follows a random walk, however, the first difference estimator is more efficient. Equality of fixed effects and first difference estimators when T=2 For the special two period case (), the fixed effects (FE) estimator and the first difference (FD) estimator are numerically equivalent. This is because the FE estimator effectively "doubles the data set" used in the FD estimator. To see this, establish that the fixed effects estimator is: Since each can be re-written as , we'll re-write the line as: Chamberlain method Gary Chamberlain's method, a generalization of the within estimator, replaces with its linear projection onto the explanatory variables. Writing the linear projection as: this results in the following equation: which can be estimated by minimum distance estimation. Hausman–Taylor method Need to have more than one time-variant regressor () and time-invariant regressor () and at least one and one that are uncorrelated with . Partition the and variables such that where and are uncorrelated with . Need . Estimating via OLS on using and as instruments yields a consistent estimate. Generalization with input uncertainty When there is input uncertainty for the data, , then the value, rather than the sum of squared residuals, should be minimized. This can be directly achieved from substitution rules: , then the values and standard deviations for and can be determined via classical ordinary least squares analysis and variance-covariance matrix. Use to test for consistency Random effects estimators may be inconsistent sometimes in the long time series limit, if the random effects are misspecified (i.e. the model chosen for the random effects is incorrect). However, the fixed effects model may still be consistent in some situations. For example, if the time series being modeled is not stationary, random effects models assuming stationarity may not be consistent in the long-series limit. One example of this is if the time series has an upward trend. Then, as the series becomes longer, the model revises estimates for the mean of earlier periods upwards, giving increasingly biased predictions of coefficients. However, a model with fixed time effects does not pool information across time, and as a result earlier estimates will not be affected. In situations like these where the fixed effects model is known to be consistent, the Durbin-Wu-Hausman test can be used to test whether the random effects model chosen is consistent. If is true, both and are consistent, but only is efficient. If is true