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A random sample of size $n$ from a bivariate distribution is denoted by $(x_r,y_r), r=1,2,3,...,n$. Show that if the regression line of $y$ on $x$ passes through the origin of its scatter diagram then $$\bar y\sum^n_{r=1} x_r^2=\bar x \sum^n_{r=1} x_r y_r$$ where $(\bar x,\bar y)$ is the mean point of the sample.

I don't really know how to begin. I am aware the line equation is $b=\frac{y}{x}=\frac{\sum xy-\frac{\sum x\sum y}{n}}{\sum x^2-\frac{(\sum x)^2}{n}}$

Not sure what to do next.

Vincenzo Tibullo
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mathnoob123
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1 Answers1

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Recall that the OLS estimators are $$ \hat{\beta}_0 = \bar{Y}_n - \hat{\beta}_1\bar{X}_n, \quad \hat{\beta}_1 = \frac{\sum X_i Y_i-n\bar{X}\bar{Y}}{\sum X_i^2 - n\bar{X}^2}, $$ because the line passes through the origin, you have that $\hat{\beta}_0 = 0 = \bar{Y} - \hat{\beta}_1\bar{X}$, thus $$ \frac{\bar{Y}}{\bar{X}} = \frac{\sum X_i Y_i-n\bar{X}\bar{Y}}{\sum X_i^2 - n\bar{X}^2}, $$ or $$ \bar{Y}\sum X_i^2 - \bar{Y}n\bar{X}^2=\bar{X}\sum X_i Y_i -n\bar{Y}\bar{X}^2, $$ hence $$ \bar{Y}\sum X_i^2=\bar{X}\sum X_i Y_i . $$

V. Vancak
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  • Can you tell me how did you decided which is the intercept? – mathnoob123 May 23 '17 at 22:40
  • It is a matter of derivation, rather than "decision". If you solve $\min_{\beta} \sum (y_i - \beta_0 - \beta_1x_i)^2$, then you'll get that the OLS of the intercept is $\bar{y}-\hat{\beta}_1\bar{x}$. – V. Vancak May 23 '17 at 22:54
  • Just one last thing. What is the variable $\beta_0$ that you used? Moreover can you please provide a proof for the intercept? (I am not that good in statistics to carry out the operation you suggested) – mathnoob123 May 23 '17 at 23:32
  • You can find here a full proof. – V. Vancak May 24 '17 at 07:10