Unbiased Estimators and Heteroskedasticity

Question

Consider a consumption model with bivariate data points $(Y_i,X_i)$, $i=1,...,n$, with $Y_i$ consumption and $X_i$ income. The univariate model is $$Y_i=\beta X_i+u_i,$$ where $E(u_i|X_i)=0, \textrm{var}(u_i|X_i)=\sigma^2$ and $\textrm{cov}(u_i,u_j|X_i)=0$.

The estimates of $\beta$ are given as follows: $$\hat{\beta_1}=\frac{\sum_{i=1}^{n}Y_i}{\sum_{i=1}^{n}X_i}$$ and $$\hat{\beta_2}=\frac{\sum_{i=1}^{n}X_iY_i}{\sum_{i=1}^{n}X_i^2}.$$ a.) Show that $\hat{\beta_1}$ and $\hat{\beta_2}$ are unbiased. Also, which estimator is 'better'.

I am not sure how to prove that both are unbiased. I think we begin by taking the expected values and then reduce it to the form of $E(\hat{\beta_1})=\beta$ and $E(\hat{\beta_2})=\beta$, but I am not sure how to do this mathematically.

To consider which is better, since both are unbiased, I would look at which has the smallest variance, but to avoid calculating this, would it be sufficient to talk about the Gauss-Markov conditions and state which one satisfies these conditions? Because the one that satisfies these conditions will have the smallest variance possible. How would I show which $\hat{\beta}$ does not satisfy the Gauss-Markov conditions? I believe that it is $\hat{\beta_1}$ but I am unsure of how to show this.

b.) If the residuals of the model are linearly related to the income so that $\textrm{var}(u_i|X_i)=\sigma^2X_i$, how could one verify if this is the case? Ie.: suggest a test.

I think you would write the OLS, take residuals and square them, then run the regression. If we assume linearity then we could use the Breusch-Pagan test with the following model: $$\hat{u_i^2}=\sigma^2+\sum_{j=1}^{k}\delta_jX_{ij}+\epsilon_i,$$ where $\sum_{j=1}^{k}\delta_jX_{ij}+\epsilon_i=0$ by the homoskedasticity assumption.

So this would give us: $H_0: \delta_1=\delta_2=...=\delta_k=0$ and $H_1: \textrm{at least one } \delta \neq 0$.

Then I think you would run an F-test since this is a joint hypothesis. We would then reject the null hypothesis if we cannot accept that all $\delta$'s equal $0$. And we would conclude that there exists heteroskedasticity in at least one of the $X$'s.

Is this correct? And how would be compute the F-statistic?

c.) If heteroskedasticity does exist, then will the estimators still be unbiased? And which estimator is now considered 'better'?

I would say that the estimators are still unbiased as the presence of heteroskedasticity affects the standard errors, not the means. So the condition for unbiasedness still holds ($E(u_i|X_i)=0$). I think the better estimate is now $\hat{\beta_1}$ because it is an optimum WLS (weighted least squares) estimator as it minimises $\textrm{var}(u_i|X_i)=\sigma^2X_i$, however I am unsure of how to prove this formally.

d.) If consumption also depends on interest $R_i$ so that the consumer may need to borrow in order to fund certain forms of expenditure), this would give the following model: $$Y_i=\beta X_i+\gamma R_i+u_i,$$ where $E(u_i|X_i, R_i)=0, \textrm{var}(u_i|X_i, R_i)=\sigma^2X_i$ and $\textrm{cov}(u_i,u_j|X_i, R_i)=0$.

What happens to the unbiasedness of the the $\hat{\beta_1}$ and $\hat{\beta_2}$ estimators? What conditions will make them unbiased?

Here, I think that again $\hat{\beta_1}$ and $\hat{\beta_2}$ are still unbiased as the $E(u_i|X_i, R_i)=0$. However, I am not sure because of the introduction of the $R_i$ term. Also, to make them unbiased would it be correct to say that the only condition that would make them unbiased is when $E(u_i|X_i)\neq0$? Or are there other conditions which can cause unbiasedness?

Any help with any of these questions will be really appreciated. Thank you!

Answer 1

Both in case of unbiasedness and variance-connected issues (like efficiency or heteroskedasticity), first of all, I would recommend plugging your equation on $Y$ into formulas for $\beta_{1} / \beta_{2}$. Thus (for $Y_{i} = \beta X_{i} + u_{i}$) we get:

$$ \beta_{1}= \beta + \frac{\sum_{i=1}^{n}u_{i}}{\sum_{i=1}^{n}X_{i}} \\ \beta_{2}= \beta + \frac{\sum_{i=1}^{n}X_{i}u_{i}}{\sum_{i=1}^{n}X_{i}^{2}} \\ $$

Since $E(u_{i}|X_{i}) = 0$, you can see clearly (Tipp: $E(u_{i})=E(E(u_{i}|X_{i})$), why both estimators are unbiased and why there is a problem if we put $Y_{i} = \beta X_{i} + \gamma R_{i} + u_{i}$ inside $\beta_{1} / \beta_{2}$ (instead of $Y_{i} = \beta X_{i} + u_{i}$).

Moreover, calculating OLS estimator ($\beta_{OLS}=(X^{'}X)^{-1}X^{'}y$) for given model, we obtain $\beta_{2} = \beta_{OLS}$. Gauss-Markov Theorem gives you that $\beta_{2}$ has lower variance (both estimators are linear in $Y$ and $\beta_{2}$ is BLUE). Alternatively, you could calculate variance of both estimators directly from definition: $$ Var(\beta_{2})=Var(\beta + \frac{\sum_{i=1}^{n}X_{i}u_{i}}{\sum_{i=1}^{n}X_{i}^{2}})=... $$ remembering that $Var(u_{i}) = E(Var(u_{i}|X_{i})) + Var(E(u_{i}|X_{i}))$. This approach might be particularly useful in c) since you can no longer use Gauss-Markov Theorem (we have heteroskedasticity!). I hope you can manage to do this; to find out which variance is lower you can use Cauchy-Schwarz inequality (https://en.wikipedia.org/wiki/Cauchy%E2%80%93Schwarz_inequality).

Last but not least, heteroskedasticity has nothing with biasedness or unbiasedness of your estimators (the only thing you should provide is $E(u_{i}|X_{i})=0$). Back to b), your approach to use Breusch - Pagan test seems correct for me.