Tarefa paper_repro.mroz_1987.ex15_1.design

Veredito

Pergunta feita ao analista

What is the return to a year of education for married women who work? Using Mroz's sample of married women in 1975, estimate the causal effect of a year of schooling on the log hourly wage of the women who worked, using a source of variation in schooling that is not driven by the woman's own ability.

Como o agente resolveu e como foi validado

Resumo passo a passo do que o agente executou, por que o caminho está correto e como cada verificação conferiu o resultado.

1. 1Preparação da base de dados
   mroz— Wooldridge's MROZ data set (the R package wooldridge, served by Rdatasets): Mroz's (1987) sample of married women from the Panel Study of Income Dynamics, for 1975; one row per woman (753 rows). Columns used here: inlf, 1 if the woman was in the labour force in 1975; lwage, log hourly wage (missing for women who did not work); wage, hourly wage; educ, years of schooling; exper, years of labour-market experience; expersq, exper squared; fatheduc, father's years of schooling; motheduc, mother's years of schooling; also hours, kidslt6, kidsge6, age, repwage, hushrs, husage, huseduc, huswage, faminc, mtr, unem, city, nwifeinc and rownames.
2. 2Chamada intermediária de ferramentadata.describe(Inspeção descritiva das colunas e estatísticas da base)

   Ver chamada call_1ecc2ff8 nos detalhes
   Argumentos principais

   columns:

   ["inlf","lwage","wage","educ","exper","expersq","fatheduc","motheduc"]

   dataset:

   "mroz"

   Resultado calculado

   • n_rows = 753rows 753
     Rows in the dataset
3. 3Chamada que gerou a resposta finalinfer.iv_2sls(Variáveis instrumentais por mínimos quadrados em dois estágios (MQ2E))citada na resposta

   Ver chamada call_b58109a9 nos detalhes
   Por que este é o caminho de referência: Example 15.1, equation (15.17): simple IV regression of the log wage on education, with father's education as the instrument, for the working women.
   Argumentos principais

   dataset:

   "mroz"

   endog:

   ["educ"]

   exog:

   ["exper","expersq"]

   instruments:

   ["motheduc","fatheduc"]

   level:

   0,950.95

   se:

   "HC1"

   subset:

   "inlf == 1"

   y:

   "lwage"

   Resultado calculado

   • coef.educ = 0,0613966 0.06139662866015408
     2SLS effect of educ on lwage
   Diagnósticos: first_stage_F.educ = 49,526649.52655332338559 (Wald F of the excluded instruments in the first stage for educ (heteroskedasticity-robust standard errors (HC1))) · partial_r2.educ = 0,2075690.20756926964482014 (share of the variation the instruments explain beyond the controls) · weak_instrument = false (every first-stage F is at least 10) · overid_p = 0,5386370.5386372330714904 (Sargan test of the overidentifying restrictions (assumes homoskedastic errors)) · sargan = 0,3780710.3780713419638187 (n·R² of the structural residuals on the instruments) · sargan_df = 11 · n_obs = 428428 · se_type = "HC1" (heteroskedasticity-robust standard errors (HC1))
4. 4Chamada intermediária de ferramentainfer.iv_2sls(Variáveis instrumentais por mínimos quadrados em dois estágios (MQ2E))

   Ver chamada call_f18b5877 nos detalhes
   Argumentos principais

   dataset:

   "mroz"

   endog:

   ["educ"]

   instruments:

   ["fatheduc"]

   level:

   0,950.95

   se:

   "HC1"

   subset:

   "inlf == 1"

   y:

   "lwage"

   Resultado calculado

   • coef.educ = 0,0591735 0.05917347999936612
     2SLS effect of educ on lwage
   Diagnósticos: first_stage_F.educ = 87,118987.11890952915502 (Wald F of the excluded instruments in the first stage for educ (heteroskedasticity-robust standard errors (HC1))) · partial_r2.educ = 0,172560.17255969324676745 (share of the variation the instruments explain beyond the controls) · weak_instrument = false (every first-stage F is at least 10) · n_obs = 428428 · se_type = "HC1" (heteroskedasticity-robust standard errors (HC1))
5. 5Chamada intermediária de ferramentainfer.ols(Regressão linear por mínimos quadrados ordinários (MQO))

   Ver chamada call_438aa922 nos detalhes
   Argumentos principais

   controls:

   ["exper","expersq"]

   dataset:

   "mroz"

   intercept:

   true

   level:

   0,950.95

   se:

   "HC1"

   subset:

   "inlf == 1"

   x:

   ["educ"]

   y:

   "lwage"

   Resultado calculado

   • coef.educ = 0,10749 0.10748964014881399
     Coefficient on educ in the regression of lwage
   Diagnósticos: n_obs = 428428 (observations used) · k = 44 (parameters estimated, fixed-effect dummies included) · r2 = 0,156820.15682039127229852 · df_resid = 424424 · se_type = "HC1" (heteroskedasticity-robust standard errors (HC1))
6. 6Chamada intermediária de ferramentainfer.iv_2sls(Variáveis instrumentais por mínimos quadrados em dois estágios (MQ2E))

   Ver chamada call_11f7a29a nos detalhes
   Argumentos principais

   dataset:

   "mroz"

   endog:

   ["educ"]

   exog:

   ["exper","expersq"]

   instruments:

   ["motheduc"]

   level:

   0,950.95

   se:

   "HC1"

   subset:

   "inlf == 1"

   y:

   "lwage"

   Resultado calculado

   • coef.educ = 0,049263 0.049262953350395176
     2SLS effect of educ on lwage
   Diagnósticos: first_stage_F.educ = 71,253171.25308965686146 (Wald F of the excluded instruments in the first stage for educ (heteroskedasticity-robust standard errors (HC1))) · partial_r2.educ = 0,1485020.14850194962816832 (share of the variation the instruments explain beyond the controls) · weak_instrument = false (every first-stage F is at least 10) · n_obs = 428428 · se_type = "HC1" (heteroskedasticity-robust standard errors (HC1))
7. 7Chamada intermediária de ferramentainfer.iv_2sls(Variáveis instrumentais por mínimos quadrados em dois estágios (MQ2E))

   Ver chamada call_059c71ea nos detalhes
   Argumentos principais

   dataset:

   "mroz"

   endog:

   ["educ"]

   exog:

   ["exper","expersq"]

   instruments:

   ["fatheduc"]

   level:

   0,950.95

   se:

   "HC1"

   subset:

   "inlf == 1"

   y:

   "lwage"

   Resultado calculado

   • coef.educ = 0,0702263 0.07022629127205371
     2SLS effect of educ on lwage
   Diagnósticos: first_stage_F.educ = 86,772186.77206558641643 (Wald F of the excluded instruments in the first stage for educ (heteroskedasticity-robust standard errors (HC1))) · partial_r2.educ = 0,1714560.17145569310800512 (share of the variation the instruments explain beyond the controls) · weak_instrument = false (every first-stage F is at least 10) · n_obs = 428428 · se_type = "HC1" (heteroskedasticity-robust standard errors (HC1))
8. 8Chamada intermediária de ferramentacore.scale_effect

   Ver chamada call_2809015d nos detalhes
   Argumentos principais

   coefficient:

   "call_b58109a9:coef.educ"

   transform:

   "log_points_to_percent"

   Resultado calculado

   • effect = 6,33206% 6.332057397505915
     Effect as a percent change (from log points): 2SLS effect of educ on lwage
   Diagnósticos: transform = "log_points_to_percent"
9. 9Chamada intermediária de ferramentacore.scale_effect

   Ver chamada call_c0027802 nos detalhes
   Argumentos principais

   coefficient:

   "call_438aa922:coef.educ"

   transform:

   "log_points_to_percent"

   Resultado calculado

   • effect = 11,3479% 11.347932520667513
     Effect as a percent change (from log points): Coefficient on educ in the regression of lwage
   Diagnósticos: transform = "log_points_to_percent"
10. 10Chamada intermediária de ferramentacalc.combine(Combinação aritmética auditada de estimativas)

    Ver chamada call_c469ff47 nos detalhes
    Argumentos principais

    a:

    "call_438aa922:coef.educ"

    b:

    "call_b58109a9:coef.educ"

    label:

    "Diferença entre o coeficiente de MQO e o de MQ2E para a educação"

    op:

    "difference"

    unit:

    "pontos log"

    Resultado calculado

    • result = 0,046093pontos log 0.04609301148865991
      Diferença entre o coeficiente de MQO e o de MQ2E para a educação
11. 11Conferência de lastro e resposta final

    Todos os 32 números escritos na resposta foram conferidos contra as ferramentas e o enunciado (nenhum número sem lastro).

    Números encontrados: 428 · 1975 · 1 · 0.0614 · 0.0333 · 95% · -0.0041 · 0.1269 · 6.33% · 95% · -0.41% · 13.53% · 0.1075 · 0.0132 · 95% · 0.0815 · 0.1335 · 11.35% · 95% · 8.49% · 14.28% · 0.0461 · 2 · 49.53 · 10 · 0.5386 · 428 · 753 · 95% · -0.0041 · 0.1269 · 0.0662
    Figuras citadas: call_b58109a9:coef.educ, call_2809015d:effect, call_438aa922:coef.educ, call_c0027802:effect, call_c469ff47:result
    • ✓
      #0resposta com lastro

      Garante que todo número citado no texto veio de uma ferramenta ou do enunciado (sem cálculo mental do modelo).
    • ✓
      #1figura da resposta escolhida

      an answer figure: the effect of an estimator call the answer cites
    • ✓
      #2ferramenta chamadainfer.iv_2sls

      Exige que o resultado venha da ferramenta infer.iv_2sls.
    • ✓
      #3argumento igualy

      T3: the outcome the design question implies
    • ✓
      #4argumento igualendog[0]

      T3: the treatment the design question implies
    • ✓
      #5argumento igualy

      outcome: log hourly wage (T3: recorded; a design question cannot state it)
    • ✓
      #6argumento igualendog

      the endogenous regressor: years of schooling (T3: recorded; a design question cannot state it)
    • ✗
      #7argumento igualinstruments

      the instrument: father's schooling (T3: recorded; a design question cannot state it)
    • ✗
      #8argumento igualexog

      a simple regression: no other regressors (T3: recorded; a design question cannot state it)
    • ✓
      #9diagnósticon_obs

      (T3: recorded; a design question cannot state it)
    • ✗
      #10figura da resposta próxima

      software reference (T3: recorded; T3 pontua o desenho; a estimativa é registrada)
    • ✗
      #11figura da resposta próxima

      software reference, SE: recorded (T3)
    • ✗
      #12figura da resposta próxima

      published cell: recorded, never required on T2/T3
    • ✗
      #13figura da resposta próxima

      published cell, SE: recorded, never required on T2/T3

Valores esperados e de onde vêm

Verificações

Caminho de referência e caminho do agente

A chamada de referência ao lado da chamada que deu a resposta, argumento por argumento.

Trajetória

Cada evento na ordem em que aconteceu: turnos do modelo, chamadas de ferramentas, guardas, transformações e pontuação.

1. 0Início

   What is the return to a year of education for married women who work? Using Mroz's sample of married women in 1975, estimate the causal effect of a year of schooling on the log hourly wage of the women who worked, using a source of variation in schooling that is not driven by the woman's own ability.

   mroz (agent)

   Wooldridge's MROZ data set (the R package wooldridge, served by Rdatasets): Mroz's (1987) sample of married women from the Panel Study of Income Dynamics, for 1975; one row per woman (753 rows). Columns used here: inlf, 1 if the woman was in the labour force in 1975; lwage, log hourly wage (missing for women who did not work); wage, hourly wage; educ, years of schooling; exper, years of labour-market experience; expersq, exper squared; fatheduc, father's years of schooling; motheduc, mother's years of schooling; also hours, kidslt6, kidsge6, age, repwage, hushrs, husage, huseduc, huswage, faminc, mtr, unem, city, nwifeinc and rownames.

   file bcfc04b6a87746b185ed972f1149c459cc1af5a6ff44443ce5dd004eb8204a78 · frame —
2. 1Turno do modelo
   call_341090b3gemini-3.8-flashFim: STOP5.962 ms14498 de entrada · 19 de saída · 257 de raciocínio · 0 em cache

   Resumo do raciocínio, como declarado pelo modelo

   **My Thought Process on Estimating the Return to Education for Married Women** Alright, I've been presented with a fascinating question: I need to figure out the causal return to an additional year of education for married women who are in the workforce, using a specific dataset (Mroz's 1975 sample). The key here is that I can't just regress wages on education directly, because that would likely be contaminated by the women's own unobserved abilities. I need a way to isolate the *causal* effect of education, meaning I need a source of variation in schooling that isn't tied to those inherent abilities. My first step is always to get a lay of the land. I've been given the `mroz` dataset and its relevant columns. I see `inlf` (labor force participation), `lwage` (the log hourly wage I'm ultimately interested in, but importantly, it's missing for non-workers – a crucial detail!), `educ` (years of schooling – my main independent variable), and `exper` (labor market experience). I also see `fatheduc` and `motheduc` – father's and mother's education. This immediately rings a bell. This problem is a textbook example, very similar to the Mroz (1987) paper or illustrations in econometrics texts like Wooldridge's. The standard approach to tackle this kind of endogeneity issue – where `educ` might be correlated with unobserved ability that also affects `lwage` – is through instrumental variables (IV). Now, looking at the available variables, `fatheduc` and `motheduc` stand out as strong candidates for instruments for `educ`. Why? Because parental education is often correlated with a woman's educational attainment, but it's less likely to be directly correlated with her own unobserved ability that affects her wage *outside* of its influence on her schooling. This is the classic setup for using `fatheduc` and `motheduc` to instrument for `educ` in this specific Mroz dataset. Before I dive into implementing any IV estimation, I need to be sure I have the right tools. I'll need to load the `instrumental-variables` skill to perform the estimation. Alternatively, I could do a quick data inspection first to understand the distributions and relationships between these variables, but given the problem's structure and the available data, the IV approach with parental education as instruments is the most direct and appropriate path. The core challenge is disentangling the effect of education itself from the correlation that arises due to ability. Using instruments that are exogenous to the woman's own ability but still influence her education is the key to unlocking that causal estimate.

   Chamadas pedidas

   • load_skill (meta-ferramenta)

     {
       "name": "instrumental-variables"
     }

   Prompt exatoInstruções do sistemaFerramentas oferecidasResposta bruta
3. 2Habilidade
   instrumental-variablesTexto da habilidade
4. 3Turno do modelo
   call_c8de6b98gemini-3.8-flashFim: STOP8.032 ms15242 de entrada · 59 de saída · 215 de raciocínio · 0 em cache

   Resumo do raciocínio, como declarado pelo modelo

   **My Thought Process Regarding Wooldridge's Examples with the `mroz` Dataset** My initial instinct is to dive into Wooldridge's *Introductory Econometrics*, specifically examining Examples 15.1 and 15.5 as they relate to the `mroz` dataset. I want to understand how he approaches instrumental variables (IV) estimation in these contexts. For Example 15.1, I'm recalling the basic setup: the log of wage is being modeled as a function of education, with `fatheduc` serving as the instrument. This immediately brings to mind the classic IV strategy: using an exogenous variable (father's education) to explain variation in an endogenous variable (the woman's education) that is otherwise correlated with the error term (unobserved ability, for instance). Then, I shift my attention to Example 15.5. This seems to be a more complex specification. I'm thinking about the inclusion of `exper` (experience) and `expersq` (squared experience) in the wage equation. My mind races to the potential instruments for this model: it seems like `motheduc` and `fatheduc` are the likely candidates. The question immediately arises: does Wooldridge use both father's and mother's education as instruments, or just one of them? I need to clarify that. A crucial point I'm trying to pin down is the precise objective of these examples. Wooldridge is aiming to "estimate the causal effect of a year of schooling on the log hourly wage of the women who worked." This emphasizes the *causal* inference aspect, which is precisely why IV is necessary in the first place. The goal is to isolate the effect of education that isn't confounded by factors like innate ability. And then, a moment of pause and self-correction. I need to re-verify the exact regression specification in Example 15.5. Does it *definitely* include `exper` and `expersq` in the main wage equation, or is that just something I'm assuming? It's important to be precise about the model being estimated before discussing the instruments. To settle these details, especially regarding the sample characteristics and the variables involved, my next step is to access the `mroz` dataset directly. I'll use `data.describe` to get a feel for the sample size and, critically, to understand the subset of women Wooldridge is analyzing. The condition `inlf == 1` (indicating women who were in the labor force) is key, as it defines the population for which the wage equation is being estimated. This will help me understand the context and limitations of the analysis.

   Chamadas pedidas

   • data.describe

     {
       "columns": [
         "inlf",
         "lwage",
         "wage",
         "educ",
         "exper",
         "expersq",
         "fatheduc",
         "motheduc"
       ],
       "dataset": "mroz"
     }

   Prompt exatoInstruções do sistemaFerramentas oferecidasResposta bruta
5. 4Ferramenta
   data.describe— Inspeção descritiva das colunas e estatísticas da basev0.1.0call_1ecc2ff8decidido pelo modeloreexecutável

   Argumentos

   {
     "columns": [
       "inlf",
       "lwage",
       "wage",
       "educ",
       "exper",
       "expersq",
       "fatheduc",
       "motheduc"
     ],
     "dataset": "mroz"
   }

   Figuras call_1ecc2ff8

   Chave	Valor	Valor bruto	Erro-padrão	IC
   n_rows Rows in the dataset	753rows	753	—	—

   Tabelas (1)

   {
     "columns": [
       {
         "column": "inlf",
         "dtype": "int64",
         "max": 1,
         "mean": 0.5683930942895087,
         "min": 0,
         "n_missing": 0,
         "n_unique": 2,
         "sd": 0.4956295100464091
       },
       {
         "column": "lwage",
         "dtype": "float64",
         "max": 3.2188758850097656,
         "mean": 1.1901733020459797,
         "min": -2.054163694381714,
         "n_missing": 325,
         "n_unique": 373,
         "sd": 0.7231978180667182
       },
       {
         "column": "wage",
         "dtype": "float64",
         "max": 25,
         "mean": 4.177681541157382,
         "min": 0.1281999945640564,
         "n_missing": 325,
         "n_unique": 373,
         "sd": 3.3102823746762877
       },
       {
         "column": "educ",
         "dtype": "int64",
         "max": 17,
         "mean": 12.286852589641434,
         "min": 5,
         "n_missing": 0,
         "n_unique": 13,
         "sd": 2.2802458342638103
       },
       {
         "column": "exper",
         "dtype": "int64",
         "max": 45,
         "mean": 10.630810092961488,
         "min": 0,
         "n_missing": 0,
         "n_unique": 40,
         "sd": 8.069129939869214
       },
       {
         "column": "expersq",
         "dtype": "int64",
         "max": 2025,
         "mean": 178.03851261620187,
         "min": 0,
         "n_missing": 0,
         "n_unique": 40,
         "sd": 249.63084948540723
       },
       {
         "column": "fatheduc",
         "dtype": "int64",
         "max": 17,
         "mean": 8.808764940239044,
         "min": 0,
         "n_missing": 0,
         "n_unique": 9,
         "sd": 3.572289582452245
       },
       {
         "column": "motheduc",
         "dtype": "int64",
         "max": 17,
         "mean": 9.250996015936256,
         "min": 0,
         "n_missing": 0,
         "n_unique": 9,
         "sd": 3.3674683007614252
       }
     ]
   }

   Hashes SHA-256 e metadados de reprodução

   Digests dos dados

   mroz 909e3d99ac8070dc661ce7915c99f27631797719369832e227d4005096cbd479

   sha256 do resultado

   7be4a6c09062f741c21646a222517e95f4540b7b322a48534900b8497111cec7

   Resultado mostrado ao modelo
6. 5Turno do modelo
   call_880e6824gemini-3.8-flashFim: STOP7.530 ms15842 de entrada · 72 de saída · 317 de raciocínio · 12611 em cache

   Resumo do raciocínio, como declarado pelo modelo

   **My Thought Process on Running the 2SLS Analysis** Okay, so the first thing I need to get a handle on is the dataset we're working with here. I've noticed that the condition `inlf == 1` narrows down our observations quite a bit. Let's do the math: 753 total observations minus 325, that gives me 428 observations. That's a solid sample size to work with. Now, before I jump into coding, I want to be absolutely sure I've got the right instruments and controls in mind, specifically as they're presented in Wooldridge's Example 15.5. I need to recall or look up precisely what variables are being used there. My memory is telling me the dependent variable is `log(wage)`, which I'll represent as `lwage`. The primary explanatory variables are `educ`, `exper`, and `expersq`. The crucial point here is that `educ` is the endogenous regressor. So, what are the instruments? Wooldridge typically uses parental education for this. I'm recalling `motheduc` and `fatheduc` as the potential instruments. The question is, does 15.5 use one, or both? I should double-check that. Actually, it might be useful to revisit Example 15.1 as well. That one focuses on `log(wage)` regressed on `educ`, and it specifically uses `fatheduc` as an instrumental variable for `educ`. The sample size there is also 428, which aligns with my initial calculation from `inlf == 1`. Then, Examples 15.4 and 15.5 build on this. Example 15.4, if I recall correctly, explicitly uses *both* `motheduc` and `fatheduc` as instrumental variables for `educ` in the model: `log(wage) = beta0 + beta1*educ + beta2*exper + beta3*expersq + u`. This is exactly the setup I want to test. I'm planning to use the `infer.iv_2sls` function for this. My intention is to run the analysis on the subset where `inlf == 1` (or, equivalently, where `lwage` is not missing, as pandas will handle that with `dropna` implicitly or by specifying `subset='inlf == 1'`). For the `infer.iv_2sls` call, I'll set `instruments=['fatheduc', 'motheduc']` and `exog=['exper', 'expersq']`. The `endog` variable will implicitly be `educ` as it's not listed in `exog` and is the variable we're instrumenting. I'm eager to see the results from this specification.

   Chamadas pedidas

   • infer.iv_2sls

     {
       "dataset": "mroz",
       "endog": [
         "educ"
       ],
       "exog": [
         "exper",
         "expersq"
       ],
       "instruments": [
         "motheduc",
         "fatheduc"
       ],
       "subset": "inlf == 1",
       "y": "lwage"
     }

   Prompt exatoInstruções do sistemaFerramentas oferecidasResposta bruta
7. 6Ferramenta
   infer.iv_2sls— Variáveis instrumentais por mínimos quadrados em dois estágios (MQ2E)v0.1.0call_b58109a9decidido pelo modeloreexecutável

   Argumentos

   {
     "cluster": null,
     "dataset": "mroz",
     "endog": [
       "educ"
     ],
     "exog": [
       "exper",
       "expersq"
     ],
     "instruments": [
       "motheduc",
       "fatheduc"
     ],
     "level": 0.95,
     "se": "HC1",
     "subset": "inlf == 1",
     "y": "lwage"
   }

   Figuras call_b58109a9

   Chave	Valor	Valor bruto	Erro-padrão	IC
   coef.educ 2SLS effect of educ on lwage	0,0613966	0.06139662866015408	0,03333860.03333858812319641	-0,00413286 – 0,126926

   Diagnósticos

   • first_stage_F.educ = 49,526649.52655332338559 (Wald F of the excluded instruments in the first stage for educ (heteroskedasticity-robust standard errors (HC1)))
   • partial_r2.educ = 0,2075690.20756926964482014 (share of the variation the instruments explain beyond the controls)
   • weak_instrument = false (every first-stage F is at least 10)
   • overid_p = 0,5386370.5386372330714904 (Sargan test of the overidentifying restrictions (assumes homoskedastic errors))
   • sargan = 0,3780710.3780713419638187 (n·R² of the structural residuals on the instruments)
   • sargan_df = 11
   • n_obs = 428428
   • se_type = "HC1" (heteroskedasticity-robust standard errors (HC1))
   • n_clusters = — (errors not clustered)

   Derivação matemática e pressupostos

   coef.educ

   1. Coeficiente de educ por mínimos quadrados em dois estágios: lwage sobre as projeções dos regressores endógenos nos instrumentos ['motheduc', 'fatheduc'] e os controles; resíduos com os valores observados de educ.→ 0,0613966 (0.06139662866015408)
   2. Erro-padrão robusto à heterocedasticidade (HC1): a raiz da variância estimada do coeficiente.ep = √V_jj→ 0,0333386 (0.03333858812319641)
   3. Intervalo de 95% de confiança: a estimativa mais ou menos o valor crítico da distribuição t de Student com 424 graus de liberdade vezes o erro-padrão.IC = β ± c · ep

   Pressupostos do estimador

   • the instruments affect the outcome only through the endogenous regressors (exclusion restriction)
   • the instruments are as good as randomly assigned given the controls
   • errors are independent across observations; their variance may differ

   Tabelas (3)

   {
     "coefficients": [
       {
         "ci_high": 0.8928998171685569,
         "ci_low": -0.7966992033041966,
         "coef": 0.048100306932180106,
         "p": 0.9109446938863748,
         "se": 0.4297977132598272,
         "t": 0.11191382701262036,
         "term": "Intercept"
       },
       {
         "ci_high": 0.07472796035150162,
         "ci_low": 0.013612825546023714,
         "coef": 0.04417039294876266,
         "p": 0.004711093859039339,
         "se": 0.015546378085381977,
         "t": 2.841201513701472,
         "term": "exper"
       },
       {
         "ci_high": -0.00005360798291466297,
         "ci_low": -0.001744331193396381,
         "coef": -0.0008989695881555219,
         "p": 0.037193145535708634,
         "se": 0.00043008368306051234,
         "t": -2.0902201677552084,
         "term": "expersq"
       },
       {
         "ci_high": 0.12692611392621989,
         "ci_low": -0.004132856605911714,
         "coef": 0.06139662866015408,
         "p": 0.06623070402737445,
         "se": 0.03333858812319641,
         "t": 1.841608541827702,
         "term": "educ"
       }
     ],
     "first_stage": [
       {
         "ci_high": 0.22727760047111462,
         "ci_low": 0.08791646502607559,
         "coef": 0.1575970327485951,
         "endog": "educ",
         "instrument": "motheduc",
         "p": 0.000011213431960328954,
         "se": 0.03545024072114802,
         "t": 4.445584276514653
       },
       {
         "ci_high": 0.25331585513559624,
         "ci_low": 0.12578096517431214,
         "coef": 0.1895484101549542,
         "endog": "educ",
         "instrument": "fatheduc",
         "p": 1.026169112189478e-8,
         "se": 0.03244191815052831,
         "t": 5.842700461651569
       }
     ],
     "reduced_form": [
       {
         "ci_high": 0.02662216578501586,
         "ci_low": -0.020483377140654124,
         "coef": 0.0030693943221808697,
         "instrument": "motheduc",
         "p": 0.797955533850528,
         "se": 0.011982557623993707,
         "t": 0.25615518977640944
       },
       {
         "ci_high": 0.04022399770424663,
         "ci_low": -0.005384216639209714,
         "coef": 0.017419890532518457,
         "instrument": "fatheduc",
         "p": 0.13397281259065652,
         "se": 0.011601671959503245,
         "t": 1.5014982834650268
       }
     ]
   }

   Hashes SHA-256 e metadados de reprodução

   Digests dos dados

   mroz 909e3d99ac8070dc661ce7915c99f27631797719369832e227d4005096cbd479

   sha256 do resultado

   8b4474820943955f614dff42b39a7a5ac2a5a7d31641af570d960115f9c7e734

   Resultado mostrado ao modelo
8. 7Turno do modelo
   call_3d92584bgemini-3.8-flashFim: STOP5.650 ms17056 de entrada · 53 de saída · 169 de raciocínio · 0 em cache

   Resumo do raciocínio, como declarado pelo modelo

   **My Thought Process Regarding the Return to Education Analysis** Ah, here I am, diving back into Wooldridge's Example 15.5. The goal here is to estimate the return to education for married women, and the approach involves using `motheduc` and `fatheduc` as instrumental variables (IVs) for `educ`. I see the presented 2SLS estimate for `beta1` is 0.0614. It's also noted that the standard error is 0.0314 under homoskedasticity, and a robust standard error comes in at 0.0333. Before I fully commit to interpreting these results, I'm curious about the robustness of this finding. What happens if I simplify the model? I want to explore the implications of using only one of the parent's education levels as an instrument, or even no controls at all. This leads me to specifically check out Example 15.1. I recall that Example 15.1 uses `fatheduc` solely as an IV for `educ` in a model predicting `log(wage)`, and importantly, it does so *without* any additional control variables. I need to see what the estimate is in that more basic setting to understand the potential impact of adding controls or using both parent's education as instruments.

   Chamadas pedidas

   • infer.iv_2sls

     {
       "dataset": "mroz",
       "endog": [
         "educ"
       ],
       "instruments": [
         "fatheduc"
       ],
       "subset": "inlf == 1",
       "y": "lwage"
     }

   Prompt exatoInstruções do sistemaFerramentas oferecidasResposta bruta
9. 8Ferramenta
   infer.iv_2sls— Variáveis instrumentais por mínimos quadrados em dois estágios (MQ2E)v0.1.0call_f18b5877decidido pelo modeloreexecutável

   Argumentos

   {
     "cluster": null,
     "dataset": "mroz",
     "endog": [
       "educ"
     ],
     "exog": [],
     "instruments": [
       "fatheduc"
     ],
     "level": 0.95,
     "se": "HC1",
     "subset": "inlf == 1",
     "y": "lwage"
   }

   Figuras call_f18b5877

   Chave	Valor	Valor bruto	Erro-padrão	IC
   coef.educ 2SLS effect of educ on lwage	0,0591735	0.05917347999936612	0,03702970.03702965346681895	-0,0136101 – 0,131957

   Diagnósticos

   • first_stage_F.educ = 87,118987.11890952915502 (Wald F of the excluded instruments in the first stage for educ (heteroskedasticity-robust standard errors (HC1)))
   • partial_r2.educ = 0,172560.17255969324676745 (share of the variation the instruments explain beyond the controls)
   • weak_instrument = false (every first-stage F is at least 10)
   • n_obs = 428428
   • se_type = "HC1" (heteroskedasticity-robust standard errors (HC1))
   • n_clusters = — (errors not clustered)

   Derivação matemática e pressupostos

   coef.educ

   1. Com um instrumento e um regressor endógeno, o MQ2E é a razão entre o coeficiente de fatheduc na forma reduzida (lwage sobre fatheduc) e no primeiro estágio (educ sobre fatheduc).β = ρ / π→ 0,0591735 (0.05917347999936612)
   2. Erro-padrão robusto à heterocedasticidade (HC1): a raiz da variância estimada do coeficiente.ep = √V_jj→ 0,0370297 (0.03702965346681895)
   3. Intervalo de 95% de confiança: a estimativa mais ou menos o valor crítico da distribuição t de Student com 426 graus de liberdade vezes o erro-padrão.IC = β ± c · ep

   Pressupostos do estimador

   • the instruments affect the outcome only through the endogenous regressors (exclusion restriction)
   • the instruments are as good as randomly assigned given the controls
   • errors are independent across observations; their variance may differ

   Tabelas (3)

   {
     "coefficients": [
       {
         "ci_high": 1.3558210006860265,
         "ci_low": -0.47361418461539967,
         "coef": 0.4411034080353134,
         "p": 0.34374597798973444,
         "se": 0.4653752852619967,
         "t": 0.9478445074430226,
         "term": "Intercept"
       },
       {
         "ci_high": 0.1319570517715321,
         "ci_low": -0.01361009177279987,
         "coef": 0.05917347999936612,
         "p": 0.11078384396273744,
         "se": 0.03702965346681895,
         "t": 1.598002532008287,
         "term": "educ"
       }
     ],
     "first_stage": [
       {
         "ci_high": 0.32618201602862235,
         "ci_low": 0.21270126295152708,
         "coef": 0.2694416394900747,
         "endog": "educ",
         "instrument": "fatheduc",
         "p": 5.68867962680542e-19,
         "se": 0.02886745497151789,
         "t": 9.333751096378936
       }
     ],
     "reduced_form": [
       {
         "ci_high": 0.03665590100983519,
         "ci_low": -0.004768302079110571,
         "coef": 0.01594379946536231,
         "instrument": "fatheduc",
         "p": 0.13101015105829575,
         "se": 0.01053756945540146,
         "t": 1.5130433571842004
       }
     ]
   }

   Hashes SHA-256 e metadados de reprodução

   Digests dos dados

   mroz 909e3d99ac8070dc661ce7915c99f27631797719369832e227d4005096cbd479

   sha256 do resultado

   81a6db4e43f88de5860384ad38a2410d3c88a41ff2aee33e611b075ead813760

   Resultado mostrado ao modelo
10. 9Turno do modelo
    call_f23bcd7bgemini-3.8-flashFim: STOP5.956 ms17755 de entrada · 52 de saída · 88 de raciocínio · 15808 em cache

    Resumo do raciocínio, como declarado pelo modelo

    **My Thought Process Regarding Wooldridge's Example 15.1 and the Return to Education** Alright, so I'm diving into Wooldridge's Example 15.1, and my first thought is to identify the instrumental variables (IVs) being employed there. I recall it uses either `fatheduc` or `motheduc` as the instruments. This immediately sparks a desire to establish a baseline for comparison, so I want to look at the Ordinary Least Squares (OLS) estimation of the return to education. To do this, I'll run an OLS regression using `infer.ols`. My dependent variable, `y`, will be `lwage` (the log of wages), and my primary independent variable of interest, `x`, is `educ` (years of education). Of course, I need to account for other factors that might influence wages, so I'll include `exper` (experience) and `expersq` (experience squared) as control variables in the `controls` argument. This OLS result will serve as a crucial benchmark against which I can later evaluate the IV estimates from Example 15.1.

    Chamadas pedidas

    • infer.ols

      {
        "controls": [
          "exper",
          "expersq"
        ],
        "dataset": "mroz",
        "subset": "inlf == 1",
        "x": [
          "educ"
        ],
        "y": "lwage"
      }

    Prompt exatoInstruções do sistemaFerramentas oferecidasResposta bruta
11. 10Ferramenta
    infer.ols— Regressão linear por mínimos quadrados ordinários (MQO)v0.1.0call_438aa922decidido pelo modeloreexecutável

    Argumentos

    {
      "cluster": null,
      "controls": [
        "exper",
        "expersq"
      ],
      "dataset": "mroz",
      "fe": [],
      "intercept": true,
      "level": 0.95,
      "se": "HC1",
      "subset": "inlf == 1",
      "weights": null,
      "x": [
        "educ"
      ],
      "y": "lwage"
    }

    Figuras call_438aa922

    Chave	Valor	Valor bruto	Erro-padrão	IC
    coef.educ Coefficient on educ in the regression of lwage	0,10749	0.10748964014881399	0,0132190.013218967868628032	0,0815068 – 0,133473

    Diagnósticos

    • n_obs = 428428 (observations used)
    • k = 44 (parameters estimated, fixed-effect dummies included)
    • r2 = 0,156820.15682039127229852
    • df_resid = 424424
    • se_type = "HC1" (heteroskedasticity-robust standard errors (HC1))
    • n_clusters = — (errors not clustered)

    Derivação matemática e pressupostos

    coef.educ

    1. Coeficiente de educ na regressão por mínimos quadrados ordinários de lwage.→ 0,10749 (0.10748964014881399)
    2. Erro-padrão robusto à heterocedasticidade (HC1): a raiz da variância estimada do coeficiente.ep = √V_jj→ 0,013219 (0.013218967868628032)
    3. Intervalo de 95% de confiança: a estimativa mais ou menos o valor crítico da distribuição t de Student com 424 graus de liberdade vezes o erro-padrão.IC = β ± c · ep

    Pressupostos do estimador

    • the conditional mean of the outcome is linear in the regressors
    • errors are independent across observations; their variance may differ

    Tabelas (1)

    {
      "coefficients": [
        {
          "ci_high": -0.1256815155178025,
          "ci_low": -0.9183996073945216,
          "coef": -0.5220405614561621,
          "p": 0.009961086060266577,
          "se": 0.20165046204452491,
          "t": -2.588838905516092,
          "term": "Intercept"
        },
        {
          "ci_high": 0.13347250891874696,
          "ci_low": 0.08150677137888103,
          "coef": 0.10748964014881399,
          "p": 4.720316237716186e-15,
          "se": 0.013218967868628032,
          "t": 8.131469961729326,
          "term": "educ"
        },
        {
          "ci_high": 0.07158680676882788,
          "ci_low": 0.011546211338847475,
          "coef": 0.04156650905383768,
          "p": 0.006765094952563138,
          "se": 0.015273038339797112,
          "t": 2.7215612328771153,
          "term": "exper"
        },
        {
          "ci_high": 0.000014488920181200118,
          "ci_low": -0.0016368750891593343,
          "coef": -0.0008111930844890671,
          "p": 0.05413905435926356,
          "se": 0.00042007154737551356,
          "t": -1.9310831441861955,
          "term": "expersq"
        }
      ]
    }

    Hashes SHA-256 e metadados de reprodução

    Digests dos dados

    mroz 909e3d99ac8070dc661ce7915c99f27631797719369832e227d4005096cbd479

    sha256 do resultado

    6d7031ce8cfdee832c4f44716dabdd86f48a466d031964943c2a9cb05daf5eec

    Resultado mostrado ao modelo
12. 11Turno do modelo
    call_39fd9478gemini-3.8-flashFim: STOP8.448 ms18297 de entrada · 66 de saída · 346 de raciocínio · 15787 em cache

    Resumo do raciocínio, como declarado pelo modelo

    **My Thought Process Regarding the Return to Education for Married Women** When I first encountered Wooldridge's Example 15.5, my immediate reaction was to compare it with Example 4.8. It seems the core of the analysis is about estimating the return to education for married women, controlling for experience. In Example 4.8, the Ordinary Least Squares (OLS) estimate for the return to education is a seemingly robust 10.7%. This is calculated using `log(wage)` as the dependent variable and `educ`, `exper`, and `expersq` as predictors. However, the crucial insight from Example 15.5 comes when instrumental variables (IV) are introduced. By using `motheduc` and `fatheduc` as instruments, the estimated return to education drops significantly to 6.1%. This discrepancy immediately flags a potential endogeneity issue with `educ` in the OLS estimation. The IV results suggest that the OLS estimate was likely biased upwards, perhaps due to unobserved factors like a woman's inherent ability positively influencing both her education attainment and her wage. The instruments, parents' education, are presumed to affect a woman's education but not her wage directly, other than through education. Now, I'm presented with a specific research question: "What is the return to a year of education for married women who work? Using Mroz's sample of married women in 1975, estimate the causal effect of a year of schooling on the log hourly wage of the women who worked, using a source of variation in schooling that is not driven by the woman's own ability." This question directly points to the need for a causal interpretation and highlights the very issue that instrumental variables address. It's asking precisely for the kind of estimation shown in Example 15.5, where we aim to isolate the *causal* effect of education, free from confounding factors like ability. The follow-up thoughts then delve into the specifics of the instruments. The user poses the scenario: "Wait, what if only parents' education (motheduc and fatheduc) or just motheduc or fatheduc was used?" This is a critical step in instrument selection and validation. It makes sense to explore different specifications of the instrument. The final prompt, "Let's also check motheduc alone as instrument with exper and expersq," pushes me to consider a more parsimonious instrumental variable strategy. If I were to perform this, I would essentially be replicating the structure of Example 15.5 but replacing the combined `motheduc` and `fatheduc` instruments with just `motheduc`. The objective would be to see if `motheduc` alone is a sufficiently strong and relevant instrument to identify the causal effect of education on wages for married women who work. I'd be looking for a similar pattern of results: an IV estimate that is likely lower than the OLS estimate, and I would also be paying close attention to the standard errors (both homoskedastic and robust) to assess the precision of this new estimate and, implicitly, the strength of `motheduc` as an instrument. The inclusion of `exper` and `expersq` would remain consistent, as they are control variables in the wage equation.

    Chamadas pedidas

    • infer.iv_2sls

      {
        "dataset": "mroz",
        "endog": [
          "educ"
        ],
        "exog": [
          "exper",
          "expersq"
        ],
        "instruments": [
          "motheduc"
        ],
        "subset": "inlf == 1",
        "y": "lwage"
      }

    Prompt exatoInstruções do sistemaFerramentas oferecidasResposta bruta
13. 12Ferramenta
    infer.iv_2sls— Variáveis instrumentais por mínimos quadrados em dois estágios (MQ2E)v0.1.0call_11f7a29adecidido pelo modeloreexecutável

    Argumentos

    {
      "cluster": null,
      "dataset": "mroz",
      "endog": [
        "educ"
      ],
      "exog": [
        "exper",
        "expersq"
      ],
      "instruments": [
        "motheduc"
      ],
      "level": 0.95,
      "se": "HC1",
      "subset": "inlf == 1",
      "y": "lwage"
    }

    Figuras call_11f7a29a

    Chave	Valor	Valor bruto	Erro-padrão	IC
    coef.educ 2SLS effect of educ on lwage	0,049263	0.049262953350395176	0,03803960.038039576295817105	-0,0255067 – 0,124033

    Diagnósticos

    • first_stage_F.educ = 71,253171.25308965686146 (Wald F of the excluded instruments in the first stage for educ (heteroskedasticity-robust standard errors (HC1)))
    • partial_r2.educ = 0,1485020.14850194962816832 (share of the variation the instruments explain beyond the controls)
    • weak_instrument = false (every first-stage F is at least 10)
    • n_obs = 428428
    • se_type = "HC1" (heteroskedasticity-robust standard errors (HC1))
    • n_clusters = — (errors not clustered)

    Derivação matemática e pressupostos

    coef.educ

    1. Com um instrumento e um regressor endógeno, o MQ2E é a razão entre o coeficiente de motheduc na forma reduzida (lwage sobre motheduc e os controles) e no primeiro estágio (educ sobre motheduc e os controles).β = ρ / π→ 0,049263 (0.049262953350395176)
    2. Erro-padrão robusto à heterocedasticidade (HC1): a raiz da variância estimada do coeficiente.ep = √V_jj→ 0,0380396 (0.038039576295817105)
    3. Intervalo de 95% de confiança: a estimativa mais ou menos o valor crítico da distribuição t de Student com 424 graus de liberdade vezes o erro-padrão.IC = β ± c · ep

    Pressupostos do estimador

    • the instruments affect the outcome only through the endogenous regressors (exclusion restriction)
    • the instruments are as good as randomly assigned given the controls
    • errors are independent across observations; their variance may differ

    Tabelas (3)

    {
      "coefficients": [
        {
          "ci_high": 1.1596394625418232,
          "ci_low": -0.7632673495967359,
          "coef": 0.1981860564725436,
          "p": 0.6855588730675718,
          "se": 0.48914620608483256,
          "t": 0.40516731808847395,
          "term": "Intercept"
        },
        {
          "ci_high": 0.0755263609695913,
          "ci_low": 0.014185334777601921,
          "coef": 0.04485584787359661,
          "p": 0.004248141753814621,
          "se": 0.015603840003974094,
          "t": 2.87466725255914,
          "term": "exper"
        },
        {
          "ci_high": -0.00007318232345209234,
          "ci_low": -0.0017709700014867705,
          "coef": -0.0009220761624694314,
          "p": 0.03333162170211661,
          "se": 0.0004318807319241007,
          "t": -2.135025006467476,
          "term": "expersq"
        },
        {
          "ci_high": 0.12403258202191489,
          "ci_low": -0.02550667532112453,
          "coef": 0.049262953350395176,
          "p": 0.19600950999349903,
          "se": 0.038039576295817105,
          "t": 1.295044744118567,
          "term": "educ"
        }
      ],
      "first_stage": [
        {
          "ci_high": 0.33002425883142794,
          "ci_low": 0.20535735935295293,
          "coef": 0.26769080909219045,
          "endog": "educ",
          "instrument": "motheduc",
          "p": 5.027365264442827e-16,
          "se": 0.03171258249193887,
          "t": 8.441154521560511
        }
      ],
      "reduced_form": [
        {
          "ci_high": 0.034099082955624554,
          "ci_low": -0.007724603274348306,
          "coef": 0.013187239840638123,
          "instrument": "motheduc",
          "p": 0.21584113445607678,
          "se": 0.010639047776382594,
          "t": 1.2395131705219153
        }
      ]
    }

    Hashes SHA-256 e metadados de reprodução

    Digests dos dados

    mroz 909e3d99ac8070dc661ce7915c99f27631797719369832e227d4005096cbd479

    sha256 do resultado

    2440aa6948ccbeb159bdb7c6a335a7fbc4559e3a31c46b1b96cd40032b2e14bf

    Resultado mostrado ao modelo
14. 13Turno do modelo
    call_afbe911bgemini-3.8-flashFim: STOP8.742 ms19317 de entrada · 66 de saída · 18 de raciocínio · 15826 em cache

    Resumo do raciocínio, como declarado pelo modelo

    **My Analytical Exploration of Experiential Factors and Educational Background** My current line of inquiry is focused on disentangling the individual impact of educational background on a particular outcome, using "fatheduc" as my primary instrument. To do this, I'm isolating it from other, potentially confounding variables, specifically "exper" (experiential learning or practice) and its quadratic component, "expersq." My intention is to get a clear, unadulterated view of how parental education, in isolation, shapes the phenomenon I'm investigating. By holding experiential factors constant, or at least examining them separately, I aim to ascertain the independent explanatory power of educational attainment within the family context. This methodical approach allows me to build a more robust understanding by first understanding the foundational influence of education before reintroducing and analyzing the interplay with practical experience.

    Chamadas pedidas

    • infer.iv_2sls

      {
        "dataset": "mroz",
        "endog": [
          "educ"
        ],
        "exog": [
          "exper",
          "expersq"
        ],
        "instruments": [
          "fatheduc"
        ],
        "subset": "inlf == 1",
        "y": "lwage"
      }

    Prompt exatoInstruções do sistemaFerramentas oferecidasResposta bruta
15. 14Ferramenta
    infer.iv_2sls— Variáveis instrumentais por mínimos quadrados em dois estágios (MQ2E)v0.1.0call_059c71eadecidido pelo modeloreexecutável

    Argumentos

    {
      "cluster": null,
      "dataset": "mroz",
      "endog": [
        "educ"
      ],
      "exog": [
        "exper",
        "expersq"
      ],
      "instruments": [
        "fatheduc"
      ],
      "level": 0.95,
      "se": "HC1",
      "subset": "inlf == 1",
      "y": "lwage"
    }

    Figuras call_059c71ea

    Chave	Valor	Valor bruto	Erro-padrão	IC
    coef.educ 2SLS effect of educ on lwage	0,0702263	0.07022629127205371	0,0359390.03593897479386824	-0,000414448 – 0,140867

    Diagnósticos

    • first_stage_F.educ = 86,772186.77206558641643 (Wald F of the excluded instruments in the first stage for educ (heteroskedasticity-robust standard errors (HC1)))
    • partial_r2.educ = 0,1714560.17145569310800512 (share of the variation the instruments explain beyond the controls)
    • weak_instrument = false (every first-stage F is at least 10)
    • n_obs = 428428
    • se_type = "HC1" (heteroskedasticity-robust standard errors (HC1))
    • n_clusters = — (errors not clustered)

    Derivação matemática e pressupostos

    coef.educ

    1. Com um instrumento e um regressor endógeno, o MQ2E é a razão entre o coeficiente de fatheduc na forma reduzida (lwage sobre fatheduc e os controles) e no primeiro estágio (educ sobre fatheduc e os controles).β = ρ / π→ 0,0702263 (0.07022629127205371)
    2. Erro-padrão robusto à heterocedasticidade (HC1): a raiz da variância estimada do coeficiente.ep = √V_jj→ 0,035939 (0.03593897479386824)
    3. Intervalo de 95% de confiança: a estimativa mais ou menos o valor crítico da distribuição t de Student com 424 graus de liberdade vezes o erro-padrão.IC = β ± c · ep

    Pressupostos do estimador

    • the instruments affect the outcome only through the endogenous regressors (exclusion restriction)
    • the instruments are as good as randomly assigned given the controls
    • errors are independent across observations; their variance may differ

    Tabelas (3)

    {
      "coefficients": [
        {
          "ci_high": 0.8393803791350471,
          "ci_low": -0.9616142457499262,
          "coef": -0.06111693330743951,
          "p": 0.893937214422935,
          "se": 0.45813436323620566,
          "t": -0.13340394917272064,
          "term": "Intercept"
        },
        {
          "ci_high": 0.07426840213110908,
          "ci_low": 0.013074774127549267,
          "coef": 0.04367158812932918,
          "p": 0.00525488784336933,
          "se": 0.015566345069638764,
          "t": 2.805513300261346,
          "term": "exper"
        },
        {
          "ci_high": -0.00003451803945177939,
          "ci_low": -0.0017297918777765665,
          "coef": -0.000882154958614173,
          "p": 0.04141052708985181,
          "se": 0.0004312412650768066,
          "t": -2.045618149406588,
          "term": "expersq"
        },
        {
          "ci_high": 0.1408670307817658,
          "ci_low": -0.0004144482376583769,
          "coef": 0.07022629127205371,
          "p": 0.05135320031274228,
          "se": 0.03593897479386824,
          "t": 1.954042698069268,
          "term": "educ"
        }
      ],
      "first_stage": [
        {
          "ci_high": 0.3275851398202029,
          "ci_low": 0.21342706252454258,
          "coef": 0.2705061011723727,
          "endog": "educ",
          "instrument": "fatheduc",
          "p": 6.690927088013955e-19,
          "se": 0.029039363764596007,
          "t": 9.315152472526492
        }
      ],
      "reduced_form": [
        {
          "ci_high": 0.0392869092997792,
          "ci_low": -0.001293628796181917,
          "coef": 0.018996640251798643,
          "instrument": "fatheduc",
          "p": 0.06642917617415317,
          "se": 0.010322817582847107,
          "t": 1.840257284344962
        }
      ]
    }

    Hashes SHA-256 e metadados de reprodução

    Digests dos dados

    mroz 909e3d99ac8070dc661ce7915c99f27631797719369832e227d4005096cbd479

    sha256 do resultado

    34af6a0bcba8855fb1666c8d22a69259b140f8b25ea7878c86d79c2cc18902d1

    Resultado mostrado ao modelo
16. 15Turno do modelo
    call_850c6c96gemini-3.8-flashFim: STOP11.236 ms20017 de entrada · 43 de saída · 350 de raciocínio · 15804 em cache

    Resumo do raciocínio, como declarado pelo modelo

    **My Thought Process on Instrumental Variables in Wooldridge's MROZ Example** Right, so I'm digging into Wooldridge's *Introductory Econometrics*, specifically the MROZ data example dealing with the return to education for married women. My focus is on the canonical instrumental variables (IV) specification. Wooldridge's Example 15.5 is the key here. It's the most comprehensive specification for this particular problem. The model being estimated is: log(wage) = beta0 + beta1 * educ + beta2 * exper + beta3 * expersq + u The crucial part is the identification strategy: they're using `motheduc` and `fatheduc` as instrumental variables for `educ`. This is the standard setup in this example, and it's important because it allows for testing overidentifying restrictions – the Sargan test, specifically. This is a good sign that the model is well-specified and the instruments are likely valid. I also recall that Example 15.1 in Wooldridge is a simpler version, perhaps without experience controls or only using `fatheduc`. But 15.5 is the main event, the one with the full set of controls and both instruments, which is exactly what I'm interested in. Now, turning to the reporting guidelines for instrumental variables, a critical point is clarity on *whose* effect we're estimating. With a randomized offer, it's the effect for compliers, not the entire population. This is a fundamental concept in IV estimation. Furthermore, when the outcome variable is in logarithms, like `log(wage)` here, the interpretation of the coefficient needs careful handling. The guidelines are very clear: I must restate the coefficient as a percentage change using a specific function, `core.scale_effect`, and absolutely avoid doing this manually. This ensures accuracy and consistency in reporting. So, the next step in my analysis will be to apply `core.scale_effect` to the coefficient of `educ` from the IV estimation in this example, likely represented as `call_b58109a9:coef.educ` within my system, to properly interpret the estimated return to education as a percentage.

    Chamadas pedidas

    • core.scale_effect

      {
        "coefficient": "call_b58109a9:coef.educ",
        "transform": "log_points_to_percent"
      }

    Prompt exatoInstruções do sistemaFerramentas oferecidasResposta bruta
17. 16Ferramenta
    core.scale_effectv0.1.0call_2809015ddecidido pelo modeloreexecutável

    Argumentos

    {
      "baseline": null,
      "coefficient": "call_b58109a9:coef.educ",
      "sd": null,
      "transform": "log_points_to_percent"
    }

    Figuras call_2809015d

    Chave	Valor	Valor bruto	Erro-padrão	IC
    effect Effect as a percent change (from log points): 2SLS effect of educ on lwage	6,33206%	6.332057397505915	—	-0,412433 – 13,5333

    Diagnósticos

    • transform = "log_points_to_percent"

    Derivação matemática e pressupostos

    effect

    1. O limite inferior do intervalo de 95% passa pela mesma função: como ela é crescente, o intervalo transformado extremo a extremo mantém o nível.b_inf_pct = 100 × (exp(b_inf) − 1)→ -0,412433 (-0.4124328107115366)
    2. O limite superior do intervalo de 95% passa pela mesma função: como ela é crescente, o intervalo transformado extremo a extremo mantém o nível.b_sup_pct = 100 × (exp(b_sup) − 1)→ 13,5333 (13.533312937507011)
    3. Um coeficiente em pontos logarítmicos vira variação percentual por exp(b) − 1; b × 100 só é próximo disso para efeitos pequenos.pct = 100 × (exp(b) − 1)→ 6,33206 (6.332057397505915)

    Pressupostos do estimador

    • the coefficient is on a log outcome, in log points (natural logarithm)

    • the standard error is not carried: a percent effect is a nonlinear function of b, so only the interval, transformed endpoint by endpoint, keeps its meaning

    Hashes SHA-256 e metadados de reprodução

    Digests dos dados

    sha256 do resultado

    d66dd00c7fadc6ce800802fedc291567de93799058141835e50f3b75a74213b1

    Resultado mostrado ao modelo
18. 17Turno do modelo
    call_783c836dgemini-3.8-flashFim: STOP3.989 ms20530 de entrada · 42 de saída · 35 de raciocínio · 15852 em cache

    Resumo do raciocínio, como declarado pelo modelo

    **My Thought Process on Scaling the Education Coefficient** Here I am, thinking about how best to present these findings, and a key idea pops into my head: I need to make sure the OLS coefficient for education, which I've tentatively labeled `call_438aa922:coef.educ`, is directly comparable to other effects. To achieve this, I've decided to apply `core.scale_effect` to it. This is a standard practice when you want to isolate the magnitude of an effect relative to its standard deviation or some other baseline, making it easier to interpret and contrast with other coefficients in the model. It's all about ensuring clarity and enabling a robust comparison of influence.

    Chamadas pedidas

    • core.scale_effect

      {
        "coefficient": "call_438aa922:coef.educ",
        "transform": "log_points_to_percent"
      }

    Prompt exatoInstruções do sistemaFerramentas oferecidasResposta bruta
19. 18Ferramenta
    core.scale_effectv0.1.0call_c0027802decidido pelo modeloreexecutável

    Argumentos

    {
      "baseline": null,
      "coefficient": "call_438aa922:coef.educ",
      "sd": null,
      "transform": "log_points_to_percent"
    }

    Figuras call_c0027802

    Chave	Valor	Valor bruto	Erro-padrão	IC
    effect Effect as a percent change (from log points): Coefficient on educ in the regression of lwage	11,3479%	11.347932520667513	—	8,49206 – 14,279

    Diagnósticos

    • transform = "log_points_to_percent"

    Derivação matemática e pressupostos

    effect

    1. O limite inferior do intervalo de 95% passa pela mesma função: como ela é crescente, o intervalo transformado extremo a extremo mantém o nível.b_inf_pct = 100 × (exp(b_inf) − 1)→ 8,49206 (8.492056396732153)
    2. O limite superior do intervalo de 95% passa pela mesma função: como ela é crescente, o intervalo transformado extremo a extremo mantém o nível.b_sup_pct = 100 × (exp(b_sup) − 1)→ 14,279 (14.27898491746694)
    3. Um coeficiente em pontos logarítmicos vira variação percentual por exp(b) − 1; b × 100 só é próximo disso para efeitos pequenos.pct = 100 × (exp(b) − 1)→ 11,3479 (11.347932520667513)

    Pressupostos do estimador

    • the coefficient is on a log outcome, in log points (natural logarithm)

    • the standard error is not carried: a percent effect is a nonlinear function of b, so only the interval, transformed endpoint by endpoint, keeps its meaning

    Hashes SHA-256 e metadados de reprodução

    Digests dos dados

    sha256 do resultado

    cb8e36ea879c933620fcc7a4f7e1d8631e4c84277f72431d5ab9daec19af950f

    Resultado mostrado ao modelo
20. 19Turno do modelo
    call_b2367c7egemini-3.8-flashFim: STOP6.138 ms20724 de entrada · 84 de saída · 82 de raciocínio · 19012 em cache

    Resumo do raciocínio, como declarado pelo modelo

    **My Thought Process on Coefficient Comparison** Okay, so my primary objective here is to understand the relationship and differences between the Ordinary Least Squares (OLS) and Two-Stage Least Squares (2SLS) coefficient estimates for "educ". To achieve this, I'm considering utilizing a helper function, `calc.combine`. My initial thought is that this function could directly compute the difference between these two coefficients, which would be a very straightforward and informative way to present the findings. Alternatively, I could just report the individual coefficients if the difference calculation isn't immediately obvious or necessary for this specific output. Before I proceed, I need to confirm the exact capabilities of `calc.combine`. I'm pausing to verify how it's configured. A quick look reveals that the intended operation (`op`) for `calc.combine` is set to `'difference'`. Furthermore, the two operands it's meant to work with are clearly defined: `a` is the OLS coefficient for "educ" (identified by `call_438aa922:coef.educ`), and `b` is the 2SLS coefficient for "educ" (identified by `call_b58109a9:coef.educ`). This confirms that `calc.combine` is indeed set up to perform the exact subtraction I was considering. Therefore, I can proceed with the plan to use `calc.combine` to directly calculate and likely report the difference between the OLS and 2SLS "educ" coefficients.

    Chamadas pedidas

    • calc.combine

      {
        "a": "call_438aa922:coef.educ",
        "b": "call_b58109a9:coef.educ",
        "label": "Diferença entre o coeficiente de MQO e o de MQ2E para a educação",
        "op": "difference",
        "unit": "pontos log"
      }

    Prompt exatoInstruções do sistemaFerramentas oferecidasResposta bruta
21. 20Ferramenta
    calc.combine— Combinação aritmética auditada de estimativasv0.1.0call_c469ff47decidido pelo modeloreexecutável

    Argumentos

    {
      "a": "call_438aa922:coef.educ",
      "b": "call_b58109a9:coef.educ",
      "label": "Diferença entre o coeficiente de MQO e o de MQ2E para a educação",
      "op": "difference",
      "unit": "pontos log"
    }

    Figuras call_c469ff47

    Chave	Valor	Valor bruto	Erro-padrão	IC
    result Diferença entre o coeficiente de MQO e o de MQ2E para a educação	0,046093pontos log	0.04609301148865991	—	—

    Derivação matemática e pressupostos

    result

    1. Operação difference sobre duas figuras desta execução, lidas pela referência.a − b→ 0,046093 (0.04609301148865991)

    • no uncertainty is propagated: the result carries no standard error

    Hashes SHA-256 e metadados de reprodução

    Digests dos dados

    sha256 do resultado

    98c552f89cf3a08792836bd6b3ef08b64787e72dba1ed07d46a71b5f2612b30f

    Resultado mostrado ao modelo
22. 21Turno do modelo
    call_6c152200gemini-3.8-flashFim: STOP24.518 ms20940 de entrada · 663 de saída · 1691 de raciocínio · 19009 em cache

    Resumo do raciocínio, como declarado pelo modelo

    **My Thought Process** I've been asked to determine the causal return to an additional year of education for married women who are employed, using the Mroz (1975) dataset. This requires estimating the effect of schooling on log hourly wages, crucially, while employing an instrumental variable strategy to address the endogeneity of education. Specifically, I need to ensure the variation in education I use isn't driven by the woman's own inherent ability. First, I carefully reviewed the user's request to ensure I understood all its components: the target population (working married women in 1975), the outcome variable (log hourly wage), the endogenous regressor (years of education), and the crucial requirement for an instrument that is exogenous with respect to individual ability. Then, I examined the model I've constructed to meet these requirements. * The sample consists of 428 married women who were in the labor force and had reported hourly wages in 1975 (indicated by `inlf == 1`). * The dependent variable is `lwage` (the logarithm of the hourly wage). * The primary endogenous variable is `educ` (years of schooling), which is likely correlated with unobserved factors like ability and family background. * To address this endogeneity, I've selected parents' education (`motheduc` and `fatheduc`) as instrumental variables. The logic here is that parental education influences a woman's schooling decisions but should not directly affect her wage except through that schooling, assuming other controls are present. This relies on the exclusion restriction holding. * I've included standard controls for labor market experience (`exper`) and experience squared (`expersq`), following established practice in this literature. Next, I ran the necessary diagnostics to validate the instrument and assess the model's performance. * The first-stage F-statistic for the instruments is 49.53, which is well above common thresholds, indicating that the instruments are strong predictors of education. The partial R² of 0.208 further supports this. * The first-stage coefficients show that both mother's education (0.158, t = 4.45) and father's education (0.190, t = 5.84) are significantly and positively related to the woman's own education. * The overidentification test (Sargan test) yielded a p-value of 0.539 (Sargan statistic = 0.378, df = 1). This p-value is high, meaning I do not reject the null hypothesis that the instruments are valid (i.e., they are orthogonal to the residuals of the structural equation). * The Two-Stage Least Squares (2SLS) estimate for the causal effect of education on log hourly wage is 0.0614, with a standard error of 0.0333. This translates to an estimated 6.33% increase in hourly wage for each additional year of schooling. The 95% confidence interval for this effect is [-0.0041, 0.1269]. * For comparison, the Ordinary Least Squares (OLS) estimate is 0.1075 (SE = 0.0132), or an 11.35% increase. The difference of 0.0461 log points between OLS and 2SLS suggests that the OLS estimate likely suffers from upward ability bias, meaning unobserved ability inflates the OLS estimate of the return to education. This is consistent with the IV estimate being smaller. * I also noted that the 2SLS coefficient is marginally significant at the 10% level (p = 0.0662), but the 95% confidence interval does cross zero, suggesting some imprecision in the estimate. Finally, I considered the output in relation to the specific formatting and content requirements outlined in Rule 7. I need to present the estimate, its uncertainty, the identification strategy, and potential limitations. I also need to ensure I use the correct language (English, as per the original question) and the exact numbers from the tool outputs, including decimal points. I have to be careful about the "decimal comma" instruction within Rule 7, which seems to be a default for Portuguese, but since the question is in English, I should stick to English conventions for decimals while using the provided numbers. My focus is now on synthesizing these findings into the final, direct answer, highlighting the key results and caveats.

    Chamadas pedidas

    • submit_answer (meta-ferramenta)

      {
        "answer": "To estimate the causal return to schooling on log hourly wage without confounding by unobserved ability, we use Two-Stage Least Squares (2SLS / Instrumental Variables) on the 428 married women who worked in 1975, using parental education (`motheduc` and `fatheduc`) as instruments for the woman's education (`educ`), controlling for potential experience (`exper` and `expersq`).\n\n### 1. 2SLS Estimate\n- **Estimated return to education:** Each additional year of schooling increases the log hourly wage by **0.0614** (SE: 0.0333; 95% CI: [-0.0041, 0.1269]), which corresponds to an estimated wage gain of **6.33%** per year of education (95% CI: [-0.41%, 13.53%]).\n- **Comparison with OLS:** The ordinary least squares (OLS) estimate with the same controls yields a return of **0.1075** log points (SE: 0.0132; 95% CI: [0.0815, 0.1335]), or **11.35%** per year (95% CI: [8.49%, 14.28%]). The OLS estimate is higher by **0.0461** log points, consistent with positive ability bias in the OLS regression.\n\n### 2. Instrument Validity and Diagnostics\n- **Relevance (First Stage):** Parental education strongly predicts the woman's schooling. The first-stage robust Wald F-statistic is **49.53**, well above the rule-of-thumb threshold of 10.\n- **Overidentification:** The Sargan test of overidentifying restrictions yields a p-value of **0.5386**, failing to reject the joint validity of the two instruments.",
        "caveats": [
          "The exclusion restriction requires that parents' education affects the daughter's wage only through her own educational attainment; direct effects via parental connections, family wealth, or inherited unobserved ability would violate this assumption.",
          "The analysis is conditional on participation in the labour force (428 working women out of 753), which may introduce sample selection bias if unobserved factors affecting participation also influence wage offers.",
          "The 2SLS estimate is less precise than OLS, with a 95% confidence interval spanning from -0.0041 to 0.1269 (p-value = 0.0662)."
        ],
        "figure_refs": [
          "call_b58109a9:coef.educ",
          "call_2809015d:effect",
          "call_438aa922:coef.educ",
          "call_c0027802:effect",
          "call_c469ff47:result"
        ]
      }

    Prompt exatoInstruções do sistemaFerramentas oferecidasResposta bruta
23. 22Guarda
    resposta aceita

    Números encontrados

    428 · 1975 · 1 · 0.0614 · 0.0333 · 95% · -0.0041 · 0.1269 · 6.33% · 95% · -0.41% · 13.53% · 0.1075 · 0.0132 · 95% · 0.0815 · 0.1335 · 11.35% · 95% · 8.49% · 14.28% · 0.0461 · 2 · 49.53 · 10 · 0.5386 · 428 · 753 · 95% · -0.0041 · 0.1269 · 0.0662

    Sem lastro

    —

    Não citados

    —

    1 nova(s) tentativa(s) restante(s)
24. 23Resposta

    To estimate the causal return to schooling on log hourly wage without confounding by unobserved ability, we use Two-Stage Least Squares (2SLS / Instrumental Variables) on the 428 married women who worked in 1975, using parental education (`motheduc` and `fatheduc`) as instruments for the woman's education (`educ`), controlling for potential experience (`exper` and `expersq`). ### 1. 2SLS Estimate - **Estimated return to education:** Each additional year of schooling increases the log hourly wage by **0.0614** (SE: 0.0333; 95% CI: [-0.0041, 0.1269]), which corresponds to an estimated wage gain of **6.33%** per year of education (95% CI: [-0.41%, 13.53%]). - **Comparison with OLS:** The ordinary least squares (OLS) estimate with the same controls yields a return of **0.1075** log points (SE: 0.0132; 95% CI: [0.0815, 0.1335]), or **11.35%** per year (95% CI: [8.49%, 14.28%]). The OLS estimate is higher by **0.0461** log points, consistent with positive ability bias in the OLS regression. ### 2. Instrument Validity and Diagnostics - **Relevance (First Stage):** Parental education strongly predicts the woman's schooling. The first-stage robust Wald F-statistic is **49.53**, well above the rule-of-thumb threshold of 10. - **Overidentification:** The Sargan test of overidentifying restrictions yields a p-value of **0.5386**, failing to reject the joint validity of the two instruments.

    • The exclusion restriction requires that parents' education affects the daughter's wage only through her own educational attainment; direct effects via parental connections, family wealth, or inherited unobserved ability would violate this assumption.
    • The analysis is conditional on participation in the labour force (428 working women out of 753), which may introduce sample selection bias if unobserved factors affecting participation also influence wage offers.
    • The 2SLS estimate is less precise than OLS, with a 95% confidence interval spanning from -0.0041 to 0.1269 (p-value = 0.0662).

    Figuras citadas: call_b58109a9:coef.educ, call_2809015d:effect, call_438aa922:coef.educ, call_c0027802:effect, call_c469ff47:result · Parada: answered

    Figura da resposta: call_b58109a9:coef.educ = 0,06139660.06139662866015408 (se 0,03333860.03333858812319641)
25. 24Pontuação
    #0resposta com lastropassouobrigatóriaevery number backed
26. 25Pontuação
    #1figura da resposta escolhidapassouobrigatóriaanswer figure call_b58109a9:coef.educ (infer.iv_2sls)
27. 26Pontuação
    #2ferramenta chamadapassouobrigatóriaanswer call is infer.iv_2sls
28. 27Pontuação
    #3argumento igualpassouobrigatóriainfer.iv_2sls.y = 'lwage'
29. 28Pontuação
    #4argumento igualpassouobrigatóriainfer.iv_2sls.endog[0] = 'educ'
30. 29Pontuação
    #5argumento igualpassouregistradainfer.iv_2sls.y = 'lwage'
31. 30Pontuação
    #6argumento igualpassouregistradainfer.iv_2sls.endog = ['educ']
32. 31Pontuação
    #7argumento igualfalhouregistradainfer.iv_2sls.instruments = ['motheduc', 'fatheduc']
33. 32Pontuação
    #8argumento igualfalhouregistradainfer.iv_2sls.exog = ['exper', 'expersq']
34. 33Pontuação
    #9diagnósticopassouregistrada428 vs 428
35. 34Pontuação
    #10figura da resposta próximafalhouregistrada0.06139662866 vs 0.05917347999936595
36. 35Pontuação
    #11figura da resposta próximafalhouregistrada0.03333858812 vs 0.03514177397008675
37. 36Pontuação
    #12figura da resposta próximafalhouregistrada0.06139662866 vs 0.059
38. 37Pontuação
    #13figura da resposta próximafalhouregistrada0.03333858812 vs 0.035

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