<p>Population Regression Function (PRF) is the relation between conditional expectation of population $Y_{i}$ conditional on population $X_{i}$, versus the corresponding $X_{i}$, which is also a random variable (to be precise, a matrix of random variables) with some specific given values, say $x$, that is, $E(Y_{i}|X_{i}=x)= f(X_{i})$, which is PRF. For linear regression, $f(X_{i})=\beta_{0}+\beta_{1}X_{i}$.</p>

<p>For a specific value $X_{i}=x$ in population, there could be multiple corresponding $Y_{i}$. (Similarly, imagine when we have a sample dataset and we sort by $X$, we can find multiple corresponding $Y$). Thus, using the relation we get above, we have: $Y_{i}=E(Y_{i}|X_{i}=x)+\mu_{i}$ (Note: it's $\mu$, the deviation of population $Y_{i}$ from its conditional expectation value, rather than $\widehat{\mu }$, which is an estimated value from a specific sample). In other words, PRF captures the perfect, or ideal situation where if $X=x$ happens, we get $Y=y$. But remember both $Y_{i}$ and $X_{i}$  are random variables that may deviate from the perfect situation. Given $X_{i}=x$, the model uses an error term $\mu_{i}$ to capture the deviation caused by other random factors other than $X_{i}$.</p>

<p>Sample Regression Function (SRF), on the other hand, is the estimated relation between estimated $Y_{i}$, denoted as $\widehat{Y_{i}}$, meaning it is calculated by SRF, that is, $\widehat f(X_{i})=\widehat\beta_{0}+\widehat\beta_{1}X_{i}$, and $X_{i}$. (Note that $X_{i}$ and $Y_{i}$ here are actual values from sample, rather than from population, although we usually use the same denotations). $\widehat\beta_{0}$ and $\widehat\beta_{1}$ mean they are calculated or estimated from sample, rather than the actual relation between population $Y_{i}$ and $X_{i}$. Also, $\widehat f(X_{i})$, $\widehat\beta_{0}$ and $\widehat\beta_{1}$ are estimated parameters/relation based on sample, in order to "guess" the true parameters/relation in population. The estimated error term $\widehat{\mu}$, is the difference between the sample $Y_{i}$ and the calculated value $\widehat{Y_{i}}$ using the model we specify, and is the estimator of population ${\mu}$.</p>

CC BY-SA 3.0
2017-09-15T20:35:23.537Z
2017-09-15T23:36:04.340Z
2017-09-15T23:36:04.340Z
HarmlessEcon
<p>I've been confused about the exact difference between PRF and SRF is. </p>

<p>I know that it has to do with other variables that influence the regression that is not accounted for, but what is the difference in the models?</p>

<hr>

<p><sub> 
PRF: Population Regression Function<br>
SRF: Sample Regression Function.</sub></p>

CC BY-SA 3.0
2017-09-12T06:18:48.530Z
What is the u hat? The estimated error term?
luchonacho
CC BY-SA 3.0
2017-09-15T03:31:52.740Z
The u hat is the error term
ttylalexa
<econometrics>
What is the exact difference between the $\hat{u}$ in PRF vs SRF?
ttylalexa
# What is the exact difference between the $\hat{u}$ in PRF vs SRF?

(Asked by: username_0 on 2017-09-11)

I've been confused about the exact difference between PRF and SRF is.

I know that it has to do with other variables that influence the regression that is not accounted for, but what is the difference in the models?

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PRF: Population Regression Function  
SRF: Sample Regression Function.

**Tags:** econometrics

> **Comments:**
> **username_1**: What is the u hat? The estimated error term?  
> **username_0**: The u hat is the error term  

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## 1 answers

### Answer by username_2 on 2017-09-15. Score: 4

Population Regression Function (PRF) is the relation between conditional expectation of population $Y\_{i}$ conditional on population $X\_{i}$, versus the corresponding $X\_{i}$, which is also a random variable (to be precise, a matrix of random variables) with some specific given values, say $x$, that is, $E(Y\_{i}|X\_{i}=x)= f(X\_{i})$, which is PRF. For linear regression, $f(X\_{i})=\beta\_{0}+\beta\_{1}X\_{i}$.

For a specific value $X\_{i}=x$ in population, there could be multiple corresponding $Y\_{i}$. (Similarly, imagine when we have a sample dataset and we sort by $X$, we can find multiple corresponding $Y$). Thus, using the relation we get above, we have: $Y\_{i}=E(Y\_{i}|X\_{i}=x)+\mu\_{i}$ (Note: it's $\mu$, the deviation of population $Y\_{i}$ from its conditional expectation value, rather than $\widehat{\mu }$, which is an estimated value from a specific sample). In other words, PRF captures the perfect, or ideal situation where if $X=x$ happens, we get $Y=y$. But remember both $Y\_{i}$ and $X\_{i}$ are random variables that may deviate from the perfect situation. Given $X\_{i}=x$, the model uses an error term $\mu\_{i}$ to capture the deviation caused by other random factors other than $X\_{i}$.

Sample Regression Function (SRF), on the other hand, is the estimated relation between estimated $Y\_{i}$, denoted as $\widehat{Y\_{i}}$, meaning it is calculated by SRF, that is, $\widehat f(X\_{i})=\widehat\beta\_{0}+\widehat\beta\_{1}X\_{i}$, and $X\_{i}$. (Note that $X\_{i}$ and $Y\_{i}$ here are actual values from sample, rather than from population, although we usually use the same denotations). $\widehat\beta\_{0}$ and $\widehat\beta\_{1}$ mean they are calculated or estimated from sample, rather than the actual relation between population $Y\_{i}$ and $X\_{i}$. Also, $\widehat f(X\_{i})$, $\widehat\beta\_{0}$ and $\widehat\beta\_{1}$ are estimated parameters/relation based on sample, in order to "guess" the true parameters/relation in population. The estimated error term $\widehat{\mu}$, is the difference between the sample $Y\_{i}$ and the calculated value $\widehat{Y\_{i}}$ using the model we specify, and is the estimator of population ${\mu}$.


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