Welcome to the Future: Forecasting with Time Series

In the previous chapters of the Time Series Models section, you learned how to identify patterns, check for stationarity, and fit models like AR, MA, and ARMA. But why do we do all that work? Most of the time, it’s because we want to know what happens next.

In this chapter, we focus on Predicted Values (our best guess for the future) and Confidence Intervals (how sure we are about that guess). If you've ever looked at a hurricane path map where the "cone of uncertainty" gets wider as the storm moves further away, you already understand the core intuition of this chapter!

Don't worry if the math looks a bit scary at first. We’ll break it down step-by-step so you can approach these Exam SRM questions with confidence.


1. Point Forecasts: Making the Best Guess

A Point Forecast is a single number representing our best estimate of a future value in a time series. In SRM, we usually denote a forecast made at time \(T\) for a future time \(T+h\) as \(\hat{y}_{T+h}\).

The Golden Rule of Forecasting

When we move from the past (where we have data) into the future (where we don't), we follow one simple rule for the error term (\(\epsilon\)):
We assume future random errors are zero.

Why? Because the expected value of white noise is zero. We can't predict the "random shocks," so our best bet is to assume they won't push the series in any specific direction.

How to Calculate Forecasts for Different Models

1. AR(1) Models:
The formula for an AR(1) model is \(y_{t} = \phi_0 + \phi_1 y_{t-1} + \epsilon_t\).
To predict one step ahead (\(h=1\)):
\(\hat{y}_{T+1} = \phi_0 + \phi_1 y_T\)
To predict two steps ahead (\(h=2\)):
\(\hat{y}_{T+2} = \phi_0 + \phi_1 \hat{y}_{T+1}\)

2. MA(1) Models:
The formula is \(y_t = c + \epsilon_t + \theta_1 \epsilon_{t-1}\).
To predict one step ahead:
\(\hat{y}_{T+1} = c + \theta_1 \epsilon_T\) (where \(\epsilon_T\) is the residual from the last observed point).
To predict two steps ahead:
\(\hat{y}_{T+2} = c\).
Wait, why did the \(\theta\) term disappear? Because at two steps ahead, we are dealing with \(\epsilon_{T+1}\), which we assume is zero!

Key Takeaway: For MA(q) models, the forecast reverts exactly to the mean after \(q\) steps. For AR(p) models, the forecast gradually decays toward the mean over time.


2. Prediction Intervals: Measuring the Uncertainty

Even the best model won't be perfectly right. A Prediction Interval (often called a Confidence Interval in this context) gives us a range where the future value is likely to fall.

The Anatomy of the Interval

Just like in linear regression, the interval is built as:
Forecast \(\pm\) (Critical Value \(\times\) Standard Error)

For a 95% interval, the formula looks like:
\(\hat{y}_{T+h} \pm 1.96 \times \sqrt{Var(e_{T+h})}\)

Where \(e_{T+h}\) is the forecast error (the difference between the actual future value and our predicted value).

Why the "Cone" Grows

Think of it this way: You can probably guess what the temperature will be 1 hour from now with high accuracy. But guessing the temperature exactly 10 days from now is much harder.

In time series:

  • The Standard Error almost always increases as the lead time (\(h\)) increases.
  • As we forecast further into the future, we have more "unknown" random shocks accumulating.
  • This results in a widening interval (the "fan" or "cone" shape).

Quick Review: - Point forecast = The "middle" of our future range. - Prediction interval = The "spread" reflecting our uncertainty. - As \(h\) increases, uncertainty increases.


3. Calculating the Variance of the Forecast Error

This is where students often get stuck, but there is a pattern to the madness! Let's look at an AR(1) model: \(y_t = \phi y_{t-1} + \epsilon_t\). (Assuming mean is 0 for simplicity).

One-step ahead (h=1):

The error is just the next random shock: \(\epsilon_{T+1}\).
The variance is simply \(\sigma^2_\epsilon\).

Two-steps ahead (h=2):

The error involves the shock from step 1 and the shock from step 2.
The variance is \(\sigma^2_\epsilon (1 + \phi^2)\).

h-steps ahead:

The variance is \(\sigma^2_\epsilon \sum_{j=0}^{h-1} \psi_j^2\).
Note: \(\psi\) (psi) weights represent the model rewritten as an Infinite MA process. For SRM, you mainly need to know that as \(h \rightarrow \infty\), the variance of the forecast error approaches the process variance (the total variance of the time series).

Did you know? If the time series is stationary, the prediction interval won't grow forever. It eventually levels off at a maximum width based on the total variance of the series.


4. Common Mistakes to Avoid

Mistake 1: Forgetting the Constant.
When calculating \(\hat{y}_{T+1}\), don't forget the intercept (\(\phi_0\) or \(c\)). If you leave it out, your forecast will be biased toward zero.

Mistake 2: Confusing the "Residual" with the "Error".
In MA(q) models, you use the calculated residuals from the data you already have. For future steps, you use 0. Don't try to "guess" a future error!

Mistake 3: Using the wrong Z-score.
For a 95% interval, use 1.96. For a 90% interval, use 1.645. Always double-check what the question asks for!


5. Summary and Key Takeaways

The "Big Picture" for Exam SRM:

  • Forecasting is iterative: Use the current value to predict the next, then use that prediction to predict the one after that.
  • Mean Reversion: In stationary models, long-term forecasts will always settle at the mean of the process.
  • Errors Accumulate: Prediction intervals get wider as you look further ahead because there are more future "shocks" (\(\epsilon\)) that we cannot see yet.
  • MA Model Limitation: MA models lose their memory quickly. After \(q\) steps, the forecast is just the mean, and the prediction interval becomes constant.

You've got this! Practice a few AR(1) forecast calculations manually, and the patterns will start to feel like second nature.