# Introduction to Econometrics

## Transcribed Text

Intro to Econometrics 1. For two generic random variables Z. and W, the Law of Iterated Expectations (LIE) says that Ez(E|W;|Zi] E(Z) You are given the following regression model: and you are told that 0. a) Does A1 hold? b) Is 3 the causal effect of X; on Y,! c) Show that E(a) = 0. Interpret what it means. d) Show that Cor(a), X1) 0. Interpret what it means. 2. Consider two randorn variables T; and G. To answer the following questions a simple sketch is enough. a) Draw n scatter plot such that Gi) is positive and R² is high and close to 1. b) Draw n scatter plot such that Cov(T). Gi) is positive and R² is low. c) Draw n scatter plot such that Cov(T).Gi) is negative and R² is high and close to 1. d) Draw a scatter plot such that Cov(T¿.G, = 0. What would be the value of the R² in this case? Explain in detail. (Hinf: the the OLS estimator for the slope of a simple linear regression to ansaber) 3. Suppose that the true relationship between the variables Y1 and X, is given by: with 14, - W; Zis where W;, Z; are additional variables that you are ignoring and that end up in the error term. a) Suppose that Cov(Xi, W;) > 0 and that Cov(Xi, Zi) > 0. Is it possible that A1 holds in this situation? Please explain why using both formality and intuition. b) Suppose that Cov(Xi,W;) > 0 and that Coo(Xi, Zi) < 0. Is it possible that the regressor is uncorrelated with the error term? Can you say precisely when this is the case? c) Does =0 imply that A1 holds?

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