Online Quiz 4
Total: 20 Marks
Half (or 0.5) marks are the minimum unit and the correction in two digits is required in the answers. No mark if the answer cannot be identified.
IV Regression
We want to estimate a supply equation for young construction workers in Australia: π»ππ’πi =
π½) + π½+ππππ0 + π½1πΈππ’π0 + π½5π΄ππ0 + π’i, in which π»ππ’πi is the supply of labour, ππππ0 is hourly wage, and πΈππ’π0 is years of education. Suppose large outliers are unlikely. All variables are i.i.d. draws from their joint distribution and have nonzero finite fourth moments.
(1, 1 mark) Explain why this supply equation cannot be consistently estimated by OLS regression.
(2, 1 mark) One of your friends argued that πΈπ₯πππ0 and its square, πΈπ₯πππ1, to be instruments for ππππ0. Explain how these variables satisfy the condition of instruments.
(3, 1 mark) How to check weak instruments in this study? (4, 1 mark) Is the supply equation identified? Explain.
(5, 1 mark) Can we statistically test the assumption that the instruments are exogenous in this study?
(6, 1 mark) Describe the steps (or STATA command) you would take to obtain IV regression.
(7, 1 mark) If the two conditions for a valid instrument hold, what is the assumption for your IV regression.
(8, 1 mark) Suppose π¦0 = π½+ + π½1 π₯0 + π’0 and π§0 is a valid instrumental variable for the endogenous variable π₯0. π’0 is the error term. Show that πΈ(π¦0 |π§0) = π½+ + π½1 πΈ(π₯0|π§0) if
πΈ(π’0|π§0) = 0.
(9, 1 mark) Show π½1 could be expressed as a fraction of the conditional expectation in (8) for the two cases with π§0 = 1 and π§0 = 0.
(10, 1 mark) Explain how πΈ(π₯0|π§0 = 1) − πΈ(π₯0|π§0 = 0) could be viewed as a measure of the strength of the instrumental variable π§0.
The condition π§0 = 1 defines a subset of the population, and the sample average of the subset of observations (i.e. π¦A+) is a consistent estimator for the population average, πΈ(π¦0|π§0 = 1). Replacing each element by its consistent estimator gives us the Wald Estimator (π½BπΆπ΄πΏπ· =
π¦AH–π¦AJ), in honour of Abraham Wald.
π₯Μ H–π₯Μ J
Time Series
Suppose we have a stationary process: π¦π‘ = π½) + π½+ π¦π‘–+ + π’π‘ and π’π‘ follows the standard normal distribution.
(11, 1 mark) Explain what is the meaning of stationarity.
(12, 2 marks) Show the expected value and variance of π¦π‘.
(13, 1 mark) π 1 is always increased whenever we include the lags and can we include the lags as much as possible?
(14, 1 mark) How to choose the number of lags π in an π΄π (π)?
(15, 1 mark) Are the forecasts from the time series model the OLS predicted values? Why?
(16, 2 marks) Compute the 1st and 2nd autocovariance of π¦π‘. (17, 1 marks) Compute the 1st and 2nd autocorrelation of π¦π‘. (18, 1 mark) What is the difference between BIC and AIC?
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