Assignment 3

Answers for these questions come from both the course content and the readings. Each question is worth one point, for a total of 20 points. Passing this module requires at least 65% (13 out of 20 correct). Email your answers to one of the course instructors (either Steve Martin or Shaoxiong Wang or Anastasiia Nosach) when you are finished.

Question 1 True or false: The time-dummy hedonic index is a biased estimator of the constant-quality geometric index.

Question 2 True or false: A hedonic price index requires no assumptions to be a constant-quality index once the characteristics of the goods being sold are known.

Question 3 A linear hedonic index equals the Jevons price index in which circumstances.

When the average characteristics of goods selling in period 1 equals the average characteristics of goods selling in period 0.
When the overlap conditions is satisfied.
When the parameters in the hedonic model do not change over time.
a and c.
None of the above.

Question 4 Which quality adjustment requires the greatest number of assumptions?

Stratification.
Hedonics using the characteristics approach.
Hedonics using the time-dummy approach.
Hedonics using the double imputation approach.

Question 5 True or false: A repeat-sale index can be motivated/derived as a type of hedonic index.

Question 6 The size of container for salmon cat food decreased between period 0 and period 1 in the examples with R for hedonic price indices. If the other types of cat food stayed the same, what can be said about the hedonic indices?

They overstate the decrease in price between period 0 and period 1.
They understate the increase in price between period 0 and period 1.
They overstate the increase in price between period 0 and period 1.
The hedonic model understates the increase and the stratified model overstates the increase.
None of the above.

The next four questions have to do with the following type of quality adjustment. A common strategy to build a constant-quality index when packaging for a product changes is to divide the price by the size of the package (e.g., price per liter of dish soap), so that the price relative for product \(i\) becomes \(p_{i1} / p_{i0} \times s_{i0} / s_{i1}\), where \(s_{it}\) is the size of the package in period \(t\). An index based on this approach can be calculated with a linear regression, as

\[\begin{align*} E(\rho | i, s, t) = \alpha_i + t \log(I) + \log(s). \end{align*}\]

Question 7 True or false: Adjusting price by package size is a type of time-dummy hedonic index.

Question 8 What is the conditional independence assumption that makes this quality adjustment work?

The only difference between the same product in period 0 and period 1 is the package size.
Products in period 1 are a good counterfactual for products in period 0.
Each product sells in both periods.
The only difference between products over time is due to package size.
Conditional independence is not needed.

Question 9 What extra assumption is made so that simply dividing price by package size adjusts for changes in quality, rather than needing to estimating the hedonic model?

The parameters in the hedonic model do not change over time.
The error is the hedonic model is homoskedastic.
The elasticity between price and package size is one.
No extra assumptions are made.

Question 10 What is the overlap assumption that is required to calculate this index?

A transaction must occur for every product-package size combination in both period 0 and period 1.
Each product needs to sell in both periods.
Each product needs to sell in both periods and package size must change for at least one product.
No overlap assumption is needed.

Question 11 True or false: The time-dummy and hedonic imputation price indices always give different values.

The next two questions deal with a variation on the time-dummy hedonic index based on conditional medians, so that \(\text{med}(\rho | X, t) = \alpha_{0} + t (\alpha_{1} - \alpha_{0}) + X \beta\). With conditional independence, \(\alpha_{1} - \alpha_{0} = \log(\text{med}(p(1)) / \text{med}(p(0)))\).

Question 12 Why is the index from the median time-dummy index a poor measure for a constant-quality index?

It compares what the median good in period 1 would sell for in period 1 to what the median good in period 0 would sell for in period 0, and hence does not compare the same goods over time.
It is a biased estimator of a constant-quality index.
It is not possible to construct a maximum likelihood estimator for an index based on medians.
It does not satisfy the monotonicity axiom from the axiomatic approach to price indices.
There is nothing wrong with the median time-dummy index.

Question 13 True or false: As the median is less sensitive to outliers than the mean, the median time-dummy index finds application in settings where there may be outlier price observations.

The next four questions use the following data on transactions for two products over two periods.

Period	Product	Price 1	Price 2	Price 3
0	A	2	2	…
1	A	1	2	…
0	B	4	…	…
1	B	2	4	8

Question 14 What is the value of the Jevons index when all the price information is pooled together for both periods?

100
98.3
104.7
106.4
89.8

Question 15 What is the value of the stratified index, where each product belongs to a separate stratum and all strata have equal weight, calculated using a Jevons index?

104.7
98.3
84.1
110.6
101.3

Question 16 Why does the stratified index show a decrease in price while the pooled index shows an increase in price?

The pooled index is a constant-quality index, whereas the stratified index is biased downwards.
The pooled index uses all the price data, whereas the stratified index does not.
Product B sells for a higher price than product A, and relatively more product B sold in period 2, but only product A changed price in period 2.
None of the above.

Question 17 True or false: If each product does not change over time, the stratified index is a constant- quality index.

Question 18 True or false: The repeat-sales index is a type of matched-model index.

Question 19 True or false: It is possible to mix stratification with hedonics to relax overlap.

Question 20 It can be shown that \[\begin{align*} \underbrace{E(\rho | t = 1) - E(\rho | t = 0)}_{\text{transaction-price index}} =& \underbrace{E(\rho(1) | t = 0) - E(\rho(0) | t = 0)}_{\text{Laspeyres-like constant-quality index}} \\ &+ \underbrace{E(\rho(1) | t = 1) - E(\rho(1) | t = 0)}_{\text{selection bias}}. \end{align*}\]

True or false: Would potential prices in period 0 being correlated with time, such that goods that sell in period 1 would have sold for less in period 0 than the goods that sold period 0, affect using transaction prices to identify a Laspeyres-like constant-quality index?