<p>This should help you figure out how to compute RSS for the different models.</p>

<p>Let's begin with what we have:</p>

<p>We know that</p>

<ul>
<li>$y$ is an $n \times 1$ vector that has observations of the endogenous variable</li>
<li>$x$ is an $n \times 2$ matrix of observations of the exogenous variables</li>
<li>$\beta$ is a $2 \times 1$ matrix of coefficients</li>
</ul>

<p>Unfortunately (or fortunately depending on how you look at it), we don't know what $x$ and $y$ look like in this equation. Instead we are given $(x' x)$, $(x' y)$, and $(y' y)$. </p>

<p>Using your comment about what RSS actually is -- Namely, that</p>

<p>$$\text{RSS} = (y_{\text{observed}} - y_{\text{estimated}})' (y_{\text{observed}} - y_{\text{estimated}})$$</p>

<p>First notice that $y_{\text{estimated}}$ is simply $x \beta$. Then replacing this in the previous formula gives us</p>

<p>\begin{align*}
  \text{RSS} &amp;= (y_{\text{observed}} - y_{\text{estimated}})' (y_{\text{observed}} - y_{\text{estimated}}) \\
  &amp;= (y_{\text{observed}} - x \beta)' (y_{\text{observed}} - x \beta) \\
  &amp;= y'y - 2y'x\beta + x \beta \beta' x'
\end{align*}</p>

<p>Now this is almost what we want. We want things to ultimately be in terms of $(y'y)$, $(x'x)$, $(x' y)$, and $\beta$. The first term is exactly $y'y$, but other terms look like they have pieces we want and we just have to find a way to get them.</p>

<p>Now, notice that each of the elements in our equation above is simply a scalar (aka sizes are such that they are $1 \times 1$). The determinant of a scalar is itself, then by using properties of determinants we can say:</p>

<p>\begin{align*}
  2y'x \beta &amp;= \det(2y'x\beta) = \det(2 \beta' x' y) \\
  x \beta \beta' x' &amp;= \det(x \beta \beta' x') = \det(\beta' x' x \beta)
\end{align*}</p>

<p>I will leave the linear algebra itself as an exercise.</p>

CC BY-SA 3.0
2015-11-04T18:21:41.927Z
2015-11-04T18:21:41.927Z
cc7768
<p>A bivariate regression is fitted to 20 sample observations on y and X. I know the following:
$X'X=\begin{pmatrix}
                             20 &amp; 10\\
                             10 &amp; 30 \\
                             \end{pmatrix}$, 
$X'y=\begin{pmatrix}
                             30\\
                             40 \\
                             \end{pmatrix}$,
$y'y=75$.</p>

<p>I received that $\beta=\begin{pmatrix}
                             1.4\\
                             1.3\\
                             \end{pmatrix}$.</p>

<p>Then a new observation was obtained: $X=2$, $Y=4$.
I should perform a Chow test of parameter constancy.
In fact I know the formula for the test: $F=\frac{(RSS_{pooled}-RSS_{1}-RSS_{2})/(k+1)}{(RSS_{1}-RSS_{2})/(n-2k-2)}$. Unfortunately, I have no idea how to calculate the RSS for any of these regressions. Help is needed. </p>

CC BY-SA 3.0
2015-11-04T17:32:46.647Z
Do you know what RSS is?
cc7768
CC BY-SA 3.0
2015-11-04T17:54:24.373Z
@cc7768 yes. RSS=sum of (y-y_estimated)^2
Pichen'ka
<econometrics>
Chow test of parameter constancy
Pichen'ka
# Chow test of parameter constancy

(Asked by: username_0 on 2015-11-04)

A bivariate regression is fitted to 20 sample observations on y and X. I know the following:
$X'X=\begin{pmatrix}
20 & 10\\
10 & 30 \\
\end{pmatrix}$,
$X'y=\begin{pmatrix}
30\\
40 \\
\end{pmatrix}$,
$y'y=75$.

I received that $\beta=\begin{pmatrix}
1.4\\
1.3\\
\end{pmatrix}$.

Then a new observation was obtained: $X=2$, $Y=4$.
I should perform a Chow test of parameter constancy.
In fact I know the formula for the test: $F=\frac{(RSS\_{pooled}-RSS\_{1}-RSS\_{2})/(k+1)}{(RSS\_{1}-RSS\_{2})/(n-2k-2)}$. Unfortunately, I have no idea how to calculate the RSS for any of these regressions. Help is needed.

**Tags:** econometrics

> **Comments:**
> **username_1**: Do you know what RSS is?  
> **username_0**: @username_1 yes. RSS=sum of (y-y_estimated)^2  

---

## 1 answers

### Answer by username_1 on 2015-11-04. Score: 1 | Accepted answer

This should help you figure out how to compute RSS for the different models.

Let's begin with what we have:

We know that

* $y$ is an $n \times 1$ vector that has observations of the endogenous variable
* $x$ is an $n \times 2$ matrix of observations of the exogenous variables
* $\beta$ is a $2 \times 1$ matrix of coefficients

Unfortunately (or fortunately depending on how you look at it), we don't know what $x$ and $y$ look like in this equation. Instead we are given $(x' x)$, $(x' y)$, and $(y' y)$.

Using your comment about what RSS actually is -- Namely, that

$$\text{RSS} = (y\_{\text{observed}} - y\_{\text{estimated}})' (y\_{\text{observed}} - y\_{\text{estimated}})$$

First notice that $y\_{\text{estimated}}$ is simply $x \beta$. Then replacing this in the previous formula gives us

\begin{align\*}
\text{RSS} &= (y\_{\text{observed}} - y\_{\text{estimated}})' (y\_{\text{observed}} - y\_{\text{estimated}}) \\
&= (y\_{\text{observed}} - x \beta)' (y\_{\text{observed}} - x \beta) \\
&= y'y - 2y'x\beta + x \beta \beta' x'
\end{align\*}

Now this is almost what we want. We want things to ultimately be in terms of $(y'y)$, $(x'x)$, $(x' y)$, and $\beta$. The first term is exactly $y'y$, but other terms look like they have pieces we want and we just have to find a way to get them.

Now, notice that each of the elements in our equation above is simply a scalar (aka sizes are such that they are $1 \times 1$). The determinant of a scalar is itself, then by using properties of determinants we can say:

\begin{align\*}
2y'x \beta &= \det(2y'x\beta) = \det(2 \beta' x' y) \\
x \beta \beta' x' &= \det(x \beta \beta' x') = \det(\beta' x' x \beta)
\end{align\*}

I will leave the linear algebra itself as an exercise.


---

