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Henrique P.
Econmoist with a Master's in Public Policy and solid experience with econometrics and other quantitative methods
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Linear Algebra
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Question:

What is an idempotent matrix?

Henrique P.
Answer:

An idempotent matrix is one squared matrix (lets say $$X$$) for which the following relationship is always true: $$X X = X$$ (i.e. if one multiplies an idempotent matrix by itself, the result is the same original matrix.)

Econometrics
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Question:

What does monotonicity stands for in Consumers' Theory?

Henrique P.
Answer:

Most of the results from consumer's theory in Microeconomics are based on assumptions that shapes consumers' behavior and preferences. When these assumptions holds, consumers are said to have "well-behaved indifference curves". Monotonicity is one of these assumptions. It means that consumers always prefers to consume more of a determined good if they do not have to reduce the consumption of any other good. It is usually formalized as: $$(x_1, x_2) \succ (y_1 , y_2)$$ if $$x_i > y_i$$ for at least one i and $$x_j \nless y_j$$ for $$j \neq i$$.

Microeconomics
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Question:

What is the implication for linear regression models (using Ordinary Least Squares) when the assumption of "exogenous explanatory variables" does not hold?

Henrique P.
Answer:

Regression models are based on assumptions that are necessary to guarantee that the estimates are unbiased and consistent as well as efficient. One of these assumptions is the "exogeneity of explanatory variables" (also referred as zero conditional mean of the error term $$(E(u|X) = 0)$$. This is a crucial assumption for estimating unbiased and consistent parameters using the method of Ordinary Least Squares (OLS). The violation of this assumption implies unbiased estimates even under large samples (meaning that the estimates are also inconsistent). Three main causes can lead to endogenous explanatory variables: 1) Omitting an important explanatory variable; 2) Measurement errors in variables; and, 3) Reverse causality (when X affects y but y also affects X). Lets see how omitting an important variable lead to bias. Imagine that we want to estimate the following regression model: (i) $$Y_i = \beta_0 + \beta_1 x_{1i} + \beta_2 x_{2i} + u_i$$ (True Model) But, for some reason we estimate the following model: (ii) $$Y_i = \hat{\beta}_0 + \hat{\beta}_1 x_{1i} + \hat{u}_i$$ i.e. we omit $$x_2$$ (possibly because this variable is not available). The OLS formula for $$\hat{\beta}_1$$ is: $$\hat{\beta}_1 = \frac{\sum_i(x_{1i}-\bar{x_1})y_i}{\sum_i(x_{1i}-\bar{x_1})^2}$$ Substituting the true model (i) into the estimated one (ii) we have: $$\hat{\beta}_1 = \frac{\sum_i(x_{1i}-\bar{x_1})(\beta_0 + \beta_1 x_{1i} + \beta_2 x_{2i} + u_i)}{\sum_i(x_{1i}-\bar{x_1})^2} $$ $$ = \frac{\beta_0 \sum_i(x_{1i}-\bar{x_1})}{\sum_i(x_{1i}-\bar{x_1})^2} + \frac{\beta_1 \sum_i(x_{1i}-\bar{x_1}) x_{1i}}{\sum_i(x_{1i}-\bar{x_1})^2} + \frac{\beta_2 \sum_i(x_{1i}-\bar{x_1}) x_{2i}}{\sum_i(x_{1i}-\bar{x_1})^2} + \frac{\sum_i(x_{1i}-\bar{x_1}) u_{i}}{\sum_i(x_{1i}-\bar{x_1})^2} $$ $$ \beta_1 + \beta_2 corr(x_1, x_2) + \frac{\sum_i(x_{1i}-\bar{x_1}) u_{i}}{\sum_i(x_{1i}-\bar{x_1})^2} $$ Taking the conditional expectation of $$\hat{\beta}_1$$: $$ E[\hat{\beta}_1| x_1] = \beta_1 + \beta_2 corr(x_1, x_2) + \frac{\sum_i(x_{1i}-\bar{x_1}) E[u_{i} | x_1]}{\sum_i(x_{1i}-\bar{x_1})^2} = \beta_1 + \beta_2 corr(x_1, x_2) $$ So, the expectation of estimated $$\hat{\beta}_1 \neq \beta_1 $$ and the bias is given by the difference between the estimated and the true population parameter: $$Bias = \hat{\beta}_1 - \beta_1 = \beta_2 corr(x_1, x_2)$$ . Interestingly, there will be no bias in two situations: when $$\beta_2 = 0$$, or when $$corr(x_1, x_2) = 0$$. However, in both situations one can say that $$x_2$$ is not a relevant variable to estimate $$\beta_1$$. If the assumption of "exogenous explanatory variables" does not hold, one can use more advanced models such as Instrument Variables or GMM to obtain consistent estimates on large samples (however, bias will still be present in small samples). Note: Wooldridge's Introductory Econometrics is an excellent textbook for further details.

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