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Inflation Experiences and Contract Choice –
Evidence from Residential Mortgages∗
Matthew J. Botsch and Ulrike Malmendier†
December 28, 2015
Abstract We show that personal lifetime experiences of inflation significantly affect the valuation of fixed- versus variable-rate financial instruments. The experience-effect hypothesis pre- dicts that individuals who have experienced higher inflation expect nominal interest rates to increase more. Hence, if borrowing, they demand greater protection against increases in nominal interest rates. In the context of mortgage financing, we analyze how borrowers choose between fixed-rate and adjustable-rate options. We estimate that every additional percentage point of experienced inflation increases a borrower’s willingness to pay for a fixed-rate mortgage by 6 to 21 basis points of the FRM contract rate, as compared to an adjustable-rate mortgage. This experience effect has a major impact on the product mix of FRMs versus ARMs: nearly one in six households would switch to an ARM if not for the impact of inflation experiences. Simulations of counterfactual mortgage payments suggest that households who would otherwise have switched pay approximately $8,000 in year-2000, after-tax dollars for the embedded inflation protection of the FRM over their expected tenure in the house, implying significant welfare consequences.
∗We thank workshop participants at Babson, Berkeley, Bowdoin, Cornell, Duke, and Barcelona, as well as the NBER Household Finance Summer Institute and the 2015 World Congress of the Econometric Society, for helpful comments. † Botsch: Assistant Professor of Economics, Bowdoin College, mbotsch@bowdoin.edu. Malmendier: Profes- sor of Economics and of Finance, UC Berkeley, ulrike@berkeley.edu.
1 Introduction
Whether to buy a home and how to finance the purchase is one of the biggest financial decisions for many households. The dominant contract type in the United States is a 30- year, level-payment, self-amortizing, fixed-rate mortgage (FRM). Since 1982 banks also origi- nate adjustable-rate mortgages (ARMs), whose interest rates reset periodically. Despite their greater liquidity on secondary mortgage markets, FRMs are priced at a premium over ARMs, on average 170 basis points over equivalent-risk and -term ARMs between 1984 and 2013.^1 The premium reflects, at least in part, that FRMs provide insurance against nominal interest rate fluctuations. In this paper, we investigate another determinant of the demand for FRMs. We ask to what extent individuals’ lifetime experience of inflation affects their choice of mortgage financing. A growing literature on experience-based belief formation in macroeconomics and finance suggests that individuals overweight their lifetime experiences relative to the optimal Bayesian scheme. Building on the notion of availability bias proposed by Tversky and Kahneman (1974), this literature posits that outcomes that occurred during one’s lifetime are more easily accessible when forming beliefs and, as a result, receive extra weight compared to outcomes an individual is merely informed about or reads about. For example, Alesina and Fuchs- Schundeln (2007) relate the personal experience of living in (communist) Eastern Germany to political attitudes post-reunification. Weber et al. (1993) and Hertwig et al. (2004) show how doctors’ experience affect their future diagnoses. In the realm of finance, Malmendier and Nagel (2011) show that stock-market experiences predict future willingness to invest in the stock market, and Kaustia and Knüpfer (2008) argue the same for IPO experiences. Most related to our question, Malmendier and Nagel (2013) show that past inflation experiences strongly affect beliefs about future inflation. Experience-bias in inflation expectations has direct implications for the choice of mortgage contract. If individuals overweight lifetime inflation experiences, their experience drives a wedge between their and others’ assessment of the value of fixed-rate assets. Those with higher lifetime experiences of inflation will overvalue and overpay for fixed-rate mortgage
(^1) Calculations based on Freddie Mac’s Primary Mortgage Market Survey.
at the time of origination via origination year fixed effects. Specifically, consumers are willing to pay between 6 and 21 basis points of interest, ex ante , for each additional percentage point of experienced inflation, compared to other individuals in the same origination year. This behavior is consistent with the hypothesized experience-effect model, under which borrowers overweight inflation experiences. The estimates also provide an ex-ante indicator of welfare loss due to experienced inflation. With pairs of interest rates for both the chosen and the non-chosen alternative in hand, we use the structural coefficient on experienced inflation to simulate how much more likely individuals are to choose FRMs and how much more they are willing to pay for FRMs due to their individual inflation experiences, controlling for the full information set available to all mortgagors in the origination year and given the actual future path of mortgage interest rates. Our simulations show that between 15 and 20% of households in the population – approximately one in six households – were close enough to indifference between the two alternatives that we can attribute their choice of an FRM to overweighting of lifetime inflation experiences. For these ‘switching’ households, we simulate how much interest each individual actually paid and how much they would have paid, ex post , under two standardized contracts: a 30-year fully amortizing FRM, and a 30-year 1/1 ARM without caps indexed to the 1-year Treasury. We calculate the dollar cost of experience bias as the excess amount of interest paid which is attributable to the individual’s experienced inflation coefficient in the struc- tural choice equation. We estimate that, among these households, the bias costs the typical household approximately $8,000 over its expected tenure in the house in after-tax, present value terms, where the expected tenure is calculated based on the borrower’s age. These losses are concentrated among young borrowers taking out mortgages in the mid-1980s and rise with the holding period. For example, we estimate that the present discounted value of excess mortgage interest payments for switching households taking out a mortgage in 1986 was approximately $18,500 after-tax through 2013 (the most recent year for which we have interest rate data), assuming that the borrower held the mortgage until then. This estimate accounts for typical household refinancing behavior when FRM rates fall. The after-tax cost could be as low as $17,000 if the households refinanced optimally, and as high as $28,
if they do not refinance at all. In all cases, the ex-post estimates imply the potential of significant welfare loss due to experienced inflation.
Our findings contribute to the growing literature on non-standard belief formation and, specifically, experience-based learning in two ways. First, we are the first to provide structural estimates of mortgage choices and their payoff consequences under experience bias. We hope that our results provide a first stepping stone towards more complete welfare estimations. Second, we deepen the understanding of the role of experience-based inflation expectations for real-estate investment and financing decisions, providing quantitative estimates of the economic magnitude. Our paper builds on the growing literature pointing to the importance of experience ef- fects. For example, Malmendier and Nagel (2011) show that people who live through different stock-market histories differ in their level of risk-taking in the stock market. They find that individuals who have experienced low stock-market returns report lower willingness to take financial risk, are less likely to participate in the stock market, invest a lower fraction of their liquid assets in stocks if they participate, and are more pessimistic about future stock returns. Malmendier and Shen (2015) show that individual experiences of macroeconomic unemploy- ment conditions strongly affect consumption behavior — households who have experienced higher unemployment rates during their lifetime spend significantly less and are more likely to use coupons and allocate expenditure toward lower-end products. Malmendier and Nagel (2013) are able to show that experience effects work through the channel of beliefs. In the context of inflation expectations, they show that differences in lifetime experiences of inflation strongly predict differences in individuals’ subjective inflation expectations. They also find that, in the Survey of Consumer Finance, outstanding mortgage balance is strongly related to lifetime experiences of inflation, but the results on type of mortgage are weak or insignif- icant, likely due to data limitations. Malmendier and Steiny (2015) apply the same logic to cross-country differences in mortgage borrowing across Europe. Empirical findings from these papers form the foundation for our model on learning from experience effects. A more formal treatment of the underlying theory can be found in Mal- mendier et al. (2015), who illustrate the experience-effect mechanism in a simple OLG model
80% market share in the United States over recent decades (see Figure 1, discussed below). Its popularity was encouraged by the Congress’s establishment of Fannie Mae in 1938 and Freddie Mac in 1970. Their mission was to purchase long-term fixed-rate mortgages from banks which might otherwise face duration risk from holding these assets. Following the onset of the S&L crisis, the Garn-St. Germain Depository Institutions Act of 1982 allowed banks to originate adjustable-rate mortgages (ARMs). A typical ARM contract also self-amortizes over a long-term period such as 30 years, but the interest rate resets periodically according to a prespecified margin over an index, typically a one-year Treasury or a district cost-of- funds index. As a result, the monthly payments may vary from year to year. More exotic mortgage types became popular in the housing boom period of the 2000s – including “hybrid ARMs” whose interest rates are initially fixed but then become variable, and “interest-only” mortgages in which no principal is paid in early periods to keep initial payments low. Most of the analysis below will focus on the dominant contract types, FRMs and ARMs, with some comparison to mortgages with balloon payments. Figure 1 shows the time-series pattern of mortgage contract choice, and its correlation with the FRM-ARM spread, based on data for outstanding residential mortgages in 1991 and 2001 collected by the Census Bureau. Despite their greater liquidity on secondary mortgage markets, FRMs are priced at a premium over ARMs, in part because they provide insurance against nominal interest rate fluctuations. Freddie Mac’s Primary Mortgage Market Survey reports that FRMs carried an average premium of 170 basis points over equivalent credit risk and term ARMs between 1984 and 2013, with the annual average spread fluctuating between a low of 34 basis points (in 2009) and a high of 302 basis points (in 1994) over this time period (S.D. = 67 basis points).
To calculate lifetime experiences of inflation, we use annual CPI-U data. We calculate experienced inflation πs,te in year t for individuals belonging to the cohort born in year s building on the experience effects estimated in Malmendier and Nagel (2013). Using individ- uals’ self-reported inflation expectations in the Michigan Survey of Consumers, Malmendier and Nagel (2013) show that households’ lifetime experiences of inflation significantly affect their inflation expectations. While the most recent years obtain the highest weight, inflation experiences early in one’s life still obtain significant consideration, following approximately
the following linearly increasing pattern (if starting from the birth year):
πes,t ≡ ∑^ t k = s
∑^ k^ −^ s tj = s ( j − s ) ·^ πk^ (1)
This formula places the highest weight on the most recent observation, and zero weight on observations prior to an individual’s birth, and connects those endpoints linearly. As a first rough cut at the relationship between lifetime experiences and choice of mortgage financing, we plot experienced inflation and mortgage product choice in 1985-1991 and 1995- 2001 for young versus individuals, using the median mortgagor age in our data, 40, to split the cohorts. As Figure 2 shows, younger cohorts experienced higher rates of inflation in the late 1980s, and were more likely to choose fixed-rate products than older cohorts. In the late 1990s experienced inflation across younger and older cohorts converged; at the same time, mortgage product choice also converged.
Our main source of individual-level data on mortgage financing and demographics is the Residential Finance Survey (RFS), which the Census Bureau used to conduct the year after each Census year.^2 The unique features of RFS is that it consists of two cross-referenced surveys, one to households and one to their mortgage lenders. The household arm of the survey provides household demographic and income data, while the lender arm provides the terms of any outstanding loans secured by the property. The sample is drawn from the Census roster of households from previous, so it misses households that have moved over the last year. The sample scheme oversamples multi-unit properties, particularly rental properties with 5+ units, but it is otherwise designed to be representative of the stock of outstanding mortgages in the preceding Census year. We obtain microdata on the mortgages linked to owner-occupied 1-4 unit properties from the 1991 and 2001 waves of the RFS. Since the sample is of outstanding mortgages, we are missing mortgages that were refinanced, prepaid, or defaulted upon prior to the survey year. To minimize these issues and approximate a flow dataset of mortgage choice situations, we restrict the sample to mortgages which were taken out no more than six years prior to the survey year (1985-1991 and 1995-2001, respectively).^3
(^2) The RFS was discontinued prior to the 2010 Census. (^3) In the 1991 survey, origination years are only reported in intervals: 1985-86, 1987-88, and 1989-91.
the first step, we estimate a reduced-form choice model where households’ decisions depend on a region- and time-varying index of FRM-ARM spreads from Freddie Mac. In the second step, we estimate a mortgage pricing equation where the household’s FRM (ARM) interest rate depends on the FRM (ARM) interest rate index and on household-level characteristics that adjust for risk. These equations are likely to suffer from selection bias, since they are estimated over the nonrandom subsample of households that chose that alternative. We use the predicted choice probabilities from the first step to construct a semi-parametric control function that generalizes Heckman (1979). Identification of the semi-parametric selection- correction model comes from a cross-equation exclusion restriction: conditional on the FRM rate index, the ARM rate index does not directly influence the FRM rate that a household is offered, and vice versa. This lets us estimate the menu of interest rates that each household would have been offered, correcting for any selection bias. In the third step, we estimate a structural choice model of mortgage product choice over alternatives that depends on the household-level menu and on lifetime inflation experiences. We now describe this estimation methodology in more detail. We begin by assuming that a household in choice situation n derives utility Uni = x ′ niβ + εni from alternative i ∈ { FRM, ARM, Balloon }. Alternative i is chosen if Uni > Unj for all j 6 = i. Utility over alternatives depends on observed components x ′ niβ and unobserved components εni. Observed components may include attributes of the alternative, such as its cost, as well as attributes of the household that sway their decision toward one alternative or the other, such as liftime inflation experiences. Following McFadden (1974), we treat the unobserved utility components εni as independently drawn from a Type I extreme value distribution. Marley (cited by Luce and Suppes 1965) and McFadden (1974) show that the implied choice probabilities may be described by a logit formula whose likelihood function is globally concave, so the utility parameters can be easily estimated by maximum likelihood.^5 Theoretically, the mortgage payment structure preferred by a household depends on a host of demographics and proxies for risk attitudes, including age and mobility, current and expected future income, risk aversion, and beliefs about future short-term interest rates (see, (^5) Utility is ordinal rather than cardinal, so its location and scale are not identified by the model. That is, the ratios of coefficients are identified, but the levels are not. We follow the usual practice of standardizing the variance of the extreme value distribution to π^2 / 6 to estimate the coefficients.
among others, Stanton and Wallace 1998, Campbell and Cocco 2003, Chambers et al. 2009, and Koijen et al. 2009). Our main observable characteristics are the alternative-specific inter- est rate, the borrower’s income, and the borrower’s age. The explanatory variable of interest is the borrower’s lifetime experienced inflation. Writing this down in indirect utility terms, we obtain the following estimation equation (with the error term capturing any unobservables):
Uni = αit + βRRateni + βπ,iπen + βInc,iIncomen + fi ( Agen ) + εni (2)
Note that we include alternative-specific year fixed effects αit , which control for the overall desirability of a given alternative in a given year. The fixed effects capture all aspects of the economic environment at the time and all information that is common to all households and might enter the rational-expectations forecast, including the full history of past inflation. They are also essential for the interpretation of our coefficient of interest, βπ,i. In the presence of year fixed effects, a borrower’s lifetime inflation experiences should not matter, unless there is a correspondence between those experiences and borrower beliefs which differ from the baseline rational-expectations forecast. Specifically, the experience-effect hypothesis implies βπ,F RM > 0 , while the standard rational framework predicts βπ,F RM = 0. (Only differences in utility affect choice probabilities, so we normalize β · ,ARM ≡ 0 for all sociodemographic characteristics, including experienced inflation.) The main difficulty in estimating this random utility model is that the interest rates of the non-chosen alternatives are not observed. We will solve this problem by imputing the missing data. If there were no selection bias in the samples of households choosing each alternative, we could simply estimate the correlation between observed borrower characteristics and interest rates using the subsample of borrowers who chose each alternative, and then use the estimated parameters to sample from the distribution of interest rates for households that did not choose that alternative. Specifically, we would estimate the parameters γi to predict the rate offered to household n for alternative i using the following equation:
Rateni = γRP M M SRateni + z n ′ γi + vn,i (3)
vidual n , residing in Census region r in year t , derives utility from alternative i of
Uni = αit + β ˜ RP M M SRater,t,i + βπ,iπen + βInc,iIncomen + fi ( Agen ) + ˜ εni (4)
The estimation sample is borrowers aged 25-74 in the year of origination (restricted to 1985- and 1995-2001, respectively) for whom all covariates are available. Tildes indicate different coefficients or variables than in equation 2. For example, alternative i is chosen if ˜ εnj − ε ˜ ni := ( εnj + βRvnj ) − ( εni + βRvni ) < αit − αjt + βRz ′ n ( γi − γj ) + ... for all j 6 = i. The pricing errors vni are absorbed into the unobserved component of latent utility, ˜ ε. We have eliminated the missing data problem by replacing household-level interest rates Rateni with the Freddie Mac index rates P M M SRateni that do not depend on an individual household’s characteristics and are always observed. We now work backwards, estimating equation 4 first, equation 3 second, and equation 2 third. Equation 4 may be consistently estimated by standard maximum likelihood methods, since it only depends on exogenous characteristics that are observed for all households. We then use the reduced-form choice model probabilities to correct for selection bias in equation
- We adopt a semi-parametric control function approach suggested by Newey (2009) in which we estimate first-stage selection probabilities, then include polynomial functions of each individual’s selection probability in the second stage. This may be viewed as a generalization of Heckman (1979) to systems whose joint error distribution is non-normal. Identification requires a single-index restriction on the first-stage selection process (which a standard logit or probit model satisfies), additive separability of the selection function in the second stage, and an exclusion restriction. We assume that the Freddie Mac index rate for the nonchosen alternative doesn’t directly influence the rate for the chosen alternative, except via its influence on the probability of being selected. So the ARM index is absent from the FRM pricing equation, and the FRM index is absent from the ARM pricing equation. We also exclude borrower age, age^2 , and experienced inflation from the second-stage pricing equations. Finally, we impute pairs of interest rates for each household using our selection-corrected estimates of the pricing coefficients γi = [ γ 0 i, γ −′ 0 i ]′^ in equation 3 and use these to estimate the structural choice model, equation 2. The intercept γ 0 i is not separately identified from
the control function for probability of selection. We estimate it using a method suggested by Heckman (1990), by calculating the average difference between the dependent variable and the predicted values from explanatory variables excluding the intercept, Rateni − z ′− 0 n ˆ γ − 0 i , over the observations whose estimated reduced-form probabilities of choosing alternative i are closest to 1.^7
3.2 Choice Model Estimates
Table 2 presents estimates of the reduced form multinomial logit model. Each coefficient represents that attribute’s or sociodemographic characteristic’s contribution to the utility of that alternative. So, for example, ˆ β ˜ R = − 0_._ 424 in column 1, indicating that individuals derive less utility from and are less likely to choose more expensive alternatives. All columns include alternative-specific year fixed effects and control for a quadratic function of the primary owner’s age. Column 1 estimates a single price coefficient on both the FRM and the ARM initial rate indices (so only the spread matters), while columns 2-4 allows the two coefficients to differ. Column 3 normalizes βπ,Balloon = βπ,ARM , while column 4 controls for characteristics of the mortgage (seniority, whether it is a refinancing of a previous mortgage, conventional dummy, and points paid). Recall that only differences in utility matter, so we normalize β · ,ARM ≡ 0 for all household-level variables, including experienced inflation. The results indicate indicate that individuals who have higher levels of πe^ as of the year of the choice situation derive greater utility from the FRM alternative, relative to the baseline ARM alternative. Experienced inflation reduces the utility of a balloon mortgage relative to an ARM, but this effect is imprecisely estimated and not significant at standard levels. A useful normalization is to calculate the compensating interest rate differential an individual would be willing to pay to “avoid” one additional percentage point of experienced inflation. This is done by taking the total derivative of utility for alternative i and setting it equal to zero (cf. Train (2009), ch. 3):
dUni = βR∂Rateni + βπ,i∂πne = 0 (^7) The individuals in this subsample are more likely to have chosen the alternative due to observed rather than unobserved factors, so suffer from the least amount of selection bias. We use the top 10% subsample based on predicted choice probabilities for each alternative.
a positive number, +6 basis points (column 1); after correcting for selection, it is -35 basis points (column 2). This is consistent with high-risk households selecting into nonconventional, FRM mortgages, and the median pricing error, conditional on choosing an FRM, being less than zero. We find evidence of less selection bias in the ARM initial rate pricing equations (columns 3 and 4). CLAD is not particularly useful for adjusting ARM margins for household risk characteristics, as more than half of all individuals carry the same margin (2.75 percentage points). Whether or not we control for selection, conditional-median based methods only adjust the margin for mortgage seniority (junior mortgages are 25 basis points more expensive than first mortgages). In later sections of the paper, we discretize the distribution of margins into ten intervals (using the 1991 RFS reporting intervals) and estimate an ordered logit model of ARM margin on the same set of covariates shown in Table 3. This model also accounts for censoring and allows us to recover coefficients to predict risk-adjusted ARM margins. The estimation results are not shown but are available upon request. Table 4 presents estimates of the structural choice equation 2. The dependent variable is coded as 1 if the household chose an FRM and 0 if it chose an ARM. We use predicted interest rates from the pricing equations presented in Table 3 for both the chosen and the nonchosen alternative in each choice situation. Standard errors are not adjusted for the first- stage estimation. A comparison of columns 1 and 2 indicates the importance of the selection- correction in equation 3. Without selection correction, the price coefficients are the wrong signs, indicating upward sloping demand curves. After we switch to the selection-correction estimates, the signs become correct (column 2). The structural model estimates confirm our previous finding that experienced inflation influences mortgage contract choice, above and beyond the information set available to all households in a given origination year. Individuals exhibit an ex ante willingness to pay of 0_._ 223 / 1_._ 065 = 0_._ 209 , i.e., 21 additional basis points of FRM interest for every additional percentage point of experienced inflation (based on the estimates from column 2). Columns 3 and 4 present estimates of the structural model using risk-adjusted ARM mar- gins. As before, the signs on the FRM rate and ARM initial rate are the wrong sign without the selection correction. With the selection correction, these signs reverse and are correct;
however, the sign on the ARM margin remains negative (indicating that a higher margin is associated with a lower probability of choosing an FRM). Since the selection correction procedure mainly affected the coefficient on the nonconventional status dummy in the pricing equations in Table 3, we hypothesize that nonconventional status might have an additional effect on mortgage choice above and beyond its structural impact on mortgage prices. To confirm this, we re-run the last two set of estimates with nonconventional status as an addi- tional explanatory variable in Table 4, columns 5 and 6. Inclusion of this variable generates “correct,” negative demand elasticities in both specifications. The bottom line is that in all specifications, higher levels of lifetime inflation experiences are associated with a greater probability of choosing an FRM compared to other individuals in the same origination year, independently of how we estimate mortgage prices and of what variables are controlled for. This is consistent with personal experiences affecting an indi- vidual’s ex ante willingness to pay for the safety of the fixed-rate alternative. But does this translate into an ex post welfare loss?
4 Simulations
4.1 The Welfare-Relevant Treatment Effect
While the effect of experienced inflation on mortgage product shares appears to be econom- ically large, it is not obvious that this is a costly mistake. Figure 4 plots the path of the national PMMS fixed-rate and adjustable initial rate indices, and the yield on a one-year con- stant maturity Treasury plus a standard margin of 2.75 percentage points, between the years 1986 and 2013. The FRM-ARM initial rate spread is always positive but varies over time (as previously seen in Figure 1, this variation is correlated with product shares). Individuals with a sufficiently short time horizon will usually benefit from the initially low rate of an ARM, but over longer time horizons the resets could make the ARM more expensive. For example, an individual taking out an FRM in 1993 would lock in a nominal rate of 7.31% for the life of the loan. An individual taking out a 1/1 ARM with no caps on rate resets would pay the much lower initial rate of 4.58% in 1993, but this would reset to 8.06% in 1994, 8.70% in 1995, etc. Resets would keep the subsequent ARM rate above the 1993 FRM rate every year until
Scenario: 1 2 3 FRM Rate: Freddie Mac PMMS
Risk-adj. (CLAD)
Risk-adj. (CLAD) ARM Initial Rate: Freddie Mac PMMS
Risk-adj. (CLAD)
Risk-adj. (CLAD) ARM Margin: 1-year T-bill +
Seniority-adj. (CLAD)
Risk-adj. (OLOGIT) Each scenario has limitations. Scenario 1 makes no adjustment for mortgagor risk charac- teristics, but it also carries the least amount of sensitivity to researcher uncertainty about the true pricing model. Scenarios 2 and 3 make progressively greater adjustments for risk charac- teristics, at the cost of increasing sensitivity to our modeling assumptions. However, take note that our simulated ARM contract has no caps on annual or lifetime interest rate adjustments. Since many ARMs are capped, all three scenarios overstate the amount of interest rate risk in an ARM and underestimate the potential savings from choosing an ARM over an FRM. This biases against our maintained hypothesis that experienced inflation is welfare-reducing. The welfare cost for switching households is easily described using the language of potential treatments and potential outcomes. For ease of exposition, we focus on the binary choice problem and number the FRM alternative as 1 (and the ARM alternative as 0). In every choice situation n , the household faces two potential outcomes: mortgage payments under the fixed-rate alternative, Yn, 1 , and mortgage payments under the adjustable-rate alternative, Yn, 0. The observed set of mortgage payments in our data is
Yn = DnYn, 1 + (1 − Dn ) Yn, 0
and depends on an individual’s mortgage choice (“treatment status”), Dn ∈ { 0 , 1 }, indicating which alternative is chosen. The value of Dn depends on the difference in latent utility between the two alternatives from equation 2:
Dn = I{FRM is chosen in choice situation n } = I{ Un, 1 > Un, 0 } = I{−( εn 1 − εn 0 ) < x ′ n 1 β 1 − x ′ n 0 β 0 }
The FRM is chosen if the difference in observed components of latent utility exceed the difference in unobserved components. Observed latent utility may include alternative charac- teristics, such as prices, and household characteristics, including experienced inflation. These are the coefficients estimated in Table 4.^8 Under a counterfactual utility model, the same individual in the same choice situation might make a different choice. This introduces the notion of potential choices (“potential treatments”). Specifically, let Dn ( bπ ) be the choice individual n would make given experienced inflation coefficient bπ. The observed mortgage choice in our data is
Dn =
An ( βπ ) Dn ( bπ ) dbπ
where An (·) = I{ bπ = ·} and βπ is the true experienced inflation coefficient, representing the additional weight placed on πe^ beyond the full-information Bayesian optimum. The house- hold’s actual choice, under the true utility model, is Dn ( βπ ) ∈ { 0 , 1 }. The welfare-relevant counterfactual is the choice the household would have made in the same choice situation if placing no additional weight on experienced inflation: Dn (0) ∈ { 0 , 1 }. If Dn ( βπ ) = Dn (0), then “assignment” was irrelevant and experienced inflation did not influence the household’s mortgage choice. If Dn ( βπ ) 6 = Dn (0) – the two potential choices are different – then the household is nearly indifferent and would switch under the counterfactual model. Using this notation, the ex post welfare loss (or gain) for switching households may be expressed as follows: E[ Yn, 1 − Yn, 0 | Dn ( βπ ) = 1 , Dn (0) = 0] (5)
By monotonicity of the choice function Dn (·) and βπ > 0 , households only switch out of an FRM. So the average welfare loss is the expected difference between FRM and ARM payments for those households that chose a fixed-rate mortgage because of the weight they placed on their personal inflation experiences. Positive numbers represent overpayment, a welfare loss, and negative numbers represent underpayment, a welfare gain. The conditioning set restricts us to the subset of households in the population for whom experienced inflation was the
(^8) Since only differences in utility matter, we are implicitly estimating the difference in household character- istic coefficients in the binary choice model: βx, 1 − βx, 0.
