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..Statistics & Analysis

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.

  • (2 marks) Explain carefully how the estimate βˆ1 = 3.259 should be interpreted, in economic terms.
  • (5 marks) Compute the long-run propensity (LRP) of prices on investment, and also explain how this number should be interpreted.
  • (5 marks) The results shown above are not sufficient to test whether the LRP is zero. Describe which additional regression you would need to run, and which statistical test you would perform based on the results of this additional regression.
  • (2 marks) If housing prices would have a unit root, how would you need to modify your regression model in order to obtain consistent estimators?
  • (6 marks) We are concerned that housing prices might have a unit root, so we estimate the model

∆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.

  • (4 marks) For each regressor in this model, briefly argue whether you think it is endogenous or exogenous, and why.

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.

  • (4 marks) If you estimated the regression model by OLS, do you think βˆ1 would be biased towards zero or away from zero? Why?
  • (4 marks) One way to mitigate endogeneity problems is to use a proxy variable. Pretending that you were in charge of data collection, think of something that could be a good proxy variable, and describe why it is a good choice.
  • (4 marks) Another way to mitigate endogeneity problems is to use an instrumental variable. Pretending that you were in charge of data collection, think of something that could be a good instrumental variable, and describe why it is a good choice.
  • (4 marks) Assume that OLS estimation resulted in βˆ1 = 0.10 with standard error 0.04, and IV estimation resulted in βˆ1 being either 0.17 or 0.03 (whichever is consistent with your answer in part (b)) with standard error 0.05. Test whether endogeneity was actually a problem in this model.
  • 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.

    • (4 marks) Think of some observable variables that x1 and x2 could be (one suggestion for x1 and one suggestion for x2 is enough), and justify your answer.
    • (4 marks) One way to estimate these six parameters is by using two-stage least squares. Describe in detail which regressions you would need to run.

    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.

    • (6 marks) Find the reduced form of this system of equations.

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