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ECMT6007/6702: Econometric Applications Problem Set 7
发布时间：2022-11-18
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ECMT6007/6702: Econometric Applications
Problem Set 7
Semester 2 2022
Question 1.
Let hy6t denote the three-month holding yield (in per cent) from buying a six-month US T-bill at time (t − 1) and selling it at time t (three months hence) as a three-month US T-bill. Let hy3t − 1 be the three-month holding yield from buying a three-month US T-bill at time (t − 1). At time (t − 1), hy3t − 1 is known, whereas hy6t is unknown because the price of three-month US T-bills, p3t, is unknown at time (t − 1). The expectations hypothesis (EH) says that these two different three-month investments should be the same, on average. Mathematically, we can write this as a conditional expectation:
E (hy6t | It − 1 ) = hy3t − 1
where It − 1 denotes all observable information up through time t − 1. This suggests estimating the model:
hy6t = β0 + β1 hy3t − 1 + ut
and testing H0 : β 1 = 1 (we can also test β0 = 0, but we often want to allow for a term premium in buying assets with different maturities, so that β0 
0).
(i) Estimating the previous equation by OLS using the data in INTQRT .RAW (spaced every three months) gives:
h一y6t = −0.058 + 1.104 hy3t − 1
(0.070) (0.039)
T = 123, R2 = 0.866
Do you reject H0 : β 1 = 1 against H1 : β 1 
1 at the 1% significance level? Does the estimate seem practically different from one?
(ii) Another implication of the EH is that no other variables dated t − 1 or earlier should help explain hy6t once hy3t − 1 has been controlled for. Including one lag of the spread between six-month and three-month US T-bill rates gives:
h一y6t = −0.123 + 1.053 hy3t − 1 + 0.480 (r6t − 1 − r3t − 1 ) (0.067) (0.039) (0.109)
T = 123, R2 = 0.885
Now, is the coefficient on hy3t − 1 statistically different from one? Is the lagged spread term significant? According to this equation, if, at the time of t − 1, r6 is above r3, should you invest in six-month or three-month US T-bills?
(iii) The sample correlation between hy3t and hy3t − 1 is 0.914. Why might this raise some concerns with the previous analysis?
Question 2. Computer Exercise: Housing Investment
Use the data in hseinv7 .dta for this exercise.
(i) Report mean, minimum and maximum values and standard deviation of the variables in the data set.
(ii) Find the first order autocorrelation in log (invpc). Now find the first order autocorrelation after linearly detrending log (invpc). Do the same for log (price). Based on the sample auto- correlations, do you suspect either of the two series have a unit root?
[Hint: To linearly detrend the series log (invpc) run the simple regression of log(invpc) on a time trend t and keep the residual from this regression. The residuals represent the detrended series]
(iii) Carry out a formal Dickey-Fuller test of the null hypothesis that log (invpc) has a unit root
(against the alternative it is weakly dependent) using a 10% significance level. Do the tests for the case with and without a linear trend in log (invpc).
(iv) Repeat the Dickey-Fuller tests for the null that log (price) has a unit root using a 10% signifi- cance level. What do you conclude?
log (invpct) = β0 + β1 ∆ log (pricet) + β2 t + ut (1)
and report the results in the usual form. Interpret the coefficient βˆ1 and determine whether it is statistically significant (at the 5% level).
(vi) Linearly detrend log (invpct) and use the detrended version as the dependent variable in the
regression model (1) in part (v). What happens to the R2?
(vii) Now use ∆ log(invpct) as the dependent variable in the model:
∆ log (invpct) = β0 + β1 ∆ log (pricet) + β2 t + ut (2)
How do your results change from part (v)? Is the trend still significant (at the 5% level)? Why or why not? Explain.
Note: The dataset hseinv7 .dta can be downloaded from the course Canvas site. The data are an- nual observations from 1947 to 1988. The dataset has 42 observations and 4 variables. The variables
correspond to:
• year: year of the observation
• t: time index (t = 1 for the 1st observation, up to t = 42 for the last observation)
• linvpc: log(invpc), the log of real annual housing investment per capita
• lprice: log(price), the log of housing price index
