Question

asymptotic properties of OLS estimates - consistency: - Why √n? Linderberg-Levy CLT - Suppose that {z1,z2.......} is a sequence of iid random variables with expectation E[zi] = u and finite variance var[zi] = Ω < infinity then as n-> infinity, the random variables √n(sn-u_ converges in distribution to a normal distribution √n(sn-u_->N(o,Ω) Sn = sample average of random variable zi √n(B^-B) = (1/nx'x)^-1(1/√nx'u) x'u is the random variable zi in the Lindeberg-Levy CLT √n(Sn-u) -> N(o,Ω) where sn = zi * What allow us to define the expected value and the variance of (x'u) and hence apply CLT to determine that 1/√n x'u converges to normal distribution?

Answer 1

The Central Limit Theorem approximates the distribution of sample means with a normal distribution. The Lindeberg-Levy CLT allows us to define expected value and variance in the OLS estimates. The OLS estimates are consistent as shown by the convergence of the random variable to a normal distribution.

Step-by-step explanation:

The Central Limit Theorem states that as the sample size increases, the distribution of the sample means approaches a normal distribution. This means that if the original population is not normally distributed, the distribution of the sample means can still be approximated by a normal distribution. The standard error of the mean, which is equal to the population standard deviation divided by the square root of the sample size, determines how close the sample mean is to the population mean.

In the Lindeberg-Levy Central Limit Theorem, we can define the expected value and variance of (x'u) by using the properties of the OLS estimates. The sample average of the random variable zi, represented by Sn, allows us to determine that 1/√n(x'u) converges to a normal distribution. As n approaches infinity, √n(Sn-u) converges in distribution to a normal distribution with mean zero and variance Ω.

Therefore, the consistency of the OLS estimates is established by the Linderberg-Levy CLT, which shows that the random variable √n(sn-u) converges to a normal distribution with mean zero and variance Ω as n approaches infinity.