Predicted Values and Prediction Intervals

Updated for the 2026-2027 CFA® Level I curriculum.

Predicted values and prediction intervals help you turn an estimated regression equation into a practical forecast. The predicted value gives the model’s best estimate at a chosen value of X, while the prediction interval shows a reasonable range for one future observation. For CFA Level I, you need to calculate both and explain what the interval means.

Key Takeaways: Predicted Values and Prediction Intervals

• The point prediction is , calculated by substituting the specified X value into the estimated regression equation.

• A prediction interval gives a range for one future value of the dependent variable at a fixed value of X.

• The predicted value sits at the midpoint of the prediction interval.

• The standard error of the forecast, , determines how much uncertainty surrounds the point prediction.

• Larger samples generally narrow the interval because the regression coefficients are estimated more precisely.

• The interval widens as moves farther from , so forecasts near the center of the sample tend to be more precise.

• A 95% prediction interval is constructed using a method that captures the future observation in 95% of repeated samples.

• Predictions outside the observed range of X rely on the assumption that the estimated linear relationship continues beyond the available data.

What You Need to Know for CFA Level I

Work through this topic in two stages. First, calculate the predicted value by substituting the specified X value into the estimated regression equation. Then use the critical t-value and the standard error of the forecast to calculate the lower and upper prediction limits.

You should also be able to interpret the completed interval. State that it applies to one future value of the dependent variable at the specified value of X. Avoid describing it as a range for the slope, the independent variable, or the average value of the dependent variable.

Finally, remember how the interval changes. It becomes wider as moves away from the sample mean, and forecasts beyond the observed X values carry additional model risk that the calculated interval does not measure.

What Is a Predicted Value in Simple Linear Regression?

A predicted value is the estimated value of the dependent variable at a chosen value of the independent variable. You calculate it by substituting the selected X value into the estimated regression equation.

The result is a point prediction, meaning it gives one best estimate. Because actual observations vary around the fitted regression line, the point prediction is usually paired with a prediction interval.

What Is a Prediction Interval?

A prediction interval gives a range for one future observation of the dependent variable at a specified value of the independent variable. It helps you judge how far the realized value might fall from the regression model’s point prediction.

The interval reflects two sources of uncertainty. The estimated intercept and slope may differ from their true population values because they were calculated from a sample. Individual observations also vary around the true regression line, even when the population coefficients are known.

A confidence interval for the mean response reflects uncertainty in the estimated regression line. A prediction interval also includes the variation of an individual future observation, which makes it wider at the same confidence level.

How Do You Calculate a Prediction Interval?

Begin with the predicted value. Then add and subtract a margin based on the critical t-value and the standard error of the forecast.

The standard error of the forecast adjusts the standard error of estimate for the uncertainty associated with one future observation and the location of within the sample.

Where:

• = predicted value for the future observation

• = estimated intercept

• = estimated slope

• = specified value of the independent variable

• = two-tailed critical t-value with degrees of freedom

• = standard error of the forecast

• = standard error of estimate

• = sample mean of the independent variable

• = number of observations

The exam may provide directly. When you need to calculate it, work through each term inside the square root carefully. Those terms explain why the prediction interval keeps some width even in a large sample and why it expands as the selected X value moves away from the sample mean.

What Determines the Width of a Prediction Interval?

Four main inputs affect the width of the interval.

The inside the square root represents the variation of one future observation around the regression line. It remains in the formula even when the sample becomes very large. As a result, a prediction interval does not shrink to zero simply because the coefficients are estimated with greater precision.

How Should a Prediction Interval Be Interpreted?

Frame the interpretation around one future value of the dependent variable at the specified value of X.

Suppose a 95% prediction interval runs from 2.72% to 6.48% when . You can interpret this as a plausible range for the realized value of Y when X equals 4.0%. More formally, the method used to construct the interval captures the future value of Y in 95% of repeated samples.

Once the interval has been calculated, its endpoints are fixed. The future observation will eventually fall either inside or outside those limits. The 95% confidence level describes the long-run reliability of the interval-building method.

Why Is Extrapolation Risky?

Extrapolation occurs when you predict Y using an X value outside the range observed in the sample. The estimated regression line may fit the available data well while failing to describe what happens beyond that range.

For example, a relationship that appears linear within the sample could flatten, curve, or reverse outside it. The prediction interval does not account for this form of model risk. Its calculation assumes that the estimated linear relationship remains appropriate at .

Forecasts near the middle of the observed data generally deserve more confidence than forecasts near the edges. Predictions outside the sample range require especially careful judgment because the available observations provide no direct evidence about the relationship there.

Worked Example: Forecasting Revenue Growth

An analyst regresses a company’s quarterly revenue growth (, percent) on industry revenue growth (, percent) using 22 quarters of data. The estimated model is:

The regression reports a standard error of estimate of 0.85, a sample mean of , and . The analyst wants a 95% prediction interval for the company’s revenue growth in a quarter when industry growth is 4.0%.

Step 1. Calculate the predicted value.

The model predicts company revenue growth of 4.6%.

Step 2. Calculate the standard error of the forecast.

The standard error of the forecast is approximately 0.90 percentage points.

Step 3. Find the critical value.

With , the regression has degrees of freedom. The two-tailed critical t-value for a 95% interval is 2.086.

Step 4. Build the interval.

Lower limit:

Upper limit:

The 95% prediction interval runs from 2.72% to 6.48%.

Step 5. Interpret it.

When industry revenue growth is 4.0%, the model predicts company revenue growth of 4.6%. The interval from 2.72% to 6.48% gives a plausible range for the company’s realized growth rate in one future quarter.

The selected industry growth rate is 1.5 percentage points above its sample mean. That distance increases the standard error of the forecast and makes the interval wider than it would be at the center of the sample.

Had the analyst forecast at , the squared-distance term would equal zero. The prediction interval would therefore be at its narrowest for this sample.

Common Exam Traps

• Skipping the substitution. Use the specified when calculating the predicted value. Reporting alone gives the predicted value only when X equals zero.

• Treating the slope as the prediction. The slope shows the estimated change in Y for a one-unit change in X. It is not the predicted value of Y.

• Using the wrong degrees of freedom. Simple linear regression uses degrees of freedom because the intercept and slope are both estimated.

• Dropping the critical value. The margin equals . Adding and subtracting alone produces an interval that is too narrow.

• Describing the interval as a range for X. The prediction interval applies to the dependent variable at a specified value of the independent variable.

• Assuming the interval has the same width at every X value. The interval becomes wider as moves away from .

• Ignoring extrapolation risk. A calculated interval may look precise even when lies outside the sample range. The formula cannot confirm that the linear relationship still holds there.

Practice Question

An analyst estimates the following model from 30 monthly observations:

For a future observation with , the standard error of the forecast is 1.10. The two-tailed critical t-value at 95% confidence with 28 degrees of freedom is 2.048.

The 95% prediction interval for the future value of is closest to:

1. 1.20 to 5.70

2. 1.80 to 6.30

3. 2.95 to 5.15

Solution

FAQs About Predicted Values and Prediction Intervals