APPLIED ECONOMIC MODELLING Theory (Chapter 3) Instructor: Joaquim - - PowerPoint PPT Presentation

Theory (Chapter 3)

APPLIED ECONOMIC MODELLING

Instructor: Joaquim J. S. Ramalho E-mail: jjsro@iscte-iul.pt Personal Website: http://home.iscte-iul.pt/~jjsro Office: D5.10 Course Website: http://home.iscte-iul.pt/~jjsro/appliedeconomicmodelling.htm Fénix: https://fenix.iscte-iul.pt/disciplinas/03178-3/2017-2018/2- semestre

Joaquim J.S. Ramalho

Hedonic regression:

▪ Regression model that relates values (prices, wages,...) to characteristics (of assets, individuals,...) ▪ It produces estimates of the contributory value (implicit price) of each characteristic to the overall value of the item being researched ▪ It allows the decomposition of a difference in a statistic between two groups, or its change over time, into various explanatory factors

Main applications:

▪ Construction of quality-adjusted price indexes for infrequently traded heterogeneous assets (houses, artworks, collectables,...) ▪ Production of measures of discrimination or other differences across groups (wage inequality between males and females,...) ▪ Measurement of quality / productivity changes over time or across groups

Our focus: construction of quality-adjusted house price indexes

3.1. Hedonic Regression Models Definition

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Construction of house price indexes:

▪ Issues:

– Each house is a unique combination of many characteristics (houses are heterogeneous) – House prices are rarely observed

▪ Consequences:

– House price indexes cannot be constructed simply by comparing the average prices of houses sold in each time period:

» Unadjusted price indexes reflect not only pure price movements but also quality differences

– House heterogeneity has to be taken into account in order to separate the influences of quality changes from pure price movements:

» Need for a quality-adjusted price index in order to measure correctly house price inflation

3.1. Hedonic Regression Models House price indexes

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Steps:

• 1. Choose a price index (arithmetic, geometric)
• 2. Choose an hedonic function (linear, log-linear)
• 3. For each time period, regress house prices on house characteristics
• 4. Apply a method to remove the effects of quality changes and capture

pure price movements

3.1. Hedonic Regression Models House price indexes

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Notation:

▪ pit : price of house i at period t ▪ t = 0 (base period) or t = s (current period) ▪ Nt observations at period t ▪ xit : vector of characteristics of house i at period t ▪ IA : Arithmetic unadjusted price index ▪ IAq : Arithmetic quality index ▪ IAp : Arithmetic quality-adjusted price index ▪ IG : Geometric unadjusted price index ▪ IGq : Geometric quality index ▪ IGp : Geometric quality-adjusted price index

3.1. Hedonic Regression Models House price indexes

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Unadjusted price indexes (observed):

▪ Arithmetic: 𝐽𝑡

𝐵 =

1 𝑂𝑡 σ𝑗=1

𝑂𝑡 𝑞𝑗𝑡

1 𝑂0 σ𝑗=1

𝑂0 𝑞𝑗0

▪ Geometric: 𝐽𝑡

𝐻 =

ς𝑗=1

𝑂𝑡 𝑞𝑗𝑡 1 𝑂𝑡

ς𝑗=1

𝑂0 𝑞𝑗𝑡 1 𝑂0

= exp 1 𝑂𝑡 σ𝑗=1

𝑂𝑡 ln 𝑞𝑗𝑡

exp 1 𝑂0 σ𝑗=1

𝑂0 ln 𝑞𝑗0

Decomposition (cannot be observed):

▪ Arithmetic: 𝐽𝑡

𝐵 = 𝐽𝑡 𝐵𝑟𝐽𝑡 𝐵𝑞

▪ Geometric: 𝐽𝑡

𝐻 = 𝐽𝑡 𝐻𝑟𝐽𝑡 𝐻𝑞

3.1. Hedonic Regression Models Step 1: Choose a Price Index

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Regression models:

▪ Linear models, for arithmetic indexes: 𝑞𝑗𝑢 = 𝑦𝑗𝑢𝛾𝑢 + 𝑣𝑢 ▪ Log-linear models, for geometric indexes: ln 𝑞𝑗𝑢 = 𝑦𝑗𝑢𝛾𝑢 + 𝑣𝑢

𝛾𝑢:

▪ Gives an estimate of the implicit price of each house characteristic at period t ▪ Multiplied by 𝑦𝑗𝑢, measures the contribution of each house charateristic to the house price

3.1. Hedonic Regression Models Step 2: Choose an Hedonic Function

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Estimation:

▪ For each time period it is estimated a different regression model ▪ Price estimates:

– Linear model, for arithmetic indexes: Ƹ 𝑞𝑗𝑢 = 𝑦𝑗𝑢 መ 𝛾𝑢 – Log-linear model, for geometric indexes: ෣ ln 𝑞𝑗𝑢 = 𝑦𝑗𝑢 መ 𝛾𝑢

▪ Unadjusted price index estimates:

– Arithmetic: መ 𝐽𝑡

𝐵 =

1 𝑂𝑡 σ𝑗=1

𝑂𝑡

Ƹ 𝑞𝑗𝑡 1 𝑂0 σ𝑗=1

𝑂0

Ƹ 𝑞𝑗0 – Geometric: መ 𝐽𝑡

𝐻 =

exp 1 𝑂𝑡 σ𝑗=1

𝑂𝑡

෣ ln 𝑞𝑗𝑡 exp 1 𝑂0 σ𝑗=1

𝑂0

෣ ln 𝑞𝑗0

3.1. Hedonic Regression Models Step 3: Regress House Prices on House Characteristics

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Arithmetic indexes:

መ 𝐽𝑡

𝐵 =

1 𝑂𝑡 σ𝑗=1

𝑂𝑡

Ƹ 𝑞𝑗𝑡 1 𝑂0 σ𝑗=1

𝑂0

Ƹ 𝑞𝑗0 = 1 𝑂𝑡 σ𝑗=1

𝑂𝑡 𝑦𝑗𝑡 መ

𝛾𝑡 1 𝑂0 σ𝑗=1

𝑂0 𝑦𝑗0 መ

𝛾0 ▪ Laspeyres-type indexes: መ 𝐽𝑡

𝐵 =

1 𝑂𝑡 σ𝑗=1

𝑂𝑡 𝑦𝑗𝑡 መ

𝛾𝑡 1 𝑂0 σ𝑗=1

𝑂0 𝑦𝑗0 መ

𝛾𝑡 1 𝑂0 σ𝑗=1

𝑂0 𝑦𝑗0 መ

𝛾𝑡 1 𝑂0 σ𝑗=1

𝑂0 𝑦𝑗0 መ

𝛾0 = መ 𝐽𝑡

𝐵𝑟 መ

𝐽𝑡

𝐵𝑞

▪ Paasche-type indexes: መ 𝐽𝑡

𝐵 =

1 𝑂𝑡 σ𝑗=1

𝑂𝑡 𝑦𝑗𝑡 መ

𝛾0 1 𝑂0 σ𝑗=1

𝑂0 𝑦𝑗0 መ

𝛾0 1 𝑂𝑡 σ𝑗=1

𝑂𝑡 𝑦𝑗𝑡 መ

𝛾𝑡 1 𝑂𝑡 σ𝑗=1

𝑂𝑡 𝑦𝑗𝑡 መ

𝛾0 = መ 𝐽𝑡

𝐵𝑟 መ

𝐽𝑡

𝐵𝑞

3.1. Hedonic Regression Models Step 4: Decompose the Estimated Price Index

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Geometric indexes: መ 𝐽𝑡

𝐻 =

exp 1 𝑂𝑡 σ𝑗=1

𝑂𝑡

෣ ln 𝑞𝑗𝑡 exp 1 𝑂0 σ𝑗=1

𝑂0

෣ ln 𝑞𝑗0 = exp 1 𝑂𝑡 σ𝑗=1

𝑂𝑡 𝑦𝑗𝑡 መ

𝛾𝑡 exp 1 𝑂0 σ𝑗=1

𝑂0 𝑦𝑗0 መ

𝛾0

▪ Laspeyres-type indexes: መ 𝐽𝑡

𝐵 =

exp 1 𝑂𝑡 σ𝑗=1

𝑂𝑡 𝑦𝑗𝑡 መ

𝛾𝑡 exp 1 𝑂0 σ𝑗=1

𝑂0 𝑦𝑗0 መ

𝛾𝑡 exp 1 𝑂0 σ𝑗=1

𝑂0 𝑦𝑗0 መ

𝛾𝑡 exp 1 𝑂0 σ𝑗=1

𝑂0 𝑦𝑗0 መ

𝛾0 = መ 𝐽𝑡

𝐵𝑟 መ

𝐽𝑡

𝐵𝑞

▪ Paasche-type indexes: መ 𝐽𝑡

𝐵 =

exp 1 𝑂𝑡 σ𝑗=1

𝑂𝑡 𝑦𝑗𝑡 መ

𝛾0 exp 1 𝑂0 σ𝑗=1

𝑂0 𝑦𝑗0 መ

𝛾0 exp 1 𝑂𝑡 σ𝑗=1

𝑂𝑡 𝑦𝑗𝑡 መ

𝛾𝑡 exp 1 𝑂𝑡 σ𝑗=1

𝑂𝑡 𝑦𝑗𝑡 መ

𝛾0 = መ 𝐽𝑡

𝐵𝑟 መ

𝐽𝑡

𝐵𝑞

3.1. Hedonic Regression Models Step 4: Decompose the Estimated Price Index

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A similar analysis may be performed across groups Example (arithmetic, Laspeyres index):

▪ Suppose that one wants to measure wage discrimination between males (t = 0) and females (t = s) ▪ Wage differences may be due to individual characteristics (xit) or how those characteristics are valued by employers for each gender ( መ 𝛾𝑢) ▪ Two hedonic functions are estimated, one for males and another for females ▪ Index decomposition is performed in the same way as before: መ 𝐽𝑡

𝐵 =

1 𝑂𝑡 σ𝑗=1

𝑂𝑡 𝑦𝑗𝑡 መ

𝛾𝑡 1 𝑂0 σ𝑗=1

𝑂0 𝑦𝑗0 መ

𝛾𝑡 1 𝑂0 σ𝑗=1

𝑂0 𝑦𝑗0 መ

𝛾𝑡 1 𝑂0 σ𝑗=1

𝑂0 𝑦𝑗0 መ

𝛾0 = መ 𝐽𝑡

𝐵𝑟 መ

𝐽𝑡

𝐵𝑞

▪ መ 𝐽𝑡

𝐵𝑞 measures the wage gap due to gender discrimination

3.1. Hedonic Regression Models Decomposition across groups

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Program evaluation:

▪ Field of study that concerns estimating the effect of a program, policy or some other intervention (“treatment”) ▪ Examples:

– What is the effect on employment of low-skilled workers of an increase in the minimum wage? – What is the effect on earnings of going through a job training program?

In these evaluations, the sample must comprise two groups, with individuals being randomly assigned to each group:

▪ Treatment group: individuals receiving the experimental treatment ▪ Control group: individuals not receiving the experimental treatment

3.2. Program Evaluation Definitions

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Two main types of experiments to estimate the effects of treatments:

▪ Randomized controlled experiments:

– Common in fields like Psychology and Medicine:

» Example: before being approved for widespread medical use, a new drug must be subjected to experimental trials in which some patients are randomly selected to receive the drug while others are given a harmless substitute (“placebo”)

– Due to cost, ethical and pratical reasons, rare in Economics:

» Example: it would be unethical to estimate the demand elasticity for cigarettes among teenagers by selling subsidized cigarettes to randomly selected high school students

▪ Quasi-experiments or natural experiments:

– More common in Economics, with randomization often imposed by external events – Randomness is introduced by variations in individual circumstances that make it appear as if the treatment is randomly assigned

» Example: one country increases the minimum wage and its neighboring state does not (assuming uncorrelation with other determinants of employment)

3.2. Program Evaluation Experiments

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Importance of randomization:

▪ If the treatment (𝑌) is assigned at random, then 𝑌 is distributed independently of any other determinant of the outcome (𝑍) ▪ Therefore, irrespective of the variables that are in the error term (𝑣), 𝐹 𝑣𝑗|𝑌𝑗 = 0 and a regression model with only 𝑌 as explanatory variable, 𝑍

𝑗 = 𝛾0 + 𝛾1𝑌𝑗 + 𝑣𝑗

is enough to consistently estimate the treatment effect (𝛾1) → with randomization, there are no omitted variables bias ▪ If the partition by groups was based on characteristics of the individuals also relevant for explaining 𝑍, then 𝐹 𝑣|𝑌 ≠ 0 and 𝛾1 would capture not only the effect of the treatment but also of the related variables contained in the error term

3.2. Program Evaluation Randomization

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Example - effect on earnings of going through a job training program:

▪ Prior work experience will influence the chances of someone getting a job after the training program ends ▪ If individuals are randomly assigned to the control and treatment groups, then both groups have individuals with a similar work experience and, therefore, participation in the treatment and work experience are uncorrelated ▪ Hence, even if experience is omitted from the regression model (being therefore contained in 𝑣), the effect of 𝑌 on 𝑍 is still consistently estimated

3.2. Program Evaluation Randomization - Example

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Base regression model for cross-sectional data (one

• bservation per individual) and randomized experiments:

𝑍

𝑗 = 𝛾0 + 𝛾1𝑌𝑗 + 𝑣𝑗

▪ 𝑌 is:

– A binary variable (= 1 if in treatment group; = 0 if in control group), if the treatment is identical for all members of the treatment group – A level variable, if the treatment varies among those in the treatment group (e.g., number of weeks in the training program)

▪ The treatment effect is given by 𝛾1, which is the so-called differences estimator:

𝐹 𝑍

𝑗|𝑌𝑗 = 1 − 𝐹 𝑍 𝑗|𝑌𝑗 = 0 = 𝛾0 + 𝛾1 − 𝛾0 = 𝛾1

▪ 𝛾1 is the casual effect of receiving treatment (binary case) or of a unit change in the level of the treatment received (level case)

3.2. Program Evaluation Regression model for randomized controlled experiments

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With natural experiments, ‘randomization’ is imposed by external events:

▪ The control and treatment groups are not randomly and explicitly chosen by the researcher; they arise from the particular policy change or exogenous event ▪ In order to control for possible systematic differences between the control and treatment groups, two years of data are necessary:

– One before the policy change – One after the policy change

▪ In each year, we may have:

– Different individuals (pooled data) – The same individuals (panel data)

3.2. Program Evaluation Natural Experiments

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Base regression model for pooled data:

𝑍

𝑗𝑢 = 𝛾0 + 𝛾1𝑌𝑗 + 𝛾2𝑈 𝑢 + 𝛾3 𝑌𝑗 ∗ 𝑈 𝑢 + 𝑤𝑗𝑢

▪ 𝑌𝑗 = ቊ1 if in treatment group 0 if in control group ▪ 𝑈

𝑢 = ቊ1 for the post − treatment year

0 for the pre − treatment year

The treatment effect is given by 𝛾3, which is the so-called differences-in-differences estimator:

𝐹 𝑍

𝑗𝑢|𝑌𝑗 = 1, 𝑈𝑢 = 1 − 𝐹 𝑍 𝑗𝑢|𝑌𝑗 = 1, 𝑈𝑢 = 0

− ሾ ሿ 𝐹 𝑍

𝑗𝑢|𝑌𝑗 = 0, 𝑈𝑢 = 1 − 𝐹ሺ

ሻ 𝑍

𝑗𝑢|𝑌𝑗 =

0, 𝑈𝑢 = 0 = 𝛾0 + 𝛾1 + 𝛾2 + 𝛾3 − 𝛾0 + 𝛾1 − 𝛾0 + 𝛾2 − 𝛾0 = 𝛾2 + 𝛾3 − 𝛾2 = 𝛾3

3.2. Program Evaluation Natural Experiments – Pooled Data

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The model for pooled data may be simplified in the case of panel data:

▪ Post-treatment year: 𝑍

𝑗,𝑞𝑝𝑡𝑢 = 𝛾0 + 𝛾2 + 𝛾1 + 𝛾3 𝑌𝑗 + 𝑤𝑗,𝑞𝑝𝑡𝑢

▪ Pre-treatment year: 𝑍

𝑗,𝑞𝑠𝑓 = 𝛾0 + 𝛾1𝑌𝑗 + 𝑤𝑗,𝑞𝑠𝑓

▪ Difference across years: ∆𝑍

𝑗 = 𝛾2 + 𝛾3𝑌𝑗 + ∆𝑤𝑗 → this is the base panel

data regression model to be estimated

Again, the treatment effect is given by the differences-in- differences estimator 𝛾3:

𝐹 ∆𝑍

𝑗|𝑌𝑗 = 1 − 𝐹 ∆𝑍 𝑗|𝑌𝑗 = 0 = 𝛾3

3.2. Program Evaluation Natural Experiments – Panel Data

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It is possible:

▪ To use the differences-in-differences estimator with randomized controlled experiments – more efficient estimators may be obtained and eventual pre- treatment differences are eliminated ▪ To use the differences estimator in natural experiments, in case there are no systematic differences between the control and treatment groups ▪ To include additional regressors (typically, pre-treatment characteristics) in any regression in order to:

– Get more efficient estimators (more regressors reduce the variance of the error term) – Achieve consistency in cases where there are threats to the validity of the experiments, such as failure to randomize (e.g., assignment partially based on the characteristics of the individuals) – Estimate different treatment effects for different groups of individuals, by interacting 𝑌𝑗 with another variable (e.g., age) – Test for randomization (based on the regression of 𝑌𝑗 on the additional regressors)

3.2. Program Evaluation Remarks

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Potential problems with experiments in practice:

▪ Failure to randomize ▪ Failure to follow treatment protocol:

– Example: less motivated trainees may not show up for the training sessions:

» If kept in the regression analysis, treatment effects of the training program may be underestimated » If excluded from the regression analysis, treatment effects of the training program may be overestimated (only the hardest workers, who would do well in the job market regardless of the training program, are considered)

▪ Hawthorne effect - if individuals know they are participating in an experiment, that might bring forth extra efforts that can affect outcomes ▪ Small samples

3.2. Program Evaluation Remarks

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