Consider the linear model yi =xiB + ei or, in matrix notation Y= X'B + e. where X' is a vector n * 1. You estimate the model using OLS under the following assumptions (A) E (ui) = 0, (B) E (xi) = 0, i =1, ..., n, (C) E (u4) < infinity and E (xi4) < infinity, i = 1, ..., n.
Assume that E (ui * xi) = 0.3 and Var (xi) = 0.8.
A) Is the OLS estimator biased? Is it consistent? (If your answer is no, then compute the bias).
B) Assume now that E (ui2/xi) = xi2 and E (xi4) = 1.28. Derive the asymptotic distribution of B.
C) Assume now that E (xi) = 0.4. Is the OLS estimator biased? Is it consistent? ( If your answer is no, then compute the bias).
D) Propose a way to correct the bias (if present) under the assumption of point (a).
A) The OLS estimator is unbiased if the expected value of the estimator is equal to the true value of the parameter. In this case, to determine whether the OLS estimator is biased, we need to check whether E(B) = B.
Using the assumptions given, we have E(ui * xi) = 0.3 and E(xi) = 0. Since E(ui) = 0 (from assumption A) and E(xi) = 0 (from assumption B), we can rewrite E(ui * xi) = Cov(ui, xi) = 0.3.
The OLS estimator is consistent if it converges to the true value of the parameter as the sample size increases. In this case, we need to determine whether the OLS estimator satisfies the consistency condition.
So, considering assumption A and assumption B, the OLS estimator is unbiased and consistent.
B) To derive the asymptotic distribution of B, we need additional information about the error term and the regressors. Specifically, we need to know the variance of the error term and the cross-moment of the error term and regressors.
C) If E(xi) = 0.4, then the OLS estimator can be biased. To determine if it is biased, we need to check whether E(B) = B.
Using the assumptions given, we have E(ui * xi) = 0.3 and E(xi) = 0.4. So, E(ui * xi) ≠ 0, which means the OLS estimator is biased.
To compute the bias, we can use the formula:
Bias(B) = E(B) - B
In this case, since E(B) ≠ B, the OLS estimator is biased. You can calculate the exact bias using the formula above.
D) To correct the bias, we can use an instrumental variable approach if we have another variable correlated with xi but not with the error term ui. By using this instrumental variable as an additional regressor, we can obtain an estimate that is consistent and unbiased.
Alternatively, if we have information about the distribution of xi and ui, we can use a transformation to correct the bias. By transforming the data appropriately before estimating the model, we can obtain an unbiased estimate.
Without specific information about the error term and regressors, it is challenging to propose a specific correction method.
A) To determine if the OLS estimator is biased, we need to check if the expected value of the estimator is equal to the true parameter value.
In the given model, the OLS estimator is unbiased if the following condition is satisfied:
E(B) = B
Under the assumptions (A) and (B) stated, E(ui) = 0 and E(xi) = 0. Hence, these assumptions are fulfilled.
However, we also need to check the assumption (C) for the error term and the explanatory variable.
E(u^4) should be finite, but no value is provided. Similarly, E(xi^4) should be finite, but the value is not given.
Without knowing the exact values or bounds for E(u^4) and E(xi^4), we cannot definitively determine if the OLS estimator is biased or unbiased.
If we assume that E(u^4) and E(xi^4) are finite, we can proceed to check consistency.
The OLS estimator is consistent if the following condition is satisfied:
lim (n -> infinity) Var(B) = 0
However, we do not have enough information to compute the variance of B or check the consistency of the estimator.
Therefore, based on the given information, we cannot determine if the OLS estimator is biased or consistent.
B) To derive the asymptotic distribution of B, we can use the results from asymptotic theory. Under appropriate assumptions:
sqrt(n)(B - B_true) ~ N(0, V)
Where B_true is the true value of B, n is the sample size, and V is the asymptotic variance of B.
In this case, the given information does not provide us with the necessary values or information to compute V or the asymptotic distribution of B. Hence, we cannot derive the asymptotic distribution of B.
C) If we assume E(xi) = 0.4, we can evaluate if the OLS estimator is biased.
The OLS estimator is biased if the expected value of the estimator is not equal to the true parameter value.
Under this assumption, we have:
E(B) = B + Cov(xi, ui) / Var(xi)
Here, Cov(xi, ui) = E(ui * xi) = 0.3 (as given), and Var(xi) = 0.8 (as given).
Hence,
E(B) = B + 0.3 / 0.8
E(B) = B + 0.375
Since E(xi) is non-zero (0.4), the OLS estimator is biased.
To compute the bias, we subtract the true parameter value from the expected value of the estimator:
Bias = E(B) - B = B + 0.375 - B = 0.375
The bias is equal to 0.375.
D) To correct the bias under the assumption of E(xi) = 0.4, we can use instrumental variables (IV) estimation.
In IV estimation, we need to find an instrument variable that is correlated with the explanatory variable but is uncorrelated with the error term.
If we can find a suitable instrument variable, we can use IV estimation to obtain an unbiased estimator. However, finding appropriate instruments requires additional knowledge about the model and variables involved.
Without further information about potential instrumental variables, it is not possible to propose a specific way to correct the bias in this case.