ECO 365 University of Michigan Introduction to Econometrics Problems

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ECO 365

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Problem 1: Nonstationarity—the Basics

1a. Dene stationarity. Note: You can dene it using math or words (or both).

1b. If our disturbance term follows a random walk, i.e., then it's variance is . Explain how this expression of its variance shows that the disturbance is nonstationary (i.e., it violates stationarity).

1c. We previously discussed autocorrelated distrubances, e.g., an AR(1) process such that Under which circumstances would this AR(1) process become a random walk

This is just part of the assignment. For detailed assignment information. need check the attached file and go to website link.

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Problem Set 4 Nonstationarity, Causality, Instrumental Variables EC 421: Introduction to Econometrics Due before midnight (11:59pm) on Wednesday, November 25, 200 DUE Your solutions to this problem set are due before midnight on Wednesday, November 25, 2020. Your les must be uploaded to Canvas. Your answers must include two les (1) your responses to the question (e.g., a Word document) and (2) the R script you used to generate any answers in R. You can instead include a PDF le produced with RMarkdown that includes both your responses and your R script. Each student must turn in their own answers. OBJECTIVE This problem set has three purposes: (1) reinforce econometrics topics from class; (2) build your R toolset; (3) strengthen your intuition on causality and time series. Problem 1: Nonstationarity—the Basics 1a. De ne stationarity. Note: You can de ne it using math or words (or both). 1b. If our disturbance term ut follows a random walk, i.e., ut = ut−1 + εt 2 then it's variance is Var(ut ) = tσε . Explain how this expression of its variance shows that the disturbance is nonstationary (i.e., it violates stationarity). 1c. We previously discussed autocorrelated distrubances, e.g., an AR(1) process such that ut = ρut−1 + εt Under which circumstances would this AR(1) process become a random walk? 2/5 Problem 2: Nonstationarity—the Simulation In this problem, we are going to create two independent, nonstationary time series. Speci cally, we'll create two random walks. Then, we'll regress the rst random walk on the second random walk. Hint: Generating random walks is nearly identical to generating AR(1) processes, as you did in lab. 2a. (2pts) Generate the rst 40-period random walk. We will name it m . mt = mt−1 + εt where εt comes from a normal distribution with mean 0 and standard deviation 1. Here is some R to help. # Set a seed (so your results stay the same) set.seed(1234) # Generate the initial number, (this will be v[1]) m rnorm(1, mean = 0, sd = 1) # For loop to create the random walk for (t in 2:40) { # Create the 'next' observation } while you're lling in the for loop, keep in mind (1) our equation for the random walk at the beginning of this question (meaning mt depends upon mt−1 and εt ) and (2) the fact that you can reference different observations in R, e.g., m[t] refers to the tth observation m[t-1] refers to the (t − 1)th observation m[3] refers to the 3rd observation If you need more help on for loops, don't forget there are lab materials on Canvas and resources online (e.g., datamentor.io and datacamp.com have lots of resources). 2b. (2pts) Generate a second 40-period random walk called n . This part is exactly the same as (2a), but you use a different seed (i.e., set.seed(456) ) and name the variable n . 2c. (2pts) We independently generated these two time series. Ideally (from a statistical point of view), should we nd a statistically signi cant relationship between the two series? Explain. 2d. (2pts) Regress m on n . Report the results from the t test. Do they match your expectations from (2c)? 3/5 Problem 3: Causality Following the causal model seen in class, imagine that we observe the following data (which would be impossible to observe in real life): Table: Imaginary dataset i Trt. y_1 y_0 1 0 21 11 2 0 19 17 3 1 7 3 4 1 9 7 3a. (2pts) Calculate the treatment effect for each individaul (i.e., τi ). 3b. (1pts) [T/F] The treatment effect is constant across individuals. 3c. (2pts) Calculate the average treatment effect. 3d (2pts) Estimate the average treatment effect by comparing the mean of the treatment group to the mean of the control group. 3e. (2pts) Should we expect our estimator in (3d) to provide unbiased estimates? Explain. 3f. (2pts) What is the fundametal problem of causal inference? 4/5 Problem 4: Instrumental Variables Let's return to our question of the returns to education. Speci cally, we will use the dataset wages.csv , which .† We're interested in estimating β1 in Wage i = β0 + β1 Educationi + ui but we have a problem with omitted-variable bias. Instrumental variables can potentially help. 4a. (0pts) Load and inspect the dataset. 4b. (2pts) What are the two requirements for a valid instrument? 4c. (2pts) We need an instrument for (endogenous) education. Do you think the variable n_kids —the number of children—would be a valid instrument? Explain why it passes/fails ech of the two requirements for a valid instrument. 4d. (2pts) We can test the relevance of our instrument by estimating the rst stage, i.e., regressing our endogenous variable education on our (potential) instrument n_kids. Do this. Is there evidence that our potential instrument is relevant? Explain using a statistical test and interpret the coef cient. 4e. (4pts) Let's assume number of children is a valid instrument for education. Using the number of children ( n_kids ) as an instrument for education ( education ), estimate the returns to education via instrumental variables (IV). Interpret the coef cient that gives the returns to education and its signi cance. Hint: Recall that we can use iv_robust(y ~ x | z, data) from the estimatr package to get IV/2SLS estimates of the effect of x on y with the instrument z (and dataset data ). 4f. (2pts) Would you trust this estimate of the returns to education? Why? These data come from wage1 in the wooldridge package. I took a subset of variables and renamed them. 5/5
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EC 421 Introduction to Econom...


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Excellent! Definitely coming back for more study materials.

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