How to Calculate Prediction Interval in Stata

A prediction interval in statistics provides a range of values within which a future observation is expected to fall, with a certain level of confidence. In Stata, calculating prediction intervals involves using regression models and understanding the underlying statistical principles.

What is a Prediction Interval?

A prediction interval is an estimate of the range within which a future observation will fall. Unlike confidence intervals, which estimate the range of a population parameter, prediction intervals account for both the uncertainty in estimating the model parameters and the variability of individual observations.

Prediction intervals are particularly useful in fields like economics, engineering, and social sciences where forecasting future values is essential.

How to Calculate Prediction Interval in Stata

Stata provides built-in commands to calculate prediction intervals for regression models. Here's a step-by-step guide:

Prerequisites

Before calculating prediction intervals, you should have:

A dataset with dependent and independent variables
A fitted regression model
Stata installed with the necessary statistical packages

Step 1: Fit a Regression Model

First, you need to fit a regression model to your data. For example, if you have a dependent variable Y and independent variables X1 and X2:

regress Y X1 X2

Step 2: Calculate Prediction Intervals

Use the predict command with the ci option to calculate prediction intervals:

predict yhat, xb
predict lower, xb ci
predict upper, xb ci

This will create three new variables: yhat (predicted values), lower (lower bound of prediction interval), and upper (upper bound of prediction interval).

Formula Used

The prediction interval is calculated as:

π = (ŷ ± tα/2,n-p-1 * √(MSE * (1 + X' (X X')⁻¹ X)))

Where:

π = prediction interval
ŷ = predicted value
tα/2,n-p-1 = critical t-value
MSE = mean squared error
X = vector of independent variables
p = number of parameters

Step 3: Visualize Results

You can create a scatter plot with prediction intervals using:

scatter Y X1, yline(yhat) yline(lower) yline(upper)

Worked Example

Let's calculate prediction intervals for a simple linear regression model.

Dataset

Y (Dependent)	X1 (Independent)
10	1
15	2
20	3
25	4
30	5

Stata Commands

regress Y X1
predict yhat, xb
predict lower, xb ci
predict upper, xb ci

Results

The prediction intervals for each observation would be calculated based on the regression model's parameters and the formula above.

Interpreting Results

When interpreting prediction intervals in Stata:

The prediction interval provides a range where you expect a new observation to fall
Wider intervals indicate more uncertainty in predictions
Narrower intervals suggest more precise predictions
Always consider the context of your data and model assumptions

FAQ

What is the difference between a confidence interval and a prediction interval?

A confidence interval estimates the range of a population parameter, while a prediction interval estimates the range of a future observation.

How do I choose the confidence level for my prediction interval?

Common confidence levels are 90%, 95%, and 99%. Higher confidence levels result in wider intervals.

Can I calculate prediction intervals for non-linear models in Stata?

Yes, Stata supports prediction intervals for various model types including logistic regression and survival models.