However, the first-stage equation of the 2SLS regres-sion is materially different from a reduced-form pricing equation in that we do notneed to have all the cost data: we really only nee
Trang 1Table 9.2. IV estimation results based on forty-four observations.
RobustRegressors Coefficient std err t P > jt j [95% Conf interval]ln.PSugar/ 0.27 0.08 3.41 0.00 [0.43 0.11]
R 2 D 0:80 The dependent variable in this regression is ln.Q Sugar /.
are a cost of producing sugar and will therefore ordinarily affect observed pricesaccording to economic theory (and also farmers!) On the other hand, given thatfarmers are a small minority of the population and that the increase in their wages
is not likely to translate into material increases in sugar consumption, farm wagesare unlikely to materially affect the aggregate demand for sugar
The 2SLS estimation proceeds in two stages:
1st-stage regression: ln Pt D a b ln WtC 1q1tC 2q2t C 3q3tC "t;2nd-stage regression: ln Qt D a bln PbtC 1q1tC 2q2tC 3q3tC vt;where Wt is the farm wage at time t andln Pbt is the estimated log of price ob-tained from the first-stage regression Most statistical computer packages are able toperform this procedure and in doing so provide the output from both regressions.8The quarterly dummies are also included in the first-stage regression since therequirement for an instrument to be valid is that it is correlated with an endogenousvariable conditional on the included exogenous variables Demand is itself seasonal,
so that the quarterly dummies are not correlated with prices conditional on theincluded exogenous variables and hence are not valid instruments for prices, even
if they are valid instruments for themselves, i.e., can be treated as exogenous.The results of the instrumental variable estimation are presented in table 9.2.The results show a lower coefficient for the price variable The elasticity of demand
is now 0:27 and is below the previous OLS estimate Because of data availability
on farm wages some observations had to be dropped so that the data for the tworegressions are not exactly the same Nonetheless, formally a Durbin–Wu–Hausmantest could be used to test between the OLS and IV regression specifications (seeGreene 2000; Nakamura and Nakamura 1981) The central question is whether theinstruments are in fact successfully addressing the endogeneity bias problem thatmotivated our use of them Often inexperienced researchers use IV regression resultseven if the resulting estimate moves the coefficient in the direction opposite to thatexpected as a result of endogeneity bias
8 STATA, for example, provides the “ivreg” command.
Trang 2Results in IV estimations should be carefully scrutinized because they will only bereliable if the instrument chosen for the first-stage regression is a good instrument.
We know that for an instrument to be valid it must satisfy the two conditions:
t j Xt; Wt/ D 0 and (ii) EŒln.Pt/ j Xt; Wt/ ¤ 0;
where in our case Xt D 1; q1; q2; q3/ are the exogenous regressors in the demandequation and Wt is the instrument, farm wages As we described earlier, the first
of these conditions is difficult to test; however, one way to evaluate whether itholds is to examine a picture of the estimated residuals against the regressors Weshould see no systematic patterns in the graphs—whatever the value of Xt or Wtthe error term on average around those values should be mean zero Such tests can
be formalized (see, for example, the specification tests due to Ramsey (1969)) Butthere are limits to the extent to which this assumption can be tested since the modelwill, to a considerable extent, actively impose this assumption on the data in order tobest derive the IV estimates obtained A variety of potential IV results can certainly
be tested against each other and against specifications which use more instrumentsthan strictly necessary to achieve identification But the reality is that the first ofthese assumptions is ultimately quite difficult to test entirely convincingly and one
is likely to ultimately mainly rely on economic theory—at least to the extent thatthe theory robustly tells us that, for example, a cost driver will generally not affectconsumer demand behavior and so will have no reason to be correlated with theunobserved component of demand
The second condition is easier to evaluate and the most popular method is to run aregression of the potentially endogenous variable (here ln.Pt/) on all the exogenousexplanatory variables in the demand equation and also the instruments, here ln.Wt/
To see whether the second condition holds, we examine the results of the following
“first-stage” regression:
PtD a b ln WtC q1C q2C q3C "t:For the variable farm wage to be a good instrument, we want the coefficient b to
be robustly and significantly different from zero in this equation If the instrumentdoes not have explanatory power in predicting the price, the predicted price used inthe second-stage regression will be poorly correlated with the actual price given theother variables already included in the demand equation In that case, the estimatedcoefficient of the price variable in the second-stage regression will be impreciselyestimated and indeed may not be distinguishable from zero Even with “good instru-ments” in the sense that they are conditionally correlated with the variable beinginstrumented, we will expect the coefficient of an instrumented variable in an IVregression to be less precisely estimated (have a higher standard error) than the analo-gous coefficient estimated using OLS (with the latter a meaningful comparison only
if in fact the OLS estimate is a valid one) IV estimation relaxes the assumptions
Trang 3Table 9.3. First-stage regression results.
ln.Price/ Coefficient Std err t P > jt j [95% conf interval]
is significant and has a high t -statistic, indicating that it is precisely estimated.However, note that the coefficient reported is rather surprising: its sign is negative!The economic theory motivating our choice of instrument tells us that an increase incosts should translate into higher prices as the supply curve shifts leftward Indeed,our aim in selecting an instrument is intuitively to use that instrument to allow us touse only the variation in observed prices which we know is due to the variation inthe supply curve
When conundrums such as this one arise, one should address them While foreconometric theory purposes “conditional correlation” is all that is required to sup-port the use of the instrument, one should not proceed further without understandingwhy the data are behaving in an unexpected way In this case, one may want to inves-tigate, for example, whether farm wages have a trend that negatively correlates withprices and for which we did not control Figure 9.2 graphs farm wage data overtime In particular, note that there is an upward trend in wages between 1995 and
2006 during the time that sugar prices fell Clearly, while farm wages may still be
an important determinant of sugar prices, they are not likely to be a major factordriving the price of sugar down We must look elsewhere for an instrument thathelps explain the major source of variation in prices conditional on the exogenousvariables in the demand equation
To search for a good instrument we must attempt to better understand the factorsthat are driving sugar prices down over time One possibility is that other costs
in the industry are falling dramatically Alternatively, there may be an institutionalreason such as changes in the amount of subsidy offered to farmers or in the tariff orquota system that governs the supply from imports There may have been substantialentry during the period Another possibility is that the price change may be driven
by important demand factors that we have omitted thus far from our model Perhapsthe taste for sugary products changed over time? Or perhaps substitutes (e.g., high-fructose corn syrup) appeared and drove down prices?
Trang 4Figure 9.2. Farm wages plotted over time.
At this point we need to go back to our industry experts and descriptive analyses ofthe industry to attempt to find possible explanations for the major variation in pricesand in particular the price decline The data and regression results have provided
us with a puzzle which we need to solve by using industry expertise Only once wehave thought hard about what in the industry is generating our data will we generally
be able to move forward to generating econometric results we can believe It is forthis reason that we described in the introduction to this section that it is very rare for
an econometrician to be able to work in isolation late into the night with her existingdata set and generate sensible regression results without going back to think aboutthe nature and drivers of competition in an industry
That said, we generally do not need to understand everything about the setting process to obtain reliable demand estimates In particular, in a demand esti-mation exercise we are not trying to estimate the pricing equations that explain howfirms optimally choose their prices Although the first-stage regression in a 2SLSestimation may closely resemble the reduced form of the pricing equation in a struc-tural model of prices and quantities (see chapter 6), it is not quite the same We saw
price-in the previous chapters that factors affectprice-ing demand are price-included price-in the pricprice-ingequation and so are cost data However, the first-stage equation of the 2SLS regres-sion is materially different from a reduced-form pricing equation in that we do notneed to have all the cost data: we really only need one good supply-side instrument
to identify the price coefficient in a homogeneous product demand equation
9.1.2 Differentiated Products Demand Systems
Most markets do not consist of a single homogeneous product but are rather posed of similar but differentiated goods that compete for customers For instance,
com-in the market for shampoos there is not a scom-ingle type of generic shampoo Ratherthere is a variety of brands and types of shampoo which consumers do not consider
Trang 5absolutely equivalent We must take such demand characteristics into account whenattempting to estimate demand in differentiated product markets In particular, weneed to take account of the fact that consumers are choosing among different prod-ucts for which they have different relative preferences and which will usually havedifferent prices Differentiated product demand systems are therefore estimated as
a system of individual product demand equations, where the demand for a productdepends on its own price but also on the price of the other products in the market
9.1.2.1 Log-Linear Demand Models
One popular differentiated product demand system is the log-linear demand system,which is simply a set of log-linear demand functions, one for each product available
in the market We label the products in the market j D 1; : : : ; J In each case, thequantity of the good purchased potentially depends on the prices of all the goods inthe market and also income y (Deaton and Muellbauer 1980b) Formally, we havethe following system of J equations:
ln Q1t D a1 b11ln P1tC b12ln P2t C :::: C b1Jln P1J C 1ln yt 1t;
ln Q2t D a2 b21ln P1tC b22ln P2t C C b2Jln PJ tC 2ln yt 2t;::
:
ln QJ t D a2 bJ1ln P1tC bJ 2ln P2tC C bJJln PJ tC Jln yt J t:Maximizing utility subject to a budget constraint will generically provide demandequations which depend on the set of all prices and income (see, for example, Pollakand Wales 1992) Clearly, with aggregate data we might use aggregate income as therelevant variable for the demand equations (e.g., GDP) However, since many studiesfocus on a particular sector of the economy, the consumer’s problem is often recastand considered as a two-stage problem At the first stage, we posit that consumersdecide how much money to spend on a category of goods—for example, beer—and
at the second stage we posit that the chosen level of expenditure is allocated acrossthe various products that the consumer must choose between, perhaps the differentbrands of beer Under particular assumptions on the shape of the utility function,this two-stage process can be shown to be equivalent to solving a single one-stageutility-maximization problem (see Deaton and Muellbauer 1980b; Gorman 1959;Hausman et al 1994) Using the two-stage interpretation, “expenditure” may beused instead of income in the demand equations but the demand equations will then
be termed “conditional” demand equations as we are conditioning on a given level
of expenditure
A well-known example of an such an exercise is Hausman et al (1994) In fact,those authors estimate a three-level choice model where consumers choose (1) thelevel of expenditure on beer, (2) how to allocate that expenditure between threebroad categories of beer (respectively termed premium beer, popular beer, and light
Trang 6Table 9.4. Market segment conditional demand in the market for beer.
Premium Popular Light
(0.283) (0.560) (0.377)
(0.011) (0.022) (0.015)log(PPremium) 2.671 2.704 0.424
(0.123) (0.244) (0.166)log(PPopular) 0.510 2.707 0.747
(0.016) (0.034) (0.023)Number of observations = 101
Source: Table 1, Hausman et al (1994).
beer) which marketing studies had identified as market segments, and (3) how toallocate expenditure between the various brands of beer within each of the segments
At level (3), we could use the observed product level price and quantity data toestimate our differentiated product demand system However, in fact, since level (3)
is modeled as a choice of brands (e.g., Coors, Budweiser, Molsen, etc.), at levels(1), (2), and (3) we would need to use price and quantity indices constructed fromunderlying product-level data to give measures of price and quantity for each of thebrands or segments of the beer industry For example, we might use a price index withexpenditure share weights for the underlying prices within each segment s to produce
a segment-level price index, Pst DP
jwjtpjt.9Similarly, we might choose to usevolumes of liquid to help aggregate over the brands to give segment-level quantityindices.10
Estimates of the second level of their demand system using price and quantityindices are shown in table 9.4 At the second level of the choice tree, the demandsystem is a conditional demand system because the amount of money to be spent
on beer has already been chosen at stage 1
9 Expenditure shares can be defined as w j t D pj tqj t= P
j pj tqj t, where p represents prices and
q quantities.
10 Formally, Deaton and Muellbauer (1980b) show that there are “correct” price and quantity indices which can be constructed for this process to preserve the multilevel models’ equivalence to a single utility-maximization problem (under strong assumptions) In practice, the authors do not seem to have settled on a universally best choice of price and quantity indices.
Trang 7Since we are dealing with a log-linear model, the bjjcoefficients provide estimates
of the own-price elasticity of demand while the bj k(j ¤ k) parameters provideestimates of the cross-price elasticities of demand If we are using segment-leveldata, we must be careful to place the correct interpretation on the elasticities For
example, the results from table 9.4 suggest that the own-price elasticity of segment
demand is 2:6 for premium beer, 2:7 for popular beer, and 2:4 for light beer.These price elasticities could be used as important evidence toward a formal test
of the hypothesis that each beer segment is a market in itself by performing a SSNIPtest That said, generally, the price elasticity relevant for such a test would includethe indirect effect of prices through their effect on the total amount of expenditure
on beer If the price of premium beer goes up, some consumption will be reallocated
to other beer segments but the total consumption of beer might also fall as peopleeither switch to other products such as wine or reduce consumption altogether.The elasticities we can read off from the equation in this instance are conditionalelasticity estimates—they are conditional on the level of expenditure on beer Thusfor market definition, if we use expenditure levels and price indices to performmarket definition tests, we must be careful to trace through the effect of a pricechange back through its effect on total expenditure on beer To do so, Hausman et
al (1994) also estimates a single top-level equation so that the demand for beer
in total is expressed as a function of prices and income In this case, the equationestimated depended on income (GDP) and also a price index constructed to capturethe general price of beer as well as demographics, Zt:
ln QBeert D ˇ0C ˇ1ln yGDPt C ˇ2ln PtBeerC Ztı C "t:
The choice of instruments in differentiated product demand systems is genericallydifficult First, we may need a lot of them In particular, we need at least oneinstrument for every product whose price is considered potentially endogenous in
a demand function (although sometimes a given instrument may in fact be used toestimate more than one equation) Second, a natural source of instruments involvescost data However, since products are often produced in a very similar way, andcost data are often recorded less frequently than prices are set, at least in financial ormanagement accounts, we are often unable to find cost variables that are genuinelysufficiently helpful for identification of each of the demand curves Data such asexchange rates and wages are often useful in homogeneous product demand esti-mation, but fundamentally such data are not product (or here segment) specific and
so will face difficulties as instruments in the differentiated product context.The reality is that there are no entirely persuasive solutions to this problem Onepotential solution, that Hausman et al (1994) suggest, is to use prices in other cities
as instruments for the prices in a given city The logic is that if, and it is often a verybig “if,” (1) demand shocks are city specific and independent across cities and (2) costshocks are correlated across markets, then any correlation between the price in thismarket and the prices in other markets will be due to cost movements In that case, the
Trang 8prices in other cities will be valid instruments for the price in this city Obviously,these are strong assumptions For example, there must not be any effect of, say,national advertising campaigns in the demand shocks since then they would not beindependent across cities Alternatively, another potentially satisfactory instrumentwould be the price of a good that shares the costs but which is not a substitute orcomplement For example, if a product under study had costs that were each heavilyinfluenced by the oil price, then the price of another good also subject to a similarsensitivity might be used Of course, in such a situation it would be easier to use theoil price so examples where this approach would genuinely be useful are perhapshard to think of.
We will explore another option for constructing instruments once we havediscussed models based on product characteristics in a later section
9.1.2.2 Indirect Utility and Expenditure Shares Models
A log-linear demand system is easy to estimate because all the equations are linear inthe parameters However, they also impose considerable assumptions on the nature
of consumer preferences For example, they impose constant own- and cross-priceelasticities of demand In addition, there is a potentially serious internal consistencyissue that we face when estimating log-linear demand functions using aggregate data.Namely, the aggregate demand function may well depend on more than aggregateincome If we only include an aggregate income variable, estimates may suffer from
“aggregation bias.”11
Misspecification and aggregation bias is easily demonstrated by taking the log-lineardemand equation for an individual,
ln Qi t D a b ln PtC j kln pk t;
where the latter equality only holds ifPJ
kD1 j kD 0 The other parameter tions can be derived by noting that we require Pt./ D Pt.1/, where
restric-ln P / D ˛0C
JXkD1
˛kln pkC 1
2
JXkD1
JX
JX
pjqj D y; where qj D qj.p; y/;
... elasticities Forexample, the results from table 9.4 suggest that the own-price elasticity of segment
demand is 2:6 for premium beer, 2:7 for popular beer, and 2:4 for light... change Forinstance, if we double all prices and we double the income, the individual demandfor all goods remains the same In general, for any > 0, we will have
when prices increase by. .. to give measures of price and quantity for each of thebrands or segments of the beer industry For example, we might use a price index withexpenditure share weights for the underlying prices within