# Meaning of $\hat{\beta}$ of the linear regression model [closed]

In the simple linear regression model, $$\hat{\beta}$$ is the sum of independent normally distributed random variables.

Is it false because in linear regression there is $$\beta$$ and not $$\hat{\beta}$$?

• I've edited your post. Please comment if I've changed the meaning. – Dave Oct 22 '19 at 19:31
• The model doesn't say anything about $\hat b.$ When you fit it with Ordinary Least Squares, the formula for $\hat b$ exhibits it as a linear combination of the responses $y.$ When you assume those responses are independent and Normally distributed, then indeed $\hat b$ is a linear combination of independent Normal variables: to see that, please search our site for any posts on OLS regression formulas (there are a few thousand of them to choose from). – whuber Oct 22 '19 at 19:41
• Remember that we estimate $\hat{b}$ via $\hat{b} = (X^TX)^{-1}X^Ty$. This means that $\hat{b}$ is a linear combination of the values of $y$. When we assume that each $y_i$ has a normal distribution, then, yes, $\hat{b}$ is just a linear combination of normal random variables. – Dave Oct 22 '19 at 19:46
• @JaneSecuiu What is "^b" if not $\hat{b}$ as the estimate of $b$ in $y=a + bx$? – Dave Oct 23 '19 at 13:50
• ''y = β x + α + ε can be estimated by y^=a^+b^x. (Since the expected value of ε is 0.) b^ then, is [Σ(xi - x¯)(yi - y¯)]/Σ (xi-x¯)2. You may have seen this expressed as Sxy/Sxx. I would hardly call this the sum of independent normal variables. There is no imposition on x to be normal. As long as it correlates linearly with y, the regression will work.'' someone answered to a similar question this.is it false? – Jane Secuiu Oct 23 '19 at 14:36

It is false, but not for the reason you listed.

$$\hat{b}$$ is an estimate of β, the unknowable parameter. As such you have:

$$y$$ = β $$x$$ + α + ε can be estimated by $$\hat{y} = \hat{a} + \hat{b}x$$. (Since the expected value of ε is 0.)

$$\hat{b}$$ then, is [Σ($$x_i$$ - $$\bar{x}$$)($$y_i$$ - $$\bar{y}$$)$$]/$$Σ ($$x_i$$-$$\bar{x}$$)$$^2$$. You may have seen this expressed as $$S_{xy} / S_{xx}$$. I would hardly call this the sum of independent normal variables. There is no imposition on $$x$$ to be normal. As long as it correlates linearly with $$y$$, the regression will work.

• someone to a similar question said: ''Remember that we estimate b^ via b^=((X^T*X)^−1)(X^T)*y. This means that b^ is a linear combination of the values of y. When we assume that each yi has a normal distribution, then, yes, b^ is just a linear combination of normal random variables.'' is this false? – Sarah kenwich Oct 23 '19 at 14:29
• @Sarahkenwich A typical (but not mandatory) assumption of OLS regression is that the response variable is conditionally normal. That is, each $y_i$ has a normal distribution. If you make this assumption, then my comment is correct. – Dave Oct 23 '19 at 15:01
• one person says it is false, the other that it is true.i do not understand who is correct – Jane Secuiu Oct 23 '19 at 15:09

$$\hat{b} = (X^TX)^{-1}X^Ty$$

When we assume that each $$y_i$$ has a normal distribution, then $$\hat{b}$$ is a linear combination of normal random variables and, therefore, a sum of normal random variables.

It is not mandatory that we assume each $$y_i$$ to be normal. When we assume exponential (for instance) $$y_i$$, then $$\hat{b}$$ is a linear combination of exponential random variables instead of normal random variables.

However, normal $$y_i$$ is a common assumption. When we make this assumption, then $$\hat{b}$$ is a sum of normal random variables, since $$(X^TX)^{-1}X^T$$ is a row vector.

In summary, when we make the (extremely common) assumption of normal $$y_i$$, then $$\hat{b}$$ is a sum of normal random variables. If we do not make that assumption, then $$\hat{b}$$ is not a sum of normal random variables.

Depending on the level where your class is operating, the full-credit answer may be either TRUE or FALSE. I would expect a class dealing with regression on one variable to be making the assumption of normal $$y_i$$, so the full-credit answer would be TRUE.

And since this probably is a question from a class assignment, please include the self-study tag. CV doesn't usually give full answers to homework questions, though it seemed necessary to do so here to clear up some confusion.