Aplicación 2.1: Regresiones con datos de corte transversal
Demanda familiar de carne
En esta aplicación se estimará una función de demanda de carne con datos microeconómicos sobre 30 familias americanas ( i = 1, 2,…, 30):
\[Q_{i} = \beta_1 + \beta_2 P_{i} + \beta_3 Y_{i} + e_{i}\]
donde \(Q\) representa la cantidad de carne demandada, \(P\) es el precio pagado por ella en el mercado y, finalmente, \(Y\) es la renta familiar neta disponible.
Desde el punto de vista técnico, en el ejemplo se utilizarán distintas funciones y librerías que permiten la estimación de modelos estadísticos. Concretamente, en el caso del lenguaje R se usa la función lm() (https://rstudio.github.io/r-manuals/r-intro/Statistical-models-in-R.html) para el ajuste del modelo lineal propuesto, y en el caso de Python se proponen tanto el uso de la librería estándar en este lenguaje de programación para la estimación de modelos estadísticos y econométricos, statsmodels (https://www.statsmodels.org/), como la librería scikit-learn (https://scikit-learn.org/), si bien esta última está más especializada en técnicas generales de machine learning.
Code
# Lectura de librerías
library(tidyverse)
library(GGally)
library(modelsummary)
library(skimr)
# Lectura de datos
DEM_CARNE <- read_csv("data/DEM_CARNE.csv")
# Descripción de la base muestral
dim(DEM_CARNE)[1] 30 3Code
str(DEM_CARNE)spc_tbl_ [30 × 3] (S3: spec_tbl_df/tbl_df/tbl/data.frame)
$ P: num [1:30] 10.76 13.03 9.24 4.61 13.04 ...
$ Q: num [1:30] 11.63 12.03 8.92 33.91 4.56 ...
$ Y: num [1:30] 488 365 541 760 422 ...
- attr(*, "spec")=
.. cols(
.. P = col_double(),
.. Q = col_double(),
.. Y = col_double()
.. )
- attr(*, "problems")=<externalptr> Code
head(DEM_CARNE)# A tibble: 6 × 3
P Q Y
<dbl> <dbl> <dbl>
1 10.8 11.6 488.
2 13.0 12.0 365.
3 9.24 8.92 541.
4 4.61 33.9 760.
5 13.0 4.56 422.
6 7.71 17.6 578.Code
tail(DEM_CARNE)# A tibble: 6 × 3
P Q Y
<dbl> <dbl> <dbl>
1 5.48 2.90 256.
2 7.89 3.14 185.
3 8.46 15.3 359.
4 6.20 22.2 629.
5 6.74 10.0 307.
6 12.0 3.98 347.Code
# Análisis exploratorio (EDA)
# Estadística descriptiva
# Resultados estándar (R 'base')
summary(DEM_CARNE) P Q Y
Min. : 4.016 Min. : 2.903 Min. :184.8
1st Qu.: 6.556 1st Qu.: 6.288 1st Qu.:320.4
Median : 7.737 Median : 9.813 Median :385.9
Mean : 8.374 Mean :10.892 Mean :409.4
3rd Qu.:10.523 3rd Qu.:14.089 3rd Qu.:485.4
Max. :14.219 Max. :33.908 Max. :760.3 Code
# Librería skimr (https://cran.r-project.org/web/packages/skimr/)
skim(DEM_CARNE)| Name | DEM_CARNE |
| Number of rows | 30 |
| Number of columns | 3 |
| _______________________ | |
| Column type frequency: | |
| numeric | 3 |
| ________________________ | |
| Group variables | None |
Variable type: numeric
| skim_variable | n_missing | complete_rate | mean | sd | p0 | p25 | p50 | p75 | p100 | hist |
|---|---|---|---|---|---|---|---|---|---|---|
| P | 0 | 1 | 8.37 | 2.84 | 4.02 | 6.56 | 7.74 | 10.52 | 14.22 | ▅▇▃▃▃ |
| Q | 0 | 1 | 10.89 | 6.67 | 2.90 | 6.29 | 9.81 | 14.09 | 33.91 | ▇▅▃▁▁ |
| Y | 0 | 1 | 409.42 | 127.84 | 184.80 | 320.38 | 385.93 | 485.45 | 760.34 | ▂▇▃▂▁ |
Code
# Librería modelsummary (https://modelsummary.com/)
datasummary_correlation(DEM_CARNE)| P | Q | Y | |
|---|---|---|---|
| P | 1 | . | . |
| Q | -.33 | 1 | . |
| Y | -.01 | .75 | 1 |
Code
# Gráficas parciales
# Librería GGally
GGally::ggpairs(DEM_CARNE)
Code
# Librería ggplot
ggplot(DEM_CARNE, aes(x = P, y = Q)) +
geom_point() +
geom_smooth(method='lm', formula = y~x, se = FALSE) +
xlab("P (precio)") +
ylab("Q (cantidad)") +
theme_minimal()
Code
ggplot(DEM_CARNE, aes(x = Y, y = Q)) +
geom_point() +
geom_smooth(method='lm', formula = y~x, se = FALSE) +
xlab("Y (renta)") +
ylab("Q (cantidad)") +
theme_minimal()
Code
# Regresión por MCO del modelo lineal Q = beta1 + beta2*P + beta3*Y + e
# Comando lm() de R-stats (<https://rdrr.io/r/stats/lm.html>).
lin_model <- lm(formula = Q ~ P + Y, data = DEM_CARNE)
# Resultados estándar (R base)
summary(lin_model)
Call:
lm(formula = Q ~ P + Y, data = DEM_CARNE)
Residuals:
Min 1Q Median 3Q Max
-6.4429 -2.6144 -0.5625 1.7284 6.8800
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 1.345236 3.301677 0.407 0.68690
P -0.767780 0.259243 -2.962 0.00631 **
Y 0.039020 0.005762 6.772 2.84e-07 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 3.966 on 27 degrees of freedom
Multiple R-squared: 0.6709, Adjusted R-squared: 0.6465
F-statistic: 27.52 on 2 and 27 DF, p-value: 3.052e-07Code
# Resultados formateados (librería modelsummary)
modelsummary(lin_model)| (1) | |
|---|---|
| (Intercept) | 1.345 |
| (3.302) | |
| P | -0.768 |
| (0.259) | |
| Y | 0.039 |
| (0.006) | |
| Num.Obs. | 30 |
| R2 | 0.671 |
| R2 Adj. | 0.646 |
| AIC | 172.6 |
| BIC | 178.3 |
| Log.Lik. | -82.324 |
| F | 27.516 |
| RMSE | 3.76 |
Code
modelplot(lin_model)
Code
# ANEXO: Regresión por MCO del modelo propuesto en versión logarítmica
# log(Q) = beta1 + beta2*log(P) + beta3*log(Y) + e
log_model <- lm(formula = log(Q) ~ log(P) + log(Y), data = DEM_CARNE)
summary(log_model)
Call:
lm(formula = log(Q) ~ log(P) + log(Y), data = DEM_CARNE)
Residuals:
Min 1Q Median 3Q Max
-0.75112 -0.24829 0.01212 0.14647 0.67313
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -5.1704 1.4151 -3.654 0.0011 **
log(P) -0.5663 0.2148 -2.636 0.0137 *
log(Y) 1.4337 0.2287 6.270 1.04e-06 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.3941 on 27 degrees of freedom
Multiple R-squared: 0.6234, Adjusted R-squared: 0.5955
F-statistic: 22.35 on 2 and 27 DF, p-value: 1.882e-06Code
# Lectura de librerías
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import seaborn as sns
from plotnine import *
# https://www.statsmodels.org/stable/api.html
import statsmodels.api as sm
import statsmodels.formula.api as smf
# https://scikit-learn.org/stable/modules/linear_model.html
from sklearn import linear_model
from sklearn.metrics import r2_score
# Lectura de datos
DEM_CARNE = pd.read_csv("data/DEM_CARNE.csv")
# Descripción de la base muestral
DEM_CARNE.shape(30, 3)Code
DEM_CARNE.info()<class 'pandas.core.frame.DataFrame'>
RangeIndex: 30 entries, 0 to 29
Data columns (total 3 columns):
# Column Non-Null Count Dtype
--- ------ -------------- -----
0 P 30 non-null float64
1 Q 30 non-null float64
2 Y 30 non-null float64
dtypes: float64(3)
memory usage: 848.0 bytesCode
DEM_CARNE.head() P Q Y
0 10.763 11.632 487.648
1 13.033 12.029 364.877
2 9.244 8.916 541.037
3 4.605 33.908 760.343
4 13.045 4.561 421.746Code
DEM_CARNE.tail() P Q Y
25 7.890 3.138 184.798
26 8.460 15.315 359.084
27 6.195 22.240 629.378
28 6.743 10.012 306.527
29 11.977 3.982 347.488Code
# Análisis exploratorio (EDA)
# Estadística descriptiva
# Resultados estándar (Python 'base')
DEM_CARNE.describe() P Q Y
count 30.000000 30.000000 30.000000
mean 8.373533 10.891567 409.418633
std 2.841277 6.671015 127.842689
min 4.016000 2.903000 184.798000
25% 6.555500 6.287750 320.382250
50% 7.737500 9.813000 385.925500
75% 10.523250 14.089500 485.449750
max 14.219000 33.908000 760.343000Code
# Librería Skimpy (https://aeturrell.github.io/skimpy/)
from skimpy import skim
skim(DEM_CARNE)╭─────────────────────────────── skimpy summary ───────────────────────────────╮
│ Data Summary Data Types │
│ ┏━━━━━━━━━━━━━━━━━━━┳━━━━━━━━┓ ┏━━━━━━━━━━━━━┳━━━━━━━┓ │
│ ┃ dataframe ┃ Values ┃ ┃ Column Type ┃ Count ┃ │
│ ┡━━━━━━━━━━━━━━━━━━━╇━━━━━━━━┩ ┡━━━━━━━━━━━━━╇━━━━━━━┩ │
│ │ Number of rows │ 30 │ │ float64 │ 3 │ │
│ │ Number of columns │ 3 │ └─────────────┴───────┘ │
│ └───────────────────┴────────┘ │
│ number │
│ ┏━━━━━━━━┳━━━━┳━━━━━━┳━━━━━━┳━━━━━┳━━━━━┳━━━━━┳━━━━━┳━━━━━┳━━━━━━┳━━━━━━━━┓ │
│ ┃ column ┃ ┃ ┃ ┃ ┃ ┃ ┃ ┃ ┃ ┃ ┃ │
│ ┃ _name ┃ NA ┃ NA % ┃ mean ┃ sd ┃ p0 ┃ p25 ┃ p50 ┃ p75 ┃ p100 ┃ hist ┃ │
│ ┡━━━━━━━━╇━━━━╇━━━━━━╇━━━━━━╇━━━━━╇━━━━━╇━━━━━╇━━━━━╇━━━━━╇━━━━━━╇━━━━━━━━┩ │
│ │ P │ 0 │ 0 │ 8.4 │ 2.8 │ 4 │ 6.6 │ 7.7 │ 11 │ 14 │ ▅▇▇▃▃▃ │ │
│ │ Q │ 0 │ 0 │ 11 │ 6.7 │ 2.9 │ 6.3 │ 9.8 │ 14 │ 34 │ ▇▇▅▁ ▁ │ │
│ │ Y │ 0 │ 0 │ 410 │ 130 │ 180 │ 320 │ 390 │ 490 │ 760 │ ▂▇▆▅▂▁ │ │
│ └────────┴────┴──────┴──────┴─────┴─────┴─────┴─────┴─────┴──────┴────────┘ │
╰──────────────────────────────────── End ─────────────────────────────────────╯Code
# Gráficas parciales
# Librería seaborn
sns.pairplot(data = DEM_CARNE, diag_kind = 'kde');
plt.show()
Code
# Librería plotnine (ggplot en Python)
(ggplot(DEM_CARNE,aes('P','Q')) +
geom_point() +
geom_smooth(method='lm',se=False, color = "blue") +
labs(x="P (precio)",y="Q (cantidad)") + theme_bw())<Figure Size: (640 x 480)>
Code
(ggplot(DEM_CARNE,aes('Y','Q')) +
geom_point() +
geom_smooth(method='lm',se=False, color = "blue") +
labs(x="Y (renta)",y="Q (cantidad)") + theme_bw())<Figure Size: (640 x 480)>
Code
# Regresión por MCO del modelo lineal Q = beta1 + beta2*P + beta3*Y + e
# Comando `smf.ols` de la librería `statsmodels`
# (<https://www.statsmodels.org/>)
model = smf.ols(formula = "Q ~ P + Y", data = DEM_CARNE)
lin_model_1 = model.fit()
print(lin_model_1.summary()) OLS Regression Results
==============================================================================
Dep. Variable: Q R-squared: 0.671
Model: OLS Adj. R-squared: 0.646
Method: Least Squares F-statistic: 27.52
Date: Sun, 09 Feb 2025 Prob (F-statistic): 3.05e-07
Time: 13:03:20 Log-Likelihood: -82.324
No. Observations: 30 AIC: 170.6
Df Residuals: 27 BIC: 174.9
Df Model: 2
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [0.025 0.975]
------------------------------------------------------------------------------
Intercept 1.3452 3.302 0.407 0.687 -5.429 8.120
P -0.7678 0.259 -2.962 0.006 -1.300 -0.236
Y 0.0390 0.006 6.772 0.000 0.027 0.051
==============================================================================
Omnibus: 1.346 Durbin-Watson: 2.328
Prob(Omnibus): 0.510 Jarque-Bera (JB): 1.218
Skew: 0.347 Prob(JB): 0.544
Kurtosis: 2.297 Cond. No. 1.96e+03
==============================================================================
Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
[2] The condition number is large, 1.96e+03. This might indicate that there are
strong multicollinearity or other numerical problems.Code
# Comando `linear_model` de la librería `scikit-learn`
# (<https://scikit-learn.org/>)
lr = linear_model.LinearRegression()
lin_model_2 = lr.fit (X=DEM_CARNE[["P", "Y"]], y=DEM_CARNE["Q"])
print("Ordenada en el origen: \n", lin_model_2.intercept_)Ordenada en el origen:
1.3452364833219423Code
print("Efectos marginales: \n", lin_model_2.coef_)Efectos marginales:
[-0.76777964 0.03901962]Code
# ANEXO: Regresión por MCO del modelo propuesto en versión logarítmica
# log(Q) = beta1 + beta2*log(P) + beta3*log(Y) + e
model = smf.ols(formula = "np.log(Q) ~ np.log(P) + np.log(Y)", data = DEM_CARNE)
log_model = model.fit()
print(log_model.summary()) OLS Regression Results
==============================================================================
Dep. Variable: np.log(Q) R-squared: 0.623
Model: OLS Adj. R-squared: 0.595
Method: Least Squares F-statistic: 22.35
Date: Sun, 09 Feb 2025 Prob (F-statistic): 1.88e-06
Time: 13:03:21 Log-Likelihood: -13.053
No. Observations: 30 AIC: 32.11
Df Residuals: 27 BIC: 36.31
Df Model: 2
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [0.025 0.975]
------------------------------------------------------------------------------
Intercept -5.1704 1.415 -3.654 0.001 -8.074 -2.267
np.log(P) -0.5663 0.215 -2.636 0.014 -1.007 -0.126
np.log(Y) 1.4337 0.229 6.270 0.000 0.965 1.903
==============================================================================
Omnibus: 0.530 Durbin-Watson: 2.252
Prob(Omnibus): 0.767 Jarque-Bera (JB): 0.636
Skew: 0.147 Prob(JB): 0.728
Kurtosis: 2.350 Cond. No. 127.
==============================================================================
Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.