check for Serial Correlation (when errors are linked to each other).
CRITICAL WARNING: You cannot use the Durbin-Watson test for AR models. This is a classic CFA exam trap! Instead, we look at the Autocorrelations of the Residuals.
Step-by-Step Check:
1. Estimate the AR model.
2. Calculate the residuals (errors).
3. Check the correlations of these residuals. If they are significantly different from zero, your model is "misspecified" (it's missing something).
Key Takeaway: Avoid Durbin-Watson for AR models; use the t-test on residual autocorrelations instead.
4. Mean Reversion
Most stationary AR models have a Mean Reverting Level. This is the "home base" that the data eventually returns to.
Formula for Mean Reverting Level: \(\frac{b_0}{1 - b_1}\)
Analogy: Think of a rubber band. You can pull the price of a stock away from its "fair value," but the model predicts it will eventually snap back to this level.
Key Takeaway: If the current value is above the mean reverting level, the model predicts it will fall. If it's below, it predicts it will rise.
5. Random Walks and Unit Roots
What happens if \(b_1 = 1\) in our AR(1) model? We get a Random Walk.
\(x_t = x_{t-1} + \epsilon_t\)
In a random walk, the best guess for tomorrow is simply whatever the price is today. This is not stationary because the variance grows over time (the "walk" can wander anywhere).
The Unit Root
If \(b_1 = 1\), we say the model has a Unit Root. This is bad news for modeling!
To test for this, we use the Dickey-Fuller Test. Don't worry about the complex math; just remember that if a unit root exists, the series is non-stationary.
How do we fix a Unit Root?
We use a process called First-Differencing. Instead of modeling the price, we model the change in price.
\(y_t = x_t - x_{t-1}\)
By modeling the change, we often turn a non-stationary series into a stationary one.
Key Takeaway: A unit root (\(b_1 = 1\)) means the data is a Random Walk. Fix it by using "First-Differencing."
6. Seasonality
Does your data have a pattern that repeats every year? Think of retail sales peaking every December. This is Seasonality.
If you ignore seasonality in an AR model, your residuals will show a big "spike" at the seasonal lag (e.g., Lag 4 for quarterly data or Lag 12 for monthly data).
The Fix: Add a Seasonal Lag to your model. If you are looking at quarterly data, you would use \(x_{t-4}\) as an additional variable to help predict \(x_t\).
Key Takeaway: Spot seasonality by looking for significant autocorrelations at specific lags (like 4 or 12), then add that lag to the model.
7. ARCH Models (The Volatility Specialists)
ARCH stands for Autoregressive Conditional Heteroskedasticity. (Try saying that five times fast!)
In simple terms, ARCH exists when the variance of the errors in one period depends on the variance of the errors in the previous period.
Example: If the stock market is wild and volatile today, it’s likely to stay wild and volatile tomorrow.
If an ARCH effect exists, you can predict the volatility of future errors. This is incredibly useful for options pricing and Risk Management (VaR).
Key Takeaway: ARCH models help us predict when "clusters" of high volatility are likely to happen.
8. Co-integration: Moving Together
Sometimes, two different time series (like the stock prices of two similar oil companies) are both non-stationary (they wander like random walks), but they stay close to each other. This is called Co-integration.
Analogy: Imagine a man walking a dog with a leash. Both the man and the dog are wandering all over the park (non-stationary), but because of the leash, they can never get too far apart. They are co-integrated.
If two series are co-integrated, we can use regression between them without getting "spurious" (fake) results.
Key Takeaway: Co-integration means two non-stationary series share a long-term economic relationship.
Summary Checklist for Students
Before you sit for the exam, make sure you can answer these:
- Is the data changing by an amount (Linear) or a percentage (Log-Linear)?
- Is the series Covariance Stationary? (Required for AR models!)
- Does the model have a Unit Root? (Use Dickey-Fuller to check; First-Difference to fix).
- Are the residuals clean? (Check autocorrelations; don't use Durbin-Watson!).
- Is there Seasonality? (Look for spikes at lags 4 or 12).
- Are two wandering series Co-integrated? (The "Leash" analogy).
Don't give up! Time-Series is like learning a new language. At first, you just see symbols, but soon you'll see the patterns of the market. Good luck with your studies!