二元模型中预测值与期望值的区别咨询

Great question—let's break this down clearly using your linear probability model (LPM) example, since that's where your confusion is rooted.

1. What is the predicted value (Ŷ)? #

When you plug in specific values for your independent variables (like X₁=X₂=X₃=1) into your LPM Ŷ = β₁X₁ + β₂X₂ + β₃X₃, the result (β₁+β₂+β₃) is a point prediction for Y given that exact combination of X values.

Crucially, in the LPM framework, this predicted value is directly the model's estimate of the conditional expectation of Y given X—because the LPM is explicitly defined such that E(Y|X) = β₁X₁ + β₂X₂ + β₃X₃ (the error term ϵ has an expected value of 0, so it drops out when calculating the expectation).

2. What is the conditional expectation E(Y|X)? #

The conditional expectation E(Y|X=x₀) (where x₀ is a specific combination like X₁=X₂=X₃=1) is the true average value of Y for all individuals in the population who have X equal to x₀.

This is a population-level truth—we can never observe it directly, because we can't collect data on every single individual in the population. Instead, we use statistical models (like your LPM) to estimate it.

3. Clarifying your confusion about "the average of predicted values for observations with specified X in the sample" #

If your sample has multiple observations where X₁=X₂=X₃=1, every one of those observations will have the same predicted value Ŷ = β₁+β₂+β₃ (since the X values are identical and the model's coefficients are fixed). So the average of these predicted values is just the same as the single predicted value itself.

But let's expand this to a more useful distinction:

• If you're talking about the average of actual Y values for observations with X=x₀ in your sample, that's a non-parametric estimate of E(Y|X=x₀). It's just the raw average of the Ys you see for that group.
• Your model's predicted value Ŷ is a parametric estimate of E(Y|X=x₀)—it uses the linear relationship you've specified to extrapolate the average Y for that X group, even if your sample has few (or no) observations with exactly x₀.

Example to make it concrete #

Suppose your estimated LPM is Ŷ = 0.2X₁ + 0.3X₂ + 0.1X₃. For X₁=X₂=X₃=1, Ŷ=0.6. This means your model estimates that the true average Y for all population members with X=(1,1,1) is 0.6.

If your sample has 5 observations with X=(1,1,1), their actual Y values might be [1, 0, 1, 1, 0]. The sample average of these Ys is 0.6—matching your predicted value perfectly (a lucky case!). If the sample average was 0.5 instead, that's a raw, data-driven estimate of the population average, while Ŷ=0.6 is the estimate from your linear model.

Key Takeaways #

• Predicted value (Ŷ): A single estimate of E(Y|X=x₀) from your model. For a given x₀, every observation with that X combination gets the same Ŷ.
• Conditional expectation E(Y|X=x₀): The true average Y for all individuals in the population with X=x₀—this is the target we're trying to estimate.
• The "average of predicted values for specified X in the sample" is just Ŷ (since all predictions for that X are identical), but the average of actual Y values for that X is a separate, non-parametric way to estimate the conditional expectation.

内容的提问来源于stack exchange，提问作者xiong