Question 1: A distributed-lag model. To study the relationship between real housing investment per capita (pcinvt) and housing prices (pricet), I collect 42 years of annual data on each time series. I begin by considering the following model, with the variables in logarithms:
log(pcinvt) = β0 + β1 log(pricet) + β2 log(pricet−1) + β3t + ut.
Based on my sample, I get the following estimates and standard errors: βˆ0 = −1.086 (0.115), βˆ1 = 3.259 (0.960), βˆ2 = −4.487 (0.959), βˆ3 = 0.013 (0.003),
n = 41, R2 = 0.5641, R¯2 = 0.5287.
∆log(pricet) = δ0 + δ1 log(pricet−1) + vt
and find that δˆ0 = −0.0018 (0.0056), δˆ1 = −0.0661 (0.0248), n = 41, R2 = 0.0435, R¯2 = 0.0190.
Test whether a unit root is present.
Question 2: Endogeneity. In this question, we are interested in studying the effect of a job seekers’ training programme that was offered, five years ago, to people who had been unemployed for at least a year at that time. This year, we interviewed the people to whom this training had been offered, and collected the following data on each of them: whether they are currently employed (yi = 1) or not (yi = 0); whether they participated in the training programme (parti = 1) or not (parti = 0); their gender (femalei = 1 or 0), how many years of education they completed before being offered this training (educi), and which state or territory they lived in five years ago (eight dummies, making the ACT the omitted category as usual − sorry, nothing personal). We then estimate the following model:
yi = β0 + β1parti + β2femalei + β3educi
+β4NSWi + β5V ICi + β6QLDi + β7WAi + β8SAi + β9TASi + β10NTi + ui.
For the remainder of this question, assume that parti is suspected to be endogenous, and all other regressors can safely be assumed to be exogenous. This is not necessarily the correct answer to part (a), but it does make things easier.
Question 3: A system of equations. Consider the following simplified description of fresh tuna sales on Sydney’s fish market. Consumers decide how much tuna they want to buy on a day (y1, in kilograms) given the price that they observe (y2, in dollars per kilogram) and some other exogenous information x1. Conversely, sellers set the price y2 based on the demand that they observe y1, and some other exogenous information x2. (Assume that this market is so efficient that, on any given day, all sellers charge the same price.) Thus, we are dealing with a system of simultaneous equations:
We are interested in estimating all six parameters of this model.
From here on, we will no longer be using 2SLS. We will rewrite the structural form that is given above into its reduced form instead, estimate that reduced form, and hope that we can recover the structuralform parameters from our reduced-form estimators.
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